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Introduction to
Computational Fluid Dynamics
Development Application and Analysis
Dr Atul Sharma
Professor Deptt of Mech EngineeringIIT Bombay
Introduction to
Computational Fluid Dynamics
Development Application and Analysis
Introduction to Computational Fluid DynamicsDevelopment Application and Analysis
Dr Atul Sharma
copy Author 2017
This Edition Published by
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Library Congress Cataloging-in-Publication DataA catalogue record for this book is available from the British Library
Cover Image The image for C F and D in the cover page is obtained from a Computational Fluid Dynamics simulations as follows
C Curvilinear-Grid and Resultant-Velocity Contours for Fluid-Flow in a C-shaped Bend of a Pipe
F Temperature-Contours in a Heat Conduction Problem
D Vertical-Velocity Contours for a vertical-wall (moving upward like a conveyor belt) driven flow in a D shaped cavity
DisclaimerThe contents sources and the information provided by the authors in different chapters are the sole responsibility of the authors themselves including the copyright issues Editors and Publishers have no liabilities whatsoever towards these in any manner
This book is dedicated
to
Suhas V Patankar
Dr Suhas V Patankar1 (born February 22 1941) is an Indian
mechanical engineer He is a pioneer in the eld of computational
uid dynamics (CFD) and nite volume method He received his
BE from the University of Pune in 1962 MTech from IIT Bombay
in 1964 and PhD from Imperial College London in 1967
Dr Patankar is currently a Professor Emeritus in the Mechanical
Engineering Department at the University of Minnesota where he
worked for 25 years (from 1975-2000) Earlier he held teaching and
research positions at IIT Kanpur Imperial College and University
of Waterloo He is also the President of Innovative Research Inc He
has authored or co-authored four books published over 150 papers
advised 35 completed PhD theses and lectured extensively in the
USA and abroad His 1980 book Numerical heat Transfer and Fluid
Flow is considered to be a groundbreaking contribution to CFD He
is one of the most cited authors in science and engineering
For excellence in teaching Dr Patankar received the 1983 George
Taylor Distinguished Teaching Award and the 1989-90 Morse-Alumni
1Source (for the Biography) Personal Communication
v
vi
Award for Outstanding Contributions to Undergraduate Education
For his research contributions to computational heat transfer he was
given the 1991 ASME Heat Transfer Memorial Award and the 1997
Classic Paper Award He was awarded the 2008 Max Jakob Award
which is considered to be the highest international honor in the eld of
heat transfer In 2015 an International Conference on Computational
Heat Transfer (CHT-15) held at Rutgers University in New Jersey
was dedicated to Professor Patankar
Dr Patankars widespread inuence on research and engineering
education has been recognized in many ways In 2007 the Editors of
the International Journal of Heat and Mass Transfer wrote There is
no person who has made a more profound and enduring impact on the
theory and practice of numerical simulation in mechanical engineering
than Professor Patankar
FOREWORD
by Prof K Muralidhar
Indian Institute of Technology Kanpur INDIA
The subject of computational uid dynamics (CFD) emerged in the
academic context but in the past two decades it has developed to a
point where it is extensively employed as a design and analysis tool in
the industry The impact of the subject is deeply felt in diverse disci-
plines ranging from the aerospace spreading all the way to the chem-
ical industry New compact and ecient designs are enabled and it
has become that much more convenient to locate optimum conditions
for the operation of engineering systems As it stands CFD is com-
pulsory learning in several branches of engineering Many universities
now opt to develop theoretical courses around it The great utility of
this tool has inuenced adjacent disciplines and the process of solv-
ing complex dierential equations by the process of discretization has
become a staple for addressing engineering challenges
For an emerging discipline it is necessary that new books fre-
quently appear on the horizon Students can choose from the se-
lection and will stand to gain from it Such books should have a
perspective of the past and the future and it is all the more signi-
cant when the author has exclusive training in the subject Dr Atul
Sharma represents a combination of perspective training and passion
that combines well and delivers a strong text
I have personal knowledge of the fact that Dr Atul Sharma has
developed a whole suite of computer programs on CFD from scratch
He has wondered and pondered over why approximations work and
conditions under which they yield meaningful solutions His spread
vii
viii Foreword
of experience over fteen and odd years includes two and three di-
mensional Navier-Stokes equations heat transfer interfacial dynam-
ics and turbulent ow He has championed the nite volume method
which is now the industry standard The programming experience
is important because it brings in the precision needed for assembling
the jigsaw puzzle The sprinkling of computer programs in the book
is no coincidence
Atul knows the conventional method of discretizing dierential
equations but has never been satised with it He has felt an ele-
ment of discomfort going away from a physical reality into a world
of numbers connected by arithmetic operations As a result he has
developed a principle that physical laws that characterize the dier-
ential equations should be reected at every stage of discretization
and every stage of approximation indeed at every scale whether a
single tetrahedron or the sub-domain or the region as a whole This
idea permeates the book where it is shown that discretized versions
must provide reasonable answers because they continue to represent
the laws of nature
This new CFD book is comprehensive and has a stamp of origi-
nality of the author It will bring students closer to the subject and
enable them to contribute to it
Kanpur
August 2016 K Muralidhar
PREFACE
As a Post-Graduate (PG) student in 1997 I got introduced to a
course on CFD at Indian Institute of Science (IISc) Bangalore Dur-
ing the introductory course on CFD although I struggled hard
in the physical understanding and computer-programming it didnt
stop my interest and enthusiasm to do research teaching and writ-
ing a text-book on a introductory course in CFD Since then I did
my Masters project work on CFD (1997-98) introduced and taught
this course to Under-Graduate (UG) students at National Institute
of Technology Hamirpur HP (1999-2000) did my Doctoral work on
this subject at Indian Institute of Technology (IIT) Kanpur (2000-
2004) and co-authored a Chapter on nite volume method in a re-
vised edition of an edited book on computational uid ow and heat
transfer during my PhD in 2003 After joining as a faculty in 2004
I taught this course to UG as well as PG students of dierent depart-
ments at IIT Bombay as well as to the students of dierent collages in
India through CDEEP (Center for Distance Engineering Programme)
IIT Bombay from 2007-2010 Furthermore I gave series of lecture
on CFD at various collages and industries in India More recently
in 2012 I delivered a ve-days lecture and lab-session on CFD (to
around 1400 collage teachers from dierent parts of India) as a part
of a project funded by ministry of human resource development under
NMEICT (National Mission on Education through information and
communication technology)
The increasing need for the development of the customized CFD
software (app) and wide spread CFD application as well as analysis
for the design optimization and innovation of various types of engi-
neering systems always motivated me to write this book Due to the
increasing importance of CFD another motivation is to present CFD
ix
x Preface
simple enough to introduce the rst-course at early under-graduate
curriculum rather than the post-graduate curriculum The moti-
vation got strengthened by my own as well as others (students
teachers researchers and practitioners) struggle on physical under-
standing of the mathematics involved during the Partial Dierential
Equation (PDE) based algebraic-formulation (nite volume method)
along with the continuous frustrating pursuit to eectively convert
the theory of CFD into the computer-program
This led me to come up with an alternate physical-law (PDE-
free) based nite volume method as well as solution methodol-
ogy which are much easier to comprehend Mostly the success of
the programming eort is decided by the perfect execution of the
implementations-details which rarely appears in-print (existing books
and journal articles) The physical law based CFD development
approach and the implementations-details are the novelties in the
present book Furthermore using an open-source software for nu-
merical computations (Scilab) computer programs are given in this
book on the basic modules of conduction advection and convection
Indeed the reader can generalize and extend these codes for the de-
velopment of Navier-Stokes solver and generate the results presented
in the chapters towards the end of this book
I have limited the scope of this book to the numerical-techniques
which I wish to recommend for the book although there are numerous
other advanced and better methods in the published literature I do
not claim that the programming-practice presented here are the most
ecient minus they are presented here more for the ease in programming
(understand as your program) than for the computational eciency
Furthermore although I am enthusiastic about my presentation of
the physical law based CFD approach it may not be more ecient
than the alternate mathematics based CFD approach The emphasis
in this book is on the ease of understanding of the formulations and
programming-practice for the introductory course on CFD
I would hereby acknowledge that I owe my greatest debt to Prof
J Srinivasan (my ME Project supervisor at IISc Bangalore) who
did a hand-holding and wonderful job in patiently introducing me to
the fascinating world of research in uid-dynamics and heat trans-
fer I would also acknowledge the excellent training from Prof V
Introduction to CFD xi
Eswaran (my PhD supervisor at IIT Kanpur) from whom I learned
systematic way of CFD development application and analysis His
inspiring comments and suggestions on my writing-style and elabora-
tion of the rst chapter in this book is also gratefully acknowledged
I also want to record my sincere thanks to Prof Pradip Datta IISc
Bangalore and Prof Gautam Biswas IIT Kanpur minus my teachers for
CFD course Special thanks to Prof K Muralidhar IIT Kanpur for
his personal encouragement academic support and motivation (for
the book writing) He also cleared my dilemma in the proper presen-
tation of the novelty in this book I thank him also for writing the
foreword of this book
I am grateful to Prof Kannan M Moudgalya and Prof Deepak B
Phatak from IIT Bombay for giving me opportunity to teach CFD
in a distance education mode I am also grateful to Prof Amit
Agrawal (my colleague at IIT Bombay) who diligently read through
all the chapters of this book and suggested some really remarkable
changes in the book I also thank him for the research interactions
over the last decade which had substantially improved my under-
standing of thermal and uid science I would also like to thank my
PhD student Namshad T who did a great job in implementing
my ideas on generating the CFD simulation based CFD image for
the cover-page and couple of gures in the rst chapter (Fig 12
and 13) Special mention that the Scilab codes developed (for the
CFD course under NMEICT) by Vishesh Aggarwal greatly helped
me to develop the codes presented in this book his contributions is
gratefully acknowledged I thank Malhar Malushte for testing some
of the formulations presented in this book
I thank my research students (Sachin B Paramane D Datta C
M Sewatkar Vinesh H Gada Kaushal Prasad Mukul Shrivastava
Absar M Lakdawala Mausam Sarkar and Javed Shaikh) for the
diligence and sincerity they have shown in unraveling some of our
ideas on CFD Apart from these several other research students and
the students teachers and CFD practitioners who took my lectures
on CFD in the past decade have contributed to my better under-
standing of CFD and inspired me to write the book I thank all of
them for their attention and enthusiastic response during the CFD
courses The inuence of the interactions with all of them can be seen
xii Preface
throughout this book I am grateful to IIT Bombay for giving me
one year sabbatical leave to write this book The ambiance of aca-
demic freedom considerate support by the faculties and the friendly
atmosphere inside the campus of IIT Bombay has greatly contributed
in the solo eort of book writing I thanks all my colleagues at IIT
Bombay for shaping my perceptions on CFD
During this long endeavor the patient support by Mr Sunil
Saxena Director Athena Academic Ltd UK is gratefully acknowl-
edged I also thank John Wiley for coming forward to take this
book to international markets Lastly I would acknowledge that I
am greatly inspired by the writing style of J D Anderson and the
book on CFD by S V Patankar Of course you will nd that I
have heavily cross-referred the CFD book by the two most acknowl-
edged pioneers on CFD I dedicate this book to Dr S V Patankar
whose book nicely introduced me to the computational as well as
ow physics in CFD and greatly helped me to introduce a physical
approach of CFD development in this book
The culmination of the book writing process fullls a long cher-
ished dream It would not have been possible without the excel-
lent environment continuous support and complete understanding
of my wife Anubha and daughter Anshita Their kindness has greatly
helped to fulll my dream now I plan to spend more time with my
wife and daughter
Mumbai
August 2016 Atul Sharma
Contents
I Introduction and Essentials
1
1 Introduction 3
11 CFD What is it 3
111 CFD as a Scientic and Engineering Analysis
Tool 5
112 Analogy with a Video-Camera 13
12 CFD Why to study 15
13 Novelty Scope and Purpose of this Book 16
2 Introduction to CFD Development Application and
Analysis 23
21 CFD Development 23
211 Grid Generation Pre-Processor 24
212 Discretization Method Algebraic Formulation 26
213 Solution Methodology Solver 29
214 Computation of Engineering-Parameters
Post-Processor 34
215 Testing 35
22 CFD Application 35
23 CFD Analysis 38
24 Closure 39
xiii
xiv Contents
3 Essentials of Fluid-Dynamics and Heat-Transfer for
CFD 41
31 Physical Laws 42
311 FundamentalConservation Laws 43
312 Subsidiary Laws 45
32 Momentum and Energy Transport Mechanisms 46
33 Physical Law based Dierential Formulation 48
331 Continuity Equation 49
332 Transport Equation 51
34 Generalized Volumetric and Flux Terms and their Dif-
ferential Formulation 57
341 Volumetric Term 58
342 Flux-Term 58
343 Discussion 62
35 Mathematical Formulation 63
351 Dimensional Study 63
352 Non-Dimensional Study 67
36 Closure 71
4 Essentials of Numerical-Methods for CFD 73
41 Finite Dierence Method A Dierential to Algebraic
Formulation for Governing PDE and BCs 75
411 Grid Generation 75
412 Finite Dierence Method 78
413 Applications to CFD 92
42 Iterative Solution of System of LAEs for a Flow Property 93
421 Iterative Methods 94
422 Applications to CFD 98
43 Numerical Dierentiation for Local Engineering-
Parameters 105
431 Dierentiation Formulas 106
432 Applications to CFD 107
44 Numerical Integration for the Total value of
Engineering-Parameters 110
441 Integration Rules 111
442 Applications to CFD 114
45 Closure 116
Problems 116
Introduction to CFD xv
II CFD for a Cartesian-Geometry
119
5 Computational Heat Conduction 12151 Physical Law based Finite Volume Method 122
511 Energy Conservation Law for a Control Volume 123512 Algebraic Formulation 124513 Approximations 127514 Approximated Algebraic-Formulation 130515 Discussion 133
52 Finite Dierence Method for Boundary Conditions 13853 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 139531 One-Dimensional Conduction 139532 Two-Dimensional Conduction 147
54 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 154541 One-Dimensional Conduction 154542 Two-Dimensional Conduction 166
Problems 176
6 Computational Heat Advection 17961 Physical Law based Finite Volume Method 180
611 Energy Conservation Law for a Control Volume 181612 Algebraic Formulation 181613 Approximations 183614 Approximated Algebraic Formulation 191615 Discussion 191
62 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 194621 Explicit-Method 197622 Implementation Details 198623 Solution Algorithm 199
63 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 207631 Advection Scheme on a Non-Uniform Grid 207632 Explicit and Implicit Method 210633 Implementation Details 216634 Solution Algorithm 220
Problems 228
xvi Contents
7 Computational Heat Convection 22971 Physical Law based Finite Volume Method 229
711 Energy Conservation Law for a Control Volume 229712 Algebraic Formulation 231713 Approximated Algebraic Formulation 232
72 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 235721 Explicit-Method 236722 Implementation Details 237723 Solution Algorithm 238
73 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 242Problems 248
8 Computational Fluid Dynamics Physical Law based
Finite Volume Method 25181 Generalized Variables for the Combined Heat and
Fluid Flow 25282 Conservation Laws for a Control Volume 25583 Algebraic Formulation 25984 Approximations 26085 Approximated Algebraic Formulation 263
851 Mass Conservation 263852 MomentumEnergy Conservation 264
86 Closure 269
9 Computational Fluid Dynamics on a Staggered Grid 27191 Challenges in the CFD Development 273
911 Non-Linearity 273912 Equation for Pressure 273913 Pressure-Velocity Decoupling 273
92 A Staggered Grid to avoid Pressure-Velocity Decoupling27593 Physical Law based FVM for a Staggered Grid 27794 Flux based Solution Methodology on a Uniform Grid
Semi-Explicit Method 281941 Philosophy of Pressure-Correction Method 283942 Semi-Explicit Method 286943 Implementation Details 292944 Solution Algorithm 298
95 Initial and Boundary Conditions 300951 Initial Condition 300952 Boundary Condition 301
Problems 306
Introduction to CFD xvii
10 Computational Fluid Dynamics on a Co-located Grid309
101 Momentum-Interpolation Method Strategy to avoid
the Pressure-Velocity Decoupling on a Co-located Grid 310
102 Coecients of LAEs based Solution Methodology on
a Non-Uniform Grid Semi-Explicit and Semi-Implicit
Method 314
1021 Predictor Step 316
1022 Corrector Step 318
1023 Solution Algorithm 323
Problems 329
III CFD for a Complex-Geometry
331
11 Computational Heat Conduction on a Curvilinear
Grid 333
111 Curvilinear Grid Generation 333
1111 Algebraic Grid Generation 334
1112 Elliptic Grid Generation 336
112 Physical Law based Finite Volume Method 343
1121 Unsteady and Source Term 343
1122 Diusion Term 344
1123 All Terms 349
113 Computation of Geometrical Properties 349
114 Flux based Solution Methodology 352
1141 Explicit Method 354
1142 Implementation Details 354
Problems 359
12 Computational Fluid Dynamics on a Curvilinear Grid361
121 Physical Law based Finite Volume Method 361
1211 Mass Conservation 363
1212 Momentum Conservation 363
122 Solution Methodology Semi-Explicit Method 372
1221 Predictor Step 373
1222 Corrector Step 375
Problems 380
References 383
Index 389
Part I
Introduction and Essentials
The book on an introductory course in CFD starts with the two intro-
ductory chapters on CFD followed by one chapter each on essentials
of two prerequisite courses minus uid-dynamics and heat transfer and
numerical-methods The rst chapter on introduction presents an
eventful thoughtful and intuitive discussion on What is CFD and
Why to study CFD whereas How CFD works which involves lots
of details is presented separately in the second chapter The third-
chapter on essentials of uid-dynamics and heat transfer presents
physical-laws transport-mechanisms physical-law based dierential-
formulation volumetric and ux terms and mathematical formula-
tion The fourth-chapter on essentials of numerical-methods presents
nite dierence method iterative solution of the system of linear alge-
braic equation numerical dierentiation and numerical integration
The presentation for the essential of the two prerequisite courses is
customized to CFD development application and analysis
1
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
1
Introduction
The introduction of any course consists of the three basic questions
What is it Why to study How it works An eventful thoughtful
and intuitive answer to the rst two questions are presented in this
chapter The third question which involves lots of working details
on the three aspects of the Computational Fluid Dynamics (CFD)
(development application and analysis) is presented separately in
the next chapter The novelty scope and purpose of this book is
also presented at the end of this chapter
11 CFD What is it
CFD is a theoretical-method of scientic and engineering investiga-
tion concerned with the development and application of a video-
camera like tool (a software) which is used for a unied cause-and-
eect based analysis of a uid-dynamics as well as heat and mass
transfer problem
CFD is a subject where you rst learn how to develop a product
minus a software which acts like a virtual video-camera Then you learn
how to apply or use this product to generate uid-dynamics movies
Finally you learn how to analyze the detailed spatial and temporal
uid-dynamics information in the movies The analysis is done to
come up with a scientically-exciting as well as engineering-relevant
story (unied cause-and-eect study) of a uid-dynamics situation in
nature as well as industrial-applications With a continuous develop-
ment and a wider application of CFD the word uid-dynamics in
3
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
4 1 Introduction
the CFD has become more generic as it also corresponds to heat and
mass transfer as well as chemical reaction
Fluid-dynamics information are of two types scientic and en-
gineering Scientic-information corresponds to a structure of the
heat and uid ow due to a physical phenomenon in uid-dynamics
such as boundary-layer ow-separation wake-formation and vortex-
shedding The ow structures are obtained from a temporal variation
of ow-properties (such as velocity pressure temperature and many
other variables which characterizes a ow) called as scientically-
exciting movies Engineering-information corresponds to certain pa-
rameters which are important in an engineering application called
as engineering-parameters for example lift force drag force wall
shear stress pressure drop and rate of heat-transfer They are ob-
tained as the temporal variation of engineering-parameters called as
engineering-relevant movies The engineering-parameters are the ef-
fect which are caused by the ow structures minus their correlation leads
to the unied cause-and-eect study A uid-dynamics movie consist
of a time-wise varying series of pictures which give uid-dynamics
information Each ow-property and engineering-parameter results
in one scientically-exciting and engineering-relevant movie respec-
tively When the large number of both the types of movies are
played synchronized in-time the detailed spatial as well as tempo-
ral uid-dynamic information greatly helps in the unied cause-and-
eect study of a uid-dynamics problem
Alternatively CFD is a subject concerned with the development of
computer-programs for the computer-simulation and study of a natu-
ral or an engineering uid-dynamics system The development starts
with the virtual-development of the system involving a geometri-
cal information based computational development of a solid-model
for the system the system may be already-existing or yet-to-be-
developed physically Thereafter the development involves numerical
solution of the governing algebraic equations for CFD formulated
from the physical laws (conservation of mass momentum and en-
ergy) and initial as well as boundary conditions corresponding to the
system CFD study involves the analysis of the uid-dynamics and
heat-transfer results obtained from the computer-simulations of the
virtual-system which is subjected to certain governing-parameters
Introduction to CFD 5
111 CFD as a Scientic and Engineering Analysis Tool
For a better understanding of the denition of the CFD and its ap-
plication as a scientic and engineering analysis tool two example
problems are presented rst conduction heat transfer in a plate
and second uid ow across a circular pillar of a bridge The rst-
problem correspond to an engineering-application for electronic cool-
ing and the second-problem is studied for an engineering-design of
the structure subjected to uid-dynamics forces and are shown in
Fig 11 Both the problems are considered as two-dimensional and
unsteady However the results asymptotes from the unsteady to a
steady state in the rst-problem and to a periodic (dynamic-steady)
state in the second-problem
Figure 11 Schematic and the associated parameters for the two example prob-lems (a) thermal conduction module of a heat sink and (b) theclassical ow across a circular cylinder
6 1 Introduction
The rst-problem is taken from the classic book on heat-transfer
by Incropera and Dewitt (1996) shown in Fig 11(a) The g-
ure shows a periodic module of the plate where a constant heat-ux
qW = 10Wcm2 (dissipated from certain electronic devices) is rst con-
ducted to an aluminum plate and then convected to the water ow-
ing through the rectangular channel grooved in the plate The forced
convection results in a convection coecient of h = 5000Wm2K in-
side the channel The unsteady heat-transfer results in an interesting
time-wise varying conduction heat ow in the plate and instantaneous
convection heat transfer rate Qprimeconv (per unit width of the plate per-
pendicular to the plane shown in Fig 11(a)) into the water
The second-problem is encountered if you are standing over a
bridge and look below to the water passing over a pillar (of circu-
lar cross-section) of the bridge shown in Fig 11(b) as a general case
of 2D ow across a cylinder The uid-dynamics results in a beau-
tiful time-wise varying structure of the uid ow just downstream
of the pillar and certain engineering-parameters The parameters
correspond to the uid-dynamic forces acting in the horizontal (or
streamwise) and vertical (or transverse) direction called as the drag-
force FD and lift-force FL respectively The respective force is pre-
sented below in a non-dimensional form called as the drag-coecient
CD and lift-coecient CL equations shown in Fig 11(b)
If you use your video camera to capture a movie for the two
problems you will get series of pictures for the plate in the rst-
problem and the owing uid in the second-problem Since these
movies result in a pictorial information but not the ow-property
based uid-dynamic information such movies are not relevant for
the uid-dynamics and heat-transfer study Instead if you use a
CFD software for computer simulation you can capture various types
of movies The various ow-properties based scientically-exciting
movies are presented here as the pictures at certain time-instant in
Fig 12(a)-(d) and Fig 13(a1)-(d2) The former gure shows gray-
colored ooded contour for the temperature whereas the latter gure
shows gray-colored ooded contour for the pressure and vorticity and
vector-plot for the velocity For the gray-colored ooded contours a
color bar can be seen in the gures for the relative magnitude of the
variable These gures also show line contours of stream-function
Introduction to CFD 7
and heat-function as streamlines and heatlines respectively They
are the visualization techniques for a 2D steady heat and uid ow
The heatlines are analogous to streamlines and represents the direc-
tion of heat ow under steady state condition proposed by Kimura
and Bejan (1983) later presented in a heat-transfer book by Bejan
(1984) The steady-state heatlines can be seen in Fig 12(d) where
the direction for the conduction-ux is tangential at any point on the
heatlines analogous to that for the mass-ux in a streamline Thus
for a better presentation of the structure of heat and uid ow the
scientically-exciting movies are not only limited to ow-properties
but also correspond to certain ow visualization techniquesAn engineering-parameter based engineering-relevant uid dy-
namic movie is presented in Fig 12(e) for the convection heat trans-
fer rate per unit width at the various walls on the channel Qprimew at the
west-wall Qprimen at the north-wall Q
primee at the east-wall and their cumu-
lative value Qprimeconv (refer Fig 11(a)) Another engineering-relevant
movie is presented in Fig 13(e) for the lift and drag coecient Note
that the movie here corresponds to the motion of a lled symbol over
a line for the temporal variation of an engineering-parameter Thus
the engineering-relevant movies are one-dimensional lower than the
scientically-exciting movies here for the 2D problems the former
movie is presented as 1D and the latter as 2D results
1111 Heat-Conduction
For the computational set-up of the heat conduction problem shown
in Fig 11(a) the results obtained from a 2D CFD simulation of
the conduction heat transfer are shown in Fig 12 For a uni-
form initial temperature of the plate as Tinfin = 15oC Fig 12(a)-(d)
shows the isotherms and heatlines in the plate as the pictures of the
scientically-exciting movies at certain discrete time instants t = 25
5 10 and 20 sec For the same discrete time instants the temporal
variation of the position of the symbols over a line plot is presented in
Fig 12(e) The gure is shown for each of the heat transfer rate and
is obtained from a engineering-relevant 1D movie There is a tempo-
ral variation of picture in the scientically-exciting movie and that
of position of symbol (along the line plot) in the engineering-relevant
movieAccording to caloric theory of the famous French scientist An-
tonie Lavoisier heat is considered as an invisible tasteless odorless
8 1 Introduction
t(sec)
Qrsquo (W
m)
0 5 10 15 20 250
200
400
600
800
1000Qrsquo
conv
Qrsquow
Qrsquon
Qrsquoe
Qrsquoin
(e)
(b)
(c)(d)
(a)
(a) t=25 sec31
29
27
25
23
21
19
17
15
T(oC)
(c) t=10 sec
(b) t=5 sec
(d) t=20 sec
Figure 12 Thermal conduction module of a heat sink temporal variation of(a)-(d) isotherms and heatlines and (e) the convection heat transferrate per unit width at the various walls of the channels and theircumulative total value Q
primeconv For the discrete time-instant corre-
sponding to heat ow patterns in (a)-(d) temporal variation of thevarious heat transfer rates are shown as symbols in (e) (a)-(c) arethe transient and (d) represents a steady state result
weightless uid which he called caloric uid Presently the caloric
theory is overtaken by mechanical theory of heat minus heat is not a sub-
stance but a dynamical form of mechanical eect (Thomson 1851)
Nevertheless there are certain problems involving heat ow for which
Lavoisiers approach is rather useful such as the discussion on heat-
lines here
The role of heatlines for heat ow is analogous to that of stream-
lines for uid ow (Paramane and Sharma 2009) For the uid-ow
the dierence of stream-function values represents the rate of uid
ow the function remains constant on a solid wall the tangent to
a streamline represents the direction of the uid-ow (with no ow
in the normal direction) and the streamline originate or emerge at
Introduction to CFD 9
a mass source Analogously for the heat-ow the dierence of heat-
function values represents the rate of heat ow the function remains
constant on an adiabatic wall the tangent to a heatline represents the
direction of the heat-ow (with no ow in the normal direction) and
the heatline originate or emerge at the heat-source For the present
problem in Fig 11(a) the heat-source is at the inlet (or left) bound-
ary and the heat-sink is at the surface of the channel Thus the
steady-state heatlines in Fig 12(d) shows the heat ow emerging
from the inletsource and ending on the channel-wallssink Further-
more it can be seen in the gure that the heatlines which are close
to the adiabatic walls are parallel to the walls The source sink and
adiabatic-walls can be seen in Fig 11(a)
Figure 12(e) shows an asymptotic time-wise increase in the con-
vection heat transfer per unit width at the various walls of the channel
and its cumulative value Qprimeconv (= Q
primew+Q
primen+Q
primee) It can be also be seen
that the heat-transfer per unit width Qprimew at the west-wall and Q
primen at
the north-wall are larger than Qprimee at the east-wall of the channel As
time progresses the gure shows that Qprimeconv increases monotonically
and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-
der steady state condition refer Fig 11(a) The dierence between
the Qprimein and Q
primeconv under the transient condition is utilized for the
sensible heating and results in the increase in the temperature of the
plate Under the steady state condition note from Fig 12(d) that
the maximum temperature is close to 32oC minus well within the per-
missible limit for the reliable and ecient operation of the electronic
device
1112 Fluid-Dynamics
For the free-stream ow across a circular cylinderpillar at a Reynolds
number of 100 the non-dimensional results obtained from a 2D CFD
simulation are shown in Fig 13 In contrast to the previous prob-
lem which reaches steady state the present problem reaches to an un-
steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-
ent increasing time instants within one time-period of the periodic-
ow the results are presented in Fig 13(a)-(d) The gure shows
the close-up view of the pictures of the scientically-exciting movies
for ow-properties (velocity pressure and vorticity) and streamlines
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow
Introduction to
Computational Fluid Dynamics
Development Application and Analysis
Dr Atul Sharma
Professor Deptt of Mech EngineeringIIT Bombay
Introduction to
Computational Fluid Dynamics
Development Application and Analysis
Introduction to Computational Fluid DynamicsDevelopment Application and Analysis
Dr Atul Sharma
copy Author 2017
This Edition Published by
John Wiley amp Sons LtdThe Atrium Southern GateChichester West Sussex PO19 8SQ United KingdomTel +44 (0)1243 779777Fax +44 (0)1243 775878e-mail customerwileycomWeb wwwwileycom
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All rights reserved No part of this publication may be reproduced stored in a retrieval system or transmitted in any form or by any means electronic mechanical photocopying recording or otherwise except as permitted by the UK Copyright Designs and Patents Act 1988 without the prior permission of the publisher
Library Congress Cataloging-in-Publication DataA catalogue record for this book is available from the British Library
Cover Image The image for C F and D in the cover page is obtained from a Computational Fluid Dynamics simulations as follows
C Curvilinear-Grid and Resultant-Velocity Contours for Fluid-Flow in a C-shaped Bend of a Pipe
F Temperature-Contours in a Heat Conduction Problem
D Vertical-Velocity Contours for a vertical-wall (moving upward like a conveyor belt) driven flow in a D shaped cavity
DisclaimerThe contents sources and the information provided by the authors in different chapters are the sole responsibility of the authors themselves including the copyright issues Editors and Publishers have no liabilities whatsoever towards these in any manner
This book is dedicated
to
Suhas V Patankar
Dr Suhas V Patankar1 (born February 22 1941) is an Indian
mechanical engineer He is a pioneer in the eld of computational
uid dynamics (CFD) and nite volume method He received his
BE from the University of Pune in 1962 MTech from IIT Bombay
in 1964 and PhD from Imperial College London in 1967
Dr Patankar is currently a Professor Emeritus in the Mechanical
Engineering Department at the University of Minnesota where he
worked for 25 years (from 1975-2000) Earlier he held teaching and
research positions at IIT Kanpur Imperial College and University
of Waterloo He is also the President of Innovative Research Inc He
has authored or co-authored four books published over 150 papers
advised 35 completed PhD theses and lectured extensively in the
USA and abroad His 1980 book Numerical heat Transfer and Fluid
Flow is considered to be a groundbreaking contribution to CFD He
is one of the most cited authors in science and engineering
For excellence in teaching Dr Patankar received the 1983 George
Taylor Distinguished Teaching Award and the 1989-90 Morse-Alumni
1Source (for the Biography) Personal Communication
v
vi
Award for Outstanding Contributions to Undergraduate Education
For his research contributions to computational heat transfer he was
given the 1991 ASME Heat Transfer Memorial Award and the 1997
Classic Paper Award He was awarded the 2008 Max Jakob Award
which is considered to be the highest international honor in the eld of
heat transfer In 2015 an International Conference on Computational
Heat Transfer (CHT-15) held at Rutgers University in New Jersey
was dedicated to Professor Patankar
Dr Patankars widespread inuence on research and engineering
education has been recognized in many ways In 2007 the Editors of
the International Journal of Heat and Mass Transfer wrote There is
no person who has made a more profound and enduring impact on the
theory and practice of numerical simulation in mechanical engineering
than Professor Patankar
FOREWORD
by Prof K Muralidhar
Indian Institute of Technology Kanpur INDIA
The subject of computational uid dynamics (CFD) emerged in the
academic context but in the past two decades it has developed to a
point where it is extensively employed as a design and analysis tool in
the industry The impact of the subject is deeply felt in diverse disci-
plines ranging from the aerospace spreading all the way to the chem-
ical industry New compact and ecient designs are enabled and it
has become that much more convenient to locate optimum conditions
for the operation of engineering systems As it stands CFD is com-
pulsory learning in several branches of engineering Many universities
now opt to develop theoretical courses around it The great utility of
this tool has inuenced adjacent disciplines and the process of solv-
ing complex dierential equations by the process of discretization has
become a staple for addressing engineering challenges
For an emerging discipline it is necessary that new books fre-
quently appear on the horizon Students can choose from the se-
lection and will stand to gain from it Such books should have a
perspective of the past and the future and it is all the more signi-
cant when the author has exclusive training in the subject Dr Atul
Sharma represents a combination of perspective training and passion
that combines well and delivers a strong text
I have personal knowledge of the fact that Dr Atul Sharma has
developed a whole suite of computer programs on CFD from scratch
He has wondered and pondered over why approximations work and
conditions under which they yield meaningful solutions His spread
vii
viii Foreword
of experience over fteen and odd years includes two and three di-
mensional Navier-Stokes equations heat transfer interfacial dynam-
ics and turbulent ow He has championed the nite volume method
which is now the industry standard The programming experience
is important because it brings in the precision needed for assembling
the jigsaw puzzle The sprinkling of computer programs in the book
is no coincidence
Atul knows the conventional method of discretizing dierential
equations but has never been satised with it He has felt an ele-
ment of discomfort going away from a physical reality into a world
of numbers connected by arithmetic operations As a result he has
developed a principle that physical laws that characterize the dier-
ential equations should be reected at every stage of discretization
and every stage of approximation indeed at every scale whether a
single tetrahedron or the sub-domain or the region as a whole This
idea permeates the book where it is shown that discretized versions
must provide reasonable answers because they continue to represent
the laws of nature
This new CFD book is comprehensive and has a stamp of origi-
nality of the author It will bring students closer to the subject and
enable them to contribute to it
Kanpur
August 2016 K Muralidhar
PREFACE
As a Post-Graduate (PG) student in 1997 I got introduced to a
course on CFD at Indian Institute of Science (IISc) Bangalore Dur-
ing the introductory course on CFD although I struggled hard
in the physical understanding and computer-programming it didnt
stop my interest and enthusiasm to do research teaching and writ-
ing a text-book on a introductory course in CFD Since then I did
my Masters project work on CFD (1997-98) introduced and taught
this course to Under-Graduate (UG) students at National Institute
of Technology Hamirpur HP (1999-2000) did my Doctoral work on
this subject at Indian Institute of Technology (IIT) Kanpur (2000-
2004) and co-authored a Chapter on nite volume method in a re-
vised edition of an edited book on computational uid ow and heat
transfer during my PhD in 2003 After joining as a faculty in 2004
I taught this course to UG as well as PG students of dierent depart-
ments at IIT Bombay as well as to the students of dierent collages in
India through CDEEP (Center for Distance Engineering Programme)
IIT Bombay from 2007-2010 Furthermore I gave series of lecture
on CFD at various collages and industries in India More recently
in 2012 I delivered a ve-days lecture and lab-session on CFD (to
around 1400 collage teachers from dierent parts of India) as a part
of a project funded by ministry of human resource development under
NMEICT (National Mission on Education through information and
communication technology)
The increasing need for the development of the customized CFD
software (app) and wide spread CFD application as well as analysis
for the design optimization and innovation of various types of engi-
neering systems always motivated me to write this book Due to the
increasing importance of CFD another motivation is to present CFD
ix
x Preface
simple enough to introduce the rst-course at early under-graduate
curriculum rather than the post-graduate curriculum The moti-
vation got strengthened by my own as well as others (students
teachers researchers and practitioners) struggle on physical under-
standing of the mathematics involved during the Partial Dierential
Equation (PDE) based algebraic-formulation (nite volume method)
along with the continuous frustrating pursuit to eectively convert
the theory of CFD into the computer-program
This led me to come up with an alternate physical-law (PDE-
free) based nite volume method as well as solution methodol-
ogy which are much easier to comprehend Mostly the success of
the programming eort is decided by the perfect execution of the
implementations-details which rarely appears in-print (existing books
and journal articles) The physical law based CFD development
approach and the implementations-details are the novelties in the
present book Furthermore using an open-source software for nu-
merical computations (Scilab) computer programs are given in this
book on the basic modules of conduction advection and convection
Indeed the reader can generalize and extend these codes for the de-
velopment of Navier-Stokes solver and generate the results presented
in the chapters towards the end of this book
I have limited the scope of this book to the numerical-techniques
which I wish to recommend for the book although there are numerous
other advanced and better methods in the published literature I do
not claim that the programming-practice presented here are the most
ecient minus they are presented here more for the ease in programming
(understand as your program) than for the computational eciency
Furthermore although I am enthusiastic about my presentation of
the physical law based CFD approach it may not be more ecient
than the alternate mathematics based CFD approach The emphasis
in this book is on the ease of understanding of the formulations and
programming-practice for the introductory course on CFD
I would hereby acknowledge that I owe my greatest debt to Prof
J Srinivasan (my ME Project supervisor at IISc Bangalore) who
did a hand-holding and wonderful job in patiently introducing me to
the fascinating world of research in uid-dynamics and heat trans-
fer I would also acknowledge the excellent training from Prof V
Introduction to CFD xi
Eswaran (my PhD supervisor at IIT Kanpur) from whom I learned
systematic way of CFD development application and analysis His
inspiring comments and suggestions on my writing-style and elabora-
tion of the rst chapter in this book is also gratefully acknowledged
I also want to record my sincere thanks to Prof Pradip Datta IISc
Bangalore and Prof Gautam Biswas IIT Kanpur minus my teachers for
CFD course Special thanks to Prof K Muralidhar IIT Kanpur for
his personal encouragement academic support and motivation (for
the book writing) He also cleared my dilemma in the proper presen-
tation of the novelty in this book I thank him also for writing the
foreword of this book
I am grateful to Prof Kannan M Moudgalya and Prof Deepak B
Phatak from IIT Bombay for giving me opportunity to teach CFD
in a distance education mode I am also grateful to Prof Amit
Agrawal (my colleague at IIT Bombay) who diligently read through
all the chapters of this book and suggested some really remarkable
changes in the book I also thank him for the research interactions
over the last decade which had substantially improved my under-
standing of thermal and uid science I would also like to thank my
PhD student Namshad T who did a great job in implementing
my ideas on generating the CFD simulation based CFD image for
the cover-page and couple of gures in the rst chapter (Fig 12
and 13) Special mention that the Scilab codes developed (for the
CFD course under NMEICT) by Vishesh Aggarwal greatly helped
me to develop the codes presented in this book his contributions is
gratefully acknowledged I thank Malhar Malushte for testing some
of the formulations presented in this book
I thank my research students (Sachin B Paramane D Datta C
M Sewatkar Vinesh H Gada Kaushal Prasad Mukul Shrivastava
Absar M Lakdawala Mausam Sarkar and Javed Shaikh) for the
diligence and sincerity they have shown in unraveling some of our
ideas on CFD Apart from these several other research students and
the students teachers and CFD practitioners who took my lectures
on CFD in the past decade have contributed to my better under-
standing of CFD and inspired me to write the book I thank all of
them for their attention and enthusiastic response during the CFD
courses The inuence of the interactions with all of them can be seen
xii Preface
throughout this book I am grateful to IIT Bombay for giving me
one year sabbatical leave to write this book The ambiance of aca-
demic freedom considerate support by the faculties and the friendly
atmosphere inside the campus of IIT Bombay has greatly contributed
in the solo eort of book writing I thanks all my colleagues at IIT
Bombay for shaping my perceptions on CFD
During this long endeavor the patient support by Mr Sunil
Saxena Director Athena Academic Ltd UK is gratefully acknowl-
edged I also thank John Wiley for coming forward to take this
book to international markets Lastly I would acknowledge that I
am greatly inspired by the writing style of J D Anderson and the
book on CFD by S V Patankar Of course you will nd that I
have heavily cross-referred the CFD book by the two most acknowl-
edged pioneers on CFD I dedicate this book to Dr S V Patankar
whose book nicely introduced me to the computational as well as
ow physics in CFD and greatly helped me to introduce a physical
approach of CFD development in this book
The culmination of the book writing process fullls a long cher-
ished dream It would not have been possible without the excel-
lent environment continuous support and complete understanding
of my wife Anubha and daughter Anshita Their kindness has greatly
helped to fulll my dream now I plan to spend more time with my
wife and daughter
Mumbai
August 2016 Atul Sharma
Contents
I Introduction and Essentials
1
1 Introduction 3
11 CFD What is it 3
111 CFD as a Scientic and Engineering Analysis
Tool 5
112 Analogy with a Video-Camera 13
12 CFD Why to study 15
13 Novelty Scope and Purpose of this Book 16
2 Introduction to CFD Development Application and
Analysis 23
21 CFD Development 23
211 Grid Generation Pre-Processor 24
212 Discretization Method Algebraic Formulation 26
213 Solution Methodology Solver 29
214 Computation of Engineering-Parameters
Post-Processor 34
215 Testing 35
22 CFD Application 35
23 CFD Analysis 38
24 Closure 39
xiii
xiv Contents
3 Essentials of Fluid-Dynamics and Heat-Transfer for
CFD 41
31 Physical Laws 42
311 FundamentalConservation Laws 43
312 Subsidiary Laws 45
32 Momentum and Energy Transport Mechanisms 46
33 Physical Law based Dierential Formulation 48
331 Continuity Equation 49
332 Transport Equation 51
34 Generalized Volumetric and Flux Terms and their Dif-
ferential Formulation 57
341 Volumetric Term 58
342 Flux-Term 58
343 Discussion 62
35 Mathematical Formulation 63
351 Dimensional Study 63
352 Non-Dimensional Study 67
36 Closure 71
4 Essentials of Numerical-Methods for CFD 73
41 Finite Dierence Method A Dierential to Algebraic
Formulation for Governing PDE and BCs 75
411 Grid Generation 75
412 Finite Dierence Method 78
413 Applications to CFD 92
42 Iterative Solution of System of LAEs for a Flow Property 93
421 Iterative Methods 94
422 Applications to CFD 98
43 Numerical Dierentiation for Local Engineering-
Parameters 105
431 Dierentiation Formulas 106
432 Applications to CFD 107
44 Numerical Integration for the Total value of
Engineering-Parameters 110
441 Integration Rules 111
442 Applications to CFD 114
45 Closure 116
Problems 116
Introduction to CFD xv
II CFD for a Cartesian-Geometry
119
5 Computational Heat Conduction 12151 Physical Law based Finite Volume Method 122
511 Energy Conservation Law for a Control Volume 123512 Algebraic Formulation 124513 Approximations 127514 Approximated Algebraic-Formulation 130515 Discussion 133
52 Finite Dierence Method for Boundary Conditions 13853 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 139531 One-Dimensional Conduction 139532 Two-Dimensional Conduction 147
54 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 154541 One-Dimensional Conduction 154542 Two-Dimensional Conduction 166
Problems 176
6 Computational Heat Advection 17961 Physical Law based Finite Volume Method 180
611 Energy Conservation Law for a Control Volume 181612 Algebraic Formulation 181613 Approximations 183614 Approximated Algebraic Formulation 191615 Discussion 191
62 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 194621 Explicit-Method 197622 Implementation Details 198623 Solution Algorithm 199
63 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 207631 Advection Scheme on a Non-Uniform Grid 207632 Explicit and Implicit Method 210633 Implementation Details 216634 Solution Algorithm 220
Problems 228
xvi Contents
7 Computational Heat Convection 22971 Physical Law based Finite Volume Method 229
711 Energy Conservation Law for a Control Volume 229712 Algebraic Formulation 231713 Approximated Algebraic Formulation 232
72 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 235721 Explicit-Method 236722 Implementation Details 237723 Solution Algorithm 238
73 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 242Problems 248
8 Computational Fluid Dynamics Physical Law based
Finite Volume Method 25181 Generalized Variables for the Combined Heat and
Fluid Flow 25282 Conservation Laws for a Control Volume 25583 Algebraic Formulation 25984 Approximations 26085 Approximated Algebraic Formulation 263
851 Mass Conservation 263852 MomentumEnergy Conservation 264
86 Closure 269
9 Computational Fluid Dynamics on a Staggered Grid 27191 Challenges in the CFD Development 273
911 Non-Linearity 273912 Equation for Pressure 273913 Pressure-Velocity Decoupling 273
92 A Staggered Grid to avoid Pressure-Velocity Decoupling27593 Physical Law based FVM for a Staggered Grid 27794 Flux based Solution Methodology on a Uniform Grid
Semi-Explicit Method 281941 Philosophy of Pressure-Correction Method 283942 Semi-Explicit Method 286943 Implementation Details 292944 Solution Algorithm 298
95 Initial and Boundary Conditions 300951 Initial Condition 300952 Boundary Condition 301
Problems 306
Introduction to CFD xvii
10 Computational Fluid Dynamics on a Co-located Grid309
101 Momentum-Interpolation Method Strategy to avoid
the Pressure-Velocity Decoupling on a Co-located Grid 310
102 Coecients of LAEs based Solution Methodology on
a Non-Uniform Grid Semi-Explicit and Semi-Implicit
Method 314
1021 Predictor Step 316
1022 Corrector Step 318
1023 Solution Algorithm 323
Problems 329
III CFD for a Complex-Geometry
331
11 Computational Heat Conduction on a Curvilinear
Grid 333
111 Curvilinear Grid Generation 333
1111 Algebraic Grid Generation 334
1112 Elliptic Grid Generation 336
112 Physical Law based Finite Volume Method 343
1121 Unsteady and Source Term 343
1122 Diusion Term 344
1123 All Terms 349
113 Computation of Geometrical Properties 349
114 Flux based Solution Methodology 352
1141 Explicit Method 354
1142 Implementation Details 354
Problems 359
12 Computational Fluid Dynamics on a Curvilinear Grid361
121 Physical Law based Finite Volume Method 361
1211 Mass Conservation 363
1212 Momentum Conservation 363
122 Solution Methodology Semi-Explicit Method 372
1221 Predictor Step 373
1222 Corrector Step 375
Problems 380
References 383
Index 389
Part I
Introduction and Essentials
The book on an introductory course in CFD starts with the two intro-
ductory chapters on CFD followed by one chapter each on essentials
of two prerequisite courses minus uid-dynamics and heat transfer and
numerical-methods The rst chapter on introduction presents an
eventful thoughtful and intuitive discussion on What is CFD and
Why to study CFD whereas How CFD works which involves lots
of details is presented separately in the second chapter The third-
chapter on essentials of uid-dynamics and heat transfer presents
physical-laws transport-mechanisms physical-law based dierential-
formulation volumetric and ux terms and mathematical formula-
tion The fourth-chapter on essentials of numerical-methods presents
nite dierence method iterative solution of the system of linear alge-
braic equation numerical dierentiation and numerical integration
The presentation for the essential of the two prerequisite courses is
customized to CFD development application and analysis
1
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
1
Introduction
The introduction of any course consists of the three basic questions
What is it Why to study How it works An eventful thoughtful
and intuitive answer to the rst two questions are presented in this
chapter The third question which involves lots of working details
on the three aspects of the Computational Fluid Dynamics (CFD)
(development application and analysis) is presented separately in
the next chapter The novelty scope and purpose of this book is
also presented at the end of this chapter
11 CFD What is it
CFD is a theoretical-method of scientic and engineering investiga-
tion concerned with the development and application of a video-
camera like tool (a software) which is used for a unied cause-and-
eect based analysis of a uid-dynamics as well as heat and mass
transfer problem
CFD is a subject where you rst learn how to develop a product
minus a software which acts like a virtual video-camera Then you learn
how to apply or use this product to generate uid-dynamics movies
Finally you learn how to analyze the detailed spatial and temporal
uid-dynamics information in the movies The analysis is done to
come up with a scientically-exciting as well as engineering-relevant
story (unied cause-and-eect study) of a uid-dynamics situation in
nature as well as industrial-applications With a continuous develop-
ment and a wider application of CFD the word uid-dynamics in
3
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
4 1 Introduction
the CFD has become more generic as it also corresponds to heat and
mass transfer as well as chemical reaction
Fluid-dynamics information are of two types scientic and en-
gineering Scientic-information corresponds to a structure of the
heat and uid ow due to a physical phenomenon in uid-dynamics
such as boundary-layer ow-separation wake-formation and vortex-
shedding The ow structures are obtained from a temporal variation
of ow-properties (such as velocity pressure temperature and many
other variables which characterizes a ow) called as scientically-
exciting movies Engineering-information corresponds to certain pa-
rameters which are important in an engineering application called
as engineering-parameters for example lift force drag force wall
shear stress pressure drop and rate of heat-transfer They are ob-
tained as the temporal variation of engineering-parameters called as
engineering-relevant movies The engineering-parameters are the ef-
fect which are caused by the ow structures minus their correlation leads
to the unied cause-and-eect study A uid-dynamics movie consist
of a time-wise varying series of pictures which give uid-dynamics
information Each ow-property and engineering-parameter results
in one scientically-exciting and engineering-relevant movie respec-
tively When the large number of both the types of movies are
played synchronized in-time the detailed spatial as well as tempo-
ral uid-dynamic information greatly helps in the unied cause-and-
eect study of a uid-dynamics problem
Alternatively CFD is a subject concerned with the development of
computer-programs for the computer-simulation and study of a natu-
ral or an engineering uid-dynamics system The development starts
with the virtual-development of the system involving a geometri-
cal information based computational development of a solid-model
for the system the system may be already-existing or yet-to-be-
developed physically Thereafter the development involves numerical
solution of the governing algebraic equations for CFD formulated
from the physical laws (conservation of mass momentum and en-
ergy) and initial as well as boundary conditions corresponding to the
system CFD study involves the analysis of the uid-dynamics and
heat-transfer results obtained from the computer-simulations of the
virtual-system which is subjected to certain governing-parameters
Introduction to CFD 5
111 CFD as a Scientic and Engineering Analysis Tool
For a better understanding of the denition of the CFD and its ap-
plication as a scientic and engineering analysis tool two example
problems are presented rst conduction heat transfer in a plate
and second uid ow across a circular pillar of a bridge The rst-
problem correspond to an engineering-application for electronic cool-
ing and the second-problem is studied for an engineering-design of
the structure subjected to uid-dynamics forces and are shown in
Fig 11 Both the problems are considered as two-dimensional and
unsteady However the results asymptotes from the unsteady to a
steady state in the rst-problem and to a periodic (dynamic-steady)
state in the second-problem
Figure 11 Schematic and the associated parameters for the two example prob-lems (a) thermal conduction module of a heat sink and (b) theclassical ow across a circular cylinder
6 1 Introduction
The rst-problem is taken from the classic book on heat-transfer
by Incropera and Dewitt (1996) shown in Fig 11(a) The g-
ure shows a periodic module of the plate where a constant heat-ux
qW = 10Wcm2 (dissipated from certain electronic devices) is rst con-
ducted to an aluminum plate and then convected to the water ow-
ing through the rectangular channel grooved in the plate The forced
convection results in a convection coecient of h = 5000Wm2K in-
side the channel The unsteady heat-transfer results in an interesting
time-wise varying conduction heat ow in the plate and instantaneous
convection heat transfer rate Qprimeconv (per unit width of the plate per-
pendicular to the plane shown in Fig 11(a)) into the water
The second-problem is encountered if you are standing over a
bridge and look below to the water passing over a pillar (of circu-
lar cross-section) of the bridge shown in Fig 11(b) as a general case
of 2D ow across a cylinder The uid-dynamics results in a beau-
tiful time-wise varying structure of the uid ow just downstream
of the pillar and certain engineering-parameters The parameters
correspond to the uid-dynamic forces acting in the horizontal (or
streamwise) and vertical (or transverse) direction called as the drag-
force FD and lift-force FL respectively The respective force is pre-
sented below in a non-dimensional form called as the drag-coecient
CD and lift-coecient CL equations shown in Fig 11(b)
If you use your video camera to capture a movie for the two
problems you will get series of pictures for the plate in the rst-
problem and the owing uid in the second-problem Since these
movies result in a pictorial information but not the ow-property
based uid-dynamic information such movies are not relevant for
the uid-dynamics and heat-transfer study Instead if you use a
CFD software for computer simulation you can capture various types
of movies The various ow-properties based scientically-exciting
movies are presented here as the pictures at certain time-instant in
Fig 12(a)-(d) and Fig 13(a1)-(d2) The former gure shows gray-
colored ooded contour for the temperature whereas the latter gure
shows gray-colored ooded contour for the pressure and vorticity and
vector-plot for the velocity For the gray-colored ooded contours a
color bar can be seen in the gures for the relative magnitude of the
variable These gures also show line contours of stream-function
Introduction to CFD 7
and heat-function as streamlines and heatlines respectively They
are the visualization techniques for a 2D steady heat and uid ow
The heatlines are analogous to streamlines and represents the direc-
tion of heat ow under steady state condition proposed by Kimura
and Bejan (1983) later presented in a heat-transfer book by Bejan
(1984) The steady-state heatlines can be seen in Fig 12(d) where
the direction for the conduction-ux is tangential at any point on the
heatlines analogous to that for the mass-ux in a streamline Thus
for a better presentation of the structure of heat and uid ow the
scientically-exciting movies are not only limited to ow-properties
but also correspond to certain ow visualization techniquesAn engineering-parameter based engineering-relevant uid dy-
namic movie is presented in Fig 12(e) for the convection heat trans-
fer rate per unit width at the various walls on the channel Qprimew at the
west-wall Qprimen at the north-wall Q
primee at the east-wall and their cumu-
lative value Qprimeconv (refer Fig 11(a)) Another engineering-relevant
movie is presented in Fig 13(e) for the lift and drag coecient Note
that the movie here corresponds to the motion of a lled symbol over
a line for the temporal variation of an engineering-parameter Thus
the engineering-relevant movies are one-dimensional lower than the
scientically-exciting movies here for the 2D problems the former
movie is presented as 1D and the latter as 2D results
1111 Heat-Conduction
For the computational set-up of the heat conduction problem shown
in Fig 11(a) the results obtained from a 2D CFD simulation of
the conduction heat transfer are shown in Fig 12 For a uni-
form initial temperature of the plate as Tinfin = 15oC Fig 12(a)-(d)
shows the isotherms and heatlines in the plate as the pictures of the
scientically-exciting movies at certain discrete time instants t = 25
5 10 and 20 sec For the same discrete time instants the temporal
variation of the position of the symbols over a line plot is presented in
Fig 12(e) The gure is shown for each of the heat transfer rate and
is obtained from a engineering-relevant 1D movie There is a tempo-
ral variation of picture in the scientically-exciting movie and that
of position of symbol (along the line plot) in the engineering-relevant
movieAccording to caloric theory of the famous French scientist An-
tonie Lavoisier heat is considered as an invisible tasteless odorless
8 1 Introduction
t(sec)
Qrsquo (W
m)
0 5 10 15 20 250
200
400
600
800
1000Qrsquo
conv
Qrsquow
Qrsquon
Qrsquoe
Qrsquoin
(e)
(b)
(c)(d)
(a)
(a) t=25 sec31
29
27
25
23
21
19
17
15
T(oC)
(c) t=10 sec
(b) t=5 sec
(d) t=20 sec
Figure 12 Thermal conduction module of a heat sink temporal variation of(a)-(d) isotherms and heatlines and (e) the convection heat transferrate per unit width at the various walls of the channels and theircumulative total value Q
primeconv For the discrete time-instant corre-
sponding to heat ow patterns in (a)-(d) temporal variation of thevarious heat transfer rates are shown as symbols in (e) (a)-(c) arethe transient and (d) represents a steady state result
weightless uid which he called caloric uid Presently the caloric
theory is overtaken by mechanical theory of heat minus heat is not a sub-
stance but a dynamical form of mechanical eect (Thomson 1851)
Nevertheless there are certain problems involving heat ow for which
Lavoisiers approach is rather useful such as the discussion on heat-
lines here
The role of heatlines for heat ow is analogous to that of stream-
lines for uid ow (Paramane and Sharma 2009) For the uid-ow
the dierence of stream-function values represents the rate of uid
ow the function remains constant on a solid wall the tangent to
a streamline represents the direction of the uid-ow (with no ow
in the normal direction) and the streamline originate or emerge at
Introduction to CFD 9
a mass source Analogously for the heat-ow the dierence of heat-
function values represents the rate of heat ow the function remains
constant on an adiabatic wall the tangent to a heatline represents the
direction of the heat-ow (with no ow in the normal direction) and
the heatline originate or emerge at the heat-source For the present
problem in Fig 11(a) the heat-source is at the inlet (or left) bound-
ary and the heat-sink is at the surface of the channel Thus the
steady-state heatlines in Fig 12(d) shows the heat ow emerging
from the inletsource and ending on the channel-wallssink Further-
more it can be seen in the gure that the heatlines which are close
to the adiabatic walls are parallel to the walls The source sink and
adiabatic-walls can be seen in Fig 11(a)
Figure 12(e) shows an asymptotic time-wise increase in the con-
vection heat transfer per unit width at the various walls of the channel
and its cumulative value Qprimeconv (= Q
primew+Q
primen+Q
primee) It can be also be seen
that the heat-transfer per unit width Qprimew at the west-wall and Q
primen at
the north-wall are larger than Qprimee at the east-wall of the channel As
time progresses the gure shows that Qprimeconv increases monotonically
and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-
der steady state condition refer Fig 11(a) The dierence between
the Qprimein and Q
primeconv under the transient condition is utilized for the
sensible heating and results in the increase in the temperature of the
plate Under the steady state condition note from Fig 12(d) that
the maximum temperature is close to 32oC minus well within the per-
missible limit for the reliable and ecient operation of the electronic
device
1112 Fluid-Dynamics
For the free-stream ow across a circular cylinderpillar at a Reynolds
number of 100 the non-dimensional results obtained from a 2D CFD
simulation are shown in Fig 13 In contrast to the previous prob-
lem which reaches steady state the present problem reaches to an un-
steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-
ent increasing time instants within one time-period of the periodic-
ow the results are presented in Fig 13(a)-(d) The gure shows
the close-up view of the pictures of the scientically-exciting movies
for ow-properties (velocity pressure and vorticity) and streamlines
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow
Dr Atul Sharma
Professor Deptt of Mech EngineeringIIT Bombay
Introduction to
Computational Fluid Dynamics
Development Application and Analysis
Introduction to Computational Fluid DynamicsDevelopment Application and Analysis
Dr Atul Sharma
copy Author 2017
This Edition Published by
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Library Congress Cataloging-in-Publication DataA catalogue record for this book is available from the British Library
Cover Image The image for C F and D in the cover page is obtained from a Computational Fluid Dynamics simulations as follows
C Curvilinear-Grid and Resultant-Velocity Contours for Fluid-Flow in a C-shaped Bend of a Pipe
F Temperature-Contours in a Heat Conduction Problem
D Vertical-Velocity Contours for a vertical-wall (moving upward like a conveyor belt) driven flow in a D shaped cavity
DisclaimerThe contents sources and the information provided by the authors in different chapters are the sole responsibility of the authors themselves including the copyright issues Editors and Publishers have no liabilities whatsoever towards these in any manner
This book is dedicated
to
Suhas V Patankar
Dr Suhas V Patankar1 (born February 22 1941) is an Indian
mechanical engineer He is a pioneer in the eld of computational
uid dynamics (CFD) and nite volume method He received his
BE from the University of Pune in 1962 MTech from IIT Bombay
in 1964 and PhD from Imperial College London in 1967
Dr Patankar is currently a Professor Emeritus in the Mechanical
Engineering Department at the University of Minnesota where he
worked for 25 years (from 1975-2000) Earlier he held teaching and
research positions at IIT Kanpur Imperial College and University
of Waterloo He is also the President of Innovative Research Inc He
has authored or co-authored four books published over 150 papers
advised 35 completed PhD theses and lectured extensively in the
USA and abroad His 1980 book Numerical heat Transfer and Fluid
Flow is considered to be a groundbreaking contribution to CFD He
is one of the most cited authors in science and engineering
For excellence in teaching Dr Patankar received the 1983 George
Taylor Distinguished Teaching Award and the 1989-90 Morse-Alumni
1Source (for the Biography) Personal Communication
v
vi
Award for Outstanding Contributions to Undergraduate Education
For his research contributions to computational heat transfer he was
given the 1991 ASME Heat Transfer Memorial Award and the 1997
Classic Paper Award He was awarded the 2008 Max Jakob Award
which is considered to be the highest international honor in the eld of
heat transfer In 2015 an International Conference on Computational
Heat Transfer (CHT-15) held at Rutgers University in New Jersey
was dedicated to Professor Patankar
Dr Patankars widespread inuence on research and engineering
education has been recognized in many ways In 2007 the Editors of
the International Journal of Heat and Mass Transfer wrote There is
no person who has made a more profound and enduring impact on the
theory and practice of numerical simulation in mechanical engineering
than Professor Patankar
FOREWORD
by Prof K Muralidhar
Indian Institute of Technology Kanpur INDIA
The subject of computational uid dynamics (CFD) emerged in the
academic context but in the past two decades it has developed to a
point where it is extensively employed as a design and analysis tool in
the industry The impact of the subject is deeply felt in diverse disci-
plines ranging from the aerospace spreading all the way to the chem-
ical industry New compact and ecient designs are enabled and it
has become that much more convenient to locate optimum conditions
for the operation of engineering systems As it stands CFD is com-
pulsory learning in several branches of engineering Many universities
now opt to develop theoretical courses around it The great utility of
this tool has inuenced adjacent disciplines and the process of solv-
ing complex dierential equations by the process of discretization has
become a staple for addressing engineering challenges
For an emerging discipline it is necessary that new books fre-
quently appear on the horizon Students can choose from the se-
lection and will stand to gain from it Such books should have a
perspective of the past and the future and it is all the more signi-
cant when the author has exclusive training in the subject Dr Atul
Sharma represents a combination of perspective training and passion
that combines well and delivers a strong text
I have personal knowledge of the fact that Dr Atul Sharma has
developed a whole suite of computer programs on CFD from scratch
He has wondered and pondered over why approximations work and
conditions under which they yield meaningful solutions His spread
vii
viii Foreword
of experience over fteen and odd years includes two and three di-
mensional Navier-Stokes equations heat transfer interfacial dynam-
ics and turbulent ow He has championed the nite volume method
which is now the industry standard The programming experience
is important because it brings in the precision needed for assembling
the jigsaw puzzle The sprinkling of computer programs in the book
is no coincidence
Atul knows the conventional method of discretizing dierential
equations but has never been satised with it He has felt an ele-
ment of discomfort going away from a physical reality into a world
of numbers connected by arithmetic operations As a result he has
developed a principle that physical laws that characterize the dier-
ential equations should be reected at every stage of discretization
and every stage of approximation indeed at every scale whether a
single tetrahedron or the sub-domain or the region as a whole This
idea permeates the book where it is shown that discretized versions
must provide reasonable answers because they continue to represent
the laws of nature
This new CFD book is comprehensive and has a stamp of origi-
nality of the author It will bring students closer to the subject and
enable them to contribute to it
Kanpur
August 2016 K Muralidhar
PREFACE
As a Post-Graduate (PG) student in 1997 I got introduced to a
course on CFD at Indian Institute of Science (IISc) Bangalore Dur-
ing the introductory course on CFD although I struggled hard
in the physical understanding and computer-programming it didnt
stop my interest and enthusiasm to do research teaching and writ-
ing a text-book on a introductory course in CFD Since then I did
my Masters project work on CFD (1997-98) introduced and taught
this course to Under-Graduate (UG) students at National Institute
of Technology Hamirpur HP (1999-2000) did my Doctoral work on
this subject at Indian Institute of Technology (IIT) Kanpur (2000-
2004) and co-authored a Chapter on nite volume method in a re-
vised edition of an edited book on computational uid ow and heat
transfer during my PhD in 2003 After joining as a faculty in 2004
I taught this course to UG as well as PG students of dierent depart-
ments at IIT Bombay as well as to the students of dierent collages in
India through CDEEP (Center for Distance Engineering Programme)
IIT Bombay from 2007-2010 Furthermore I gave series of lecture
on CFD at various collages and industries in India More recently
in 2012 I delivered a ve-days lecture and lab-session on CFD (to
around 1400 collage teachers from dierent parts of India) as a part
of a project funded by ministry of human resource development under
NMEICT (National Mission on Education through information and
communication technology)
The increasing need for the development of the customized CFD
software (app) and wide spread CFD application as well as analysis
for the design optimization and innovation of various types of engi-
neering systems always motivated me to write this book Due to the
increasing importance of CFD another motivation is to present CFD
ix
x Preface
simple enough to introduce the rst-course at early under-graduate
curriculum rather than the post-graduate curriculum The moti-
vation got strengthened by my own as well as others (students
teachers researchers and practitioners) struggle on physical under-
standing of the mathematics involved during the Partial Dierential
Equation (PDE) based algebraic-formulation (nite volume method)
along with the continuous frustrating pursuit to eectively convert
the theory of CFD into the computer-program
This led me to come up with an alternate physical-law (PDE-
free) based nite volume method as well as solution methodol-
ogy which are much easier to comprehend Mostly the success of
the programming eort is decided by the perfect execution of the
implementations-details which rarely appears in-print (existing books
and journal articles) The physical law based CFD development
approach and the implementations-details are the novelties in the
present book Furthermore using an open-source software for nu-
merical computations (Scilab) computer programs are given in this
book on the basic modules of conduction advection and convection
Indeed the reader can generalize and extend these codes for the de-
velopment of Navier-Stokes solver and generate the results presented
in the chapters towards the end of this book
I have limited the scope of this book to the numerical-techniques
which I wish to recommend for the book although there are numerous
other advanced and better methods in the published literature I do
not claim that the programming-practice presented here are the most
ecient minus they are presented here more for the ease in programming
(understand as your program) than for the computational eciency
Furthermore although I am enthusiastic about my presentation of
the physical law based CFD approach it may not be more ecient
than the alternate mathematics based CFD approach The emphasis
in this book is on the ease of understanding of the formulations and
programming-practice for the introductory course on CFD
I would hereby acknowledge that I owe my greatest debt to Prof
J Srinivasan (my ME Project supervisor at IISc Bangalore) who
did a hand-holding and wonderful job in patiently introducing me to
the fascinating world of research in uid-dynamics and heat trans-
fer I would also acknowledge the excellent training from Prof V
Introduction to CFD xi
Eswaran (my PhD supervisor at IIT Kanpur) from whom I learned
systematic way of CFD development application and analysis His
inspiring comments and suggestions on my writing-style and elabora-
tion of the rst chapter in this book is also gratefully acknowledged
I also want to record my sincere thanks to Prof Pradip Datta IISc
Bangalore and Prof Gautam Biswas IIT Kanpur minus my teachers for
CFD course Special thanks to Prof K Muralidhar IIT Kanpur for
his personal encouragement academic support and motivation (for
the book writing) He also cleared my dilemma in the proper presen-
tation of the novelty in this book I thank him also for writing the
foreword of this book
I am grateful to Prof Kannan M Moudgalya and Prof Deepak B
Phatak from IIT Bombay for giving me opportunity to teach CFD
in a distance education mode I am also grateful to Prof Amit
Agrawal (my colleague at IIT Bombay) who diligently read through
all the chapters of this book and suggested some really remarkable
changes in the book I also thank him for the research interactions
over the last decade which had substantially improved my under-
standing of thermal and uid science I would also like to thank my
PhD student Namshad T who did a great job in implementing
my ideas on generating the CFD simulation based CFD image for
the cover-page and couple of gures in the rst chapter (Fig 12
and 13) Special mention that the Scilab codes developed (for the
CFD course under NMEICT) by Vishesh Aggarwal greatly helped
me to develop the codes presented in this book his contributions is
gratefully acknowledged I thank Malhar Malushte for testing some
of the formulations presented in this book
I thank my research students (Sachin B Paramane D Datta C
M Sewatkar Vinesh H Gada Kaushal Prasad Mukul Shrivastava
Absar M Lakdawala Mausam Sarkar and Javed Shaikh) for the
diligence and sincerity they have shown in unraveling some of our
ideas on CFD Apart from these several other research students and
the students teachers and CFD practitioners who took my lectures
on CFD in the past decade have contributed to my better under-
standing of CFD and inspired me to write the book I thank all of
them for their attention and enthusiastic response during the CFD
courses The inuence of the interactions with all of them can be seen
xii Preface
throughout this book I am grateful to IIT Bombay for giving me
one year sabbatical leave to write this book The ambiance of aca-
demic freedom considerate support by the faculties and the friendly
atmosphere inside the campus of IIT Bombay has greatly contributed
in the solo eort of book writing I thanks all my colleagues at IIT
Bombay for shaping my perceptions on CFD
During this long endeavor the patient support by Mr Sunil
Saxena Director Athena Academic Ltd UK is gratefully acknowl-
edged I also thank John Wiley for coming forward to take this
book to international markets Lastly I would acknowledge that I
am greatly inspired by the writing style of J D Anderson and the
book on CFD by S V Patankar Of course you will nd that I
have heavily cross-referred the CFD book by the two most acknowl-
edged pioneers on CFD I dedicate this book to Dr S V Patankar
whose book nicely introduced me to the computational as well as
ow physics in CFD and greatly helped me to introduce a physical
approach of CFD development in this book
The culmination of the book writing process fullls a long cher-
ished dream It would not have been possible without the excel-
lent environment continuous support and complete understanding
of my wife Anubha and daughter Anshita Their kindness has greatly
helped to fulll my dream now I plan to spend more time with my
wife and daughter
Mumbai
August 2016 Atul Sharma
Contents
I Introduction and Essentials
1
1 Introduction 3
11 CFD What is it 3
111 CFD as a Scientic and Engineering Analysis
Tool 5
112 Analogy with a Video-Camera 13
12 CFD Why to study 15
13 Novelty Scope and Purpose of this Book 16
2 Introduction to CFD Development Application and
Analysis 23
21 CFD Development 23
211 Grid Generation Pre-Processor 24
212 Discretization Method Algebraic Formulation 26
213 Solution Methodology Solver 29
214 Computation of Engineering-Parameters
Post-Processor 34
215 Testing 35
22 CFD Application 35
23 CFD Analysis 38
24 Closure 39
xiii
xiv Contents
3 Essentials of Fluid-Dynamics and Heat-Transfer for
CFD 41
31 Physical Laws 42
311 FundamentalConservation Laws 43
312 Subsidiary Laws 45
32 Momentum and Energy Transport Mechanisms 46
33 Physical Law based Dierential Formulation 48
331 Continuity Equation 49
332 Transport Equation 51
34 Generalized Volumetric and Flux Terms and their Dif-
ferential Formulation 57
341 Volumetric Term 58
342 Flux-Term 58
343 Discussion 62
35 Mathematical Formulation 63
351 Dimensional Study 63
352 Non-Dimensional Study 67
36 Closure 71
4 Essentials of Numerical-Methods for CFD 73
41 Finite Dierence Method A Dierential to Algebraic
Formulation for Governing PDE and BCs 75
411 Grid Generation 75
412 Finite Dierence Method 78
413 Applications to CFD 92
42 Iterative Solution of System of LAEs for a Flow Property 93
421 Iterative Methods 94
422 Applications to CFD 98
43 Numerical Dierentiation for Local Engineering-
Parameters 105
431 Dierentiation Formulas 106
432 Applications to CFD 107
44 Numerical Integration for the Total value of
Engineering-Parameters 110
441 Integration Rules 111
442 Applications to CFD 114
45 Closure 116
Problems 116
Introduction to CFD xv
II CFD for a Cartesian-Geometry
119
5 Computational Heat Conduction 12151 Physical Law based Finite Volume Method 122
511 Energy Conservation Law for a Control Volume 123512 Algebraic Formulation 124513 Approximations 127514 Approximated Algebraic-Formulation 130515 Discussion 133
52 Finite Dierence Method for Boundary Conditions 13853 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 139531 One-Dimensional Conduction 139532 Two-Dimensional Conduction 147
54 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 154541 One-Dimensional Conduction 154542 Two-Dimensional Conduction 166
Problems 176
6 Computational Heat Advection 17961 Physical Law based Finite Volume Method 180
611 Energy Conservation Law for a Control Volume 181612 Algebraic Formulation 181613 Approximations 183614 Approximated Algebraic Formulation 191615 Discussion 191
62 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 194621 Explicit-Method 197622 Implementation Details 198623 Solution Algorithm 199
63 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 207631 Advection Scheme on a Non-Uniform Grid 207632 Explicit and Implicit Method 210633 Implementation Details 216634 Solution Algorithm 220
Problems 228
xvi Contents
7 Computational Heat Convection 22971 Physical Law based Finite Volume Method 229
711 Energy Conservation Law for a Control Volume 229712 Algebraic Formulation 231713 Approximated Algebraic Formulation 232
72 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 235721 Explicit-Method 236722 Implementation Details 237723 Solution Algorithm 238
73 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 242Problems 248
8 Computational Fluid Dynamics Physical Law based
Finite Volume Method 25181 Generalized Variables for the Combined Heat and
Fluid Flow 25282 Conservation Laws for a Control Volume 25583 Algebraic Formulation 25984 Approximations 26085 Approximated Algebraic Formulation 263
851 Mass Conservation 263852 MomentumEnergy Conservation 264
86 Closure 269
9 Computational Fluid Dynamics on a Staggered Grid 27191 Challenges in the CFD Development 273
911 Non-Linearity 273912 Equation for Pressure 273913 Pressure-Velocity Decoupling 273
92 A Staggered Grid to avoid Pressure-Velocity Decoupling27593 Physical Law based FVM for a Staggered Grid 27794 Flux based Solution Methodology on a Uniform Grid
Semi-Explicit Method 281941 Philosophy of Pressure-Correction Method 283942 Semi-Explicit Method 286943 Implementation Details 292944 Solution Algorithm 298
95 Initial and Boundary Conditions 300951 Initial Condition 300952 Boundary Condition 301
Problems 306
Introduction to CFD xvii
10 Computational Fluid Dynamics on a Co-located Grid309
101 Momentum-Interpolation Method Strategy to avoid
the Pressure-Velocity Decoupling on a Co-located Grid 310
102 Coecients of LAEs based Solution Methodology on
a Non-Uniform Grid Semi-Explicit and Semi-Implicit
Method 314
1021 Predictor Step 316
1022 Corrector Step 318
1023 Solution Algorithm 323
Problems 329
III CFD for a Complex-Geometry
331
11 Computational Heat Conduction on a Curvilinear
Grid 333
111 Curvilinear Grid Generation 333
1111 Algebraic Grid Generation 334
1112 Elliptic Grid Generation 336
112 Physical Law based Finite Volume Method 343
1121 Unsteady and Source Term 343
1122 Diusion Term 344
1123 All Terms 349
113 Computation of Geometrical Properties 349
114 Flux based Solution Methodology 352
1141 Explicit Method 354
1142 Implementation Details 354
Problems 359
12 Computational Fluid Dynamics on a Curvilinear Grid361
121 Physical Law based Finite Volume Method 361
1211 Mass Conservation 363
1212 Momentum Conservation 363
122 Solution Methodology Semi-Explicit Method 372
1221 Predictor Step 373
1222 Corrector Step 375
Problems 380
References 383
Index 389
Part I
Introduction and Essentials
The book on an introductory course in CFD starts with the two intro-
ductory chapters on CFD followed by one chapter each on essentials
of two prerequisite courses minus uid-dynamics and heat transfer and
numerical-methods The rst chapter on introduction presents an
eventful thoughtful and intuitive discussion on What is CFD and
Why to study CFD whereas How CFD works which involves lots
of details is presented separately in the second chapter The third-
chapter on essentials of uid-dynamics and heat transfer presents
physical-laws transport-mechanisms physical-law based dierential-
formulation volumetric and ux terms and mathematical formula-
tion The fourth-chapter on essentials of numerical-methods presents
nite dierence method iterative solution of the system of linear alge-
braic equation numerical dierentiation and numerical integration
The presentation for the essential of the two prerequisite courses is
customized to CFD development application and analysis
1
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
1
Introduction
The introduction of any course consists of the three basic questions
What is it Why to study How it works An eventful thoughtful
and intuitive answer to the rst two questions are presented in this
chapter The third question which involves lots of working details
on the three aspects of the Computational Fluid Dynamics (CFD)
(development application and analysis) is presented separately in
the next chapter The novelty scope and purpose of this book is
also presented at the end of this chapter
11 CFD What is it
CFD is a theoretical-method of scientic and engineering investiga-
tion concerned with the development and application of a video-
camera like tool (a software) which is used for a unied cause-and-
eect based analysis of a uid-dynamics as well as heat and mass
transfer problem
CFD is a subject where you rst learn how to develop a product
minus a software which acts like a virtual video-camera Then you learn
how to apply or use this product to generate uid-dynamics movies
Finally you learn how to analyze the detailed spatial and temporal
uid-dynamics information in the movies The analysis is done to
come up with a scientically-exciting as well as engineering-relevant
story (unied cause-and-eect study) of a uid-dynamics situation in
nature as well as industrial-applications With a continuous develop-
ment and a wider application of CFD the word uid-dynamics in
3
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
4 1 Introduction
the CFD has become more generic as it also corresponds to heat and
mass transfer as well as chemical reaction
Fluid-dynamics information are of two types scientic and en-
gineering Scientic-information corresponds to a structure of the
heat and uid ow due to a physical phenomenon in uid-dynamics
such as boundary-layer ow-separation wake-formation and vortex-
shedding The ow structures are obtained from a temporal variation
of ow-properties (such as velocity pressure temperature and many
other variables which characterizes a ow) called as scientically-
exciting movies Engineering-information corresponds to certain pa-
rameters which are important in an engineering application called
as engineering-parameters for example lift force drag force wall
shear stress pressure drop and rate of heat-transfer They are ob-
tained as the temporal variation of engineering-parameters called as
engineering-relevant movies The engineering-parameters are the ef-
fect which are caused by the ow structures minus their correlation leads
to the unied cause-and-eect study A uid-dynamics movie consist
of a time-wise varying series of pictures which give uid-dynamics
information Each ow-property and engineering-parameter results
in one scientically-exciting and engineering-relevant movie respec-
tively When the large number of both the types of movies are
played synchronized in-time the detailed spatial as well as tempo-
ral uid-dynamic information greatly helps in the unied cause-and-
eect study of a uid-dynamics problem
Alternatively CFD is a subject concerned with the development of
computer-programs for the computer-simulation and study of a natu-
ral or an engineering uid-dynamics system The development starts
with the virtual-development of the system involving a geometri-
cal information based computational development of a solid-model
for the system the system may be already-existing or yet-to-be-
developed physically Thereafter the development involves numerical
solution of the governing algebraic equations for CFD formulated
from the physical laws (conservation of mass momentum and en-
ergy) and initial as well as boundary conditions corresponding to the
system CFD study involves the analysis of the uid-dynamics and
heat-transfer results obtained from the computer-simulations of the
virtual-system which is subjected to certain governing-parameters
Introduction to CFD 5
111 CFD as a Scientic and Engineering Analysis Tool
For a better understanding of the denition of the CFD and its ap-
plication as a scientic and engineering analysis tool two example
problems are presented rst conduction heat transfer in a plate
and second uid ow across a circular pillar of a bridge The rst-
problem correspond to an engineering-application for electronic cool-
ing and the second-problem is studied for an engineering-design of
the structure subjected to uid-dynamics forces and are shown in
Fig 11 Both the problems are considered as two-dimensional and
unsteady However the results asymptotes from the unsteady to a
steady state in the rst-problem and to a periodic (dynamic-steady)
state in the second-problem
Figure 11 Schematic and the associated parameters for the two example prob-lems (a) thermal conduction module of a heat sink and (b) theclassical ow across a circular cylinder
6 1 Introduction
The rst-problem is taken from the classic book on heat-transfer
by Incropera and Dewitt (1996) shown in Fig 11(a) The g-
ure shows a periodic module of the plate where a constant heat-ux
qW = 10Wcm2 (dissipated from certain electronic devices) is rst con-
ducted to an aluminum plate and then convected to the water ow-
ing through the rectangular channel grooved in the plate The forced
convection results in a convection coecient of h = 5000Wm2K in-
side the channel The unsteady heat-transfer results in an interesting
time-wise varying conduction heat ow in the plate and instantaneous
convection heat transfer rate Qprimeconv (per unit width of the plate per-
pendicular to the plane shown in Fig 11(a)) into the water
The second-problem is encountered if you are standing over a
bridge and look below to the water passing over a pillar (of circu-
lar cross-section) of the bridge shown in Fig 11(b) as a general case
of 2D ow across a cylinder The uid-dynamics results in a beau-
tiful time-wise varying structure of the uid ow just downstream
of the pillar and certain engineering-parameters The parameters
correspond to the uid-dynamic forces acting in the horizontal (or
streamwise) and vertical (or transverse) direction called as the drag-
force FD and lift-force FL respectively The respective force is pre-
sented below in a non-dimensional form called as the drag-coecient
CD and lift-coecient CL equations shown in Fig 11(b)
If you use your video camera to capture a movie for the two
problems you will get series of pictures for the plate in the rst-
problem and the owing uid in the second-problem Since these
movies result in a pictorial information but not the ow-property
based uid-dynamic information such movies are not relevant for
the uid-dynamics and heat-transfer study Instead if you use a
CFD software for computer simulation you can capture various types
of movies The various ow-properties based scientically-exciting
movies are presented here as the pictures at certain time-instant in
Fig 12(a)-(d) and Fig 13(a1)-(d2) The former gure shows gray-
colored ooded contour for the temperature whereas the latter gure
shows gray-colored ooded contour for the pressure and vorticity and
vector-plot for the velocity For the gray-colored ooded contours a
color bar can be seen in the gures for the relative magnitude of the
variable These gures also show line contours of stream-function
Introduction to CFD 7
and heat-function as streamlines and heatlines respectively They
are the visualization techniques for a 2D steady heat and uid ow
The heatlines are analogous to streamlines and represents the direc-
tion of heat ow under steady state condition proposed by Kimura
and Bejan (1983) later presented in a heat-transfer book by Bejan
(1984) The steady-state heatlines can be seen in Fig 12(d) where
the direction for the conduction-ux is tangential at any point on the
heatlines analogous to that for the mass-ux in a streamline Thus
for a better presentation of the structure of heat and uid ow the
scientically-exciting movies are not only limited to ow-properties
but also correspond to certain ow visualization techniquesAn engineering-parameter based engineering-relevant uid dy-
namic movie is presented in Fig 12(e) for the convection heat trans-
fer rate per unit width at the various walls on the channel Qprimew at the
west-wall Qprimen at the north-wall Q
primee at the east-wall and their cumu-
lative value Qprimeconv (refer Fig 11(a)) Another engineering-relevant
movie is presented in Fig 13(e) for the lift and drag coecient Note
that the movie here corresponds to the motion of a lled symbol over
a line for the temporal variation of an engineering-parameter Thus
the engineering-relevant movies are one-dimensional lower than the
scientically-exciting movies here for the 2D problems the former
movie is presented as 1D and the latter as 2D results
1111 Heat-Conduction
For the computational set-up of the heat conduction problem shown
in Fig 11(a) the results obtained from a 2D CFD simulation of
the conduction heat transfer are shown in Fig 12 For a uni-
form initial temperature of the plate as Tinfin = 15oC Fig 12(a)-(d)
shows the isotherms and heatlines in the plate as the pictures of the
scientically-exciting movies at certain discrete time instants t = 25
5 10 and 20 sec For the same discrete time instants the temporal
variation of the position of the symbols over a line plot is presented in
Fig 12(e) The gure is shown for each of the heat transfer rate and
is obtained from a engineering-relevant 1D movie There is a tempo-
ral variation of picture in the scientically-exciting movie and that
of position of symbol (along the line plot) in the engineering-relevant
movieAccording to caloric theory of the famous French scientist An-
tonie Lavoisier heat is considered as an invisible tasteless odorless
8 1 Introduction
t(sec)
Qrsquo (W
m)
0 5 10 15 20 250
200
400
600
800
1000Qrsquo
conv
Qrsquow
Qrsquon
Qrsquoe
Qrsquoin
(e)
(b)
(c)(d)
(a)
(a) t=25 sec31
29
27
25
23
21
19
17
15
T(oC)
(c) t=10 sec
(b) t=5 sec
(d) t=20 sec
Figure 12 Thermal conduction module of a heat sink temporal variation of(a)-(d) isotherms and heatlines and (e) the convection heat transferrate per unit width at the various walls of the channels and theircumulative total value Q
primeconv For the discrete time-instant corre-
sponding to heat ow patterns in (a)-(d) temporal variation of thevarious heat transfer rates are shown as symbols in (e) (a)-(c) arethe transient and (d) represents a steady state result
weightless uid which he called caloric uid Presently the caloric
theory is overtaken by mechanical theory of heat minus heat is not a sub-
stance but a dynamical form of mechanical eect (Thomson 1851)
Nevertheless there are certain problems involving heat ow for which
Lavoisiers approach is rather useful such as the discussion on heat-
lines here
The role of heatlines for heat ow is analogous to that of stream-
lines for uid ow (Paramane and Sharma 2009) For the uid-ow
the dierence of stream-function values represents the rate of uid
ow the function remains constant on a solid wall the tangent to
a streamline represents the direction of the uid-ow (with no ow
in the normal direction) and the streamline originate or emerge at
Introduction to CFD 9
a mass source Analogously for the heat-ow the dierence of heat-
function values represents the rate of heat ow the function remains
constant on an adiabatic wall the tangent to a heatline represents the
direction of the heat-ow (with no ow in the normal direction) and
the heatline originate or emerge at the heat-source For the present
problem in Fig 11(a) the heat-source is at the inlet (or left) bound-
ary and the heat-sink is at the surface of the channel Thus the
steady-state heatlines in Fig 12(d) shows the heat ow emerging
from the inletsource and ending on the channel-wallssink Further-
more it can be seen in the gure that the heatlines which are close
to the adiabatic walls are parallel to the walls The source sink and
adiabatic-walls can be seen in Fig 11(a)
Figure 12(e) shows an asymptotic time-wise increase in the con-
vection heat transfer per unit width at the various walls of the channel
and its cumulative value Qprimeconv (= Q
primew+Q
primen+Q
primee) It can be also be seen
that the heat-transfer per unit width Qprimew at the west-wall and Q
primen at
the north-wall are larger than Qprimee at the east-wall of the channel As
time progresses the gure shows that Qprimeconv increases monotonically
and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-
der steady state condition refer Fig 11(a) The dierence between
the Qprimein and Q
primeconv under the transient condition is utilized for the
sensible heating and results in the increase in the temperature of the
plate Under the steady state condition note from Fig 12(d) that
the maximum temperature is close to 32oC minus well within the per-
missible limit for the reliable and ecient operation of the electronic
device
1112 Fluid-Dynamics
For the free-stream ow across a circular cylinderpillar at a Reynolds
number of 100 the non-dimensional results obtained from a 2D CFD
simulation are shown in Fig 13 In contrast to the previous prob-
lem which reaches steady state the present problem reaches to an un-
steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-
ent increasing time instants within one time-period of the periodic-
ow the results are presented in Fig 13(a)-(d) The gure shows
the close-up view of the pictures of the scientically-exciting movies
for ow-properties (velocity pressure and vorticity) and streamlines
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow
Introduction to Computational Fluid DynamicsDevelopment Application and Analysis
Dr Atul Sharma
copy Author 2017
This Edition Published by
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All rights reserved No part of this publication may be reproduced stored in a retrieval system or transmitted in any form or by any means electronic mechanical photocopying recording or otherwise except as permitted by the UK Copyright Designs and Patents Act 1988 without the prior permission of the publisher
Library Congress Cataloging-in-Publication DataA catalogue record for this book is available from the British Library
Cover Image The image for C F and D in the cover page is obtained from a Computational Fluid Dynamics simulations as follows
C Curvilinear-Grid and Resultant-Velocity Contours for Fluid-Flow in a C-shaped Bend of a Pipe
F Temperature-Contours in a Heat Conduction Problem
D Vertical-Velocity Contours for a vertical-wall (moving upward like a conveyor belt) driven flow in a D shaped cavity
DisclaimerThe contents sources and the information provided by the authors in different chapters are the sole responsibility of the authors themselves including the copyright issues Editors and Publishers have no liabilities whatsoever towards these in any manner
This book is dedicated
to
Suhas V Patankar
Dr Suhas V Patankar1 (born February 22 1941) is an Indian
mechanical engineer He is a pioneer in the eld of computational
uid dynamics (CFD) and nite volume method He received his
BE from the University of Pune in 1962 MTech from IIT Bombay
in 1964 and PhD from Imperial College London in 1967
Dr Patankar is currently a Professor Emeritus in the Mechanical
Engineering Department at the University of Minnesota where he
worked for 25 years (from 1975-2000) Earlier he held teaching and
research positions at IIT Kanpur Imperial College and University
of Waterloo He is also the President of Innovative Research Inc He
has authored or co-authored four books published over 150 papers
advised 35 completed PhD theses and lectured extensively in the
USA and abroad His 1980 book Numerical heat Transfer and Fluid
Flow is considered to be a groundbreaking contribution to CFD He
is one of the most cited authors in science and engineering
For excellence in teaching Dr Patankar received the 1983 George
Taylor Distinguished Teaching Award and the 1989-90 Morse-Alumni
1Source (for the Biography) Personal Communication
v
vi
Award for Outstanding Contributions to Undergraduate Education
For his research contributions to computational heat transfer he was
given the 1991 ASME Heat Transfer Memorial Award and the 1997
Classic Paper Award He was awarded the 2008 Max Jakob Award
which is considered to be the highest international honor in the eld of
heat transfer In 2015 an International Conference on Computational
Heat Transfer (CHT-15) held at Rutgers University in New Jersey
was dedicated to Professor Patankar
Dr Patankars widespread inuence on research and engineering
education has been recognized in many ways In 2007 the Editors of
the International Journal of Heat and Mass Transfer wrote There is
no person who has made a more profound and enduring impact on the
theory and practice of numerical simulation in mechanical engineering
than Professor Patankar
FOREWORD
by Prof K Muralidhar
Indian Institute of Technology Kanpur INDIA
The subject of computational uid dynamics (CFD) emerged in the
academic context but in the past two decades it has developed to a
point where it is extensively employed as a design and analysis tool in
the industry The impact of the subject is deeply felt in diverse disci-
plines ranging from the aerospace spreading all the way to the chem-
ical industry New compact and ecient designs are enabled and it
has become that much more convenient to locate optimum conditions
for the operation of engineering systems As it stands CFD is com-
pulsory learning in several branches of engineering Many universities
now opt to develop theoretical courses around it The great utility of
this tool has inuenced adjacent disciplines and the process of solv-
ing complex dierential equations by the process of discretization has
become a staple for addressing engineering challenges
For an emerging discipline it is necessary that new books fre-
quently appear on the horizon Students can choose from the se-
lection and will stand to gain from it Such books should have a
perspective of the past and the future and it is all the more signi-
cant when the author has exclusive training in the subject Dr Atul
Sharma represents a combination of perspective training and passion
that combines well and delivers a strong text
I have personal knowledge of the fact that Dr Atul Sharma has
developed a whole suite of computer programs on CFD from scratch
He has wondered and pondered over why approximations work and
conditions under which they yield meaningful solutions His spread
vii
viii Foreword
of experience over fteen and odd years includes two and three di-
mensional Navier-Stokes equations heat transfer interfacial dynam-
ics and turbulent ow He has championed the nite volume method
which is now the industry standard The programming experience
is important because it brings in the precision needed for assembling
the jigsaw puzzle The sprinkling of computer programs in the book
is no coincidence
Atul knows the conventional method of discretizing dierential
equations but has never been satised with it He has felt an ele-
ment of discomfort going away from a physical reality into a world
of numbers connected by arithmetic operations As a result he has
developed a principle that physical laws that characterize the dier-
ential equations should be reected at every stage of discretization
and every stage of approximation indeed at every scale whether a
single tetrahedron or the sub-domain or the region as a whole This
idea permeates the book where it is shown that discretized versions
must provide reasonable answers because they continue to represent
the laws of nature
This new CFD book is comprehensive and has a stamp of origi-
nality of the author It will bring students closer to the subject and
enable them to contribute to it
Kanpur
August 2016 K Muralidhar
PREFACE
As a Post-Graduate (PG) student in 1997 I got introduced to a
course on CFD at Indian Institute of Science (IISc) Bangalore Dur-
ing the introductory course on CFD although I struggled hard
in the physical understanding and computer-programming it didnt
stop my interest and enthusiasm to do research teaching and writ-
ing a text-book on a introductory course in CFD Since then I did
my Masters project work on CFD (1997-98) introduced and taught
this course to Under-Graduate (UG) students at National Institute
of Technology Hamirpur HP (1999-2000) did my Doctoral work on
this subject at Indian Institute of Technology (IIT) Kanpur (2000-
2004) and co-authored a Chapter on nite volume method in a re-
vised edition of an edited book on computational uid ow and heat
transfer during my PhD in 2003 After joining as a faculty in 2004
I taught this course to UG as well as PG students of dierent depart-
ments at IIT Bombay as well as to the students of dierent collages in
India through CDEEP (Center for Distance Engineering Programme)
IIT Bombay from 2007-2010 Furthermore I gave series of lecture
on CFD at various collages and industries in India More recently
in 2012 I delivered a ve-days lecture and lab-session on CFD (to
around 1400 collage teachers from dierent parts of India) as a part
of a project funded by ministry of human resource development under
NMEICT (National Mission on Education through information and
communication technology)
The increasing need for the development of the customized CFD
software (app) and wide spread CFD application as well as analysis
for the design optimization and innovation of various types of engi-
neering systems always motivated me to write this book Due to the
increasing importance of CFD another motivation is to present CFD
ix
x Preface
simple enough to introduce the rst-course at early under-graduate
curriculum rather than the post-graduate curriculum The moti-
vation got strengthened by my own as well as others (students
teachers researchers and practitioners) struggle on physical under-
standing of the mathematics involved during the Partial Dierential
Equation (PDE) based algebraic-formulation (nite volume method)
along with the continuous frustrating pursuit to eectively convert
the theory of CFD into the computer-program
This led me to come up with an alternate physical-law (PDE-
free) based nite volume method as well as solution methodol-
ogy which are much easier to comprehend Mostly the success of
the programming eort is decided by the perfect execution of the
implementations-details which rarely appears in-print (existing books
and journal articles) The physical law based CFD development
approach and the implementations-details are the novelties in the
present book Furthermore using an open-source software for nu-
merical computations (Scilab) computer programs are given in this
book on the basic modules of conduction advection and convection
Indeed the reader can generalize and extend these codes for the de-
velopment of Navier-Stokes solver and generate the results presented
in the chapters towards the end of this book
I have limited the scope of this book to the numerical-techniques
which I wish to recommend for the book although there are numerous
other advanced and better methods in the published literature I do
not claim that the programming-practice presented here are the most
ecient minus they are presented here more for the ease in programming
(understand as your program) than for the computational eciency
Furthermore although I am enthusiastic about my presentation of
the physical law based CFD approach it may not be more ecient
than the alternate mathematics based CFD approach The emphasis
in this book is on the ease of understanding of the formulations and
programming-practice for the introductory course on CFD
I would hereby acknowledge that I owe my greatest debt to Prof
J Srinivasan (my ME Project supervisor at IISc Bangalore) who
did a hand-holding and wonderful job in patiently introducing me to
the fascinating world of research in uid-dynamics and heat trans-
fer I would also acknowledge the excellent training from Prof V
Introduction to CFD xi
Eswaran (my PhD supervisor at IIT Kanpur) from whom I learned
systematic way of CFD development application and analysis His
inspiring comments and suggestions on my writing-style and elabora-
tion of the rst chapter in this book is also gratefully acknowledged
I also want to record my sincere thanks to Prof Pradip Datta IISc
Bangalore and Prof Gautam Biswas IIT Kanpur minus my teachers for
CFD course Special thanks to Prof K Muralidhar IIT Kanpur for
his personal encouragement academic support and motivation (for
the book writing) He also cleared my dilemma in the proper presen-
tation of the novelty in this book I thank him also for writing the
foreword of this book
I am grateful to Prof Kannan M Moudgalya and Prof Deepak B
Phatak from IIT Bombay for giving me opportunity to teach CFD
in a distance education mode I am also grateful to Prof Amit
Agrawal (my colleague at IIT Bombay) who diligently read through
all the chapters of this book and suggested some really remarkable
changes in the book I also thank him for the research interactions
over the last decade which had substantially improved my under-
standing of thermal and uid science I would also like to thank my
PhD student Namshad T who did a great job in implementing
my ideas on generating the CFD simulation based CFD image for
the cover-page and couple of gures in the rst chapter (Fig 12
and 13) Special mention that the Scilab codes developed (for the
CFD course under NMEICT) by Vishesh Aggarwal greatly helped
me to develop the codes presented in this book his contributions is
gratefully acknowledged I thank Malhar Malushte for testing some
of the formulations presented in this book
I thank my research students (Sachin B Paramane D Datta C
M Sewatkar Vinesh H Gada Kaushal Prasad Mukul Shrivastava
Absar M Lakdawala Mausam Sarkar and Javed Shaikh) for the
diligence and sincerity they have shown in unraveling some of our
ideas on CFD Apart from these several other research students and
the students teachers and CFD practitioners who took my lectures
on CFD in the past decade have contributed to my better under-
standing of CFD and inspired me to write the book I thank all of
them for their attention and enthusiastic response during the CFD
courses The inuence of the interactions with all of them can be seen
xii Preface
throughout this book I am grateful to IIT Bombay for giving me
one year sabbatical leave to write this book The ambiance of aca-
demic freedom considerate support by the faculties and the friendly
atmosphere inside the campus of IIT Bombay has greatly contributed
in the solo eort of book writing I thanks all my colleagues at IIT
Bombay for shaping my perceptions on CFD
During this long endeavor the patient support by Mr Sunil
Saxena Director Athena Academic Ltd UK is gratefully acknowl-
edged I also thank John Wiley for coming forward to take this
book to international markets Lastly I would acknowledge that I
am greatly inspired by the writing style of J D Anderson and the
book on CFD by S V Patankar Of course you will nd that I
have heavily cross-referred the CFD book by the two most acknowl-
edged pioneers on CFD I dedicate this book to Dr S V Patankar
whose book nicely introduced me to the computational as well as
ow physics in CFD and greatly helped me to introduce a physical
approach of CFD development in this book
The culmination of the book writing process fullls a long cher-
ished dream It would not have been possible without the excel-
lent environment continuous support and complete understanding
of my wife Anubha and daughter Anshita Their kindness has greatly
helped to fulll my dream now I plan to spend more time with my
wife and daughter
Mumbai
August 2016 Atul Sharma
Contents
I Introduction and Essentials
1
1 Introduction 3
11 CFD What is it 3
111 CFD as a Scientic and Engineering Analysis
Tool 5
112 Analogy with a Video-Camera 13
12 CFD Why to study 15
13 Novelty Scope and Purpose of this Book 16
2 Introduction to CFD Development Application and
Analysis 23
21 CFD Development 23
211 Grid Generation Pre-Processor 24
212 Discretization Method Algebraic Formulation 26
213 Solution Methodology Solver 29
214 Computation of Engineering-Parameters
Post-Processor 34
215 Testing 35
22 CFD Application 35
23 CFD Analysis 38
24 Closure 39
xiii
xiv Contents
3 Essentials of Fluid-Dynamics and Heat-Transfer for
CFD 41
31 Physical Laws 42
311 FundamentalConservation Laws 43
312 Subsidiary Laws 45
32 Momentum and Energy Transport Mechanisms 46
33 Physical Law based Dierential Formulation 48
331 Continuity Equation 49
332 Transport Equation 51
34 Generalized Volumetric and Flux Terms and their Dif-
ferential Formulation 57
341 Volumetric Term 58
342 Flux-Term 58
343 Discussion 62
35 Mathematical Formulation 63
351 Dimensional Study 63
352 Non-Dimensional Study 67
36 Closure 71
4 Essentials of Numerical-Methods for CFD 73
41 Finite Dierence Method A Dierential to Algebraic
Formulation for Governing PDE and BCs 75
411 Grid Generation 75
412 Finite Dierence Method 78
413 Applications to CFD 92
42 Iterative Solution of System of LAEs for a Flow Property 93
421 Iterative Methods 94
422 Applications to CFD 98
43 Numerical Dierentiation for Local Engineering-
Parameters 105
431 Dierentiation Formulas 106
432 Applications to CFD 107
44 Numerical Integration for the Total value of
Engineering-Parameters 110
441 Integration Rules 111
442 Applications to CFD 114
45 Closure 116
Problems 116
Introduction to CFD xv
II CFD for a Cartesian-Geometry
119
5 Computational Heat Conduction 12151 Physical Law based Finite Volume Method 122
511 Energy Conservation Law for a Control Volume 123512 Algebraic Formulation 124513 Approximations 127514 Approximated Algebraic-Formulation 130515 Discussion 133
52 Finite Dierence Method for Boundary Conditions 13853 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 139531 One-Dimensional Conduction 139532 Two-Dimensional Conduction 147
54 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 154541 One-Dimensional Conduction 154542 Two-Dimensional Conduction 166
Problems 176
6 Computational Heat Advection 17961 Physical Law based Finite Volume Method 180
611 Energy Conservation Law for a Control Volume 181612 Algebraic Formulation 181613 Approximations 183614 Approximated Algebraic Formulation 191615 Discussion 191
62 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 194621 Explicit-Method 197622 Implementation Details 198623 Solution Algorithm 199
63 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 207631 Advection Scheme on a Non-Uniform Grid 207632 Explicit and Implicit Method 210633 Implementation Details 216634 Solution Algorithm 220
Problems 228
xvi Contents
7 Computational Heat Convection 22971 Physical Law based Finite Volume Method 229
711 Energy Conservation Law for a Control Volume 229712 Algebraic Formulation 231713 Approximated Algebraic Formulation 232
72 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 235721 Explicit-Method 236722 Implementation Details 237723 Solution Algorithm 238
73 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 242Problems 248
8 Computational Fluid Dynamics Physical Law based
Finite Volume Method 25181 Generalized Variables for the Combined Heat and
Fluid Flow 25282 Conservation Laws for a Control Volume 25583 Algebraic Formulation 25984 Approximations 26085 Approximated Algebraic Formulation 263
851 Mass Conservation 263852 MomentumEnergy Conservation 264
86 Closure 269
9 Computational Fluid Dynamics on a Staggered Grid 27191 Challenges in the CFD Development 273
911 Non-Linearity 273912 Equation for Pressure 273913 Pressure-Velocity Decoupling 273
92 A Staggered Grid to avoid Pressure-Velocity Decoupling27593 Physical Law based FVM for a Staggered Grid 27794 Flux based Solution Methodology on a Uniform Grid
Semi-Explicit Method 281941 Philosophy of Pressure-Correction Method 283942 Semi-Explicit Method 286943 Implementation Details 292944 Solution Algorithm 298
95 Initial and Boundary Conditions 300951 Initial Condition 300952 Boundary Condition 301
Problems 306
Introduction to CFD xvii
10 Computational Fluid Dynamics on a Co-located Grid309
101 Momentum-Interpolation Method Strategy to avoid
the Pressure-Velocity Decoupling on a Co-located Grid 310
102 Coecients of LAEs based Solution Methodology on
a Non-Uniform Grid Semi-Explicit and Semi-Implicit
Method 314
1021 Predictor Step 316
1022 Corrector Step 318
1023 Solution Algorithm 323
Problems 329
III CFD for a Complex-Geometry
331
11 Computational Heat Conduction on a Curvilinear
Grid 333
111 Curvilinear Grid Generation 333
1111 Algebraic Grid Generation 334
1112 Elliptic Grid Generation 336
112 Physical Law based Finite Volume Method 343
1121 Unsteady and Source Term 343
1122 Diusion Term 344
1123 All Terms 349
113 Computation of Geometrical Properties 349
114 Flux based Solution Methodology 352
1141 Explicit Method 354
1142 Implementation Details 354
Problems 359
12 Computational Fluid Dynamics on a Curvilinear Grid361
121 Physical Law based Finite Volume Method 361
1211 Mass Conservation 363
1212 Momentum Conservation 363
122 Solution Methodology Semi-Explicit Method 372
1221 Predictor Step 373
1222 Corrector Step 375
Problems 380
References 383
Index 389
Part I
Introduction and Essentials
The book on an introductory course in CFD starts with the two intro-
ductory chapters on CFD followed by one chapter each on essentials
of two prerequisite courses minus uid-dynamics and heat transfer and
numerical-methods The rst chapter on introduction presents an
eventful thoughtful and intuitive discussion on What is CFD and
Why to study CFD whereas How CFD works which involves lots
of details is presented separately in the second chapter The third-
chapter on essentials of uid-dynamics and heat transfer presents
physical-laws transport-mechanisms physical-law based dierential-
formulation volumetric and ux terms and mathematical formula-
tion The fourth-chapter on essentials of numerical-methods presents
nite dierence method iterative solution of the system of linear alge-
braic equation numerical dierentiation and numerical integration
The presentation for the essential of the two prerequisite courses is
customized to CFD development application and analysis
1
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
1
Introduction
The introduction of any course consists of the three basic questions
What is it Why to study How it works An eventful thoughtful
and intuitive answer to the rst two questions are presented in this
chapter The third question which involves lots of working details
on the three aspects of the Computational Fluid Dynamics (CFD)
(development application and analysis) is presented separately in
the next chapter The novelty scope and purpose of this book is
also presented at the end of this chapter
11 CFD What is it
CFD is a theoretical-method of scientic and engineering investiga-
tion concerned with the development and application of a video-
camera like tool (a software) which is used for a unied cause-and-
eect based analysis of a uid-dynamics as well as heat and mass
transfer problem
CFD is a subject where you rst learn how to develop a product
minus a software which acts like a virtual video-camera Then you learn
how to apply or use this product to generate uid-dynamics movies
Finally you learn how to analyze the detailed spatial and temporal
uid-dynamics information in the movies The analysis is done to
come up with a scientically-exciting as well as engineering-relevant
story (unied cause-and-eect study) of a uid-dynamics situation in
nature as well as industrial-applications With a continuous develop-
ment and a wider application of CFD the word uid-dynamics in
3
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
4 1 Introduction
the CFD has become more generic as it also corresponds to heat and
mass transfer as well as chemical reaction
Fluid-dynamics information are of two types scientic and en-
gineering Scientic-information corresponds to a structure of the
heat and uid ow due to a physical phenomenon in uid-dynamics
such as boundary-layer ow-separation wake-formation and vortex-
shedding The ow structures are obtained from a temporal variation
of ow-properties (such as velocity pressure temperature and many
other variables which characterizes a ow) called as scientically-
exciting movies Engineering-information corresponds to certain pa-
rameters which are important in an engineering application called
as engineering-parameters for example lift force drag force wall
shear stress pressure drop and rate of heat-transfer They are ob-
tained as the temporal variation of engineering-parameters called as
engineering-relevant movies The engineering-parameters are the ef-
fect which are caused by the ow structures minus their correlation leads
to the unied cause-and-eect study A uid-dynamics movie consist
of a time-wise varying series of pictures which give uid-dynamics
information Each ow-property and engineering-parameter results
in one scientically-exciting and engineering-relevant movie respec-
tively When the large number of both the types of movies are
played synchronized in-time the detailed spatial as well as tempo-
ral uid-dynamic information greatly helps in the unied cause-and-
eect study of a uid-dynamics problem
Alternatively CFD is a subject concerned with the development of
computer-programs for the computer-simulation and study of a natu-
ral or an engineering uid-dynamics system The development starts
with the virtual-development of the system involving a geometri-
cal information based computational development of a solid-model
for the system the system may be already-existing or yet-to-be-
developed physically Thereafter the development involves numerical
solution of the governing algebraic equations for CFD formulated
from the physical laws (conservation of mass momentum and en-
ergy) and initial as well as boundary conditions corresponding to the
system CFD study involves the analysis of the uid-dynamics and
heat-transfer results obtained from the computer-simulations of the
virtual-system which is subjected to certain governing-parameters
Introduction to CFD 5
111 CFD as a Scientic and Engineering Analysis Tool
For a better understanding of the denition of the CFD and its ap-
plication as a scientic and engineering analysis tool two example
problems are presented rst conduction heat transfer in a plate
and second uid ow across a circular pillar of a bridge The rst-
problem correspond to an engineering-application for electronic cool-
ing and the second-problem is studied for an engineering-design of
the structure subjected to uid-dynamics forces and are shown in
Fig 11 Both the problems are considered as two-dimensional and
unsteady However the results asymptotes from the unsteady to a
steady state in the rst-problem and to a periodic (dynamic-steady)
state in the second-problem
Figure 11 Schematic and the associated parameters for the two example prob-lems (a) thermal conduction module of a heat sink and (b) theclassical ow across a circular cylinder
6 1 Introduction
The rst-problem is taken from the classic book on heat-transfer
by Incropera and Dewitt (1996) shown in Fig 11(a) The g-
ure shows a periodic module of the plate where a constant heat-ux
qW = 10Wcm2 (dissipated from certain electronic devices) is rst con-
ducted to an aluminum plate and then convected to the water ow-
ing through the rectangular channel grooved in the plate The forced
convection results in a convection coecient of h = 5000Wm2K in-
side the channel The unsteady heat-transfer results in an interesting
time-wise varying conduction heat ow in the plate and instantaneous
convection heat transfer rate Qprimeconv (per unit width of the plate per-
pendicular to the plane shown in Fig 11(a)) into the water
The second-problem is encountered if you are standing over a
bridge and look below to the water passing over a pillar (of circu-
lar cross-section) of the bridge shown in Fig 11(b) as a general case
of 2D ow across a cylinder The uid-dynamics results in a beau-
tiful time-wise varying structure of the uid ow just downstream
of the pillar and certain engineering-parameters The parameters
correspond to the uid-dynamic forces acting in the horizontal (or
streamwise) and vertical (or transverse) direction called as the drag-
force FD and lift-force FL respectively The respective force is pre-
sented below in a non-dimensional form called as the drag-coecient
CD and lift-coecient CL equations shown in Fig 11(b)
If you use your video camera to capture a movie for the two
problems you will get series of pictures for the plate in the rst-
problem and the owing uid in the second-problem Since these
movies result in a pictorial information but not the ow-property
based uid-dynamic information such movies are not relevant for
the uid-dynamics and heat-transfer study Instead if you use a
CFD software for computer simulation you can capture various types
of movies The various ow-properties based scientically-exciting
movies are presented here as the pictures at certain time-instant in
Fig 12(a)-(d) and Fig 13(a1)-(d2) The former gure shows gray-
colored ooded contour for the temperature whereas the latter gure
shows gray-colored ooded contour for the pressure and vorticity and
vector-plot for the velocity For the gray-colored ooded contours a
color bar can be seen in the gures for the relative magnitude of the
variable These gures also show line contours of stream-function
Introduction to CFD 7
and heat-function as streamlines and heatlines respectively They
are the visualization techniques for a 2D steady heat and uid ow
The heatlines are analogous to streamlines and represents the direc-
tion of heat ow under steady state condition proposed by Kimura
and Bejan (1983) later presented in a heat-transfer book by Bejan
(1984) The steady-state heatlines can be seen in Fig 12(d) where
the direction for the conduction-ux is tangential at any point on the
heatlines analogous to that for the mass-ux in a streamline Thus
for a better presentation of the structure of heat and uid ow the
scientically-exciting movies are not only limited to ow-properties
but also correspond to certain ow visualization techniquesAn engineering-parameter based engineering-relevant uid dy-
namic movie is presented in Fig 12(e) for the convection heat trans-
fer rate per unit width at the various walls on the channel Qprimew at the
west-wall Qprimen at the north-wall Q
primee at the east-wall and their cumu-
lative value Qprimeconv (refer Fig 11(a)) Another engineering-relevant
movie is presented in Fig 13(e) for the lift and drag coecient Note
that the movie here corresponds to the motion of a lled symbol over
a line for the temporal variation of an engineering-parameter Thus
the engineering-relevant movies are one-dimensional lower than the
scientically-exciting movies here for the 2D problems the former
movie is presented as 1D and the latter as 2D results
1111 Heat-Conduction
For the computational set-up of the heat conduction problem shown
in Fig 11(a) the results obtained from a 2D CFD simulation of
the conduction heat transfer are shown in Fig 12 For a uni-
form initial temperature of the plate as Tinfin = 15oC Fig 12(a)-(d)
shows the isotherms and heatlines in the plate as the pictures of the
scientically-exciting movies at certain discrete time instants t = 25
5 10 and 20 sec For the same discrete time instants the temporal
variation of the position of the symbols over a line plot is presented in
Fig 12(e) The gure is shown for each of the heat transfer rate and
is obtained from a engineering-relevant 1D movie There is a tempo-
ral variation of picture in the scientically-exciting movie and that
of position of symbol (along the line plot) in the engineering-relevant
movieAccording to caloric theory of the famous French scientist An-
tonie Lavoisier heat is considered as an invisible tasteless odorless
8 1 Introduction
t(sec)
Qrsquo (W
m)
0 5 10 15 20 250
200
400
600
800
1000Qrsquo
conv
Qrsquow
Qrsquon
Qrsquoe
Qrsquoin
(e)
(b)
(c)(d)
(a)
(a) t=25 sec31
29
27
25
23
21
19
17
15
T(oC)
(c) t=10 sec
(b) t=5 sec
(d) t=20 sec
Figure 12 Thermal conduction module of a heat sink temporal variation of(a)-(d) isotherms and heatlines and (e) the convection heat transferrate per unit width at the various walls of the channels and theircumulative total value Q
primeconv For the discrete time-instant corre-
sponding to heat ow patterns in (a)-(d) temporal variation of thevarious heat transfer rates are shown as symbols in (e) (a)-(c) arethe transient and (d) represents a steady state result
weightless uid which he called caloric uid Presently the caloric
theory is overtaken by mechanical theory of heat minus heat is not a sub-
stance but a dynamical form of mechanical eect (Thomson 1851)
Nevertheless there are certain problems involving heat ow for which
Lavoisiers approach is rather useful such as the discussion on heat-
lines here
The role of heatlines for heat ow is analogous to that of stream-
lines for uid ow (Paramane and Sharma 2009) For the uid-ow
the dierence of stream-function values represents the rate of uid
ow the function remains constant on a solid wall the tangent to
a streamline represents the direction of the uid-ow (with no ow
in the normal direction) and the streamline originate or emerge at
Introduction to CFD 9
a mass source Analogously for the heat-ow the dierence of heat-
function values represents the rate of heat ow the function remains
constant on an adiabatic wall the tangent to a heatline represents the
direction of the heat-ow (with no ow in the normal direction) and
the heatline originate or emerge at the heat-source For the present
problem in Fig 11(a) the heat-source is at the inlet (or left) bound-
ary and the heat-sink is at the surface of the channel Thus the
steady-state heatlines in Fig 12(d) shows the heat ow emerging
from the inletsource and ending on the channel-wallssink Further-
more it can be seen in the gure that the heatlines which are close
to the adiabatic walls are parallel to the walls The source sink and
adiabatic-walls can be seen in Fig 11(a)
Figure 12(e) shows an asymptotic time-wise increase in the con-
vection heat transfer per unit width at the various walls of the channel
and its cumulative value Qprimeconv (= Q
primew+Q
primen+Q
primee) It can be also be seen
that the heat-transfer per unit width Qprimew at the west-wall and Q
primen at
the north-wall are larger than Qprimee at the east-wall of the channel As
time progresses the gure shows that Qprimeconv increases monotonically
and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-
der steady state condition refer Fig 11(a) The dierence between
the Qprimein and Q
primeconv under the transient condition is utilized for the
sensible heating and results in the increase in the temperature of the
plate Under the steady state condition note from Fig 12(d) that
the maximum temperature is close to 32oC minus well within the per-
missible limit for the reliable and ecient operation of the electronic
device
1112 Fluid-Dynamics
For the free-stream ow across a circular cylinderpillar at a Reynolds
number of 100 the non-dimensional results obtained from a 2D CFD
simulation are shown in Fig 13 In contrast to the previous prob-
lem which reaches steady state the present problem reaches to an un-
steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-
ent increasing time instants within one time-period of the periodic-
ow the results are presented in Fig 13(a)-(d) The gure shows
the close-up view of the pictures of the scientically-exciting movies
for ow-properties (velocity pressure and vorticity) and streamlines
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow
This book is dedicated
to
Suhas V Patankar
Dr Suhas V Patankar1 (born February 22 1941) is an Indian
mechanical engineer He is a pioneer in the eld of computational
uid dynamics (CFD) and nite volume method He received his
BE from the University of Pune in 1962 MTech from IIT Bombay
in 1964 and PhD from Imperial College London in 1967
Dr Patankar is currently a Professor Emeritus in the Mechanical
Engineering Department at the University of Minnesota where he
worked for 25 years (from 1975-2000) Earlier he held teaching and
research positions at IIT Kanpur Imperial College and University
of Waterloo He is also the President of Innovative Research Inc He
has authored or co-authored four books published over 150 papers
advised 35 completed PhD theses and lectured extensively in the
USA and abroad His 1980 book Numerical heat Transfer and Fluid
Flow is considered to be a groundbreaking contribution to CFD He
is one of the most cited authors in science and engineering
For excellence in teaching Dr Patankar received the 1983 George
Taylor Distinguished Teaching Award and the 1989-90 Morse-Alumni
1Source (for the Biography) Personal Communication
v
vi
Award for Outstanding Contributions to Undergraduate Education
For his research contributions to computational heat transfer he was
given the 1991 ASME Heat Transfer Memorial Award and the 1997
Classic Paper Award He was awarded the 2008 Max Jakob Award
which is considered to be the highest international honor in the eld of
heat transfer In 2015 an International Conference on Computational
Heat Transfer (CHT-15) held at Rutgers University in New Jersey
was dedicated to Professor Patankar
Dr Patankars widespread inuence on research and engineering
education has been recognized in many ways In 2007 the Editors of
the International Journal of Heat and Mass Transfer wrote There is
no person who has made a more profound and enduring impact on the
theory and practice of numerical simulation in mechanical engineering
than Professor Patankar
FOREWORD
by Prof K Muralidhar
Indian Institute of Technology Kanpur INDIA
The subject of computational uid dynamics (CFD) emerged in the
academic context but in the past two decades it has developed to a
point where it is extensively employed as a design and analysis tool in
the industry The impact of the subject is deeply felt in diverse disci-
plines ranging from the aerospace spreading all the way to the chem-
ical industry New compact and ecient designs are enabled and it
has become that much more convenient to locate optimum conditions
for the operation of engineering systems As it stands CFD is com-
pulsory learning in several branches of engineering Many universities
now opt to develop theoretical courses around it The great utility of
this tool has inuenced adjacent disciplines and the process of solv-
ing complex dierential equations by the process of discretization has
become a staple for addressing engineering challenges
For an emerging discipline it is necessary that new books fre-
quently appear on the horizon Students can choose from the se-
lection and will stand to gain from it Such books should have a
perspective of the past and the future and it is all the more signi-
cant when the author has exclusive training in the subject Dr Atul
Sharma represents a combination of perspective training and passion
that combines well and delivers a strong text
I have personal knowledge of the fact that Dr Atul Sharma has
developed a whole suite of computer programs on CFD from scratch
He has wondered and pondered over why approximations work and
conditions under which they yield meaningful solutions His spread
vii
viii Foreword
of experience over fteen and odd years includes two and three di-
mensional Navier-Stokes equations heat transfer interfacial dynam-
ics and turbulent ow He has championed the nite volume method
which is now the industry standard The programming experience
is important because it brings in the precision needed for assembling
the jigsaw puzzle The sprinkling of computer programs in the book
is no coincidence
Atul knows the conventional method of discretizing dierential
equations but has never been satised with it He has felt an ele-
ment of discomfort going away from a physical reality into a world
of numbers connected by arithmetic operations As a result he has
developed a principle that physical laws that characterize the dier-
ential equations should be reected at every stage of discretization
and every stage of approximation indeed at every scale whether a
single tetrahedron or the sub-domain or the region as a whole This
idea permeates the book where it is shown that discretized versions
must provide reasonable answers because they continue to represent
the laws of nature
This new CFD book is comprehensive and has a stamp of origi-
nality of the author It will bring students closer to the subject and
enable them to contribute to it
Kanpur
August 2016 K Muralidhar
PREFACE
As a Post-Graduate (PG) student in 1997 I got introduced to a
course on CFD at Indian Institute of Science (IISc) Bangalore Dur-
ing the introductory course on CFD although I struggled hard
in the physical understanding and computer-programming it didnt
stop my interest and enthusiasm to do research teaching and writ-
ing a text-book on a introductory course in CFD Since then I did
my Masters project work on CFD (1997-98) introduced and taught
this course to Under-Graduate (UG) students at National Institute
of Technology Hamirpur HP (1999-2000) did my Doctoral work on
this subject at Indian Institute of Technology (IIT) Kanpur (2000-
2004) and co-authored a Chapter on nite volume method in a re-
vised edition of an edited book on computational uid ow and heat
transfer during my PhD in 2003 After joining as a faculty in 2004
I taught this course to UG as well as PG students of dierent depart-
ments at IIT Bombay as well as to the students of dierent collages in
India through CDEEP (Center for Distance Engineering Programme)
IIT Bombay from 2007-2010 Furthermore I gave series of lecture
on CFD at various collages and industries in India More recently
in 2012 I delivered a ve-days lecture and lab-session on CFD (to
around 1400 collage teachers from dierent parts of India) as a part
of a project funded by ministry of human resource development under
NMEICT (National Mission on Education through information and
communication technology)
The increasing need for the development of the customized CFD
software (app) and wide spread CFD application as well as analysis
for the design optimization and innovation of various types of engi-
neering systems always motivated me to write this book Due to the
increasing importance of CFD another motivation is to present CFD
ix
x Preface
simple enough to introduce the rst-course at early under-graduate
curriculum rather than the post-graduate curriculum The moti-
vation got strengthened by my own as well as others (students
teachers researchers and practitioners) struggle on physical under-
standing of the mathematics involved during the Partial Dierential
Equation (PDE) based algebraic-formulation (nite volume method)
along with the continuous frustrating pursuit to eectively convert
the theory of CFD into the computer-program
This led me to come up with an alternate physical-law (PDE-
free) based nite volume method as well as solution methodol-
ogy which are much easier to comprehend Mostly the success of
the programming eort is decided by the perfect execution of the
implementations-details which rarely appears in-print (existing books
and journal articles) The physical law based CFD development
approach and the implementations-details are the novelties in the
present book Furthermore using an open-source software for nu-
merical computations (Scilab) computer programs are given in this
book on the basic modules of conduction advection and convection
Indeed the reader can generalize and extend these codes for the de-
velopment of Navier-Stokes solver and generate the results presented
in the chapters towards the end of this book
I have limited the scope of this book to the numerical-techniques
which I wish to recommend for the book although there are numerous
other advanced and better methods in the published literature I do
not claim that the programming-practice presented here are the most
ecient minus they are presented here more for the ease in programming
(understand as your program) than for the computational eciency
Furthermore although I am enthusiastic about my presentation of
the physical law based CFD approach it may not be more ecient
than the alternate mathematics based CFD approach The emphasis
in this book is on the ease of understanding of the formulations and
programming-practice for the introductory course on CFD
I would hereby acknowledge that I owe my greatest debt to Prof
J Srinivasan (my ME Project supervisor at IISc Bangalore) who
did a hand-holding and wonderful job in patiently introducing me to
the fascinating world of research in uid-dynamics and heat trans-
fer I would also acknowledge the excellent training from Prof V
Introduction to CFD xi
Eswaran (my PhD supervisor at IIT Kanpur) from whom I learned
systematic way of CFD development application and analysis His
inspiring comments and suggestions on my writing-style and elabora-
tion of the rst chapter in this book is also gratefully acknowledged
I also want to record my sincere thanks to Prof Pradip Datta IISc
Bangalore and Prof Gautam Biswas IIT Kanpur minus my teachers for
CFD course Special thanks to Prof K Muralidhar IIT Kanpur for
his personal encouragement academic support and motivation (for
the book writing) He also cleared my dilemma in the proper presen-
tation of the novelty in this book I thank him also for writing the
foreword of this book
I am grateful to Prof Kannan M Moudgalya and Prof Deepak B
Phatak from IIT Bombay for giving me opportunity to teach CFD
in a distance education mode I am also grateful to Prof Amit
Agrawal (my colleague at IIT Bombay) who diligently read through
all the chapters of this book and suggested some really remarkable
changes in the book I also thank him for the research interactions
over the last decade which had substantially improved my under-
standing of thermal and uid science I would also like to thank my
PhD student Namshad T who did a great job in implementing
my ideas on generating the CFD simulation based CFD image for
the cover-page and couple of gures in the rst chapter (Fig 12
and 13) Special mention that the Scilab codes developed (for the
CFD course under NMEICT) by Vishesh Aggarwal greatly helped
me to develop the codes presented in this book his contributions is
gratefully acknowledged I thank Malhar Malushte for testing some
of the formulations presented in this book
I thank my research students (Sachin B Paramane D Datta C
M Sewatkar Vinesh H Gada Kaushal Prasad Mukul Shrivastava
Absar M Lakdawala Mausam Sarkar and Javed Shaikh) for the
diligence and sincerity they have shown in unraveling some of our
ideas on CFD Apart from these several other research students and
the students teachers and CFD practitioners who took my lectures
on CFD in the past decade have contributed to my better under-
standing of CFD and inspired me to write the book I thank all of
them for their attention and enthusiastic response during the CFD
courses The inuence of the interactions with all of them can be seen
xii Preface
throughout this book I am grateful to IIT Bombay for giving me
one year sabbatical leave to write this book The ambiance of aca-
demic freedom considerate support by the faculties and the friendly
atmosphere inside the campus of IIT Bombay has greatly contributed
in the solo eort of book writing I thanks all my colleagues at IIT
Bombay for shaping my perceptions on CFD
During this long endeavor the patient support by Mr Sunil
Saxena Director Athena Academic Ltd UK is gratefully acknowl-
edged I also thank John Wiley for coming forward to take this
book to international markets Lastly I would acknowledge that I
am greatly inspired by the writing style of J D Anderson and the
book on CFD by S V Patankar Of course you will nd that I
have heavily cross-referred the CFD book by the two most acknowl-
edged pioneers on CFD I dedicate this book to Dr S V Patankar
whose book nicely introduced me to the computational as well as
ow physics in CFD and greatly helped me to introduce a physical
approach of CFD development in this book
The culmination of the book writing process fullls a long cher-
ished dream It would not have been possible without the excel-
lent environment continuous support and complete understanding
of my wife Anubha and daughter Anshita Their kindness has greatly
helped to fulll my dream now I plan to spend more time with my
wife and daughter
Mumbai
August 2016 Atul Sharma
Contents
I Introduction and Essentials
1
1 Introduction 3
11 CFD What is it 3
111 CFD as a Scientic and Engineering Analysis
Tool 5
112 Analogy with a Video-Camera 13
12 CFD Why to study 15
13 Novelty Scope and Purpose of this Book 16
2 Introduction to CFD Development Application and
Analysis 23
21 CFD Development 23
211 Grid Generation Pre-Processor 24
212 Discretization Method Algebraic Formulation 26
213 Solution Methodology Solver 29
214 Computation of Engineering-Parameters
Post-Processor 34
215 Testing 35
22 CFD Application 35
23 CFD Analysis 38
24 Closure 39
xiii
xiv Contents
3 Essentials of Fluid-Dynamics and Heat-Transfer for
CFD 41
31 Physical Laws 42
311 FundamentalConservation Laws 43
312 Subsidiary Laws 45
32 Momentum and Energy Transport Mechanisms 46
33 Physical Law based Dierential Formulation 48
331 Continuity Equation 49
332 Transport Equation 51
34 Generalized Volumetric and Flux Terms and their Dif-
ferential Formulation 57
341 Volumetric Term 58
342 Flux-Term 58
343 Discussion 62
35 Mathematical Formulation 63
351 Dimensional Study 63
352 Non-Dimensional Study 67
36 Closure 71
4 Essentials of Numerical-Methods for CFD 73
41 Finite Dierence Method A Dierential to Algebraic
Formulation for Governing PDE and BCs 75
411 Grid Generation 75
412 Finite Dierence Method 78
413 Applications to CFD 92
42 Iterative Solution of System of LAEs for a Flow Property 93
421 Iterative Methods 94
422 Applications to CFD 98
43 Numerical Dierentiation for Local Engineering-
Parameters 105
431 Dierentiation Formulas 106
432 Applications to CFD 107
44 Numerical Integration for the Total value of
Engineering-Parameters 110
441 Integration Rules 111
442 Applications to CFD 114
45 Closure 116
Problems 116
Introduction to CFD xv
II CFD for a Cartesian-Geometry
119
5 Computational Heat Conduction 12151 Physical Law based Finite Volume Method 122
511 Energy Conservation Law for a Control Volume 123512 Algebraic Formulation 124513 Approximations 127514 Approximated Algebraic-Formulation 130515 Discussion 133
52 Finite Dierence Method for Boundary Conditions 13853 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 139531 One-Dimensional Conduction 139532 Two-Dimensional Conduction 147
54 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 154541 One-Dimensional Conduction 154542 Two-Dimensional Conduction 166
Problems 176
6 Computational Heat Advection 17961 Physical Law based Finite Volume Method 180
611 Energy Conservation Law for a Control Volume 181612 Algebraic Formulation 181613 Approximations 183614 Approximated Algebraic Formulation 191615 Discussion 191
62 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 194621 Explicit-Method 197622 Implementation Details 198623 Solution Algorithm 199
63 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 207631 Advection Scheme on a Non-Uniform Grid 207632 Explicit and Implicit Method 210633 Implementation Details 216634 Solution Algorithm 220
Problems 228
xvi Contents
7 Computational Heat Convection 22971 Physical Law based Finite Volume Method 229
711 Energy Conservation Law for a Control Volume 229712 Algebraic Formulation 231713 Approximated Algebraic Formulation 232
72 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 235721 Explicit-Method 236722 Implementation Details 237723 Solution Algorithm 238
73 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 242Problems 248
8 Computational Fluid Dynamics Physical Law based
Finite Volume Method 25181 Generalized Variables for the Combined Heat and
Fluid Flow 25282 Conservation Laws for a Control Volume 25583 Algebraic Formulation 25984 Approximations 26085 Approximated Algebraic Formulation 263
851 Mass Conservation 263852 MomentumEnergy Conservation 264
86 Closure 269
9 Computational Fluid Dynamics on a Staggered Grid 27191 Challenges in the CFD Development 273
911 Non-Linearity 273912 Equation for Pressure 273913 Pressure-Velocity Decoupling 273
92 A Staggered Grid to avoid Pressure-Velocity Decoupling27593 Physical Law based FVM for a Staggered Grid 27794 Flux based Solution Methodology on a Uniform Grid
Semi-Explicit Method 281941 Philosophy of Pressure-Correction Method 283942 Semi-Explicit Method 286943 Implementation Details 292944 Solution Algorithm 298
95 Initial and Boundary Conditions 300951 Initial Condition 300952 Boundary Condition 301
Problems 306
Introduction to CFD xvii
10 Computational Fluid Dynamics on a Co-located Grid309
101 Momentum-Interpolation Method Strategy to avoid
the Pressure-Velocity Decoupling on a Co-located Grid 310
102 Coecients of LAEs based Solution Methodology on
a Non-Uniform Grid Semi-Explicit and Semi-Implicit
Method 314
1021 Predictor Step 316
1022 Corrector Step 318
1023 Solution Algorithm 323
Problems 329
III CFD for a Complex-Geometry
331
11 Computational Heat Conduction on a Curvilinear
Grid 333
111 Curvilinear Grid Generation 333
1111 Algebraic Grid Generation 334
1112 Elliptic Grid Generation 336
112 Physical Law based Finite Volume Method 343
1121 Unsteady and Source Term 343
1122 Diusion Term 344
1123 All Terms 349
113 Computation of Geometrical Properties 349
114 Flux based Solution Methodology 352
1141 Explicit Method 354
1142 Implementation Details 354
Problems 359
12 Computational Fluid Dynamics on a Curvilinear Grid361
121 Physical Law based Finite Volume Method 361
1211 Mass Conservation 363
1212 Momentum Conservation 363
122 Solution Methodology Semi-Explicit Method 372
1221 Predictor Step 373
1222 Corrector Step 375
Problems 380
References 383
Index 389
Part I
Introduction and Essentials
The book on an introductory course in CFD starts with the two intro-
ductory chapters on CFD followed by one chapter each on essentials
of two prerequisite courses minus uid-dynamics and heat transfer and
numerical-methods The rst chapter on introduction presents an
eventful thoughtful and intuitive discussion on What is CFD and
Why to study CFD whereas How CFD works which involves lots
of details is presented separately in the second chapter The third-
chapter on essentials of uid-dynamics and heat transfer presents
physical-laws transport-mechanisms physical-law based dierential-
formulation volumetric and ux terms and mathematical formula-
tion The fourth-chapter on essentials of numerical-methods presents
nite dierence method iterative solution of the system of linear alge-
braic equation numerical dierentiation and numerical integration
The presentation for the essential of the two prerequisite courses is
customized to CFD development application and analysis
1
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
1
Introduction
The introduction of any course consists of the three basic questions
What is it Why to study How it works An eventful thoughtful
and intuitive answer to the rst two questions are presented in this
chapter The third question which involves lots of working details
on the three aspects of the Computational Fluid Dynamics (CFD)
(development application and analysis) is presented separately in
the next chapter The novelty scope and purpose of this book is
also presented at the end of this chapter
11 CFD What is it
CFD is a theoretical-method of scientic and engineering investiga-
tion concerned with the development and application of a video-
camera like tool (a software) which is used for a unied cause-and-
eect based analysis of a uid-dynamics as well as heat and mass
transfer problem
CFD is a subject where you rst learn how to develop a product
minus a software which acts like a virtual video-camera Then you learn
how to apply or use this product to generate uid-dynamics movies
Finally you learn how to analyze the detailed spatial and temporal
uid-dynamics information in the movies The analysis is done to
come up with a scientically-exciting as well as engineering-relevant
story (unied cause-and-eect study) of a uid-dynamics situation in
nature as well as industrial-applications With a continuous develop-
ment and a wider application of CFD the word uid-dynamics in
3
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
4 1 Introduction
the CFD has become more generic as it also corresponds to heat and
mass transfer as well as chemical reaction
Fluid-dynamics information are of two types scientic and en-
gineering Scientic-information corresponds to a structure of the
heat and uid ow due to a physical phenomenon in uid-dynamics
such as boundary-layer ow-separation wake-formation and vortex-
shedding The ow structures are obtained from a temporal variation
of ow-properties (such as velocity pressure temperature and many
other variables which characterizes a ow) called as scientically-
exciting movies Engineering-information corresponds to certain pa-
rameters which are important in an engineering application called
as engineering-parameters for example lift force drag force wall
shear stress pressure drop and rate of heat-transfer They are ob-
tained as the temporal variation of engineering-parameters called as
engineering-relevant movies The engineering-parameters are the ef-
fect which are caused by the ow structures minus their correlation leads
to the unied cause-and-eect study A uid-dynamics movie consist
of a time-wise varying series of pictures which give uid-dynamics
information Each ow-property and engineering-parameter results
in one scientically-exciting and engineering-relevant movie respec-
tively When the large number of both the types of movies are
played synchronized in-time the detailed spatial as well as tempo-
ral uid-dynamic information greatly helps in the unied cause-and-
eect study of a uid-dynamics problem
Alternatively CFD is a subject concerned with the development of
computer-programs for the computer-simulation and study of a natu-
ral or an engineering uid-dynamics system The development starts
with the virtual-development of the system involving a geometri-
cal information based computational development of a solid-model
for the system the system may be already-existing or yet-to-be-
developed physically Thereafter the development involves numerical
solution of the governing algebraic equations for CFD formulated
from the physical laws (conservation of mass momentum and en-
ergy) and initial as well as boundary conditions corresponding to the
system CFD study involves the analysis of the uid-dynamics and
heat-transfer results obtained from the computer-simulations of the
virtual-system which is subjected to certain governing-parameters
Introduction to CFD 5
111 CFD as a Scientic and Engineering Analysis Tool
For a better understanding of the denition of the CFD and its ap-
plication as a scientic and engineering analysis tool two example
problems are presented rst conduction heat transfer in a plate
and second uid ow across a circular pillar of a bridge The rst-
problem correspond to an engineering-application for electronic cool-
ing and the second-problem is studied for an engineering-design of
the structure subjected to uid-dynamics forces and are shown in
Fig 11 Both the problems are considered as two-dimensional and
unsteady However the results asymptotes from the unsteady to a
steady state in the rst-problem and to a periodic (dynamic-steady)
state in the second-problem
Figure 11 Schematic and the associated parameters for the two example prob-lems (a) thermal conduction module of a heat sink and (b) theclassical ow across a circular cylinder
6 1 Introduction
The rst-problem is taken from the classic book on heat-transfer
by Incropera and Dewitt (1996) shown in Fig 11(a) The g-
ure shows a periodic module of the plate where a constant heat-ux
qW = 10Wcm2 (dissipated from certain electronic devices) is rst con-
ducted to an aluminum plate and then convected to the water ow-
ing through the rectangular channel grooved in the plate The forced
convection results in a convection coecient of h = 5000Wm2K in-
side the channel The unsteady heat-transfer results in an interesting
time-wise varying conduction heat ow in the plate and instantaneous
convection heat transfer rate Qprimeconv (per unit width of the plate per-
pendicular to the plane shown in Fig 11(a)) into the water
The second-problem is encountered if you are standing over a
bridge and look below to the water passing over a pillar (of circu-
lar cross-section) of the bridge shown in Fig 11(b) as a general case
of 2D ow across a cylinder The uid-dynamics results in a beau-
tiful time-wise varying structure of the uid ow just downstream
of the pillar and certain engineering-parameters The parameters
correspond to the uid-dynamic forces acting in the horizontal (or
streamwise) and vertical (or transverse) direction called as the drag-
force FD and lift-force FL respectively The respective force is pre-
sented below in a non-dimensional form called as the drag-coecient
CD and lift-coecient CL equations shown in Fig 11(b)
If you use your video camera to capture a movie for the two
problems you will get series of pictures for the plate in the rst-
problem and the owing uid in the second-problem Since these
movies result in a pictorial information but not the ow-property
based uid-dynamic information such movies are not relevant for
the uid-dynamics and heat-transfer study Instead if you use a
CFD software for computer simulation you can capture various types
of movies The various ow-properties based scientically-exciting
movies are presented here as the pictures at certain time-instant in
Fig 12(a)-(d) and Fig 13(a1)-(d2) The former gure shows gray-
colored ooded contour for the temperature whereas the latter gure
shows gray-colored ooded contour for the pressure and vorticity and
vector-plot for the velocity For the gray-colored ooded contours a
color bar can be seen in the gures for the relative magnitude of the
variable These gures also show line contours of stream-function
Introduction to CFD 7
and heat-function as streamlines and heatlines respectively They
are the visualization techniques for a 2D steady heat and uid ow
The heatlines are analogous to streamlines and represents the direc-
tion of heat ow under steady state condition proposed by Kimura
and Bejan (1983) later presented in a heat-transfer book by Bejan
(1984) The steady-state heatlines can be seen in Fig 12(d) where
the direction for the conduction-ux is tangential at any point on the
heatlines analogous to that for the mass-ux in a streamline Thus
for a better presentation of the structure of heat and uid ow the
scientically-exciting movies are not only limited to ow-properties
but also correspond to certain ow visualization techniquesAn engineering-parameter based engineering-relevant uid dy-
namic movie is presented in Fig 12(e) for the convection heat trans-
fer rate per unit width at the various walls on the channel Qprimew at the
west-wall Qprimen at the north-wall Q
primee at the east-wall and their cumu-
lative value Qprimeconv (refer Fig 11(a)) Another engineering-relevant
movie is presented in Fig 13(e) for the lift and drag coecient Note
that the movie here corresponds to the motion of a lled symbol over
a line for the temporal variation of an engineering-parameter Thus
the engineering-relevant movies are one-dimensional lower than the
scientically-exciting movies here for the 2D problems the former
movie is presented as 1D and the latter as 2D results
1111 Heat-Conduction
For the computational set-up of the heat conduction problem shown
in Fig 11(a) the results obtained from a 2D CFD simulation of
the conduction heat transfer are shown in Fig 12 For a uni-
form initial temperature of the plate as Tinfin = 15oC Fig 12(a)-(d)
shows the isotherms and heatlines in the plate as the pictures of the
scientically-exciting movies at certain discrete time instants t = 25
5 10 and 20 sec For the same discrete time instants the temporal
variation of the position of the symbols over a line plot is presented in
Fig 12(e) The gure is shown for each of the heat transfer rate and
is obtained from a engineering-relevant 1D movie There is a tempo-
ral variation of picture in the scientically-exciting movie and that
of position of symbol (along the line plot) in the engineering-relevant
movieAccording to caloric theory of the famous French scientist An-
tonie Lavoisier heat is considered as an invisible tasteless odorless
8 1 Introduction
t(sec)
Qrsquo (W
m)
0 5 10 15 20 250
200
400
600
800
1000Qrsquo
conv
Qrsquow
Qrsquon
Qrsquoe
Qrsquoin
(e)
(b)
(c)(d)
(a)
(a) t=25 sec31
29
27
25
23
21
19
17
15
T(oC)
(c) t=10 sec
(b) t=5 sec
(d) t=20 sec
Figure 12 Thermal conduction module of a heat sink temporal variation of(a)-(d) isotherms and heatlines and (e) the convection heat transferrate per unit width at the various walls of the channels and theircumulative total value Q
primeconv For the discrete time-instant corre-
sponding to heat ow patterns in (a)-(d) temporal variation of thevarious heat transfer rates are shown as symbols in (e) (a)-(c) arethe transient and (d) represents a steady state result
weightless uid which he called caloric uid Presently the caloric
theory is overtaken by mechanical theory of heat minus heat is not a sub-
stance but a dynamical form of mechanical eect (Thomson 1851)
Nevertheless there are certain problems involving heat ow for which
Lavoisiers approach is rather useful such as the discussion on heat-
lines here
The role of heatlines for heat ow is analogous to that of stream-
lines for uid ow (Paramane and Sharma 2009) For the uid-ow
the dierence of stream-function values represents the rate of uid
ow the function remains constant on a solid wall the tangent to
a streamline represents the direction of the uid-ow (with no ow
in the normal direction) and the streamline originate or emerge at
Introduction to CFD 9
a mass source Analogously for the heat-ow the dierence of heat-
function values represents the rate of heat ow the function remains
constant on an adiabatic wall the tangent to a heatline represents the
direction of the heat-ow (with no ow in the normal direction) and
the heatline originate or emerge at the heat-source For the present
problem in Fig 11(a) the heat-source is at the inlet (or left) bound-
ary and the heat-sink is at the surface of the channel Thus the
steady-state heatlines in Fig 12(d) shows the heat ow emerging
from the inletsource and ending on the channel-wallssink Further-
more it can be seen in the gure that the heatlines which are close
to the adiabatic walls are parallel to the walls The source sink and
adiabatic-walls can be seen in Fig 11(a)
Figure 12(e) shows an asymptotic time-wise increase in the con-
vection heat transfer per unit width at the various walls of the channel
and its cumulative value Qprimeconv (= Q
primew+Q
primen+Q
primee) It can be also be seen
that the heat-transfer per unit width Qprimew at the west-wall and Q
primen at
the north-wall are larger than Qprimee at the east-wall of the channel As
time progresses the gure shows that Qprimeconv increases monotonically
and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-
der steady state condition refer Fig 11(a) The dierence between
the Qprimein and Q
primeconv under the transient condition is utilized for the
sensible heating and results in the increase in the temperature of the
plate Under the steady state condition note from Fig 12(d) that
the maximum temperature is close to 32oC minus well within the per-
missible limit for the reliable and ecient operation of the electronic
device
1112 Fluid-Dynamics
For the free-stream ow across a circular cylinderpillar at a Reynolds
number of 100 the non-dimensional results obtained from a 2D CFD
simulation are shown in Fig 13 In contrast to the previous prob-
lem which reaches steady state the present problem reaches to an un-
steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-
ent increasing time instants within one time-period of the periodic-
ow the results are presented in Fig 13(a)-(d) The gure shows
the close-up view of the pictures of the scientically-exciting movies
for ow-properties (velocity pressure and vorticity) and streamlines
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow
vi
Award for Outstanding Contributions to Undergraduate Education
For his research contributions to computational heat transfer he was
given the 1991 ASME Heat Transfer Memorial Award and the 1997
Classic Paper Award He was awarded the 2008 Max Jakob Award
which is considered to be the highest international honor in the eld of
heat transfer In 2015 an International Conference on Computational
Heat Transfer (CHT-15) held at Rutgers University in New Jersey
was dedicated to Professor Patankar
Dr Patankars widespread inuence on research and engineering
education has been recognized in many ways In 2007 the Editors of
the International Journal of Heat and Mass Transfer wrote There is
no person who has made a more profound and enduring impact on the
theory and practice of numerical simulation in mechanical engineering
than Professor Patankar
FOREWORD
by Prof K Muralidhar
Indian Institute of Technology Kanpur INDIA
The subject of computational uid dynamics (CFD) emerged in the
academic context but in the past two decades it has developed to a
point where it is extensively employed as a design and analysis tool in
the industry The impact of the subject is deeply felt in diverse disci-
plines ranging from the aerospace spreading all the way to the chem-
ical industry New compact and ecient designs are enabled and it
has become that much more convenient to locate optimum conditions
for the operation of engineering systems As it stands CFD is com-
pulsory learning in several branches of engineering Many universities
now opt to develop theoretical courses around it The great utility of
this tool has inuenced adjacent disciplines and the process of solv-
ing complex dierential equations by the process of discretization has
become a staple for addressing engineering challenges
For an emerging discipline it is necessary that new books fre-
quently appear on the horizon Students can choose from the se-
lection and will stand to gain from it Such books should have a
perspective of the past and the future and it is all the more signi-
cant when the author has exclusive training in the subject Dr Atul
Sharma represents a combination of perspective training and passion
that combines well and delivers a strong text
I have personal knowledge of the fact that Dr Atul Sharma has
developed a whole suite of computer programs on CFD from scratch
He has wondered and pondered over why approximations work and
conditions under which they yield meaningful solutions His spread
vii
viii Foreword
of experience over fteen and odd years includes two and three di-
mensional Navier-Stokes equations heat transfer interfacial dynam-
ics and turbulent ow He has championed the nite volume method
which is now the industry standard The programming experience
is important because it brings in the precision needed for assembling
the jigsaw puzzle The sprinkling of computer programs in the book
is no coincidence
Atul knows the conventional method of discretizing dierential
equations but has never been satised with it He has felt an ele-
ment of discomfort going away from a physical reality into a world
of numbers connected by arithmetic operations As a result he has
developed a principle that physical laws that characterize the dier-
ential equations should be reected at every stage of discretization
and every stage of approximation indeed at every scale whether a
single tetrahedron or the sub-domain or the region as a whole This
idea permeates the book where it is shown that discretized versions
must provide reasonable answers because they continue to represent
the laws of nature
This new CFD book is comprehensive and has a stamp of origi-
nality of the author It will bring students closer to the subject and
enable them to contribute to it
Kanpur
August 2016 K Muralidhar
PREFACE
As a Post-Graduate (PG) student in 1997 I got introduced to a
course on CFD at Indian Institute of Science (IISc) Bangalore Dur-
ing the introductory course on CFD although I struggled hard
in the physical understanding and computer-programming it didnt
stop my interest and enthusiasm to do research teaching and writ-
ing a text-book on a introductory course in CFD Since then I did
my Masters project work on CFD (1997-98) introduced and taught
this course to Under-Graduate (UG) students at National Institute
of Technology Hamirpur HP (1999-2000) did my Doctoral work on
this subject at Indian Institute of Technology (IIT) Kanpur (2000-
2004) and co-authored a Chapter on nite volume method in a re-
vised edition of an edited book on computational uid ow and heat
transfer during my PhD in 2003 After joining as a faculty in 2004
I taught this course to UG as well as PG students of dierent depart-
ments at IIT Bombay as well as to the students of dierent collages in
India through CDEEP (Center for Distance Engineering Programme)
IIT Bombay from 2007-2010 Furthermore I gave series of lecture
on CFD at various collages and industries in India More recently
in 2012 I delivered a ve-days lecture and lab-session on CFD (to
around 1400 collage teachers from dierent parts of India) as a part
of a project funded by ministry of human resource development under
NMEICT (National Mission on Education through information and
communication technology)
The increasing need for the development of the customized CFD
software (app) and wide spread CFD application as well as analysis
for the design optimization and innovation of various types of engi-
neering systems always motivated me to write this book Due to the
increasing importance of CFD another motivation is to present CFD
ix
x Preface
simple enough to introduce the rst-course at early under-graduate
curriculum rather than the post-graduate curriculum The moti-
vation got strengthened by my own as well as others (students
teachers researchers and practitioners) struggle on physical under-
standing of the mathematics involved during the Partial Dierential
Equation (PDE) based algebraic-formulation (nite volume method)
along with the continuous frustrating pursuit to eectively convert
the theory of CFD into the computer-program
This led me to come up with an alternate physical-law (PDE-
free) based nite volume method as well as solution methodol-
ogy which are much easier to comprehend Mostly the success of
the programming eort is decided by the perfect execution of the
implementations-details which rarely appears in-print (existing books
and journal articles) The physical law based CFD development
approach and the implementations-details are the novelties in the
present book Furthermore using an open-source software for nu-
merical computations (Scilab) computer programs are given in this
book on the basic modules of conduction advection and convection
Indeed the reader can generalize and extend these codes for the de-
velopment of Navier-Stokes solver and generate the results presented
in the chapters towards the end of this book
I have limited the scope of this book to the numerical-techniques
which I wish to recommend for the book although there are numerous
other advanced and better methods in the published literature I do
not claim that the programming-practice presented here are the most
ecient minus they are presented here more for the ease in programming
(understand as your program) than for the computational eciency
Furthermore although I am enthusiastic about my presentation of
the physical law based CFD approach it may not be more ecient
than the alternate mathematics based CFD approach The emphasis
in this book is on the ease of understanding of the formulations and
programming-practice for the introductory course on CFD
I would hereby acknowledge that I owe my greatest debt to Prof
J Srinivasan (my ME Project supervisor at IISc Bangalore) who
did a hand-holding and wonderful job in patiently introducing me to
the fascinating world of research in uid-dynamics and heat trans-
fer I would also acknowledge the excellent training from Prof V
Introduction to CFD xi
Eswaran (my PhD supervisor at IIT Kanpur) from whom I learned
systematic way of CFD development application and analysis His
inspiring comments and suggestions on my writing-style and elabora-
tion of the rst chapter in this book is also gratefully acknowledged
I also want to record my sincere thanks to Prof Pradip Datta IISc
Bangalore and Prof Gautam Biswas IIT Kanpur minus my teachers for
CFD course Special thanks to Prof K Muralidhar IIT Kanpur for
his personal encouragement academic support and motivation (for
the book writing) He also cleared my dilemma in the proper presen-
tation of the novelty in this book I thank him also for writing the
foreword of this book
I am grateful to Prof Kannan M Moudgalya and Prof Deepak B
Phatak from IIT Bombay for giving me opportunity to teach CFD
in a distance education mode I am also grateful to Prof Amit
Agrawal (my colleague at IIT Bombay) who diligently read through
all the chapters of this book and suggested some really remarkable
changes in the book I also thank him for the research interactions
over the last decade which had substantially improved my under-
standing of thermal and uid science I would also like to thank my
PhD student Namshad T who did a great job in implementing
my ideas on generating the CFD simulation based CFD image for
the cover-page and couple of gures in the rst chapter (Fig 12
and 13) Special mention that the Scilab codes developed (for the
CFD course under NMEICT) by Vishesh Aggarwal greatly helped
me to develop the codes presented in this book his contributions is
gratefully acknowledged I thank Malhar Malushte for testing some
of the formulations presented in this book
I thank my research students (Sachin B Paramane D Datta C
M Sewatkar Vinesh H Gada Kaushal Prasad Mukul Shrivastava
Absar M Lakdawala Mausam Sarkar and Javed Shaikh) for the
diligence and sincerity they have shown in unraveling some of our
ideas on CFD Apart from these several other research students and
the students teachers and CFD practitioners who took my lectures
on CFD in the past decade have contributed to my better under-
standing of CFD and inspired me to write the book I thank all of
them for their attention and enthusiastic response during the CFD
courses The inuence of the interactions with all of them can be seen
xii Preface
throughout this book I am grateful to IIT Bombay for giving me
one year sabbatical leave to write this book The ambiance of aca-
demic freedom considerate support by the faculties and the friendly
atmosphere inside the campus of IIT Bombay has greatly contributed
in the solo eort of book writing I thanks all my colleagues at IIT
Bombay for shaping my perceptions on CFD
During this long endeavor the patient support by Mr Sunil
Saxena Director Athena Academic Ltd UK is gratefully acknowl-
edged I also thank John Wiley for coming forward to take this
book to international markets Lastly I would acknowledge that I
am greatly inspired by the writing style of J D Anderson and the
book on CFD by S V Patankar Of course you will nd that I
have heavily cross-referred the CFD book by the two most acknowl-
edged pioneers on CFD I dedicate this book to Dr S V Patankar
whose book nicely introduced me to the computational as well as
ow physics in CFD and greatly helped me to introduce a physical
approach of CFD development in this book
The culmination of the book writing process fullls a long cher-
ished dream It would not have been possible without the excel-
lent environment continuous support and complete understanding
of my wife Anubha and daughter Anshita Their kindness has greatly
helped to fulll my dream now I plan to spend more time with my
wife and daughter
Mumbai
August 2016 Atul Sharma
Contents
I Introduction and Essentials
1
1 Introduction 3
11 CFD What is it 3
111 CFD as a Scientic and Engineering Analysis
Tool 5
112 Analogy with a Video-Camera 13
12 CFD Why to study 15
13 Novelty Scope and Purpose of this Book 16
2 Introduction to CFD Development Application and
Analysis 23
21 CFD Development 23
211 Grid Generation Pre-Processor 24
212 Discretization Method Algebraic Formulation 26
213 Solution Methodology Solver 29
214 Computation of Engineering-Parameters
Post-Processor 34
215 Testing 35
22 CFD Application 35
23 CFD Analysis 38
24 Closure 39
xiii
xiv Contents
3 Essentials of Fluid-Dynamics and Heat-Transfer for
CFD 41
31 Physical Laws 42
311 FundamentalConservation Laws 43
312 Subsidiary Laws 45
32 Momentum and Energy Transport Mechanisms 46
33 Physical Law based Dierential Formulation 48
331 Continuity Equation 49
332 Transport Equation 51
34 Generalized Volumetric and Flux Terms and their Dif-
ferential Formulation 57
341 Volumetric Term 58
342 Flux-Term 58
343 Discussion 62
35 Mathematical Formulation 63
351 Dimensional Study 63
352 Non-Dimensional Study 67
36 Closure 71
4 Essentials of Numerical-Methods for CFD 73
41 Finite Dierence Method A Dierential to Algebraic
Formulation for Governing PDE and BCs 75
411 Grid Generation 75
412 Finite Dierence Method 78
413 Applications to CFD 92
42 Iterative Solution of System of LAEs for a Flow Property 93
421 Iterative Methods 94
422 Applications to CFD 98
43 Numerical Dierentiation for Local Engineering-
Parameters 105
431 Dierentiation Formulas 106
432 Applications to CFD 107
44 Numerical Integration for the Total value of
Engineering-Parameters 110
441 Integration Rules 111
442 Applications to CFD 114
45 Closure 116
Problems 116
Introduction to CFD xv
II CFD for a Cartesian-Geometry
119
5 Computational Heat Conduction 12151 Physical Law based Finite Volume Method 122
511 Energy Conservation Law for a Control Volume 123512 Algebraic Formulation 124513 Approximations 127514 Approximated Algebraic-Formulation 130515 Discussion 133
52 Finite Dierence Method for Boundary Conditions 13853 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 139531 One-Dimensional Conduction 139532 Two-Dimensional Conduction 147
54 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 154541 One-Dimensional Conduction 154542 Two-Dimensional Conduction 166
Problems 176
6 Computational Heat Advection 17961 Physical Law based Finite Volume Method 180
611 Energy Conservation Law for a Control Volume 181612 Algebraic Formulation 181613 Approximations 183614 Approximated Algebraic Formulation 191615 Discussion 191
62 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 194621 Explicit-Method 197622 Implementation Details 198623 Solution Algorithm 199
63 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 207631 Advection Scheme on a Non-Uniform Grid 207632 Explicit and Implicit Method 210633 Implementation Details 216634 Solution Algorithm 220
Problems 228
xvi Contents
7 Computational Heat Convection 22971 Physical Law based Finite Volume Method 229
711 Energy Conservation Law for a Control Volume 229712 Algebraic Formulation 231713 Approximated Algebraic Formulation 232
72 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 235721 Explicit-Method 236722 Implementation Details 237723 Solution Algorithm 238
73 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 242Problems 248
8 Computational Fluid Dynamics Physical Law based
Finite Volume Method 25181 Generalized Variables for the Combined Heat and
Fluid Flow 25282 Conservation Laws for a Control Volume 25583 Algebraic Formulation 25984 Approximations 26085 Approximated Algebraic Formulation 263
851 Mass Conservation 263852 MomentumEnergy Conservation 264
86 Closure 269
9 Computational Fluid Dynamics on a Staggered Grid 27191 Challenges in the CFD Development 273
911 Non-Linearity 273912 Equation for Pressure 273913 Pressure-Velocity Decoupling 273
92 A Staggered Grid to avoid Pressure-Velocity Decoupling27593 Physical Law based FVM for a Staggered Grid 27794 Flux based Solution Methodology on a Uniform Grid
Semi-Explicit Method 281941 Philosophy of Pressure-Correction Method 283942 Semi-Explicit Method 286943 Implementation Details 292944 Solution Algorithm 298
95 Initial and Boundary Conditions 300951 Initial Condition 300952 Boundary Condition 301
Problems 306
Introduction to CFD xvii
10 Computational Fluid Dynamics on a Co-located Grid309
101 Momentum-Interpolation Method Strategy to avoid
the Pressure-Velocity Decoupling on a Co-located Grid 310
102 Coecients of LAEs based Solution Methodology on
a Non-Uniform Grid Semi-Explicit and Semi-Implicit
Method 314
1021 Predictor Step 316
1022 Corrector Step 318
1023 Solution Algorithm 323
Problems 329
III CFD for a Complex-Geometry
331
11 Computational Heat Conduction on a Curvilinear
Grid 333
111 Curvilinear Grid Generation 333
1111 Algebraic Grid Generation 334
1112 Elliptic Grid Generation 336
112 Physical Law based Finite Volume Method 343
1121 Unsteady and Source Term 343
1122 Diusion Term 344
1123 All Terms 349
113 Computation of Geometrical Properties 349
114 Flux based Solution Methodology 352
1141 Explicit Method 354
1142 Implementation Details 354
Problems 359
12 Computational Fluid Dynamics on a Curvilinear Grid361
121 Physical Law based Finite Volume Method 361
1211 Mass Conservation 363
1212 Momentum Conservation 363
122 Solution Methodology Semi-Explicit Method 372
1221 Predictor Step 373
1222 Corrector Step 375
Problems 380
References 383
Index 389
Part I
Introduction and Essentials
The book on an introductory course in CFD starts with the two intro-
ductory chapters on CFD followed by one chapter each on essentials
of two prerequisite courses minus uid-dynamics and heat transfer and
numerical-methods The rst chapter on introduction presents an
eventful thoughtful and intuitive discussion on What is CFD and
Why to study CFD whereas How CFD works which involves lots
of details is presented separately in the second chapter The third-
chapter on essentials of uid-dynamics and heat transfer presents
physical-laws transport-mechanisms physical-law based dierential-
formulation volumetric and ux terms and mathematical formula-
tion The fourth-chapter on essentials of numerical-methods presents
nite dierence method iterative solution of the system of linear alge-
braic equation numerical dierentiation and numerical integration
The presentation for the essential of the two prerequisite courses is
customized to CFD development application and analysis
1
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
1
Introduction
The introduction of any course consists of the three basic questions
What is it Why to study How it works An eventful thoughtful
and intuitive answer to the rst two questions are presented in this
chapter The third question which involves lots of working details
on the three aspects of the Computational Fluid Dynamics (CFD)
(development application and analysis) is presented separately in
the next chapter The novelty scope and purpose of this book is
also presented at the end of this chapter
11 CFD What is it
CFD is a theoretical-method of scientic and engineering investiga-
tion concerned with the development and application of a video-
camera like tool (a software) which is used for a unied cause-and-
eect based analysis of a uid-dynamics as well as heat and mass
transfer problem
CFD is a subject where you rst learn how to develop a product
minus a software which acts like a virtual video-camera Then you learn
how to apply or use this product to generate uid-dynamics movies
Finally you learn how to analyze the detailed spatial and temporal
uid-dynamics information in the movies The analysis is done to
come up with a scientically-exciting as well as engineering-relevant
story (unied cause-and-eect study) of a uid-dynamics situation in
nature as well as industrial-applications With a continuous develop-
ment and a wider application of CFD the word uid-dynamics in
3
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
4 1 Introduction
the CFD has become more generic as it also corresponds to heat and
mass transfer as well as chemical reaction
Fluid-dynamics information are of two types scientic and en-
gineering Scientic-information corresponds to a structure of the
heat and uid ow due to a physical phenomenon in uid-dynamics
such as boundary-layer ow-separation wake-formation and vortex-
shedding The ow structures are obtained from a temporal variation
of ow-properties (such as velocity pressure temperature and many
other variables which characterizes a ow) called as scientically-
exciting movies Engineering-information corresponds to certain pa-
rameters which are important in an engineering application called
as engineering-parameters for example lift force drag force wall
shear stress pressure drop and rate of heat-transfer They are ob-
tained as the temporal variation of engineering-parameters called as
engineering-relevant movies The engineering-parameters are the ef-
fect which are caused by the ow structures minus their correlation leads
to the unied cause-and-eect study A uid-dynamics movie consist
of a time-wise varying series of pictures which give uid-dynamics
information Each ow-property and engineering-parameter results
in one scientically-exciting and engineering-relevant movie respec-
tively When the large number of both the types of movies are
played synchronized in-time the detailed spatial as well as tempo-
ral uid-dynamic information greatly helps in the unied cause-and-
eect study of a uid-dynamics problem
Alternatively CFD is a subject concerned with the development of
computer-programs for the computer-simulation and study of a natu-
ral or an engineering uid-dynamics system The development starts
with the virtual-development of the system involving a geometri-
cal information based computational development of a solid-model
for the system the system may be already-existing or yet-to-be-
developed physically Thereafter the development involves numerical
solution of the governing algebraic equations for CFD formulated
from the physical laws (conservation of mass momentum and en-
ergy) and initial as well as boundary conditions corresponding to the
system CFD study involves the analysis of the uid-dynamics and
heat-transfer results obtained from the computer-simulations of the
virtual-system which is subjected to certain governing-parameters
Introduction to CFD 5
111 CFD as a Scientic and Engineering Analysis Tool
For a better understanding of the denition of the CFD and its ap-
plication as a scientic and engineering analysis tool two example
problems are presented rst conduction heat transfer in a plate
and second uid ow across a circular pillar of a bridge The rst-
problem correspond to an engineering-application for electronic cool-
ing and the second-problem is studied for an engineering-design of
the structure subjected to uid-dynamics forces and are shown in
Fig 11 Both the problems are considered as two-dimensional and
unsteady However the results asymptotes from the unsteady to a
steady state in the rst-problem and to a periodic (dynamic-steady)
state in the second-problem
Figure 11 Schematic and the associated parameters for the two example prob-lems (a) thermal conduction module of a heat sink and (b) theclassical ow across a circular cylinder
6 1 Introduction
The rst-problem is taken from the classic book on heat-transfer
by Incropera and Dewitt (1996) shown in Fig 11(a) The g-
ure shows a periodic module of the plate where a constant heat-ux
qW = 10Wcm2 (dissipated from certain electronic devices) is rst con-
ducted to an aluminum plate and then convected to the water ow-
ing through the rectangular channel grooved in the plate The forced
convection results in a convection coecient of h = 5000Wm2K in-
side the channel The unsteady heat-transfer results in an interesting
time-wise varying conduction heat ow in the plate and instantaneous
convection heat transfer rate Qprimeconv (per unit width of the plate per-
pendicular to the plane shown in Fig 11(a)) into the water
The second-problem is encountered if you are standing over a
bridge and look below to the water passing over a pillar (of circu-
lar cross-section) of the bridge shown in Fig 11(b) as a general case
of 2D ow across a cylinder The uid-dynamics results in a beau-
tiful time-wise varying structure of the uid ow just downstream
of the pillar and certain engineering-parameters The parameters
correspond to the uid-dynamic forces acting in the horizontal (or
streamwise) and vertical (or transverse) direction called as the drag-
force FD and lift-force FL respectively The respective force is pre-
sented below in a non-dimensional form called as the drag-coecient
CD and lift-coecient CL equations shown in Fig 11(b)
If you use your video camera to capture a movie for the two
problems you will get series of pictures for the plate in the rst-
problem and the owing uid in the second-problem Since these
movies result in a pictorial information but not the ow-property
based uid-dynamic information such movies are not relevant for
the uid-dynamics and heat-transfer study Instead if you use a
CFD software for computer simulation you can capture various types
of movies The various ow-properties based scientically-exciting
movies are presented here as the pictures at certain time-instant in
Fig 12(a)-(d) and Fig 13(a1)-(d2) The former gure shows gray-
colored ooded contour for the temperature whereas the latter gure
shows gray-colored ooded contour for the pressure and vorticity and
vector-plot for the velocity For the gray-colored ooded contours a
color bar can be seen in the gures for the relative magnitude of the
variable These gures also show line contours of stream-function
Introduction to CFD 7
and heat-function as streamlines and heatlines respectively They
are the visualization techniques for a 2D steady heat and uid ow
The heatlines are analogous to streamlines and represents the direc-
tion of heat ow under steady state condition proposed by Kimura
and Bejan (1983) later presented in a heat-transfer book by Bejan
(1984) The steady-state heatlines can be seen in Fig 12(d) where
the direction for the conduction-ux is tangential at any point on the
heatlines analogous to that for the mass-ux in a streamline Thus
for a better presentation of the structure of heat and uid ow the
scientically-exciting movies are not only limited to ow-properties
but also correspond to certain ow visualization techniquesAn engineering-parameter based engineering-relevant uid dy-
namic movie is presented in Fig 12(e) for the convection heat trans-
fer rate per unit width at the various walls on the channel Qprimew at the
west-wall Qprimen at the north-wall Q
primee at the east-wall and their cumu-
lative value Qprimeconv (refer Fig 11(a)) Another engineering-relevant
movie is presented in Fig 13(e) for the lift and drag coecient Note
that the movie here corresponds to the motion of a lled symbol over
a line for the temporal variation of an engineering-parameter Thus
the engineering-relevant movies are one-dimensional lower than the
scientically-exciting movies here for the 2D problems the former
movie is presented as 1D and the latter as 2D results
1111 Heat-Conduction
For the computational set-up of the heat conduction problem shown
in Fig 11(a) the results obtained from a 2D CFD simulation of
the conduction heat transfer are shown in Fig 12 For a uni-
form initial temperature of the plate as Tinfin = 15oC Fig 12(a)-(d)
shows the isotherms and heatlines in the plate as the pictures of the
scientically-exciting movies at certain discrete time instants t = 25
5 10 and 20 sec For the same discrete time instants the temporal
variation of the position of the symbols over a line plot is presented in
Fig 12(e) The gure is shown for each of the heat transfer rate and
is obtained from a engineering-relevant 1D movie There is a tempo-
ral variation of picture in the scientically-exciting movie and that
of position of symbol (along the line plot) in the engineering-relevant
movieAccording to caloric theory of the famous French scientist An-
tonie Lavoisier heat is considered as an invisible tasteless odorless
8 1 Introduction
t(sec)
Qrsquo (W
m)
0 5 10 15 20 250
200
400
600
800
1000Qrsquo
conv
Qrsquow
Qrsquon
Qrsquoe
Qrsquoin
(e)
(b)
(c)(d)
(a)
(a) t=25 sec31
29
27
25
23
21
19
17
15
T(oC)
(c) t=10 sec
(b) t=5 sec
(d) t=20 sec
Figure 12 Thermal conduction module of a heat sink temporal variation of(a)-(d) isotherms and heatlines and (e) the convection heat transferrate per unit width at the various walls of the channels and theircumulative total value Q
primeconv For the discrete time-instant corre-
sponding to heat ow patterns in (a)-(d) temporal variation of thevarious heat transfer rates are shown as symbols in (e) (a)-(c) arethe transient and (d) represents a steady state result
weightless uid which he called caloric uid Presently the caloric
theory is overtaken by mechanical theory of heat minus heat is not a sub-
stance but a dynamical form of mechanical eect (Thomson 1851)
Nevertheless there are certain problems involving heat ow for which
Lavoisiers approach is rather useful such as the discussion on heat-
lines here
The role of heatlines for heat ow is analogous to that of stream-
lines for uid ow (Paramane and Sharma 2009) For the uid-ow
the dierence of stream-function values represents the rate of uid
ow the function remains constant on a solid wall the tangent to
a streamline represents the direction of the uid-ow (with no ow
in the normal direction) and the streamline originate or emerge at
Introduction to CFD 9
a mass source Analogously for the heat-ow the dierence of heat-
function values represents the rate of heat ow the function remains
constant on an adiabatic wall the tangent to a heatline represents the
direction of the heat-ow (with no ow in the normal direction) and
the heatline originate or emerge at the heat-source For the present
problem in Fig 11(a) the heat-source is at the inlet (or left) bound-
ary and the heat-sink is at the surface of the channel Thus the
steady-state heatlines in Fig 12(d) shows the heat ow emerging
from the inletsource and ending on the channel-wallssink Further-
more it can be seen in the gure that the heatlines which are close
to the adiabatic walls are parallel to the walls The source sink and
adiabatic-walls can be seen in Fig 11(a)
Figure 12(e) shows an asymptotic time-wise increase in the con-
vection heat transfer per unit width at the various walls of the channel
and its cumulative value Qprimeconv (= Q
primew+Q
primen+Q
primee) It can be also be seen
that the heat-transfer per unit width Qprimew at the west-wall and Q
primen at
the north-wall are larger than Qprimee at the east-wall of the channel As
time progresses the gure shows that Qprimeconv increases monotonically
and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-
der steady state condition refer Fig 11(a) The dierence between
the Qprimein and Q
primeconv under the transient condition is utilized for the
sensible heating and results in the increase in the temperature of the
plate Under the steady state condition note from Fig 12(d) that
the maximum temperature is close to 32oC minus well within the per-
missible limit for the reliable and ecient operation of the electronic
device
1112 Fluid-Dynamics
For the free-stream ow across a circular cylinderpillar at a Reynolds
number of 100 the non-dimensional results obtained from a 2D CFD
simulation are shown in Fig 13 In contrast to the previous prob-
lem which reaches steady state the present problem reaches to an un-
steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-
ent increasing time instants within one time-period of the periodic-
ow the results are presented in Fig 13(a)-(d) The gure shows
the close-up view of the pictures of the scientically-exciting movies
for ow-properties (velocity pressure and vorticity) and streamlines
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow
FOREWORD
by Prof K Muralidhar
Indian Institute of Technology Kanpur INDIA
The subject of computational uid dynamics (CFD) emerged in the
academic context but in the past two decades it has developed to a
point where it is extensively employed as a design and analysis tool in
the industry The impact of the subject is deeply felt in diverse disci-
plines ranging from the aerospace spreading all the way to the chem-
ical industry New compact and ecient designs are enabled and it
has become that much more convenient to locate optimum conditions
for the operation of engineering systems As it stands CFD is com-
pulsory learning in several branches of engineering Many universities
now opt to develop theoretical courses around it The great utility of
this tool has inuenced adjacent disciplines and the process of solv-
ing complex dierential equations by the process of discretization has
become a staple for addressing engineering challenges
For an emerging discipline it is necessary that new books fre-
quently appear on the horizon Students can choose from the se-
lection and will stand to gain from it Such books should have a
perspective of the past and the future and it is all the more signi-
cant when the author has exclusive training in the subject Dr Atul
Sharma represents a combination of perspective training and passion
that combines well and delivers a strong text
I have personal knowledge of the fact that Dr Atul Sharma has
developed a whole suite of computer programs on CFD from scratch
He has wondered and pondered over why approximations work and
conditions under which they yield meaningful solutions His spread
vii
viii Foreword
of experience over fteen and odd years includes two and three di-
mensional Navier-Stokes equations heat transfer interfacial dynam-
ics and turbulent ow He has championed the nite volume method
which is now the industry standard The programming experience
is important because it brings in the precision needed for assembling
the jigsaw puzzle The sprinkling of computer programs in the book
is no coincidence
Atul knows the conventional method of discretizing dierential
equations but has never been satised with it He has felt an ele-
ment of discomfort going away from a physical reality into a world
of numbers connected by arithmetic operations As a result he has
developed a principle that physical laws that characterize the dier-
ential equations should be reected at every stage of discretization
and every stage of approximation indeed at every scale whether a
single tetrahedron or the sub-domain or the region as a whole This
idea permeates the book where it is shown that discretized versions
must provide reasonable answers because they continue to represent
the laws of nature
This new CFD book is comprehensive and has a stamp of origi-
nality of the author It will bring students closer to the subject and
enable them to contribute to it
Kanpur
August 2016 K Muralidhar
PREFACE
As a Post-Graduate (PG) student in 1997 I got introduced to a
course on CFD at Indian Institute of Science (IISc) Bangalore Dur-
ing the introductory course on CFD although I struggled hard
in the physical understanding and computer-programming it didnt
stop my interest and enthusiasm to do research teaching and writ-
ing a text-book on a introductory course in CFD Since then I did
my Masters project work on CFD (1997-98) introduced and taught
this course to Under-Graduate (UG) students at National Institute
of Technology Hamirpur HP (1999-2000) did my Doctoral work on
this subject at Indian Institute of Technology (IIT) Kanpur (2000-
2004) and co-authored a Chapter on nite volume method in a re-
vised edition of an edited book on computational uid ow and heat
transfer during my PhD in 2003 After joining as a faculty in 2004
I taught this course to UG as well as PG students of dierent depart-
ments at IIT Bombay as well as to the students of dierent collages in
India through CDEEP (Center for Distance Engineering Programme)
IIT Bombay from 2007-2010 Furthermore I gave series of lecture
on CFD at various collages and industries in India More recently
in 2012 I delivered a ve-days lecture and lab-session on CFD (to
around 1400 collage teachers from dierent parts of India) as a part
of a project funded by ministry of human resource development under
NMEICT (National Mission on Education through information and
communication technology)
The increasing need for the development of the customized CFD
software (app) and wide spread CFD application as well as analysis
for the design optimization and innovation of various types of engi-
neering systems always motivated me to write this book Due to the
increasing importance of CFD another motivation is to present CFD
ix
x Preface
simple enough to introduce the rst-course at early under-graduate
curriculum rather than the post-graduate curriculum The moti-
vation got strengthened by my own as well as others (students
teachers researchers and practitioners) struggle on physical under-
standing of the mathematics involved during the Partial Dierential
Equation (PDE) based algebraic-formulation (nite volume method)
along with the continuous frustrating pursuit to eectively convert
the theory of CFD into the computer-program
This led me to come up with an alternate physical-law (PDE-
free) based nite volume method as well as solution methodol-
ogy which are much easier to comprehend Mostly the success of
the programming eort is decided by the perfect execution of the
implementations-details which rarely appears in-print (existing books
and journal articles) The physical law based CFD development
approach and the implementations-details are the novelties in the
present book Furthermore using an open-source software for nu-
merical computations (Scilab) computer programs are given in this
book on the basic modules of conduction advection and convection
Indeed the reader can generalize and extend these codes for the de-
velopment of Navier-Stokes solver and generate the results presented
in the chapters towards the end of this book
I have limited the scope of this book to the numerical-techniques
which I wish to recommend for the book although there are numerous
other advanced and better methods in the published literature I do
not claim that the programming-practice presented here are the most
ecient minus they are presented here more for the ease in programming
(understand as your program) than for the computational eciency
Furthermore although I am enthusiastic about my presentation of
the physical law based CFD approach it may not be more ecient
than the alternate mathematics based CFD approach The emphasis
in this book is on the ease of understanding of the formulations and
programming-practice for the introductory course on CFD
I would hereby acknowledge that I owe my greatest debt to Prof
J Srinivasan (my ME Project supervisor at IISc Bangalore) who
did a hand-holding and wonderful job in patiently introducing me to
the fascinating world of research in uid-dynamics and heat trans-
fer I would also acknowledge the excellent training from Prof V
Introduction to CFD xi
Eswaran (my PhD supervisor at IIT Kanpur) from whom I learned
systematic way of CFD development application and analysis His
inspiring comments and suggestions on my writing-style and elabora-
tion of the rst chapter in this book is also gratefully acknowledged
I also want to record my sincere thanks to Prof Pradip Datta IISc
Bangalore and Prof Gautam Biswas IIT Kanpur minus my teachers for
CFD course Special thanks to Prof K Muralidhar IIT Kanpur for
his personal encouragement academic support and motivation (for
the book writing) He also cleared my dilemma in the proper presen-
tation of the novelty in this book I thank him also for writing the
foreword of this book
I am grateful to Prof Kannan M Moudgalya and Prof Deepak B
Phatak from IIT Bombay for giving me opportunity to teach CFD
in a distance education mode I am also grateful to Prof Amit
Agrawal (my colleague at IIT Bombay) who diligently read through
all the chapters of this book and suggested some really remarkable
changes in the book I also thank him for the research interactions
over the last decade which had substantially improved my under-
standing of thermal and uid science I would also like to thank my
PhD student Namshad T who did a great job in implementing
my ideas on generating the CFD simulation based CFD image for
the cover-page and couple of gures in the rst chapter (Fig 12
and 13) Special mention that the Scilab codes developed (for the
CFD course under NMEICT) by Vishesh Aggarwal greatly helped
me to develop the codes presented in this book his contributions is
gratefully acknowledged I thank Malhar Malushte for testing some
of the formulations presented in this book
I thank my research students (Sachin B Paramane D Datta C
M Sewatkar Vinesh H Gada Kaushal Prasad Mukul Shrivastava
Absar M Lakdawala Mausam Sarkar and Javed Shaikh) for the
diligence and sincerity they have shown in unraveling some of our
ideas on CFD Apart from these several other research students and
the students teachers and CFD practitioners who took my lectures
on CFD in the past decade have contributed to my better under-
standing of CFD and inspired me to write the book I thank all of
them for their attention and enthusiastic response during the CFD
courses The inuence of the interactions with all of them can be seen
xii Preface
throughout this book I am grateful to IIT Bombay for giving me
one year sabbatical leave to write this book The ambiance of aca-
demic freedom considerate support by the faculties and the friendly
atmosphere inside the campus of IIT Bombay has greatly contributed
in the solo eort of book writing I thanks all my colleagues at IIT
Bombay for shaping my perceptions on CFD
During this long endeavor the patient support by Mr Sunil
Saxena Director Athena Academic Ltd UK is gratefully acknowl-
edged I also thank John Wiley for coming forward to take this
book to international markets Lastly I would acknowledge that I
am greatly inspired by the writing style of J D Anderson and the
book on CFD by S V Patankar Of course you will nd that I
have heavily cross-referred the CFD book by the two most acknowl-
edged pioneers on CFD I dedicate this book to Dr S V Patankar
whose book nicely introduced me to the computational as well as
ow physics in CFD and greatly helped me to introduce a physical
approach of CFD development in this book
The culmination of the book writing process fullls a long cher-
ished dream It would not have been possible without the excel-
lent environment continuous support and complete understanding
of my wife Anubha and daughter Anshita Their kindness has greatly
helped to fulll my dream now I plan to spend more time with my
wife and daughter
Mumbai
August 2016 Atul Sharma
Contents
I Introduction and Essentials
1
1 Introduction 3
11 CFD What is it 3
111 CFD as a Scientic and Engineering Analysis
Tool 5
112 Analogy with a Video-Camera 13
12 CFD Why to study 15
13 Novelty Scope and Purpose of this Book 16
2 Introduction to CFD Development Application and
Analysis 23
21 CFD Development 23
211 Grid Generation Pre-Processor 24
212 Discretization Method Algebraic Formulation 26
213 Solution Methodology Solver 29
214 Computation of Engineering-Parameters
Post-Processor 34
215 Testing 35
22 CFD Application 35
23 CFD Analysis 38
24 Closure 39
xiii
xiv Contents
3 Essentials of Fluid-Dynamics and Heat-Transfer for
CFD 41
31 Physical Laws 42
311 FundamentalConservation Laws 43
312 Subsidiary Laws 45
32 Momentum and Energy Transport Mechanisms 46
33 Physical Law based Dierential Formulation 48
331 Continuity Equation 49
332 Transport Equation 51
34 Generalized Volumetric and Flux Terms and their Dif-
ferential Formulation 57
341 Volumetric Term 58
342 Flux-Term 58
343 Discussion 62
35 Mathematical Formulation 63
351 Dimensional Study 63
352 Non-Dimensional Study 67
36 Closure 71
4 Essentials of Numerical-Methods for CFD 73
41 Finite Dierence Method A Dierential to Algebraic
Formulation for Governing PDE and BCs 75
411 Grid Generation 75
412 Finite Dierence Method 78
413 Applications to CFD 92
42 Iterative Solution of System of LAEs for a Flow Property 93
421 Iterative Methods 94
422 Applications to CFD 98
43 Numerical Dierentiation for Local Engineering-
Parameters 105
431 Dierentiation Formulas 106
432 Applications to CFD 107
44 Numerical Integration for the Total value of
Engineering-Parameters 110
441 Integration Rules 111
442 Applications to CFD 114
45 Closure 116
Problems 116
Introduction to CFD xv
II CFD for a Cartesian-Geometry
119
5 Computational Heat Conduction 12151 Physical Law based Finite Volume Method 122
511 Energy Conservation Law for a Control Volume 123512 Algebraic Formulation 124513 Approximations 127514 Approximated Algebraic-Formulation 130515 Discussion 133
52 Finite Dierence Method for Boundary Conditions 13853 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 139531 One-Dimensional Conduction 139532 Two-Dimensional Conduction 147
54 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 154541 One-Dimensional Conduction 154542 Two-Dimensional Conduction 166
Problems 176
6 Computational Heat Advection 17961 Physical Law based Finite Volume Method 180
611 Energy Conservation Law for a Control Volume 181612 Algebraic Formulation 181613 Approximations 183614 Approximated Algebraic Formulation 191615 Discussion 191
62 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 194621 Explicit-Method 197622 Implementation Details 198623 Solution Algorithm 199
63 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 207631 Advection Scheme on a Non-Uniform Grid 207632 Explicit and Implicit Method 210633 Implementation Details 216634 Solution Algorithm 220
Problems 228
xvi Contents
7 Computational Heat Convection 22971 Physical Law based Finite Volume Method 229
711 Energy Conservation Law for a Control Volume 229712 Algebraic Formulation 231713 Approximated Algebraic Formulation 232
72 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 235721 Explicit-Method 236722 Implementation Details 237723 Solution Algorithm 238
73 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 242Problems 248
8 Computational Fluid Dynamics Physical Law based
Finite Volume Method 25181 Generalized Variables for the Combined Heat and
Fluid Flow 25282 Conservation Laws for a Control Volume 25583 Algebraic Formulation 25984 Approximations 26085 Approximated Algebraic Formulation 263
851 Mass Conservation 263852 MomentumEnergy Conservation 264
86 Closure 269
9 Computational Fluid Dynamics on a Staggered Grid 27191 Challenges in the CFD Development 273
911 Non-Linearity 273912 Equation for Pressure 273913 Pressure-Velocity Decoupling 273
92 A Staggered Grid to avoid Pressure-Velocity Decoupling27593 Physical Law based FVM for a Staggered Grid 27794 Flux based Solution Methodology on a Uniform Grid
Semi-Explicit Method 281941 Philosophy of Pressure-Correction Method 283942 Semi-Explicit Method 286943 Implementation Details 292944 Solution Algorithm 298
95 Initial and Boundary Conditions 300951 Initial Condition 300952 Boundary Condition 301
Problems 306
Introduction to CFD xvii
10 Computational Fluid Dynamics on a Co-located Grid309
101 Momentum-Interpolation Method Strategy to avoid
the Pressure-Velocity Decoupling on a Co-located Grid 310
102 Coecients of LAEs based Solution Methodology on
a Non-Uniform Grid Semi-Explicit and Semi-Implicit
Method 314
1021 Predictor Step 316
1022 Corrector Step 318
1023 Solution Algorithm 323
Problems 329
III CFD for a Complex-Geometry
331
11 Computational Heat Conduction on a Curvilinear
Grid 333
111 Curvilinear Grid Generation 333
1111 Algebraic Grid Generation 334
1112 Elliptic Grid Generation 336
112 Physical Law based Finite Volume Method 343
1121 Unsteady and Source Term 343
1122 Diusion Term 344
1123 All Terms 349
113 Computation of Geometrical Properties 349
114 Flux based Solution Methodology 352
1141 Explicit Method 354
1142 Implementation Details 354
Problems 359
12 Computational Fluid Dynamics on a Curvilinear Grid361
121 Physical Law based Finite Volume Method 361
1211 Mass Conservation 363
1212 Momentum Conservation 363
122 Solution Methodology Semi-Explicit Method 372
1221 Predictor Step 373
1222 Corrector Step 375
Problems 380
References 383
Index 389
Part I
Introduction and Essentials
The book on an introductory course in CFD starts with the two intro-
ductory chapters on CFD followed by one chapter each on essentials
of two prerequisite courses minus uid-dynamics and heat transfer and
numerical-methods The rst chapter on introduction presents an
eventful thoughtful and intuitive discussion on What is CFD and
Why to study CFD whereas How CFD works which involves lots
of details is presented separately in the second chapter The third-
chapter on essentials of uid-dynamics and heat transfer presents
physical-laws transport-mechanisms physical-law based dierential-
formulation volumetric and ux terms and mathematical formula-
tion The fourth-chapter on essentials of numerical-methods presents
nite dierence method iterative solution of the system of linear alge-
braic equation numerical dierentiation and numerical integration
The presentation for the essential of the two prerequisite courses is
customized to CFD development application and analysis
1
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
1
Introduction
The introduction of any course consists of the three basic questions
What is it Why to study How it works An eventful thoughtful
and intuitive answer to the rst two questions are presented in this
chapter The third question which involves lots of working details
on the three aspects of the Computational Fluid Dynamics (CFD)
(development application and analysis) is presented separately in
the next chapter The novelty scope and purpose of this book is
also presented at the end of this chapter
11 CFD What is it
CFD is a theoretical-method of scientic and engineering investiga-
tion concerned with the development and application of a video-
camera like tool (a software) which is used for a unied cause-and-
eect based analysis of a uid-dynamics as well as heat and mass
transfer problem
CFD is a subject where you rst learn how to develop a product
minus a software which acts like a virtual video-camera Then you learn
how to apply or use this product to generate uid-dynamics movies
Finally you learn how to analyze the detailed spatial and temporal
uid-dynamics information in the movies The analysis is done to
come up with a scientically-exciting as well as engineering-relevant
story (unied cause-and-eect study) of a uid-dynamics situation in
nature as well as industrial-applications With a continuous develop-
ment and a wider application of CFD the word uid-dynamics in
3
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
4 1 Introduction
the CFD has become more generic as it also corresponds to heat and
mass transfer as well as chemical reaction
Fluid-dynamics information are of two types scientic and en-
gineering Scientic-information corresponds to a structure of the
heat and uid ow due to a physical phenomenon in uid-dynamics
such as boundary-layer ow-separation wake-formation and vortex-
shedding The ow structures are obtained from a temporal variation
of ow-properties (such as velocity pressure temperature and many
other variables which characterizes a ow) called as scientically-
exciting movies Engineering-information corresponds to certain pa-
rameters which are important in an engineering application called
as engineering-parameters for example lift force drag force wall
shear stress pressure drop and rate of heat-transfer They are ob-
tained as the temporal variation of engineering-parameters called as
engineering-relevant movies The engineering-parameters are the ef-
fect which are caused by the ow structures minus their correlation leads
to the unied cause-and-eect study A uid-dynamics movie consist
of a time-wise varying series of pictures which give uid-dynamics
information Each ow-property and engineering-parameter results
in one scientically-exciting and engineering-relevant movie respec-
tively When the large number of both the types of movies are
played synchronized in-time the detailed spatial as well as tempo-
ral uid-dynamic information greatly helps in the unied cause-and-
eect study of a uid-dynamics problem
Alternatively CFD is a subject concerned with the development of
computer-programs for the computer-simulation and study of a natu-
ral or an engineering uid-dynamics system The development starts
with the virtual-development of the system involving a geometri-
cal information based computational development of a solid-model
for the system the system may be already-existing or yet-to-be-
developed physically Thereafter the development involves numerical
solution of the governing algebraic equations for CFD formulated
from the physical laws (conservation of mass momentum and en-
ergy) and initial as well as boundary conditions corresponding to the
system CFD study involves the analysis of the uid-dynamics and
heat-transfer results obtained from the computer-simulations of the
virtual-system which is subjected to certain governing-parameters
Introduction to CFD 5
111 CFD as a Scientic and Engineering Analysis Tool
For a better understanding of the denition of the CFD and its ap-
plication as a scientic and engineering analysis tool two example
problems are presented rst conduction heat transfer in a plate
and second uid ow across a circular pillar of a bridge The rst-
problem correspond to an engineering-application for electronic cool-
ing and the second-problem is studied for an engineering-design of
the structure subjected to uid-dynamics forces and are shown in
Fig 11 Both the problems are considered as two-dimensional and
unsteady However the results asymptotes from the unsteady to a
steady state in the rst-problem and to a periodic (dynamic-steady)
state in the second-problem
Figure 11 Schematic and the associated parameters for the two example prob-lems (a) thermal conduction module of a heat sink and (b) theclassical ow across a circular cylinder
6 1 Introduction
The rst-problem is taken from the classic book on heat-transfer
by Incropera and Dewitt (1996) shown in Fig 11(a) The g-
ure shows a periodic module of the plate where a constant heat-ux
qW = 10Wcm2 (dissipated from certain electronic devices) is rst con-
ducted to an aluminum plate and then convected to the water ow-
ing through the rectangular channel grooved in the plate The forced
convection results in a convection coecient of h = 5000Wm2K in-
side the channel The unsteady heat-transfer results in an interesting
time-wise varying conduction heat ow in the plate and instantaneous
convection heat transfer rate Qprimeconv (per unit width of the plate per-
pendicular to the plane shown in Fig 11(a)) into the water
The second-problem is encountered if you are standing over a
bridge and look below to the water passing over a pillar (of circu-
lar cross-section) of the bridge shown in Fig 11(b) as a general case
of 2D ow across a cylinder The uid-dynamics results in a beau-
tiful time-wise varying structure of the uid ow just downstream
of the pillar and certain engineering-parameters The parameters
correspond to the uid-dynamic forces acting in the horizontal (or
streamwise) and vertical (or transverse) direction called as the drag-
force FD and lift-force FL respectively The respective force is pre-
sented below in a non-dimensional form called as the drag-coecient
CD and lift-coecient CL equations shown in Fig 11(b)
If you use your video camera to capture a movie for the two
problems you will get series of pictures for the plate in the rst-
problem and the owing uid in the second-problem Since these
movies result in a pictorial information but not the ow-property
based uid-dynamic information such movies are not relevant for
the uid-dynamics and heat-transfer study Instead if you use a
CFD software for computer simulation you can capture various types
of movies The various ow-properties based scientically-exciting
movies are presented here as the pictures at certain time-instant in
Fig 12(a)-(d) and Fig 13(a1)-(d2) The former gure shows gray-
colored ooded contour for the temperature whereas the latter gure
shows gray-colored ooded contour for the pressure and vorticity and
vector-plot for the velocity For the gray-colored ooded contours a
color bar can be seen in the gures for the relative magnitude of the
variable These gures also show line contours of stream-function
Introduction to CFD 7
and heat-function as streamlines and heatlines respectively They
are the visualization techniques for a 2D steady heat and uid ow
The heatlines are analogous to streamlines and represents the direc-
tion of heat ow under steady state condition proposed by Kimura
and Bejan (1983) later presented in a heat-transfer book by Bejan
(1984) The steady-state heatlines can be seen in Fig 12(d) where
the direction for the conduction-ux is tangential at any point on the
heatlines analogous to that for the mass-ux in a streamline Thus
for a better presentation of the structure of heat and uid ow the
scientically-exciting movies are not only limited to ow-properties
but also correspond to certain ow visualization techniquesAn engineering-parameter based engineering-relevant uid dy-
namic movie is presented in Fig 12(e) for the convection heat trans-
fer rate per unit width at the various walls on the channel Qprimew at the
west-wall Qprimen at the north-wall Q
primee at the east-wall and their cumu-
lative value Qprimeconv (refer Fig 11(a)) Another engineering-relevant
movie is presented in Fig 13(e) for the lift and drag coecient Note
that the movie here corresponds to the motion of a lled symbol over
a line for the temporal variation of an engineering-parameter Thus
the engineering-relevant movies are one-dimensional lower than the
scientically-exciting movies here for the 2D problems the former
movie is presented as 1D and the latter as 2D results
1111 Heat-Conduction
For the computational set-up of the heat conduction problem shown
in Fig 11(a) the results obtained from a 2D CFD simulation of
the conduction heat transfer are shown in Fig 12 For a uni-
form initial temperature of the plate as Tinfin = 15oC Fig 12(a)-(d)
shows the isotherms and heatlines in the plate as the pictures of the
scientically-exciting movies at certain discrete time instants t = 25
5 10 and 20 sec For the same discrete time instants the temporal
variation of the position of the symbols over a line plot is presented in
Fig 12(e) The gure is shown for each of the heat transfer rate and
is obtained from a engineering-relevant 1D movie There is a tempo-
ral variation of picture in the scientically-exciting movie and that
of position of symbol (along the line plot) in the engineering-relevant
movieAccording to caloric theory of the famous French scientist An-
tonie Lavoisier heat is considered as an invisible tasteless odorless
8 1 Introduction
t(sec)
Qrsquo (W
m)
0 5 10 15 20 250
200
400
600
800
1000Qrsquo
conv
Qrsquow
Qrsquon
Qrsquoe
Qrsquoin
(e)
(b)
(c)(d)
(a)
(a) t=25 sec31
29
27
25
23
21
19
17
15
T(oC)
(c) t=10 sec
(b) t=5 sec
(d) t=20 sec
Figure 12 Thermal conduction module of a heat sink temporal variation of(a)-(d) isotherms and heatlines and (e) the convection heat transferrate per unit width at the various walls of the channels and theircumulative total value Q
primeconv For the discrete time-instant corre-
sponding to heat ow patterns in (a)-(d) temporal variation of thevarious heat transfer rates are shown as symbols in (e) (a)-(c) arethe transient and (d) represents a steady state result
weightless uid which he called caloric uid Presently the caloric
theory is overtaken by mechanical theory of heat minus heat is not a sub-
stance but a dynamical form of mechanical eect (Thomson 1851)
Nevertheless there are certain problems involving heat ow for which
Lavoisiers approach is rather useful such as the discussion on heat-
lines here
The role of heatlines for heat ow is analogous to that of stream-
lines for uid ow (Paramane and Sharma 2009) For the uid-ow
the dierence of stream-function values represents the rate of uid
ow the function remains constant on a solid wall the tangent to
a streamline represents the direction of the uid-ow (with no ow
in the normal direction) and the streamline originate or emerge at
Introduction to CFD 9
a mass source Analogously for the heat-ow the dierence of heat-
function values represents the rate of heat ow the function remains
constant on an adiabatic wall the tangent to a heatline represents the
direction of the heat-ow (with no ow in the normal direction) and
the heatline originate or emerge at the heat-source For the present
problem in Fig 11(a) the heat-source is at the inlet (or left) bound-
ary and the heat-sink is at the surface of the channel Thus the
steady-state heatlines in Fig 12(d) shows the heat ow emerging
from the inletsource and ending on the channel-wallssink Further-
more it can be seen in the gure that the heatlines which are close
to the adiabatic walls are parallel to the walls The source sink and
adiabatic-walls can be seen in Fig 11(a)
Figure 12(e) shows an asymptotic time-wise increase in the con-
vection heat transfer per unit width at the various walls of the channel
and its cumulative value Qprimeconv (= Q
primew+Q
primen+Q
primee) It can be also be seen
that the heat-transfer per unit width Qprimew at the west-wall and Q
primen at
the north-wall are larger than Qprimee at the east-wall of the channel As
time progresses the gure shows that Qprimeconv increases monotonically
and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-
der steady state condition refer Fig 11(a) The dierence between
the Qprimein and Q
primeconv under the transient condition is utilized for the
sensible heating and results in the increase in the temperature of the
plate Under the steady state condition note from Fig 12(d) that
the maximum temperature is close to 32oC minus well within the per-
missible limit for the reliable and ecient operation of the electronic
device
1112 Fluid-Dynamics
For the free-stream ow across a circular cylinderpillar at a Reynolds
number of 100 the non-dimensional results obtained from a 2D CFD
simulation are shown in Fig 13 In contrast to the previous prob-
lem which reaches steady state the present problem reaches to an un-
steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-
ent increasing time instants within one time-period of the periodic-
ow the results are presented in Fig 13(a)-(d) The gure shows
the close-up view of the pictures of the scientically-exciting movies
for ow-properties (velocity pressure and vorticity) and streamlines
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow
viii Foreword
of experience over fteen and odd years includes two and three di-
mensional Navier-Stokes equations heat transfer interfacial dynam-
ics and turbulent ow He has championed the nite volume method
which is now the industry standard The programming experience
is important because it brings in the precision needed for assembling
the jigsaw puzzle The sprinkling of computer programs in the book
is no coincidence
Atul knows the conventional method of discretizing dierential
equations but has never been satised with it He has felt an ele-
ment of discomfort going away from a physical reality into a world
of numbers connected by arithmetic operations As a result he has
developed a principle that physical laws that characterize the dier-
ential equations should be reected at every stage of discretization
and every stage of approximation indeed at every scale whether a
single tetrahedron or the sub-domain or the region as a whole This
idea permeates the book where it is shown that discretized versions
must provide reasonable answers because they continue to represent
the laws of nature
This new CFD book is comprehensive and has a stamp of origi-
nality of the author It will bring students closer to the subject and
enable them to contribute to it
Kanpur
August 2016 K Muralidhar
PREFACE
As a Post-Graduate (PG) student in 1997 I got introduced to a
course on CFD at Indian Institute of Science (IISc) Bangalore Dur-
ing the introductory course on CFD although I struggled hard
in the physical understanding and computer-programming it didnt
stop my interest and enthusiasm to do research teaching and writ-
ing a text-book on a introductory course in CFD Since then I did
my Masters project work on CFD (1997-98) introduced and taught
this course to Under-Graduate (UG) students at National Institute
of Technology Hamirpur HP (1999-2000) did my Doctoral work on
this subject at Indian Institute of Technology (IIT) Kanpur (2000-
2004) and co-authored a Chapter on nite volume method in a re-
vised edition of an edited book on computational uid ow and heat
transfer during my PhD in 2003 After joining as a faculty in 2004
I taught this course to UG as well as PG students of dierent depart-
ments at IIT Bombay as well as to the students of dierent collages in
India through CDEEP (Center for Distance Engineering Programme)
IIT Bombay from 2007-2010 Furthermore I gave series of lecture
on CFD at various collages and industries in India More recently
in 2012 I delivered a ve-days lecture and lab-session on CFD (to
around 1400 collage teachers from dierent parts of India) as a part
of a project funded by ministry of human resource development under
NMEICT (National Mission on Education through information and
communication technology)
The increasing need for the development of the customized CFD
software (app) and wide spread CFD application as well as analysis
for the design optimization and innovation of various types of engi-
neering systems always motivated me to write this book Due to the
increasing importance of CFD another motivation is to present CFD
ix
x Preface
simple enough to introduce the rst-course at early under-graduate
curriculum rather than the post-graduate curriculum The moti-
vation got strengthened by my own as well as others (students
teachers researchers and practitioners) struggle on physical under-
standing of the mathematics involved during the Partial Dierential
Equation (PDE) based algebraic-formulation (nite volume method)
along with the continuous frustrating pursuit to eectively convert
the theory of CFD into the computer-program
This led me to come up with an alternate physical-law (PDE-
free) based nite volume method as well as solution methodol-
ogy which are much easier to comprehend Mostly the success of
the programming eort is decided by the perfect execution of the
implementations-details which rarely appears in-print (existing books
and journal articles) The physical law based CFD development
approach and the implementations-details are the novelties in the
present book Furthermore using an open-source software for nu-
merical computations (Scilab) computer programs are given in this
book on the basic modules of conduction advection and convection
Indeed the reader can generalize and extend these codes for the de-
velopment of Navier-Stokes solver and generate the results presented
in the chapters towards the end of this book
I have limited the scope of this book to the numerical-techniques
which I wish to recommend for the book although there are numerous
other advanced and better methods in the published literature I do
not claim that the programming-practice presented here are the most
ecient minus they are presented here more for the ease in programming
(understand as your program) than for the computational eciency
Furthermore although I am enthusiastic about my presentation of
the physical law based CFD approach it may not be more ecient
than the alternate mathematics based CFD approach The emphasis
in this book is on the ease of understanding of the formulations and
programming-practice for the introductory course on CFD
I would hereby acknowledge that I owe my greatest debt to Prof
J Srinivasan (my ME Project supervisor at IISc Bangalore) who
did a hand-holding and wonderful job in patiently introducing me to
the fascinating world of research in uid-dynamics and heat trans-
fer I would also acknowledge the excellent training from Prof V
Introduction to CFD xi
Eswaran (my PhD supervisor at IIT Kanpur) from whom I learned
systematic way of CFD development application and analysis His
inspiring comments and suggestions on my writing-style and elabora-
tion of the rst chapter in this book is also gratefully acknowledged
I also want to record my sincere thanks to Prof Pradip Datta IISc
Bangalore and Prof Gautam Biswas IIT Kanpur minus my teachers for
CFD course Special thanks to Prof K Muralidhar IIT Kanpur for
his personal encouragement academic support and motivation (for
the book writing) He also cleared my dilemma in the proper presen-
tation of the novelty in this book I thank him also for writing the
foreword of this book
I am grateful to Prof Kannan M Moudgalya and Prof Deepak B
Phatak from IIT Bombay for giving me opportunity to teach CFD
in a distance education mode I am also grateful to Prof Amit
Agrawal (my colleague at IIT Bombay) who diligently read through
all the chapters of this book and suggested some really remarkable
changes in the book I also thank him for the research interactions
over the last decade which had substantially improved my under-
standing of thermal and uid science I would also like to thank my
PhD student Namshad T who did a great job in implementing
my ideas on generating the CFD simulation based CFD image for
the cover-page and couple of gures in the rst chapter (Fig 12
and 13) Special mention that the Scilab codes developed (for the
CFD course under NMEICT) by Vishesh Aggarwal greatly helped
me to develop the codes presented in this book his contributions is
gratefully acknowledged I thank Malhar Malushte for testing some
of the formulations presented in this book
I thank my research students (Sachin B Paramane D Datta C
M Sewatkar Vinesh H Gada Kaushal Prasad Mukul Shrivastava
Absar M Lakdawala Mausam Sarkar and Javed Shaikh) for the
diligence and sincerity they have shown in unraveling some of our
ideas on CFD Apart from these several other research students and
the students teachers and CFD practitioners who took my lectures
on CFD in the past decade have contributed to my better under-
standing of CFD and inspired me to write the book I thank all of
them for their attention and enthusiastic response during the CFD
courses The inuence of the interactions with all of them can be seen
xii Preface
throughout this book I am grateful to IIT Bombay for giving me
one year sabbatical leave to write this book The ambiance of aca-
demic freedom considerate support by the faculties and the friendly
atmosphere inside the campus of IIT Bombay has greatly contributed
in the solo eort of book writing I thanks all my colleagues at IIT
Bombay for shaping my perceptions on CFD
During this long endeavor the patient support by Mr Sunil
Saxena Director Athena Academic Ltd UK is gratefully acknowl-
edged I also thank John Wiley for coming forward to take this
book to international markets Lastly I would acknowledge that I
am greatly inspired by the writing style of J D Anderson and the
book on CFD by S V Patankar Of course you will nd that I
have heavily cross-referred the CFD book by the two most acknowl-
edged pioneers on CFD I dedicate this book to Dr S V Patankar
whose book nicely introduced me to the computational as well as
ow physics in CFD and greatly helped me to introduce a physical
approach of CFD development in this book
The culmination of the book writing process fullls a long cher-
ished dream It would not have been possible without the excel-
lent environment continuous support and complete understanding
of my wife Anubha and daughter Anshita Their kindness has greatly
helped to fulll my dream now I plan to spend more time with my
wife and daughter
Mumbai
August 2016 Atul Sharma
Contents
I Introduction and Essentials
1
1 Introduction 3
11 CFD What is it 3
111 CFD as a Scientic and Engineering Analysis
Tool 5
112 Analogy with a Video-Camera 13
12 CFD Why to study 15
13 Novelty Scope and Purpose of this Book 16
2 Introduction to CFD Development Application and
Analysis 23
21 CFD Development 23
211 Grid Generation Pre-Processor 24
212 Discretization Method Algebraic Formulation 26
213 Solution Methodology Solver 29
214 Computation of Engineering-Parameters
Post-Processor 34
215 Testing 35
22 CFD Application 35
23 CFD Analysis 38
24 Closure 39
xiii
xiv Contents
3 Essentials of Fluid-Dynamics and Heat-Transfer for
CFD 41
31 Physical Laws 42
311 FundamentalConservation Laws 43
312 Subsidiary Laws 45
32 Momentum and Energy Transport Mechanisms 46
33 Physical Law based Dierential Formulation 48
331 Continuity Equation 49
332 Transport Equation 51
34 Generalized Volumetric and Flux Terms and their Dif-
ferential Formulation 57
341 Volumetric Term 58
342 Flux-Term 58
343 Discussion 62
35 Mathematical Formulation 63
351 Dimensional Study 63
352 Non-Dimensional Study 67
36 Closure 71
4 Essentials of Numerical-Methods for CFD 73
41 Finite Dierence Method A Dierential to Algebraic
Formulation for Governing PDE and BCs 75
411 Grid Generation 75
412 Finite Dierence Method 78
413 Applications to CFD 92
42 Iterative Solution of System of LAEs for a Flow Property 93
421 Iterative Methods 94
422 Applications to CFD 98
43 Numerical Dierentiation for Local Engineering-
Parameters 105
431 Dierentiation Formulas 106
432 Applications to CFD 107
44 Numerical Integration for the Total value of
Engineering-Parameters 110
441 Integration Rules 111
442 Applications to CFD 114
45 Closure 116
Problems 116
Introduction to CFD xv
II CFD for a Cartesian-Geometry
119
5 Computational Heat Conduction 12151 Physical Law based Finite Volume Method 122
511 Energy Conservation Law for a Control Volume 123512 Algebraic Formulation 124513 Approximations 127514 Approximated Algebraic-Formulation 130515 Discussion 133
52 Finite Dierence Method for Boundary Conditions 13853 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 139531 One-Dimensional Conduction 139532 Two-Dimensional Conduction 147
54 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 154541 One-Dimensional Conduction 154542 Two-Dimensional Conduction 166
Problems 176
6 Computational Heat Advection 17961 Physical Law based Finite Volume Method 180
611 Energy Conservation Law for a Control Volume 181612 Algebraic Formulation 181613 Approximations 183614 Approximated Algebraic Formulation 191615 Discussion 191
62 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 194621 Explicit-Method 197622 Implementation Details 198623 Solution Algorithm 199
63 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 207631 Advection Scheme on a Non-Uniform Grid 207632 Explicit and Implicit Method 210633 Implementation Details 216634 Solution Algorithm 220
Problems 228
xvi Contents
7 Computational Heat Convection 22971 Physical Law based Finite Volume Method 229
711 Energy Conservation Law for a Control Volume 229712 Algebraic Formulation 231713 Approximated Algebraic Formulation 232
72 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 235721 Explicit-Method 236722 Implementation Details 237723 Solution Algorithm 238
73 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 242Problems 248
8 Computational Fluid Dynamics Physical Law based
Finite Volume Method 25181 Generalized Variables for the Combined Heat and
Fluid Flow 25282 Conservation Laws for a Control Volume 25583 Algebraic Formulation 25984 Approximations 26085 Approximated Algebraic Formulation 263
851 Mass Conservation 263852 MomentumEnergy Conservation 264
86 Closure 269
9 Computational Fluid Dynamics on a Staggered Grid 27191 Challenges in the CFD Development 273
911 Non-Linearity 273912 Equation for Pressure 273913 Pressure-Velocity Decoupling 273
92 A Staggered Grid to avoid Pressure-Velocity Decoupling27593 Physical Law based FVM for a Staggered Grid 27794 Flux based Solution Methodology on a Uniform Grid
Semi-Explicit Method 281941 Philosophy of Pressure-Correction Method 283942 Semi-Explicit Method 286943 Implementation Details 292944 Solution Algorithm 298
95 Initial and Boundary Conditions 300951 Initial Condition 300952 Boundary Condition 301
Problems 306
Introduction to CFD xvii
10 Computational Fluid Dynamics on a Co-located Grid309
101 Momentum-Interpolation Method Strategy to avoid
the Pressure-Velocity Decoupling on a Co-located Grid 310
102 Coecients of LAEs based Solution Methodology on
a Non-Uniform Grid Semi-Explicit and Semi-Implicit
Method 314
1021 Predictor Step 316
1022 Corrector Step 318
1023 Solution Algorithm 323
Problems 329
III CFD for a Complex-Geometry
331
11 Computational Heat Conduction on a Curvilinear
Grid 333
111 Curvilinear Grid Generation 333
1111 Algebraic Grid Generation 334
1112 Elliptic Grid Generation 336
112 Physical Law based Finite Volume Method 343
1121 Unsteady and Source Term 343
1122 Diusion Term 344
1123 All Terms 349
113 Computation of Geometrical Properties 349
114 Flux based Solution Methodology 352
1141 Explicit Method 354
1142 Implementation Details 354
Problems 359
12 Computational Fluid Dynamics on a Curvilinear Grid361
121 Physical Law based Finite Volume Method 361
1211 Mass Conservation 363
1212 Momentum Conservation 363
122 Solution Methodology Semi-Explicit Method 372
1221 Predictor Step 373
1222 Corrector Step 375
Problems 380
References 383
Index 389
Part I
Introduction and Essentials
The book on an introductory course in CFD starts with the two intro-
ductory chapters on CFD followed by one chapter each on essentials
of two prerequisite courses minus uid-dynamics and heat transfer and
numerical-methods The rst chapter on introduction presents an
eventful thoughtful and intuitive discussion on What is CFD and
Why to study CFD whereas How CFD works which involves lots
of details is presented separately in the second chapter The third-
chapter on essentials of uid-dynamics and heat transfer presents
physical-laws transport-mechanisms physical-law based dierential-
formulation volumetric and ux terms and mathematical formula-
tion The fourth-chapter on essentials of numerical-methods presents
nite dierence method iterative solution of the system of linear alge-
braic equation numerical dierentiation and numerical integration
The presentation for the essential of the two prerequisite courses is
customized to CFD development application and analysis
1
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
1
Introduction
The introduction of any course consists of the three basic questions
What is it Why to study How it works An eventful thoughtful
and intuitive answer to the rst two questions are presented in this
chapter The third question which involves lots of working details
on the three aspects of the Computational Fluid Dynamics (CFD)
(development application and analysis) is presented separately in
the next chapter The novelty scope and purpose of this book is
also presented at the end of this chapter
11 CFD What is it
CFD is a theoretical-method of scientic and engineering investiga-
tion concerned with the development and application of a video-
camera like tool (a software) which is used for a unied cause-and-
eect based analysis of a uid-dynamics as well as heat and mass
transfer problem
CFD is a subject where you rst learn how to develop a product
minus a software which acts like a virtual video-camera Then you learn
how to apply or use this product to generate uid-dynamics movies
Finally you learn how to analyze the detailed spatial and temporal
uid-dynamics information in the movies The analysis is done to
come up with a scientically-exciting as well as engineering-relevant
story (unied cause-and-eect study) of a uid-dynamics situation in
nature as well as industrial-applications With a continuous develop-
ment and a wider application of CFD the word uid-dynamics in
3
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
4 1 Introduction
the CFD has become more generic as it also corresponds to heat and
mass transfer as well as chemical reaction
Fluid-dynamics information are of two types scientic and en-
gineering Scientic-information corresponds to a structure of the
heat and uid ow due to a physical phenomenon in uid-dynamics
such as boundary-layer ow-separation wake-formation and vortex-
shedding The ow structures are obtained from a temporal variation
of ow-properties (such as velocity pressure temperature and many
other variables which characterizes a ow) called as scientically-
exciting movies Engineering-information corresponds to certain pa-
rameters which are important in an engineering application called
as engineering-parameters for example lift force drag force wall
shear stress pressure drop and rate of heat-transfer They are ob-
tained as the temporal variation of engineering-parameters called as
engineering-relevant movies The engineering-parameters are the ef-
fect which are caused by the ow structures minus their correlation leads
to the unied cause-and-eect study A uid-dynamics movie consist
of a time-wise varying series of pictures which give uid-dynamics
information Each ow-property and engineering-parameter results
in one scientically-exciting and engineering-relevant movie respec-
tively When the large number of both the types of movies are
played synchronized in-time the detailed spatial as well as tempo-
ral uid-dynamic information greatly helps in the unied cause-and-
eect study of a uid-dynamics problem
Alternatively CFD is a subject concerned with the development of
computer-programs for the computer-simulation and study of a natu-
ral or an engineering uid-dynamics system The development starts
with the virtual-development of the system involving a geometri-
cal information based computational development of a solid-model
for the system the system may be already-existing or yet-to-be-
developed physically Thereafter the development involves numerical
solution of the governing algebraic equations for CFD formulated
from the physical laws (conservation of mass momentum and en-
ergy) and initial as well as boundary conditions corresponding to the
system CFD study involves the analysis of the uid-dynamics and
heat-transfer results obtained from the computer-simulations of the
virtual-system which is subjected to certain governing-parameters
Introduction to CFD 5
111 CFD as a Scientic and Engineering Analysis Tool
For a better understanding of the denition of the CFD and its ap-
plication as a scientic and engineering analysis tool two example
problems are presented rst conduction heat transfer in a plate
and second uid ow across a circular pillar of a bridge The rst-
problem correspond to an engineering-application for electronic cool-
ing and the second-problem is studied for an engineering-design of
the structure subjected to uid-dynamics forces and are shown in
Fig 11 Both the problems are considered as two-dimensional and
unsteady However the results asymptotes from the unsteady to a
steady state in the rst-problem and to a periodic (dynamic-steady)
state in the second-problem
Figure 11 Schematic and the associated parameters for the two example prob-lems (a) thermal conduction module of a heat sink and (b) theclassical ow across a circular cylinder
6 1 Introduction
The rst-problem is taken from the classic book on heat-transfer
by Incropera and Dewitt (1996) shown in Fig 11(a) The g-
ure shows a periodic module of the plate where a constant heat-ux
qW = 10Wcm2 (dissipated from certain electronic devices) is rst con-
ducted to an aluminum plate and then convected to the water ow-
ing through the rectangular channel grooved in the plate The forced
convection results in a convection coecient of h = 5000Wm2K in-
side the channel The unsteady heat-transfer results in an interesting
time-wise varying conduction heat ow in the plate and instantaneous
convection heat transfer rate Qprimeconv (per unit width of the plate per-
pendicular to the plane shown in Fig 11(a)) into the water
The second-problem is encountered if you are standing over a
bridge and look below to the water passing over a pillar (of circu-
lar cross-section) of the bridge shown in Fig 11(b) as a general case
of 2D ow across a cylinder The uid-dynamics results in a beau-
tiful time-wise varying structure of the uid ow just downstream
of the pillar and certain engineering-parameters The parameters
correspond to the uid-dynamic forces acting in the horizontal (or
streamwise) and vertical (or transverse) direction called as the drag-
force FD and lift-force FL respectively The respective force is pre-
sented below in a non-dimensional form called as the drag-coecient
CD and lift-coecient CL equations shown in Fig 11(b)
If you use your video camera to capture a movie for the two
problems you will get series of pictures for the plate in the rst-
problem and the owing uid in the second-problem Since these
movies result in a pictorial information but not the ow-property
based uid-dynamic information such movies are not relevant for
the uid-dynamics and heat-transfer study Instead if you use a
CFD software for computer simulation you can capture various types
of movies The various ow-properties based scientically-exciting
movies are presented here as the pictures at certain time-instant in
Fig 12(a)-(d) and Fig 13(a1)-(d2) The former gure shows gray-
colored ooded contour for the temperature whereas the latter gure
shows gray-colored ooded contour for the pressure and vorticity and
vector-plot for the velocity For the gray-colored ooded contours a
color bar can be seen in the gures for the relative magnitude of the
variable These gures also show line contours of stream-function
Introduction to CFD 7
and heat-function as streamlines and heatlines respectively They
are the visualization techniques for a 2D steady heat and uid ow
The heatlines are analogous to streamlines and represents the direc-
tion of heat ow under steady state condition proposed by Kimura
and Bejan (1983) later presented in a heat-transfer book by Bejan
(1984) The steady-state heatlines can be seen in Fig 12(d) where
the direction for the conduction-ux is tangential at any point on the
heatlines analogous to that for the mass-ux in a streamline Thus
for a better presentation of the structure of heat and uid ow the
scientically-exciting movies are not only limited to ow-properties
but also correspond to certain ow visualization techniquesAn engineering-parameter based engineering-relevant uid dy-
namic movie is presented in Fig 12(e) for the convection heat trans-
fer rate per unit width at the various walls on the channel Qprimew at the
west-wall Qprimen at the north-wall Q
primee at the east-wall and their cumu-
lative value Qprimeconv (refer Fig 11(a)) Another engineering-relevant
movie is presented in Fig 13(e) for the lift and drag coecient Note
that the movie here corresponds to the motion of a lled symbol over
a line for the temporal variation of an engineering-parameter Thus
the engineering-relevant movies are one-dimensional lower than the
scientically-exciting movies here for the 2D problems the former
movie is presented as 1D and the latter as 2D results
1111 Heat-Conduction
For the computational set-up of the heat conduction problem shown
in Fig 11(a) the results obtained from a 2D CFD simulation of
the conduction heat transfer are shown in Fig 12 For a uni-
form initial temperature of the plate as Tinfin = 15oC Fig 12(a)-(d)
shows the isotherms and heatlines in the plate as the pictures of the
scientically-exciting movies at certain discrete time instants t = 25
5 10 and 20 sec For the same discrete time instants the temporal
variation of the position of the symbols over a line plot is presented in
Fig 12(e) The gure is shown for each of the heat transfer rate and
is obtained from a engineering-relevant 1D movie There is a tempo-
ral variation of picture in the scientically-exciting movie and that
of position of symbol (along the line plot) in the engineering-relevant
movieAccording to caloric theory of the famous French scientist An-
tonie Lavoisier heat is considered as an invisible tasteless odorless
8 1 Introduction
t(sec)
Qrsquo (W
m)
0 5 10 15 20 250
200
400
600
800
1000Qrsquo
conv
Qrsquow
Qrsquon
Qrsquoe
Qrsquoin
(e)
(b)
(c)(d)
(a)
(a) t=25 sec31
29
27
25
23
21
19
17
15
T(oC)
(c) t=10 sec
(b) t=5 sec
(d) t=20 sec
Figure 12 Thermal conduction module of a heat sink temporal variation of(a)-(d) isotherms and heatlines and (e) the convection heat transferrate per unit width at the various walls of the channels and theircumulative total value Q
primeconv For the discrete time-instant corre-
sponding to heat ow patterns in (a)-(d) temporal variation of thevarious heat transfer rates are shown as symbols in (e) (a)-(c) arethe transient and (d) represents a steady state result
weightless uid which he called caloric uid Presently the caloric
theory is overtaken by mechanical theory of heat minus heat is not a sub-
stance but a dynamical form of mechanical eect (Thomson 1851)
Nevertheless there are certain problems involving heat ow for which
Lavoisiers approach is rather useful such as the discussion on heat-
lines here
The role of heatlines for heat ow is analogous to that of stream-
lines for uid ow (Paramane and Sharma 2009) For the uid-ow
the dierence of stream-function values represents the rate of uid
ow the function remains constant on a solid wall the tangent to
a streamline represents the direction of the uid-ow (with no ow
in the normal direction) and the streamline originate or emerge at
Introduction to CFD 9
a mass source Analogously for the heat-ow the dierence of heat-
function values represents the rate of heat ow the function remains
constant on an adiabatic wall the tangent to a heatline represents the
direction of the heat-ow (with no ow in the normal direction) and
the heatline originate or emerge at the heat-source For the present
problem in Fig 11(a) the heat-source is at the inlet (or left) bound-
ary and the heat-sink is at the surface of the channel Thus the
steady-state heatlines in Fig 12(d) shows the heat ow emerging
from the inletsource and ending on the channel-wallssink Further-
more it can be seen in the gure that the heatlines which are close
to the adiabatic walls are parallel to the walls The source sink and
adiabatic-walls can be seen in Fig 11(a)
Figure 12(e) shows an asymptotic time-wise increase in the con-
vection heat transfer per unit width at the various walls of the channel
and its cumulative value Qprimeconv (= Q
primew+Q
primen+Q
primee) It can be also be seen
that the heat-transfer per unit width Qprimew at the west-wall and Q
primen at
the north-wall are larger than Qprimee at the east-wall of the channel As
time progresses the gure shows that Qprimeconv increases monotonically
and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-
der steady state condition refer Fig 11(a) The dierence between
the Qprimein and Q
primeconv under the transient condition is utilized for the
sensible heating and results in the increase in the temperature of the
plate Under the steady state condition note from Fig 12(d) that
the maximum temperature is close to 32oC minus well within the per-
missible limit for the reliable and ecient operation of the electronic
device
1112 Fluid-Dynamics
For the free-stream ow across a circular cylinderpillar at a Reynolds
number of 100 the non-dimensional results obtained from a 2D CFD
simulation are shown in Fig 13 In contrast to the previous prob-
lem which reaches steady state the present problem reaches to an un-
steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-
ent increasing time instants within one time-period of the periodic-
ow the results are presented in Fig 13(a)-(d) The gure shows
the close-up view of the pictures of the scientically-exciting movies
for ow-properties (velocity pressure and vorticity) and streamlines
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow
PREFACE
As a Post-Graduate (PG) student in 1997 I got introduced to a
course on CFD at Indian Institute of Science (IISc) Bangalore Dur-
ing the introductory course on CFD although I struggled hard
in the physical understanding and computer-programming it didnt
stop my interest and enthusiasm to do research teaching and writ-
ing a text-book on a introductory course in CFD Since then I did
my Masters project work on CFD (1997-98) introduced and taught
this course to Under-Graduate (UG) students at National Institute
of Technology Hamirpur HP (1999-2000) did my Doctoral work on
this subject at Indian Institute of Technology (IIT) Kanpur (2000-
2004) and co-authored a Chapter on nite volume method in a re-
vised edition of an edited book on computational uid ow and heat
transfer during my PhD in 2003 After joining as a faculty in 2004
I taught this course to UG as well as PG students of dierent depart-
ments at IIT Bombay as well as to the students of dierent collages in
India through CDEEP (Center for Distance Engineering Programme)
IIT Bombay from 2007-2010 Furthermore I gave series of lecture
on CFD at various collages and industries in India More recently
in 2012 I delivered a ve-days lecture and lab-session on CFD (to
around 1400 collage teachers from dierent parts of India) as a part
of a project funded by ministry of human resource development under
NMEICT (National Mission on Education through information and
communication technology)
The increasing need for the development of the customized CFD
software (app) and wide spread CFD application as well as analysis
for the design optimization and innovation of various types of engi-
neering systems always motivated me to write this book Due to the
increasing importance of CFD another motivation is to present CFD
ix
x Preface
simple enough to introduce the rst-course at early under-graduate
curriculum rather than the post-graduate curriculum The moti-
vation got strengthened by my own as well as others (students
teachers researchers and practitioners) struggle on physical under-
standing of the mathematics involved during the Partial Dierential
Equation (PDE) based algebraic-formulation (nite volume method)
along with the continuous frustrating pursuit to eectively convert
the theory of CFD into the computer-program
This led me to come up with an alternate physical-law (PDE-
free) based nite volume method as well as solution methodol-
ogy which are much easier to comprehend Mostly the success of
the programming eort is decided by the perfect execution of the
implementations-details which rarely appears in-print (existing books
and journal articles) The physical law based CFD development
approach and the implementations-details are the novelties in the
present book Furthermore using an open-source software for nu-
merical computations (Scilab) computer programs are given in this
book on the basic modules of conduction advection and convection
Indeed the reader can generalize and extend these codes for the de-
velopment of Navier-Stokes solver and generate the results presented
in the chapters towards the end of this book
I have limited the scope of this book to the numerical-techniques
which I wish to recommend for the book although there are numerous
other advanced and better methods in the published literature I do
not claim that the programming-practice presented here are the most
ecient minus they are presented here more for the ease in programming
(understand as your program) than for the computational eciency
Furthermore although I am enthusiastic about my presentation of
the physical law based CFD approach it may not be more ecient
than the alternate mathematics based CFD approach The emphasis
in this book is on the ease of understanding of the formulations and
programming-practice for the introductory course on CFD
I would hereby acknowledge that I owe my greatest debt to Prof
J Srinivasan (my ME Project supervisor at IISc Bangalore) who
did a hand-holding and wonderful job in patiently introducing me to
the fascinating world of research in uid-dynamics and heat trans-
fer I would also acknowledge the excellent training from Prof V
Introduction to CFD xi
Eswaran (my PhD supervisor at IIT Kanpur) from whom I learned
systematic way of CFD development application and analysis His
inspiring comments and suggestions on my writing-style and elabora-
tion of the rst chapter in this book is also gratefully acknowledged
I also want to record my sincere thanks to Prof Pradip Datta IISc
Bangalore and Prof Gautam Biswas IIT Kanpur minus my teachers for
CFD course Special thanks to Prof K Muralidhar IIT Kanpur for
his personal encouragement academic support and motivation (for
the book writing) He also cleared my dilemma in the proper presen-
tation of the novelty in this book I thank him also for writing the
foreword of this book
I am grateful to Prof Kannan M Moudgalya and Prof Deepak B
Phatak from IIT Bombay for giving me opportunity to teach CFD
in a distance education mode I am also grateful to Prof Amit
Agrawal (my colleague at IIT Bombay) who diligently read through
all the chapters of this book and suggested some really remarkable
changes in the book I also thank him for the research interactions
over the last decade which had substantially improved my under-
standing of thermal and uid science I would also like to thank my
PhD student Namshad T who did a great job in implementing
my ideas on generating the CFD simulation based CFD image for
the cover-page and couple of gures in the rst chapter (Fig 12
and 13) Special mention that the Scilab codes developed (for the
CFD course under NMEICT) by Vishesh Aggarwal greatly helped
me to develop the codes presented in this book his contributions is
gratefully acknowledged I thank Malhar Malushte for testing some
of the formulations presented in this book
I thank my research students (Sachin B Paramane D Datta C
M Sewatkar Vinesh H Gada Kaushal Prasad Mukul Shrivastava
Absar M Lakdawala Mausam Sarkar and Javed Shaikh) for the
diligence and sincerity they have shown in unraveling some of our
ideas on CFD Apart from these several other research students and
the students teachers and CFD practitioners who took my lectures
on CFD in the past decade have contributed to my better under-
standing of CFD and inspired me to write the book I thank all of
them for their attention and enthusiastic response during the CFD
courses The inuence of the interactions with all of them can be seen
xii Preface
throughout this book I am grateful to IIT Bombay for giving me
one year sabbatical leave to write this book The ambiance of aca-
demic freedom considerate support by the faculties and the friendly
atmosphere inside the campus of IIT Bombay has greatly contributed
in the solo eort of book writing I thanks all my colleagues at IIT
Bombay for shaping my perceptions on CFD
During this long endeavor the patient support by Mr Sunil
Saxena Director Athena Academic Ltd UK is gratefully acknowl-
edged I also thank John Wiley for coming forward to take this
book to international markets Lastly I would acknowledge that I
am greatly inspired by the writing style of J D Anderson and the
book on CFD by S V Patankar Of course you will nd that I
have heavily cross-referred the CFD book by the two most acknowl-
edged pioneers on CFD I dedicate this book to Dr S V Patankar
whose book nicely introduced me to the computational as well as
ow physics in CFD and greatly helped me to introduce a physical
approach of CFD development in this book
The culmination of the book writing process fullls a long cher-
ished dream It would not have been possible without the excel-
lent environment continuous support and complete understanding
of my wife Anubha and daughter Anshita Their kindness has greatly
helped to fulll my dream now I plan to spend more time with my
wife and daughter
Mumbai
August 2016 Atul Sharma
Contents
I Introduction and Essentials
1
1 Introduction 3
11 CFD What is it 3
111 CFD as a Scientic and Engineering Analysis
Tool 5
112 Analogy with a Video-Camera 13
12 CFD Why to study 15
13 Novelty Scope and Purpose of this Book 16
2 Introduction to CFD Development Application and
Analysis 23
21 CFD Development 23
211 Grid Generation Pre-Processor 24
212 Discretization Method Algebraic Formulation 26
213 Solution Methodology Solver 29
214 Computation of Engineering-Parameters
Post-Processor 34
215 Testing 35
22 CFD Application 35
23 CFD Analysis 38
24 Closure 39
xiii
xiv Contents
3 Essentials of Fluid-Dynamics and Heat-Transfer for
CFD 41
31 Physical Laws 42
311 FundamentalConservation Laws 43
312 Subsidiary Laws 45
32 Momentum and Energy Transport Mechanisms 46
33 Physical Law based Dierential Formulation 48
331 Continuity Equation 49
332 Transport Equation 51
34 Generalized Volumetric and Flux Terms and their Dif-
ferential Formulation 57
341 Volumetric Term 58
342 Flux-Term 58
343 Discussion 62
35 Mathematical Formulation 63
351 Dimensional Study 63
352 Non-Dimensional Study 67
36 Closure 71
4 Essentials of Numerical-Methods for CFD 73
41 Finite Dierence Method A Dierential to Algebraic
Formulation for Governing PDE and BCs 75
411 Grid Generation 75
412 Finite Dierence Method 78
413 Applications to CFD 92
42 Iterative Solution of System of LAEs for a Flow Property 93
421 Iterative Methods 94
422 Applications to CFD 98
43 Numerical Dierentiation for Local Engineering-
Parameters 105
431 Dierentiation Formulas 106
432 Applications to CFD 107
44 Numerical Integration for the Total value of
Engineering-Parameters 110
441 Integration Rules 111
442 Applications to CFD 114
45 Closure 116
Problems 116
Introduction to CFD xv
II CFD for a Cartesian-Geometry
119
5 Computational Heat Conduction 12151 Physical Law based Finite Volume Method 122
511 Energy Conservation Law for a Control Volume 123512 Algebraic Formulation 124513 Approximations 127514 Approximated Algebraic-Formulation 130515 Discussion 133
52 Finite Dierence Method for Boundary Conditions 13853 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 139531 One-Dimensional Conduction 139532 Two-Dimensional Conduction 147
54 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 154541 One-Dimensional Conduction 154542 Two-Dimensional Conduction 166
Problems 176
6 Computational Heat Advection 17961 Physical Law based Finite Volume Method 180
611 Energy Conservation Law for a Control Volume 181612 Algebraic Formulation 181613 Approximations 183614 Approximated Algebraic Formulation 191615 Discussion 191
62 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 194621 Explicit-Method 197622 Implementation Details 198623 Solution Algorithm 199
63 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 207631 Advection Scheme on a Non-Uniform Grid 207632 Explicit and Implicit Method 210633 Implementation Details 216634 Solution Algorithm 220
Problems 228
xvi Contents
7 Computational Heat Convection 22971 Physical Law based Finite Volume Method 229
711 Energy Conservation Law for a Control Volume 229712 Algebraic Formulation 231713 Approximated Algebraic Formulation 232
72 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 235721 Explicit-Method 236722 Implementation Details 237723 Solution Algorithm 238
73 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 242Problems 248
8 Computational Fluid Dynamics Physical Law based
Finite Volume Method 25181 Generalized Variables for the Combined Heat and
Fluid Flow 25282 Conservation Laws for a Control Volume 25583 Algebraic Formulation 25984 Approximations 26085 Approximated Algebraic Formulation 263
851 Mass Conservation 263852 MomentumEnergy Conservation 264
86 Closure 269
9 Computational Fluid Dynamics on a Staggered Grid 27191 Challenges in the CFD Development 273
911 Non-Linearity 273912 Equation for Pressure 273913 Pressure-Velocity Decoupling 273
92 A Staggered Grid to avoid Pressure-Velocity Decoupling27593 Physical Law based FVM for a Staggered Grid 27794 Flux based Solution Methodology on a Uniform Grid
Semi-Explicit Method 281941 Philosophy of Pressure-Correction Method 283942 Semi-Explicit Method 286943 Implementation Details 292944 Solution Algorithm 298
95 Initial and Boundary Conditions 300951 Initial Condition 300952 Boundary Condition 301
Problems 306
Introduction to CFD xvii
10 Computational Fluid Dynamics on a Co-located Grid309
101 Momentum-Interpolation Method Strategy to avoid
the Pressure-Velocity Decoupling on a Co-located Grid 310
102 Coecients of LAEs based Solution Methodology on
a Non-Uniform Grid Semi-Explicit and Semi-Implicit
Method 314
1021 Predictor Step 316
1022 Corrector Step 318
1023 Solution Algorithm 323
Problems 329
III CFD for a Complex-Geometry
331
11 Computational Heat Conduction on a Curvilinear
Grid 333
111 Curvilinear Grid Generation 333
1111 Algebraic Grid Generation 334
1112 Elliptic Grid Generation 336
112 Physical Law based Finite Volume Method 343
1121 Unsteady and Source Term 343
1122 Diusion Term 344
1123 All Terms 349
113 Computation of Geometrical Properties 349
114 Flux based Solution Methodology 352
1141 Explicit Method 354
1142 Implementation Details 354
Problems 359
12 Computational Fluid Dynamics on a Curvilinear Grid361
121 Physical Law based Finite Volume Method 361
1211 Mass Conservation 363
1212 Momentum Conservation 363
122 Solution Methodology Semi-Explicit Method 372
1221 Predictor Step 373
1222 Corrector Step 375
Problems 380
References 383
Index 389
Part I
Introduction and Essentials
The book on an introductory course in CFD starts with the two intro-
ductory chapters on CFD followed by one chapter each on essentials
of two prerequisite courses minus uid-dynamics and heat transfer and
numerical-methods The rst chapter on introduction presents an
eventful thoughtful and intuitive discussion on What is CFD and
Why to study CFD whereas How CFD works which involves lots
of details is presented separately in the second chapter The third-
chapter on essentials of uid-dynamics and heat transfer presents
physical-laws transport-mechanisms physical-law based dierential-
formulation volumetric and ux terms and mathematical formula-
tion The fourth-chapter on essentials of numerical-methods presents
nite dierence method iterative solution of the system of linear alge-
braic equation numerical dierentiation and numerical integration
The presentation for the essential of the two prerequisite courses is
customized to CFD development application and analysis
1
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
1
Introduction
The introduction of any course consists of the three basic questions
What is it Why to study How it works An eventful thoughtful
and intuitive answer to the rst two questions are presented in this
chapter The third question which involves lots of working details
on the three aspects of the Computational Fluid Dynamics (CFD)
(development application and analysis) is presented separately in
the next chapter The novelty scope and purpose of this book is
also presented at the end of this chapter
11 CFD What is it
CFD is a theoretical-method of scientic and engineering investiga-
tion concerned with the development and application of a video-
camera like tool (a software) which is used for a unied cause-and-
eect based analysis of a uid-dynamics as well as heat and mass
transfer problem
CFD is a subject where you rst learn how to develop a product
minus a software which acts like a virtual video-camera Then you learn
how to apply or use this product to generate uid-dynamics movies
Finally you learn how to analyze the detailed spatial and temporal
uid-dynamics information in the movies The analysis is done to
come up with a scientically-exciting as well as engineering-relevant
story (unied cause-and-eect study) of a uid-dynamics situation in
nature as well as industrial-applications With a continuous develop-
ment and a wider application of CFD the word uid-dynamics in
3
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
4 1 Introduction
the CFD has become more generic as it also corresponds to heat and
mass transfer as well as chemical reaction
Fluid-dynamics information are of two types scientic and en-
gineering Scientic-information corresponds to a structure of the
heat and uid ow due to a physical phenomenon in uid-dynamics
such as boundary-layer ow-separation wake-formation and vortex-
shedding The ow structures are obtained from a temporal variation
of ow-properties (such as velocity pressure temperature and many
other variables which characterizes a ow) called as scientically-
exciting movies Engineering-information corresponds to certain pa-
rameters which are important in an engineering application called
as engineering-parameters for example lift force drag force wall
shear stress pressure drop and rate of heat-transfer They are ob-
tained as the temporal variation of engineering-parameters called as
engineering-relevant movies The engineering-parameters are the ef-
fect which are caused by the ow structures minus their correlation leads
to the unied cause-and-eect study A uid-dynamics movie consist
of a time-wise varying series of pictures which give uid-dynamics
information Each ow-property and engineering-parameter results
in one scientically-exciting and engineering-relevant movie respec-
tively When the large number of both the types of movies are
played synchronized in-time the detailed spatial as well as tempo-
ral uid-dynamic information greatly helps in the unied cause-and-
eect study of a uid-dynamics problem
Alternatively CFD is a subject concerned with the development of
computer-programs for the computer-simulation and study of a natu-
ral or an engineering uid-dynamics system The development starts
with the virtual-development of the system involving a geometri-
cal information based computational development of a solid-model
for the system the system may be already-existing or yet-to-be-
developed physically Thereafter the development involves numerical
solution of the governing algebraic equations for CFD formulated
from the physical laws (conservation of mass momentum and en-
ergy) and initial as well as boundary conditions corresponding to the
system CFD study involves the analysis of the uid-dynamics and
heat-transfer results obtained from the computer-simulations of the
virtual-system which is subjected to certain governing-parameters
Introduction to CFD 5
111 CFD as a Scientic and Engineering Analysis Tool
For a better understanding of the denition of the CFD and its ap-
plication as a scientic and engineering analysis tool two example
problems are presented rst conduction heat transfer in a plate
and second uid ow across a circular pillar of a bridge The rst-
problem correspond to an engineering-application for electronic cool-
ing and the second-problem is studied for an engineering-design of
the structure subjected to uid-dynamics forces and are shown in
Fig 11 Both the problems are considered as two-dimensional and
unsteady However the results asymptotes from the unsteady to a
steady state in the rst-problem and to a periodic (dynamic-steady)
state in the second-problem
Figure 11 Schematic and the associated parameters for the two example prob-lems (a) thermal conduction module of a heat sink and (b) theclassical ow across a circular cylinder
6 1 Introduction
The rst-problem is taken from the classic book on heat-transfer
by Incropera and Dewitt (1996) shown in Fig 11(a) The g-
ure shows a periodic module of the plate where a constant heat-ux
qW = 10Wcm2 (dissipated from certain electronic devices) is rst con-
ducted to an aluminum plate and then convected to the water ow-
ing through the rectangular channel grooved in the plate The forced
convection results in a convection coecient of h = 5000Wm2K in-
side the channel The unsteady heat-transfer results in an interesting
time-wise varying conduction heat ow in the plate and instantaneous
convection heat transfer rate Qprimeconv (per unit width of the plate per-
pendicular to the plane shown in Fig 11(a)) into the water
The second-problem is encountered if you are standing over a
bridge and look below to the water passing over a pillar (of circu-
lar cross-section) of the bridge shown in Fig 11(b) as a general case
of 2D ow across a cylinder The uid-dynamics results in a beau-
tiful time-wise varying structure of the uid ow just downstream
of the pillar and certain engineering-parameters The parameters
correspond to the uid-dynamic forces acting in the horizontal (or
streamwise) and vertical (or transverse) direction called as the drag-
force FD and lift-force FL respectively The respective force is pre-
sented below in a non-dimensional form called as the drag-coecient
CD and lift-coecient CL equations shown in Fig 11(b)
If you use your video camera to capture a movie for the two
problems you will get series of pictures for the plate in the rst-
problem and the owing uid in the second-problem Since these
movies result in a pictorial information but not the ow-property
based uid-dynamic information such movies are not relevant for
the uid-dynamics and heat-transfer study Instead if you use a
CFD software for computer simulation you can capture various types
of movies The various ow-properties based scientically-exciting
movies are presented here as the pictures at certain time-instant in
Fig 12(a)-(d) and Fig 13(a1)-(d2) The former gure shows gray-
colored ooded contour for the temperature whereas the latter gure
shows gray-colored ooded contour for the pressure and vorticity and
vector-plot for the velocity For the gray-colored ooded contours a
color bar can be seen in the gures for the relative magnitude of the
variable These gures also show line contours of stream-function
Introduction to CFD 7
and heat-function as streamlines and heatlines respectively They
are the visualization techniques for a 2D steady heat and uid ow
The heatlines are analogous to streamlines and represents the direc-
tion of heat ow under steady state condition proposed by Kimura
and Bejan (1983) later presented in a heat-transfer book by Bejan
(1984) The steady-state heatlines can be seen in Fig 12(d) where
the direction for the conduction-ux is tangential at any point on the
heatlines analogous to that for the mass-ux in a streamline Thus
for a better presentation of the structure of heat and uid ow the
scientically-exciting movies are not only limited to ow-properties
but also correspond to certain ow visualization techniquesAn engineering-parameter based engineering-relevant uid dy-
namic movie is presented in Fig 12(e) for the convection heat trans-
fer rate per unit width at the various walls on the channel Qprimew at the
west-wall Qprimen at the north-wall Q
primee at the east-wall and their cumu-
lative value Qprimeconv (refer Fig 11(a)) Another engineering-relevant
movie is presented in Fig 13(e) for the lift and drag coecient Note
that the movie here corresponds to the motion of a lled symbol over
a line for the temporal variation of an engineering-parameter Thus
the engineering-relevant movies are one-dimensional lower than the
scientically-exciting movies here for the 2D problems the former
movie is presented as 1D and the latter as 2D results
1111 Heat-Conduction
For the computational set-up of the heat conduction problem shown
in Fig 11(a) the results obtained from a 2D CFD simulation of
the conduction heat transfer are shown in Fig 12 For a uni-
form initial temperature of the plate as Tinfin = 15oC Fig 12(a)-(d)
shows the isotherms and heatlines in the plate as the pictures of the
scientically-exciting movies at certain discrete time instants t = 25
5 10 and 20 sec For the same discrete time instants the temporal
variation of the position of the symbols over a line plot is presented in
Fig 12(e) The gure is shown for each of the heat transfer rate and
is obtained from a engineering-relevant 1D movie There is a tempo-
ral variation of picture in the scientically-exciting movie and that
of position of symbol (along the line plot) in the engineering-relevant
movieAccording to caloric theory of the famous French scientist An-
tonie Lavoisier heat is considered as an invisible tasteless odorless
8 1 Introduction
t(sec)
Qrsquo (W
m)
0 5 10 15 20 250
200
400
600
800
1000Qrsquo
conv
Qrsquow
Qrsquon
Qrsquoe
Qrsquoin
(e)
(b)
(c)(d)
(a)
(a) t=25 sec31
29
27
25
23
21
19
17
15
T(oC)
(c) t=10 sec
(b) t=5 sec
(d) t=20 sec
Figure 12 Thermal conduction module of a heat sink temporal variation of(a)-(d) isotherms and heatlines and (e) the convection heat transferrate per unit width at the various walls of the channels and theircumulative total value Q
primeconv For the discrete time-instant corre-
sponding to heat ow patterns in (a)-(d) temporal variation of thevarious heat transfer rates are shown as symbols in (e) (a)-(c) arethe transient and (d) represents a steady state result
weightless uid which he called caloric uid Presently the caloric
theory is overtaken by mechanical theory of heat minus heat is not a sub-
stance but a dynamical form of mechanical eect (Thomson 1851)
Nevertheless there are certain problems involving heat ow for which
Lavoisiers approach is rather useful such as the discussion on heat-
lines here
The role of heatlines for heat ow is analogous to that of stream-
lines for uid ow (Paramane and Sharma 2009) For the uid-ow
the dierence of stream-function values represents the rate of uid
ow the function remains constant on a solid wall the tangent to
a streamline represents the direction of the uid-ow (with no ow
in the normal direction) and the streamline originate or emerge at
Introduction to CFD 9
a mass source Analogously for the heat-ow the dierence of heat-
function values represents the rate of heat ow the function remains
constant on an adiabatic wall the tangent to a heatline represents the
direction of the heat-ow (with no ow in the normal direction) and
the heatline originate or emerge at the heat-source For the present
problem in Fig 11(a) the heat-source is at the inlet (or left) bound-
ary and the heat-sink is at the surface of the channel Thus the
steady-state heatlines in Fig 12(d) shows the heat ow emerging
from the inletsource and ending on the channel-wallssink Further-
more it can be seen in the gure that the heatlines which are close
to the adiabatic walls are parallel to the walls The source sink and
adiabatic-walls can be seen in Fig 11(a)
Figure 12(e) shows an asymptotic time-wise increase in the con-
vection heat transfer per unit width at the various walls of the channel
and its cumulative value Qprimeconv (= Q
primew+Q
primen+Q
primee) It can be also be seen
that the heat-transfer per unit width Qprimew at the west-wall and Q
primen at
the north-wall are larger than Qprimee at the east-wall of the channel As
time progresses the gure shows that Qprimeconv increases monotonically
and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-
der steady state condition refer Fig 11(a) The dierence between
the Qprimein and Q
primeconv under the transient condition is utilized for the
sensible heating and results in the increase in the temperature of the
plate Under the steady state condition note from Fig 12(d) that
the maximum temperature is close to 32oC minus well within the per-
missible limit for the reliable and ecient operation of the electronic
device
1112 Fluid-Dynamics
For the free-stream ow across a circular cylinderpillar at a Reynolds
number of 100 the non-dimensional results obtained from a 2D CFD
simulation are shown in Fig 13 In contrast to the previous prob-
lem which reaches steady state the present problem reaches to an un-
steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-
ent increasing time instants within one time-period of the periodic-
ow the results are presented in Fig 13(a)-(d) The gure shows
the close-up view of the pictures of the scientically-exciting movies
for ow-properties (velocity pressure and vorticity) and streamlines
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow
x Preface
simple enough to introduce the rst-course at early under-graduate
curriculum rather than the post-graduate curriculum The moti-
vation got strengthened by my own as well as others (students
teachers researchers and practitioners) struggle on physical under-
standing of the mathematics involved during the Partial Dierential
Equation (PDE) based algebraic-formulation (nite volume method)
along with the continuous frustrating pursuit to eectively convert
the theory of CFD into the computer-program
This led me to come up with an alternate physical-law (PDE-
free) based nite volume method as well as solution methodol-
ogy which are much easier to comprehend Mostly the success of
the programming eort is decided by the perfect execution of the
implementations-details which rarely appears in-print (existing books
and journal articles) The physical law based CFD development
approach and the implementations-details are the novelties in the
present book Furthermore using an open-source software for nu-
merical computations (Scilab) computer programs are given in this
book on the basic modules of conduction advection and convection
Indeed the reader can generalize and extend these codes for the de-
velopment of Navier-Stokes solver and generate the results presented
in the chapters towards the end of this book
I have limited the scope of this book to the numerical-techniques
which I wish to recommend for the book although there are numerous
other advanced and better methods in the published literature I do
not claim that the programming-practice presented here are the most
ecient minus they are presented here more for the ease in programming
(understand as your program) than for the computational eciency
Furthermore although I am enthusiastic about my presentation of
the physical law based CFD approach it may not be more ecient
than the alternate mathematics based CFD approach The emphasis
in this book is on the ease of understanding of the formulations and
programming-practice for the introductory course on CFD
I would hereby acknowledge that I owe my greatest debt to Prof
J Srinivasan (my ME Project supervisor at IISc Bangalore) who
did a hand-holding and wonderful job in patiently introducing me to
the fascinating world of research in uid-dynamics and heat trans-
fer I would also acknowledge the excellent training from Prof V
Introduction to CFD xi
Eswaran (my PhD supervisor at IIT Kanpur) from whom I learned
systematic way of CFD development application and analysis His
inspiring comments and suggestions on my writing-style and elabora-
tion of the rst chapter in this book is also gratefully acknowledged
I also want to record my sincere thanks to Prof Pradip Datta IISc
Bangalore and Prof Gautam Biswas IIT Kanpur minus my teachers for
CFD course Special thanks to Prof K Muralidhar IIT Kanpur for
his personal encouragement academic support and motivation (for
the book writing) He also cleared my dilemma in the proper presen-
tation of the novelty in this book I thank him also for writing the
foreword of this book
I am grateful to Prof Kannan M Moudgalya and Prof Deepak B
Phatak from IIT Bombay for giving me opportunity to teach CFD
in a distance education mode I am also grateful to Prof Amit
Agrawal (my colleague at IIT Bombay) who diligently read through
all the chapters of this book and suggested some really remarkable
changes in the book I also thank him for the research interactions
over the last decade which had substantially improved my under-
standing of thermal and uid science I would also like to thank my
PhD student Namshad T who did a great job in implementing
my ideas on generating the CFD simulation based CFD image for
the cover-page and couple of gures in the rst chapter (Fig 12
and 13) Special mention that the Scilab codes developed (for the
CFD course under NMEICT) by Vishesh Aggarwal greatly helped
me to develop the codes presented in this book his contributions is
gratefully acknowledged I thank Malhar Malushte for testing some
of the formulations presented in this book
I thank my research students (Sachin B Paramane D Datta C
M Sewatkar Vinesh H Gada Kaushal Prasad Mukul Shrivastava
Absar M Lakdawala Mausam Sarkar and Javed Shaikh) for the
diligence and sincerity they have shown in unraveling some of our
ideas on CFD Apart from these several other research students and
the students teachers and CFD practitioners who took my lectures
on CFD in the past decade have contributed to my better under-
standing of CFD and inspired me to write the book I thank all of
them for their attention and enthusiastic response during the CFD
courses The inuence of the interactions with all of them can be seen
xii Preface
throughout this book I am grateful to IIT Bombay for giving me
one year sabbatical leave to write this book The ambiance of aca-
demic freedom considerate support by the faculties and the friendly
atmosphere inside the campus of IIT Bombay has greatly contributed
in the solo eort of book writing I thanks all my colleagues at IIT
Bombay for shaping my perceptions on CFD
During this long endeavor the patient support by Mr Sunil
Saxena Director Athena Academic Ltd UK is gratefully acknowl-
edged I also thank John Wiley for coming forward to take this
book to international markets Lastly I would acknowledge that I
am greatly inspired by the writing style of J D Anderson and the
book on CFD by S V Patankar Of course you will nd that I
have heavily cross-referred the CFD book by the two most acknowl-
edged pioneers on CFD I dedicate this book to Dr S V Patankar
whose book nicely introduced me to the computational as well as
ow physics in CFD and greatly helped me to introduce a physical
approach of CFD development in this book
The culmination of the book writing process fullls a long cher-
ished dream It would not have been possible without the excel-
lent environment continuous support and complete understanding
of my wife Anubha and daughter Anshita Their kindness has greatly
helped to fulll my dream now I plan to spend more time with my
wife and daughter
Mumbai
August 2016 Atul Sharma
Contents
I Introduction and Essentials
1
1 Introduction 3
11 CFD What is it 3
111 CFD as a Scientic and Engineering Analysis
Tool 5
112 Analogy with a Video-Camera 13
12 CFD Why to study 15
13 Novelty Scope and Purpose of this Book 16
2 Introduction to CFD Development Application and
Analysis 23
21 CFD Development 23
211 Grid Generation Pre-Processor 24
212 Discretization Method Algebraic Formulation 26
213 Solution Methodology Solver 29
214 Computation of Engineering-Parameters
Post-Processor 34
215 Testing 35
22 CFD Application 35
23 CFD Analysis 38
24 Closure 39
xiii
xiv Contents
3 Essentials of Fluid-Dynamics and Heat-Transfer for
CFD 41
31 Physical Laws 42
311 FundamentalConservation Laws 43
312 Subsidiary Laws 45
32 Momentum and Energy Transport Mechanisms 46
33 Physical Law based Dierential Formulation 48
331 Continuity Equation 49
332 Transport Equation 51
34 Generalized Volumetric and Flux Terms and their Dif-
ferential Formulation 57
341 Volumetric Term 58
342 Flux-Term 58
343 Discussion 62
35 Mathematical Formulation 63
351 Dimensional Study 63
352 Non-Dimensional Study 67
36 Closure 71
4 Essentials of Numerical-Methods for CFD 73
41 Finite Dierence Method A Dierential to Algebraic
Formulation for Governing PDE and BCs 75
411 Grid Generation 75
412 Finite Dierence Method 78
413 Applications to CFD 92
42 Iterative Solution of System of LAEs for a Flow Property 93
421 Iterative Methods 94
422 Applications to CFD 98
43 Numerical Dierentiation for Local Engineering-
Parameters 105
431 Dierentiation Formulas 106
432 Applications to CFD 107
44 Numerical Integration for the Total value of
Engineering-Parameters 110
441 Integration Rules 111
442 Applications to CFD 114
45 Closure 116
Problems 116
Introduction to CFD xv
II CFD for a Cartesian-Geometry
119
5 Computational Heat Conduction 12151 Physical Law based Finite Volume Method 122
511 Energy Conservation Law for a Control Volume 123512 Algebraic Formulation 124513 Approximations 127514 Approximated Algebraic-Formulation 130515 Discussion 133
52 Finite Dierence Method for Boundary Conditions 13853 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 139531 One-Dimensional Conduction 139532 Two-Dimensional Conduction 147
54 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 154541 One-Dimensional Conduction 154542 Two-Dimensional Conduction 166
Problems 176
6 Computational Heat Advection 17961 Physical Law based Finite Volume Method 180
611 Energy Conservation Law for a Control Volume 181612 Algebraic Formulation 181613 Approximations 183614 Approximated Algebraic Formulation 191615 Discussion 191
62 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 194621 Explicit-Method 197622 Implementation Details 198623 Solution Algorithm 199
63 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 207631 Advection Scheme on a Non-Uniform Grid 207632 Explicit and Implicit Method 210633 Implementation Details 216634 Solution Algorithm 220
Problems 228
xvi Contents
7 Computational Heat Convection 22971 Physical Law based Finite Volume Method 229
711 Energy Conservation Law for a Control Volume 229712 Algebraic Formulation 231713 Approximated Algebraic Formulation 232
72 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 235721 Explicit-Method 236722 Implementation Details 237723 Solution Algorithm 238
73 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 242Problems 248
8 Computational Fluid Dynamics Physical Law based
Finite Volume Method 25181 Generalized Variables for the Combined Heat and
Fluid Flow 25282 Conservation Laws for a Control Volume 25583 Algebraic Formulation 25984 Approximations 26085 Approximated Algebraic Formulation 263
851 Mass Conservation 263852 MomentumEnergy Conservation 264
86 Closure 269
9 Computational Fluid Dynamics on a Staggered Grid 27191 Challenges in the CFD Development 273
911 Non-Linearity 273912 Equation for Pressure 273913 Pressure-Velocity Decoupling 273
92 A Staggered Grid to avoid Pressure-Velocity Decoupling27593 Physical Law based FVM for a Staggered Grid 27794 Flux based Solution Methodology on a Uniform Grid
Semi-Explicit Method 281941 Philosophy of Pressure-Correction Method 283942 Semi-Explicit Method 286943 Implementation Details 292944 Solution Algorithm 298
95 Initial and Boundary Conditions 300951 Initial Condition 300952 Boundary Condition 301
Problems 306
Introduction to CFD xvii
10 Computational Fluid Dynamics on a Co-located Grid309
101 Momentum-Interpolation Method Strategy to avoid
the Pressure-Velocity Decoupling on a Co-located Grid 310
102 Coecients of LAEs based Solution Methodology on
a Non-Uniform Grid Semi-Explicit and Semi-Implicit
Method 314
1021 Predictor Step 316
1022 Corrector Step 318
1023 Solution Algorithm 323
Problems 329
III CFD for a Complex-Geometry
331
11 Computational Heat Conduction on a Curvilinear
Grid 333
111 Curvilinear Grid Generation 333
1111 Algebraic Grid Generation 334
1112 Elliptic Grid Generation 336
112 Physical Law based Finite Volume Method 343
1121 Unsteady and Source Term 343
1122 Diusion Term 344
1123 All Terms 349
113 Computation of Geometrical Properties 349
114 Flux based Solution Methodology 352
1141 Explicit Method 354
1142 Implementation Details 354
Problems 359
12 Computational Fluid Dynamics on a Curvilinear Grid361
121 Physical Law based Finite Volume Method 361
1211 Mass Conservation 363
1212 Momentum Conservation 363
122 Solution Methodology Semi-Explicit Method 372
1221 Predictor Step 373
1222 Corrector Step 375
Problems 380
References 383
Index 389
Part I
Introduction and Essentials
The book on an introductory course in CFD starts with the two intro-
ductory chapters on CFD followed by one chapter each on essentials
of two prerequisite courses minus uid-dynamics and heat transfer and
numerical-methods The rst chapter on introduction presents an
eventful thoughtful and intuitive discussion on What is CFD and
Why to study CFD whereas How CFD works which involves lots
of details is presented separately in the second chapter The third-
chapter on essentials of uid-dynamics and heat transfer presents
physical-laws transport-mechanisms physical-law based dierential-
formulation volumetric and ux terms and mathematical formula-
tion The fourth-chapter on essentials of numerical-methods presents
nite dierence method iterative solution of the system of linear alge-
braic equation numerical dierentiation and numerical integration
The presentation for the essential of the two prerequisite courses is
customized to CFD development application and analysis
1
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
1
Introduction
The introduction of any course consists of the three basic questions
What is it Why to study How it works An eventful thoughtful
and intuitive answer to the rst two questions are presented in this
chapter The third question which involves lots of working details
on the three aspects of the Computational Fluid Dynamics (CFD)
(development application and analysis) is presented separately in
the next chapter The novelty scope and purpose of this book is
also presented at the end of this chapter
11 CFD What is it
CFD is a theoretical-method of scientic and engineering investiga-
tion concerned with the development and application of a video-
camera like tool (a software) which is used for a unied cause-and-
eect based analysis of a uid-dynamics as well as heat and mass
transfer problem
CFD is a subject where you rst learn how to develop a product
minus a software which acts like a virtual video-camera Then you learn
how to apply or use this product to generate uid-dynamics movies
Finally you learn how to analyze the detailed spatial and temporal
uid-dynamics information in the movies The analysis is done to
come up with a scientically-exciting as well as engineering-relevant
story (unied cause-and-eect study) of a uid-dynamics situation in
nature as well as industrial-applications With a continuous develop-
ment and a wider application of CFD the word uid-dynamics in
3
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
4 1 Introduction
the CFD has become more generic as it also corresponds to heat and
mass transfer as well as chemical reaction
Fluid-dynamics information are of two types scientic and en-
gineering Scientic-information corresponds to a structure of the
heat and uid ow due to a physical phenomenon in uid-dynamics
such as boundary-layer ow-separation wake-formation and vortex-
shedding The ow structures are obtained from a temporal variation
of ow-properties (such as velocity pressure temperature and many
other variables which characterizes a ow) called as scientically-
exciting movies Engineering-information corresponds to certain pa-
rameters which are important in an engineering application called
as engineering-parameters for example lift force drag force wall
shear stress pressure drop and rate of heat-transfer They are ob-
tained as the temporal variation of engineering-parameters called as
engineering-relevant movies The engineering-parameters are the ef-
fect which are caused by the ow structures minus their correlation leads
to the unied cause-and-eect study A uid-dynamics movie consist
of a time-wise varying series of pictures which give uid-dynamics
information Each ow-property and engineering-parameter results
in one scientically-exciting and engineering-relevant movie respec-
tively When the large number of both the types of movies are
played synchronized in-time the detailed spatial as well as tempo-
ral uid-dynamic information greatly helps in the unied cause-and-
eect study of a uid-dynamics problem
Alternatively CFD is a subject concerned with the development of
computer-programs for the computer-simulation and study of a natu-
ral or an engineering uid-dynamics system The development starts
with the virtual-development of the system involving a geometri-
cal information based computational development of a solid-model
for the system the system may be already-existing or yet-to-be-
developed physically Thereafter the development involves numerical
solution of the governing algebraic equations for CFD formulated
from the physical laws (conservation of mass momentum and en-
ergy) and initial as well as boundary conditions corresponding to the
system CFD study involves the analysis of the uid-dynamics and
heat-transfer results obtained from the computer-simulations of the
virtual-system which is subjected to certain governing-parameters
Introduction to CFD 5
111 CFD as a Scientic and Engineering Analysis Tool
For a better understanding of the denition of the CFD and its ap-
plication as a scientic and engineering analysis tool two example
problems are presented rst conduction heat transfer in a plate
and second uid ow across a circular pillar of a bridge The rst-
problem correspond to an engineering-application for electronic cool-
ing and the second-problem is studied for an engineering-design of
the structure subjected to uid-dynamics forces and are shown in
Fig 11 Both the problems are considered as two-dimensional and
unsteady However the results asymptotes from the unsteady to a
steady state in the rst-problem and to a periodic (dynamic-steady)
state in the second-problem
Figure 11 Schematic and the associated parameters for the two example prob-lems (a) thermal conduction module of a heat sink and (b) theclassical ow across a circular cylinder
6 1 Introduction
The rst-problem is taken from the classic book on heat-transfer
by Incropera and Dewitt (1996) shown in Fig 11(a) The g-
ure shows a periodic module of the plate where a constant heat-ux
qW = 10Wcm2 (dissipated from certain electronic devices) is rst con-
ducted to an aluminum plate and then convected to the water ow-
ing through the rectangular channel grooved in the plate The forced
convection results in a convection coecient of h = 5000Wm2K in-
side the channel The unsteady heat-transfer results in an interesting
time-wise varying conduction heat ow in the plate and instantaneous
convection heat transfer rate Qprimeconv (per unit width of the plate per-
pendicular to the plane shown in Fig 11(a)) into the water
The second-problem is encountered if you are standing over a
bridge and look below to the water passing over a pillar (of circu-
lar cross-section) of the bridge shown in Fig 11(b) as a general case
of 2D ow across a cylinder The uid-dynamics results in a beau-
tiful time-wise varying structure of the uid ow just downstream
of the pillar and certain engineering-parameters The parameters
correspond to the uid-dynamic forces acting in the horizontal (or
streamwise) and vertical (or transverse) direction called as the drag-
force FD and lift-force FL respectively The respective force is pre-
sented below in a non-dimensional form called as the drag-coecient
CD and lift-coecient CL equations shown in Fig 11(b)
If you use your video camera to capture a movie for the two
problems you will get series of pictures for the plate in the rst-
problem and the owing uid in the second-problem Since these
movies result in a pictorial information but not the ow-property
based uid-dynamic information such movies are not relevant for
the uid-dynamics and heat-transfer study Instead if you use a
CFD software for computer simulation you can capture various types
of movies The various ow-properties based scientically-exciting
movies are presented here as the pictures at certain time-instant in
Fig 12(a)-(d) and Fig 13(a1)-(d2) The former gure shows gray-
colored ooded contour for the temperature whereas the latter gure
shows gray-colored ooded contour for the pressure and vorticity and
vector-plot for the velocity For the gray-colored ooded contours a
color bar can be seen in the gures for the relative magnitude of the
variable These gures also show line contours of stream-function
Introduction to CFD 7
and heat-function as streamlines and heatlines respectively They
are the visualization techniques for a 2D steady heat and uid ow
The heatlines are analogous to streamlines and represents the direc-
tion of heat ow under steady state condition proposed by Kimura
and Bejan (1983) later presented in a heat-transfer book by Bejan
(1984) The steady-state heatlines can be seen in Fig 12(d) where
the direction for the conduction-ux is tangential at any point on the
heatlines analogous to that for the mass-ux in a streamline Thus
for a better presentation of the structure of heat and uid ow the
scientically-exciting movies are not only limited to ow-properties
but also correspond to certain ow visualization techniquesAn engineering-parameter based engineering-relevant uid dy-
namic movie is presented in Fig 12(e) for the convection heat trans-
fer rate per unit width at the various walls on the channel Qprimew at the
west-wall Qprimen at the north-wall Q
primee at the east-wall and their cumu-
lative value Qprimeconv (refer Fig 11(a)) Another engineering-relevant
movie is presented in Fig 13(e) for the lift and drag coecient Note
that the movie here corresponds to the motion of a lled symbol over
a line for the temporal variation of an engineering-parameter Thus
the engineering-relevant movies are one-dimensional lower than the
scientically-exciting movies here for the 2D problems the former
movie is presented as 1D and the latter as 2D results
1111 Heat-Conduction
For the computational set-up of the heat conduction problem shown
in Fig 11(a) the results obtained from a 2D CFD simulation of
the conduction heat transfer are shown in Fig 12 For a uni-
form initial temperature of the plate as Tinfin = 15oC Fig 12(a)-(d)
shows the isotherms and heatlines in the plate as the pictures of the
scientically-exciting movies at certain discrete time instants t = 25
5 10 and 20 sec For the same discrete time instants the temporal
variation of the position of the symbols over a line plot is presented in
Fig 12(e) The gure is shown for each of the heat transfer rate and
is obtained from a engineering-relevant 1D movie There is a tempo-
ral variation of picture in the scientically-exciting movie and that
of position of symbol (along the line plot) in the engineering-relevant
movieAccording to caloric theory of the famous French scientist An-
tonie Lavoisier heat is considered as an invisible tasteless odorless
8 1 Introduction
t(sec)
Qrsquo (W
m)
0 5 10 15 20 250
200
400
600
800
1000Qrsquo
conv
Qrsquow
Qrsquon
Qrsquoe
Qrsquoin
(e)
(b)
(c)(d)
(a)
(a) t=25 sec31
29
27
25
23
21
19
17
15
T(oC)
(c) t=10 sec
(b) t=5 sec
(d) t=20 sec
Figure 12 Thermal conduction module of a heat sink temporal variation of(a)-(d) isotherms and heatlines and (e) the convection heat transferrate per unit width at the various walls of the channels and theircumulative total value Q
primeconv For the discrete time-instant corre-
sponding to heat ow patterns in (a)-(d) temporal variation of thevarious heat transfer rates are shown as symbols in (e) (a)-(c) arethe transient and (d) represents a steady state result
weightless uid which he called caloric uid Presently the caloric
theory is overtaken by mechanical theory of heat minus heat is not a sub-
stance but a dynamical form of mechanical eect (Thomson 1851)
Nevertheless there are certain problems involving heat ow for which
Lavoisiers approach is rather useful such as the discussion on heat-
lines here
The role of heatlines for heat ow is analogous to that of stream-
lines for uid ow (Paramane and Sharma 2009) For the uid-ow
the dierence of stream-function values represents the rate of uid
ow the function remains constant on a solid wall the tangent to
a streamline represents the direction of the uid-ow (with no ow
in the normal direction) and the streamline originate or emerge at
Introduction to CFD 9
a mass source Analogously for the heat-ow the dierence of heat-
function values represents the rate of heat ow the function remains
constant on an adiabatic wall the tangent to a heatline represents the
direction of the heat-ow (with no ow in the normal direction) and
the heatline originate or emerge at the heat-source For the present
problem in Fig 11(a) the heat-source is at the inlet (or left) bound-
ary and the heat-sink is at the surface of the channel Thus the
steady-state heatlines in Fig 12(d) shows the heat ow emerging
from the inletsource and ending on the channel-wallssink Further-
more it can be seen in the gure that the heatlines which are close
to the adiabatic walls are parallel to the walls The source sink and
adiabatic-walls can be seen in Fig 11(a)
Figure 12(e) shows an asymptotic time-wise increase in the con-
vection heat transfer per unit width at the various walls of the channel
and its cumulative value Qprimeconv (= Q
primew+Q
primen+Q
primee) It can be also be seen
that the heat-transfer per unit width Qprimew at the west-wall and Q
primen at
the north-wall are larger than Qprimee at the east-wall of the channel As
time progresses the gure shows that Qprimeconv increases monotonically
and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-
der steady state condition refer Fig 11(a) The dierence between
the Qprimein and Q
primeconv under the transient condition is utilized for the
sensible heating and results in the increase in the temperature of the
plate Under the steady state condition note from Fig 12(d) that
the maximum temperature is close to 32oC minus well within the per-
missible limit for the reliable and ecient operation of the electronic
device
1112 Fluid-Dynamics
For the free-stream ow across a circular cylinderpillar at a Reynolds
number of 100 the non-dimensional results obtained from a 2D CFD
simulation are shown in Fig 13 In contrast to the previous prob-
lem which reaches steady state the present problem reaches to an un-
steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-
ent increasing time instants within one time-period of the periodic-
ow the results are presented in Fig 13(a)-(d) The gure shows
the close-up view of the pictures of the scientically-exciting movies
for ow-properties (velocity pressure and vorticity) and streamlines
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow
Introduction to CFD xi
Eswaran (my PhD supervisor at IIT Kanpur) from whom I learned
systematic way of CFD development application and analysis His
inspiring comments and suggestions on my writing-style and elabora-
tion of the rst chapter in this book is also gratefully acknowledged
I also want to record my sincere thanks to Prof Pradip Datta IISc
Bangalore and Prof Gautam Biswas IIT Kanpur minus my teachers for
CFD course Special thanks to Prof K Muralidhar IIT Kanpur for
his personal encouragement academic support and motivation (for
the book writing) He also cleared my dilemma in the proper presen-
tation of the novelty in this book I thank him also for writing the
foreword of this book
I am grateful to Prof Kannan M Moudgalya and Prof Deepak B
Phatak from IIT Bombay for giving me opportunity to teach CFD
in a distance education mode I am also grateful to Prof Amit
Agrawal (my colleague at IIT Bombay) who diligently read through
all the chapters of this book and suggested some really remarkable
changes in the book I also thank him for the research interactions
over the last decade which had substantially improved my under-
standing of thermal and uid science I would also like to thank my
PhD student Namshad T who did a great job in implementing
my ideas on generating the CFD simulation based CFD image for
the cover-page and couple of gures in the rst chapter (Fig 12
and 13) Special mention that the Scilab codes developed (for the
CFD course under NMEICT) by Vishesh Aggarwal greatly helped
me to develop the codes presented in this book his contributions is
gratefully acknowledged I thank Malhar Malushte for testing some
of the formulations presented in this book
I thank my research students (Sachin B Paramane D Datta C
M Sewatkar Vinesh H Gada Kaushal Prasad Mukul Shrivastava
Absar M Lakdawala Mausam Sarkar and Javed Shaikh) for the
diligence and sincerity they have shown in unraveling some of our
ideas on CFD Apart from these several other research students and
the students teachers and CFD practitioners who took my lectures
on CFD in the past decade have contributed to my better under-
standing of CFD and inspired me to write the book I thank all of
them for their attention and enthusiastic response during the CFD
courses The inuence of the interactions with all of them can be seen
xii Preface
throughout this book I am grateful to IIT Bombay for giving me
one year sabbatical leave to write this book The ambiance of aca-
demic freedom considerate support by the faculties and the friendly
atmosphere inside the campus of IIT Bombay has greatly contributed
in the solo eort of book writing I thanks all my colleagues at IIT
Bombay for shaping my perceptions on CFD
During this long endeavor the patient support by Mr Sunil
Saxena Director Athena Academic Ltd UK is gratefully acknowl-
edged I also thank John Wiley for coming forward to take this
book to international markets Lastly I would acknowledge that I
am greatly inspired by the writing style of J D Anderson and the
book on CFD by S V Patankar Of course you will nd that I
have heavily cross-referred the CFD book by the two most acknowl-
edged pioneers on CFD I dedicate this book to Dr S V Patankar
whose book nicely introduced me to the computational as well as
ow physics in CFD and greatly helped me to introduce a physical
approach of CFD development in this book
The culmination of the book writing process fullls a long cher-
ished dream It would not have been possible without the excel-
lent environment continuous support and complete understanding
of my wife Anubha and daughter Anshita Their kindness has greatly
helped to fulll my dream now I plan to spend more time with my
wife and daughter
Mumbai
August 2016 Atul Sharma
Contents
I Introduction and Essentials
1
1 Introduction 3
11 CFD What is it 3
111 CFD as a Scientic and Engineering Analysis
Tool 5
112 Analogy with a Video-Camera 13
12 CFD Why to study 15
13 Novelty Scope and Purpose of this Book 16
2 Introduction to CFD Development Application and
Analysis 23
21 CFD Development 23
211 Grid Generation Pre-Processor 24
212 Discretization Method Algebraic Formulation 26
213 Solution Methodology Solver 29
214 Computation of Engineering-Parameters
Post-Processor 34
215 Testing 35
22 CFD Application 35
23 CFD Analysis 38
24 Closure 39
xiii
xiv Contents
3 Essentials of Fluid-Dynamics and Heat-Transfer for
CFD 41
31 Physical Laws 42
311 FundamentalConservation Laws 43
312 Subsidiary Laws 45
32 Momentum and Energy Transport Mechanisms 46
33 Physical Law based Dierential Formulation 48
331 Continuity Equation 49
332 Transport Equation 51
34 Generalized Volumetric and Flux Terms and their Dif-
ferential Formulation 57
341 Volumetric Term 58
342 Flux-Term 58
343 Discussion 62
35 Mathematical Formulation 63
351 Dimensional Study 63
352 Non-Dimensional Study 67
36 Closure 71
4 Essentials of Numerical-Methods for CFD 73
41 Finite Dierence Method A Dierential to Algebraic
Formulation for Governing PDE and BCs 75
411 Grid Generation 75
412 Finite Dierence Method 78
413 Applications to CFD 92
42 Iterative Solution of System of LAEs for a Flow Property 93
421 Iterative Methods 94
422 Applications to CFD 98
43 Numerical Dierentiation for Local Engineering-
Parameters 105
431 Dierentiation Formulas 106
432 Applications to CFD 107
44 Numerical Integration for the Total value of
Engineering-Parameters 110
441 Integration Rules 111
442 Applications to CFD 114
45 Closure 116
Problems 116
Introduction to CFD xv
II CFD for a Cartesian-Geometry
119
5 Computational Heat Conduction 12151 Physical Law based Finite Volume Method 122
511 Energy Conservation Law for a Control Volume 123512 Algebraic Formulation 124513 Approximations 127514 Approximated Algebraic-Formulation 130515 Discussion 133
52 Finite Dierence Method for Boundary Conditions 13853 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 139531 One-Dimensional Conduction 139532 Two-Dimensional Conduction 147
54 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 154541 One-Dimensional Conduction 154542 Two-Dimensional Conduction 166
Problems 176
6 Computational Heat Advection 17961 Physical Law based Finite Volume Method 180
611 Energy Conservation Law for a Control Volume 181612 Algebraic Formulation 181613 Approximations 183614 Approximated Algebraic Formulation 191615 Discussion 191
62 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 194621 Explicit-Method 197622 Implementation Details 198623 Solution Algorithm 199
63 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 207631 Advection Scheme on a Non-Uniform Grid 207632 Explicit and Implicit Method 210633 Implementation Details 216634 Solution Algorithm 220
Problems 228
xvi Contents
7 Computational Heat Convection 22971 Physical Law based Finite Volume Method 229
711 Energy Conservation Law for a Control Volume 229712 Algebraic Formulation 231713 Approximated Algebraic Formulation 232
72 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 235721 Explicit-Method 236722 Implementation Details 237723 Solution Algorithm 238
73 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 242Problems 248
8 Computational Fluid Dynamics Physical Law based
Finite Volume Method 25181 Generalized Variables for the Combined Heat and
Fluid Flow 25282 Conservation Laws for a Control Volume 25583 Algebraic Formulation 25984 Approximations 26085 Approximated Algebraic Formulation 263
851 Mass Conservation 263852 MomentumEnergy Conservation 264
86 Closure 269
9 Computational Fluid Dynamics on a Staggered Grid 27191 Challenges in the CFD Development 273
911 Non-Linearity 273912 Equation for Pressure 273913 Pressure-Velocity Decoupling 273
92 A Staggered Grid to avoid Pressure-Velocity Decoupling27593 Physical Law based FVM for a Staggered Grid 27794 Flux based Solution Methodology on a Uniform Grid
Semi-Explicit Method 281941 Philosophy of Pressure-Correction Method 283942 Semi-Explicit Method 286943 Implementation Details 292944 Solution Algorithm 298
95 Initial and Boundary Conditions 300951 Initial Condition 300952 Boundary Condition 301
Problems 306
Introduction to CFD xvii
10 Computational Fluid Dynamics on a Co-located Grid309
101 Momentum-Interpolation Method Strategy to avoid
the Pressure-Velocity Decoupling on a Co-located Grid 310
102 Coecients of LAEs based Solution Methodology on
a Non-Uniform Grid Semi-Explicit and Semi-Implicit
Method 314
1021 Predictor Step 316
1022 Corrector Step 318
1023 Solution Algorithm 323
Problems 329
III CFD for a Complex-Geometry
331
11 Computational Heat Conduction on a Curvilinear
Grid 333
111 Curvilinear Grid Generation 333
1111 Algebraic Grid Generation 334
1112 Elliptic Grid Generation 336
112 Physical Law based Finite Volume Method 343
1121 Unsteady and Source Term 343
1122 Diusion Term 344
1123 All Terms 349
113 Computation of Geometrical Properties 349
114 Flux based Solution Methodology 352
1141 Explicit Method 354
1142 Implementation Details 354
Problems 359
12 Computational Fluid Dynamics on a Curvilinear Grid361
121 Physical Law based Finite Volume Method 361
1211 Mass Conservation 363
1212 Momentum Conservation 363
122 Solution Methodology Semi-Explicit Method 372
1221 Predictor Step 373
1222 Corrector Step 375
Problems 380
References 383
Index 389
Part I
Introduction and Essentials
The book on an introductory course in CFD starts with the two intro-
ductory chapters on CFD followed by one chapter each on essentials
of two prerequisite courses minus uid-dynamics and heat transfer and
numerical-methods The rst chapter on introduction presents an
eventful thoughtful and intuitive discussion on What is CFD and
Why to study CFD whereas How CFD works which involves lots
of details is presented separately in the second chapter The third-
chapter on essentials of uid-dynamics and heat transfer presents
physical-laws transport-mechanisms physical-law based dierential-
formulation volumetric and ux terms and mathematical formula-
tion The fourth-chapter on essentials of numerical-methods presents
nite dierence method iterative solution of the system of linear alge-
braic equation numerical dierentiation and numerical integration
The presentation for the essential of the two prerequisite courses is
customized to CFD development application and analysis
1
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
1
Introduction
The introduction of any course consists of the three basic questions
What is it Why to study How it works An eventful thoughtful
and intuitive answer to the rst two questions are presented in this
chapter The third question which involves lots of working details
on the three aspects of the Computational Fluid Dynamics (CFD)
(development application and analysis) is presented separately in
the next chapter The novelty scope and purpose of this book is
also presented at the end of this chapter
11 CFD What is it
CFD is a theoretical-method of scientic and engineering investiga-
tion concerned with the development and application of a video-
camera like tool (a software) which is used for a unied cause-and-
eect based analysis of a uid-dynamics as well as heat and mass
transfer problem
CFD is a subject where you rst learn how to develop a product
minus a software which acts like a virtual video-camera Then you learn
how to apply or use this product to generate uid-dynamics movies
Finally you learn how to analyze the detailed spatial and temporal
uid-dynamics information in the movies The analysis is done to
come up with a scientically-exciting as well as engineering-relevant
story (unied cause-and-eect study) of a uid-dynamics situation in
nature as well as industrial-applications With a continuous develop-
ment and a wider application of CFD the word uid-dynamics in
3
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
4 1 Introduction
the CFD has become more generic as it also corresponds to heat and
mass transfer as well as chemical reaction
Fluid-dynamics information are of two types scientic and en-
gineering Scientic-information corresponds to a structure of the
heat and uid ow due to a physical phenomenon in uid-dynamics
such as boundary-layer ow-separation wake-formation and vortex-
shedding The ow structures are obtained from a temporal variation
of ow-properties (such as velocity pressure temperature and many
other variables which characterizes a ow) called as scientically-
exciting movies Engineering-information corresponds to certain pa-
rameters which are important in an engineering application called
as engineering-parameters for example lift force drag force wall
shear stress pressure drop and rate of heat-transfer They are ob-
tained as the temporal variation of engineering-parameters called as
engineering-relevant movies The engineering-parameters are the ef-
fect which are caused by the ow structures minus their correlation leads
to the unied cause-and-eect study A uid-dynamics movie consist
of a time-wise varying series of pictures which give uid-dynamics
information Each ow-property and engineering-parameter results
in one scientically-exciting and engineering-relevant movie respec-
tively When the large number of both the types of movies are
played synchronized in-time the detailed spatial as well as tempo-
ral uid-dynamic information greatly helps in the unied cause-and-
eect study of a uid-dynamics problem
Alternatively CFD is a subject concerned with the development of
computer-programs for the computer-simulation and study of a natu-
ral or an engineering uid-dynamics system The development starts
with the virtual-development of the system involving a geometri-
cal information based computational development of a solid-model
for the system the system may be already-existing or yet-to-be-
developed physically Thereafter the development involves numerical
solution of the governing algebraic equations for CFD formulated
from the physical laws (conservation of mass momentum and en-
ergy) and initial as well as boundary conditions corresponding to the
system CFD study involves the analysis of the uid-dynamics and
heat-transfer results obtained from the computer-simulations of the
virtual-system which is subjected to certain governing-parameters
Introduction to CFD 5
111 CFD as a Scientic and Engineering Analysis Tool
For a better understanding of the denition of the CFD and its ap-
plication as a scientic and engineering analysis tool two example
problems are presented rst conduction heat transfer in a plate
and second uid ow across a circular pillar of a bridge The rst-
problem correspond to an engineering-application for electronic cool-
ing and the second-problem is studied for an engineering-design of
the structure subjected to uid-dynamics forces and are shown in
Fig 11 Both the problems are considered as two-dimensional and
unsteady However the results asymptotes from the unsteady to a
steady state in the rst-problem and to a periodic (dynamic-steady)
state in the second-problem
Figure 11 Schematic and the associated parameters for the two example prob-lems (a) thermal conduction module of a heat sink and (b) theclassical ow across a circular cylinder
6 1 Introduction
The rst-problem is taken from the classic book on heat-transfer
by Incropera and Dewitt (1996) shown in Fig 11(a) The g-
ure shows a periodic module of the plate where a constant heat-ux
qW = 10Wcm2 (dissipated from certain electronic devices) is rst con-
ducted to an aluminum plate and then convected to the water ow-
ing through the rectangular channel grooved in the plate The forced
convection results in a convection coecient of h = 5000Wm2K in-
side the channel The unsteady heat-transfer results in an interesting
time-wise varying conduction heat ow in the plate and instantaneous
convection heat transfer rate Qprimeconv (per unit width of the plate per-
pendicular to the plane shown in Fig 11(a)) into the water
The second-problem is encountered if you are standing over a
bridge and look below to the water passing over a pillar (of circu-
lar cross-section) of the bridge shown in Fig 11(b) as a general case
of 2D ow across a cylinder The uid-dynamics results in a beau-
tiful time-wise varying structure of the uid ow just downstream
of the pillar and certain engineering-parameters The parameters
correspond to the uid-dynamic forces acting in the horizontal (or
streamwise) and vertical (or transverse) direction called as the drag-
force FD and lift-force FL respectively The respective force is pre-
sented below in a non-dimensional form called as the drag-coecient
CD and lift-coecient CL equations shown in Fig 11(b)
If you use your video camera to capture a movie for the two
problems you will get series of pictures for the plate in the rst-
problem and the owing uid in the second-problem Since these
movies result in a pictorial information but not the ow-property
based uid-dynamic information such movies are not relevant for
the uid-dynamics and heat-transfer study Instead if you use a
CFD software for computer simulation you can capture various types
of movies The various ow-properties based scientically-exciting
movies are presented here as the pictures at certain time-instant in
Fig 12(a)-(d) and Fig 13(a1)-(d2) The former gure shows gray-
colored ooded contour for the temperature whereas the latter gure
shows gray-colored ooded contour for the pressure and vorticity and
vector-plot for the velocity For the gray-colored ooded contours a
color bar can be seen in the gures for the relative magnitude of the
variable These gures also show line contours of stream-function
Introduction to CFD 7
and heat-function as streamlines and heatlines respectively They
are the visualization techniques for a 2D steady heat and uid ow
The heatlines are analogous to streamlines and represents the direc-
tion of heat ow under steady state condition proposed by Kimura
and Bejan (1983) later presented in a heat-transfer book by Bejan
(1984) The steady-state heatlines can be seen in Fig 12(d) where
the direction for the conduction-ux is tangential at any point on the
heatlines analogous to that for the mass-ux in a streamline Thus
for a better presentation of the structure of heat and uid ow the
scientically-exciting movies are not only limited to ow-properties
but also correspond to certain ow visualization techniquesAn engineering-parameter based engineering-relevant uid dy-
namic movie is presented in Fig 12(e) for the convection heat trans-
fer rate per unit width at the various walls on the channel Qprimew at the
west-wall Qprimen at the north-wall Q
primee at the east-wall and their cumu-
lative value Qprimeconv (refer Fig 11(a)) Another engineering-relevant
movie is presented in Fig 13(e) for the lift and drag coecient Note
that the movie here corresponds to the motion of a lled symbol over
a line for the temporal variation of an engineering-parameter Thus
the engineering-relevant movies are one-dimensional lower than the
scientically-exciting movies here for the 2D problems the former
movie is presented as 1D and the latter as 2D results
1111 Heat-Conduction
For the computational set-up of the heat conduction problem shown
in Fig 11(a) the results obtained from a 2D CFD simulation of
the conduction heat transfer are shown in Fig 12 For a uni-
form initial temperature of the plate as Tinfin = 15oC Fig 12(a)-(d)
shows the isotherms and heatlines in the plate as the pictures of the
scientically-exciting movies at certain discrete time instants t = 25
5 10 and 20 sec For the same discrete time instants the temporal
variation of the position of the symbols over a line plot is presented in
Fig 12(e) The gure is shown for each of the heat transfer rate and
is obtained from a engineering-relevant 1D movie There is a tempo-
ral variation of picture in the scientically-exciting movie and that
of position of symbol (along the line plot) in the engineering-relevant
movieAccording to caloric theory of the famous French scientist An-
tonie Lavoisier heat is considered as an invisible tasteless odorless
8 1 Introduction
t(sec)
Qrsquo (W
m)
0 5 10 15 20 250
200
400
600
800
1000Qrsquo
conv
Qrsquow
Qrsquon
Qrsquoe
Qrsquoin
(e)
(b)
(c)(d)
(a)
(a) t=25 sec31
29
27
25
23
21
19
17
15
T(oC)
(c) t=10 sec
(b) t=5 sec
(d) t=20 sec
Figure 12 Thermal conduction module of a heat sink temporal variation of(a)-(d) isotherms and heatlines and (e) the convection heat transferrate per unit width at the various walls of the channels and theircumulative total value Q
primeconv For the discrete time-instant corre-
sponding to heat ow patterns in (a)-(d) temporal variation of thevarious heat transfer rates are shown as symbols in (e) (a)-(c) arethe transient and (d) represents a steady state result
weightless uid which he called caloric uid Presently the caloric
theory is overtaken by mechanical theory of heat minus heat is not a sub-
stance but a dynamical form of mechanical eect (Thomson 1851)
Nevertheless there are certain problems involving heat ow for which
Lavoisiers approach is rather useful such as the discussion on heat-
lines here
The role of heatlines for heat ow is analogous to that of stream-
lines for uid ow (Paramane and Sharma 2009) For the uid-ow
the dierence of stream-function values represents the rate of uid
ow the function remains constant on a solid wall the tangent to
a streamline represents the direction of the uid-ow (with no ow
in the normal direction) and the streamline originate or emerge at
Introduction to CFD 9
a mass source Analogously for the heat-ow the dierence of heat-
function values represents the rate of heat ow the function remains
constant on an adiabatic wall the tangent to a heatline represents the
direction of the heat-ow (with no ow in the normal direction) and
the heatline originate or emerge at the heat-source For the present
problem in Fig 11(a) the heat-source is at the inlet (or left) bound-
ary and the heat-sink is at the surface of the channel Thus the
steady-state heatlines in Fig 12(d) shows the heat ow emerging
from the inletsource and ending on the channel-wallssink Further-
more it can be seen in the gure that the heatlines which are close
to the adiabatic walls are parallel to the walls The source sink and
adiabatic-walls can be seen in Fig 11(a)
Figure 12(e) shows an asymptotic time-wise increase in the con-
vection heat transfer per unit width at the various walls of the channel
and its cumulative value Qprimeconv (= Q
primew+Q
primen+Q
primee) It can be also be seen
that the heat-transfer per unit width Qprimew at the west-wall and Q
primen at
the north-wall are larger than Qprimee at the east-wall of the channel As
time progresses the gure shows that Qprimeconv increases monotonically
and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-
der steady state condition refer Fig 11(a) The dierence between
the Qprimein and Q
primeconv under the transient condition is utilized for the
sensible heating and results in the increase in the temperature of the
plate Under the steady state condition note from Fig 12(d) that
the maximum temperature is close to 32oC minus well within the per-
missible limit for the reliable and ecient operation of the electronic
device
1112 Fluid-Dynamics
For the free-stream ow across a circular cylinderpillar at a Reynolds
number of 100 the non-dimensional results obtained from a 2D CFD
simulation are shown in Fig 13 In contrast to the previous prob-
lem which reaches steady state the present problem reaches to an un-
steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-
ent increasing time instants within one time-period of the periodic-
ow the results are presented in Fig 13(a)-(d) The gure shows
the close-up view of the pictures of the scientically-exciting movies
for ow-properties (velocity pressure and vorticity) and streamlines
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow
xii Preface
throughout this book I am grateful to IIT Bombay for giving me
one year sabbatical leave to write this book The ambiance of aca-
demic freedom considerate support by the faculties and the friendly
atmosphere inside the campus of IIT Bombay has greatly contributed
in the solo eort of book writing I thanks all my colleagues at IIT
Bombay for shaping my perceptions on CFD
During this long endeavor the patient support by Mr Sunil
Saxena Director Athena Academic Ltd UK is gratefully acknowl-
edged I also thank John Wiley for coming forward to take this
book to international markets Lastly I would acknowledge that I
am greatly inspired by the writing style of J D Anderson and the
book on CFD by S V Patankar Of course you will nd that I
have heavily cross-referred the CFD book by the two most acknowl-
edged pioneers on CFD I dedicate this book to Dr S V Patankar
whose book nicely introduced me to the computational as well as
ow physics in CFD and greatly helped me to introduce a physical
approach of CFD development in this book
The culmination of the book writing process fullls a long cher-
ished dream It would not have been possible without the excel-
lent environment continuous support and complete understanding
of my wife Anubha and daughter Anshita Their kindness has greatly
helped to fulll my dream now I plan to spend more time with my
wife and daughter
Mumbai
August 2016 Atul Sharma
Contents
I Introduction and Essentials
1
1 Introduction 3
11 CFD What is it 3
111 CFD as a Scientic and Engineering Analysis
Tool 5
112 Analogy with a Video-Camera 13
12 CFD Why to study 15
13 Novelty Scope and Purpose of this Book 16
2 Introduction to CFD Development Application and
Analysis 23
21 CFD Development 23
211 Grid Generation Pre-Processor 24
212 Discretization Method Algebraic Formulation 26
213 Solution Methodology Solver 29
214 Computation of Engineering-Parameters
Post-Processor 34
215 Testing 35
22 CFD Application 35
23 CFD Analysis 38
24 Closure 39
xiii
xiv Contents
3 Essentials of Fluid-Dynamics and Heat-Transfer for
CFD 41
31 Physical Laws 42
311 FundamentalConservation Laws 43
312 Subsidiary Laws 45
32 Momentum and Energy Transport Mechanisms 46
33 Physical Law based Dierential Formulation 48
331 Continuity Equation 49
332 Transport Equation 51
34 Generalized Volumetric and Flux Terms and their Dif-
ferential Formulation 57
341 Volumetric Term 58
342 Flux-Term 58
343 Discussion 62
35 Mathematical Formulation 63
351 Dimensional Study 63
352 Non-Dimensional Study 67
36 Closure 71
4 Essentials of Numerical-Methods for CFD 73
41 Finite Dierence Method A Dierential to Algebraic
Formulation for Governing PDE and BCs 75
411 Grid Generation 75
412 Finite Dierence Method 78
413 Applications to CFD 92
42 Iterative Solution of System of LAEs for a Flow Property 93
421 Iterative Methods 94
422 Applications to CFD 98
43 Numerical Dierentiation for Local Engineering-
Parameters 105
431 Dierentiation Formulas 106
432 Applications to CFD 107
44 Numerical Integration for the Total value of
Engineering-Parameters 110
441 Integration Rules 111
442 Applications to CFD 114
45 Closure 116
Problems 116
Introduction to CFD xv
II CFD for a Cartesian-Geometry
119
5 Computational Heat Conduction 12151 Physical Law based Finite Volume Method 122
511 Energy Conservation Law for a Control Volume 123512 Algebraic Formulation 124513 Approximations 127514 Approximated Algebraic-Formulation 130515 Discussion 133
52 Finite Dierence Method for Boundary Conditions 13853 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 139531 One-Dimensional Conduction 139532 Two-Dimensional Conduction 147
54 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 154541 One-Dimensional Conduction 154542 Two-Dimensional Conduction 166
Problems 176
6 Computational Heat Advection 17961 Physical Law based Finite Volume Method 180
611 Energy Conservation Law for a Control Volume 181612 Algebraic Formulation 181613 Approximations 183614 Approximated Algebraic Formulation 191615 Discussion 191
62 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 194621 Explicit-Method 197622 Implementation Details 198623 Solution Algorithm 199
63 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 207631 Advection Scheme on a Non-Uniform Grid 207632 Explicit and Implicit Method 210633 Implementation Details 216634 Solution Algorithm 220
Problems 228
xvi Contents
7 Computational Heat Convection 22971 Physical Law based Finite Volume Method 229
711 Energy Conservation Law for a Control Volume 229712 Algebraic Formulation 231713 Approximated Algebraic Formulation 232
72 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 235721 Explicit-Method 236722 Implementation Details 237723 Solution Algorithm 238
73 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 242Problems 248
8 Computational Fluid Dynamics Physical Law based
Finite Volume Method 25181 Generalized Variables for the Combined Heat and
Fluid Flow 25282 Conservation Laws for a Control Volume 25583 Algebraic Formulation 25984 Approximations 26085 Approximated Algebraic Formulation 263
851 Mass Conservation 263852 MomentumEnergy Conservation 264
86 Closure 269
9 Computational Fluid Dynamics on a Staggered Grid 27191 Challenges in the CFD Development 273
911 Non-Linearity 273912 Equation for Pressure 273913 Pressure-Velocity Decoupling 273
92 A Staggered Grid to avoid Pressure-Velocity Decoupling27593 Physical Law based FVM for a Staggered Grid 27794 Flux based Solution Methodology on a Uniform Grid
Semi-Explicit Method 281941 Philosophy of Pressure-Correction Method 283942 Semi-Explicit Method 286943 Implementation Details 292944 Solution Algorithm 298
95 Initial and Boundary Conditions 300951 Initial Condition 300952 Boundary Condition 301
Problems 306
Introduction to CFD xvii
10 Computational Fluid Dynamics on a Co-located Grid309
101 Momentum-Interpolation Method Strategy to avoid
the Pressure-Velocity Decoupling on a Co-located Grid 310
102 Coecients of LAEs based Solution Methodology on
a Non-Uniform Grid Semi-Explicit and Semi-Implicit
Method 314
1021 Predictor Step 316
1022 Corrector Step 318
1023 Solution Algorithm 323
Problems 329
III CFD for a Complex-Geometry
331
11 Computational Heat Conduction on a Curvilinear
Grid 333
111 Curvilinear Grid Generation 333
1111 Algebraic Grid Generation 334
1112 Elliptic Grid Generation 336
112 Physical Law based Finite Volume Method 343
1121 Unsteady and Source Term 343
1122 Diusion Term 344
1123 All Terms 349
113 Computation of Geometrical Properties 349
114 Flux based Solution Methodology 352
1141 Explicit Method 354
1142 Implementation Details 354
Problems 359
12 Computational Fluid Dynamics on a Curvilinear Grid361
121 Physical Law based Finite Volume Method 361
1211 Mass Conservation 363
1212 Momentum Conservation 363
122 Solution Methodology Semi-Explicit Method 372
1221 Predictor Step 373
1222 Corrector Step 375
Problems 380
References 383
Index 389
Part I
Introduction and Essentials
The book on an introductory course in CFD starts with the two intro-
ductory chapters on CFD followed by one chapter each on essentials
of two prerequisite courses minus uid-dynamics and heat transfer and
numerical-methods The rst chapter on introduction presents an
eventful thoughtful and intuitive discussion on What is CFD and
Why to study CFD whereas How CFD works which involves lots
of details is presented separately in the second chapter The third-
chapter on essentials of uid-dynamics and heat transfer presents
physical-laws transport-mechanisms physical-law based dierential-
formulation volumetric and ux terms and mathematical formula-
tion The fourth-chapter on essentials of numerical-methods presents
nite dierence method iterative solution of the system of linear alge-
braic equation numerical dierentiation and numerical integration
The presentation for the essential of the two prerequisite courses is
customized to CFD development application and analysis
1
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
1
Introduction
The introduction of any course consists of the three basic questions
What is it Why to study How it works An eventful thoughtful
and intuitive answer to the rst two questions are presented in this
chapter The third question which involves lots of working details
on the three aspects of the Computational Fluid Dynamics (CFD)
(development application and analysis) is presented separately in
the next chapter The novelty scope and purpose of this book is
also presented at the end of this chapter
11 CFD What is it
CFD is a theoretical-method of scientic and engineering investiga-
tion concerned with the development and application of a video-
camera like tool (a software) which is used for a unied cause-and-
eect based analysis of a uid-dynamics as well as heat and mass
transfer problem
CFD is a subject where you rst learn how to develop a product
minus a software which acts like a virtual video-camera Then you learn
how to apply or use this product to generate uid-dynamics movies
Finally you learn how to analyze the detailed spatial and temporal
uid-dynamics information in the movies The analysis is done to
come up with a scientically-exciting as well as engineering-relevant
story (unied cause-and-eect study) of a uid-dynamics situation in
nature as well as industrial-applications With a continuous develop-
ment and a wider application of CFD the word uid-dynamics in
3
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
4 1 Introduction
the CFD has become more generic as it also corresponds to heat and
mass transfer as well as chemical reaction
Fluid-dynamics information are of two types scientic and en-
gineering Scientic-information corresponds to a structure of the
heat and uid ow due to a physical phenomenon in uid-dynamics
such as boundary-layer ow-separation wake-formation and vortex-
shedding The ow structures are obtained from a temporal variation
of ow-properties (such as velocity pressure temperature and many
other variables which characterizes a ow) called as scientically-
exciting movies Engineering-information corresponds to certain pa-
rameters which are important in an engineering application called
as engineering-parameters for example lift force drag force wall
shear stress pressure drop and rate of heat-transfer They are ob-
tained as the temporal variation of engineering-parameters called as
engineering-relevant movies The engineering-parameters are the ef-
fect which are caused by the ow structures minus their correlation leads
to the unied cause-and-eect study A uid-dynamics movie consist
of a time-wise varying series of pictures which give uid-dynamics
information Each ow-property and engineering-parameter results
in one scientically-exciting and engineering-relevant movie respec-
tively When the large number of both the types of movies are
played synchronized in-time the detailed spatial as well as tempo-
ral uid-dynamic information greatly helps in the unied cause-and-
eect study of a uid-dynamics problem
Alternatively CFD is a subject concerned with the development of
computer-programs for the computer-simulation and study of a natu-
ral or an engineering uid-dynamics system The development starts
with the virtual-development of the system involving a geometri-
cal information based computational development of a solid-model
for the system the system may be already-existing or yet-to-be-
developed physically Thereafter the development involves numerical
solution of the governing algebraic equations for CFD formulated
from the physical laws (conservation of mass momentum and en-
ergy) and initial as well as boundary conditions corresponding to the
system CFD study involves the analysis of the uid-dynamics and
heat-transfer results obtained from the computer-simulations of the
virtual-system which is subjected to certain governing-parameters
Introduction to CFD 5
111 CFD as a Scientic and Engineering Analysis Tool
For a better understanding of the denition of the CFD and its ap-
plication as a scientic and engineering analysis tool two example
problems are presented rst conduction heat transfer in a plate
and second uid ow across a circular pillar of a bridge The rst-
problem correspond to an engineering-application for electronic cool-
ing and the second-problem is studied for an engineering-design of
the structure subjected to uid-dynamics forces and are shown in
Fig 11 Both the problems are considered as two-dimensional and
unsteady However the results asymptotes from the unsteady to a
steady state in the rst-problem and to a periodic (dynamic-steady)
state in the second-problem
Figure 11 Schematic and the associated parameters for the two example prob-lems (a) thermal conduction module of a heat sink and (b) theclassical ow across a circular cylinder
6 1 Introduction
The rst-problem is taken from the classic book on heat-transfer
by Incropera and Dewitt (1996) shown in Fig 11(a) The g-
ure shows a periodic module of the plate where a constant heat-ux
qW = 10Wcm2 (dissipated from certain electronic devices) is rst con-
ducted to an aluminum plate and then convected to the water ow-
ing through the rectangular channel grooved in the plate The forced
convection results in a convection coecient of h = 5000Wm2K in-
side the channel The unsteady heat-transfer results in an interesting
time-wise varying conduction heat ow in the plate and instantaneous
convection heat transfer rate Qprimeconv (per unit width of the plate per-
pendicular to the plane shown in Fig 11(a)) into the water
The second-problem is encountered if you are standing over a
bridge and look below to the water passing over a pillar (of circu-
lar cross-section) of the bridge shown in Fig 11(b) as a general case
of 2D ow across a cylinder The uid-dynamics results in a beau-
tiful time-wise varying structure of the uid ow just downstream
of the pillar and certain engineering-parameters The parameters
correspond to the uid-dynamic forces acting in the horizontal (or
streamwise) and vertical (or transverse) direction called as the drag-
force FD and lift-force FL respectively The respective force is pre-
sented below in a non-dimensional form called as the drag-coecient
CD and lift-coecient CL equations shown in Fig 11(b)
If you use your video camera to capture a movie for the two
problems you will get series of pictures for the plate in the rst-
problem and the owing uid in the second-problem Since these
movies result in a pictorial information but not the ow-property
based uid-dynamic information such movies are not relevant for
the uid-dynamics and heat-transfer study Instead if you use a
CFD software for computer simulation you can capture various types
of movies The various ow-properties based scientically-exciting
movies are presented here as the pictures at certain time-instant in
Fig 12(a)-(d) and Fig 13(a1)-(d2) The former gure shows gray-
colored ooded contour for the temperature whereas the latter gure
shows gray-colored ooded contour for the pressure and vorticity and
vector-plot for the velocity For the gray-colored ooded contours a
color bar can be seen in the gures for the relative magnitude of the
variable These gures also show line contours of stream-function
Introduction to CFD 7
and heat-function as streamlines and heatlines respectively They
are the visualization techniques for a 2D steady heat and uid ow
The heatlines are analogous to streamlines and represents the direc-
tion of heat ow under steady state condition proposed by Kimura
and Bejan (1983) later presented in a heat-transfer book by Bejan
(1984) The steady-state heatlines can be seen in Fig 12(d) where
the direction for the conduction-ux is tangential at any point on the
heatlines analogous to that for the mass-ux in a streamline Thus
for a better presentation of the structure of heat and uid ow the
scientically-exciting movies are not only limited to ow-properties
but also correspond to certain ow visualization techniquesAn engineering-parameter based engineering-relevant uid dy-
namic movie is presented in Fig 12(e) for the convection heat trans-
fer rate per unit width at the various walls on the channel Qprimew at the
west-wall Qprimen at the north-wall Q
primee at the east-wall and their cumu-
lative value Qprimeconv (refer Fig 11(a)) Another engineering-relevant
movie is presented in Fig 13(e) for the lift and drag coecient Note
that the movie here corresponds to the motion of a lled symbol over
a line for the temporal variation of an engineering-parameter Thus
the engineering-relevant movies are one-dimensional lower than the
scientically-exciting movies here for the 2D problems the former
movie is presented as 1D and the latter as 2D results
1111 Heat-Conduction
For the computational set-up of the heat conduction problem shown
in Fig 11(a) the results obtained from a 2D CFD simulation of
the conduction heat transfer are shown in Fig 12 For a uni-
form initial temperature of the plate as Tinfin = 15oC Fig 12(a)-(d)
shows the isotherms and heatlines in the plate as the pictures of the
scientically-exciting movies at certain discrete time instants t = 25
5 10 and 20 sec For the same discrete time instants the temporal
variation of the position of the symbols over a line plot is presented in
Fig 12(e) The gure is shown for each of the heat transfer rate and
is obtained from a engineering-relevant 1D movie There is a tempo-
ral variation of picture in the scientically-exciting movie and that
of position of symbol (along the line plot) in the engineering-relevant
movieAccording to caloric theory of the famous French scientist An-
tonie Lavoisier heat is considered as an invisible tasteless odorless
8 1 Introduction
t(sec)
Qrsquo (W
m)
0 5 10 15 20 250
200
400
600
800
1000Qrsquo
conv
Qrsquow
Qrsquon
Qrsquoe
Qrsquoin
(e)
(b)
(c)(d)
(a)
(a) t=25 sec31
29
27
25
23
21
19
17
15
T(oC)
(c) t=10 sec
(b) t=5 sec
(d) t=20 sec
Figure 12 Thermal conduction module of a heat sink temporal variation of(a)-(d) isotherms and heatlines and (e) the convection heat transferrate per unit width at the various walls of the channels and theircumulative total value Q
primeconv For the discrete time-instant corre-
sponding to heat ow patterns in (a)-(d) temporal variation of thevarious heat transfer rates are shown as symbols in (e) (a)-(c) arethe transient and (d) represents a steady state result
weightless uid which he called caloric uid Presently the caloric
theory is overtaken by mechanical theory of heat minus heat is not a sub-
stance but a dynamical form of mechanical eect (Thomson 1851)
Nevertheless there are certain problems involving heat ow for which
Lavoisiers approach is rather useful such as the discussion on heat-
lines here
The role of heatlines for heat ow is analogous to that of stream-
lines for uid ow (Paramane and Sharma 2009) For the uid-ow
the dierence of stream-function values represents the rate of uid
ow the function remains constant on a solid wall the tangent to
a streamline represents the direction of the uid-ow (with no ow
in the normal direction) and the streamline originate or emerge at
Introduction to CFD 9
a mass source Analogously for the heat-ow the dierence of heat-
function values represents the rate of heat ow the function remains
constant on an adiabatic wall the tangent to a heatline represents the
direction of the heat-ow (with no ow in the normal direction) and
the heatline originate or emerge at the heat-source For the present
problem in Fig 11(a) the heat-source is at the inlet (or left) bound-
ary and the heat-sink is at the surface of the channel Thus the
steady-state heatlines in Fig 12(d) shows the heat ow emerging
from the inletsource and ending on the channel-wallssink Further-
more it can be seen in the gure that the heatlines which are close
to the adiabatic walls are parallel to the walls The source sink and
adiabatic-walls can be seen in Fig 11(a)
Figure 12(e) shows an asymptotic time-wise increase in the con-
vection heat transfer per unit width at the various walls of the channel
and its cumulative value Qprimeconv (= Q
primew+Q
primen+Q
primee) It can be also be seen
that the heat-transfer per unit width Qprimew at the west-wall and Q
primen at
the north-wall are larger than Qprimee at the east-wall of the channel As
time progresses the gure shows that Qprimeconv increases monotonically
and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-
der steady state condition refer Fig 11(a) The dierence between
the Qprimein and Q
primeconv under the transient condition is utilized for the
sensible heating and results in the increase in the temperature of the
plate Under the steady state condition note from Fig 12(d) that
the maximum temperature is close to 32oC minus well within the per-
missible limit for the reliable and ecient operation of the electronic
device
1112 Fluid-Dynamics
For the free-stream ow across a circular cylinderpillar at a Reynolds
number of 100 the non-dimensional results obtained from a 2D CFD
simulation are shown in Fig 13 In contrast to the previous prob-
lem which reaches steady state the present problem reaches to an un-
steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-
ent increasing time instants within one time-period of the periodic-
ow the results are presented in Fig 13(a)-(d) The gure shows
the close-up view of the pictures of the scientically-exciting movies
for ow-properties (velocity pressure and vorticity) and streamlines
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow
Contents
I Introduction and Essentials
1
1 Introduction 3
11 CFD What is it 3
111 CFD as a Scientic and Engineering Analysis
Tool 5
112 Analogy with a Video-Camera 13
12 CFD Why to study 15
13 Novelty Scope and Purpose of this Book 16
2 Introduction to CFD Development Application and
Analysis 23
21 CFD Development 23
211 Grid Generation Pre-Processor 24
212 Discretization Method Algebraic Formulation 26
213 Solution Methodology Solver 29
214 Computation of Engineering-Parameters
Post-Processor 34
215 Testing 35
22 CFD Application 35
23 CFD Analysis 38
24 Closure 39
xiii
xiv Contents
3 Essentials of Fluid-Dynamics and Heat-Transfer for
CFD 41
31 Physical Laws 42
311 FundamentalConservation Laws 43
312 Subsidiary Laws 45
32 Momentum and Energy Transport Mechanisms 46
33 Physical Law based Dierential Formulation 48
331 Continuity Equation 49
332 Transport Equation 51
34 Generalized Volumetric and Flux Terms and their Dif-
ferential Formulation 57
341 Volumetric Term 58
342 Flux-Term 58
343 Discussion 62
35 Mathematical Formulation 63
351 Dimensional Study 63
352 Non-Dimensional Study 67
36 Closure 71
4 Essentials of Numerical-Methods for CFD 73
41 Finite Dierence Method A Dierential to Algebraic
Formulation for Governing PDE and BCs 75
411 Grid Generation 75
412 Finite Dierence Method 78
413 Applications to CFD 92
42 Iterative Solution of System of LAEs for a Flow Property 93
421 Iterative Methods 94
422 Applications to CFD 98
43 Numerical Dierentiation for Local Engineering-
Parameters 105
431 Dierentiation Formulas 106
432 Applications to CFD 107
44 Numerical Integration for the Total value of
Engineering-Parameters 110
441 Integration Rules 111
442 Applications to CFD 114
45 Closure 116
Problems 116
Introduction to CFD xv
II CFD for a Cartesian-Geometry
119
5 Computational Heat Conduction 12151 Physical Law based Finite Volume Method 122
511 Energy Conservation Law for a Control Volume 123512 Algebraic Formulation 124513 Approximations 127514 Approximated Algebraic-Formulation 130515 Discussion 133
52 Finite Dierence Method for Boundary Conditions 13853 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 139531 One-Dimensional Conduction 139532 Two-Dimensional Conduction 147
54 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 154541 One-Dimensional Conduction 154542 Two-Dimensional Conduction 166
Problems 176
6 Computational Heat Advection 17961 Physical Law based Finite Volume Method 180
611 Energy Conservation Law for a Control Volume 181612 Algebraic Formulation 181613 Approximations 183614 Approximated Algebraic Formulation 191615 Discussion 191
62 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 194621 Explicit-Method 197622 Implementation Details 198623 Solution Algorithm 199
63 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 207631 Advection Scheme on a Non-Uniform Grid 207632 Explicit and Implicit Method 210633 Implementation Details 216634 Solution Algorithm 220
Problems 228
xvi Contents
7 Computational Heat Convection 22971 Physical Law based Finite Volume Method 229
711 Energy Conservation Law for a Control Volume 229712 Algebraic Formulation 231713 Approximated Algebraic Formulation 232
72 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 235721 Explicit-Method 236722 Implementation Details 237723 Solution Algorithm 238
73 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 242Problems 248
8 Computational Fluid Dynamics Physical Law based
Finite Volume Method 25181 Generalized Variables for the Combined Heat and
Fluid Flow 25282 Conservation Laws for a Control Volume 25583 Algebraic Formulation 25984 Approximations 26085 Approximated Algebraic Formulation 263
851 Mass Conservation 263852 MomentumEnergy Conservation 264
86 Closure 269
9 Computational Fluid Dynamics on a Staggered Grid 27191 Challenges in the CFD Development 273
911 Non-Linearity 273912 Equation for Pressure 273913 Pressure-Velocity Decoupling 273
92 A Staggered Grid to avoid Pressure-Velocity Decoupling27593 Physical Law based FVM for a Staggered Grid 27794 Flux based Solution Methodology on a Uniform Grid
Semi-Explicit Method 281941 Philosophy of Pressure-Correction Method 283942 Semi-Explicit Method 286943 Implementation Details 292944 Solution Algorithm 298
95 Initial and Boundary Conditions 300951 Initial Condition 300952 Boundary Condition 301
Problems 306
Introduction to CFD xvii
10 Computational Fluid Dynamics on a Co-located Grid309
101 Momentum-Interpolation Method Strategy to avoid
the Pressure-Velocity Decoupling on a Co-located Grid 310
102 Coecients of LAEs based Solution Methodology on
a Non-Uniform Grid Semi-Explicit and Semi-Implicit
Method 314
1021 Predictor Step 316
1022 Corrector Step 318
1023 Solution Algorithm 323
Problems 329
III CFD for a Complex-Geometry
331
11 Computational Heat Conduction on a Curvilinear
Grid 333
111 Curvilinear Grid Generation 333
1111 Algebraic Grid Generation 334
1112 Elliptic Grid Generation 336
112 Physical Law based Finite Volume Method 343
1121 Unsteady and Source Term 343
1122 Diusion Term 344
1123 All Terms 349
113 Computation of Geometrical Properties 349
114 Flux based Solution Methodology 352
1141 Explicit Method 354
1142 Implementation Details 354
Problems 359
12 Computational Fluid Dynamics on a Curvilinear Grid361
121 Physical Law based Finite Volume Method 361
1211 Mass Conservation 363
1212 Momentum Conservation 363
122 Solution Methodology Semi-Explicit Method 372
1221 Predictor Step 373
1222 Corrector Step 375
Problems 380
References 383
Index 389
Part I
Introduction and Essentials
The book on an introductory course in CFD starts with the two intro-
ductory chapters on CFD followed by one chapter each on essentials
of two prerequisite courses minus uid-dynamics and heat transfer and
numerical-methods The rst chapter on introduction presents an
eventful thoughtful and intuitive discussion on What is CFD and
Why to study CFD whereas How CFD works which involves lots
of details is presented separately in the second chapter The third-
chapter on essentials of uid-dynamics and heat transfer presents
physical-laws transport-mechanisms physical-law based dierential-
formulation volumetric and ux terms and mathematical formula-
tion The fourth-chapter on essentials of numerical-methods presents
nite dierence method iterative solution of the system of linear alge-
braic equation numerical dierentiation and numerical integration
The presentation for the essential of the two prerequisite courses is
customized to CFD development application and analysis
1
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
1
Introduction
The introduction of any course consists of the three basic questions
What is it Why to study How it works An eventful thoughtful
and intuitive answer to the rst two questions are presented in this
chapter The third question which involves lots of working details
on the three aspects of the Computational Fluid Dynamics (CFD)
(development application and analysis) is presented separately in
the next chapter The novelty scope and purpose of this book is
also presented at the end of this chapter
11 CFD What is it
CFD is a theoretical-method of scientic and engineering investiga-
tion concerned with the development and application of a video-
camera like tool (a software) which is used for a unied cause-and-
eect based analysis of a uid-dynamics as well as heat and mass
transfer problem
CFD is a subject where you rst learn how to develop a product
minus a software which acts like a virtual video-camera Then you learn
how to apply or use this product to generate uid-dynamics movies
Finally you learn how to analyze the detailed spatial and temporal
uid-dynamics information in the movies The analysis is done to
come up with a scientically-exciting as well as engineering-relevant
story (unied cause-and-eect study) of a uid-dynamics situation in
nature as well as industrial-applications With a continuous develop-
ment and a wider application of CFD the word uid-dynamics in
3
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
4 1 Introduction
the CFD has become more generic as it also corresponds to heat and
mass transfer as well as chemical reaction
Fluid-dynamics information are of two types scientic and en-
gineering Scientic-information corresponds to a structure of the
heat and uid ow due to a physical phenomenon in uid-dynamics
such as boundary-layer ow-separation wake-formation and vortex-
shedding The ow structures are obtained from a temporal variation
of ow-properties (such as velocity pressure temperature and many
other variables which characterizes a ow) called as scientically-
exciting movies Engineering-information corresponds to certain pa-
rameters which are important in an engineering application called
as engineering-parameters for example lift force drag force wall
shear stress pressure drop and rate of heat-transfer They are ob-
tained as the temporal variation of engineering-parameters called as
engineering-relevant movies The engineering-parameters are the ef-
fect which are caused by the ow structures minus their correlation leads
to the unied cause-and-eect study A uid-dynamics movie consist
of a time-wise varying series of pictures which give uid-dynamics
information Each ow-property and engineering-parameter results
in one scientically-exciting and engineering-relevant movie respec-
tively When the large number of both the types of movies are
played synchronized in-time the detailed spatial as well as tempo-
ral uid-dynamic information greatly helps in the unied cause-and-
eect study of a uid-dynamics problem
Alternatively CFD is a subject concerned with the development of
computer-programs for the computer-simulation and study of a natu-
ral or an engineering uid-dynamics system The development starts
with the virtual-development of the system involving a geometri-
cal information based computational development of a solid-model
for the system the system may be already-existing or yet-to-be-
developed physically Thereafter the development involves numerical
solution of the governing algebraic equations for CFD formulated
from the physical laws (conservation of mass momentum and en-
ergy) and initial as well as boundary conditions corresponding to the
system CFD study involves the analysis of the uid-dynamics and
heat-transfer results obtained from the computer-simulations of the
virtual-system which is subjected to certain governing-parameters
Introduction to CFD 5
111 CFD as a Scientic and Engineering Analysis Tool
For a better understanding of the denition of the CFD and its ap-
plication as a scientic and engineering analysis tool two example
problems are presented rst conduction heat transfer in a plate
and second uid ow across a circular pillar of a bridge The rst-
problem correspond to an engineering-application for electronic cool-
ing and the second-problem is studied for an engineering-design of
the structure subjected to uid-dynamics forces and are shown in
Fig 11 Both the problems are considered as two-dimensional and
unsteady However the results asymptotes from the unsteady to a
steady state in the rst-problem and to a periodic (dynamic-steady)
state in the second-problem
Figure 11 Schematic and the associated parameters for the two example prob-lems (a) thermal conduction module of a heat sink and (b) theclassical ow across a circular cylinder
6 1 Introduction
The rst-problem is taken from the classic book on heat-transfer
by Incropera and Dewitt (1996) shown in Fig 11(a) The g-
ure shows a periodic module of the plate where a constant heat-ux
qW = 10Wcm2 (dissipated from certain electronic devices) is rst con-
ducted to an aluminum plate and then convected to the water ow-
ing through the rectangular channel grooved in the plate The forced
convection results in a convection coecient of h = 5000Wm2K in-
side the channel The unsteady heat-transfer results in an interesting
time-wise varying conduction heat ow in the plate and instantaneous
convection heat transfer rate Qprimeconv (per unit width of the plate per-
pendicular to the plane shown in Fig 11(a)) into the water
The second-problem is encountered if you are standing over a
bridge and look below to the water passing over a pillar (of circu-
lar cross-section) of the bridge shown in Fig 11(b) as a general case
of 2D ow across a cylinder The uid-dynamics results in a beau-
tiful time-wise varying structure of the uid ow just downstream
of the pillar and certain engineering-parameters The parameters
correspond to the uid-dynamic forces acting in the horizontal (or
streamwise) and vertical (or transverse) direction called as the drag-
force FD and lift-force FL respectively The respective force is pre-
sented below in a non-dimensional form called as the drag-coecient
CD and lift-coecient CL equations shown in Fig 11(b)
If you use your video camera to capture a movie for the two
problems you will get series of pictures for the plate in the rst-
problem and the owing uid in the second-problem Since these
movies result in a pictorial information but not the ow-property
based uid-dynamic information such movies are not relevant for
the uid-dynamics and heat-transfer study Instead if you use a
CFD software for computer simulation you can capture various types
of movies The various ow-properties based scientically-exciting
movies are presented here as the pictures at certain time-instant in
Fig 12(a)-(d) and Fig 13(a1)-(d2) The former gure shows gray-
colored ooded contour for the temperature whereas the latter gure
shows gray-colored ooded contour for the pressure and vorticity and
vector-plot for the velocity For the gray-colored ooded contours a
color bar can be seen in the gures for the relative magnitude of the
variable These gures also show line contours of stream-function
Introduction to CFD 7
and heat-function as streamlines and heatlines respectively They
are the visualization techniques for a 2D steady heat and uid ow
The heatlines are analogous to streamlines and represents the direc-
tion of heat ow under steady state condition proposed by Kimura
and Bejan (1983) later presented in a heat-transfer book by Bejan
(1984) The steady-state heatlines can be seen in Fig 12(d) where
the direction for the conduction-ux is tangential at any point on the
heatlines analogous to that for the mass-ux in a streamline Thus
for a better presentation of the structure of heat and uid ow the
scientically-exciting movies are not only limited to ow-properties
but also correspond to certain ow visualization techniquesAn engineering-parameter based engineering-relevant uid dy-
namic movie is presented in Fig 12(e) for the convection heat trans-
fer rate per unit width at the various walls on the channel Qprimew at the
west-wall Qprimen at the north-wall Q
primee at the east-wall and their cumu-
lative value Qprimeconv (refer Fig 11(a)) Another engineering-relevant
movie is presented in Fig 13(e) for the lift and drag coecient Note
that the movie here corresponds to the motion of a lled symbol over
a line for the temporal variation of an engineering-parameter Thus
the engineering-relevant movies are one-dimensional lower than the
scientically-exciting movies here for the 2D problems the former
movie is presented as 1D and the latter as 2D results
1111 Heat-Conduction
For the computational set-up of the heat conduction problem shown
in Fig 11(a) the results obtained from a 2D CFD simulation of
the conduction heat transfer are shown in Fig 12 For a uni-
form initial temperature of the plate as Tinfin = 15oC Fig 12(a)-(d)
shows the isotherms and heatlines in the plate as the pictures of the
scientically-exciting movies at certain discrete time instants t = 25
5 10 and 20 sec For the same discrete time instants the temporal
variation of the position of the symbols over a line plot is presented in
Fig 12(e) The gure is shown for each of the heat transfer rate and
is obtained from a engineering-relevant 1D movie There is a tempo-
ral variation of picture in the scientically-exciting movie and that
of position of symbol (along the line plot) in the engineering-relevant
movieAccording to caloric theory of the famous French scientist An-
tonie Lavoisier heat is considered as an invisible tasteless odorless
8 1 Introduction
t(sec)
Qrsquo (W
m)
0 5 10 15 20 250
200
400
600
800
1000Qrsquo
conv
Qrsquow
Qrsquon
Qrsquoe
Qrsquoin
(e)
(b)
(c)(d)
(a)
(a) t=25 sec31
29
27
25
23
21
19
17
15
T(oC)
(c) t=10 sec
(b) t=5 sec
(d) t=20 sec
Figure 12 Thermal conduction module of a heat sink temporal variation of(a)-(d) isotherms and heatlines and (e) the convection heat transferrate per unit width at the various walls of the channels and theircumulative total value Q
primeconv For the discrete time-instant corre-
sponding to heat ow patterns in (a)-(d) temporal variation of thevarious heat transfer rates are shown as symbols in (e) (a)-(c) arethe transient and (d) represents a steady state result
weightless uid which he called caloric uid Presently the caloric
theory is overtaken by mechanical theory of heat minus heat is not a sub-
stance but a dynamical form of mechanical eect (Thomson 1851)
Nevertheless there are certain problems involving heat ow for which
Lavoisiers approach is rather useful such as the discussion on heat-
lines here
The role of heatlines for heat ow is analogous to that of stream-
lines for uid ow (Paramane and Sharma 2009) For the uid-ow
the dierence of stream-function values represents the rate of uid
ow the function remains constant on a solid wall the tangent to
a streamline represents the direction of the uid-ow (with no ow
in the normal direction) and the streamline originate or emerge at
Introduction to CFD 9
a mass source Analogously for the heat-ow the dierence of heat-
function values represents the rate of heat ow the function remains
constant on an adiabatic wall the tangent to a heatline represents the
direction of the heat-ow (with no ow in the normal direction) and
the heatline originate or emerge at the heat-source For the present
problem in Fig 11(a) the heat-source is at the inlet (or left) bound-
ary and the heat-sink is at the surface of the channel Thus the
steady-state heatlines in Fig 12(d) shows the heat ow emerging
from the inletsource and ending on the channel-wallssink Further-
more it can be seen in the gure that the heatlines which are close
to the adiabatic walls are parallel to the walls The source sink and
adiabatic-walls can be seen in Fig 11(a)
Figure 12(e) shows an asymptotic time-wise increase in the con-
vection heat transfer per unit width at the various walls of the channel
and its cumulative value Qprimeconv (= Q
primew+Q
primen+Q
primee) It can be also be seen
that the heat-transfer per unit width Qprimew at the west-wall and Q
primen at
the north-wall are larger than Qprimee at the east-wall of the channel As
time progresses the gure shows that Qprimeconv increases monotonically
and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-
der steady state condition refer Fig 11(a) The dierence between
the Qprimein and Q
primeconv under the transient condition is utilized for the
sensible heating and results in the increase in the temperature of the
plate Under the steady state condition note from Fig 12(d) that
the maximum temperature is close to 32oC minus well within the per-
missible limit for the reliable and ecient operation of the electronic
device
1112 Fluid-Dynamics
For the free-stream ow across a circular cylinderpillar at a Reynolds
number of 100 the non-dimensional results obtained from a 2D CFD
simulation are shown in Fig 13 In contrast to the previous prob-
lem which reaches steady state the present problem reaches to an un-
steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-
ent increasing time instants within one time-period of the periodic-
ow the results are presented in Fig 13(a)-(d) The gure shows
the close-up view of the pictures of the scientically-exciting movies
for ow-properties (velocity pressure and vorticity) and streamlines
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow
xiv Contents
3 Essentials of Fluid-Dynamics and Heat-Transfer for
CFD 41
31 Physical Laws 42
311 FundamentalConservation Laws 43
312 Subsidiary Laws 45
32 Momentum and Energy Transport Mechanisms 46
33 Physical Law based Dierential Formulation 48
331 Continuity Equation 49
332 Transport Equation 51
34 Generalized Volumetric and Flux Terms and their Dif-
ferential Formulation 57
341 Volumetric Term 58
342 Flux-Term 58
343 Discussion 62
35 Mathematical Formulation 63
351 Dimensional Study 63
352 Non-Dimensional Study 67
36 Closure 71
4 Essentials of Numerical-Methods for CFD 73
41 Finite Dierence Method A Dierential to Algebraic
Formulation for Governing PDE and BCs 75
411 Grid Generation 75
412 Finite Dierence Method 78
413 Applications to CFD 92
42 Iterative Solution of System of LAEs for a Flow Property 93
421 Iterative Methods 94
422 Applications to CFD 98
43 Numerical Dierentiation for Local Engineering-
Parameters 105
431 Dierentiation Formulas 106
432 Applications to CFD 107
44 Numerical Integration for the Total value of
Engineering-Parameters 110
441 Integration Rules 111
442 Applications to CFD 114
45 Closure 116
Problems 116
Introduction to CFD xv
II CFD for a Cartesian-Geometry
119
5 Computational Heat Conduction 12151 Physical Law based Finite Volume Method 122
511 Energy Conservation Law for a Control Volume 123512 Algebraic Formulation 124513 Approximations 127514 Approximated Algebraic-Formulation 130515 Discussion 133
52 Finite Dierence Method for Boundary Conditions 13853 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 139531 One-Dimensional Conduction 139532 Two-Dimensional Conduction 147
54 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 154541 One-Dimensional Conduction 154542 Two-Dimensional Conduction 166
Problems 176
6 Computational Heat Advection 17961 Physical Law based Finite Volume Method 180
611 Energy Conservation Law for a Control Volume 181612 Algebraic Formulation 181613 Approximations 183614 Approximated Algebraic Formulation 191615 Discussion 191
62 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 194621 Explicit-Method 197622 Implementation Details 198623 Solution Algorithm 199
63 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 207631 Advection Scheme on a Non-Uniform Grid 207632 Explicit and Implicit Method 210633 Implementation Details 216634 Solution Algorithm 220
Problems 228
xvi Contents
7 Computational Heat Convection 22971 Physical Law based Finite Volume Method 229
711 Energy Conservation Law for a Control Volume 229712 Algebraic Formulation 231713 Approximated Algebraic Formulation 232
72 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 235721 Explicit-Method 236722 Implementation Details 237723 Solution Algorithm 238
73 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 242Problems 248
8 Computational Fluid Dynamics Physical Law based
Finite Volume Method 25181 Generalized Variables for the Combined Heat and
Fluid Flow 25282 Conservation Laws for a Control Volume 25583 Algebraic Formulation 25984 Approximations 26085 Approximated Algebraic Formulation 263
851 Mass Conservation 263852 MomentumEnergy Conservation 264
86 Closure 269
9 Computational Fluid Dynamics on a Staggered Grid 27191 Challenges in the CFD Development 273
911 Non-Linearity 273912 Equation for Pressure 273913 Pressure-Velocity Decoupling 273
92 A Staggered Grid to avoid Pressure-Velocity Decoupling27593 Physical Law based FVM for a Staggered Grid 27794 Flux based Solution Methodology on a Uniform Grid
Semi-Explicit Method 281941 Philosophy of Pressure-Correction Method 283942 Semi-Explicit Method 286943 Implementation Details 292944 Solution Algorithm 298
95 Initial and Boundary Conditions 300951 Initial Condition 300952 Boundary Condition 301
Problems 306
Introduction to CFD xvii
10 Computational Fluid Dynamics on a Co-located Grid309
101 Momentum-Interpolation Method Strategy to avoid
the Pressure-Velocity Decoupling on a Co-located Grid 310
102 Coecients of LAEs based Solution Methodology on
a Non-Uniform Grid Semi-Explicit and Semi-Implicit
Method 314
1021 Predictor Step 316
1022 Corrector Step 318
1023 Solution Algorithm 323
Problems 329
III CFD for a Complex-Geometry
331
11 Computational Heat Conduction on a Curvilinear
Grid 333
111 Curvilinear Grid Generation 333
1111 Algebraic Grid Generation 334
1112 Elliptic Grid Generation 336
112 Physical Law based Finite Volume Method 343
1121 Unsteady and Source Term 343
1122 Diusion Term 344
1123 All Terms 349
113 Computation of Geometrical Properties 349
114 Flux based Solution Methodology 352
1141 Explicit Method 354
1142 Implementation Details 354
Problems 359
12 Computational Fluid Dynamics on a Curvilinear Grid361
121 Physical Law based Finite Volume Method 361
1211 Mass Conservation 363
1212 Momentum Conservation 363
122 Solution Methodology Semi-Explicit Method 372
1221 Predictor Step 373
1222 Corrector Step 375
Problems 380
References 383
Index 389
Part I
Introduction and Essentials
The book on an introductory course in CFD starts with the two intro-
ductory chapters on CFD followed by one chapter each on essentials
of two prerequisite courses minus uid-dynamics and heat transfer and
numerical-methods The rst chapter on introduction presents an
eventful thoughtful and intuitive discussion on What is CFD and
Why to study CFD whereas How CFD works which involves lots
of details is presented separately in the second chapter The third-
chapter on essentials of uid-dynamics and heat transfer presents
physical-laws transport-mechanisms physical-law based dierential-
formulation volumetric and ux terms and mathematical formula-
tion The fourth-chapter on essentials of numerical-methods presents
nite dierence method iterative solution of the system of linear alge-
braic equation numerical dierentiation and numerical integration
The presentation for the essential of the two prerequisite courses is
customized to CFD development application and analysis
1
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
1
Introduction
The introduction of any course consists of the three basic questions
What is it Why to study How it works An eventful thoughtful
and intuitive answer to the rst two questions are presented in this
chapter The third question which involves lots of working details
on the three aspects of the Computational Fluid Dynamics (CFD)
(development application and analysis) is presented separately in
the next chapter The novelty scope and purpose of this book is
also presented at the end of this chapter
11 CFD What is it
CFD is a theoretical-method of scientic and engineering investiga-
tion concerned with the development and application of a video-
camera like tool (a software) which is used for a unied cause-and-
eect based analysis of a uid-dynamics as well as heat and mass
transfer problem
CFD is a subject where you rst learn how to develop a product
minus a software which acts like a virtual video-camera Then you learn
how to apply or use this product to generate uid-dynamics movies
Finally you learn how to analyze the detailed spatial and temporal
uid-dynamics information in the movies The analysis is done to
come up with a scientically-exciting as well as engineering-relevant
story (unied cause-and-eect study) of a uid-dynamics situation in
nature as well as industrial-applications With a continuous develop-
ment and a wider application of CFD the word uid-dynamics in
3
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
4 1 Introduction
the CFD has become more generic as it also corresponds to heat and
mass transfer as well as chemical reaction
Fluid-dynamics information are of two types scientic and en-
gineering Scientic-information corresponds to a structure of the
heat and uid ow due to a physical phenomenon in uid-dynamics
such as boundary-layer ow-separation wake-formation and vortex-
shedding The ow structures are obtained from a temporal variation
of ow-properties (such as velocity pressure temperature and many
other variables which characterizes a ow) called as scientically-
exciting movies Engineering-information corresponds to certain pa-
rameters which are important in an engineering application called
as engineering-parameters for example lift force drag force wall
shear stress pressure drop and rate of heat-transfer They are ob-
tained as the temporal variation of engineering-parameters called as
engineering-relevant movies The engineering-parameters are the ef-
fect which are caused by the ow structures minus their correlation leads
to the unied cause-and-eect study A uid-dynamics movie consist
of a time-wise varying series of pictures which give uid-dynamics
information Each ow-property and engineering-parameter results
in one scientically-exciting and engineering-relevant movie respec-
tively When the large number of both the types of movies are
played synchronized in-time the detailed spatial as well as tempo-
ral uid-dynamic information greatly helps in the unied cause-and-
eect study of a uid-dynamics problem
Alternatively CFD is a subject concerned with the development of
computer-programs for the computer-simulation and study of a natu-
ral or an engineering uid-dynamics system The development starts
with the virtual-development of the system involving a geometri-
cal information based computational development of a solid-model
for the system the system may be already-existing or yet-to-be-
developed physically Thereafter the development involves numerical
solution of the governing algebraic equations for CFD formulated
from the physical laws (conservation of mass momentum and en-
ergy) and initial as well as boundary conditions corresponding to the
system CFD study involves the analysis of the uid-dynamics and
heat-transfer results obtained from the computer-simulations of the
virtual-system which is subjected to certain governing-parameters
Introduction to CFD 5
111 CFD as a Scientic and Engineering Analysis Tool
For a better understanding of the denition of the CFD and its ap-
plication as a scientic and engineering analysis tool two example
problems are presented rst conduction heat transfer in a plate
and second uid ow across a circular pillar of a bridge The rst-
problem correspond to an engineering-application for electronic cool-
ing and the second-problem is studied for an engineering-design of
the structure subjected to uid-dynamics forces and are shown in
Fig 11 Both the problems are considered as two-dimensional and
unsteady However the results asymptotes from the unsteady to a
steady state in the rst-problem and to a periodic (dynamic-steady)
state in the second-problem
Figure 11 Schematic and the associated parameters for the two example prob-lems (a) thermal conduction module of a heat sink and (b) theclassical ow across a circular cylinder
6 1 Introduction
The rst-problem is taken from the classic book on heat-transfer
by Incropera and Dewitt (1996) shown in Fig 11(a) The g-
ure shows a periodic module of the plate where a constant heat-ux
qW = 10Wcm2 (dissipated from certain electronic devices) is rst con-
ducted to an aluminum plate and then convected to the water ow-
ing through the rectangular channel grooved in the plate The forced
convection results in a convection coecient of h = 5000Wm2K in-
side the channel The unsteady heat-transfer results in an interesting
time-wise varying conduction heat ow in the plate and instantaneous
convection heat transfer rate Qprimeconv (per unit width of the plate per-
pendicular to the plane shown in Fig 11(a)) into the water
The second-problem is encountered if you are standing over a
bridge and look below to the water passing over a pillar (of circu-
lar cross-section) of the bridge shown in Fig 11(b) as a general case
of 2D ow across a cylinder The uid-dynamics results in a beau-
tiful time-wise varying structure of the uid ow just downstream
of the pillar and certain engineering-parameters The parameters
correspond to the uid-dynamic forces acting in the horizontal (or
streamwise) and vertical (or transverse) direction called as the drag-
force FD and lift-force FL respectively The respective force is pre-
sented below in a non-dimensional form called as the drag-coecient
CD and lift-coecient CL equations shown in Fig 11(b)
If you use your video camera to capture a movie for the two
problems you will get series of pictures for the plate in the rst-
problem and the owing uid in the second-problem Since these
movies result in a pictorial information but not the ow-property
based uid-dynamic information such movies are not relevant for
the uid-dynamics and heat-transfer study Instead if you use a
CFD software for computer simulation you can capture various types
of movies The various ow-properties based scientically-exciting
movies are presented here as the pictures at certain time-instant in
Fig 12(a)-(d) and Fig 13(a1)-(d2) The former gure shows gray-
colored ooded contour for the temperature whereas the latter gure
shows gray-colored ooded contour for the pressure and vorticity and
vector-plot for the velocity For the gray-colored ooded contours a
color bar can be seen in the gures for the relative magnitude of the
variable These gures also show line contours of stream-function
Introduction to CFD 7
and heat-function as streamlines and heatlines respectively They
are the visualization techniques for a 2D steady heat and uid ow
The heatlines are analogous to streamlines and represents the direc-
tion of heat ow under steady state condition proposed by Kimura
and Bejan (1983) later presented in a heat-transfer book by Bejan
(1984) The steady-state heatlines can be seen in Fig 12(d) where
the direction for the conduction-ux is tangential at any point on the
heatlines analogous to that for the mass-ux in a streamline Thus
for a better presentation of the structure of heat and uid ow the
scientically-exciting movies are not only limited to ow-properties
but also correspond to certain ow visualization techniquesAn engineering-parameter based engineering-relevant uid dy-
namic movie is presented in Fig 12(e) for the convection heat trans-
fer rate per unit width at the various walls on the channel Qprimew at the
west-wall Qprimen at the north-wall Q
primee at the east-wall and their cumu-
lative value Qprimeconv (refer Fig 11(a)) Another engineering-relevant
movie is presented in Fig 13(e) for the lift and drag coecient Note
that the movie here corresponds to the motion of a lled symbol over
a line for the temporal variation of an engineering-parameter Thus
the engineering-relevant movies are one-dimensional lower than the
scientically-exciting movies here for the 2D problems the former
movie is presented as 1D and the latter as 2D results
1111 Heat-Conduction
For the computational set-up of the heat conduction problem shown
in Fig 11(a) the results obtained from a 2D CFD simulation of
the conduction heat transfer are shown in Fig 12 For a uni-
form initial temperature of the plate as Tinfin = 15oC Fig 12(a)-(d)
shows the isotherms and heatlines in the plate as the pictures of the
scientically-exciting movies at certain discrete time instants t = 25
5 10 and 20 sec For the same discrete time instants the temporal
variation of the position of the symbols over a line plot is presented in
Fig 12(e) The gure is shown for each of the heat transfer rate and
is obtained from a engineering-relevant 1D movie There is a tempo-
ral variation of picture in the scientically-exciting movie and that
of position of symbol (along the line plot) in the engineering-relevant
movieAccording to caloric theory of the famous French scientist An-
tonie Lavoisier heat is considered as an invisible tasteless odorless
8 1 Introduction
t(sec)
Qrsquo (W
m)
0 5 10 15 20 250
200
400
600
800
1000Qrsquo
conv
Qrsquow
Qrsquon
Qrsquoe
Qrsquoin
(e)
(b)
(c)(d)
(a)
(a) t=25 sec31
29
27
25
23
21
19
17
15
T(oC)
(c) t=10 sec
(b) t=5 sec
(d) t=20 sec
Figure 12 Thermal conduction module of a heat sink temporal variation of(a)-(d) isotherms and heatlines and (e) the convection heat transferrate per unit width at the various walls of the channels and theircumulative total value Q
primeconv For the discrete time-instant corre-
sponding to heat ow patterns in (a)-(d) temporal variation of thevarious heat transfer rates are shown as symbols in (e) (a)-(c) arethe transient and (d) represents a steady state result
weightless uid which he called caloric uid Presently the caloric
theory is overtaken by mechanical theory of heat minus heat is not a sub-
stance but a dynamical form of mechanical eect (Thomson 1851)
Nevertheless there are certain problems involving heat ow for which
Lavoisiers approach is rather useful such as the discussion on heat-
lines here
The role of heatlines for heat ow is analogous to that of stream-
lines for uid ow (Paramane and Sharma 2009) For the uid-ow
the dierence of stream-function values represents the rate of uid
ow the function remains constant on a solid wall the tangent to
a streamline represents the direction of the uid-ow (with no ow
in the normal direction) and the streamline originate or emerge at
Introduction to CFD 9
a mass source Analogously for the heat-ow the dierence of heat-
function values represents the rate of heat ow the function remains
constant on an adiabatic wall the tangent to a heatline represents the
direction of the heat-ow (with no ow in the normal direction) and
the heatline originate or emerge at the heat-source For the present
problem in Fig 11(a) the heat-source is at the inlet (or left) bound-
ary and the heat-sink is at the surface of the channel Thus the
steady-state heatlines in Fig 12(d) shows the heat ow emerging
from the inletsource and ending on the channel-wallssink Further-
more it can be seen in the gure that the heatlines which are close
to the adiabatic walls are parallel to the walls The source sink and
adiabatic-walls can be seen in Fig 11(a)
Figure 12(e) shows an asymptotic time-wise increase in the con-
vection heat transfer per unit width at the various walls of the channel
and its cumulative value Qprimeconv (= Q
primew+Q
primen+Q
primee) It can be also be seen
that the heat-transfer per unit width Qprimew at the west-wall and Q
primen at
the north-wall are larger than Qprimee at the east-wall of the channel As
time progresses the gure shows that Qprimeconv increases monotonically
and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-
der steady state condition refer Fig 11(a) The dierence between
the Qprimein and Q
primeconv under the transient condition is utilized for the
sensible heating and results in the increase in the temperature of the
plate Under the steady state condition note from Fig 12(d) that
the maximum temperature is close to 32oC minus well within the per-
missible limit for the reliable and ecient operation of the electronic
device
1112 Fluid-Dynamics
For the free-stream ow across a circular cylinderpillar at a Reynolds
number of 100 the non-dimensional results obtained from a 2D CFD
simulation are shown in Fig 13 In contrast to the previous prob-
lem which reaches steady state the present problem reaches to an un-
steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-
ent increasing time instants within one time-period of the periodic-
ow the results are presented in Fig 13(a)-(d) The gure shows
the close-up view of the pictures of the scientically-exciting movies
for ow-properties (velocity pressure and vorticity) and streamlines
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow
Introduction to CFD xv
II CFD for a Cartesian-Geometry
119
5 Computational Heat Conduction 12151 Physical Law based Finite Volume Method 122
511 Energy Conservation Law for a Control Volume 123512 Algebraic Formulation 124513 Approximations 127514 Approximated Algebraic-Formulation 130515 Discussion 133
52 Finite Dierence Method for Boundary Conditions 13853 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 139531 One-Dimensional Conduction 139532 Two-Dimensional Conduction 147
54 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 154541 One-Dimensional Conduction 154542 Two-Dimensional Conduction 166
Problems 176
6 Computational Heat Advection 17961 Physical Law based Finite Volume Method 180
611 Energy Conservation Law for a Control Volume 181612 Algebraic Formulation 181613 Approximations 183614 Approximated Algebraic Formulation 191615 Discussion 191
62 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 194621 Explicit-Method 197622 Implementation Details 198623 Solution Algorithm 199
63 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 207631 Advection Scheme on a Non-Uniform Grid 207632 Explicit and Implicit Method 210633 Implementation Details 216634 Solution Algorithm 220
Problems 228
xvi Contents
7 Computational Heat Convection 22971 Physical Law based Finite Volume Method 229
711 Energy Conservation Law for a Control Volume 229712 Algebraic Formulation 231713 Approximated Algebraic Formulation 232
72 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 235721 Explicit-Method 236722 Implementation Details 237723 Solution Algorithm 238
73 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 242Problems 248
8 Computational Fluid Dynamics Physical Law based
Finite Volume Method 25181 Generalized Variables for the Combined Heat and
Fluid Flow 25282 Conservation Laws for a Control Volume 25583 Algebraic Formulation 25984 Approximations 26085 Approximated Algebraic Formulation 263
851 Mass Conservation 263852 MomentumEnergy Conservation 264
86 Closure 269
9 Computational Fluid Dynamics on a Staggered Grid 27191 Challenges in the CFD Development 273
911 Non-Linearity 273912 Equation for Pressure 273913 Pressure-Velocity Decoupling 273
92 A Staggered Grid to avoid Pressure-Velocity Decoupling27593 Physical Law based FVM for a Staggered Grid 27794 Flux based Solution Methodology on a Uniform Grid
Semi-Explicit Method 281941 Philosophy of Pressure-Correction Method 283942 Semi-Explicit Method 286943 Implementation Details 292944 Solution Algorithm 298
95 Initial and Boundary Conditions 300951 Initial Condition 300952 Boundary Condition 301
Problems 306
Introduction to CFD xvii
10 Computational Fluid Dynamics on a Co-located Grid309
101 Momentum-Interpolation Method Strategy to avoid
the Pressure-Velocity Decoupling on a Co-located Grid 310
102 Coecients of LAEs based Solution Methodology on
a Non-Uniform Grid Semi-Explicit and Semi-Implicit
Method 314
1021 Predictor Step 316
1022 Corrector Step 318
1023 Solution Algorithm 323
Problems 329
III CFD for a Complex-Geometry
331
11 Computational Heat Conduction on a Curvilinear
Grid 333
111 Curvilinear Grid Generation 333
1111 Algebraic Grid Generation 334
1112 Elliptic Grid Generation 336
112 Physical Law based Finite Volume Method 343
1121 Unsteady and Source Term 343
1122 Diusion Term 344
1123 All Terms 349
113 Computation of Geometrical Properties 349
114 Flux based Solution Methodology 352
1141 Explicit Method 354
1142 Implementation Details 354
Problems 359
12 Computational Fluid Dynamics on a Curvilinear Grid361
121 Physical Law based Finite Volume Method 361
1211 Mass Conservation 363
1212 Momentum Conservation 363
122 Solution Methodology Semi-Explicit Method 372
1221 Predictor Step 373
1222 Corrector Step 375
Problems 380
References 383
Index 389
Part I
Introduction and Essentials
The book on an introductory course in CFD starts with the two intro-
ductory chapters on CFD followed by one chapter each on essentials
of two prerequisite courses minus uid-dynamics and heat transfer and
numerical-methods The rst chapter on introduction presents an
eventful thoughtful and intuitive discussion on What is CFD and
Why to study CFD whereas How CFD works which involves lots
of details is presented separately in the second chapter The third-
chapter on essentials of uid-dynamics and heat transfer presents
physical-laws transport-mechanisms physical-law based dierential-
formulation volumetric and ux terms and mathematical formula-
tion The fourth-chapter on essentials of numerical-methods presents
nite dierence method iterative solution of the system of linear alge-
braic equation numerical dierentiation and numerical integration
The presentation for the essential of the two prerequisite courses is
customized to CFD development application and analysis
1
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
1
Introduction
The introduction of any course consists of the three basic questions
What is it Why to study How it works An eventful thoughtful
and intuitive answer to the rst two questions are presented in this
chapter The third question which involves lots of working details
on the three aspects of the Computational Fluid Dynamics (CFD)
(development application and analysis) is presented separately in
the next chapter The novelty scope and purpose of this book is
also presented at the end of this chapter
11 CFD What is it
CFD is a theoretical-method of scientic and engineering investiga-
tion concerned with the development and application of a video-
camera like tool (a software) which is used for a unied cause-and-
eect based analysis of a uid-dynamics as well as heat and mass
transfer problem
CFD is a subject where you rst learn how to develop a product
minus a software which acts like a virtual video-camera Then you learn
how to apply or use this product to generate uid-dynamics movies
Finally you learn how to analyze the detailed spatial and temporal
uid-dynamics information in the movies The analysis is done to
come up with a scientically-exciting as well as engineering-relevant
story (unied cause-and-eect study) of a uid-dynamics situation in
nature as well as industrial-applications With a continuous develop-
ment and a wider application of CFD the word uid-dynamics in
3
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
4 1 Introduction
the CFD has become more generic as it also corresponds to heat and
mass transfer as well as chemical reaction
Fluid-dynamics information are of two types scientic and en-
gineering Scientic-information corresponds to a structure of the
heat and uid ow due to a physical phenomenon in uid-dynamics
such as boundary-layer ow-separation wake-formation and vortex-
shedding The ow structures are obtained from a temporal variation
of ow-properties (such as velocity pressure temperature and many
other variables which characterizes a ow) called as scientically-
exciting movies Engineering-information corresponds to certain pa-
rameters which are important in an engineering application called
as engineering-parameters for example lift force drag force wall
shear stress pressure drop and rate of heat-transfer They are ob-
tained as the temporal variation of engineering-parameters called as
engineering-relevant movies The engineering-parameters are the ef-
fect which are caused by the ow structures minus their correlation leads
to the unied cause-and-eect study A uid-dynamics movie consist
of a time-wise varying series of pictures which give uid-dynamics
information Each ow-property and engineering-parameter results
in one scientically-exciting and engineering-relevant movie respec-
tively When the large number of both the types of movies are
played synchronized in-time the detailed spatial as well as tempo-
ral uid-dynamic information greatly helps in the unied cause-and-
eect study of a uid-dynamics problem
Alternatively CFD is a subject concerned with the development of
computer-programs for the computer-simulation and study of a natu-
ral or an engineering uid-dynamics system The development starts
with the virtual-development of the system involving a geometri-
cal information based computational development of a solid-model
for the system the system may be already-existing or yet-to-be-
developed physically Thereafter the development involves numerical
solution of the governing algebraic equations for CFD formulated
from the physical laws (conservation of mass momentum and en-
ergy) and initial as well as boundary conditions corresponding to the
system CFD study involves the analysis of the uid-dynamics and
heat-transfer results obtained from the computer-simulations of the
virtual-system which is subjected to certain governing-parameters
Introduction to CFD 5
111 CFD as a Scientic and Engineering Analysis Tool
For a better understanding of the denition of the CFD and its ap-
plication as a scientic and engineering analysis tool two example
problems are presented rst conduction heat transfer in a plate
and second uid ow across a circular pillar of a bridge The rst-
problem correspond to an engineering-application for electronic cool-
ing and the second-problem is studied for an engineering-design of
the structure subjected to uid-dynamics forces and are shown in
Fig 11 Both the problems are considered as two-dimensional and
unsteady However the results asymptotes from the unsteady to a
steady state in the rst-problem and to a periodic (dynamic-steady)
state in the second-problem
Figure 11 Schematic and the associated parameters for the two example prob-lems (a) thermal conduction module of a heat sink and (b) theclassical ow across a circular cylinder
6 1 Introduction
The rst-problem is taken from the classic book on heat-transfer
by Incropera and Dewitt (1996) shown in Fig 11(a) The g-
ure shows a periodic module of the plate where a constant heat-ux
qW = 10Wcm2 (dissipated from certain electronic devices) is rst con-
ducted to an aluminum plate and then convected to the water ow-
ing through the rectangular channel grooved in the plate The forced
convection results in a convection coecient of h = 5000Wm2K in-
side the channel The unsteady heat-transfer results in an interesting
time-wise varying conduction heat ow in the plate and instantaneous
convection heat transfer rate Qprimeconv (per unit width of the plate per-
pendicular to the plane shown in Fig 11(a)) into the water
The second-problem is encountered if you are standing over a
bridge and look below to the water passing over a pillar (of circu-
lar cross-section) of the bridge shown in Fig 11(b) as a general case
of 2D ow across a cylinder The uid-dynamics results in a beau-
tiful time-wise varying structure of the uid ow just downstream
of the pillar and certain engineering-parameters The parameters
correspond to the uid-dynamic forces acting in the horizontal (or
streamwise) and vertical (or transverse) direction called as the drag-
force FD and lift-force FL respectively The respective force is pre-
sented below in a non-dimensional form called as the drag-coecient
CD and lift-coecient CL equations shown in Fig 11(b)
If you use your video camera to capture a movie for the two
problems you will get series of pictures for the plate in the rst-
problem and the owing uid in the second-problem Since these
movies result in a pictorial information but not the ow-property
based uid-dynamic information such movies are not relevant for
the uid-dynamics and heat-transfer study Instead if you use a
CFD software for computer simulation you can capture various types
of movies The various ow-properties based scientically-exciting
movies are presented here as the pictures at certain time-instant in
Fig 12(a)-(d) and Fig 13(a1)-(d2) The former gure shows gray-
colored ooded contour for the temperature whereas the latter gure
shows gray-colored ooded contour for the pressure and vorticity and
vector-plot for the velocity For the gray-colored ooded contours a
color bar can be seen in the gures for the relative magnitude of the
variable These gures also show line contours of stream-function
Introduction to CFD 7
and heat-function as streamlines and heatlines respectively They
are the visualization techniques for a 2D steady heat and uid ow
The heatlines are analogous to streamlines and represents the direc-
tion of heat ow under steady state condition proposed by Kimura
and Bejan (1983) later presented in a heat-transfer book by Bejan
(1984) The steady-state heatlines can be seen in Fig 12(d) where
the direction for the conduction-ux is tangential at any point on the
heatlines analogous to that for the mass-ux in a streamline Thus
for a better presentation of the structure of heat and uid ow the
scientically-exciting movies are not only limited to ow-properties
but also correspond to certain ow visualization techniquesAn engineering-parameter based engineering-relevant uid dy-
namic movie is presented in Fig 12(e) for the convection heat trans-
fer rate per unit width at the various walls on the channel Qprimew at the
west-wall Qprimen at the north-wall Q
primee at the east-wall and their cumu-
lative value Qprimeconv (refer Fig 11(a)) Another engineering-relevant
movie is presented in Fig 13(e) for the lift and drag coecient Note
that the movie here corresponds to the motion of a lled symbol over
a line for the temporal variation of an engineering-parameter Thus
the engineering-relevant movies are one-dimensional lower than the
scientically-exciting movies here for the 2D problems the former
movie is presented as 1D and the latter as 2D results
1111 Heat-Conduction
For the computational set-up of the heat conduction problem shown
in Fig 11(a) the results obtained from a 2D CFD simulation of
the conduction heat transfer are shown in Fig 12 For a uni-
form initial temperature of the plate as Tinfin = 15oC Fig 12(a)-(d)
shows the isotherms and heatlines in the plate as the pictures of the
scientically-exciting movies at certain discrete time instants t = 25
5 10 and 20 sec For the same discrete time instants the temporal
variation of the position of the symbols over a line plot is presented in
Fig 12(e) The gure is shown for each of the heat transfer rate and
is obtained from a engineering-relevant 1D movie There is a tempo-
ral variation of picture in the scientically-exciting movie and that
of position of symbol (along the line plot) in the engineering-relevant
movieAccording to caloric theory of the famous French scientist An-
tonie Lavoisier heat is considered as an invisible tasteless odorless
8 1 Introduction
t(sec)
Qrsquo (W
m)
0 5 10 15 20 250
200
400
600
800
1000Qrsquo
conv
Qrsquow
Qrsquon
Qrsquoe
Qrsquoin
(e)
(b)
(c)(d)
(a)
(a) t=25 sec31
29
27
25
23
21
19
17
15
T(oC)
(c) t=10 sec
(b) t=5 sec
(d) t=20 sec
Figure 12 Thermal conduction module of a heat sink temporal variation of(a)-(d) isotherms and heatlines and (e) the convection heat transferrate per unit width at the various walls of the channels and theircumulative total value Q
primeconv For the discrete time-instant corre-
sponding to heat ow patterns in (a)-(d) temporal variation of thevarious heat transfer rates are shown as symbols in (e) (a)-(c) arethe transient and (d) represents a steady state result
weightless uid which he called caloric uid Presently the caloric
theory is overtaken by mechanical theory of heat minus heat is not a sub-
stance but a dynamical form of mechanical eect (Thomson 1851)
Nevertheless there are certain problems involving heat ow for which
Lavoisiers approach is rather useful such as the discussion on heat-
lines here
The role of heatlines for heat ow is analogous to that of stream-
lines for uid ow (Paramane and Sharma 2009) For the uid-ow
the dierence of stream-function values represents the rate of uid
ow the function remains constant on a solid wall the tangent to
a streamline represents the direction of the uid-ow (with no ow
in the normal direction) and the streamline originate or emerge at
Introduction to CFD 9
a mass source Analogously for the heat-ow the dierence of heat-
function values represents the rate of heat ow the function remains
constant on an adiabatic wall the tangent to a heatline represents the
direction of the heat-ow (with no ow in the normal direction) and
the heatline originate or emerge at the heat-source For the present
problem in Fig 11(a) the heat-source is at the inlet (or left) bound-
ary and the heat-sink is at the surface of the channel Thus the
steady-state heatlines in Fig 12(d) shows the heat ow emerging
from the inletsource and ending on the channel-wallssink Further-
more it can be seen in the gure that the heatlines which are close
to the adiabatic walls are parallel to the walls The source sink and
adiabatic-walls can be seen in Fig 11(a)
Figure 12(e) shows an asymptotic time-wise increase in the con-
vection heat transfer per unit width at the various walls of the channel
and its cumulative value Qprimeconv (= Q
primew+Q
primen+Q
primee) It can be also be seen
that the heat-transfer per unit width Qprimew at the west-wall and Q
primen at
the north-wall are larger than Qprimee at the east-wall of the channel As
time progresses the gure shows that Qprimeconv increases monotonically
and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-
der steady state condition refer Fig 11(a) The dierence between
the Qprimein and Q
primeconv under the transient condition is utilized for the
sensible heating and results in the increase in the temperature of the
plate Under the steady state condition note from Fig 12(d) that
the maximum temperature is close to 32oC minus well within the per-
missible limit for the reliable and ecient operation of the electronic
device
1112 Fluid-Dynamics
For the free-stream ow across a circular cylinderpillar at a Reynolds
number of 100 the non-dimensional results obtained from a 2D CFD
simulation are shown in Fig 13 In contrast to the previous prob-
lem which reaches steady state the present problem reaches to an un-
steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-
ent increasing time instants within one time-period of the periodic-
ow the results are presented in Fig 13(a)-(d) The gure shows
the close-up view of the pictures of the scientically-exciting movies
for ow-properties (velocity pressure and vorticity) and streamlines
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow
xvi Contents
7 Computational Heat Convection 22971 Physical Law based Finite Volume Method 229
711 Energy Conservation Law for a Control Volume 229712 Algebraic Formulation 231713 Approximated Algebraic Formulation 232
72 Flux based Solution Methodology on a Uniform Grid
Explicit-Method 235721 Explicit-Method 236722 Implementation Details 237723 Solution Algorithm 238
73 Coecients of LAEs based Solution Methodology on a
Non-Uniform Grid Explicit and Implicit Method 242Problems 248
8 Computational Fluid Dynamics Physical Law based
Finite Volume Method 25181 Generalized Variables for the Combined Heat and
Fluid Flow 25282 Conservation Laws for a Control Volume 25583 Algebraic Formulation 25984 Approximations 26085 Approximated Algebraic Formulation 263
851 Mass Conservation 263852 MomentumEnergy Conservation 264
86 Closure 269
9 Computational Fluid Dynamics on a Staggered Grid 27191 Challenges in the CFD Development 273
911 Non-Linearity 273912 Equation for Pressure 273913 Pressure-Velocity Decoupling 273
92 A Staggered Grid to avoid Pressure-Velocity Decoupling27593 Physical Law based FVM for a Staggered Grid 27794 Flux based Solution Methodology on a Uniform Grid
Semi-Explicit Method 281941 Philosophy of Pressure-Correction Method 283942 Semi-Explicit Method 286943 Implementation Details 292944 Solution Algorithm 298
95 Initial and Boundary Conditions 300951 Initial Condition 300952 Boundary Condition 301
Problems 306
Introduction to CFD xvii
10 Computational Fluid Dynamics on a Co-located Grid309
101 Momentum-Interpolation Method Strategy to avoid
the Pressure-Velocity Decoupling on a Co-located Grid 310
102 Coecients of LAEs based Solution Methodology on
a Non-Uniform Grid Semi-Explicit and Semi-Implicit
Method 314
1021 Predictor Step 316
1022 Corrector Step 318
1023 Solution Algorithm 323
Problems 329
III CFD for a Complex-Geometry
331
11 Computational Heat Conduction on a Curvilinear
Grid 333
111 Curvilinear Grid Generation 333
1111 Algebraic Grid Generation 334
1112 Elliptic Grid Generation 336
112 Physical Law based Finite Volume Method 343
1121 Unsteady and Source Term 343
1122 Diusion Term 344
1123 All Terms 349
113 Computation of Geometrical Properties 349
114 Flux based Solution Methodology 352
1141 Explicit Method 354
1142 Implementation Details 354
Problems 359
12 Computational Fluid Dynamics on a Curvilinear Grid361
121 Physical Law based Finite Volume Method 361
1211 Mass Conservation 363
1212 Momentum Conservation 363
122 Solution Methodology Semi-Explicit Method 372
1221 Predictor Step 373
1222 Corrector Step 375
Problems 380
References 383
Index 389
Part I
Introduction and Essentials
The book on an introductory course in CFD starts with the two intro-
ductory chapters on CFD followed by one chapter each on essentials
of two prerequisite courses minus uid-dynamics and heat transfer and
numerical-methods The rst chapter on introduction presents an
eventful thoughtful and intuitive discussion on What is CFD and
Why to study CFD whereas How CFD works which involves lots
of details is presented separately in the second chapter The third-
chapter on essentials of uid-dynamics and heat transfer presents
physical-laws transport-mechanisms physical-law based dierential-
formulation volumetric and ux terms and mathematical formula-
tion The fourth-chapter on essentials of numerical-methods presents
nite dierence method iterative solution of the system of linear alge-
braic equation numerical dierentiation and numerical integration
The presentation for the essential of the two prerequisite courses is
customized to CFD development application and analysis
1
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
1
Introduction
The introduction of any course consists of the three basic questions
What is it Why to study How it works An eventful thoughtful
and intuitive answer to the rst two questions are presented in this
chapter The third question which involves lots of working details
on the three aspects of the Computational Fluid Dynamics (CFD)
(development application and analysis) is presented separately in
the next chapter The novelty scope and purpose of this book is
also presented at the end of this chapter
11 CFD What is it
CFD is a theoretical-method of scientic and engineering investiga-
tion concerned with the development and application of a video-
camera like tool (a software) which is used for a unied cause-and-
eect based analysis of a uid-dynamics as well as heat and mass
transfer problem
CFD is a subject where you rst learn how to develop a product
minus a software which acts like a virtual video-camera Then you learn
how to apply or use this product to generate uid-dynamics movies
Finally you learn how to analyze the detailed spatial and temporal
uid-dynamics information in the movies The analysis is done to
come up with a scientically-exciting as well as engineering-relevant
story (unied cause-and-eect study) of a uid-dynamics situation in
nature as well as industrial-applications With a continuous develop-
ment and a wider application of CFD the word uid-dynamics in
3
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
4 1 Introduction
the CFD has become more generic as it also corresponds to heat and
mass transfer as well as chemical reaction
Fluid-dynamics information are of two types scientic and en-
gineering Scientic-information corresponds to a structure of the
heat and uid ow due to a physical phenomenon in uid-dynamics
such as boundary-layer ow-separation wake-formation and vortex-
shedding The ow structures are obtained from a temporal variation
of ow-properties (such as velocity pressure temperature and many
other variables which characterizes a ow) called as scientically-
exciting movies Engineering-information corresponds to certain pa-
rameters which are important in an engineering application called
as engineering-parameters for example lift force drag force wall
shear stress pressure drop and rate of heat-transfer They are ob-
tained as the temporal variation of engineering-parameters called as
engineering-relevant movies The engineering-parameters are the ef-
fect which are caused by the ow structures minus their correlation leads
to the unied cause-and-eect study A uid-dynamics movie consist
of a time-wise varying series of pictures which give uid-dynamics
information Each ow-property and engineering-parameter results
in one scientically-exciting and engineering-relevant movie respec-
tively When the large number of both the types of movies are
played synchronized in-time the detailed spatial as well as tempo-
ral uid-dynamic information greatly helps in the unied cause-and-
eect study of a uid-dynamics problem
Alternatively CFD is a subject concerned with the development of
computer-programs for the computer-simulation and study of a natu-
ral or an engineering uid-dynamics system The development starts
with the virtual-development of the system involving a geometri-
cal information based computational development of a solid-model
for the system the system may be already-existing or yet-to-be-
developed physically Thereafter the development involves numerical
solution of the governing algebraic equations for CFD formulated
from the physical laws (conservation of mass momentum and en-
ergy) and initial as well as boundary conditions corresponding to the
system CFD study involves the analysis of the uid-dynamics and
heat-transfer results obtained from the computer-simulations of the
virtual-system which is subjected to certain governing-parameters
Introduction to CFD 5
111 CFD as a Scientic and Engineering Analysis Tool
For a better understanding of the denition of the CFD and its ap-
plication as a scientic and engineering analysis tool two example
problems are presented rst conduction heat transfer in a plate
and second uid ow across a circular pillar of a bridge The rst-
problem correspond to an engineering-application for electronic cool-
ing and the second-problem is studied for an engineering-design of
the structure subjected to uid-dynamics forces and are shown in
Fig 11 Both the problems are considered as two-dimensional and
unsteady However the results asymptotes from the unsteady to a
steady state in the rst-problem and to a periodic (dynamic-steady)
state in the second-problem
Figure 11 Schematic and the associated parameters for the two example prob-lems (a) thermal conduction module of a heat sink and (b) theclassical ow across a circular cylinder
6 1 Introduction
The rst-problem is taken from the classic book on heat-transfer
by Incropera and Dewitt (1996) shown in Fig 11(a) The g-
ure shows a periodic module of the plate where a constant heat-ux
qW = 10Wcm2 (dissipated from certain electronic devices) is rst con-
ducted to an aluminum plate and then convected to the water ow-
ing through the rectangular channel grooved in the plate The forced
convection results in a convection coecient of h = 5000Wm2K in-
side the channel The unsteady heat-transfer results in an interesting
time-wise varying conduction heat ow in the plate and instantaneous
convection heat transfer rate Qprimeconv (per unit width of the plate per-
pendicular to the plane shown in Fig 11(a)) into the water
The second-problem is encountered if you are standing over a
bridge and look below to the water passing over a pillar (of circu-
lar cross-section) of the bridge shown in Fig 11(b) as a general case
of 2D ow across a cylinder The uid-dynamics results in a beau-
tiful time-wise varying structure of the uid ow just downstream
of the pillar and certain engineering-parameters The parameters
correspond to the uid-dynamic forces acting in the horizontal (or
streamwise) and vertical (or transverse) direction called as the drag-
force FD and lift-force FL respectively The respective force is pre-
sented below in a non-dimensional form called as the drag-coecient
CD and lift-coecient CL equations shown in Fig 11(b)
If you use your video camera to capture a movie for the two
problems you will get series of pictures for the plate in the rst-
problem and the owing uid in the second-problem Since these
movies result in a pictorial information but not the ow-property
based uid-dynamic information such movies are not relevant for
the uid-dynamics and heat-transfer study Instead if you use a
CFD software for computer simulation you can capture various types
of movies The various ow-properties based scientically-exciting
movies are presented here as the pictures at certain time-instant in
Fig 12(a)-(d) and Fig 13(a1)-(d2) The former gure shows gray-
colored ooded contour for the temperature whereas the latter gure
shows gray-colored ooded contour for the pressure and vorticity and
vector-plot for the velocity For the gray-colored ooded contours a
color bar can be seen in the gures for the relative magnitude of the
variable These gures also show line contours of stream-function
Introduction to CFD 7
and heat-function as streamlines and heatlines respectively They
are the visualization techniques for a 2D steady heat and uid ow
The heatlines are analogous to streamlines and represents the direc-
tion of heat ow under steady state condition proposed by Kimura
and Bejan (1983) later presented in a heat-transfer book by Bejan
(1984) The steady-state heatlines can be seen in Fig 12(d) where
the direction for the conduction-ux is tangential at any point on the
heatlines analogous to that for the mass-ux in a streamline Thus
for a better presentation of the structure of heat and uid ow the
scientically-exciting movies are not only limited to ow-properties
but also correspond to certain ow visualization techniquesAn engineering-parameter based engineering-relevant uid dy-
namic movie is presented in Fig 12(e) for the convection heat trans-
fer rate per unit width at the various walls on the channel Qprimew at the
west-wall Qprimen at the north-wall Q
primee at the east-wall and their cumu-
lative value Qprimeconv (refer Fig 11(a)) Another engineering-relevant
movie is presented in Fig 13(e) for the lift and drag coecient Note
that the movie here corresponds to the motion of a lled symbol over
a line for the temporal variation of an engineering-parameter Thus
the engineering-relevant movies are one-dimensional lower than the
scientically-exciting movies here for the 2D problems the former
movie is presented as 1D and the latter as 2D results
1111 Heat-Conduction
For the computational set-up of the heat conduction problem shown
in Fig 11(a) the results obtained from a 2D CFD simulation of
the conduction heat transfer are shown in Fig 12 For a uni-
form initial temperature of the plate as Tinfin = 15oC Fig 12(a)-(d)
shows the isotherms and heatlines in the plate as the pictures of the
scientically-exciting movies at certain discrete time instants t = 25
5 10 and 20 sec For the same discrete time instants the temporal
variation of the position of the symbols over a line plot is presented in
Fig 12(e) The gure is shown for each of the heat transfer rate and
is obtained from a engineering-relevant 1D movie There is a tempo-
ral variation of picture in the scientically-exciting movie and that
of position of symbol (along the line plot) in the engineering-relevant
movieAccording to caloric theory of the famous French scientist An-
tonie Lavoisier heat is considered as an invisible tasteless odorless
8 1 Introduction
t(sec)
Qrsquo (W
m)
0 5 10 15 20 250
200
400
600
800
1000Qrsquo
conv
Qrsquow
Qrsquon
Qrsquoe
Qrsquoin
(e)
(b)
(c)(d)
(a)
(a) t=25 sec31
29
27
25
23
21
19
17
15
T(oC)
(c) t=10 sec
(b) t=5 sec
(d) t=20 sec
Figure 12 Thermal conduction module of a heat sink temporal variation of(a)-(d) isotherms and heatlines and (e) the convection heat transferrate per unit width at the various walls of the channels and theircumulative total value Q
primeconv For the discrete time-instant corre-
sponding to heat ow patterns in (a)-(d) temporal variation of thevarious heat transfer rates are shown as symbols in (e) (a)-(c) arethe transient and (d) represents a steady state result
weightless uid which he called caloric uid Presently the caloric
theory is overtaken by mechanical theory of heat minus heat is not a sub-
stance but a dynamical form of mechanical eect (Thomson 1851)
Nevertheless there are certain problems involving heat ow for which
Lavoisiers approach is rather useful such as the discussion on heat-
lines here
The role of heatlines for heat ow is analogous to that of stream-
lines for uid ow (Paramane and Sharma 2009) For the uid-ow
the dierence of stream-function values represents the rate of uid
ow the function remains constant on a solid wall the tangent to
a streamline represents the direction of the uid-ow (with no ow
in the normal direction) and the streamline originate or emerge at
Introduction to CFD 9
a mass source Analogously for the heat-ow the dierence of heat-
function values represents the rate of heat ow the function remains
constant on an adiabatic wall the tangent to a heatline represents the
direction of the heat-ow (with no ow in the normal direction) and
the heatline originate or emerge at the heat-source For the present
problem in Fig 11(a) the heat-source is at the inlet (or left) bound-
ary and the heat-sink is at the surface of the channel Thus the
steady-state heatlines in Fig 12(d) shows the heat ow emerging
from the inletsource and ending on the channel-wallssink Further-
more it can be seen in the gure that the heatlines which are close
to the adiabatic walls are parallel to the walls The source sink and
adiabatic-walls can be seen in Fig 11(a)
Figure 12(e) shows an asymptotic time-wise increase in the con-
vection heat transfer per unit width at the various walls of the channel
and its cumulative value Qprimeconv (= Q
primew+Q
primen+Q
primee) It can be also be seen
that the heat-transfer per unit width Qprimew at the west-wall and Q
primen at
the north-wall are larger than Qprimee at the east-wall of the channel As
time progresses the gure shows that Qprimeconv increases monotonically
and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-
der steady state condition refer Fig 11(a) The dierence between
the Qprimein and Q
primeconv under the transient condition is utilized for the
sensible heating and results in the increase in the temperature of the
plate Under the steady state condition note from Fig 12(d) that
the maximum temperature is close to 32oC minus well within the per-
missible limit for the reliable and ecient operation of the electronic
device
1112 Fluid-Dynamics
For the free-stream ow across a circular cylinderpillar at a Reynolds
number of 100 the non-dimensional results obtained from a 2D CFD
simulation are shown in Fig 13 In contrast to the previous prob-
lem which reaches steady state the present problem reaches to an un-
steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-
ent increasing time instants within one time-period of the periodic-
ow the results are presented in Fig 13(a)-(d) The gure shows
the close-up view of the pictures of the scientically-exciting movies
for ow-properties (velocity pressure and vorticity) and streamlines
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow
Introduction to CFD xvii
10 Computational Fluid Dynamics on a Co-located Grid309
101 Momentum-Interpolation Method Strategy to avoid
the Pressure-Velocity Decoupling on a Co-located Grid 310
102 Coecients of LAEs based Solution Methodology on
a Non-Uniform Grid Semi-Explicit and Semi-Implicit
Method 314
1021 Predictor Step 316
1022 Corrector Step 318
1023 Solution Algorithm 323
Problems 329
III CFD for a Complex-Geometry
331
11 Computational Heat Conduction on a Curvilinear
Grid 333
111 Curvilinear Grid Generation 333
1111 Algebraic Grid Generation 334
1112 Elliptic Grid Generation 336
112 Physical Law based Finite Volume Method 343
1121 Unsteady and Source Term 343
1122 Diusion Term 344
1123 All Terms 349
113 Computation of Geometrical Properties 349
114 Flux based Solution Methodology 352
1141 Explicit Method 354
1142 Implementation Details 354
Problems 359
12 Computational Fluid Dynamics on a Curvilinear Grid361
121 Physical Law based Finite Volume Method 361
1211 Mass Conservation 363
1212 Momentum Conservation 363
122 Solution Methodology Semi-Explicit Method 372
1221 Predictor Step 373
1222 Corrector Step 375
Problems 380
References 383
Index 389
Part I
Introduction and Essentials
The book on an introductory course in CFD starts with the two intro-
ductory chapters on CFD followed by one chapter each on essentials
of two prerequisite courses minus uid-dynamics and heat transfer and
numerical-methods The rst chapter on introduction presents an
eventful thoughtful and intuitive discussion on What is CFD and
Why to study CFD whereas How CFD works which involves lots
of details is presented separately in the second chapter The third-
chapter on essentials of uid-dynamics and heat transfer presents
physical-laws transport-mechanisms physical-law based dierential-
formulation volumetric and ux terms and mathematical formula-
tion The fourth-chapter on essentials of numerical-methods presents
nite dierence method iterative solution of the system of linear alge-
braic equation numerical dierentiation and numerical integration
The presentation for the essential of the two prerequisite courses is
customized to CFD development application and analysis
1
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
1
Introduction
The introduction of any course consists of the three basic questions
What is it Why to study How it works An eventful thoughtful
and intuitive answer to the rst two questions are presented in this
chapter The third question which involves lots of working details
on the three aspects of the Computational Fluid Dynamics (CFD)
(development application and analysis) is presented separately in
the next chapter The novelty scope and purpose of this book is
also presented at the end of this chapter
11 CFD What is it
CFD is a theoretical-method of scientic and engineering investiga-
tion concerned with the development and application of a video-
camera like tool (a software) which is used for a unied cause-and-
eect based analysis of a uid-dynamics as well as heat and mass
transfer problem
CFD is a subject where you rst learn how to develop a product
minus a software which acts like a virtual video-camera Then you learn
how to apply or use this product to generate uid-dynamics movies
Finally you learn how to analyze the detailed spatial and temporal
uid-dynamics information in the movies The analysis is done to
come up with a scientically-exciting as well as engineering-relevant
story (unied cause-and-eect study) of a uid-dynamics situation in
nature as well as industrial-applications With a continuous develop-
ment and a wider application of CFD the word uid-dynamics in
3
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
4 1 Introduction
the CFD has become more generic as it also corresponds to heat and
mass transfer as well as chemical reaction
Fluid-dynamics information are of two types scientic and en-
gineering Scientic-information corresponds to a structure of the
heat and uid ow due to a physical phenomenon in uid-dynamics
such as boundary-layer ow-separation wake-formation and vortex-
shedding The ow structures are obtained from a temporal variation
of ow-properties (such as velocity pressure temperature and many
other variables which characterizes a ow) called as scientically-
exciting movies Engineering-information corresponds to certain pa-
rameters which are important in an engineering application called
as engineering-parameters for example lift force drag force wall
shear stress pressure drop and rate of heat-transfer They are ob-
tained as the temporal variation of engineering-parameters called as
engineering-relevant movies The engineering-parameters are the ef-
fect which are caused by the ow structures minus their correlation leads
to the unied cause-and-eect study A uid-dynamics movie consist
of a time-wise varying series of pictures which give uid-dynamics
information Each ow-property and engineering-parameter results
in one scientically-exciting and engineering-relevant movie respec-
tively When the large number of both the types of movies are
played synchronized in-time the detailed spatial as well as tempo-
ral uid-dynamic information greatly helps in the unied cause-and-
eect study of a uid-dynamics problem
Alternatively CFD is a subject concerned with the development of
computer-programs for the computer-simulation and study of a natu-
ral or an engineering uid-dynamics system The development starts
with the virtual-development of the system involving a geometri-
cal information based computational development of a solid-model
for the system the system may be already-existing or yet-to-be-
developed physically Thereafter the development involves numerical
solution of the governing algebraic equations for CFD formulated
from the physical laws (conservation of mass momentum and en-
ergy) and initial as well as boundary conditions corresponding to the
system CFD study involves the analysis of the uid-dynamics and
heat-transfer results obtained from the computer-simulations of the
virtual-system which is subjected to certain governing-parameters
Introduction to CFD 5
111 CFD as a Scientic and Engineering Analysis Tool
For a better understanding of the denition of the CFD and its ap-
plication as a scientic and engineering analysis tool two example
problems are presented rst conduction heat transfer in a plate
and second uid ow across a circular pillar of a bridge The rst-
problem correspond to an engineering-application for electronic cool-
ing and the second-problem is studied for an engineering-design of
the structure subjected to uid-dynamics forces and are shown in
Fig 11 Both the problems are considered as two-dimensional and
unsteady However the results asymptotes from the unsteady to a
steady state in the rst-problem and to a periodic (dynamic-steady)
state in the second-problem
Figure 11 Schematic and the associated parameters for the two example prob-lems (a) thermal conduction module of a heat sink and (b) theclassical ow across a circular cylinder
6 1 Introduction
The rst-problem is taken from the classic book on heat-transfer
by Incropera and Dewitt (1996) shown in Fig 11(a) The g-
ure shows a periodic module of the plate where a constant heat-ux
qW = 10Wcm2 (dissipated from certain electronic devices) is rst con-
ducted to an aluminum plate and then convected to the water ow-
ing through the rectangular channel grooved in the plate The forced
convection results in a convection coecient of h = 5000Wm2K in-
side the channel The unsteady heat-transfer results in an interesting
time-wise varying conduction heat ow in the plate and instantaneous
convection heat transfer rate Qprimeconv (per unit width of the plate per-
pendicular to the plane shown in Fig 11(a)) into the water
The second-problem is encountered if you are standing over a
bridge and look below to the water passing over a pillar (of circu-
lar cross-section) of the bridge shown in Fig 11(b) as a general case
of 2D ow across a cylinder The uid-dynamics results in a beau-
tiful time-wise varying structure of the uid ow just downstream
of the pillar and certain engineering-parameters The parameters
correspond to the uid-dynamic forces acting in the horizontal (or
streamwise) and vertical (or transverse) direction called as the drag-
force FD and lift-force FL respectively The respective force is pre-
sented below in a non-dimensional form called as the drag-coecient
CD and lift-coecient CL equations shown in Fig 11(b)
If you use your video camera to capture a movie for the two
problems you will get series of pictures for the plate in the rst-
problem and the owing uid in the second-problem Since these
movies result in a pictorial information but not the ow-property
based uid-dynamic information such movies are not relevant for
the uid-dynamics and heat-transfer study Instead if you use a
CFD software for computer simulation you can capture various types
of movies The various ow-properties based scientically-exciting
movies are presented here as the pictures at certain time-instant in
Fig 12(a)-(d) and Fig 13(a1)-(d2) The former gure shows gray-
colored ooded contour for the temperature whereas the latter gure
shows gray-colored ooded contour for the pressure and vorticity and
vector-plot for the velocity For the gray-colored ooded contours a
color bar can be seen in the gures for the relative magnitude of the
variable These gures also show line contours of stream-function
Introduction to CFD 7
and heat-function as streamlines and heatlines respectively They
are the visualization techniques for a 2D steady heat and uid ow
The heatlines are analogous to streamlines and represents the direc-
tion of heat ow under steady state condition proposed by Kimura
and Bejan (1983) later presented in a heat-transfer book by Bejan
(1984) The steady-state heatlines can be seen in Fig 12(d) where
the direction for the conduction-ux is tangential at any point on the
heatlines analogous to that for the mass-ux in a streamline Thus
for a better presentation of the structure of heat and uid ow the
scientically-exciting movies are not only limited to ow-properties
but also correspond to certain ow visualization techniquesAn engineering-parameter based engineering-relevant uid dy-
namic movie is presented in Fig 12(e) for the convection heat trans-
fer rate per unit width at the various walls on the channel Qprimew at the
west-wall Qprimen at the north-wall Q
primee at the east-wall and their cumu-
lative value Qprimeconv (refer Fig 11(a)) Another engineering-relevant
movie is presented in Fig 13(e) for the lift and drag coecient Note
that the movie here corresponds to the motion of a lled symbol over
a line for the temporal variation of an engineering-parameter Thus
the engineering-relevant movies are one-dimensional lower than the
scientically-exciting movies here for the 2D problems the former
movie is presented as 1D and the latter as 2D results
1111 Heat-Conduction
For the computational set-up of the heat conduction problem shown
in Fig 11(a) the results obtained from a 2D CFD simulation of
the conduction heat transfer are shown in Fig 12 For a uni-
form initial temperature of the plate as Tinfin = 15oC Fig 12(a)-(d)
shows the isotherms and heatlines in the plate as the pictures of the
scientically-exciting movies at certain discrete time instants t = 25
5 10 and 20 sec For the same discrete time instants the temporal
variation of the position of the symbols over a line plot is presented in
Fig 12(e) The gure is shown for each of the heat transfer rate and
is obtained from a engineering-relevant 1D movie There is a tempo-
ral variation of picture in the scientically-exciting movie and that
of position of symbol (along the line plot) in the engineering-relevant
movieAccording to caloric theory of the famous French scientist An-
tonie Lavoisier heat is considered as an invisible tasteless odorless
8 1 Introduction
t(sec)
Qrsquo (W
m)
0 5 10 15 20 250
200
400
600
800
1000Qrsquo
conv
Qrsquow
Qrsquon
Qrsquoe
Qrsquoin
(e)
(b)
(c)(d)
(a)
(a) t=25 sec31
29
27
25
23
21
19
17
15
T(oC)
(c) t=10 sec
(b) t=5 sec
(d) t=20 sec
Figure 12 Thermal conduction module of a heat sink temporal variation of(a)-(d) isotherms and heatlines and (e) the convection heat transferrate per unit width at the various walls of the channels and theircumulative total value Q
primeconv For the discrete time-instant corre-
sponding to heat ow patterns in (a)-(d) temporal variation of thevarious heat transfer rates are shown as symbols in (e) (a)-(c) arethe transient and (d) represents a steady state result
weightless uid which he called caloric uid Presently the caloric
theory is overtaken by mechanical theory of heat minus heat is not a sub-
stance but a dynamical form of mechanical eect (Thomson 1851)
Nevertheless there are certain problems involving heat ow for which
Lavoisiers approach is rather useful such as the discussion on heat-
lines here
The role of heatlines for heat ow is analogous to that of stream-
lines for uid ow (Paramane and Sharma 2009) For the uid-ow
the dierence of stream-function values represents the rate of uid
ow the function remains constant on a solid wall the tangent to
a streamline represents the direction of the uid-ow (with no ow
in the normal direction) and the streamline originate or emerge at
Introduction to CFD 9
a mass source Analogously for the heat-ow the dierence of heat-
function values represents the rate of heat ow the function remains
constant on an adiabatic wall the tangent to a heatline represents the
direction of the heat-ow (with no ow in the normal direction) and
the heatline originate or emerge at the heat-source For the present
problem in Fig 11(a) the heat-source is at the inlet (or left) bound-
ary and the heat-sink is at the surface of the channel Thus the
steady-state heatlines in Fig 12(d) shows the heat ow emerging
from the inletsource and ending on the channel-wallssink Further-
more it can be seen in the gure that the heatlines which are close
to the adiabatic walls are parallel to the walls The source sink and
adiabatic-walls can be seen in Fig 11(a)
Figure 12(e) shows an asymptotic time-wise increase in the con-
vection heat transfer per unit width at the various walls of the channel
and its cumulative value Qprimeconv (= Q
primew+Q
primen+Q
primee) It can be also be seen
that the heat-transfer per unit width Qprimew at the west-wall and Q
primen at
the north-wall are larger than Qprimee at the east-wall of the channel As
time progresses the gure shows that Qprimeconv increases monotonically
and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-
der steady state condition refer Fig 11(a) The dierence between
the Qprimein and Q
primeconv under the transient condition is utilized for the
sensible heating and results in the increase in the temperature of the
plate Under the steady state condition note from Fig 12(d) that
the maximum temperature is close to 32oC minus well within the per-
missible limit for the reliable and ecient operation of the electronic
device
1112 Fluid-Dynamics
For the free-stream ow across a circular cylinderpillar at a Reynolds
number of 100 the non-dimensional results obtained from a 2D CFD
simulation are shown in Fig 13 In contrast to the previous prob-
lem which reaches steady state the present problem reaches to an un-
steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-
ent increasing time instants within one time-period of the periodic-
ow the results are presented in Fig 13(a)-(d) The gure shows
the close-up view of the pictures of the scientically-exciting movies
for ow-properties (velocity pressure and vorticity) and streamlines
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow
12 Computational Fluid Dynamics on a Curvilinear Grid361
121 Physical Law based Finite Volume Method 361
1211 Mass Conservation 363
1212 Momentum Conservation 363
122 Solution Methodology Semi-Explicit Method 372
1221 Predictor Step 373
1222 Corrector Step 375
Problems 380
References 383
Index 389
Part I
Introduction and Essentials
The book on an introductory course in CFD starts with the two intro-
ductory chapters on CFD followed by one chapter each on essentials
of two prerequisite courses minus uid-dynamics and heat transfer and
numerical-methods The rst chapter on introduction presents an
eventful thoughtful and intuitive discussion on What is CFD and
Why to study CFD whereas How CFD works which involves lots
of details is presented separately in the second chapter The third-
chapter on essentials of uid-dynamics and heat transfer presents
physical-laws transport-mechanisms physical-law based dierential-
formulation volumetric and ux terms and mathematical formula-
tion The fourth-chapter on essentials of numerical-methods presents
nite dierence method iterative solution of the system of linear alge-
braic equation numerical dierentiation and numerical integration
The presentation for the essential of the two prerequisite courses is
customized to CFD development application and analysis
1
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
1
Introduction
The introduction of any course consists of the three basic questions
What is it Why to study How it works An eventful thoughtful
and intuitive answer to the rst two questions are presented in this
chapter The third question which involves lots of working details
on the three aspects of the Computational Fluid Dynamics (CFD)
(development application and analysis) is presented separately in
the next chapter The novelty scope and purpose of this book is
also presented at the end of this chapter
11 CFD What is it
CFD is a theoretical-method of scientic and engineering investiga-
tion concerned with the development and application of a video-
camera like tool (a software) which is used for a unied cause-and-
eect based analysis of a uid-dynamics as well as heat and mass
transfer problem
CFD is a subject where you rst learn how to develop a product
minus a software which acts like a virtual video-camera Then you learn
how to apply or use this product to generate uid-dynamics movies
Finally you learn how to analyze the detailed spatial and temporal
uid-dynamics information in the movies The analysis is done to
come up with a scientically-exciting as well as engineering-relevant
story (unied cause-and-eect study) of a uid-dynamics situation in
nature as well as industrial-applications With a continuous develop-
ment and a wider application of CFD the word uid-dynamics in
3
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
4 1 Introduction
the CFD has become more generic as it also corresponds to heat and
mass transfer as well as chemical reaction
Fluid-dynamics information are of two types scientic and en-
gineering Scientic-information corresponds to a structure of the
heat and uid ow due to a physical phenomenon in uid-dynamics
such as boundary-layer ow-separation wake-formation and vortex-
shedding The ow structures are obtained from a temporal variation
of ow-properties (such as velocity pressure temperature and many
other variables which characterizes a ow) called as scientically-
exciting movies Engineering-information corresponds to certain pa-
rameters which are important in an engineering application called
as engineering-parameters for example lift force drag force wall
shear stress pressure drop and rate of heat-transfer They are ob-
tained as the temporal variation of engineering-parameters called as
engineering-relevant movies The engineering-parameters are the ef-
fect which are caused by the ow structures minus their correlation leads
to the unied cause-and-eect study A uid-dynamics movie consist
of a time-wise varying series of pictures which give uid-dynamics
information Each ow-property and engineering-parameter results
in one scientically-exciting and engineering-relevant movie respec-
tively When the large number of both the types of movies are
played synchronized in-time the detailed spatial as well as tempo-
ral uid-dynamic information greatly helps in the unied cause-and-
eect study of a uid-dynamics problem
Alternatively CFD is a subject concerned with the development of
computer-programs for the computer-simulation and study of a natu-
ral or an engineering uid-dynamics system The development starts
with the virtual-development of the system involving a geometri-
cal information based computational development of a solid-model
for the system the system may be already-existing or yet-to-be-
developed physically Thereafter the development involves numerical
solution of the governing algebraic equations for CFD formulated
from the physical laws (conservation of mass momentum and en-
ergy) and initial as well as boundary conditions corresponding to the
system CFD study involves the analysis of the uid-dynamics and
heat-transfer results obtained from the computer-simulations of the
virtual-system which is subjected to certain governing-parameters
Introduction to CFD 5
111 CFD as a Scientic and Engineering Analysis Tool
For a better understanding of the denition of the CFD and its ap-
plication as a scientic and engineering analysis tool two example
problems are presented rst conduction heat transfer in a plate
and second uid ow across a circular pillar of a bridge The rst-
problem correspond to an engineering-application for electronic cool-
ing and the second-problem is studied for an engineering-design of
the structure subjected to uid-dynamics forces and are shown in
Fig 11 Both the problems are considered as two-dimensional and
unsteady However the results asymptotes from the unsteady to a
steady state in the rst-problem and to a periodic (dynamic-steady)
state in the second-problem
Figure 11 Schematic and the associated parameters for the two example prob-lems (a) thermal conduction module of a heat sink and (b) theclassical ow across a circular cylinder
6 1 Introduction
The rst-problem is taken from the classic book on heat-transfer
by Incropera and Dewitt (1996) shown in Fig 11(a) The g-
ure shows a periodic module of the plate where a constant heat-ux
qW = 10Wcm2 (dissipated from certain electronic devices) is rst con-
ducted to an aluminum plate and then convected to the water ow-
ing through the rectangular channel grooved in the plate The forced
convection results in a convection coecient of h = 5000Wm2K in-
side the channel The unsteady heat-transfer results in an interesting
time-wise varying conduction heat ow in the plate and instantaneous
convection heat transfer rate Qprimeconv (per unit width of the plate per-
pendicular to the plane shown in Fig 11(a)) into the water
The second-problem is encountered if you are standing over a
bridge and look below to the water passing over a pillar (of circu-
lar cross-section) of the bridge shown in Fig 11(b) as a general case
of 2D ow across a cylinder The uid-dynamics results in a beau-
tiful time-wise varying structure of the uid ow just downstream
of the pillar and certain engineering-parameters The parameters
correspond to the uid-dynamic forces acting in the horizontal (or
streamwise) and vertical (or transverse) direction called as the drag-
force FD and lift-force FL respectively The respective force is pre-
sented below in a non-dimensional form called as the drag-coecient
CD and lift-coecient CL equations shown in Fig 11(b)
If you use your video camera to capture a movie for the two
problems you will get series of pictures for the plate in the rst-
problem and the owing uid in the second-problem Since these
movies result in a pictorial information but not the ow-property
based uid-dynamic information such movies are not relevant for
the uid-dynamics and heat-transfer study Instead if you use a
CFD software for computer simulation you can capture various types
of movies The various ow-properties based scientically-exciting
movies are presented here as the pictures at certain time-instant in
Fig 12(a)-(d) and Fig 13(a1)-(d2) The former gure shows gray-
colored ooded contour for the temperature whereas the latter gure
shows gray-colored ooded contour for the pressure and vorticity and
vector-plot for the velocity For the gray-colored ooded contours a
color bar can be seen in the gures for the relative magnitude of the
variable These gures also show line contours of stream-function
Introduction to CFD 7
and heat-function as streamlines and heatlines respectively They
are the visualization techniques for a 2D steady heat and uid ow
The heatlines are analogous to streamlines and represents the direc-
tion of heat ow under steady state condition proposed by Kimura
and Bejan (1983) later presented in a heat-transfer book by Bejan
(1984) The steady-state heatlines can be seen in Fig 12(d) where
the direction for the conduction-ux is tangential at any point on the
heatlines analogous to that for the mass-ux in a streamline Thus
for a better presentation of the structure of heat and uid ow the
scientically-exciting movies are not only limited to ow-properties
but also correspond to certain ow visualization techniquesAn engineering-parameter based engineering-relevant uid dy-
namic movie is presented in Fig 12(e) for the convection heat trans-
fer rate per unit width at the various walls on the channel Qprimew at the
west-wall Qprimen at the north-wall Q
primee at the east-wall and their cumu-
lative value Qprimeconv (refer Fig 11(a)) Another engineering-relevant
movie is presented in Fig 13(e) for the lift and drag coecient Note
that the movie here corresponds to the motion of a lled symbol over
a line for the temporal variation of an engineering-parameter Thus
the engineering-relevant movies are one-dimensional lower than the
scientically-exciting movies here for the 2D problems the former
movie is presented as 1D and the latter as 2D results
1111 Heat-Conduction
For the computational set-up of the heat conduction problem shown
in Fig 11(a) the results obtained from a 2D CFD simulation of
the conduction heat transfer are shown in Fig 12 For a uni-
form initial temperature of the plate as Tinfin = 15oC Fig 12(a)-(d)
shows the isotherms and heatlines in the plate as the pictures of the
scientically-exciting movies at certain discrete time instants t = 25
5 10 and 20 sec For the same discrete time instants the temporal
variation of the position of the symbols over a line plot is presented in
Fig 12(e) The gure is shown for each of the heat transfer rate and
is obtained from a engineering-relevant 1D movie There is a tempo-
ral variation of picture in the scientically-exciting movie and that
of position of symbol (along the line plot) in the engineering-relevant
movieAccording to caloric theory of the famous French scientist An-
tonie Lavoisier heat is considered as an invisible tasteless odorless
8 1 Introduction
t(sec)
Qrsquo (W
m)
0 5 10 15 20 250
200
400
600
800
1000Qrsquo
conv
Qrsquow
Qrsquon
Qrsquoe
Qrsquoin
(e)
(b)
(c)(d)
(a)
(a) t=25 sec31
29
27
25
23
21
19
17
15
T(oC)
(c) t=10 sec
(b) t=5 sec
(d) t=20 sec
Figure 12 Thermal conduction module of a heat sink temporal variation of(a)-(d) isotherms and heatlines and (e) the convection heat transferrate per unit width at the various walls of the channels and theircumulative total value Q
primeconv For the discrete time-instant corre-
sponding to heat ow patterns in (a)-(d) temporal variation of thevarious heat transfer rates are shown as symbols in (e) (a)-(c) arethe transient and (d) represents a steady state result
weightless uid which he called caloric uid Presently the caloric
theory is overtaken by mechanical theory of heat minus heat is not a sub-
stance but a dynamical form of mechanical eect (Thomson 1851)
Nevertheless there are certain problems involving heat ow for which
Lavoisiers approach is rather useful such as the discussion on heat-
lines here
The role of heatlines for heat ow is analogous to that of stream-
lines for uid ow (Paramane and Sharma 2009) For the uid-ow
the dierence of stream-function values represents the rate of uid
ow the function remains constant on a solid wall the tangent to
a streamline represents the direction of the uid-ow (with no ow
in the normal direction) and the streamline originate or emerge at
Introduction to CFD 9
a mass source Analogously for the heat-ow the dierence of heat-
function values represents the rate of heat ow the function remains
constant on an adiabatic wall the tangent to a heatline represents the
direction of the heat-ow (with no ow in the normal direction) and
the heatline originate or emerge at the heat-source For the present
problem in Fig 11(a) the heat-source is at the inlet (or left) bound-
ary and the heat-sink is at the surface of the channel Thus the
steady-state heatlines in Fig 12(d) shows the heat ow emerging
from the inletsource and ending on the channel-wallssink Further-
more it can be seen in the gure that the heatlines which are close
to the adiabatic walls are parallel to the walls The source sink and
adiabatic-walls can be seen in Fig 11(a)
Figure 12(e) shows an asymptotic time-wise increase in the con-
vection heat transfer per unit width at the various walls of the channel
and its cumulative value Qprimeconv (= Q
primew+Q
primen+Q
primee) It can be also be seen
that the heat-transfer per unit width Qprimew at the west-wall and Q
primen at
the north-wall are larger than Qprimee at the east-wall of the channel As
time progresses the gure shows that Qprimeconv increases monotonically
and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-
der steady state condition refer Fig 11(a) The dierence between
the Qprimein and Q
primeconv under the transient condition is utilized for the
sensible heating and results in the increase in the temperature of the
plate Under the steady state condition note from Fig 12(d) that
the maximum temperature is close to 32oC minus well within the per-
missible limit for the reliable and ecient operation of the electronic
device
1112 Fluid-Dynamics
For the free-stream ow across a circular cylinderpillar at a Reynolds
number of 100 the non-dimensional results obtained from a 2D CFD
simulation are shown in Fig 13 In contrast to the previous prob-
lem which reaches steady state the present problem reaches to an un-
steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-
ent increasing time instants within one time-period of the periodic-
ow the results are presented in Fig 13(a)-(d) The gure shows
the close-up view of the pictures of the scientically-exciting movies
for ow-properties (velocity pressure and vorticity) and streamlines
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow
Part I
Introduction and Essentials
The book on an introductory course in CFD starts with the two intro-
ductory chapters on CFD followed by one chapter each on essentials
of two prerequisite courses minus uid-dynamics and heat transfer and
numerical-methods The rst chapter on introduction presents an
eventful thoughtful and intuitive discussion on What is CFD and
Why to study CFD whereas How CFD works which involves lots
of details is presented separately in the second chapter The third-
chapter on essentials of uid-dynamics and heat transfer presents
physical-laws transport-mechanisms physical-law based dierential-
formulation volumetric and ux terms and mathematical formula-
tion The fourth-chapter on essentials of numerical-methods presents
nite dierence method iterative solution of the system of linear alge-
braic equation numerical dierentiation and numerical integration
The presentation for the essential of the two prerequisite courses is
customized to CFD development application and analysis
1
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
1
Introduction
The introduction of any course consists of the three basic questions
What is it Why to study How it works An eventful thoughtful
and intuitive answer to the rst two questions are presented in this
chapter The third question which involves lots of working details
on the three aspects of the Computational Fluid Dynamics (CFD)
(development application and analysis) is presented separately in
the next chapter The novelty scope and purpose of this book is
also presented at the end of this chapter
11 CFD What is it
CFD is a theoretical-method of scientic and engineering investiga-
tion concerned with the development and application of a video-
camera like tool (a software) which is used for a unied cause-and-
eect based analysis of a uid-dynamics as well as heat and mass
transfer problem
CFD is a subject where you rst learn how to develop a product
minus a software which acts like a virtual video-camera Then you learn
how to apply or use this product to generate uid-dynamics movies
Finally you learn how to analyze the detailed spatial and temporal
uid-dynamics information in the movies The analysis is done to
come up with a scientically-exciting as well as engineering-relevant
story (unied cause-and-eect study) of a uid-dynamics situation in
nature as well as industrial-applications With a continuous develop-
ment and a wider application of CFD the word uid-dynamics in
3
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
4 1 Introduction
the CFD has become more generic as it also corresponds to heat and
mass transfer as well as chemical reaction
Fluid-dynamics information are of two types scientic and en-
gineering Scientic-information corresponds to a structure of the
heat and uid ow due to a physical phenomenon in uid-dynamics
such as boundary-layer ow-separation wake-formation and vortex-
shedding The ow structures are obtained from a temporal variation
of ow-properties (such as velocity pressure temperature and many
other variables which characterizes a ow) called as scientically-
exciting movies Engineering-information corresponds to certain pa-
rameters which are important in an engineering application called
as engineering-parameters for example lift force drag force wall
shear stress pressure drop and rate of heat-transfer They are ob-
tained as the temporal variation of engineering-parameters called as
engineering-relevant movies The engineering-parameters are the ef-
fect which are caused by the ow structures minus their correlation leads
to the unied cause-and-eect study A uid-dynamics movie consist
of a time-wise varying series of pictures which give uid-dynamics
information Each ow-property and engineering-parameter results
in one scientically-exciting and engineering-relevant movie respec-
tively When the large number of both the types of movies are
played synchronized in-time the detailed spatial as well as tempo-
ral uid-dynamic information greatly helps in the unied cause-and-
eect study of a uid-dynamics problem
Alternatively CFD is a subject concerned with the development of
computer-programs for the computer-simulation and study of a natu-
ral or an engineering uid-dynamics system The development starts
with the virtual-development of the system involving a geometri-
cal information based computational development of a solid-model
for the system the system may be already-existing or yet-to-be-
developed physically Thereafter the development involves numerical
solution of the governing algebraic equations for CFD formulated
from the physical laws (conservation of mass momentum and en-
ergy) and initial as well as boundary conditions corresponding to the
system CFD study involves the analysis of the uid-dynamics and
heat-transfer results obtained from the computer-simulations of the
virtual-system which is subjected to certain governing-parameters
Introduction to CFD 5
111 CFD as a Scientic and Engineering Analysis Tool
For a better understanding of the denition of the CFD and its ap-
plication as a scientic and engineering analysis tool two example
problems are presented rst conduction heat transfer in a plate
and second uid ow across a circular pillar of a bridge The rst-
problem correspond to an engineering-application for electronic cool-
ing and the second-problem is studied for an engineering-design of
the structure subjected to uid-dynamics forces and are shown in
Fig 11 Both the problems are considered as two-dimensional and
unsteady However the results asymptotes from the unsteady to a
steady state in the rst-problem and to a periodic (dynamic-steady)
state in the second-problem
Figure 11 Schematic and the associated parameters for the two example prob-lems (a) thermal conduction module of a heat sink and (b) theclassical ow across a circular cylinder
6 1 Introduction
The rst-problem is taken from the classic book on heat-transfer
by Incropera and Dewitt (1996) shown in Fig 11(a) The g-
ure shows a periodic module of the plate where a constant heat-ux
qW = 10Wcm2 (dissipated from certain electronic devices) is rst con-
ducted to an aluminum plate and then convected to the water ow-
ing through the rectangular channel grooved in the plate The forced
convection results in a convection coecient of h = 5000Wm2K in-
side the channel The unsteady heat-transfer results in an interesting
time-wise varying conduction heat ow in the plate and instantaneous
convection heat transfer rate Qprimeconv (per unit width of the plate per-
pendicular to the plane shown in Fig 11(a)) into the water
The second-problem is encountered if you are standing over a
bridge and look below to the water passing over a pillar (of circu-
lar cross-section) of the bridge shown in Fig 11(b) as a general case
of 2D ow across a cylinder The uid-dynamics results in a beau-
tiful time-wise varying structure of the uid ow just downstream
of the pillar and certain engineering-parameters The parameters
correspond to the uid-dynamic forces acting in the horizontal (or
streamwise) and vertical (or transverse) direction called as the drag-
force FD and lift-force FL respectively The respective force is pre-
sented below in a non-dimensional form called as the drag-coecient
CD and lift-coecient CL equations shown in Fig 11(b)
If you use your video camera to capture a movie for the two
problems you will get series of pictures for the plate in the rst-
problem and the owing uid in the second-problem Since these
movies result in a pictorial information but not the ow-property
based uid-dynamic information such movies are not relevant for
the uid-dynamics and heat-transfer study Instead if you use a
CFD software for computer simulation you can capture various types
of movies The various ow-properties based scientically-exciting
movies are presented here as the pictures at certain time-instant in
Fig 12(a)-(d) and Fig 13(a1)-(d2) The former gure shows gray-
colored ooded contour for the temperature whereas the latter gure
shows gray-colored ooded contour for the pressure and vorticity and
vector-plot for the velocity For the gray-colored ooded contours a
color bar can be seen in the gures for the relative magnitude of the
variable These gures also show line contours of stream-function
Introduction to CFD 7
and heat-function as streamlines and heatlines respectively They
are the visualization techniques for a 2D steady heat and uid ow
The heatlines are analogous to streamlines and represents the direc-
tion of heat ow under steady state condition proposed by Kimura
and Bejan (1983) later presented in a heat-transfer book by Bejan
(1984) The steady-state heatlines can be seen in Fig 12(d) where
the direction for the conduction-ux is tangential at any point on the
heatlines analogous to that for the mass-ux in a streamline Thus
for a better presentation of the structure of heat and uid ow the
scientically-exciting movies are not only limited to ow-properties
but also correspond to certain ow visualization techniquesAn engineering-parameter based engineering-relevant uid dy-
namic movie is presented in Fig 12(e) for the convection heat trans-
fer rate per unit width at the various walls on the channel Qprimew at the
west-wall Qprimen at the north-wall Q
primee at the east-wall and their cumu-
lative value Qprimeconv (refer Fig 11(a)) Another engineering-relevant
movie is presented in Fig 13(e) for the lift and drag coecient Note
that the movie here corresponds to the motion of a lled symbol over
a line for the temporal variation of an engineering-parameter Thus
the engineering-relevant movies are one-dimensional lower than the
scientically-exciting movies here for the 2D problems the former
movie is presented as 1D and the latter as 2D results
1111 Heat-Conduction
For the computational set-up of the heat conduction problem shown
in Fig 11(a) the results obtained from a 2D CFD simulation of
the conduction heat transfer are shown in Fig 12 For a uni-
form initial temperature of the plate as Tinfin = 15oC Fig 12(a)-(d)
shows the isotherms and heatlines in the plate as the pictures of the
scientically-exciting movies at certain discrete time instants t = 25
5 10 and 20 sec For the same discrete time instants the temporal
variation of the position of the symbols over a line plot is presented in
Fig 12(e) The gure is shown for each of the heat transfer rate and
is obtained from a engineering-relevant 1D movie There is a tempo-
ral variation of picture in the scientically-exciting movie and that
of position of symbol (along the line plot) in the engineering-relevant
movieAccording to caloric theory of the famous French scientist An-
tonie Lavoisier heat is considered as an invisible tasteless odorless
8 1 Introduction
t(sec)
Qrsquo (W
m)
0 5 10 15 20 250
200
400
600
800
1000Qrsquo
conv
Qrsquow
Qrsquon
Qrsquoe
Qrsquoin
(e)
(b)
(c)(d)
(a)
(a) t=25 sec31
29
27
25
23
21
19
17
15
T(oC)
(c) t=10 sec
(b) t=5 sec
(d) t=20 sec
Figure 12 Thermal conduction module of a heat sink temporal variation of(a)-(d) isotherms and heatlines and (e) the convection heat transferrate per unit width at the various walls of the channels and theircumulative total value Q
primeconv For the discrete time-instant corre-
sponding to heat ow patterns in (a)-(d) temporal variation of thevarious heat transfer rates are shown as symbols in (e) (a)-(c) arethe transient and (d) represents a steady state result
weightless uid which he called caloric uid Presently the caloric
theory is overtaken by mechanical theory of heat minus heat is not a sub-
stance but a dynamical form of mechanical eect (Thomson 1851)
Nevertheless there are certain problems involving heat ow for which
Lavoisiers approach is rather useful such as the discussion on heat-
lines here
The role of heatlines for heat ow is analogous to that of stream-
lines for uid ow (Paramane and Sharma 2009) For the uid-ow
the dierence of stream-function values represents the rate of uid
ow the function remains constant on a solid wall the tangent to
a streamline represents the direction of the uid-ow (with no ow
in the normal direction) and the streamline originate or emerge at
Introduction to CFD 9
a mass source Analogously for the heat-ow the dierence of heat-
function values represents the rate of heat ow the function remains
constant on an adiabatic wall the tangent to a heatline represents the
direction of the heat-ow (with no ow in the normal direction) and
the heatline originate or emerge at the heat-source For the present
problem in Fig 11(a) the heat-source is at the inlet (or left) bound-
ary and the heat-sink is at the surface of the channel Thus the
steady-state heatlines in Fig 12(d) shows the heat ow emerging
from the inletsource and ending on the channel-wallssink Further-
more it can be seen in the gure that the heatlines which are close
to the adiabatic walls are parallel to the walls The source sink and
adiabatic-walls can be seen in Fig 11(a)
Figure 12(e) shows an asymptotic time-wise increase in the con-
vection heat transfer per unit width at the various walls of the channel
and its cumulative value Qprimeconv (= Q
primew+Q
primen+Q
primee) It can be also be seen
that the heat-transfer per unit width Qprimew at the west-wall and Q
primen at
the north-wall are larger than Qprimee at the east-wall of the channel As
time progresses the gure shows that Qprimeconv increases monotonically
and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-
der steady state condition refer Fig 11(a) The dierence between
the Qprimein and Q
primeconv under the transient condition is utilized for the
sensible heating and results in the increase in the temperature of the
plate Under the steady state condition note from Fig 12(d) that
the maximum temperature is close to 32oC minus well within the per-
missible limit for the reliable and ecient operation of the electronic
device
1112 Fluid-Dynamics
For the free-stream ow across a circular cylinderpillar at a Reynolds
number of 100 the non-dimensional results obtained from a 2D CFD
simulation are shown in Fig 13 In contrast to the previous prob-
lem which reaches steady state the present problem reaches to an un-
steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-
ent increasing time instants within one time-period of the periodic-
ow the results are presented in Fig 13(a)-(d) The gure shows
the close-up view of the pictures of the scientically-exciting movies
for ow-properties (velocity pressure and vorticity) and streamlines
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow
1
Introduction
The introduction of any course consists of the three basic questions
What is it Why to study How it works An eventful thoughtful
and intuitive answer to the rst two questions are presented in this
chapter The third question which involves lots of working details
on the three aspects of the Computational Fluid Dynamics (CFD)
(development application and analysis) is presented separately in
the next chapter The novelty scope and purpose of this book is
also presented at the end of this chapter
11 CFD What is it
CFD is a theoretical-method of scientic and engineering investiga-
tion concerned with the development and application of a video-
camera like tool (a software) which is used for a unied cause-and-
eect based analysis of a uid-dynamics as well as heat and mass
transfer problem
CFD is a subject where you rst learn how to develop a product
minus a software which acts like a virtual video-camera Then you learn
how to apply or use this product to generate uid-dynamics movies
Finally you learn how to analyze the detailed spatial and temporal
uid-dynamics information in the movies The analysis is done to
come up with a scientically-exciting as well as engineering-relevant
story (unied cause-and-eect study) of a uid-dynamics situation in
nature as well as industrial-applications With a continuous develop-
ment and a wider application of CFD the word uid-dynamics in
3
Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd
4 1 Introduction
the CFD has become more generic as it also corresponds to heat and
mass transfer as well as chemical reaction
Fluid-dynamics information are of two types scientic and en-
gineering Scientic-information corresponds to a structure of the
heat and uid ow due to a physical phenomenon in uid-dynamics
such as boundary-layer ow-separation wake-formation and vortex-
shedding The ow structures are obtained from a temporal variation
of ow-properties (such as velocity pressure temperature and many
other variables which characterizes a ow) called as scientically-
exciting movies Engineering-information corresponds to certain pa-
rameters which are important in an engineering application called
as engineering-parameters for example lift force drag force wall
shear stress pressure drop and rate of heat-transfer They are ob-
tained as the temporal variation of engineering-parameters called as
engineering-relevant movies The engineering-parameters are the ef-
fect which are caused by the ow structures minus their correlation leads
to the unied cause-and-eect study A uid-dynamics movie consist
of a time-wise varying series of pictures which give uid-dynamics
information Each ow-property and engineering-parameter results
in one scientically-exciting and engineering-relevant movie respec-
tively When the large number of both the types of movies are
played synchronized in-time the detailed spatial as well as tempo-
ral uid-dynamic information greatly helps in the unied cause-and-
eect study of a uid-dynamics problem
Alternatively CFD is a subject concerned with the development of
computer-programs for the computer-simulation and study of a natu-
ral or an engineering uid-dynamics system The development starts
with the virtual-development of the system involving a geometri-
cal information based computational development of a solid-model
for the system the system may be already-existing or yet-to-be-
developed physically Thereafter the development involves numerical
solution of the governing algebraic equations for CFD formulated
from the physical laws (conservation of mass momentum and en-
ergy) and initial as well as boundary conditions corresponding to the
system CFD study involves the analysis of the uid-dynamics and
heat-transfer results obtained from the computer-simulations of the
virtual-system which is subjected to certain governing-parameters
Introduction to CFD 5
111 CFD as a Scientic and Engineering Analysis Tool
For a better understanding of the denition of the CFD and its ap-
plication as a scientic and engineering analysis tool two example
problems are presented rst conduction heat transfer in a plate
and second uid ow across a circular pillar of a bridge The rst-
problem correspond to an engineering-application for electronic cool-
ing and the second-problem is studied for an engineering-design of
the structure subjected to uid-dynamics forces and are shown in
Fig 11 Both the problems are considered as two-dimensional and
unsteady However the results asymptotes from the unsteady to a
steady state in the rst-problem and to a periodic (dynamic-steady)
state in the second-problem
Figure 11 Schematic and the associated parameters for the two example prob-lems (a) thermal conduction module of a heat sink and (b) theclassical ow across a circular cylinder
6 1 Introduction
The rst-problem is taken from the classic book on heat-transfer
by Incropera and Dewitt (1996) shown in Fig 11(a) The g-
ure shows a periodic module of the plate where a constant heat-ux
qW = 10Wcm2 (dissipated from certain electronic devices) is rst con-
ducted to an aluminum plate and then convected to the water ow-
ing through the rectangular channel grooved in the plate The forced
convection results in a convection coecient of h = 5000Wm2K in-
side the channel The unsteady heat-transfer results in an interesting
time-wise varying conduction heat ow in the plate and instantaneous
convection heat transfer rate Qprimeconv (per unit width of the plate per-
pendicular to the plane shown in Fig 11(a)) into the water
The second-problem is encountered if you are standing over a
bridge and look below to the water passing over a pillar (of circu-
lar cross-section) of the bridge shown in Fig 11(b) as a general case
of 2D ow across a cylinder The uid-dynamics results in a beau-
tiful time-wise varying structure of the uid ow just downstream
of the pillar and certain engineering-parameters The parameters
correspond to the uid-dynamic forces acting in the horizontal (or
streamwise) and vertical (or transverse) direction called as the drag-
force FD and lift-force FL respectively The respective force is pre-
sented below in a non-dimensional form called as the drag-coecient
CD and lift-coecient CL equations shown in Fig 11(b)
If you use your video camera to capture a movie for the two
problems you will get series of pictures for the plate in the rst-
problem and the owing uid in the second-problem Since these
movies result in a pictorial information but not the ow-property
based uid-dynamic information such movies are not relevant for
the uid-dynamics and heat-transfer study Instead if you use a
CFD software for computer simulation you can capture various types
of movies The various ow-properties based scientically-exciting
movies are presented here as the pictures at certain time-instant in
Fig 12(a)-(d) and Fig 13(a1)-(d2) The former gure shows gray-
colored ooded contour for the temperature whereas the latter gure
shows gray-colored ooded contour for the pressure and vorticity and
vector-plot for the velocity For the gray-colored ooded contours a
color bar can be seen in the gures for the relative magnitude of the
variable These gures also show line contours of stream-function
Introduction to CFD 7
and heat-function as streamlines and heatlines respectively They
are the visualization techniques for a 2D steady heat and uid ow
The heatlines are analogous to streamlines and represents the direc-
tion of heat ow under steady state condition proposed by Kimura
and Bejan (1983) later presented in a heat-transfer book by Bejan
(1984) The steady-state heatlines can be seen in Fig 12(d) where
the direction for the conduction-ux is tangential at any point on the
heatlines analogous to that for the mass-ux in a streamline Thus
for a better presentation of the structure of heat and uid ow the
scientically-exciting movies are not only limited to ow-properties
but also correspond to certain ow visualization techniquesAn engineering-parameter based engineering-relevant uid dy-
namic movie is presented in Fig 12(e) for the convection heat trans-
fer rate per unit width at the various walls on the channel Qprimew at the
west-wall Qprimen at the north-wall Q
primee at the east-wall and their cumu-
lative value Qprimeconv (refer Fig 11(a)) Another engineering-relevant
movie is presented in Fig 13(e) for the lift and drag coecient Note
that the movie here corresponds to the motion of a lled symbol over
a line for the temporal variation of an engineering-parameter Thus
the engineering-relevant movies are one-dimensional lower than the
scientically-exciting movies here for the 2D problems the former
movie is presented as 1D and the latter as 2D results
1111 Heat-Conduction
For the computational set-up of the heat conduction problem shown
in Fig 11(a) the results obtained from a 2D CFD simulation of
the conduction heat transfer are shown in Fig 12 For a uni-
form initial temperature of the plate as Tinfin = 15oC Fig 12(a)-(d)
shows the isotherms and heatlines in the plate as the pictures of the
scientically-exciting movies at certain discrete time instants t = 25
5 10 and 20 sec For the same discrete time instants the temporal
variation of the position of the symbols over a line plot is presented in
Fig 12(e) The gure is shown for each of the heat transfer rate and
is obtained from a engineering-relevant 1D movie There is a tempo-
ral variation of picture in the scientically-exciting movie and that
of position of symbol (along the line plot) in the engineering-relevant
movieAccording to caloric theory of the famous French scientist An-
tonie Lavoisier heat is considered as an invisible tasteless odorless
8 1 Introduction
t(sec)
Qrsquo (W
m)
0 5 10 15 20 250
200
400
600
800
1000Qrsquo
conv
Qrsquow
Qrsquon
Qrsquoe
Qrsquoin
(e)
(b)
(c)(d)
(a)
(a) t=25 sec31
29
27
25
23
21
19
17
15
T(oC)
(c) t=10 sec
(b) t=5 sec
(d) t=20 sec
Figure 12 Thermal conduction module of a heat sink temporal variation of(a)-(d) isotherms and heatlines and (e) the convection heat transferrate per unit width at the various walls of the channels and theircumulative total value Q
primeconv For the discrete time-instant corre-
sponding to heat ow patterns in (a)-(d) temporal variation of thevarious heat transfer rates are shown as symbols in (e) (a)-(c) arethe transient and (d) represents a steady state result
weightless uid which he called caloric uid Presently the caloric
theory is overtaken by mechanical theory of heat minus heat is not a sub-
stance but a dynamical form of mechanical eect (Thomson 1851)
Nevertheless there are certain problems involving heat ow for which
Lavoisiers approach is rather useful such as the discussion on heat-
lines here
The role of heatlines for heat ow is analogous to that of stream-
lines for uid ow (Paramane and Sharma 2009) For the uid-ow
the dierence of stream-function values represents the rate of uid
ow the function remains constant on a solid wall the tangent to
a streamline represents the direction of the uid-ow (with no ow
in the normal direction) and the streamline originate or emerge at
Introduction to CFD 9
a mass source Analogously for the heat-ow the dierence of heat-
function values represents the rate of heat ow the function remains
constant on an adiabatic wall the tangent to a heatline represents the
direction of the heat-ow (with no ow in the normal direction) and
the heatline originate or emerge at the heat-source For the present
problem in Fig 11(a) the heat-source is at the inlet (or left) bound-
ary and the heat-sink is at the surface of the channel Thus the
steady-state heatlines in Fig 12(d) shows the heat ow emerging
from the inletsource and ending on the channel-wallssink Further-
more it can be seen in the gure that the heatlines which are close
to the adiabatic walls are parallel to the walls The source sink and
adiabatic-walls can be seen in Fig 11(a)
Figure 12(e) shows an asymptotic time-wise increase in the con-
vection heat transfer per unit width at the various walls of the channel
and its cumulative value Qprimeconv (= Q
primew+Q
primen+Q
primee) It can be also be seen
that the heat-transfer per unit width Qprimew at the west-wall and Q
primen at
the north-wall are larger than Qprimee at the east-wall of the channel As
time progresses the gure shows that Qprimeconv increases monotonically
and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-
der steady state condition refer Fig 11(a) The dierence between
the Qprimein and Q
primeconv under the transient condition is utilized for the
sensible heating and results in the increase in the temperature of the
plate Under the steady state condition note from Fig 12(d) that
the maximum temperature is close to 32oC minus well within the per-
missible limit for the reliable and ecient operation of the electronic
device
1112 Fluid-Dynamics
For the free-stream ow across a circular cylinderpillar at a Reynolds
number of 100 the non-dimensional results obtained from a 2D CFD
simulation are shown in Fig 13 In contrast to the previous prob-
lem which reaches steady state the present problem reaches to an un-
steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-
ent increasing time instants within one time-period of the periodic-
ow the results are presented in Fig 13(a)-(d) The gure shows
the close-up view of the pictures of the scientically-exciting movies
for ow-properties (velocity pressure and vorticity) and streamlines
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow
4 1 Introduction
the CFD has become more generic as it also corresponds to heat and
mass transfer as well as chemical reaction
Fluid-dynamics information are of two types scientic and en-
gineering Scientic-information corresponds to a structure of the
heat and uid ow due to a physical phenomenon in uid-dynamics
such as boundary-layer ow-separation wake-formation and vortex-
shedding The ow structures are obtained from a temporal variation
of ow-properties (such as velocity pressure temperature and many
other variables which characterizes a ow) called as scientically-
exciting movies Engineering-information corresponds to certain pa-
rameters which are important in an engineering application called
as engineering-parameters for example lift force drag force wall
shear stress pressure drop and rate of heat-transfer They are ob-
tained as the temporal variation of engineering-parameters called as
engineering-relevant movies The engineering-parameters are the ef-
fect which are caused by the ow structures minus their correlation leads
to the unied cause-and-eect study A uid-dynamics movie consist
of a time-wise varying series of pictures which give uid-dynamics
information Each ow-property and engineering-parameter results
in one scientically-exciting and engineering-relevant movie respec-
tively When the large number of both the types of movies are
played synchronized in-time the detailed spatial as well as tempo-
ral uid-dynamic information greatly helps in the unied cause-and-
eect study of a uid-dynamics problem
Alternatively CFD is a subject concerned with the development of
computer-programs for the computer-simulation and study of a natu-
ral or an engineering uid-dynamics system The development starts
with the virtual-development of the system involving a geometri-
cal information based computational development of a solid-model
for the system the system may be already-existing or yet-to-be-
developed physically Thereafter the development involves numerical
solution of the governing algebraic equations for CFD formulated
from the physical laws (conservation of mass momentum and en-
ergy) and initial as well as boundary conditions corresponding to the
system CFD study involves the analysis of the uid-dynamics and
heat-transfer results obtained from the computer-simulations of the
virtual-system which is subjected to certain governing-parameters
Introduction to CFD 5
111 CFD as a Scientic and Engineering Analysis Tool
For a better understanding of the denition of the CFD and its ap-
plication as a scientic and engineering analysis tool two example
problems are presented rst conduction heat transfer in a plate
and second uid ow across a circular pillar of a bridge The rst-
problem correspond to an engineering-application for electronic cool-
ing and the second-problem is studied for an engineering-design of
the structure subjected to uid-dynamics forces and are shown in
Fig 11 Both the problems are considered as two-dimensional and
unsteady However the results asymptotes from the unsteady to a
steady state in the rst-problem and to a periodic (dynamic-steady)
state in the second-problem
Figure 11 Schematic and the associated parameters for the two example prob-lems (a) thermal conduction module of a heat sink and (b) theclassical ow across a circular cylinder
6 1 Introduction
The rst-problem is taken from the classic book on heat-transfer
by Incropera and Dewitt (1996) shown in Fig 11(a) The g-
ure shows a periodic module of the plate where a constant heat-ux
qW = 10Wcm2 (dissipated from certain electronic devices) is rst con-
ducted to an aluminum plate and then convected to the water ow-
ing through the rectangular channel grooved in the plate The forced
convection results in a convection coecient of h = 5000Wm2K in-
side the channel The unsteady heat-transfer results in an interesting
time-wise varying conduction heat ow in the plate and instantaneous
convection heat transfer rate Qprimeconv (per unit width of the plate per-
pendicular to the plane shown in Fig 11(a)) into the water
The second-problem is encountered if you are standing over a
bridge and look below to the water passing over a pillar (of circu-
lar cross-section) of the bridge shown in Fig 11(b) as a general case
of 2D ow across a cylinder The uid-dynamics results in a beau-
tiful time-wise varying structure of the uid ow just downstream
of the pillar and certain engineering-parameters The parameters
correspond to the uid-dynamic forces acting in the horizontal (or
streamwise) and vertical (or transverse) direction called as the drag-
force FD and lift-force FL respectively The respective force is pre-
sented below in a non-dimensional form called as the drag-coecient
CD and lift-coecient CL equations shown in Fig 11(b)
If you use your video camera to capture a movie for the two
problems you will get series of pictures for the plate in the rst-
problem and the owing uid in the second-problem Since these
movies result in a pictorial information but not the ow-property
based uid-dynamic information such movies are not relevant for
the uid-dynamics and heat-transfer study Instead if you use a
CFD software for computer simulation you can capture various types
of movies The various ow-properties based scientically-exciting
movies are presented here as the pictures at certain time-instant in
Fig 12(a)-(d) and Fig 13(a1)-(d2) The former gure shows gray-
colored ooded contour for the temperature whereas the latter gure
shows gray-colored ooded contour for the pressure and vorticity and
vector-plot for the velocity For the gray-colored ooded contours a
color bar can be seen in the gures for the relative magnitude of the
variable These gures also show line contours of stream-function
Introduction to CFD 7
and heat-function as streamlines and heatlines respectively They
are the visualization techniques for a 2D steady heat and uid ow
The heatlines are analogous to streamlines and represents the direc-
tion of heat ow under steady state condition proposed by Kimura
and Bejan (1983) later presented in a heat-transfer book by Bejan
(1984) The steady-state heatlines can be seen in Fig 12(d) where
the direction for the conduction-ux is tangential at any point on the
heatlines analogous to that for the mass-ux in a streamline Thus
for a better presentation of the structure of heat and uid ow the
scientically-exciting movies are not only limited to ow-properties
but also correspond to certain ow visualization techniquesAn engineering-parameter based engineering-relevant uid dy-
namic movie is presented in Fig 12(e) for the convection heat trans-
fer rate per unit width at the various walls on the channel Qprimew at the
west-wall Qprimen at the north-wall Q
primee at the east-wall and their cumu-
lative value Qprimeconv (refer Fig 11(a)) Another engineering-relevant
movie is presented in Fig 13(e) for the lift and drag coecient Note
that the movie here corresponds to the motion of a lled symbol over
a line for the temporal variation of an engineering-parameter Thus
the engineering-relevant movies are one-dimensional lower than the
scientically-exciting movies here for the 2D problems the former
movie is presented as 1D and the latter as 2D results
1111 Heat-Conduction
For the computational set-up of the heat conduction problem shown
in Fig 11(a) the results obtained from a 2D CFD simulation of
the conduction heat transfer are shown in Fig 12 For a uni-
form initial temperature of the plate as Tinfin = 15oC Fig 12(a)-(d)
shows the isotherms and heatlines in the plate as the pictures of the
scientically-exciting movies at certain discrete time instants t = 25
5 10 and 20 sec For the same discrete time instants the temporal
variation of the position of the symbols over a line plot is presented in
Fig 12(e) The gure is shown for each of the heat transfer rate and
is obtained from a engineering-relevant 1D movie There is a tempo-
ral variation of picture in the scientically-exciting movie and that
of position of symbol (along the line plot) in the engineering-relevant
movieAccording to caloric theory of the famous French scientist An-
tonie Lavoisier heat is considered as an invisible tasteless odorless
8 1 Introduction
t(sec)
Qrsquo (W
m)
0 5 10 15 20 250
200
400
600
800
1000Qrsquo
conv
Qrsquow
Qrsquon
Qrsquoe
Qrsquoin
(e)
(b)
(c)(d)
(a)
(a) t=25 sec31
29
27
25
23
21
19
17
15
T(oC)
(c) t=10 sec
(b) t=5 sec
(d) t=20 sec
Figure 12 Thermal conduction module of a heat sink temporal variation of(a)-(d) isotherms and heatlines and (e) the convection heat transferrate per unit width at the various walls of the channels and theircumulative total value Q
primeconv For the discrete time-instant corre-
sponding to heat ow patterns in (a)-(d) temporal variation of thevarious heat transfer rates are shown as symbols in (e) (a)-(c) arethe transient and (d) represents a steady state result
weightless uid which he called caloric uid Presently the caloric
theory is overtaken by mechanical theory of heat minus heat is not a sub-
stance but a dynamical form of mechanical eect (Thomson 1851)
Nevertheless there are certain problems involving heat ow for which
Lavoisiers approach is rather useful such as the discussion on heat-
lines here
The role of heatlines for heat ow is analogous to that of stream-
lines for uid ow (Paramane and Sharma 2009) For the uid-ow
the dierence of stream-function values represents the rate of uid
ow the function remains constant on a solid wall the tangent to
a streamline represents the direction of the uid-ow (with no ow
in the normal direction) and the streamline originate or emerge at
Introduction to CFD 9
a mass source Analogously for the heat-ow the dierence of heat-
function values represents the rate of heat ow the function remains
constant on an adiabatic wall the tangent to a heatline represents the
direction of the heat-ow (with no ow in the normal direction) and
the heatline originate or emerge at the heat-source For the present
problem in Fig 11(a) the heat-source is at the inlet (or left) bound-
ary and the heat-sink is at the surface of the channel Thus the
steady-state heatlines in Fig 12(d) shows the heat ow emerging
from the inletsource and ending on the channel-wallssink Further-
more it can be seen in the gure that the heatlines which are close
to the adiabatic walls are parallel to the walls The source sink and
adiabatic-walls can be seen in Fig 11(a)
Figure 12(e) shows an asymptotic time-wise increase in the con-
vection heat transfer per unit width at the various walls of the channel
and its cumulative value Qprimeconv (= Q
primew+Q
primen+Q
primee) It can be also be seen
that the heat-transfer per unit width Qprimew at the west-wall and Q
primen at
the north-wall are larger than Qprimee at the east-wall of the channel As
time progresses the gure shows that Qprimeconv increases monotonically
and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-
der steady state condition refer Fig 11(a) The dierence between
the Qprimein and Q
primeconv under the transient condition is utilized for the
sensible heating and results in the increase in the temperature of the
plate Under the steady state condition note from Fig 12(d) that
the maximum temperature is close to 32oC minus well within the per-
missible limit for the reliable and ecient operation of the electronic
device
1112 Fluid-Dynamics
For the free-stream ow across a circular cylinderpillar at a Reynolds
number of 100 the non-dimensional results obtained from a 2D CFD
simulation are shown in Fig 13 In contrast to the previous prob-
lem which reaches steady state the present problem reaches to an un-
steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-
ent increasing time instants within one time-period of the periodic-
ow the results are presented in Fig 13(a)-(d) The gure shows
the close-up view of the pictures of the scientically-exciting movies
for ow-properties (velocity pressure and vorticity) and streamlines
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow
Introduction to CFD 5
111 CFD as a Scientic and Engineering Analysis Tool
For a better understanding of the denition of the CFD and its ap-
plication as a scientic and engineering analysis tool two example
problems are presented rst conduction heat transfer in a plate
and second uid ow across a circular pillar of a bridge The rst-
problem correspond to an engineering-application for electronic cool-
ing and the second-problem is studied for an engineering-design of
the structure subjected to uid-dynamics forces and are shown in
Fig 11 Both the problems are considered as two-dimensional and
unsteady However the results asymptotes from the unsteady to a
steady state in the rst-problem and to a periodic (dynamic-steady)
state in the second-problem
Figure 11 Schematic and the associated parameters for the two example prob-lems (a) thermal conduction module of a heat sink and (b) theclassical ow across a circular cylinder
6 1 Introduction
The rst-problem is taken from the classic book on heat-transfer
by Incropera and Dewitt (1996) shown in Fig 11(a) The g-
ure shows a periodic module of the plate where a constant heat-ux
qW = 10Wcm2 (dissipated from certain electronic devices) is rst con-
ducted to an aluminum plate and then convected to the water ow-
ing through the rectangular channel grooved in the plate The forced
convection results in a convection coecient of h = 5000Wm2K in-
side the channel The unsteady heat-transfer results in an interesting
time-wise varying conduction heat ow in the plate and instantaneous
convection heat transfer rate Qprimeconv (per unit width of the plate per-
pendicular to the plane shown in Fig 11(a)) into the water
The second-problem is encountered if you are standing over a
bridge and look below to the water passing over a pillar (of circu-
lar cross-section) of the bridge shown in Fig 11(b) as a general case
of 2D ow across a cylinder The uid-dynamics results in a beau-
tiful time-wise varying structure of the uid ow just downstream
of the pillar and certain engineering-parameters The parameters
correspond to the uid-dynamic forces acting in the horizontal (or
streamwise) and vertical (or transverse) direction called as the drag-
force FD and lift-force FL respectively The respective force is pre-
sented below in a non-dimensional form called as the drag-coecient
CD and lift-coecient CL equations shown in Fig 11(b)
If you use your video camera to capture a movie for the two
problems you will get series of pictures for the plate in the rst-
problem and the owing uid in the second-problem Since these
movies result in a pictorial information but not the ow-property
based uid-dynamic information such movies are not relevant for
the uid-dynamics and heat-transfer study Instead if you use a
CFD software for computer simulation you can capture various types
of movies The various ow-properties based scientically-exciting
movies are presented here as the pictures at certain time-instant in
Fig 12(a)-(d) and Fig 13(a1)-(d2) The former gure shows gray-
colored ooded contour for the temperature whereas the latter gure
shows gray-colored ooded contour for the pressure and vorticity and
vector-plot for the velocity For the gray-colored ooded contours a
color bar can be seen in the gures for the relative magnitude of the
variable These gures also show line contours of stream-function
Introduction to CFD 7
and heat-function as streamlines and heatlines respectively They
are the visualization techniques for a 2D steady heat and uid ow
The heatlines are analogous to streamlines and represents the direc-
tion of heat ow under steady state condition proposed by Kimura
and Bejan (1983) later presented in a heat-transfer book by Bejan
(1984) The steady-state heatlines can be seen in Fig 12(d) where
the direction for the conduction-ux is tangential at any point on the
heatlines analogous to that for the mass-ux in a streamline Thus
for a better presentation of the structure of heat and uid ow the
scientically-exciting movies are not only limited to ow-properties
but also correspond to certain ow visualization techniquesAn engineering-parameter based engineering-relevant uid dy-
namic movie is presented in Fig 12(e) for the convection heat trans-
fer rate per unit width at the various walls on the channel Qprimew at the
west-wall Qprimen at the north-wall Q
primee at the east-wall and their cumu-
lative value Qprimeconv (refer Fig 11(a)) Another engineering-relevant
movie is presented in Fig 13(e) for the lift and drag coecient Note
that the movie here corresponds to the motion of a lled symbol over
a line for the temporal variation of an engineering-parameter Thus
the engineering-relevant movies are one-dimensional lower than the
scientically-exciting movies here for the 2D problems the former
movie is presented as 1D and the latter as 2D results
1111 Heat-Conduction
For the computational set-up of the heat conduction problem shown
in Fig 11(a) the results obtained from a 2D CFD simulation of
the conduction heat transfer are shown in Fig 12 For a uni-
form initial temperature of the plate as Tinfin = 15oC Fig 12(a)-(d)
shows the isotherms and heatlines in the plate as the pictures of the
scientically-exciting movies at certain discrete time instants t = 25
5 10 and 20 sec For the same discrete time instants the temporal
variation of the position of the symbols over a line plot is presented in
Fig 12(e) The gure is shown for each of the heat transfer rate and
is obtained from a engineering-relevant 1D movie There is a tempo-
ral variation of picture in the scientically-exciting movie and that
of position of symbol (along the line plot) in the engineering-relevant
movieAccording to caloric theory of the famous French scientist An-
tonie Lavoisier heat is considered as an invisible tasteless odorless
8 1 Introduction
t(sec)
Qrsquo (W
m)
0 5 10 15 20 250
200
400
600
800
1000Qrsquo
conv
Qrsquow
Qrsquon
Qrsquoe
Qrsquoin
(e)
(b)
(c)(d)
(a)
(a) t=25 sec31
29
27
25
23
21
19
17
15
T(oC)
(c) t=10 sec
(b) t=5 sec
(d) t=20 sec
Figure 12 Thermal conduction module of a heat sink temporal variation of(a)-(d) isotherms and heatlines and (e) the convection heat transferrate per unit width at the various walls of the channels and theircumulative total value Q
primeconv For the discrete time-instant corre-
sponding to heat ow patterns in (a)-(d) temporal variation of thevarious heat transfer rates are shown as symbols in (e) (a)-(c) arethe transient and (d) represents a steady state result
weightless uid which he called caloric uid Presently the caloric
theory is overtaken by mechanical theory of heat minus heat is not a sub-
stance but a dynamical form of mechanical eect (Thomson 1851)
Nevertheless there are certain problems involving heat ow for which
Lavoisiers approach is rather useful such as the discussion on heat-
lines here
The role of heatlines for heat ow is analogous to that of stream-
lines for uid ow (Paramane and Sharma 2009) For the uid-ow
the dierence of stream-function values represents the rate of uid
ow the function remains constant on a solid wall the tangent to
a streamline represents the direction of the uid-ow (with no ow
in the normal direction) and the streamline originate or emerge at
Introduction to CFD 9
a mass source Analogously for the heat-ow the dierence of heat-
function values represents the rate of heat ow the function remains
constant on an adiabatic wall the tangent to a heatline represents the
direction of the heat-ow (with no ow in the normal direction) and
the heatline originate or emerge at the heat-source For the present
problem in Fig 11(a) the heat-source is at the inlet (or left) bound-
ary and the heat-sink is at the surface of the channel Thus the
steady-state heatlines in Fig 12(d) shows the heat ow emerging
from the inletsource and ending on the channel-wallssink Further-
more it can be seen in the gure that the heatlines which are close
to the adiabatic walls are parallel to the walls The source sink and
adiabatic-walls can be seen in Fig 11(a)
Figure 12(e) shows an asymptotic time-wise increase in the con-
vection heat transfer per unit width at the various walls of the channel
and its cumulative value Qprimeconv (= Q
primew+Q
primen+Q
primee) It can be also be seen
that the heat-transfer per unit width Qprimew at the west-wall and Q
primen at
the north-wall are larger than Qprimee at the east-wall of the channel As
time progresses the gure shows that Qprimeconv increases monotonically
and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-
der steady state condition refer Fig 11(a) The dierence between
the Qprimein and Q
primeconv under the transient condition is utilized for the
sensible heating and results in the increase in the temperature of the
plate Under the steady state condition note from Fig 12(d) that
the maximum temperature is close to 32oC minus well within the per-
missible limit for the reliable and ecient operation of the electronic
device
1112 Fluid-Dynamics
For the free-stream ow across a circular cylinderpillar at a Reynolds
number of 100 the non-dimensional results obtained from a 2D CFD
simulation are shown in Fig 13 In contrast to the previous prob-
lem which reaches steady state the present problem reaches to an un-
steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-
ent increasing time instants within one time-period of the periodic-
ow the results are presented in Fig 13(a)-(d) The gure shows
the close-up view of the pictures of the scientically-exciting movies
for ow-properties (velocity pressure and vorticity) and streamlines
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow
6 1 Introduction
The rst-problem is taken from the classic book on heat-transfer
by Incropera and Dewitt (1996) shown in Fig 11(a) The g-
ure shows a periodic module of the plate where a constant heat-ux
qW = 10Wcm2 (dissipated from certain electronic devices) is rst con-
ducted to an aluminum plate and then convected to the water ow-
ing through the rectangular channel grooved in the plate The forced
convection results in a convection coecient of h = 5000Wm2K in-
side the channel The unsteady heat-transfer results in an interesting
time-wise varying conduction heat ow in the plate and instantaneous
convection heat transfer rate Qprimeconv (per unit width of the plate per-
pendicular to the plane shown in Fig 11(a)) into the water
The second-problem is encountered if you are standing over a
bridge and look below to the water passing over a pillar (of circu-
lar cross-section) of the bridge shown in Fig 11(b) as a general case
of 2D ow across a cylinder The uid-dynamics results in a beau-
tiful time-wise varying structure of the uid ow just downstream
of the pillar and certain engineering-parameters The parameters
correspond to the uid-dynamic forces acting in the horizontal (or
streamwise) and vertical (or transverse) direction called as the drag-
force FD and lift-force FL respectively The respective force is pre-
sented below in a non-dimensional form called as the drag-coecient
CD and lift-coecient CL equations shown in Fig 11(b)
If you use your video camera to capture a movie for the two
problems you will get series of pictures for the plate in the rst-
problem and the owing uid in the second-problem Since these
movies result in a pictorial information but not the ow-property
based uid-dynamic information such movies are not relevant for
the uid-dynamics and heat-transfer study Instead if you use a
CFD software for computer simulation you can capture various types
of movies The various ow-properties based scientically-exciting
movies are presented here as the pictures at certain time-instant in
Fig 12(a)-(d) and Fig 13(a1)-(d2) The former gure shows gray-
colored ooded contour for the temperature whereas the latter gure
shows gray-colored ooded contour for the pressure and vorticity and
vector-plot for the velocity For the gray-colored ooded contours a
color bar can be seen in the gures for the relative magnitude of the
variable These gures also show line contours of stream-function
Introduction to CFD 7
and heat-function as streamlines and heatlines respectively They
are the visualization techniques for a 2D steady heat and uid ow
The heatlines are analogous to streamlines and represents the direc-
tion of heat ow under steady state condition proposed by Kimura
and Bejan (1983) later presented in a heat-transfer book by Bejan
(1984) The steady-state heatlines can be seen in Fig 12(d) where
the direction for the conduction-ux is tangential at any point on the
heatlines analogous to that for the mass-ux in a streamline Thus
for a better presentation of the structure of heat and uid ow the
scientically-exciting movies are not only limited to ow-properties
but also correspond to certain ow visualization techniquesAn engineering-parameter based engineering-relevant uid dy-
namic movie is presented in Fig 12(e) for the convection heat trans-
fer rate per unit width at the various walls on the channel Qprimew at the
west-wall Qprimen at the north-wall Q
primee at the east-wall and their cumu-
lative value Qprimeconv (refer Fig 11(a)) Another engineering-relevant
movie is presented in Fig 13(e) for the lift and drag coecient Note
that the movie here corresponds to the motion of a lled symbol over
a line for the temporal variation of an engineering-parameter Thus
the engineering-relevant movies are one-dimensional lower than the
scientically-exciting movies here for the 2D problems the former
movie is presented as 1D and the latter as 2D results
1111 Heat-Conduction
For the computational set-up of the heat conduction problem shown
in Fig 11(a) the results obtained from a 2D CFD simulation of
the conduction heat transfer are shown in Fig 12 For a uni-
form initial temperature of the plate as Tinfin = 15oC Fig 12(a)-(d)
shows the isotherms and heatlines in the plate as the pictures of the
scientically-exciting movies at certain discrete time instants t = 25
5 10 and 20 sec For the same discrete time instants the temporal
variation of the position of the symbols over a line plot is presented in
Fig 12(e) The gure is shown for each of the heat transfer rate and
is obtained from a engineering-relevant 1D movie There is a tempo-
ral variation of picture in the scientically-exciting movie and that
of position of symbol (along the line plot) in the engineering-relevant
movieAccording to caloric theory of the famous French scientist An-
tonie Lavoisier heat is considered as an invisible tasteless odorless
8 1 Introduction
t(sec)
Qrsquo (W
m)
0 5 10 15 20 250
200
400
600
800
1000Qrsquo
conv
Qrsquow
Qrsquon
Qrsquoe
Qrsquoin
(e)
(b)
(c)(d)
(a)
(a) t=25 sec31
29
27
25
23
21
19
17
15
T(oC)
(c) t=10 sec
(b) t=5 sec
(d) t=20 sec
Figure 12 Thermal conduction module of a heat sink temporal variation of(a)-(d) isotherms and heatlines and (e) the convection heat transferrate per unit width at the various walls of the channels and theircumulative total value Q
primeconv For the discrete time-instant corre-
sponding to heat ow patterns in (a)-(d) temporal variation of thevarious heat transfer rates are shown as symbols in (e) (a)-(c) arethe transient and (d) represents a steady state result
weightless uid which he called caloric uid Presently the caloric
theory is overtaken by mechanical theory of heat minus heat is not a sub-
stance but a dynamical form of mechanical eect (Thomson 1851)
Nevertheless there are certain problems involving heat ow for which
Lavoisiers approach is rather useful such as the discussion on heat-
lines here
The role of heatlines for heat ow is analogous to that of stream-
lines for uid ow (Paramane and Sharma 2009) For the uid-ow
the dierence of stream-function values represents the rate of uid
ow the function remains constant on a solid wall the tangent to
a streamline represents the direction of the uid-ow (with no ow
in the normal direction) and the streamline originate or emerge at
Introduction to CFD 9
a mass source Analogously for the heat-ow the dierence of heat-
function values represents the rate of heat ow the function remains
constant on an adiabatic wall the tangent to a heatline represents the
direction of the heat-ow (with no ow in the normal direction) and
the heatline originate or emerge at the heat-source For the present
problem in Fig 11(a) the heat-source is at the inlet (or left) bound-
ary and the heat-sink is at the surface of the channel Thus the
steady-state heatlines in Fig 12(d) shows the heat ow emerging
from the inletsource and ending on the channel-wallssink Further-
more it can be seen in the gure that the heatlines which are close
to the adiabatic walls are parallel to the walls The source sink and
adiabatic-walls can be seen in Fig 11(a)
Figure 12(e) shows an asymptotic time-wise increase in the con-
vection heat transfer per unit width at the various walls of the channel
and its cumulative value Qprimeconv (= Q
primew+Q
primen+Q
primee) It can be also be seen
that the heat-transfer per unit width Qprimew at the west-wall and Q
primen at
the north-wall are larger than Qprimee at the east-wall of the channel As
time progresses the gure shows that Qprimeconv increases monotonically
and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-
der steady state condition refer Fig 11(a) The dierence between
the Qprimein and Q
primeconv under the transient condition is utilized for the
sensible heating and results in the increase in the temperature of the
plate Under the steady state condition note from Fig 12(d) that
the maximum temperature is close to 32oC minus well within the per-
missible limit for the reliable and ecient operation of the electronic
device
1112 Fluid-Dynamics
For the free-stream ow across a circular cylinderpillar at a Reynolds
number of 100 the non-dimensional results obtained from a 2D CFD
simulation are shown in Fig 13 In contrast to the previous prob-
lem which reaches steady state the present problem reaches to an un-
steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-
ent increasing time instants within one time-period of the periodic-
ow the results are presented in Fig 13(a)-(d) The gure shows
the close-up view of the pictures of the scientically-exciting movies
for ow-properties (velocity pressure and vorticity) and streamlines
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow
Introduction to CFD 7
and heat-function as streamlines and heatlines respectively They
are the visualization techniques for a 2D steady heat and uid ow
The heatlines are analogous to streamlines and represents the direc-
tion of heat ow under steady state condition proposed by Kimura
and Bejan (1983) later presented in a heat-transfer book by Bejan
(1984) The steady-state heatlines can be seen in Fig 12(d) where
the direction for the conduction-ux is tangential at any point on the
heatlines analogous to that for the mass-ux in a streamline Thus
for a better presentation of the structure of heat and uid ow the
scientically-exciting movies are not only limited to ow-properties
but also correspond to certain ow visualization techniquesAn engineering-parameter based engineering-relevant uid dy-
namic movie is presented in Fig 12(e) for the convection heat trans-
fer rate per unit width at the various walls on the channel Qprimew at the
west-wall Qprimen at the north-wall Q
primee at the east-wall and their cumu-
lative value Qprimeconv (refer Fig 11(a)) Another engineering-relevant
movie is presented in Fig 13(e) for the lift and drag coecient Note
that the movie here corresponds to the motion of a lled symbol over
a line for the temporal variation of an engineering-parameter Thus
the engineering-relevant movies are one-dimensional lower than the
scientically-exciting movies here for the 2D problems the former
movie is presented as 1D and the latter as 2D results
1111 Heat-Conduction
For the computational set-up of the heat conduction problem shown
in Fig 11(a) the results obtained from a 2D CFD simulation of
the conduction heat transfer are shown in Fig 12 For a uni-
form initial temperature of the plate as Tinfin = 15oC Fig 12(a)-(d)
shows the isotherms and heatlines in the plate as the pictures of the
scientically-exciting movies at certain discrete time instants t = 25
5 10 and 20 sec For the same discrete time instants the temporal
variation of the position of the symbols over a line plot is presented in
Fig 12(e) The gure is shown for each of the heat transfer rate and
is obtained from a engineering-relevant 1D movie There is a tempo-
ral variation of picture in the scientically-exciting movie and that
of position of symbol (along the line plot) in the engineering-relevant
movieAccording to caloric theory of the famous French scientist An-
tonie Lavoisier heat is considered as an invisible tasteless odorless
8 1 Introduction
t(sec)
Qrsquo (W
m)
0 5 10 15 20 250
200
400
600
800
1000Qrsquo
conv
Qrsquow
Qrsquon
Qrsquoe
Qrsquoin
(e)
(b)
(c)(d)
(a)
(a) t=25 sec31
29
27
25
23
21
19
17
15
T(oC)
(c) t=10 sec
(b) t=5 sec
(d) t=20 sec
Figure 12 Thermal conduction module of a heat sink temporal variation of(a)-(d) isotherms and heatlines and (e) the convection heat transferrate per unit width at the various walls of the channels and theircumulative total value Q
primeconv For the discrete time-instant corre-
sponding to heat ow patterns in (a)-(d) temporal variation of thevarious heat transfer rates are shown as symbols in (e) (a)-(c) arethe transient and (d) represents a steady state result
weightless uid which he called caloric uid Presently the caloric
theory is overtaken by mechanical theory of heat minus heat is not a sub-
stance but a dynamical form of mechanical eect (Thomson 1851)
Nevertheless there are certain problems involving heat ow for which
Lavoisiers approach is rather useful such as the discussion on heat-
lines here
The role of heatlines for heat ow is analogous to that of stream-
lines for uid ow (Paramane and Sharma 2009) For the uid-ow
the dierence of stream-function values represents the rate of uid
ow the function remains constant on a solid wall the tangent to
a streamline represents the direction of the uid-ow (with no ow
in the normal direction) and the streamline originate or emerge at
Introduction to CFD 9
a mass source Analogously for the heat-ow the dierence of heat-
function values represents the rate of heat ow the function remains
constant on an adiabatic wall the tangent to a heatline represents the
direction of the heat-ow (with no ow in the normal direction) and
the heatline originate or emerge at the heat-source For the present
problem in Fig 11(a) the heat-source is at the inlet (or left) bound-
ary and the heat-sink is at the surface of the channel Thus the
steady-state heatlines in Fig 12(d) shows the heat ow emerging
from the inletsource and ending on the channel-wallssink Further-
more it can be seen in the gure that the heatlines which are close
to the adiabatic walls are parallel to the walls The source sink and
adiabatic-walls can be seen in Fig 11(a)
Figure 12(e) shows an asymptotic time-wise increase in the con-
vection heat transfer per unit width at the various walls of the channel
and its cumulative value Qprimeconv (= Q
primew+Q
primen+Q
primee) It can be also be seen
that the heat-transfer per unit width Qprimew at the west-wall and Q
primen at
the north-wall are larger than Qprimee at the east-wall of the channel As
time progresses the gure shows that Qprimeconv increases monotonically
and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-
der steady state condition refer Fig 11(a) The dierence between
the Qprimein and Q
primeconv under the transient condition is utilized for the
sensible heating and results in the increase in the temperature of the
plate Under the steady state condition note from Fig 12(d) that
the maximum temperature is close to 32oC minus well within the per-
missible limit for the reliable and ecient operation of the electronic
device
1112 Fluid-Dynamics
For the free-stream ow across a circular cylinderpillar at a Reynolds
number of 100 the non-dimensional results obtained from a 2D CFD
simulation are shown in Fig 13 In contrast to the previous prob-
lem which reaches steady state the present problem reaches to an un-
steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-
ent increasing time instants within one time-period of the periodic-
ow the results are presented in Fig 13(a)-(d) The gure shows
the close-up view of the pictures of the scientically-exciting movies
for ow-properties (velocity pressure and vorticity) and streamlines
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow
8 1 Introduction
t(sec)
Qrsquo (W
m)
0 5 10 15 20 250
200
400
600
800
1000Qrsquo
conv
Qrsquow
Qrsquon
Qrsquoe
Qrsquoin
(e)
(b)
(c)(d)
(a)
(a) t=25 sec31
29
27
25
23
21
19
17
15
T(oC)
(c) t=10 sec
(b) t=5 sec
(d) t=20 sec
Figure 12 Thermal conduction module of a heat sink temporal variation of(a)-(d) isotherms and heatlines and (e) the convection heat transferrate per unit width at the various walls of the channels and theircumulative total value Q
primeconv For the discrete time-instant corre-
sponding to heat ow patterns in (a)-(d) temporal variation of thevarious heat transfer rates are shown as symbols in (e) (a)-(c) arethe transient and (d) represents a steady state result
weightless uid which he called caloric uid Presently the caloric
theory is overtaken by mechanical theory of heat minus heat is not a sub-
stance but a dynamical form of mechanical eect (Thomson 1851)
Nevertheless there are certain problems involving heat ow for which
Lavoisiers approach is rather useful such as the discussion on heat-
lines here
The role of heatlines for heat ow is analogous to that of stream-
lines for uid ow (Paramane and Sharma 2009) For the uid-ow
the dierence of stream-function values represents the rate of uid
ow the function remains constant on a solid wall the tangent to
a streamline represents the direction of the uid-ow (with no ow
in the normal direction) and the streamline originate or emerge at
Introduction to CFD 9
a mass source Analogously for the heat-ow the dierence of heat-
function values represents the rate of heat ow the function remains
constant on an adiabatic wall the tangent to a heatline represents the
direction of the heat-ow (with no ow in the normal direction) and
the heatline originate or emerge at the heat-source For the present
problem in Fig 11(a) the heat-source is at the inlet (or left) bound-
ary and the heat-sink is at the surface of the channel Thus the
steady-state heatlines in Fig 12(d) shows the heat ow emerging
from the inletsource and ending on the channel-wallssink Further-
more it can be seen in the gure that the heatlines which are close
to the adiabatic walls are parallel to the walls The source sink and
adiabatic-walls can be seen in Fig 11(a)
Figure 12(e) shows an asymptotic time-wise increase in the con-
vection heat transfer per unit width at the various walls of the channel
and its cumulative value Qprimeconv (= Q
primew+Q
primen+Q
primee) It can be also be seen
that the heat-transfer per unit width Qprimew at the west-wall and Q
primen at
the north-wall are larger than Qprimee at the east-wall of the channel As
time progresses the gure shows that Qprimeconv increases monotonically
and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-
der steady state condition refer Fig 11(a) The dierence between
the Qprimein and Q
primeconv under the transient condition is utilized for the
sensible heating and results in the increase in the temperature of the
plate Under the steady state condition note from Fig 12(d) that
the maximum temperature is close to 32oC minus well within the per-
missible limit for the reliable and ecient operation of the electronic
device
1112 Fluid-Dynamics
For the free-stream ow across a circular cylinderpillar at a Reynolds
number of 100 the non-dimensional results obtained from a 2D CFD
simulation are shown in Fig 13 In contrast to the previous prob-
lem which reaches steady state the present problem reaches to an un-
steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-
ent increasing time instants within one time-period of the periodic-
ow the results are presented in Fig 13(a)-(d) The gure shows
the close-up view of the pictures of the scientically-exciting movies
for ow-properties (velocity pressure and vorticity) and streamlines
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow
Introduction to CFD 9
a mass source Analogously for the heat-ow the dierence of heat-
function values represents the rate of heat ow the function remains
constant on an adiabatic wall the tangent to a heatline represents the
direction of the heat-ow (with no ow in the normal direction) and
the heatline originate or emerge at the heat-source For the present
problem in Fig 11(a) the heat-source is at the inlet (or left) bound-
ary and the heat-sink is at the surface of the channel Thus the
steady-state heatlines in Fig 12(d) shows the heat ow emerging
from the inletsource and ending on the channel-wallssink Further-
more it can be seen in the gure that the heatlines which are close
to the adiabatic walls are parallel to the walls The source sink and
adiabatic-walls can be seen in Fig 11(a)
Figure 12(e) shows an asymptotic time-wise increase in the con-
vection heat transfer per unit width at the various walls of the channel
and its cumulative value Qprimeconv (= Q
primew+Q
primen+Q
primee) It can be also be seen
that the heat-transfer per unit width Qprimew at the west-wall and Q
primen at
the north-wall are larger than Qprimee at the east-wall of the channel As
time progresses the gure shows that Qprimeconv increases monotonically
and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-
der steady state condition refer Fig 11(a) The dierence between
the Qprimein and Q
primeconv under the transient condition is utilized for the
sensible heating and results in the increase in the temperature of the
plate Under the steady state condition note from Fig 12(d) that
the maximum temperature is close to 32oC minus well within the per-
missible limit for the reliable and ecient operation of the electronic
device
1112 Fluid-Dynamics
For the free-stream ow across a circular cylinderpillar at a Reynolds
number of 100 the non-dimensional results obtained from a 2D CFD
simulation are shown in Fig 13 In contrast to the previous prob-
lem which reaches steady state the present problem reaches to an un-
steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-
ent increasing time instants within one time-period of the periodic-
ow the results are presented in Fig 13(a)-(d) The gure shows
the close-up view of the pictures of the scientically-exciting movies
for ow-properties (velocity pressure and vorticity) and streamlines
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow
10 1 Introduction
Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow