rans simulations of axisymmetric bodies in turning
TRANSCRIPT
RANS SIMULATIONS OF AXISYMMETRIC BODIES IN TURNING
by
Jian Tao Zhang
B.Sc. in Engineering, University of New Brunswick, 2008 NSERC Scholar
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
Master of Science
In the Graduate Academic Unit of Mechanical Engineering
Supervisor(s): Dr. Andrew G. Gerber, PhD, Mechanical Engineering Dr. A. Gordon Holloway, PhD, Mechanical Engineering
Examining Board: Dr. Joseph W. Hall, PhD, Mechanical Engineering, Chair Dr. George D. Watt, PhD, DRDC-Atlantic
External Examiner: Dr. William Cook, PhD, Chemical Engineering
This thesis is accepted by the Dean of Graduate Studies
T H E UNIVERSITY OF N E W B R U N S W I C K
April 2010
©Jian Tao Zhang, 2010
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Canada
To my wife, Lu Lu, our child, Eric, and my family.
11
Abstract
A numerical approach has been applied to estimate the hydrodynamic forces and
moments exerted on DARPA SUBOFF submarine hull without appendages during
a turning manoeuvre. Calculations were performed based on solving the Reynolds
Averaged Navier Stokes (RANS) equations using the Computational Fluid Dynam
ics (CFD) code ANSYS-CFX. Two turbulent models - the Shear Stress Transport
model (SST) and the Baseline Reynolds Stress model (BSL-RSM) were applied. A
meshing script was developed to construct hybrid meshes rapidly for a range of turn
ing parameters.
Plots of wall shear stress lines and circumferential distribution of pressure on the
hull surface were presented. Overall force and moment coefficients were calculated
from the simulation results and compared to experimental data for Re = 6.5 x 106.
m
Acknowledgments
This work would not have been possible without the help and assistance of many
people. I would like to thank my co-supervisors, Dr. Andrew Gerber and Dr. Gor
don Holloway both of the University of New Brunswick, and Dr. George Watt of
Defence Research and Development Canada Atlantic, for their continual support and
contributions during this project. I would particularly like to thank Dr. Gerber,
Dr. Holloway and Dr. Watt for their effort in reviewing this document.
I would like to acknowledge my coworkers in the CFD Lab for the many technical
discussions which helped advance my project, and more importantly making my time
at UNB something I will always remember. Special thanks to Jordan Maxwell, who
is continuing this project for his Master degree, for his help and ideas.
My gratitude is extended to the Natural Sciences and Engineering Research Council
of Canada and Defence Research and Development Canada Atlantic for their finan
cial support.
I would like to thank my family, especially my wife, Lu Lu, for your continual
support. You make me stronger every step, and this mission would not have been
possible without your support.
iv
Table of Contents
Dedication ii
Abstract iii
Acknowledgments iv
Table of Contents vii
List of Tables viii
List of Figures xiii
Nomenclature xiv
1 Introduction 1
1.1 Background 1
1.1.1 Motivation 1
1.1.2 Submarine in a Steady Turn 2
1.2 Proposed Research 4
1.2.1 Research Program Objectives 4
1.2.2 Short Term Objectives 5
2 Literature Review 6
2.1 Previous CFD Studies for Axisymmetric Bodies in Translation . . . . 6
v
2.2 Experiments for Axisymmetric Bodies in Turning 7
2.3 Related CFD Studies for Axisymmetric Bodies in Turning 10
3 Theory 12
3.1 Governing Equations of Fluid Flow in CFD 12
3.2 Turbulence Models in CFD 13
3.2.1 The SST Turbulence Model 15
3.2.2 The BSL-RSM Turbulence Model 16
3.3 Rotating Frame of Reference 17
3.4 Solution Methodology 18
4 Verification and Validation against Experimental Data on a Turning
Submarine 19
4.1 CFD Model for a turning Axisymmetric Body 19
4.1.1 Flow Conditions and Submarine Geometry 19
4.1.2 RANS Simulation 21
4.1.3 Fluid Boundary Conditions 21
4.2 Unstructured Mesh 22
4.3 Structured Mesh Topology (Final) 24
4.3.1 Structured Near Body 26
4.3.2 Structured Far Field Mesh 29
4.4 RANS Calculation 33
4.4.1 Verification of RANS Calculations 33
4.4.1.1 Preliminary Unstructured Mesh 33
4.4.1.2 Structured Mesh 35
4.4.2 Validation of RANS Calculations 38
4.4.3 Other Validation of Results 41
vi
5 Results for an Axisymmetric Body (SUBOFF) in Turning 45
5.1 Pivot point location for SUBOFF 45
5.2 Mesh Topology for SUBOFF in Tight Turning 48
5.3 RANS Calculation using the Mesh Template 52
5.3.1 Fluid Boundary Conditions 52
5.3.2 Wake Diffusion in Computational Domain in Tight Turning . 52
5.3.3 Convergence of RANS Calculation 53
5.3.4 Observations on Results 54
6 Conclusion 63
6.1 Conclusion 63
6.2 Future Work 64
References 65
A Private Communication with D R D C Atlantic 68
B Script File (script_suboff_with_Sting.gif) to Generate Meshes for
SUBOFF-with-Sting 70
C Script File (script_suboff_no_Sting_tight_turn.gif) to Generate Meshes
for SUBOFF-no-Sting in Tight Turn 71
D Script File (script_suboff_no_Sting_large_turn.gif) to Generate Meshes
for SUBOFF-no-Sting in Large Turn 72
E Script File (script_PROCEDURES.gif) to define the subroutines
used in other scripts 73
Vita 74
vii
List of Tables
2.1 Experimental force and moment coefficients on the truncated SUB-
OFF hull at Re = 6.5 x 106 in steady turning (BH=bare hull, FA=fully
appended, U=uncertainty.) 9
4.1 BSL-RSM RANS predictions of force and moment coefficients on the
truncated SUBOFF hull at R = 12.485 m, r = 0.1237 rad/s and
Re = 6.5 x 106 in steady turning 40
5.1 Nondimensional coefficients used in submarine equations of motion
for SUBOFF (Config. 2 Fully Appended) 47
5.2 RANS predictions of hydrodynamic force and moment coefficients for
the SUBOFF hull at r = 0.1237 rad/s and Re = 6.5 x 106 using the
BSL-RSM turbulence model and the 3.2 million node meshes 56
vin
List of Figures
1 The standard hydrodynamic notation on an axisymmetric body. . . . xiv
1.1 Orientation of a submarine and its rudder in a steady turn (top view
of the plane of rotation) 4
4.1 Profile of SUBOFF hull to scale 20
4.2 SUBOFF unstructured mesh with 1.3 million nodes at (3 = 10.2°,
R = 12.48 m, r = 0.1237 rad/s and Re = 6.5 x 106. (a)Far field
boundaries of computational domain; (b)Symmetry plane; (c)Mesh
around hull on symmetry plane; (d)Transverse plane at X/£ — 0.2. . . 23
4.3 Inflation layer of mesh around hull surface (SUBOFF unstructured
mesh with 1.3 million nodes at (5 = 10.2°, R = 12.48 m, r = 0.1237
rad/s and Re = 6.5 x 106) 24
4.4 SUBOFF-with-sting mesh with 3.6 million nodes at (5 — 10.2° and
R = 12.48 m using the mesh script. (a)Inlet plane; (b)Symmetry
plane; (c)Outlet plane; (d)Opening planes 27
4.5 Surface meshes on the SUBOFF hull and sting with 3.6 million nodes
using the mesh script. (a)Overall view; (b)Detailed views near the
nose and tail of the hull 28
4.6 Highlighted transverse planes showing the radial node distribution of
SUBOFF-with-sting mesh with 3.6 million nodes using the mesh script. 29
ix
4.7 The near body structured mesh on the SUBOFF-with-sting symmetry
plane with 3.6 million nodes using the mesh script. (a)Structured
mesh near body and wake region; (b)Structured mesh near stagnation
region; (c)Structured mesh around two backwards facing steps due to
presence of the sting 30
4.8 The SUBOFF-with-sting mesh on the symmetry plane at different
drift angles with 3.6 million nodes at R/£ = 2.9 using the mesh script.
(a)At 0° drift angle; (b)At 10° drift angle; (c)At 20° drift angle; (d)At
30° drift angle 32
4.9 The SUBOFF-with-sting mesh on the symmetry plane with 3.6 mil
lion nodes at j3 = 10.2° and different turning radius using the mesh
script. (a)At R/£ = 3.3; (b)At R/£ = 5.0; (c)At R/£ = 10.0; (d)At
R/£ = 1000.0 33
4.10 RANS predictions of lateral force, Y, at different mesh resolution
using two turbulence model for SUBOFF (hull) turning at [5 = 10.2°,
R = 12.48 m, r = 0.1237 rad/s and Re = 6.5 x 106 with SST and
BSL-RSM turbulence model using unstructured meshes 34
4.11 Convergence history showing the maximum normalized Residuals for
SUBOFF-with-sting (hull) turning at (3 = 10.2°, R = 12.48 m, r =
0.1237 rad/s and Re = 6.5 x 106 with BSL-RSM turbulence model us
ing 3 structured mesh resolutions. (a)3.6 million nodes; (b)6.3 million
nodes; (c)9.9 million nodes 36
4.12 RANS predictions of forces and moments for SUBOFF-with-sting
(hull) turning at f3 = 10.2°, R = 12.48 m, r = 0.1237 rad/s and
Re — 6.5 x 106 using BSL-RSM turbulence model and the structured
meshes with 3.6, 6.3 and 9.9 million nodes. (a)Axial and lateral forces
on SUBOFF hull; (b)Yawing moment on SUBOFF hull 37
x
4.13 Prediction history for SUBOFF-with-sting (hull) turning at /3 = 10.2°,
R = 12.48 m, r = 0.1237 rad/s and Re = 6.5 x 106 using BSL-RSM
turbulence model and the structured meshes with 3.6, 6.3 and 9.9
million nodes, (a)Tail plane circulation; (b)Lateral force on SUBOFF
hull 37
4.14 Mirrored SUBOFF-with-sting mesh with 7.2 million nodes at j3 =
10.2° and R = 12.48 m, highlighted with inlet and outlet plane. . . . 38
4.15 BSL-RSM RANS predictions of shear stress lines on hull surface for
SUBOFF (hull) turning at /3 = 10.2°, R = 12.48 m, r = 0.1237 rad/s
and Re = 6.5 x 106 using the structured mesh. (a)Windward side,
(b)Leeward side 39
4.16 BSL-RSM RANS predictions of total pressure on transverse planes
at several locations, A/£=0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0
and 1.2, for SUBOFF (hull) turning at Q = 10.2°, R = 12.48 m,
r = 0.1237 rad/s and Re = 6.5 x 106 using the structured mesh. . . . 39
4.17 Axial force coefficient for both experiment and BSL-RSM RANS pre
dictions for the SUBOFF-with-sting (hull) turning at R = 12.48 m,
r = 0.1237 rad/s and Re = 6.5 x 106 41
4.18 Lateral force coefficient for both experiment and BSL-RSM RANS
predictions for the SUBOFF-with-sting (hull) turning at R = 12.48
m, r = 0.1237 rad/s and Re = 6.5 x 106 42
4.19 Yawing moment coefficient for both experiment and BSL-RSM RANS
predictions for the SUBOFF-with-sting (hull) turning at R = 12.48
m, r = 0.1237 rad/s and Re = 6.5 x 106 42
4.20 Comparison of BSL-RSM RANS predictions of Cp to experimental
measurement on SUBOFF-with-sting hull at 0 = 10.2°, R = 12.48 m,
r = 0.1237 rad/s and Re = 6.5 x 106 at X/£ = 0.63 44
XI
4.21 Comparison of BSL-RSM RANS predictions of Cp to experimental
measurement on SUBOFF-with-sting hull at (5 = 10.2°, R = 12.48 m,
r = 0.1237 rad/s and i2e = 6.5 x 106 at X/£ = 0.84 44
5.1 The turning variables for SUBOFF in a steady turn 49
5.2 The computational domain for SUBOFF in a tight turn at (5 = 21.6°
and R/£ = 1 50
5.3 The symmetry plane of SUBOFF mesh in a tight turn with 1.7 million
nodes at /? = 21.6° and R/l = 1 using the tight turn mesh script. . . 50
5.4 Structured mesh of SUBOFF in a tight turn with 1.7 million nodes
using the mesh script. (a)Surface mesh near the tail of SUBOFF;
(b)Structured mesh around tail on symmetry plane; (c)Structured
mesh near body and wake region on symmetry plan 51
5.5 Polylines originating at the CR on the symmetry plane for SUBOFF
in a tight turn at /3 = 21.6° and R/t, = 1 53
5.6 Static pressure distribution at polylines passing the CR on the symme
try plane for SUBOFF in a tight turn at (3 = 21.6°, r = 0.1237 rad/s,
R/t = 1 and Re = 6.5 x 106 using the BSL-RSM turbulence model
and the 3.2 million node mesh. (a)Upstream of SUBOFF; (b)Wake
of SUBOFF; c)Close view for wake of SUBOFF 57
5.7 Velocity (in the Rotating Domain) distribution at polylines passing
the CR on the symmetry plane for SUBOFF in a tight turn at (5 =
21.6°, r = 0.1237 rad/s, R/l = 1 and Re = 6.5 x 106 using the BSL-
RSM turbulence model and the 3.2 million node mesh. (a)Overall
view; (b)Close view 58
xn
5.8 History showing values of Normalized Residuals and force/moment
predictions for SUBOFF (hull) turning at Q = 21.6°, R/l = 1, r =
0.1237 rad/s and Re = 6.5 x 106 using the BSL-RSM turbulence
model and the 3.2 million node mesh. (a)Maximum value of Residual;
(b)RMS value of Residual; (c)Force X,Y and Moment N prediction. . 59
5.9 Isosurface of Normalized U Momentum Residual with value of lx lO - 4
for SUBOFF (hull) turning at /? = 21.6°, R/l = 1, r = 0.1237 rad/s
and Re = 6.5 x 106 using the BSL-RSM turbulence model and the
3.2 million node mesh. a)Overall view; b)Detailed view near the tail
of the hull 60
5.10 Comparison of BSL-RSM RANS predictions of Cp on SUBOFF hull
at r = 0.353 rad/s and Re = 6.5xl06 at X/l = 0.1,0.3,0.5,0.7 and
0.9. (a)/? = 21.6°, R/l = 1; (b)/? = 7.1°, R/l = 3; (c)/? = 8.0°,
R/l = oo 61
5.11 Contours of x vorticity and wall shear lines for SUBOFF from the
RANS calculation using the BSL-RSM turbulence model and 3.2 mil
lion node meshes, (a)turning with (3 = 21.6° and R/l = 1, r = 0.353
rad/s and Re = 6.5 x 106; (b)turning with j3 = 7.1° and R/l = 3,
r = 0.353 rad/s and Re = 6.5 x 106; (c)translating with (3 — 8° and
Re — 6.5 x 106 (obtained with the Chapter 4 template but without
sting) 62
xm
List of Symbols, Nomenclature
0" f x,u,X
la
,NCJ) Q,M
z,w, Z
y,v,Y
Figure 1: The standard hydrodynamic notation on an axisymmetric body.
CB Center of buoyancy
CR Center of rotation
C„ lnTT2\ Static pressure coefficient: Cp = (Ps — P^/^pU
Maximum hull diameter, m
Hull length, m
m Mass of submarine, kilogram
K, M, N Rolling, pitching and yawing moment, Nm
K', M', N' Rolling, pitching and yawing moment coefficients normalized by ^pU2£3
xiv
A^ Normalized coefficient used in representing A" as a function of ur
N'v Normalized coefficient used in representing A" as a function of uv
N'5r Normalized coefficient used in representing A" as a function of u25r
Ps Static pressure, Pa
p, q, r Turning rates in body axes, s _ 1
r' Normalized turning rate: r' = r£/U
R Turning radius, m
Re Reynolds number: Re = pUl/[i
u, v, w Velocity in body axes, m/s
v{x) Velocity v along axial axis: v(x) = v + rx, m/s
U Overall velocity at the body axes origin: U = \/u2 + v2 + w2, m/s
V Submarine volume, m3
x,y,z Body-fixed axes
x'P Pivot point location of submarine normalized by £
X, F, Z Axial, lateral and normal force, A
X', Y', Z' Axial, lateral and normal force coefficients normalized by ^pU2£2
Y/, Normalized coefficient used in representing Y as a function of ur
Y/} Normalized coefficient used in representing Y as a function of uv
xv
Yg Normalized coefficient used in representing Y as a function of u25r
a Angle of incidence, degree
P Angle of drift at the reference piont, degree
fl(x) Local angle of drift: fl(x) = ta,n~1(—v(x)/u), degree
5 Identity matrix
Sr Deflection of rudder, degree
X Longitudinal distance from nose towards tail, m
fi Dynamic viscosity of sea water, O.OOlOBA^s/m2
p Density of sea water, 1030fc(?/m3
xvi
Chapter 1
Introduction
1.1 Background
All streamlined underwater vehicles experience significant hydrodynamic forces and
moments while manoeuvring. These hydrodynamically induced forces and moments
affect performance, stability, and recoverability in emergency situations. Some of the
forces and moments are not well understood, especially during extreme six degree-
of-freedom (DOF) manoeuvres, and are difficult to predict using estimation tools
currently available. Improved hydrodynamic force predictions in extreme cases will
improve design, safe operation, and control of underwater vehicles. This is especially
important for submarines which harbour human life.
1.1.1 Motivation
Computer manoeuvering simulations, using mathematical models of the hydrody
namic forces, are used for assessing performance and establishing safe operating
procedures. Force estimation models allow for fast simulations and therefore a
great many simulations can be carried out to empirically establish a safe operating
envelope. The hydrodynamic forces on underwater vehicles are estimated by build-
1
ing them up on one vehicle component at a time, starting with the streamlined hull
force, adding on appendage and propulsion forces, and then estimating interference
between them. Classical force estimation methods are fairly good at appendage
forces predictions, because they have sharp trailing edges which fix flow separation,
but are inaccurate for predicting hull forces which are strongly dependent on un
known flow separation line locations along the smooth hull. There is a great need
for improved hydrodynamic force estimation on the streamlined axisymmetric hull.
CFD (Computational Fluid Dynamics) has been proven in many cases to be a more
efficient way to obtain detailed flow field information over the entire domain than
obtaining it experimentally. However, the validation of the CFD calculation with
experimental data is important to ensure confidence in the predictions. Translation
and rotation each accounts for half of the six DOF of a manoeuvering underwater
vehicle. A previous study [1] of an axisymmetric hull in translation has shown that
CFD can predict the lateral forces and moments of a submarine shaped axisymmetric
hull body in a translating motion within experimental uncertainty up to 25 ° angle
of drift. With sufficient validation against experimental data, CFD can be used
to establish a database of flow field information to assist in developing quick and
relatively accurate mathematical force estimation methods to meet the requirement
for the design and control of submarines. The objective of the present work is to
extend this to the case of an axisymmetric hull in turning. This will be developed
further in Section 1.2.1.
1.1.2 Submarine in a Steady Turn
When a submarine carries out a turning manoeuvre clockwise, the rudder is de
flected to the starboard side (negative 5r) to produce a lateral force at the stern.
The yawing moment induced by this force causes the submarine to rotate in a hori-
2
zontal plane, so that the bow of the submarine points into the starboard turn, and
a positive drift angle, (3 (measured at the center of buoyancy), is established. This
situation is depicted in Figure 1.1. Increasing the rudder deflection will cause a
larger drift angle and a tighter turn. With the rudder in a fixed position, the hy-
drodynamic forces and moments on the submarine are eventually balanced with its
inertial forces and moments and any propulsion forces, and the resultant force passes
through the center of rotation. Under these conditions the submarine establishes a
steady turn.
When a submarine is in a turning maneuver, there is a distribution of drift angle
developed along the length of the hull, defined as fJ(x) = ta.n~1(—v(x)/u), where
v(x) = v 4- rx. A special point on the hull centerline, called the pivot point (xp) is
located at v(x) = 0. It is the point where a line drawn from CR intersects the body
x axis perpendicularly, usually near the front of the hull, as shown in Figure 1.1.
The rudder induced sideslip causes the local drift angle to have a different sign on
either side of the pivot point. The magnitude of local drift angle is relatively small
along the front of the hull, but is large near the stern plane [2]. This variation of
the local drift angle reduces the yawing moment.
The present work is mostly concerned with the hydrodynamic forces on an axisym-
metric hull during a steady turn, any transient result is beyond the scope of this
work.
3
CR
R
x u Pivot Point
I ku \
O ' U
Figure 1.1: Orientation of a submarine and its rudder in a steady turn (top view of the plane of rotation).
1.2 Proposed Research
1.2.1 Research Program Objectives
The primary objective of the proposed research is to improve the use of CFD to create
an enhanced hydrodynamic database of flow field information acting on streamlined
axisymmetric bodies of revolution in a steady turn. The database will consist of
results from CFD simulations with a variety of axisymmetric body profiles, and a
range of drift angles and turning radii. The database will be validated with experi
mental data where possible.
The CFD generated database will provide detailed hydrodynamic information over
the entire fluid domain in the vicinity of the body at a level of detail not readily
available from experiments. The validation with experimental data will ensure
^-Reference PoiTtt ^
Rudder C7~~
4
confidence in the use of the CFD calculations. This enhanced database will be used
in a follow-on study to develop a semi-empirical mathematical model, for quick and
accurate force estimation, for turning axisymmetric bodies.
1.2.2 Short Term Objectives
From the experience of a previous study for an axisymmetric hull in translation [1],
the current study for the turning case should also take the following step to meet
the primary objectives,
1. create an improved CFD model which is capable of simulating an axisymmetric
body in a steady turn with constant turning radius and a fixed drift angle at
the reference point.
2. develop a mesh topology which provides adequate discretization of the compu
tational domain, and applying proper boundary condition assumptions to the
CFD model.
3. extend the CFD model to a wide range of turning radii and drift angle.
A mesh generation script to accommodate different parameters, such as the turning
radius, the drift angle, and the mesh resolutions, will be developed to facilitate the
discretization of the computational domain in an intelligent, fast manner and also
with high quality.
The CFD simulations will be first verified to minimize the numerical errors associated
with the discretization of the computational domain, and then it will be validated
against experimental data.
5
Chapter 2
Literature Review
2.1 Previous CFD Studies for Axisymmetric Bod
ies in Translation
Various CFD studies simulating axisymmetric bodies in translation using RANS
calculations have been published. By using a structured quasi-axisymmetric mesh,
in 1999 Watt et. al. [3] were able to predict the lateral force and pitching moment
on the DRDC-STR hull well within the experimental uncertainty at model scale
Reynolds number of 20 million and drift angles of 0°, 10° and 20°, but they under
predicted the lateral force at 30° by about 10%.
Sung et. al. [4] predicted the lateral forces on the Series-58 Model 4621 hull at
Reynolds numbers up to 11.7 million and drift angles up to 18° in 2004. The results
agreed well with experimental data at low drift angle smaller than 15°, but also
tended to under predict the lateral force at high drift angles.
A comprehensive CFD study was performed on several axisymmetric shapes at drift
angles up to 30° at the University of New Brunswick, in collaboration with Defence
6
Research and Development Canada (DRDC) Atlantic in 2006. Baker et. al. [5]
developed a meshing script so that the adequate discretization of the entire fluid
domain, especially inside the lee side vortex and near wall shear layers at moderate
to high drift angle, was generated for those CFD simulations. A comprehensive CFD
database for translation was completed for a wide range of drift angles, Reynolds
numbers and axisymmetric body shapes using the meshes generated by the script.
The results were very encouraging. The overall force and moment predictions were
within the experimental uncertainty for drift angles up to 25°. Following this, Jeans
et. al. [1] developed mathematical expressions for predicting both the lateral force
and yawing moment and their distributions using the CFD database. In addition,
the performance of two turbulence models, the k — u> based SST model and the BSL-
RSM model, were examined in the study. It found that the BSL-RSM turbulence
model reduced the error in overall force predictions by 50% compared to the SST
model.
2.2 Experiments for Axisymmetric Bodies in Turn
ing
Captive-model experiments on scale models can provide hydrodynamic data which is
suitable for the validation of CFD simulations. At the present time, the only avail
able experimental work for a submarine in a turning maneuver is from the SUBOFF
project. It was conducted by Etebari et al [6] in 2007 at the Rotating Arm Basin
at Naval Surface Warfare Center Carderock Division (NSWCCD). The SUBOFF
model is designed and constructed in a manner that can be used to support both
experimental and computational studies. All tests used a truncated 4.0132 m long
(metric length) and 0.508 m diameter scale model with a sting support 0.14 m in
diameter attached to the tail. It should be noted that the original length of the
7
SUBOFF hull is I = 4.356 m which means approximately 8% of the hull length was
truncated to accomodate the sting.
All tests were run at 3 knots (1.543 m/s) with a turning radius of 12.48 m and a
Reynolds number of 6.5 million. Hydrodynamic forces and moments in three direc
tions were measured for bare hull and fully appended cases for drift angles up to 18°,
using a forward and aft set of three block gages referenced to the center of reference
of the model. The reference point was at 2.013 m aft of the nose along the model
centerline [9]. The drift angle here, f3, refers to the angle between the hull axis and
the tangent to the turning circle at the reference point, as shown in Figure 1.1. The
force and moment coefficients at each drift angle in degrees are listed in Table 2.1 for
two configurations: bare hull (BH) and fully appended (FA). The total uncertainties
of these coefficients in the experiments, which were the summation of all identified
errors, are also given in Table 2.1. There were a total of six drift angles tested for
the bare hull configuration and one drift angle for the fully appended configuration.
The axial and lateral force and the yawing moment vary with drift angle, but the
other hydrodynamic force (normal) and moments (pitching and rolling) do not vary
significantly in the tests, as shown in Table 2.1. Also it should be noted that the
value of the normal force is much smaller than the other two forces, same for the
pitching and rolling moments is much smaller than the yawing moment.
In the SUBOFF tests the 3 - D velocity field was measured in a cross plane at two
axial locations using the Stereo Particle Image Velocimetry (SPIV) system, so that
normalized vorticity contours and surface streamlines on these cross planes were able
to be visualized. The regions of boundary layer attachment and separation were
indicated from the streamline plots. In addition 23 static pressure measurements
8
were made on the leeward side of the hull surface at cross planes having X/£ =
0.63 & 0.84. The published data of the measured pressures were corrected for
depth.
Table 2.1: Experimental force and moment coefficients on the truncated SUBOFF hull at Re = 6.5 x 106 in steady turning (BH=bare hull, FA=fully appended, U=uncertainty.).
BH
FA
BH
FA
fH°)
3.8
6.2
10.2
12.2
14.3
16.5
8.2
P(°)
3.8
6.2
10.2
12.2
14.3
16.5
8.2
X'
-1.52E-03
-1.69E-03
-2.07E-03
-2.21E-03
-2.29E-03
-2.26E-03
-3.18E-03
K'
1.92E-06
7.85E-07
6.99E-07
-6.54E-07
2.57E-07
1.42E-06
-4.70E-05
Ux>
8.43E-05
6.54E-05
1.03E-04
1.08E-04
8.72E-05
1.82E-04
9.96E-05
UK>
7.53E-06
7.36E-06
7.66E-06
7.59E-06
7.50E-06
9.41E-06
9.69E-06
Y'
1.30E-03
1.93E-03
3.71E-03
4.83E-03
6.15E-03
7.32E-03
5.04E-03
M'
-4.52E-05
-5.17E-05
-5.95E-05
-6.51E-05
-6.51E-05
-4.45E-05
6.21E-05
UY>
1.76E-04
1.41E-04
1.81E-04
1.92E-04
1.57E-04
3.58E-04
1.46E-04
uM,
6.07E-05
4.39E-05
6.44E-05
6.76E-05
4.70E-05
1.16E-04
3.96E-05
Z'
-1.46E-04
-1.33E-04
-1.42E-04
-1.00E-04
-1.38E-04
-2.63E-05
3.57E-04
N'
4.67E-04
7.73E-04
1.25E-03
1.43E-03
1.58E-03
1.72E-03
7.58E-04
uz.
2.90E-04
2.10E-04
3.08E-04
3.23E-04
2.25E-04
5.51E-04
1.90E-04
uN>
3.54E-05
2.88E-05
3.80E-05
3.96E-05
2.91E-05
7.12E-05
3.11E-05
9
2.3 Related CFD Studies for Axisymmetric Bod
ies in Turning
A series of CFD simulations was performed by Sung et. al. [7] in 1996 on the model
undergoing a constant radius turning. Sung et. al. predicted the forces and mo
ments acting on axisymmetric bodies at various angles of attack and drift in steady
turns, using structured meshes at several mesh resolutions and an in-house CFD code
named IFLOW. A number of numerical approaches have been implemented in his
simulations, including multiblock, multigrid and local refinement methods. Multi-
block structures made the grid generation process easier for complex geometries,
for example, the control surfaces of submarines. The simulations were performed
using both the standard k — LU and the modified Baldwin-Lomax turbulence model.
Modifications in the Baldwin-Lomax turbulence model were made to overcome the
difficulties in formulating the eddy viscosity in the outer boundary layer, arising
from the boundary layer thickening. The reported convergence in the solution was
in the order of 10~4 (RMS residual), and the simulation results were found to be
mesh independent since the change of calculated flow variables were less than 1%
after doubling the mesh resolution.
The prediction of force and moment coefficients from their turning simulations were
compared with the unpublished experimental data obtained by Bedel, et. al. in the
DTMB rotating arm facility, and the results were within the experimental uncer
tainty of 20%. Sung et. al. found that at a constant turning radius and zero pitch
angle, the magnitude of the axial force increased with increasing drift angle, as well
as the lateral force and yawing moment. The effect of turning radius was also ex
amined. It was found that at zero angle of incidence and drift, the magnitude of
the lateral force and yawing moment decreased with increasing turning radius.
10
In 2006, an experimental and computational study modeling axisymmetric bodies
in turning with curved hulls in rectilinear flow was conducted by Gregory et. al. [8]
using the method developed by Gurzhienko in 1934. The body axis was deformed
in a manner that preserved the variation of the local drift angle which is a function
of axial position along the straight body in turning. A velocity gradient with the
higher velocity on the concave side of the body was added to the approaching flow
to account for the distortion of the free stream in this transformation.
Both the curved hull and straight hull simulation were performed using a commercial
RANS code ANSYS-FLUENT. The hull was simplified to ease the construction for
the curved hull in both his numerical and experimental work. A mesh dependent
study was performed for the CFD simulation results. To provide the validation
for the CFD results using the curved body, a wind tunnel experimental program
was conducted by Gregory et. al. subsequently. Extensive measurements were per
formed such as the surface static pressure, skin friction on the curved hull, and
surface streamlines etc.
The results suggested that the curved body simulation can accurately produce sur
face flow features and flow separation behavior. However, the force and moment
predictions were considered inaccurate with an error as large as 20% using the curved
body as compared to the straight body in turning.
11
Chapter 3
Theory
3.1 Governing Equations of Fluid Flow in CFD
The governing equations of fluid flow in CFD are mathematical conservation state
ments of mass and momentum, the Navier-Stokcs equations, which describe fluid
substances in both laminar and turbulent flows. However, to avoid the Direct
Numerical Simulation (DNS) of turbulent flows, which for practical applications,
requires computing power far beyond what is available today, turbulent models have
been developed to account for the effects of turbulence. The majority of these tur
bulence models must be used with appropriately averaged Navier-Stokes equations.
The present CFD simulations are based on the Reynolds Averaged Navier-Stokes
(RANS) equations which can be used to predict time averaged fluid flow conditions
and incorporate the fluid state of turbulence into the predictions.
For an incompressible fluid and steady state calculation, the mass conservation can
be written in vector form as [10]:
div(pU) = 0 (3.1)
12
where U is the mean flow velocity vector. The Reynolds averaged momentum
equation [10] in vector form is:
dpU div(pU®U) = div(—p5)+div <p grad U + (grad U)T \—div(pu® U)+SM
(3.2)
where 5 is the identity matrix and u is the fluctuating velocity component. The
quantity pu <g> u is known as the Reynolds stress tensor, and this additional term
represents a time averaged momentum transfer due to turbulent velocity fluctua
tions. The source term SM accounts for contributions where Eq. 3.2 is applied in a
rotating frame of reference.
The equation sets 3.1 and 3.2 are called the RANS equations. These partial dif
ferential equations have no general analytical solutions due to their complexity, but
can be solved numerically when properly discretized onto a mesh. The most com
mon discretization approach used in commercial CFD codes is the Finite Volume
Method (FVM). This method performs volume integrations of each term in the
Navier-Stokes equations over a finite volume defined by the mesh. Details can be
found in many textbooks on the subject [10].
3.2 Turbulence Models in CFD
Turbulent fluctuations are difficult to predict directly in CFD because the turbu
lent length scales are generally much smaller than the practical finite volume mesh
resolution. In turbulent flow, an instantaneous velocity component can be divided
into a mean component, U, and a fluctuating component, u. Substitution of the
mean and fluctuation values into the instantaneous form of the Navier-Stokes equa
tions, and applying averaging rules to the resulting terms, leads to Equations 3.1
13
and 3.2. The unknown Reynolds stresses term in Eq. 3.2 is the key addition to
the Navier-Stokes equations following time averaging and needs to be predicted by
solving additional sets of equations. The process of modeling the Reynolds stresses
is called "turbulence modeling". Two common classes of turbulence models used
in todays commercial CFD codes for the computation of the Reynolds stresses are
eddy-viscosity models and Reynolds stress models. The eddy viscosity models as
sume that the Reynolds stresses are proportional to mean flow velocity gradients
and generally assume the turbulence is locally isotropic.
Two-equation turbulence models, such as the standard k — UJ and k — e models used
in this work, use the eddy viscosity hypothesis to relate the Reynolds stresses to the
mean flow velocity gradients as follows [11]:
—pu§§u——-pk5 — --iJ,tdiv(U5)+iit grad U + (grad U)T (3.3)
where k is the turbulent kinetic energy, and fit is the eddy viscosity. In conjunc
tion with two transport equations for turbulence quantities, k and w o r e , the eddy
viscosity /xt can be computed and applied to all transport equations to incorporate
the influence of turbulence. The eddy viscosity in terms of k and e is p,t = c^p—.
Note that in the two equations approach the \it is added to the laminar diffusion
coefficients.
Unlike the eddy viscosity models, the Reynolds stress models require a separate set
of equations for the transport of Reynolds stresses in the fluid, so that the individ
ual stress components of the Reynolds stress tensor are resolved to obtain a better
estimate of k and hence fxt (where needed) and for direct substitution into the mo
mentum equations.
14
Two turbulence models were implemented in CFD simulations for the present re
search: the k — UJ based Shear Stress Transport (SST) model and the Baseline
Reynolds Stress (BSL-RSM) model.
3.2.1 The SST Turbulence Model
The SST model was designed to predict the onset and the amount of flow separation
under adverse pressure gradients with high accuracy. It is a blend between two
turbulence models, namely: the standard k — u and k — e models.
The SST model uses blending functions sensitive to the proximity of the flow to a
bounding wall to switch between the standard k — u and k — e models, dividing
the flow into the near surface and outer region respectively. The near surface
region employs the standard UJ equation and the outer region employs the e equation.
The e equation is reformulated in terms of ui for developing the final form. The
final w-equation is the same equation as in the Baseline k — ui model developed by
Menter [11]:
d(pu) LU + div(pUto) = a^Y^k — fhpu + div
dt yr ' °k /J, H I (grad u>)
C W 3 /
+(1 - Fi)2p (grad k) [grad u) (3.4) a2co
where a, (3 and a are all constants. The fc-equation is:
^ ' + div(pUk) = div ji -\ ) (grad k) ak3j
+ Pk- (3'pku (3.5) dt
where P& is the production rate of turbulence. In the SST model, the eddy viscosity
is formulated by introducing a limiter:
15
\H = -, ?TFT 3 - 6
max(aiui, 0^2)
where UJ is the turbulent frequency, a\ is a constant, and S is an invariant measure
of the strain rate. F^ is a blending function, which is a function of the wall distance
and restricts the limiter to the wall boundary layer, as defined in Equation 3.7.
F2 = tanh(argl) (3.7)
with:
(2Vk 500*A arg2 = max\ ——, —5— (3.8)
3.2.2 The BSL-RSM Turbulence Model
Another turbulence model utilized in the current CFD simulations is the Baseline
Reynolds Stress (BSL-RSM) model. When the flow is complex and specifically
anisotropic, the Reynolds stresses are poorly represented by Equation 3.3. The
exact transport of the Reynolds stress requires an individual equation for each com
ponent [11]:
V div(pU <8>u<g>u) = —pP + -0pukS -pU + div (A* + 77) grad u®u
(3.9)
where 5* is the turbulent Prandtl number, II is the constitutive relation for the
pressure-strain correlation, and P is the exact production term of Reynolds stress
given by:
P = u ® u(grad U)1 + (grad U)u <8> u (3.10)
16
The turbulent frequency, to, is also required to be solved as given in Eq. 3.4. In
resolving conditions close to walls similar techniques as used in the SST model
are employed. The Reynolds stresses as calculated with Eq. 3.9 are used also, in
conjunction with Eq. 3.4, to calculate the eddy viscosity:
IH = P- (3.11)
This eddy viscosity is employed in equations other than momentum for robustness,
but employs an isotropic assumption as a consequence. It should be noted that the
computational cost associated with the BSL-RSM model is significantly increased
compared with the SST model. This is mainly caused by solving 7 transport equa
tions in the BSL-RSM model instead of 2.
3.3 Rotating Frame of Reference
Rotating frames of reference have been implemented extensively in CFD to handle
applications where rotating boundaries are involved, for example in turbomachines.
This approach can also be used to simulate the flow over an axisymmetric body
in turning. In such cases additional terms need to be added to the momentum
equations. Additional forces are presented in a rotating frame of reference: the
Coriolis force and the centrifugal force. Two forces result in additional momentum
terms [12], Scor = —2pu5 x U and Scfg = —p£> x (w x r ) respectively, added directly
to the right hand side of Eq. 3.9, where r is the location vector.
The velocity in the rotating frame of reference is [12]:
Ur = Ua - Q x R (3.12)
17
where u; is the angular velocity, R is the local radius vector, and Ua is velocity
vector in the stationary frame. Similarly, the total pressure in the rotating frame
for incompressible flow becomes:
Ptot = Pstatic + 2 ^ \pr ' Urj (3.13)
Finally it should be noted that RSM equations have additional corrections employed
for the rotating frame case. This is outlined in reference [13].
3.4 Solution Methodology
The RANS simulations were run using a commercial CFD code ANSYS-CFX using
the finite volume/finite element method. The governing equations of fluid flow are
solved (p, u, v, w) by an implicit coupled solver. The additive correction algebraic
multigrid procedure was used to accelerate the solution. The default high resolution
discretization scheme was first implemented to solve the conservation equations in
the solver for the preliminary work. Subsequently a fully second order discretization
scheme was implemented to minimize numerical diffusion.
18
Chapter 4
Verification and Validation against
Experimental Data on a Turning
Submarine
4.1 CFD Model for a turning Axisymmetric Body
This section describes results obtained utilizing the commercial CFD code ANSYS-
CFX to predict the hydrodynamic forces and moments acting on the submarine
model DARPA (Defense Advanced Research Projects Agency) SUBOFF (bare hull)
in a steady turn. A preliminary study (using unstructured mesh) was performed
and provided insight to the flow field, which was used to improve subsequent CFD
models. Later a mesh template (structured mesh) was developed to provide the
adequate discretization of flow domain in the CFD model.
4.1.1 Flow Conditions and Submarine Geometry
The initial investigation started with a CFD simulation of the SUBOFF bare hull
configuration. The simulations were conducted with the same conditions as the
19
rotating arm experiment [14], with a constant turning rate of 0.1237 rad/s at a
radius of 12.48 m and one of the drift angles in the experiment - positive 10.2° , ex
cept that the reference point was slightly off between the experiment and the CFD
simulations. The reference point was taken to be 2.013 m aft of the nose in the
experiment, but at 2.009 m aft of the nose (where the CB is located) throughout
the present CFD work. This discrepancy resulted from a initial lack of information
on the experimental conditions. The turning radius is measured from the reference
point of the model and the drift angle at this point taken as is positive when the nose
is pointing towards the center of rotation for a clock wise turn. These simulation
parameters, chosen on a basis of the experimental work conducted in Naval Surface
Warfare Center Carderock Division (NSWCCD) Rotating Arm Facility, allow for
validation process of the CFD results.
The hull profile of SUBOFF is shown in Figure 4.1. The original hull length was
£ = 4.356 m without the sting, and this is the body length used in the prelimi
nary calculations. The reference point is located at \/£ = 0.4612. The SUBOFF
hull has a rounded nose, a constant radius mid-body and a tapered tail. Notice
that there is a rounded cap at the end of the tail. The experimental model used a
hull truncated at 92% of its length to accept a sting in the rotating arm experiments.
01
0 05
o
0 05
-0 1
20
r-~~^0 1 02 03 04 05 06 07 0
M
Figure 4.1: Profile of SUBOFF hull to scale.
4.1.2 RANS Simulation
Steady-state simulations were conducted for a steady turn. Two turbulence mod
els were tested in the simulations, the Shear Stress Transport (SST) model and the
Baseline Reynolds Stress (BSL-RSM) model. The default high resolution discretiza
tion scheme was employed first using unstructured meshes, and later a fully second
order discretization scheme was employed using structured meshes along with dou
ble precision calculations to minimize possible round-off error that may occur in the
high aspect ratio elements found in the viscous sub layer. Sea water was the working
fluid with a dynamic viscosity of 1.06 x 10~3 kg/m/s and a density of 1030 kg/m3,
so that the resulting Reynolds number was 6.5 x 106 to match with the rotating arm
experiment [6].
4.1.3 Fluid Boundary Conditions
The computational domain is set to rotate using a frame of reference attached to the
SUBOFF body axis. The CFX rotating frame of reference setup is commonly em
ployed with applications such as turbo-machinery and allows for applying boundary
conditions in either Cartesian Inertial or Cylindrical Coordinates. Some quanti
ties such as pressure can be applied in the absolute or rotating frames. The far
field boundaries are kept two body lengths, £, away from the hull, as shown in Fig
ure 4.2(a). The hull surface is centered in the domain and set to a wall with a no-slip
condition. The inlet condition in the rotating frame is set at a plane upstream of
the hull nose with a normal flow direction specification. The value of the normal
flow speed at the inlet varies linearly based on the radius measured from the center
of rotation and the constant turning rate. The outlet condition (the plane behind
the SUBOFF tail) is set to a zero average static pressure.
A symmetry condition is applied to the plane passing through the symmetry plane of
21
the hull that is normal to the axis of rotation, Figure 4.2(a), 4.2(b) and 4.2(d). The
remaining boundary conditions in the domain have an opening with static pressure
for entrainment settings, which has no predetermined flow direction, since the flow
can either enter or leave the boundary as dictated by the solution. The curvature
of these opening planes has been kept tangential to the flow direction, (i.e., at a
constant radius) which gives the most accurate calculations for this type of boundary
condition [1].
4.2 Unstructured Mesh
Initially a relatively coarse mesh with 1.3 million nodes was generated using CFX-
Mesh and is shown in Figure 4.2. CFX-Mesh produces a mesh containing unstruc
tured tetrahedral, prism, and pyramid elements within the computational domain.
It can be seen in Figure 4.2 that mesh refinement is applied at critical locations
of the flow field, such as stagnation and separation zones near the nose and tail,
Figure 4.2(c), and around the main body, Figure 4.2(b) and 4.2(d), where the flow
can have large gradients due to flow separation from the body.
As shown in Figure 4.3, prismatic elements were inflated from the surface of the
model up to a distance of 5 cm with an expansion factor of 1.2. The purpose of the
inflation layer is to resolve the boundary layer flows near the walls of the axisymmet-
ric hull. The height of first layer of prismatic elements was kept at 1 x 10~6 m to meet
the requirement of y-\- value less than one according to the solver manual [13]. This
allows for appropriate resolution of the velocity gradients in the viscous sub-layer.
This is important to the accurate prediction of such things as flow separation. Equa
tion 4.1 shows such an estimation of the first node spacing for a desired y+ value [13].
22
. ^ . f ^ "
(a) (b)
\v%
(c) (d)
Figure 4.2: SUBOFF unstructured mesh with 1.3 million nodes at j3 = 10.2°, R = 12.48 m, r = 0.1237 rad/s and i?e - 6.5 x 106. (a)Far field boundaries of computational domain; (b)Symmetry plane; (c)Mesh around hull on symmetry plane; (d)Transverse plane at \/£ = 0.2.
Ay = eAy+V80Re -13/14 (4.1)
where Ay is the mesh spacing between the wall and the first node away from the
wall, / is the length of the hull, Ay+ is the desired y+ value at the first node, Rei
is the Reynolds number based on the hull length. For example, for a Reynolds
number of 6.5 x 106 and a desired y+ value of 0.1, it requires the first node to be
23
kept away from the wall at 1.8 x 10 6 m. This equation provided some guidance
in estimating where the first node should be placed in the mesh development process.
Figure 4.3: Inflation layer of mesh around hull surface (SUBOFF unstructured mesh with 1.3 million nodes at /? = 10.2°, R = 12.48 m, r = 0.1237 rad/s and Re = 6.5 x 106).
Outside of the inflation region, the rest of the fluid domain was filled initially with
tetrahedral elements. A number of refinements were subsequently made to the 1.3
million node model just described. The local refinements applied are described more
in later sections, but ultimately allowed for solutions where the maximum residual in
the mass and momentum equations was below 1 x 10~4. The unstructured meshes
were used in the preliminary study to provide insight to the flow field, which assisted
the development of a mesh script described in the next section.
4.3 Structured Mesh Topology (Final)
One objective of this work is to establish a RANS calculation database that will be
used to develop a semi-empirical model for quick and accurate force and moment
estimation of an axisymmetric body in turning. A comprehensive CFD simulation
24
with a variety of turning parameters, such as: drift angles, turning radius, and hull
shapes is required for developing such a semi-empirical model. The discretization
of flow domain needs to be adequate to resolve the critical flow features, and be
able to adapt to different turning parameters. In the mean time the mesh needs to
be rapidly built for each turning parameter. A mesh generation script is therefore
developed to meet this requirement. It will help to keep discretization in a similar
fashion at different turning parameters with precise mesh controls, and allow for
a fully second order discretization scheme to be implemented, maximally reducing
numerical diffusion error.
The results obtained using the unstructured meshes have shown (see Section 4.4.2)
that the predictions of the hydrodynamic forces and moments are not mesh indepen
dent even with a fine mesh of 18.5 million nodes although most force and moment
predictions are within the experiment uncertainty. The exception to this agreement
is the yawing moment prediction and this could be partially due to the absence of the
experimental sting structure in the preliminary CFD model. Therefore a mesh with
sting configuration and the development of an improved mesh topology utilizing a
more structured mesh in the domain will be employed (See Figure 4.4). The mesh
generation script will be able to build the mesh with and without a sting configura
tion. A validation using the with-sting mesh can be performed to check the accuracy
of predictions of the CFD model. Later comprehensive CFD simulations using the
without-sting mesh can be conducted to obtain a database of hydrodynamic forces
and moments acting on axisymmetric bodies in turning.
The final mesh is constructed based on 3 blocks: a structured near body mesh, a
structured far field mesh and an unstructured intermediate mesh which provides a
smooth transition between the very dense near body mesh and the less dense far field
25
mesh, as shown in Figure 4.4(b). The primary benefit of using a structured mesh
is that it is easy to align with the flow direction, which greatly minimizes numerical
diffusion error. Another advantage of using a structured mesh over an unstructured
mesh is that the memory required in the solver is about two times less with the
same number of nodes [16]. It is a goal to keep the node number of the final mesh
relatively low when mesh independent results are obtained.
Effective use of element placement is required in the mesh script development. This
involves placing refined hex elements in critical regions such as the boundary layer
and the leeward side vortex, where the flow undergoes large variations, and coarse
hex elements on the far boundaries which are adequate.
The overall mesh topology is illustrated in Figure 4.4 with each of the boundary con
ditions highlighted individually. The axisymmetric hull is centered in the computa
tion domain, and the outer boundaries are kept 21 away from the hull approximately.
The same boundary conditions as in the preliminary CFD model simulations were
applied to each of these surfaces. At the inlet a normal flow speed varies based on
the turning radius (Figure 4.4(a)). The turbulence intensity at the inlet was set
to 1%. A zero average static pressure was set to the outlet plane (Figure 4.4(c)).
A symmetry condition is applied to the plane normal to the rotation axis (Fig
ure 4.4(b)). An opening boundary condition with static pressure for entrainment
settings was set to the remaining planes (Figure 4.4(d)).
4.3.1 Structured Near Body
The near body mesh is started with a surface mesh on the hull and sting surfaces
constructed by hexahedra elements. An illustration of the axial and circumferential
node distribution on the hull surface is shown in Figure 4.5. The axial spacing was
26
x — - . \
(d)
Figure 4.4: SUBOFF-with-stmg mesh with 3.6 million nodes at (3 = 10.2° and R = 12.48 m using the mesh script. (a)Inlet plane, (b)Symmetry plane; (c)Outlet plane; (d)Opening planes.
specified and distributed smoothly at several areas to provide adequate discretiza
tion based on the variation of velocity and pressure gradients in that area. For
example, the distribution of nodal spacing is denser at the nose because of the stag
nation region and at the tapered tail where the size of the leeside vortex is largest.
The axial mid-body spacing is kept constant. The circumferential spacing is kept
equal because with varying the circumferential distribution of nodal spacing from
windward side to leeward side had no effect on the overall force predictions.
Once the surface mesh is made, it is expanded out radially at a constant rate of
1.08 to complete the near body mesh. Transverse planes at several axial locations
are constructed to define the radial node distribution, as shown in Figure 4.6. An
illustration of the radial node distribution at the symmetry plane is also shown in
(a)
(c)
27
Figure 4.7. The structured near body region covers approximately a radius of one
and half hull diameters at mid-body so that the boundary layer and the leeside vor
tex are within this structured region. The radii of the structured region becomes
greater at the tail and the wake, especially on the leeside, due to the growing sizes
of the leeside vortex.
(b)
Figure 4.5: Surface meshes on the SUBOFF hull and sting with 3.6 million nodes using the mesh script. (a)Overall view; (b)Detailed views near the nose and tail of the hull.
28
Note that two backwards facing steps exist at both ends of the sting, as shown in
Figure 4.7(c). It is well known that the recirculation zone behind a backwards facing
step can cause convergence difficulties in the RANS solver if the discretization in that
region is not sufficient. This is particularly a concern if a strict goal of utilizing fully
second order numerics and reducing RMS residuals below 1 x 10~5 for all equations.
This requires the axial density of the mesh around these two regions to be as high
as 120 nodal points over a length of six step heights [17] which is the typical size of
the recirculation zone trailing a backwards facing step. The structured mesh m the
trailing wake aligns with the flow direction to reduce the numerical diffusion. The
mesh spacing in the wake expands to a maximum value from the sting end to outlet.
Figure 4.6: Highlighted transverse planes showing the radial node distribution of SUBOFF-with-sting mesh with 3.6 million nodes using the mesh script.
4.3.2 Structured Far Field Mesh
The structured far field mesh is aligned with the flow direction which provides re
alistic opening boundary condition and reduces numerical diffusion, as shown in
29
(a)
(b)
(c)
Figure 4 7 The near body structured mesh on the SUBOFF-with-sting symmetry plane with 3 6 million nodes using the mesh script (a)Structured mesh near body and wake region, (b)Structured mesh near stagnation region, (c)Structured mesh around two backwards facing steps due to presence of the sting
30
Figure 4.8 and Figure 4.9. The gradient of flow properties in the far field is much
less than near the body. The mesh in the far field region is greatly coarsened
compared with near body mesh to save the computational cost. Figure 4.8 and
Figure 4.9 illustrate how the structured and unstructured meshes adapt their shapes
to different drift angle from 0° to 30° at a constant turning radius R/£ = 2.9, and
different normalized turning radii from R/£ = 3.3 to 1000 at a constant drift angle re
spectively. The mesh script is able to generate the meshes accounting for the change
of shapes of three blocks due to the change of drift angles and turning radius with
out the need of user interaction. The script is designed to automatically calculate
the boundaries of three blocks depending on the input drift angle and turning radius.
In Figure 4.8, both the near body and far field meshes maintain their node numbers
unchanged while the drift angle increases, and the unstructured transition region
becomes larger in width due to the widened near body structured mesh. An effort
is also made to keep the structured wake region wide enough so that it covers the
earlier separation due to increased drift angle. In Figure 4.9, the structured mesh
both near body and far field retains its alignment with flow direction and its node
number remains unchanged while the turning radius increases. The opening bound
aries are kept aligned with flow direction reducing numerical diffusion.
The mesh script always tends to keep the far boundaries two body lengths away from
the hull. Therefore there is a limitation of turning radius for the current mesh script
that the user should be aware of: the minimum turning radius can only be set to just
over two body lengths to keep the inner domain boundary, that closest to the center
of rotation, far enough from the hull. As turning radius increases, the computational
domain becomes more rectangular and less curved in shape. At a very large turning
radius, the local drift angle along the hull becomes close to constant and the hull
31
\ " \
(a) (b)
~ ~ ' " "N ?-*..,. *" *~~~ \ v
(c) (d)
Figure 4.8: The SUBOFF-with-sting mesh on the symmetry plane at different drift angles with 3.6 million nodes at R/£ = 2.9 using the mesh script, (a)At 0° drift angle; (b)At 10° drift angle; (c)At 20° drift angle; (d)At 30° drift angle.
essentially is translating. Using the mesh script with an extremely large turning
radius parameter {Rjt = 104), a series of RANS simulations for the axisymmetric
body in translation can be carried out. These results can be validated against
experimental translation data of which there is much more available than for the
rotating case. This provides an extra way of assessing discretization error for the
current mesh.
32
(a) (b)
v \
(c) (d)
Figure 4.9: The SUBOFF-with-sting mesh on the symmetry plane with 3.6 million nodes at (3 = 10.2° and different turning radius using the mesh script. (a)At R/£ = 3.3; (b)At R/£ = 5.0; (c)At R/£ = 10.0; (d)At R/£ = 1000.0.
4.4 RANS Calculation
4.4.1 Verification of RANS Calculations
4.4.1.1 Pre l iminary Uns t ruc tu r ed Mesh
The magnitude of the Y' predictions using unstructured meshes increases as the
mesh is refined, as shown in Figure 4.10. The SST turbulence model under predicts
the BSL-RSM turbulence model predictions of Y' consistently by 23% (at each mesh
density level). This result is consistent with Watt et. al. [15] results for translated
motion, where the SST model was found to consistently under predict the normal
force compared to the BSL-RSM model. In that study it was found that reduced
33
energy loss in the wake predicted by the SST model lead to smaller Y' predictions.
The improvement in force prediction with the BSL-RSM model is likely a result of
solving the six Reynolds stresses equations and the dissipation rate to account for
anisotropies in the Reynolds stress tensor in the three dimensional wake regions of
the flow.
100.0
95.0 -- _ _ * — — * > 92.885
90.0 > ^ ^ ^ * 9 1 2 1 1 91.8182 __ n / 89.1365 -#-Lateral-SST
. X 82.806 Force [N] gQ Q , -*-Lateral-RSM
75.0 ^ —~ —4—— — *** 7 5 24 4 C . , / i n 74.266 74.7429
70.0 * 72.4149
65.0
60.0
67.6062
2 4 6 8 10 12 14 16 18 20
# Node [million]
Figure 4.10: RANS predictions of lateral force, Y, at different mesh resolution using two turbulence model for SUBOFF (hull) turning at /? = 10.2°, R = 12.48 m, r = 0.1237 rad/s and Re = 6.5 x 106 with SST and BSL-RSM turbulence model using unstructured meshes.
Results from this preliminary work (Figrure 4.10) suggest that a mesh independent
solution has not yet been obtained even with a fine mesh of 18.5 million nodes for
both turbulence models using unstructured meshes. This result points to the need
of developing a mesh script with much more control of local element distributions
and to use structured nodes efficiently to keep the total node number reasonably
34
small.
4.4.1.2 Structured Mesh
The mesh script is developed with the capacity to scale meshes uniformly in all three
directions for a structured mesh with a single user control parameter. This advanced
feature guarantees that the verification process can be carried out with meaningful
results. To verify that the RANS solution is mesh insensitive, RANS simulations
were completed for the SUBOFF hull with sting at R = 12.48 m, (3 = 10.2° and
Re = 6.5 x 106 using structured meshes with 3.6, 6.3 and 9.9 million nodes over the
half body. For each simulation, the maximum normalized residuals in the RANS
solution variables were reached below 1 x 10~4 except for the 6.3 million node case
where the u, v-momentum residuals were only reduced to 1 x 10~3, as shown in Fig
ure 4.11. The maximum residuals for each simulation reduced more than one order
of magnitude from time step 100 to the end of convergence. A comparison of lateral
force predictions from these simulations is shown in Figure 4.12. The difference
in force predictions is found to be as low as 0.01% using the mesh with doubling
the number of nodes. Unlike the preliminary CFD model work, where the RANS
calculation results were mesh dependent even with a very fine mesh of 18.5 million
nodes, a mesh independent solution was obtained using the mesh template with only
the intermediate sized mesh.
In addition to the lateral force convergence history plots, the transverse plane cir
culation at the tail was also monitored through the iteration steps, as shown in
Figure 4.13(a). The circulation at the tail plane has great influence on the forces on
the hull which is of direct interest to the present work. It can be seen that the value
of the tail plane circulation becomes constant after approximately 150 time steps
for all three mesh resolutions. A similar result can be found in the force prediction
35
history that the overall forces on the hull do not change any after 150 time steps.
Note that the difference m the value of the tail plane circulation is found to be only
0.2% with the mesh doubling the number of nodes. There is some innaccuracy
in calculating the circulation quantity since the vorticity components used are not
conserved variables in the solution. Otherwise the error would be expected to be
lower.
Hun subofi"10 2withsting20 001 Momentum and Mass
kVi4
Run suboff 10 2 w!thst)ng?S 002 Momentum and Mass
- \ , -v./Vv> .,
Accumulated Time Step
MAX P Mass MAX U Mom MAX V Mom MAX W Mom
Accumulated Time Step MAX P Mass MAX U Mom » MAX V Mom
(a) (b) Run suboff 10.2 wrthstmg30 003
Momentum and Mass
X _
50 100 150 200 250 300 Accumulated Time Step
-lAXPMass MAX U Mom - MAX V Mom MAX W Mora
(c)
Figure 4.11: Convergence history showing the maximum normalized Residuals for SUBOFF-with-sting (hull) turning at (3 = 10.2°, R = 12.48 m, r = 0.1237 rad/s and Re = 6.5 x 106 with BSL-RSM turbulence model using 3 structured mesh resolutions. (a)3.6 million nodes; (b)6.3 million nodes, (c)9.9 million nodes.
A full body RANS simulation was conducted to check if the assumption of symme-
36
90.0
80.0
70 0
60 0
50 0
40.0
30.0
20 0
0
80.448 80.302
45.230 45.136
2 4 6 8
# Node [million]
80.294
45.091
10 12
-V-Latera l
••*$••" Axial
E
e « E o 2
190 0
170.0
150.0
130.0
110.0
90.0
70.0
50 0
30.0
0
131.396 131.705
2 4 6 8
# Node [million]
131.709
10 12
^•—Yawing
(a) (b)
Figure 4.12: RANS predictions of forces and moments for SUBOFF-with-sting (hull) turning at 0 = 10.2°, R = 12.48 m, r = 0.1237 rad/s and Re = 6.5 x 106 using BSL-RSM turbulence model and the structured meshes with 3.6, 6.3 and 9.9 million nodes. (a)Axial and lateral forces on SUBOFF hull; (b)Yawing moment on SUBOFF hull.
0.444 43.5
0 442 ^ 43
0.44 ^
0.438 \ - V - 3 6M \ v 41 5
0.436 \ \ - » . S 3 M a \ \ n ! 9 M "- 41
42 5
^ 42
0.434
0.432
0.43
40.5
40
39.5
0 100 200 300 100 200 300
Accumulated Time Step Accumulated Time Step
(a) (b)
Figure 4.13: Prediction history for SUBOFF-with-sting (hull) turning at (5 = 10.2°, R = 12.48 m, r = 0.1237 rad/s and Re = 6.5 x 106 using BSL-RSM turbulence model and the structured meshes with 3.6, 6.3 and 9.9 million nodes, (a)Tail plane circulation; (b)Lateral force on SUBOFF hull.
try flow is correct in these submarine turning maneuvers. The full domain mesh,
obtained by reflecting the half domain, as shown in Figure 4.14. The resulting
force predictions were very close to those using the symmetry boundary condition.
37
The percent difference in the X, Y and N predictions were 0.04%, 0.01% and 0.3%,
respectively. The Z force value was nearly zero at -0.000137 N which indicates very
good symmetry.
Figure 4.14: Mirrored SUBOFF-with-sting mesh with 7.2 million nodes at /3 = 10.2° and R = 12.48 m, highlighted with inlet and outlet plane.
4.4.2 Validation of RANS Calculations
The wall shear stress lines on the SUBOFF hull are shown in Figure 4.15. The flow
attachment regions on the windward side are clearly shown on the plot, where the
streamlines diverge. On the leeward side it can be seen that the flow separates (i.e.,
where the streamlines converge) and reattaches (where the streamlines diverge). It
is not clear in Figure 4.15 if there are any secondary flow separations or reattach
ments regions. A quantitative investigation of primary and secondary separation
is required in the future by examining where the local minimum value of wall shear
stress occurs.
38
(a) (b)
Figure 4.15: BSL-RSM RANS predictions of shear stress lines on hull surface for SUBOFF (hull) turning at (3 = 10.2°, R = 12.48 m, r = 0.1237 rad/s and Re = 6.5 x 106 using the structured mesh. (a)Windward side, (b)Leeward side.
Figure 4.16 shows a total pressure contour plot on transverse planes at several axial
locations, and the leeward side wall shear stress lines on the hull surface. At 10.2°
drift angle, the cross component of the flow generates the separation and rolls up
into vortex pair. It is well known that the recirculation within the separation region
produces significant pressure differentials.
Figure 4.16: BSL-RSM RANS predictions of total pressure on transverse planes at several locations, A/£=0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0 and 1.2, for SUBOFF (hull) turning at 0 = 10.2°, R = 12.48 m, r = 0.1237 rad/s and Re = 6.5 x 106 using the structured mesh.
39
The RANS predictions using the structured 6.3 million node mesh for the SUBOFF-
with-sting (hull) turning are tabulated in the Table 4.1.
Table 4.1: BSL-RSM RANS predictions of force and moment coefficients on the truncated SUBOFF hull at R = 12.485 m, r = 0.1237 rad/s and Re = 6.5 x 106 in steady turning.
P(°)
3.8
6.2
10.2
12.2
14.3
16.5
X'
-1.41E-03
-1.62E-03
-1.94E-03
-2.07E-03
-2.17E-03
-2.25E-03
Y'
1.21E-03
1.85E-03
3.44E-03
4.50E-03
5.76E-03
7.27E-03
N'
3.61E-04
7.52E-04
1.30E-04
1.51E-03
1.69E-03
1.88E-03
The RANS predictions using the structured meshes generated by the mesh script
with 6.3 million nodes are compared with the experimental data for validation pur
poses, as shown in Figures 4.17 through 4.19. Both RANS predictions are shown in
these figures, the preliminary CFD results using the 18.5 million node unstructured
mesh and the results using the new mesh script with-sting. Although discrepancies
between the newest RANS prediction and experimental data suggest room for fur
ther improvement, the results indicate a good match of the trends with experimental
data for drift angles up to 16°. Figure 4.17 shows that the newest RANS predictions
of the axial force coefficients are within experimental uncertainty for drift angle up
to 16°. In Figure 4.18, the newest RANS simulations under predict the lateral
force coefficient for drift angles from 10° to 14°. The newest RANS prediction of
the yawing moment coefficient for 10.2° drift angle has been greatly improved using
the mesh script compared with the preliminary CFD model work, as shown in Fig-
40
ure 4.19, although the difference to experimental data increases at a higher angle of
drift. Note that the reference point was offset by 4 mm (in a 4.356 m hull) between
the experiment and the RANS simulation. The yawing moment calculations were
found to have 0.2% difference using different referecnce point.
0.00E+00
-5.00E-04
-1.00E-03
-1.50E-03
-2.00E-03
-2.50E-03
-3.00E-03
X" p [degree]
8 10 12 14 16 18
• Experiment BH
% Unstructrued (18 5M Full Body)
• Structured (6 3M Half Body)
Figure 4.17: Axial force coefficient for both experiment and BSL-RSM RANS predictions for the SUBOFF-with-sting (hull) turning atR= 12.48 m, r = 0.1237 rad/s and Re = 6.5 x 106.
4.4.3 Other Validation of Results
Another source of validation from the SUBOFF experiments is a comparison of the
static pressure circumferential distribution along the hull surface at two axial loca
tions A/£=0.63 and 0.84 as shown in Figure 4.20 and Figure 4.21 respectively. The
pressure is plotted in terms of static pressure coefficient, Cp, normalized using the
untruncated hull length and velocity v at the body axes origin. The static pressure
measurements in the experiment were obtained using pressure taps located primar
ily on the leeward side, beginning from 90° (top of the hull) to 270° (bottom of the
hull) [14]. Differential pressure transducers were used in the pressure measurement
41
9.00E-03
8.00E-03
7.00E-03
6.00E-03
5.00E-03
4.00E-03
3.00E-03
2.00E-03
1.00E-03
O.OOE+00
Y"
• Experiment BH
% Unstructrued (18 5M Full Body)
• Structrued (6 3M Half Body)
p [degree]
10 12 14 16 18
Figure 4.18: Lateral force coefficient for both experiment and BSL-RSM RANS predictions for the SUBOFF-with-sting (hull) turning at R = 12.48 m, r = 0.1237 rad/s and Re = 6.5 x 106.
2.00E-03
1.80E-03
1.60E-03
1.40E-03
1.20E-03
1.00E-03
8.00E-04
6.00E-04
4.00E-04
2.00E-04
0.O0E+00
N1
• Experiment BH
i Unstructrued (18 5M Full Body)
is Structrued (6 3M Half Body)
P [degree]
10 12 14 16 18
Figure 4.19: Yawing moment coefficient for both experiment and BSL-RSM RANS predictions for the SUBOFF-with-sting (hull) turning at R = 12.48 m, r = 0.1237 rad/s and Re = 6.5 x 106.
42
with the reference pressure being the static pressure at each gage at zero model
speed [9]. Note that the pressure data from the RANS calculation using the mesh
script is only available for the half of the hull surface, this is due to the fact that
the mesh script used a symmetry plane to save computational cost. At A/£=0.63
(Figure 4.20), the static pressure tends to drop quickly from the stagnation point
(at 0°) to the top (at 90°) due to acceleration of the flow. On the leeward side
the static pressure is recovered, and it can be seen in Figure 4.20 there is no flow
separation occurring at A/^=0.63 since no sudden pressure drop is observed on the
leeward side (from 90° to 180°). At A/£=0.84 (Figure 4.21), the static pressure
recovery starts earlier (at 60°) than at A/£=0.63 due to a thicker boundary layer,
the pressure recovery on the leeward side stops where the flow separates (at 140°).
A secondary leeward side pressure recovery happens after (at 160°) due to the flow
reattachment. Noted that the experimental circumferential pressure distribution at
these two locations is symmetrical about the symmetry plane which provides sup
port for modeling the flow with a symmetry condition.
For BSL-RSM RANS predictions of the static pressure coefficient, larger values at
both measurement locations are obtained, compared to experimental results. The
shape of the two curves agrees well with experimental results. The discrepancies
in Cp between the RANS predictions and experiment at both locations represent
1.3 and 2.1 cm of hydro-static pressure, respectively. It is possible that a steady
wave pattern on the surface of water caused by the sting assembly could effect the
hydro-static pressure at model depth and cause these discrepancies [9].
43
0 10
0 00
-0 10
-0 20
-0 30
-0 40
-0 50
KIL= 0.63, p = 10.2°
l-^A.
* * * t .
• Experiment
Unstructrued (18 5M Full Body)
k Structrued (6 3M Half Body)
1 - 1 - T - W
Tap Location (degree)
Figure 4.20: Comparison of BSL-RSM RANS predictions of Cp to experimental measurement on SUBOFF-with-sting hull at [3 = 10.2°, R = 12.48 m, r = 0.1237 rad/s and Re = 6.5 x 106 at \/£ = 0.63.
-0 50
A/L = 0.84, P = 10.2° Experiment
Unstructrued (18 5M Full Body)
Structrued (6 3M Half Body)
CO (D O) CN U7 0O CN CN CM CO CO (O
Tap Location (degree)
Figure 4.21: Comparison of BSL-RSM RANS predictions of Cp to experimental measurement on SUBOFF-with-sting hull at (3 = 10.2°, R = 12.48 m, r = 0.1237 rad/s and Re = 6.5 x 106 at X/£ = 0.84.
44
Chapter 5
Results for an Axisymmetric Body
(SUBOFF) in Turning
5.1 Pivot point location for SUBOFF
Standard submarine equations of motion have been described in detail by Feld-
man [18]. From his lateral force and yawing moment equations, the unsteady v and
r terms can be eliminated since only the steady turn is of current interest. The
terms associated with the CG coordinates can be neglected since the CG is close
to the reference point. Hence, the linearized horizontal plane equations of motion
from Feldman's lateral force and yawing moment equations are simplified as:
m • ur = £ £3 Y'r ur + ^ I2 Y'v uv + ^-f Y'5r u28r (5.1)
0 = P-P N'r ur + P- £3 N'v xw + P-P N'Sr uHr (5.2)
where u, v are the velocities at the reference point. Rearranging equations 5.1
and 5.2:
45
where
Let
Nrr + N5ru5r V = -N„
NrYv-(Yr-m)Nv[ r)
Yr-
Yv:
Y5r =
Nr~-
Nv =
NSr =
2 r
=(- e Y' 2 "
- - P2 Y'
-p-tK
-\?K
^3K
Cx YSrNv - N5rYv
NrYv - {Yr - m)Nv
Then Equations (5.3) and (5.4) becomes:
r = Ci{u5r)
46
NrCX + NSr
v = — (udr) (5.7)
As submarines float neutrally in the water during a steady turn, that gives m = pV,
where p is the density of water. The coefficients used in submarine equations of
motion (5.1)-(5.7) for SUBOFF are tabulated in Table 5.1 and can be found in the
reference [19].
Table 5.1: Nondimensional coefficients used in submarine equations of motion for SUBOFF (Config. 2 Fully Appended).
Y' r
0.005251
K
-0.027834
^5r
0.005929
K
-0.004444
K
-0.013648
K -0.002217
v \m 24.98991
The turning radius, pivot point and turning rate are related by:
r (5.8)
xp = Rsin(P) (5.9)
where j3 is the drift angle measured at the reference point. We have U = y/u2 + v2,
and substituting Equations (5.6) and (5.7) into (5.8), it gives:
R = 1 + NrCl+Nlr
C\5r
By neglecting the second order term, it becomes:
(5.10)
R C\5r
(5.11)
47
The drift angle at the reference point can be calculated by:
D = tan"1 (-^)
^ t m - ( w c - ; ^ ) (5.i2)
From Equations (5.11) and (5.12), it has been found that the pivot point is at a
fixed location along the submarine central axis for any value of turning radius, and
it is at x = 1.609 m for SUBOFF. The normalized turning radius has been plotted
against the drift angle at the reference point as well as rudder angle, as shown in
Figure 5.1. When the turning radius is close to 5 £, the drift angle at the reference
point is about 5.1° for SUBOFF. As the turning radius approaches infinity the drift
angle tends to zero. The relation between the drift angle and the turning radius,
described in Equation (5.12), will be employed in the mesh script for SUBOFF in
turning.
5.2 Mesh Topology for SUBOFF in Tight Turning
When a submarine is undergoing a tight turn (e.g., the turning radius is equal to
the hull length (R/£ = 1)), the center of rotation is within 2£ of the hull surface.
Maintaining a mesh topology similar to that described in previous sections will have
convergence issues in the RANS simulation, particularly at the inner boundary clos
est to the center of rotation. It has been found by numerical testing that neither
an opening nor a free-slip wall boundary condition is suitable at the inner curved
boundary when using the same mesh topology as in previous chapters. Therefore
modifications to the mesh topology have been made for SUBOFF in tight turns. As
48
10 20 30 40
{degree/
Figure 5.1: The turning variables for SUBOFF in a steady turn.
shown in Figure 5.2, the revised mesh uses a full cylinder with the center located
at the rotation axis, and opening boundary conditions are applied on the side and
top planes (there is no inlet and outlet anymore). The mesh is, first, constructed
by removing the sting presented in Figure 4.4, and filling up the center and the gap
between inlet and outlet planes (Figure 4.4) with tetrahedral elements as shown in
Figure 5.3. This modified mesh script generates meshes for a turning radius ranging
from £ to 3 £. Simulations of SUBOFF in tight turns using the mesh generated by
the modified script arc presented in a later section.
The modified surface mesh near the tail uses the same topology as near the nose
(Figure 4.5), as shown in Figure 4.4(a). The sting has been removed from the hull
in the modified mesh, and the mesh is expanded out radially to complete the near
tail mesh, as shown in Figure 5.4(b). Figure 5.4(c) shows the structured near body
and wake region, and a transition layer connects the near body region and the center
8
7
6
5
4
3
2
1
0
49
Figure 5.2: The computational domain for SUBOFF in a tight turn at ft = 21.6° and R/£=l.
Figure 5.3: The symmetry plane of SUBOFF mesh in a tight turn with 1.7 million nodes at (5 = 21.6° and R/£ = 1 using the tight turn mesh script.
50
of the computational domain.
(a) (b)
(c)
Figure 5.4. Structured mesh of SUBOFF in a tight turn with 1.7 million nodes using the mesh script. (a)Surface mesh near the tail of SUBOFF; (b)Structured mesh around tail on symmetry plane; (c)Structured mesh near body and wake region on symmetry plan.
51
5.3 RANS Calculation using the Mesh Template
5.3.1 Fluid Boundary Conditions
The computational domain is set to rotate using a rotating frame of reference, and
three types of boundary condition are applied in the RANS simulation: a no-slip
wall condition applied to the hull surface, a symmetry condition applied to the plane
that cuts through the submarine and opening conditions applied to the rest of the
surfaces.
5.3.2 Wake Diffusion in Computational Domain in Tight
Turning
One concern with using a cylindrical domain in the RANS simulation is that the
SUBOFF wake could possibly be carried over to the upstream inflow condition which
would compromise the flow solution. The examination of the inflow upstream of
SUBOFF was done by looking at radial pressure and velocity distributions at several
locations through the domain. Polylines 1 to 8 originating at the CR on the symme
try plane were created in a clockwise direction from the upstream to the downstream
of SUBOFF, and spaced equally as shown in Figure 5.5.
The static pressure distributions at those polylines have been plotted in Figure 5.6.
The origin of the horizontal axis is located at the center of rotation. It can be
seen that the pressures along the polyline 2~4 are very close to 0 Pa with small
fluctuation (± 2 Pa), as shown in Figure 5.6(a). The pressure fluctuates up to 8
Pa at polyline 1 (approximately 0.3 I ahead of the nose) because of the upstream
influence of the hull. Figure 5.6(c) shows how the pressure varies inside the wake of
SUBOFF. As is apparent the pressure varies significantly immediately after the hull
(Polyline 8) but rapidly diffuses toward Polyline 5. Combined with the upstream
52
PoiyisiH1- 5
Polyhne 8
Polype 4
Polyline 3
Poiyjsro 2
Potyhre- ?
Polyhne 8
Poiyme 1
Figure 5.5: Polylines originating at the CR on the symmetry plane for SUBOFF in a tight turn at (3 = 21.6° and R/£ = 1.
profiles (Polyline 2~4), it is apparent that the wake has been significantly diffused.
The velocity (in the Rotating Domain) distribution have also been plotted at those
polylines, as shown if Figure 5.7. The fluctuation of velocity is not significant for
the upstream of the hull. It appears the modified mesh topology and boundary
condition settings are suitable for this type of turning problem.
5.3.3 Convergence of RANS Calculation
The SUBOFF hull has a smooth body and a rounded tail, and as such there are
no fixed separation lines over the entire hull. This makes unsteady flow separation
more probable. This is especially true when the turning radius is small, i.e. the
drift angle is large. For example, at attack angles above 30° the crossflow dominates
for axisymmetric bodies and the vortices are unsteady [1]. Note that the local drift
53
angle at the tail is f3tcai = 44° for tight turning cases when R/i = 1, which makes
steady-state convergence very difficult in these cases since the flow in some regions
will be inherently unsteady, and possibly asymmetrical (i.e., symmetry plane may
not be appropriate).
The convergence history of the residuals of the variables in the RANS solution have
been plotted for SUBOFF at its tightest turn (R/l = 1), as shown in Figure 5.8(a)
and 5.8(b). It can be seen that the solver has picked up some unsteadiness in the
flow as the value of residuals changes periodically at a certain frequency. A number
of different mesh resolutions were applied in critical flow regions to see if this un
steadiness could be removed without success. The maximum U-Momentum residual
fluctuates between 4 x 10~4 and 1 x 10~"3 after 100 iteration steps. The values of
other residuals fluctuate with a smaller magnitude as well, as shown in Figure 5.8.
This apparent unsteadiness could be a result of vortex shedding happening at the
SUBOFF tail, as the maximum residuals were identified and located only in the tail
wake region, as shown in Figure 5.9 with an isosurface plot. It should be noted that
the force and moment prediction changes its magnitude only by 0.1% (Figure 5.8(c))
when the residuals are going through a cyclical pattern of changes. In future studies
utilizing this mesh it will be neccessary if unsteady simulations should be applied,
and if the additional accuracy is worth the additional computational expense.
5.3.4 Observations on Results
Two RANS simulations of SUBOFF in turning for R/£ = 1 and 3, and one simula
tion in translation were conducted using the meshes generated by the mesh script,
and the results are tabulated in Table 5.2. The resolution of meshes used in these
RANS simulations (near the hull surface) are the same as those described in the pre-
54
vious chapter, but with a total nodal count reduced from 6.3 million to 3.2 million
because of absence of the sting. Therefore, the results to be shown subsequently
are assumed to be mesh independent solutions based on the mesh sensitivity study
done in the previous chapter since the local mesh topology, and mesh resolution in
the vicinity of the hull, are essentially equivalent to the 6.3 million node with-sting
mesh. However this assumption should be checked in future work with a more
formal mesh sensitivity study for tight turns.
The Reynolds number of the simulations was at 6.5 million. As the R/£ increases
from 1 to oo, the axial and lateral force coefficients on the hull decrease significantly
from -5.41 x 10"3 to -1.11 x 10"3 and from 1.49 x 10"2 to 1.55 x 10"3, respec
tively, as shown in Table 5.2. The yawing moment coefficient however increases
from 3.43 x 10~4 to 1.59 x 10~3. A comparison of static pressure circumferential
distribution along the hull surface is made for three cases: R/£ = 1, 3 and oo at
several axial locations A/7=0.1, 0.3, 0.5, 0.7 and 0.9 as shown in Figure 5.10. Signif
icant differences in circumferential pressure distribution can be noted for three cases
from the figures. For the tight turn (R/£ = 1) case, significant pressure drop is
observed: at 145° for A/Z=0.7, which may indicate the presence of flow separation.
For a more mild turn (R/£ = 3) case, no flow separations are found until at 100° for
\/l=0.9. Because of the flow separation associated with a tight turn, the energy
loss in the wake region is responsible for the lateral force on the body [1], and this
is the main reason that the force coefficient increases significantly as turning radius
decreases as shown in Table 5.2.
The wall shear lines and vorticity contours from the RANS calculation of SUBOFF
in both turning and translation are plotted and shown in Figure 5.11. The drift
angles at the reference point are 21.6°, 7.1° and 8° for turning and translation (cal-
55
Table 5.2: RANS predictions of hydrodynamic force and moment coefficients for the SUBOFF hull at r = 0.1237 rad/s and Re = 6.5 x 106 using the BSL-RSM turbulence model and the 3.2 million node meshes.
SUBOFF
n°) 21.649
7.064
8.0
X'
-5.14E-03
-1.67E-03
-1.11E-03
Y'
1.49E-02
1.96E-03
1.55E-03
N'
3.43E-04
9.23E-04
1.59E-03
Re (xlO6)
6.5
6.5
6.5
R/i.
1
3
oo
culated using the earlier template but without the sting) respectively. It can be
seen that the flow is signihcantly different for each case, in both the wall shear line
pattern and the vortex shape. The results are generally in-line with that expected
based on previous studies for translating hulls [1].
56
0-
0.
55
v Polyline 1
*» Polyline 2
Polyline 3
X Polyline 4
Radius from CR |m]
(a)
« 0.
£ 3 CM V] <U
CL. u
-** t/3
100
50
0 •
-50 0
-100
-150
-200 -250
-300
-350
10 12 14 16
Radius from CR |mj
Polyline 5
Polyline 6
Polyline 7
Polyline 8
(b)
30
-30
Polyline 5
Polyline 6
Polyline 7
Polyline 8
Radius from CR [m]
(c)
Figure 5.6: Static pressure distribution at polylines passing the CR on the symmetry plane for SUBOFF in a tight turn at (3 = 21.6°, r = 0.1237 rad/s, R/i = 1 and Re = 6.5 x 106 using the BSL-RSM turbulence model and the 3.2 million node mesh. (a)Upstream of SUBOFF; (b)Wake of SUBOFF; c)Close view for wake of SUBOFF.
57
6
5 v Polyline 1
J Polyline 2
S 4 Polyline 3
\ Polyline 4
u X Polyline 5
,© 2 - Polyline 6
4£ Polyline 7
* 3
2.4
4 6 8 10 12 14 16
Radius from CR [m]
(a)
3 3.5 4 4.5 5 5 5 6 6.5
Radius from CR [m]
(b)
Polyline 8
o Polyline 1 • — > 2 2 j o t) Polyline 2 S 2 -• Polyline 3
^ , 1 . 8 x Polyline 4
• - . „ V Polyline 5
© s Polyline 6 « 1 4
£»> -^t Polyline 7 1 -2 ^ M p P ^ Polyline 8
Figure 5.7: Velocity (in the Rotating Domain) distribution at polylines passing the CR on the symmetry plane for SUBOFF in a tight turn at /3 = 21.6°, r = 0.1237 rad/s, R/£ = 1 and Re = 6.5 x 106 using the BSL-RSM turbulence model and the 3.2 million node mesh. (a)Overall view; (b)Close view.
58
v X/KAWA ""A V \ V / \ ; V \ H ^ I V V ^ V ^ V \
' v ^ < x / /> V ^ A / % ^ W V \ / V \ / N A A / \ A A A A A - ^
(a) 1 L
(b)
(c)
Figure 5 8: History showing values of Normalized Residuals and force/moment predictions for SUBOFF (hull) turning at 0 = 21 6°, R/£ = 1, r = 0.1237 rad/s and Re = 6.5 x 106 using the BSL-RSM turbulence model and the 3.2 million node mesh. (a)Maximum value of Residual, (b)RMS value of Residual; (c)Force X,Y and Moment N prediction.
59
U Mnrr Rss-dia^ isost rfacs
5 }&ti 04
U Mom Res dual
B 3 99Se04
3 244e 04
(a)
(b)
Figure 5 9 Isosurface of Normalized U Momentum Residual with value of 1x10 4 for SUBOFF (hull) turning at [3 = 21 6°, R/£ = 1, r = 0 1237 rad/s and Re = 6 5 x 106
using the BSL-RSM turbulence model and the 3 2 million node mesh a)Overall view, b)Detailed view near the tail of the hull
60
a. o
0 60
0 40
0 20
0 00
-0 20 -
-0 40
-0 60
-0 80
-1 00
" ' ^ s , i , j ( ^ i s ( i ^
«
Circumferentail Angle (degrees)
(a)
(b)
' A/L=0 1 WL=0 3 WL=0 5 WL=0 7 WL=0 9
0 15 -
0 10
0 05 -
0 00
O -0 05
-0 10
-0 15
C
""""'"*"M"-.^4
3 O O O O O CO (O CD CM W
Circumferentail Angle (degrees)
<
o 00
• WL=0 1 WL=0 3
WL=0 5 WL=0 7
WL=0 9
0 20
0 15
0 10
0 05
0 00
-0 05
-0 10
-0 15
-0 20
* WL=0 1 A/L=0 3 WL=0 5 WL=0 7 WL=0 9
Circumferentail Angle (degrees)
(c)
Figure 5 10 Comparison of BSL-RSM RANS predictions of Cp on SUBOFF hull at r = 0 353 rad/s and Re = 6 5xl06 at \/£ = 0 1,0 3,0 5, 0 7 and 0 9 (a)/? = 21 6°, R/£ = 1, (b)/3 = 7 1°, R/£ = 3, (c)/? = 8 0°, R/t = oo
61
\ \ . N.;;
"".V
(b)
(c)
Figure 5.11: Contours of x vorticity and wall shear lines for SUBOFF from the RANS calculation using the BSL-RSM turbulence model and 3.2 million node meshes. (a)turning with (3 = 21.6° and R/£ = 1, r = 0.353 rad/s and Re = 6.5 x 106; (b)turning with /3 = 7.1° and R/t = 3, r = 0.353 rad/s and Re = 6.5 x 106; (c)translating with (3 = 8° and Re = 6.5 x 106 (obtained with the Chapter 4 template but without sting).
62
Chapter 6
Conclusion
6.1 Conclusion
This research is an extension of a CFD study for streamlined axisymmetric hulls
in translation to include rotation. The present research supports, by developing a
means to create high quality meshes, the generation of a CFD database which can
be used to assist in developing an improved force estimation method for an axisym
metric hull in turning.
A CFD study of a streamlined axisymmetric hull in turning was performed on the
DARPA SUBOFF hull using the RANS solver ANSYS-CFX software package. High
quality meshes were generated by developing a scripted algorithm using the Pointwise
software package to ensure adequate discretization of the computational domain, and
to allow a fully second order accurate numerical scheme to be applied in the calcula
tions. Overall force and moment predictions using the improved CFD model show
good agreement of the trends with experimental data for drift angle up to 16°. A
larger difference was found between RANS predictions and the experimental data at
extreme angles for yawing moment, and this needs to be further investigated. Static
63
pressure circumferential distributions around the hull surface show good agreement
with the trends in the experimental data at two axial locations, though a small offset
is present.
The mesh template was revised to produce high quality discretization for RANS
calculations with the SUBOFF hull for R ranging from 11 and 31. This mesh
template also provides practical topology suitable for other axisymmetric hulls for
both turning and translating.
6.2 Future Work
A formal mesh sensitivity study for the tight turning case should be carried out.
The mesh templates developed in this research were based only on the SUBOFF
geometry. They need to be adapted to other axisymmetric bodies to complete a
full study for a wide range of body profiles, which is straightforward. Additional
templates can be easily built using the topology designed for the SUBOFF case.
Finally, the current research has focused on turning axisymmetric body, a follow-on
step from previous research focused on translation. The next step is to model both
rotation and translation for axisymmetric bodies.
64
References
[1] Jeans, T.L., Watt, G.D., Gerber, A.G., Holloway, A.G.L., and Baker, C.R.,
High-Resolution Reynolds-Averaged Navier-Stokes Flow Predictions over Ax-
isymmetric Bodies with Tapered Tails, American Institute of Aeronautics and
Astronautics Journal, Vol. 47, No. 1, January 2009.
[2] DRDC Atlantic, Private Communication, May 2009.
[3] Watt, G.D. and Knill, K.J., A Preliminary Validation of the RANS Code
TASCflow Against Model Scale Submarine Hydrodynamic Data, The 7th An
nual Conference of the CFD Society of Canada, 1999.
[4] Sung, C.H., Rhee, B., and Koh, I.Y., Validation of Forces, Moments and Sta
bility Derivatives of a Manoeuvring Series-58 Bare Hull, The 25th Symposium
on Naval Hydrodynamics, St. John's, Canada, August 2004.
[5] Baker, C.D., A Strategic Meshing Approach to Modeling Hydrodynamic Flow
Around Streamlined Axisymmetric Shapes, Masters Thesis, Department of Me
chanical Engineering, University of New Brunswick, March 2006.
[6] Atsavapranee, P., Sung, C-H, and Ammeen, Ed., SUBOFF Rotating Arm Ex
periments, Submarine Hydrodynamic Working Group Proceedings, St. John's,
May 2008.
65
[7] Sung, C H., Fu, T C , Griffin, M., and Huang, T., Validation of Incompressible
Flow Computation of Forces and Moments on Axisymmetric Bodies Undergo
ing Constant Radius Turning, The 21th Symposium on Naval Hydrodynamics,
Trondheim, Norway, June 1996.
[8] Gregory, P., Flow Over a Body of Revolution in a Steady Turn, PhD Thesis,
University of Melbourne, 2006.
[9] NSWCCD, Private Communication, December, 2009.
[10] Versteeg, H.K. and Malalasekera, W., An Introduction to Computational Fluid
Dynamics, Pearson Prentice Hall, Toronto, 2007.
[11] ANSYS Canada Ltd., 2004, ANSYS CFX-Solver, Release 10.0: Theory - Tur
bulence and Wall Function Theory, ANSYS CFX 10.0 Help Files, pp. 68-86.
[12] ANSYS Canada Ltd., 2004, ANSYS CFX-Solver, Release 10.0: Theory - Basic
Solver Capability Theory, ANSYS CFX 10.0 Help Files, pp. 26-40.
[13] ANSYS Canada Ltd., 2004, ANSYS CFX-Solver, Release 10.0: Modelling -
Turbulence and Near-Wall Modelling, ANSYS CFX 10.0 Help Files, pp. 122-
129.
[14] Etebari, A., Atsavapranee, P., Carneal, J.B., Percival, A.S., Grant, D.J., Sung,
C.H., and Koh, I.Y., Experimental Measurements on a SUBOFF Model in a
Turning Maneuver, The 27th Symposium on Naval Hydrodynamics, Seoul, Ko
rea, October 2008.
[15] Watt, G.D., Baker, C.R. and Gerber, A.G., ANSYS CFX-10 RANS Normal
Force Predictions for the Series 58 Model 4621 Unappended Axisymmetric Sub
marine Hull in Translation, DRDC-Atlantic TM2006-037, Technical Memoran
dum, September 2006.
66
[16] ANSYS Canada Ltd., 2004, ANSYS CFX-Solver Manager, Release 10.0: CPU
and Memory Requirements, ANSYS CFX 10.0 Help Files, pp. 110.
[17] Yoder, D.A, A Backwards Facing Step Study #1, 1998, available at
http://www.grc.nasa.gov/WWW/wind/valid/backstep/backstepOl/
backstepOl.html.
[18] Feldman, J., DTNSRDC Revised Standard Submarine Equations of Motion,
DTNSRDC/SPD-0393-09, DTNSRDC Reports, June 6.
[19] Roddy, Robert F., Investigation of the stability and control characteristics of
several configurations of the DARPA SUBOFF model from Captive-Model Ex
periments, DTRC/SHD-1298-08, Departmental Reports, September 1990.
67
Appendix A
Private Communication with
DRDC Atlantic
Summarized information about the SUBOFF experiment [6]:
1) The reference point was at 2.013 m aft of the nose along the centerline.
2) The untruncated model length (4.356 m) was used for nondimensionalizing.
3) The turning radius and drift angle was calculated at the reference point.
4) The pressure measurement was performed with a differential pressure transducer
with referring to the static pressure at each gage with zero velocity.
5) The reported forces and moments are on the truncated model only (not including
the sting).
6) The sting diameter was 5.5 inch and the length of the truncated hull was 158 inch.
The sting went into the model and end in a strong back where the block gages are
attached to. The model was attached to the other side of the block gage assembly,
which was the point where the forces and moments were measured.
Appendix B
Script File
(script_suboff_with_Sting.gif) to
Generate Meshes for
SUBOFF-with-Sting
Available upon request
Appendix C
Script File
(script_suboff_no_Sting_tight_turn.gif)
to Generate Meshes for
SUBOFF-no-Sting in Tight Turn
Available upon request
Appendix D
Script File
(script_suboff_no_Sting_large_turn.gif)
to Generate Meshes for
SUBOFF-no-Sting in Large Turn
Available upon request
Appendix E
Script File
(script_PROCEDURES.gif) to
define the subroutines used in
other scripts
Available upon request
Vita
Candidate's full name:
Jian Tao Zhang
University attended:
University of New Brunswick
Bachelor of Science in Mechanical Engineering
May 2008
Publications:
Conference Presentations:
Holloway, A.G.L., Gerber, A.G., Zhang, J.T., Maxwell, J.A., and Watt, G.D., Sim
ulation and Modelling of the Flow over Axisymmetnc Submarine Hulls in Steady
Turning, The 28th Symposium on Naval Hydrodynamics, Pasadena California, USA,
September, 2010.