numerical calculations of velocity and pressure distribution around oscillating ... ·...

N A S A C O N T R A C T O R

R E P O R T

COCSI

N A S A C R - 2 3 6 8

NUMERICAL CALCULATIONS OF

VELOCITY AND PRESSURE DISTRIBUTION

AROUND OSCILLATING AIRFOILS

by Theodore Bratanow, Akin Ecer, and Michael Kobiske

Prepared by

UNIVERSITY OF WISCONSIN-MILWAUKEE

Milwaukee, Wis.

for Langley Research Center

NATIONAL A E R O N A U T I C S AND SPACE ADMINISTRATION • WASHINGTON, D. C. • FEBRUARY 1974

https://ntrs.nasa.gov/search.jsp?R=19740008591 2018-09-14T22:54:58+00:00Z

1. Report No. 2. Government Accession No.NASA CR-2368

4. Title and Subtitle

mmimmasMMg**"*7. Author(s)

THEODORE BRATANOW, AKIN ECER, AND MICHAEL KOBISKE

9. Performing Organization Name and Address

UNIVERSITY OF WISCONSIN - MILWAUKEEMILWAUKEE, WI

12. Sponsoring Agency Name and Address

NATIONAL AERONAUTICS AND SPACE ADMINISTRATIONWASHINGTON, D.C. 20546

3. Recipient's Catalog No.

5. Report DateFEBRUARY 1974

6. Performing Organization Code

8. Performing Organization Report No.

10. Work Unit No.

11. Contract or Grant No.

NGR 50-007-00113. Type of Report and Period Covered

CONTRACTOR REPORT

14. Sponsoring Agency Code

15. Supplementary Notes

TOPICAL REPORT

16. Abstract

AN ANALYTICAL PROCEDURE BASED ON THE NAVIER-STOKES EQUATIONS WAS DEVELOPED FORANALYZING AND REPRESENTING PROPERTIES OF UNSTEADY VISCOUS FLOW AROUND OSCILLATINGOBSTACLES. A VARIATIONAL FORMULATION OF THE VORTICITY TRANSPORT EQUATION WASDISCRETIZED IN FINITE ELEMENT FORM AND INTEGRATED NUMERICALLY. AT EACH TIME STEP OFTHE NUMERICAL INTEGRATION, THE VELOCITY FIELD AROUND THE OBSTACLE WAS DETERMINED FORTHE INSTANTANEOUS VORTICITY DISTRIBUTION FROM THE FINITE ELEMENT SOLUTION OF POISSON'SEQUATION. THE TIME-DEPENDENT BOUNDARY CONDITIONS AROUND THE OSCILLATING OBSTACLE WEREINTRODUCED AS EXTERNAL CONSTRAINTS, USING THE L.AGRANGIAN MULTIPLIER TECHNIQUE, ATEACH TIME STEP OF THE NUMERICAL INTEGRATION. THE PROCEDURE WAS THEN APPLIED FORDETERMINING PRESSURES AROUND OBSTACLES OSCILLATING IN UNSTEADY FLOW. THE OBTAINEDRESULTS FOR A CYLINDER AND AN AIRFOIL WERE ILLUSTRATED IN THE FORM OF STREAMLINESAND VORTICITY AND PRESSURE DISTRIBUTIONS.

17. Key Words (Suggested by Author(s))

FINITE ELEMENT ANALYSISNAVIER-STOKES EQUATIONSUNSTEADY FLOW

18. Distribution Statement

UNCLASSIFIED - UNLIMITED

Cat. ol

19. Security dassif. (of this report)

UNCLASSIFIED20. Security Classif. (of this page)

UNCLASSIFIED

21. No. of Pages

84

22. Price*

$3.75

For sale by the National Technical Information Service, Springfield, Virginia 22151

CONTENTS

Page

SUMMARY 1

INTRODUCTION 1

SYMBOLS 2

MATHEMATICAL ANALYSIS 6

Equations of Motion 6

Stream function and vorticity formulation 6

Vorticity transport equation 7

Equation for pressures 8

Boundary Conditions 8

Boundary conditions for stream functions 8

Boundary conditions for vorticity 9

Boundary conditions for pressures 9

Variational Formulation of Navier-Stokes Equations 9

Stream function equation 9


. Vorticity transport equation 10

Lagrange Multipliers 12

THE NUMERICAL SOLUTION OF THE GOVERNING EQUATIONS 13

Discretization of the Equations of Motion 13

Stream function equation 13

Vorticity transport equation 15


Solution of the System of Equations 17

Stream function solution 17

iii

Vorticity solution 18

Solution for pressures 19

Detailed Analysis of the Boundary Conditions 21

Stream function analysis 21

Vorticity analysis 21

Analysis for pressure 22

DISCUSSION AND RESULTS 22

Initial and Boundary Conditions 22

Efficiency of the Method of Solution 24

Discussion of Results 25

Conclusions 25

APPENDIX A - AREA COORDINATES 27

Definition ' • 27

Differentiation of Area Coordinates 28

Integration of the Area Coordinates 29

APPENDIX B - FINITE ELEMENT MATRICES 30

APPENDIX C - FLOW CHARTS 38

REFERENCES 52

ILLUSTRATIONS 54

IV

ILLUSTRATIONS

Figure Page

Al Area coordinate definition for arbitrary point P 27

Cl Overall procedure for the computer solution 39

C2 Computational procedure for assembly of the constant matrixfor stream functions 41

C3 Computational procedure for assembly and inversion of theconstant matrix for vorticities 43

C4 Computational procedure for assembly and inversion of theconstant matrix for pressures 44

C5 Inverse routine used for the stream function matrix 45

C6 Flow chart to obtain the final solutions 46

C7 Flow chart for the subroutine used in the final programshown in Fig. C6 . 51

1 Finite element mesh for circular cylinder calculations 54

2 Finite element mesh for the pitching and plunging airfoilanalysis 55

3 A typical triangular grid element for the finite elementanalysis of the Navier-Stokes equations 56

4 Definition of the tangential and normal directions for anypoint on the obstacle 56

5a Stability of the numerical integration of the vorticitytransport equation for a point upstream of the obstacle 57

5b Stability of the numerical integration of the vorticitytransport equation for a point near the obstacle 57

6a Streamlines around a circular cylinder (Re = 40) 58

6b Streamlines around a circular cylinder (Re = 40) 59

7 Vorticity distribution around a circular cylinder (Re = 40) 60

8 Pressure distribution around a circular cylinder (Re = 40) 61

9 Streamlines around a circular cylinder (Re = 10 ) 62

lOa Vorticity distribution around a circular cylinder (Re = 10 ) 63

lOb Vorticity distribution around a circular cylinder (Re = 10 ) 644

lla Streamlines around a circular cylinder (Re = 10 ) 65

lib Streamlines around a circular cylinder (Re = 10 ) 664

12a Vorticity distribution around a circular cylinder (Re = 10 ) 674

12b Vorticity distribution around a circular cylinder (Re = 10 ) 68

v

413 Pressure distribution around a circular cylinder (Re = 10 ) 69

14 Development of the surface pressure distribution on a circu-lar cylinder (Re = 40) 70

15 Streamlines around a NACA 0012 airfoil at a 0° angle of

attack (Re = 103) 71

16a Vorticity distribution about a NACA 0012 airfoil at a 0°

angle of attack (Re = 10 ) 72

16b Vorticity distribution about a NACA 0012 airfoil at a 0°

angle of attack (Re = 103) 73

17 Pressure distribution around a NACA 0012 airfoil at a 0°


18 Vorticity distribution around a NACA 0012 airfoil at a 0°



angle of attack (Re = 105) " 76

20 Vorticity distribution about a NACA 0012 airfoil at a 4°




22 Vorticity distribution about an oscillating NACA 0012 airfoil

(Re = 103) 79

23 Pressure distribution about an' oscillating NACA 0012 airfoil

(Re = 103) 80

vx

NUMERICAL CALCULATIONS OF VELOCITY AND PRESSUREDISTRIBUTION AROUND OSCILLATING AIRFOILS

Theodore Bratanow,* Akin Ecer,**and Michael Kobisket

University of Wisconsin-Milwaukee

SUMMARY

An analytical procedure based on the Navier-Stokes equations was devel-oped for analyzing and representing properties of unsteady viscous flowaround oscillating obstacles. A variational formulation of the vorticitytransport equation was discretized in finite element form and integrated nu-merically. At each time step of the numerical integration, the velocityfield around the obstacle was determined for the instantaneous vorticitydistribution from the finite element solution of Poisson's equation. Thetime-dependent boundary conditions around the oscillating obstacle were in-troduced as external constraints, using the Lagrangian Multiplier Technique,at each time step of the numerical integration. The procedure was then ap-plied for determining pressures around obstacles oscillating in unsteadyflow. The obtained results for a cylinder and an airfoil were illustratedin the form of streamlines and vorticity and pressure distributions.

INTRODUCTION

The objectives of the investigation were to develop a method for analy-sis of general, unsteady, viscous and incompressible flow around an oscil-lating obstacle of arbitrary shape and to determine the velocity and pres-sure distribution around this obstacle. Using the finite element method, anumerical procedure was developed and tested by investigating first the un-steady flow past a stationary- circular cylinder. This method was then ap-plied to analyze the flow around stationary and pitching airfoils. The vel-ocity and pressure distributions around these obstacles were determined.The method of analysis considers the viscous effects of the boundary layerand the overall flow characteristics in a unified mathematical frame.

Unsteady flow around a circular cylinder has been analyzed by several in-vestigators [1,2,3,4,5]. In most cases, however, the numerical procedurewas developed specifically to suit the geometry of a stationary obstacle.The problem of moving obstacles was analyzed only for a circular cylinder os-cillating around its longitudinal axis [4], where the instantaneous geometryof the obstacle remains the same. A review of existing studies of unsteady,

* Professor, ** Assistant Professor, t Research Assistant, Department ofMechanics, College of Engineering and Applied Science.

viscous flow showed discrepancies in the results between analyses and exper-iments. In the case of a circular cylinder, the experimental results forthe surface pressure distribution do not agree with theoretical results [3,6,7]. Similar discrepancies are observed for the drag coefficients and thepoints of separation.

SYMBOLS

2 2A area of a finite element, in. (m )

A rectangular matrix defined in Eq. (Bl)

A general differential operator

b_ rectangular matrix representing the boundary conditions on the obstacle

b* rectangular matrix containing matrix b_, b* = (b_ 0)

B^ row vector defining the variation of vorticity, pressure and velocit-ies over an element

c^ rectangular matrix defining the free stream boundary conditions

C_ row vector defining the variation of stream functions over an element

d_ column vector specifying the stream function values on the free streamboundary

e^ column vector specifying the vorticity on the obstacle's boundary

f f(x,y) - arbitrary function constrained by boundary conditions

F F(x,y,A.) - arbitrary function with Lagrange multipliers as defined

in Eq. (27)

F_ inverse of matrix S_*

h_ column vector defined in Eqs. (49), (57), and (65)

h* column vector containing vector h_, h* = (h_ d)

_!_ identity matrix

J^ rectangular matrix defined in Eq. (B6)

k constant members in Eqs. (B5) and (B9)

j( row vectors representing the stream function's partial derivativeswith respect to the area coordinates

M(UjV,w) - nonlinear term in the vorticity transport equation,

0^ null matrix

2 2p local static pressure, Ibf/in. (N/m )

p_ column vector representing the pressure values at the nodal points of2 2an element, Ibf/in. (N/m )

p_ column vector representing the pressure values at the nodes of the en-

tire grid, Ibf/in.2 (N/m2)

p_* cplumn_vector containing vector p_ and the Lagrange multipliers,p* = (p 6 )*- *•*- — cpj

P_ symmetric matrix defined in Eq. (65)

P* symmetric matrix containing matrix P_ and the free stream boundaryconditions

CL column vector resulting from the multiplication of column h* by asymmetric inverse

n r\f in oQ Q(x,y), Q= 2 — — -__

^ column vector in Eq. (40)

r_ rectangular matrix resulting from the multiplication of matrix b* bya symmetric inverse

S^ symmetric matrix defined by Eq. (49)

S* symmetric matrix containing matrix S^ and the free stream boundaryconditions

t time, sec.

T^ square transformation matrix defined in Eq. (B7)

' ' velocity components, in. /sec. (m/sec.)

u_,v_ velocity vectors, in. /sec. (m/sec.)

3 3V volume of a fluid element, in. (m )

ttf symmetric matrix defined by Eq. (57)

W* symmetric matrix containing matrix VY and the free stream boundary con-ditions

x y coordinate directionsz

X,Y body forces in the x and y directions respectively, Ibf (N)

3 c\V del operator, V^ = i_ -tr- + 2. ~jT~

2 2 d 9V Laplace operator, V = — — + —

£,r) tangential and radial coordinate directions respectively

^ column vector representing the Lagrange multiplier

y_ column vector representing the second derivatives of the stream func-tion at the nodal points of an element

F length of the perimeter of area A, in. (m)

A Lagrange multiplier

2 2v kinematic viscosity coefficient, in. /sec. (m /sec.)

3 3p density of the fluid, Ibm/in. (kg/m )

(j) constant members of the matrix in Eq. (B6)

_£ jKx,y) - boundary conditions of an arbitrary function

$ general quadratic functional for variational formulations

2 2ty iKx,y) - stream function defined in Eq. (6), in. /sec. (m /sec.)

_^ column vector representing stream function and velocity values at thenodal points of an element

^ column vector representing the stream function and velocity values atthe nodes of the entire grid

j|j* cx>lumn_vector containing vector ty_ and the Lagrange multipliers,r = ciangle between the horizontal line and the line normal to the surfaceof the obstacle

w vorticity at a point, sec.

03 vorticity vector, sec.*a) column vector representing the time derivative of vorticity at the

_2nodes of the entire grid, sec.

oi* column vector containing vector 01 and the Lagrange multipliers,•

oj* = (a) 6 )— v cw

L, area coordinate, in. (m)

Subscripts

b obstacle

c free stream

i i-th boundary condition

m m-th node

n n-th node

o at time t

p pressure

s stream function

v total volume

w vorticity

x,y derivative with respect to x or y respectively

£ tangential direction

ri normal direction

Superscripts

m> ' variable powers

t transpose

MATHEMATICAL ANALYSIS

The mathematical analysis of the two-dimensional unsteady viscous flowaround an obstacle of arbitrary shape will be described. The variationalformulation of the problem is based on the most general form of the Navier-Stokes equations. The finite element method was applied for the discreterepresentation of the governing equations. The equations from which veloc-ity and pressure distributions were calculated, were derived using the prin-ciple of conservation of mass and the balance of momentum in terms of streamfunctions and vorticities. The boundary conditions applied for the solutionof these equations will also be described.

Equations of Motion

The governing differential equations of motion for the analysis of two-dimensional unsteady flow of incompressible viscous fluids can be writtenas follows [8]:

9u_9t + u - + v . - Y

9x v 3y " p X "d U

9x

U

9yCi)

9v9t

+ u 9v_9x + v 9y

-p 9y

9x

v

9yC2)

TT-+TT-9x 9y (3)

Eqs. (1) and (2) are the Navier-Stokes equations and Eq. (3) is the continu-ity condition for incompressible fluids.

Stream function and vorticity formulation - The vorticity and pressuredistributions can be obtained from a simultaneous solution of Eqs. (1), (2)and (3). Although a direct solution of these equations is possible, theequation of continuity can be included in the formulation in a more naturalmanner using the concept of stream functions and vorticities. The streamfunctions can be defined in such a way that they satisfy the equation ofcontinuity. The system of Eqs. (1), (2) and (3) can then be reduced to twoequations in terms of vorticities and stream functions. The vorticity isdefined as [8] :

= V x u (4)

In the case of two-dimensional flow only one component of the vorticityexists and it is perpendicular to the plane of the flow:

9v 3u ,_,.0) = - -~— (5)3x 3y *• '

The stream functions are defined for the two-dimensional case as follows:

3* 3*u = 37 v= -£Substituting Eq. (6) into Eq. (3), the continuity equation is automaticallysatisfied. From Eq. (5) the vorticity can be written in terms of the streamfunctions as follows:

to = - V24> (7)

Vorticity transport equation - Vorticity is a measure of the angular mo-mentum of the fluid particles in motion. Eqs. (1), (2) and (3) can be com-bined and using Eqs. (4), (5) and (6), the vorticity transport equation orthe Helmholtz equation is obtained as follows [8]:

3(0 3to 3cofvT + U •*— + V - ~ — = V9t 3x 3y

7 ?3 a) . 3 to

C8)

According to this equation, the time derivative of the vorticity in space,which consists of local and convective terms, is equal to the rate of dis-sipation of the viscous terms.

Using Eq. (7), Eq. (8) can be converted into another equation with asingle variable in terms of the stream functions as follows:

The left-hand side of the equation again contains local and convective in-ertia terms, while the right-hand side describes the viscous dissipation.When the fluid motion is slow, the viscous forces are considerably greaterthan the convective inertia terms. Neglecting the convective inertia terms,the equations of motion become linear with respect to velocity and describethe creeping motion of a viscous fluid.

At high Reynolds numbers the inertia forces are more important in com-parison with the viscous forces. In this case, the viscous forces can notbe neglected because, although they are small in magnitude, they change thebasic characteristics of the equations. If the viscous terms are neglected,the complete set of boundary conditions is not necessary for a unique solu-tion of the problem. This corresponds to the Euler's equation for an ideal

fluid. A unique solution of Euler's equation is obtained by using a bound-ary condition specifying zero normal velocity at the boundary. However, thecondition for zero tangential velocity is not satisfied. Therefore, theboundary layer is still important at higher Reynolds numbers although thechanges from the ideal fluid conditions occur only in a small region aroundthe obstacle. In the conducted analysis, the Navier-Stokes equations werekept in the complete form for the cases of different Reynolds numbers andthe boundary layer effects were considered.

Equation for pressures - The pressures can be calculated from anotherequation, which can be derived again from the Navier-Stokes equations.Using Eqs. (1), (2) and (3), the pressure distribution is expressed in termsof the velocity distribution as follows [9]:

,2 a24 + -£ - - PQ (10)9x 3y

where

9v _ 8u 9vQ ~ 2 1 9x 3x 9

According to Eq. CIO), the instantaneous pressure distribution can be de-termined for a particular velocity distribution. On the other hand, pres-sures are not required for the determination of the velocity distribution.

Boundary Conditions

The boundary conditions for stream functions, vorticities and pressuresaround the obstacle and the free stream boundary conditions can be des-cribed as follows:

Boundary conditions for stream functions - For the solution of Poisson'sequation, Eq. (7), two types of boundary conditions were considered:Dirichlet type boundary conditions, specifying the stream functions, andNeumann type boundary conditions, specifying the normal slope of the streamfunctions. These two types of conditions can be summarized as follows [11]:

4> = prescribed for free stream flow on the outer rectangular boundary

V- = zero on the outer rectangular boundarydx

-5-'- = free stream velocity on the outer rectangular boundary

41 = constant on the obstacle boundary

•7- = zero on the obstacle boundary (see figure 4)ot,

Boundary conditions for vorticity - The conditions for the free streamboundary can be defined according to Kelvin's theorem. A closed boundaryin the free stream will have zero vorticity along the boundary line. On theother hand, the vorticity at the obstacle boundary can be defined in termsof the normal derivative of the tangential velocity only.

a) = zero on the outer rectangular boundary

i\

u) = ^ on the obstacle boundary3n

Boundary conditions for pressures - The boundary conditions for pressurescan be defined by specifying the static pressure at the free stream boundaryand specifying the fluid pressure around the solid boundary.

p = zero on the outer rectangular boundary

•Ji = zero on the obstacle boundarydT|

Variational Formulation ofNavier-Stokes Equations

A variational method was applied for the solution of Eqs. (7), (8) and(10). The solution of the following partial differential equation

A i|> = (D (12)

was sought, under a set of boundary conditions, where A represents the dif-ferential operator. The solution of Eq. (12) can be calculated by minimiz-ing the quadratic functional

- 2 $ = / [ i | j A ^ - 2 i J J U ) ) d A (13)

with respect to solution functions that satisfy the boundary conditions.

Stream function equation - The variational formulation of the Poisson'sequation is the well-known Dirichlet's problem [10]. As applied to thePoisson equation, Eq. (7), Eq. (13) becomes

2 * = U V ijj + 2 i|j u) dA (14)

The first term of the integral in Eq. .(14) can be rewritten using Green'stheorem as

dA (15)

The last integral in Eq. (15) vanishes because of the applied boundary condi-tions. One of the boundary conditions for Eq. (7) states that the normalderivative 9i /8x is zero on the rectangular boundary. Thus, the value ofthe integral is identically zero on the vertical sides of the rectangle. Onthe upper and lower sides, the stream functions have constant magnitudes butalternating signs. Thus, their sum is equal to zero. After the applicationof Green's theorem, Eq. (7) can be written as

8x dA (16)

The solution of Eq. (7) can now be obtained by minimizing the functional $in Eq. (16). The stream functions and the velocity distribution can be ob-tained from this formulation for a given set of vorticities.

Equation for pressures - The variational formulation of the equation forpressures, Eq. (10), is again a Dirichlet type of problem. A similar pro-cedure is followed for the variational formulation of the Poisson's equa-tion for pressures, Eq. (10). The contour integral in Eq. (15) vanishesagain, since the pressure is identically zero on the outer boundary. Theresulting variational functional is

dA - p Q p dA (17)

The parameter Q in Eq. (17) is defined for a particular distribution of thevelocity.

Vorticity transport equation - For the finite element representation ofthe vorticity transport equation, Eq. (8), a variational formulation of thisdifferential equation was developed. However, for this case, an exact var-iational formulation does not exist due to the nonlinear terms on the left-hand side of Eq. (8). The right-hand side is in the form of Poisson's equa-tion and can be written in a variational form corresponding to Dirichlet'sproblem. The solution of the right-hand side of Eq. (8) can be obtained byminimizing a functional $ , given as

10

3to9x

3to9y dA (18)

The local time derivative of the vorticity term in Eq. (8) can be formu-lated in variational form using Hamilton's principle. The discretized formof the vorticity transport equation can then be written in matrix form as

3o>(19)

where the matrix M is a nonlinear function of velocity and vorticity dis-tributions. The solution of Eq. (19) can be obtained by minimizing thefunctional <I> , given as,

^- [ (/to) - t/M to dA.dt \ / J (20)

The matrix M in Eq. (19) can be derived from a perturbation analysis of thevorticity transport equation. From a Taylor series expansion of the vel-ocities in terms of the vorticities, the velocity distribution at time tcan be written as

u = u3u

"3to

32u(Aw)

(Ato)'

2T (21)

Once the instantaneous derivatives of the velocities with respect to vor-ticities are determined, the matrix M can be calculated for any desiredaccuracy. From the equation of continuity and the definition of vorticity

u = 0 V x u = to (22)

the derivatives of velocities with respect to vorticities can be calculatedas

3uV • — = 03u

3u(23)

The solutions of the systems of equations in Eq. (23) are similar to thesolution of Eq. (7) except for a right-hand side consisting of unit vortic-ities at each nodal point of the finite element grid. Considering that thevelocity distribution is not very sensitive to the incremental changes inthe vorticity distribution, only the first terms in the Taylor series can beemployed in the analysis. The variational formulation of the vorticity

11

transport equation can then be written as follows:

9xdA (24)

The local time derivatives of the vorticities can be calculated by minimiz-ing Eq. (24).

Lagrange Multipliers

The variational formulation of the governing differential equations de-rived above does not include the boundary conditions. The time dependentboundary conditions can be introduced in a simple fashion which then becomesone of the main advantages of the variational method. The Lagrangian mul-tiplier technique was used for this purpose. Accordingly, the boundary con-ditions are added to the variational formulation as additional constraintconditions. The method of solution will now be summarized.

Considering a variational functional of the form

f(x,y) (25)

where the boundary conditions are excluded and that

iiCx.y) = 0 (26)

is a set of boundary conditions, the solution of the differential equationssatisfying the boundary conditions is sought through another functional

V^F(x,y,Ai) = f(x,y) + ) X Cx,/) (27)* •

where A. is the Lagrange multiplier. The solution can be obtained by mini-

mizing F(x,y,A.) with respect to x, y and A., as

-° -" =0 (28)

In the case of a quadratic functional, a system of linear algebraic equa-tions are obtained. The application of the Lagrange multiplier technique

12

produces a constrained minimum of the functional, which satisfies the bound-ary conditions.

New boundary conditions can be included using the Lagrangian multipliertechnique by defining only the additional constraint equations at each timestep. In this problem the obstacle boundary conditions change at each stepof the numerical integration, depending on the position and the motion ofthe obstacle. For a stationary obstacle, the Poisson's equation is solvedonly for changing terms on the right-hand side of Eq. (7). The boundaryconditions at the free stream boundary do not change with time. The appli-cation of the Lagrangian multiplier technique reduces the computational costof the solution considerably.

THE NUMERICAL SOLUTION OF THE GOVERNING EQUATIONS

When the variational formulation of the governing differential equationsis completed, they can be discretized in finite element form. The result-ing discrete systems consist of linear algebraic equations and a matrix dif-ferential equation. The details of the discretization and the applicationof the finite element method is summarized in this section,. The solutionsof the systems of equations are analyzed and the corresponding boundaryconditions are discussed.

Discretization of the Equations of Motion

Following the basic discretization procedure of the finite element anal-ysis, the obstacle and the surrounding continuum were represented by tri-angular finite elements. The finite element gridwork for the analysis offlow around a cylinder is shown in figure 1. In this case the objectiveof the particular triangulation was to represent most accurately, theboundary layer effects around the cylinder. However, in general, it is notnecessary to describe the obstacle boundary with nodal points. For example,the gridwork in figure 2 was used for the analysis of the flow around amoving airfoil, where the position of the airfoil changed at each time step.

Stream function equation - The variation of the stream functions over atriangular element was approximated by a third-order polynomial, which canuniquely be determined from the values of the stream functions and veloci-ties at the nodes of the triangles. The stream function at any point ofthe triangle can then be written as

ty = C_ A (29)

where A_ is a rectangular matrix describing the approximation functions asshown in Eq. (Bl) of Appendix B. C^ is a row vector formed by the area co-ordinates of a point and is another vector formed by the nodal veloci-ties and the stream functions of the finite element:

13

(30)

(31)

The area coordinates C- in Eq. (30) are described in detail in Appendix A[11]. 1

Vorticities, on the other hand, were considered to vary linearly over atriangular element and can be defined as follows:

= B a) (32)

Eqs. (29) and (32) can be substituted in Eq. (16) to obtain a discrete var-iational functional as follows:

A + AtCtC A dA - A C B dA (33)

In this case the integration domain A indicates an integration over the areaof the triangular element and a summation of the terms at each node from allneighboring elements.

The solution of Eq. (7) can be obtained by minimizing the variationalfunctional in Eq. (33) with respect to u>.. After setting the first deriv-

ative of $ in Eq. (33) equal to zero, a system of linear algebraic equationsis obtained as follows:

r rI \ ff^C A + AtCtC A I dA ty = / [A tC tB Ii L — x— x— — y-y— J — I L --- J

J A J A

dA CO (34)

where matrix A_ is constant for each triangle. Integrating and adding the

C C and C C terms in Eq. (34), a square matrix is obtained for each of the—x—x —y-y ttriangular elements. The matrix £ B^ is also constant for each triangularelement. Each of the matrices in Eq. (34) is given in Appendix B.

14

Vorticity transport equation - In the solution of the vorticity trans-port equation, the variation of velocities over each finite element was ap-proximated linearly as

= B u (35)

= B v (36)

Substituting Eqs.(32), (35) and (36) into the variational form of the vor-ticity transport equation, Eq. (24), the following discrete functional wasobtained:

TT-— — — ouV 8*8—x—x + BtB I to-y-y -

+ cotBtB u B c o----- x- + c o B B v B toy -

dA (37)

The functional $ in Eq. (37) was then minimized by differentiating with re-spect to 0). The resulting system of ordinary differential equations withrespect to time was written in matrix form as

B B dA 3- =at + B B u Bx + B B v B > dA CO (38)

The velocity distribution obtained from the solution of Eq. (34) togetherwith the instantaneous vorticity distribution is used to calculate the timederivative of the vorticity distribution. The coefficient matrix of theleft-hand side of the equation does not change with respect to time. Theright-hand side of the equation is calculated at each time step of the nu-merical integration. Each of the matrices in Eq. (38) is given in AppendixB.

Equation for pressures - The pressures were also assumed to vary linearlyover a finite element and were defined as follows:

15

= IE. (39)

Repeating the same analytical procedure, a variational functional was ob-tained by substituting Eqs. (35), (36) and (39) into Eq. (17) to obtain:

* = -2etB + BtB— x— x — y-y

dA - p Q B p dA~

(40)

By minimizing the functional in Eq. (40) with respect to nodal pressures,the resulting system of algebraic equations can be written as

BtB + BtB—x—x ±y—y (41)

The right-hand side of Eq. (41) can be calculated in terms of the var-iation of velocities over the finite element from Eq. (11)

Q = - 23x3y 2 2

3x 3y(42)

The second derivatives of the stream functions in Eq. (42) can be calcu-lated from the definition of _^ for a finite element in Eq. (29). By tak-ing derivatives of the stream functions in Eq. (29), the second derivativescan be written in discrete form as

Y = J K (43)

where J_ is a rectangular matrix as shown in Appendix B. The nodal valuesand the derivatives of stream functions, in terms of area coordinates, canbe defined in Eqs. (44) and (45) respectively.

(44)

-a2iir-aV2 21 2

-3V- aV" "32ij;(45)

16

Substituting Eq. (45) into Eq. (43), the second derivatives of the streamfunctions can be obtained for each element in terms of nodal stream func-tions as follows:

Y = T (46)

Matrix T_ is obtained from matrix J_ in Eq. (B6) and matrix £ A in Eq. (29).The resultant matrix T_ for each finite element is given in Appendix B.After the vector £ in Eq. (41) is defined, the final result of the integralon the right-hand side of Eq. (41) was calculated as shown in Appendix B.

Solution of the System of Equations

For the numerical solution of the governing equations, each of the Eqs.(34), (38) and (41) was written in matrix form including the boundary con-ditions. A similar procedure was used for all three equations and the pro-gramming was carried out in a uniform manner.

Stream function solution - The boundary conditions for the solution ofstream functions in Eq. (34) were included as external constraints in ma-trix form as follows:

(47)

for the free stream boundary and

= 0 (48)

for the obstacle.tain

Eqs. (47) and (48) can be combined with Eq. (34) to ob-

S c b— — s -s

c1 0 0— s — —

t^s - -

"i "

6— cs

.V

Vd— s

°-

(49)

In Eq. (49), the S_, c and d matrices are constant while b depends on the•3 > 5

position and the shape of the obstacle. Separating the time-dependent por-tion, Eq. (49) can be written as

17

b* >"

^— bs_

=

In*

£(50)

where

S* = b* =— s

b

0_h* =— s

h— s

d-s

47* =

"i "

6-cs

(51)

In this case, _6, is a column vector representing the Lagrange multipliers

corresponding to the obstacle's boundary conditions. In the case of themoving obstacle, matrix b* is defined at each step of the numerical inte-

gration. Eq. (50) can be solved to obtain ty* as

i* = -where

= S*"1 h*

r = S* b*

(52)

(53)

(54)

S* matrix is a constant for a particular grid system and has to be invertedonly once. Different types of obstacles can be defined by specifying the b^vector. Computational efforts are thus reduced considerably.

Vorticity solution - The boundary condition for the vorticities can bewritten in matrix form as

(55)

on the free stream boundary and

tb w = e—w — —w (56)

on the obstacle boundary.

The total system of equations for the solution of the vorticity vectoris then

18

W e b— — w — w

c1 0 0— w — —

tb 0 0— w — —

to

6— cw

— bw

h— w

0

e. — w

(57)

In this case IV and c matrices are again constant and can be combined as

W* b*— — w

b*1 0.~y — _

to*

6_^bw_

=

h*— w

e-w

(58)

where

W* =

W c— -w

ct 0-w — _

b* =— w

b— w

£h* =— w

h— w

0w* =*"~

•

tO

6_— cw

(59)

The time derivative of the vorticities can be calculated from Eq. (58) as

o* = a— -% - r (b* r )— w — w — w (b*— w

where

q = W-% — h*— w

- e )—w (60)

(61)

r = W* b*—w — -w (62)

Matrix W* is again constant for a particular grid system and matrix b isdefined for a particular obstacle shape.

Solution for pressures - The boundary conditions for pressure are

c* p = 0~P -

at the free stream boundary and

(63)

19

= 0 (64)

on the obstacle. The total system of equations for obtaining pressures canbe written as

P c b"- -p -p

ct 0 0"P

tb o. 2

E

6^P

_V

h-P

0_

0_

(65)

Combining matrices P_ and c

P* b*-P

b** 0

E*"

.V

=

h*-P

(66)

where

P* =

" P c "-pct 0

. -P -.

b* =-P

"b "

oh* =

r h "-P

o—

p* =E

6.-CP.

(67)

The solution of Eq. (66) for p_*is

E* = r (b* r )-P -P -P V

(68)

where

P*"1 h*- -P

(69)

r = P* b*-P - -P

(70)

Matrix ]?* is assembled and inverted only once for a specified finite elementgridwork.

20

Detailed Analysis of the Boundary Conditions

The assembly of the matrices corresponding to the boundary conditionscan now be described in further detail. The governing equations have to bedefined in a discrete form suitable for computer applications.

Stream function analysis - The c matrix in Eq. (47) represents the

boundary conditions applied at the free stream boundary. The stream func-tions and the velocities at the free stream are described depending on thegeometry and flow conditions. Vector d represents the obstacle boundary

conditions as described on page 8.

Matrix b of Eq. (48) has two parts; the first half specifies constant

stream functions on the obstacle and the second half makes the tangentialvelocity on the obstacle equal to zero. The first half of the matrix con-sists of terms for the nodes with equal values of stream functions

The second half of matrix b specifies that the tangential velocitieso

are equal to zero again at specific points on the obstacle boundary. Thetangential velocity u>- was defined from figure 4 in terms of the horizontal

and vertical velocities as

u,. = - u sin 6 + v cos 9 (72)

The angle 6 is defined by the obstacle geometry and can be included into thematrix formulation of the boundary conditions as follows:

Ur = - PC A sin 6 + £xA cos 6 1 _£ (73)

The discrete coefficients of Eq. (73) were put in matrix form as shown inAppendix B.

Vorticity analysis - The equations specifying zero values of vorticitiesat the free stream boundary were assembled in matrix form. The componentsof the c matrix were defined at points on the free stream boundary. As de-wfined in Eq. (72), the tangential component of the velocity was calculatedagain in terms of the nodal velocities and stream functions. Then matrix bcontains the area coordinates of each of the points of the obstacle. Col-umn vector e is defined at each point by:

21

where u) is obtained from Eq. (32) and u£ from Eq. (73). Substituting Eq. (6)

into Eq. (72) and taking derivatives with respect to r\, the normal gradientof the tangential velocity can be written as

2 2sin 29 + T\> cos 6 + ii» sin (xy rxx yy (75)

The second derivative of stream functions has already been presented in Eq.(46).

Analysis for pressure - Matrix c of Eq. (63), specifying zero pressure

conditions, is identical to matrix c for the vorticity boundary conditions.

In Eq. (64) matrix b is formed to satisfy the boundary conditions on the ob-

stacle surface by specifying that the normal gradient of the pressure isequal to zero. For the angle 6 calculated previously in Eq. (72), the normalgradient of pressure is

Using the discrete finite element representation of the pressures, Eq. (76)can be written as

B cos 6 - B sin 9 I p (77)-y —x J £- *• J

Each of the three terms in Eq. (77), defining the boundary conditions at onepoint, is inserted into the system of equations corresponding to the nodes ofthe surrounding triangle.

DISCUSSION AND RESULTS

The analytical procedure was tested by calculating first the unsteadyviscous flow past a circular cylinder. The procedure was then applied forthe analysis of flow around an airfoil. The obtained results and some of thedifficulties related to the initial and boundary conditions will now be des-cribed. Accuracy and stability of the solution and the numerical integrationtechnique, along with convergence rates and computer time, will also bediscussed.

Initial and Boundary Conditions

The boundary conditions were employed in a discretized form. Several al-ternatives for satisfying the boundary conditions on the obstacle surface

22

were tested using different discretized constraint equations. This investi-gation was carried out to decide which set of constraint equations yieldedthe best accuracy.

For the stream functions, the obstacle boundary conditions were definedby specifying that the tangential velocities are equal to zero and that thestream functions are equal to each other at several points on the obstacleboundary. The same boundary conditions were also defined by specifying thatthe horizontal and vertical velocities are equal to zero at the same points.The number and positions of the points at which the boundary conditions arespecified were tested and compared. It was found that the direct applica-tion of zero tangential velocity condition was as accurate as specifying thatboth horizontal and vertical velocities are equal to zero.

For the analysis of vorticities, the boundary conditions were defined bysetting the vorticity equal to the normal derivative of the tangential vel-ocity at the obstacle boundary. The accuracy of the boundary conditions wastested by imposing an additional boundary condition at the leading edge ofthe cylinder and the airfoil at zero angle of attack. When the vorticitywas artificially set equal to zero at the leading edge, the obtained resultsfrom the numerical integration were close to the results obtained withoutthis additional condition.

Another important feature of the developed procedure is the descriptionof the initial vorticity distribution. At the beginning of the numerical in-tegration, the vorticity distribution was assumed to be equal to zero for theentire velocity field. With advancing time, vorticities develop on the ob-stacle surface due to the vorticity boundary conditions and then diffusethroughout the flow region. In the initial stages, the disturbances intro-duced by these boundary conditions are quite high in magnitude. These dis-turbances cause instability in the numerical integration of the vorticitytransport equation. To eliminate this problem, an underrelaxation factor wasapplied for the first few steps of the numerical integration. A small per-centage of the vorticities, generated at the obstacle surface, was includedin the equations. Once these vorticities diffused away from the obstacleboundaries, the underrelaxation factor was removed and the integration wascontinued with the full value of the surface vorticity distribution.

Another computational difficulty encountered during the solution was re-lated to the position of the points representing the obstacle on the finiteelement gridwork. Singularities develop in the solution routine, when thenodes describing the obstacle are concentrated in any one finite element tri-angle. This problem is a result of the number of boundary conditions spec-ified in each finite element. Since the stream functions vary as a third or-der polynomial, up to three boundary conditions can be specified in each tri-angle. However, since .there are two boundary conditions applied to each ob-stacle node in the stream function analysis, only one obstacle node can beplaced in a single triangle. Similarly, since vorticities and pressures varylinearly over each finite element and only one type of boundary condition isapplied in the solution, again only one node can be considered for each tri-angle. If more than one node is specified in a triangle, singularities arise

23

in the matrices shown in Eqs. (51), (59) and (67). A similar problem occursif a boundary condition is duplicated. For example, if the vertical and tan-gential velocities are specified as the boundary conditions, these two condi-tions may be equivalent at some points on the obstacle.

Efficiency of the Method of Solution

The systems of linear algebraic equations, Eqs. (49), (57) and (65),were solved by direct inversion of the coefficient matrices shown in Eqs.(52), (60) and (68). Since three unknowns are considered at each node of thegridwork for the stream function solution, the order of the largest matrix isequal to three times the total number of nodes plus the number of boundaryconditions at the free stream. Although the resulting matrix is relativelylarge, it is well-conditioned and the inverse of this matrix was obtainedquite accurately by the method of partitioning. The accuracy of the resultswas checked from the symmetric properties of the flow around a circularcylinder.

For a UNIVAC 1108 computer with 65K core storage, the computation timeneeded to invert the largest matrix (458 x 458) was slightly over 12 minutes.A significant portion of this time was used up in transferring data betweenthe low speed drum and the high speed core. This was necessary because thecore was able to hold only one-fourth of the total matrix at any one time.

Since the major portion of the system of equations was solved by the in-version of matrices S*, W*, and P*, only the constraint equations represent-ing the obstacle's boundary conditions were solved at each step of the numer-ical integration. The computer time needed for the solution of several sam-ple problems, presented in this analysis, is summarized below:

Re At No. Time Steps Comp. Time

40 1.00 110 7.5 min.

1000 0.10 120 8.5 min.

10000 0.01 300 15.5 min.

Again, most of the time used by the computer was for the transfer of datafrom drum storage to core storage.

An important source of numerical errors in the analysis was due to thenumerical integration procedure. The accuracy and stability of the integra-tion was checked by comparing results from different integration techniquesand dif ferent integration time steps. Three different-'numerical integrationroutines were tested, and the results were found to be quite accurate. Thetime increment employed for each of these integration methods was propor-tional to the inverse of the Reynolds number. At higher Reynolds numbers,although it required more steps to reach a steady state, the total number ofsteps corresponded to a shorter real time, due to the convection terms.

24

Figure 5 shows the convergence to a steady state with different timeincrements.

Discussion of Results

The results shown in figures 6 through 23 include contour plots ofstream functions and vorticity and pressure distributions around a station-ary circular cylinder and a stationary and oscillating airfoil.

The sequence of streamline plots in figures 6, 9 and 11 shows the dev-elopment of the wake behind a cylinder. As time advances, the wake regionbecomes larger, as does .the angle at which the flow separates. Streamlinesaround a stationary airfoil are shown in figure 15. At the beginning of theintegration, the flow is attached to the airfoil. As time advances, vortic-ities are generated, forcing the flow away from the surface of the airfoil.This process continues until the vorticities have moved past the airfoil in-to its wake. It should be noted that these results remain symmetric through-out the entire integration time sequence, which is one indication of the ac-curacy and stability of the solution. At higher Reynolds numbers, unsymmet-rical results are also obtained.

The plots of the vorticity distribution shown in figures 7, 10, 12, 16and 18 also show symmetric results. The flow around an airfoil, stationaryor oscillating, is shown in figures 20 and 22, respectively. Since the ini-tial vorticity distribution was assumed to be zero, the first few plots showthe vorticities located near the obstacle. The following plots show the dif-fusion of the vorticities in the surrounding area.

The pressure contours, or isobars, are shown in figures 8, 13, 17, 19, 21and 23. The highest pressure is at the leading edge of the cylinder and theairfoil. The surface pressure distribution in figure 14 shows that the pres-sure decreases towards the trailing edge until reverse flow occurs. In thisregion, the pressure is of an unsteady nature.

Conclusions

The advantages of the developed mathematical procedure will now be sum-marized. The analysis of the fluid flow around any arbitrarily shaped obsta-cle can be facilitated by specifying a series of points on the boundary ofthe obstacle. The solution routines of the analytical procedure are effi-cient. Since the time-dependent boundary conditions were kept separately,the remaining portion of the system of equations, which amounts to the bulkof computations, was assembled in matrix form and inverted only once. Thevelocity distribution was calculated directly at each time step, without us-ing iteration. The major portion of the computational cost involved in theanalysis was due to core-to-drum transfers of data. Further programmingefforts to minimize the data handling costs can decrease the computationalcost considerably.

25

The illustrated results give a good indication of the type of flow prob-lems for which the method can be applied. The cylinder problem was analyzedfor testing the method. Numerous solutions for the cylinder already exist.The obtained results were compared and good correlation was achieved. A re-fined evaluation of the method and the accuracy of the obtained results canbe achieved through a more rigorous error analysis. The numerical errorscan be related to the physical problem in terms of the flow characteristics,the obstacle shape and the finite element representation of the problem;such as the finite element gridwork and numerical integration technique.The use of higher order finite elements can also lead to an improvement ofthe present method.

26

APPENDIX A

AREA COORDINATES

Definition

Area coordinates are useful for integration of polynomials over an area[12, 13]. In this analysis, area coordinates were employed to approximatea variational functional to be integrated over a finite element region.

Using area coordinates, the position of a point P inside the triangle infigure Al can be specified in terms of the ratio of the area of the subtri-angles to the area of the triangle,as:

The ratios for AI, A™, A are defined

A(Al)

Figure Al. Area coordinate definition for arbitrary point P.

The £ 's are defined as area coordinates. The three area coordinates aremalso related by the expression

(A2)

The area coordinates can be written in terms of rectangular coordi-nates by satisfying the conditions at the nodes and using Eq. (A2) as

27

y

1

~ x lyl1

X2

y2

1

Vy3

1

(A3)

Inversion of the square matrix in Eq. (A3) leads to the definition of areacoordinates in terms of the rectangular coordinates:

12A

X23 X32 X273 X2

X13 X3yl X173

y!2 X21 Xly2 X2yl

y

1

(A4)

where

x = x - xran m n y = y ~ ymn m n (AS)

Differentiation of Area Coordinates

The derivatives of the area coordinates in x and y directions can be cal-culated using Eq. (A4) as

Bt

— x

9x

1bT_

12A

y23

y31

_y!2_

(A6)

28

32

B1-y 3yi_2A 13 (A7)

9y

Integration of the Area Coordinates

The simplicity of the integration of polynomials in terms of area coor-dinates is one of the important advantages of area coordinates. In generalform the integral can be written as [12]:

n m! n! P!(m+n+p+2)! (A8)

Eq. (A8) was used extensively in deriving the matrices in Appendix B.

29

APPENDIX B

FINITE ELEMENT MATRICES

A =

1

0

0

3

3

0

0

0

0

2

0

0

0

x21

-X1 3

0

0

0

0

3

0

0

0

12

y31

0

0

0

0

-K

0

1

0

0

0

3

3

0

0

2

0 0

0 0

0 0

0 0

0 0

x -y32 23

-x y21 12

0 0

0 0

_lx .12 X2 2 72

0

0

1

0

0

0

0

3

3

2

0

0

0

0

0

0

0

x1 3

-X32

3

"2 X3

0

0

0

0

0

0

0

-y.

y* 1

32

31

23

(Bl)

where:

x = x - xran m n

. = y yran 'm 'n

30

K X

Ti ^

K Xi i

P-I CM

K X%o o

~ ~

K X1 1

<M rg

« <*

<o o

K X

' '

'"it '"X

X X

•• M

a iK X

rt rt•« N

K Xt 1M «

i i« m

s" £

K X

5" £

m MK X

M ft

2 2

K X

00 COO O

e*t *>

a 5;5 £

^ ^K X1 1

^ ^

5 £

m <viK X

•« c*

*iS ,xm ,-.

00 00

W« <N«

\o \o1 1

•• fl<

K X

S ^M m

X X0 «

X X

•* •»

5 &

X X

00 00o o

"i ^1 1

^ ^15 ^

n M

K Xi i

£ £

00 00

- „

X X*

s $5 £

"<M '«•>

X X

^ ^~CN

sO vO

X X

•o >o1 1

"v "vr00 00o o

X X1 1

A £X Xf*i (N

N m

X X

s" -fm N

X X

(M (M

X X

<N IN

MM w«

* -0

X X

5~ ^

X X

X X(M fM

«N (N

•* m

X X*

- -

*• M

X X

X X(N (M

¥ ¥

• i£ £

c^ M

— 1 •»X X

X X

X X

X XrM (M

X X

X X(N r*j

\O <A

M mX X

s $X X

w NP^ (N

X X

X X

w —CM (M

W« (MM

00 00

^ ^1 1

^S -b

EN r-;

M 2

M *-

TN (N

« >£>

x X

-. «X X

sO O

X X

(N <M

« ^X XI1* CM

x- "X X

X X*

5 5"m M

X X00 CO

"It ^»

1 1

5 £fs r-4

x" ?T

X >,« 0

X X

(N f4

m M

X X

— M

X >s

7 7«- NmOO 00

y

f-<4-1

t/1

II | | m v> C4

•i ^ »iS 4 "ti ~t>

CN 01 (M (M n d «t

^^ + +— M w « CM

N « -*1 — i m

— (VI (NX «H --H

-. 00 00

XN

<N1

X Xoo oo ••'.

•!

' •,

31

A180

12

3

3

3

3

1

2

2

1

1

3

12

3

2

1

3

3

1

2

1

3

3

12

1

2

2

1

3

3

1

(B3)

B dA = (B4)

32

V BtB + BtB + BtB u B + 8*8 v B > dA|_-x-x -y-y J x y f

k,d24

2u + u + u1 2 3

u + 2u + u1 2 3

u + u + 2u1 2 3

vk_,_ d4A

y23

yai

y12

ke24

2v + v + v1 2 3

v + 2v + v1 2 3

v + v + 2v1 2 3

vk+ e

4A

x32

X13

X21

where

[y to + y U) + y to \\ 23 1 31 2 12 3 j

(B5)

k = - [ x a j + x o ) + x a )3 2 1 1 3 2 2 1 3

J =14A2

2

2

X32

2x y32 23

2

2

X1 3

2x y13 31

2

2

X21

2x y21 12

where

2y y 2y y 2y y3 1 2 3 3 1 1 2 2 3 1 2

2x x 2x x 2x x1 3 3 2 1 3 2 1 3 2 2 1

<f> = x y + x y1 1 3 1 2 2 1 3 1

) = x y + x y2 3 2 1 2 2 1 2 3

> = x y + x y3 3 2 3 1 1 3 2 3

2<f> 24)

(B6)

33

en~ Xen CM

X -«* X

<N X

'

X« CM

X X01 •-•

CM ** <O

£ 3?t

X*OI «

X X

*"• X

X* -Ol« ^

X •»N £1

01

** X

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CM « -t

CM X*1 to

I

CM

X

X X

+

X*

X*

0)

X*

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00 tO^1

CM«

X«M «1 CM0)

*» X"x 7

1

0>** X

** xCM CM *O

<M K1 tO

1

X

X X

CM CM *-

X XfM tO

1

01

Xft 0»

X X

CM<M X

x" **«•» X

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— XX oi

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^X

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+

>rCM v+

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x

to~t

CM ~

X.1 CM CM

<M XCM 0> CM

X i

1

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X i

i

0t

XCM M

X •»— XCM tO

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<M +

CM

X

m ^*» X-^ to

X 1

—x

0» -A*

**» x**X** t

1

«nX

x *rfM +

01 —X X

•* »OO 1

X X

00 1

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i x

en i

1

X*> CM

X X^

i

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0> ~*x x

x x1 1

xCM en

X X

xCM

X01

1

xen ™

X X

1

en

XrM oi

— X

>* 'i

>rx x

en 01

¥ -f

X

Xm X

^ 3r

01 01 CM

so* X

ftX

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X X-1 01

CM 01 01

X X(M to

i

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0t OI

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CM CM mX XfM rO

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X X

CM

Xen n— • CM

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OI

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0>X

t

„01 01

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oi -* fMX I

X*01 M

« CM

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CM

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CM

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x -e-CM X

x »o1

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(M +

01 CM

x -e-01

< 3?OO 1

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x00

1

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«

xW C4

fl" t

X0)CM

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1

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00 to*I

„CM enX m

fsl oi CM1 XCM CM

X

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CM toI i

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en

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X X

^X X

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1

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— CMX X

£ to"

CM

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*

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CM 0>X X

Xoo to

1

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A

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X

CM

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X XCM «

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1 I

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xen CMr* 0t

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x xCM tOI i

CM

CM en M

X XsO ««•

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01** mx -e-

00 1

CM 0>

x -e-m

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enX

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X K

CM 1

X00

1

0>

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x^ -e-

/" "f'

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X •*

x

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01

X

x -e-01 M

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1

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,— 1

X*mXX

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i-H

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rH

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i•-IX

CM X

CM

XX

CM

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m

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(N CN VO

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01

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1 bO<st

ftX

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enX

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bo

bOX

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bo

CM en enX

LO

XCM N CM en en er

kj* X X kj> X XI I I I I I

e n e n e n r H ' H f H c M c M C Mc M C M c M e n e o e n r - H » - < r - i

x x x x x x x x xP H i - t r H C M C M C M e n e n e n

CN CN \O CM VO CM CM

CM

CD

•Htn

X

+

^-

X

II

CD

t/)OO

X

LO

4rH CM I-H

X

LO

I

CM CM CM

X

LO

X kj^ X kj> X kJ*

L O < ; ^ L O < ; X L O < ;i — I C M C M i - I C M C M i — C C M

I + + I + + I +

II•r

Xi

XI I

CD

•Ht/1

T

c M C M c Mm en en

X X^ X ~r-i_ r-t _ _ CM CM CM

X X X X X

\ O C M C M \ O C M C M \ O C M C M

<CM

0)rH<U

36

or-H09

(NX

(Nr—I

X

CNJ<^>

X(N

X

37

APPENDIX C

FLOW CHARTS

Several computer programs were prepared in obtaining the presented re-sults. The first step of the numerical procedure was to determine the mat-.rices describing stream functions, vorticities and pressure distributionsfor the finite element mesh without the obstacle; the square matrices S_*, W*and P* shown in Eqs. (51), (59) and (67) were assembled and inverted. Sincethese matrices remain constant, they were inverted only once and then storedfor use in the solution program. After these matrices were inverted, thematrices representing the boundary conditions in Eqs. (52), (60) and (68)were added to give the final solution. The overall procedure is summarizedin figure Cl.

The matrices defining the finite element mesh were assembled from indi-vidual element matrices. These individual matrices were derived from Eqs.(B2), (B4) and (BIO) for stream functions, vorticities and pressures re-spectively. It should be noted that the order of the element matrix forstream functions was three times larger than those for either vorticitiesor pressures, since at each node there were three unknowns; stream functionand two velocities. For the final matrices S*, W* and P*, the order of thelatter two was equal to the number of nodes in the finite element mesh,while the order of S_* was three times larger. Since the entire matrix Sj*could not be stored in the computer at one time, it was assembled one quad-rant at a time and stored on magnetic tape as shown in figure C2. Thesmaller matrices, for vorticity and pressure, were assembled at the sametime in conjunction with the free stream boundary conditions, as shown infigures C3 and C4 respectively.

After the matrices were inverted and stored on tape, the solution wasobtained using another computer program. This program sets up the matricesfor the obstacle boundary conditions and the column vectors correspondingto the right-hand sides of Eqs. (49), (57) and (65), and then calculatesthe velocity, pressure and vorticity distributions. These distributions canbe plotted and their development with respect to time can be recorded at anytime instant. The flow chart for this final program is given in figure C6.

38

( Start )

1 I

AssembleS*, W*, P*

Find Inverses

S*"1, W*"1, P*"

Assemble theh* and b* matrices— s — s

Do multiplicationsfor i|>*

Assemble theh* and b* matrices— w — w

Do multiplicationsfor ca*

Do additions for0) - co ,, + w* At—new —old —

No

Fig. Cl. Overall procedure for the computer solution.

39

Fig. Cl. Continued.

Assemble theh* and b* matrices- p - p

Do multiplicationsfor p_*

Plot ip*, w, £*

Stop )

40

( Start )

Read mesh coordinatesand triangle nodes

Do once for eachquadrant

IInitialize the

S* matrix

Do for alltriangles

Pick out the nodes and theircoordinates for each triangle

Calculate the area ofeach triangle

Assemble the (A) matrixfrom eqn. (Bl)

IAssemble the C* C + C* C—x —x -^y —y

matrix from eqn. (B2)

Multiply these two matricesL /"/"»C- .-,for A C C ) A-y -y -

Fig. C2. Computational procedure for assembly of the constant matrix forstream functions.

41

Yes

Put this matrix intoits proper place in S*

J

Do for all nodes onthe free stream boundary

Pick up one node

Yes

Put one's in three placesin the S* matrix

Store the completed/S_* matrix on tape \

I

Fig. C2. Continued.

42

( Start ~)


Initialize the W*matrix

Do for alltriangles


Calculate the area foreach triangle

Assemble the matrixshown in eqn. (B4)

Put the elements of eqn. (B4)into the proper place in

matrix W*

Erase those rows and columns inW* for the nodes on the free

stream boundary

Store W*" onmagnetic tape

C Stop

Fig. C3. Computational procedures for assembly and inversion of the con-stant matrix for vorticities.

43

( Start ~)


Initialize the P*matrix

Do for alltriangles

\


Calculate the area foreach triangle

Assemble the matrixshown in eqn. CB10)

IPut the elements of eqn. (BIO)into the proper place in

matrix P*

I

I

Erase those rows and columnsin P* for the nodes on the

free stream boundary

( Stop )

Fig. C4. Computational procedure for assembly and inversion of the con-stant matrix for pressures.

44

Do multiplications for

S^l £12 and store


.§21 £u and store


-1Subtract (§21 S_u S

from £22

Invert for-1 -1

£22 = (§22 - §21 §J1

and store


£21 = £22 ('§.21 §.ll)and store


Zl2 = (§,11 £

and store


f.12 (§21 §.ll)

Subtract £12 (§21 *Lnfrom Sij and store

Fig. C5. Inverse routine for the stream function matrix.

45

Read the node coordinatesand triangle numbers

Read the coordinatesof the obstacle's nodes

Do for eachobstacle node

Calculate sin 6 andcos 6 as shown in Fig. 4

Do for each triangley—

*Find each triangle's

nodes and their coordinates

1Calculate the area of

each triangle

Calculate the area coordinatesfrom eqn. (A4)

No

Yes

Save this triangle's numberand these area coordinates


\

I

Find the triangle for eachnode and its coordinates

Fig. C6. Flow chart to obtain the final solutions.

46

Assemble eqn. (B9) and putit in the proper place in

the b_* matrix

_LI Assemble eqns. (30) and (Bl) )

Multiply together and putin the proper place in

the b* matrix

Negate it and put in thenext column of the b* matrix

Initialize the h matrix

Assemble the d^

matrix from eqn. (47)

J_Do for each triangle y— —

Fig. C6. Continued.

Find each triangle's nodesand their coordinates

Calculate the areaof each triangle

Assemble eqn. (Bl)for the A matrix

_LAssemble eqn. (B2) and

multiply it by A_ , then by A_

IMultiply the result by

each triangle's vorticitiesand put in the properplace in the h* matrix

J_| Do 4 times ^— — — -»,

/Read one quadrant

\of the S* matrix"J

Multiply by the correct halfof the b* and h* matrices

~ ~"

Addthe

the result intocr and r matrices

_J

"47

Fig. C6. Continued.

Call the subroutinefor if*

IWrite and/or plot |* [

Initialize the h* matrix

|Do for each triangle^

Find each triangle's nodesand their coordinates

Calculate the areaof each triangle

Find the vorticities andvelocities for this triangle

Assemble eqn. (B5) and putin the proper place in the

h* matrix—w

IInitialize theb* matrix


Find the triangle and areacoordinates for each obstacle node

Put the area coordinates inthe proper place in the

b* matrix

Assemble the e

matrix from eqn. (74)L I

Read the complete

W* matrix

i.Do multiplications

for eqns. (61) and (62/

48

G K-

Call the subroutine for u*

No

Yes

Initialize the h* matrix-P

[Do for each triangle) •*

T ,Find each triangle's nodes

and their coordinates

ICalculate the area of

each triangle

Assemble the T matrixfrom eqn. (B7)

Assemble the £ matrixfrom eqn. (31)

Multiply the T and <l>matrices for eqn. (36)

Assemble eqn. (B8) and put inthe proper place in the h* matrix- J

Initialize the b* matrix_ -P _

| Do for each obstacle node ^ •.

VFind the triangle, its nodesand their coordinates for

each obstacle node

I - •• — ' ••'Calculate the area of each triangle~~Z I

Assemble eqn. (77) and putin the proper place in the

b* matrix-p

*Read the complete P* matrix

Fig. C6. Continued.

Do multiplications foreqns. (69) and (70)

49

No

Call the subroutinefor P*

Write and/or plot

»*• Integration Routine

Write and/or plot0)

SolutioConverged

9

( Stop )

Fig. C6. Continued.

50

(Enter J


(b** rj and (.B*1 3)

IInvert for

(b** rf1

IDo multiplications for


r (b (b

Subtract the resultfrom £ for the final

solution

^Return J

Fig. C7.fig. C6.

Flow chart for the subroutine used in the final program shown in

51

REFERENCES

1. Son, J.S. and Hanratty, T.J.: Numerical Solution for the Flow Around aCylinder at Reynolds Numbers of 40, 200, and 500. J. Fluid Mech., Vol.35, Feb. 1969, p. 369.

2. Sykes, D.M.: The Supersonic and Low-Speed Flows Past Circular Cylindersof Finite Length Supported at One End. J. Fluid Mech., Vol. 12, Mar.1962, p. 367.

3. Takami, H. and Keller, H.B.: Steady Two-Dimensional Viscous Flow of anIncompressible Fluid Past a Circular Cylinder. Phys. Fluids Suppl. II,1969, pp. 51-56.

4. Okajima, A.: Numerical Solution for the Analysis of Unsteady ViscousFlow Around an Oscillating Cylinder. Recent Advances in Matrix Methodsof Structural Analysis and Design, R.H. Gallagher, Y. Yamada, and J.T.Oden, Eds., The Univ. of Ala. Press, Ala., 1971, p. 809.

5. Achenbach, E.: Distribution of Local Pressure and Skin Friction Arounda Circular Cylinder in Cross-Flow Up to Re = SxlO6. J. Fluid Mech.,Vol. 34, Dec. 1968, p. 625.

6. Thoman, D.C. and Szewczyk,A.A.: Time-Dependent Viscous Flow Over aCircular Cylinder. Phys. Fluids Suppl. II, Vol. 12, No. 12, Dec.1969, p. 76.

7. Dennis, S.C.R. and Staniforth, A.N.: A Numerical Method for Calculatingthe Initial Flow Past a Cylinder in a Viscous Fluid. Proc. of the Conf.on Numerical Methods in Fluid Dyn., held at the Univ. of Calif., Berkel-ey, Sep. 1970, M. Holt, ed., p. 343.

8. Schlichting, H.: Boundary Layer Theory. McGraw-Hill Book Co. Inc.,New York, 1968.

9. Fromm, J.E.: A Method for Computing Nonsteady, Incompressible, ViscousFluid Flows. Los Alamos Scientific Laboratory Report LA-2910, Sep.1963.

10. Mikhlin, S.G.: Variational Methods in Mathematical Physics. The Mac-millan Co., New York, 1964.

11. Cheng, S.I. and Rimon, Y.: Numerical Solution of a Uniform Flow over aSphere at Intermediate Reynolds Numbers. Phys. Fluids, Vol. 12, No. 5,May 1969, p. 949.

12. Holand, I.: The Finite Element Method in Plane Stress Analysis. FiniteElement Methods in Stress Analysis, I. Holand and K. Bell, eds., TapirBook Co., Norway, 1969, p. 43.

52

13. Zienkiewicz, O.C. and Cheung, Y.K: The Finite Element Method in Struc-tural and Continuum Mechanics. McGraw-Hill Publishing Company, Limited,Great Britain, 1967.

14. Kawaguti, M.: Numerical Solution of'the Navier-Stokes Equations for theFlow Around a Circular Cylinder at Reynolds Number 40. J. Phys. Soc.Japan, Vol. 8, No. 6, 1953, p. 747.

53

t/1o

rtr—I

O1—IrtO

f-l

•Hi—IX

•HCJ

fHO

g

•P

I0)

0)+J•HC

(30• Htu

54

tflX

I—Ictf

!H•Hrt

C•HW)

c•H,C

o-p•Hft

.8

C<D

0)

+J•HC

•H

bO•HU,

55

y1 1

* V V

Fig. 3. A typical triangular grid element for the finite element analysisof the Navier-Stokes equations.

Fig. 4. Definition of the tangential and normal directions for any pointon the obstacle. Sines and cosines for point P- are defined in terms ofpoints PI and P_.

56

o

f-tO

.0004

.0002

0

-.0002

-.0004

-.0006

-.0008

-.0010

o<D

•HO•H4->f-tO

20 40 60 80

Time, sec.

100 120

Fig. 5a. Stability of the numerical integration of the vor-ticity transport equation for a point upstream of the ob-stacle (Re = 40).

.16 .

.12 -

.08 -

.04 .

0

-.04

At = 1.0

At = 1.5

0 20 40 60Time, sec.

80 100 120

Fig. 5b. Stability of the numerical integration of the vor-ticity transport equation for a point near the obstacle(Re = 40).

57

o

id

59

r -\ r ~\

r

61

aT3

62

r

•oX

00•Hu.

63

oCO

Ifld

aTJ

sIfl

64

I*ID73C

a•o

65

I

5 .2

cd

13

66

\

01

13

67

rt

•O

swc

•ax

68

2.a

1.5-

i.or

0.5-

-O.S

-l.Q

Finite Element Analysis

Ref. 6 (Finite Difference)

Ref. 14 (Finite Difference)

O

O t = 110.o

45 90 135 180

Angle from stagnation point,degrees

Fig. 14. Development of the surface pressure distributionon a circular cylinder (Re = 40).

70

0)oi

Ort

soort-Prt

•Ho

OO

u

Ofnrt

§0)

co

•HUH

71

r\V

0)ci

ort

o0>

1—IbOc

oort

•HO

•HnJ

CN^-tOo

irt

o43rt

o•H-p3

tn•H

•Ho•H

rt

W)•H

72

<D

onj•M•P

o0)

oO

•HO

CM

Oo

3O

(3O•H

CO•H

X•P•HO

•H•P?HO

•HPU

73

a)

rt•M

M-iO

0)

5oO

•Prt

•Hctf

CM

oo

-a

ofn

O•H

tn•HT3

03

<L>

0,

•Hu.

74

O

g

a-a

> 00

§

75

Irt

•O

t/J O</)O 0)f-l <-*

CX DO

§

76

o

O —i

77

" O «~

a. <4-<o

78

Fig. 22. Vorticity distribution about an oscillating NACA 0012 airfoil (Re = 103), (t = 0.01 -0.05), (a = 0° - 2.5°).

79

Fig. 23. Pressure distribution about :in oscillating NAG\ 0012 airfoil (lie = 103j , (t = 0.01 -0.05), (u = 0° - 2.5°).

80 NASA-Langley, 1974 CR-2368

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