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Computational Fluid Dynamics of
Compressible Flows
MECH5304
21 April 2010
Dr. Edgar A. Matida Department of Mechanical &
Aerospace Engineering
Term project report
Student: Aymen Sakka
Carleton ID: 100828756
Page 2
Abstract
The flying distance of a golf ball is influenced not only by its material, but also by the aerodynamics of
the dimples on its surface. By using ANSYS CFX, the aerodynamics characteristics and to the drag forces
over the golf balls were studied. The drag coefficient variation with different Re numbers was studied.
The validation is done by comparison to a flow over a smooth sphere. The results qualitatively agreed
with the literature.
Introduction
Although poorly documented, golf is believed to have originated in the early 1400s [1]. It was first played
as a very casual game for which no standard rules existed. A wooden ball was used in conjunction with
wooden clubs prior to 1618[1], when the “featherie" (a ball made of stitched leather and tightly packed
with feathers) was introduced. The featherie was favored for its more forgiving feel on the hands of
players when it was struck and was used until 1848 when the invention of the “Gutta” surpassed the
“feathery” in both durability and cost. The “Gutta” was made of gutta-percha packing material which was
not brittle and became soft and moldable at 100°C.
The Gutta's pliability made it necessary to roll the ball on a “smoothing board” in order to maintain its
shape and keep it free of imperfections which were created during normal play of the game. The smooth
Gutta was used for only a few years before players began to realize that balls that had not been well
maintained and had many nicks and scratches had a much more favourable flight. Thus began the practice
of hammering the Gutta with a sharp-edged hammer in a regular pattern to increase the consistency of the
ball's play.
In 1898 the first “Balata” ball was created by wrapping rubber thread around a solid rubber core which
was then covered by a solid layer of rubber that later became known as the “ball cover”. The Balata was
the first sign of a modern age of golf technology for it allowed molds to be used to create consistent cover
patterns. In 1908 makers discovered the superiority of a regular “dimple” pattern over the haphazard grid
pattern favoured by players at the time. Dimples are small indentations on the exterior of the golf ball.
They are typically round in shape and vary in diameter from 2-5mm in diameter and are about .2mm
deep. Modern golf balls pack anywhere from 300-450 dimples of varying size arranged in a regular
pattern on the outside of every ball [3]. Dimples have been one of the most influential developments in
golf ball design because they alter the dynamics of the balls flight in such a way that gives golfers a
significant amount of control over the height and shape of their shots.
b 0.725 mm
c 3.5868 mm
k 0.7 mm
Figure 1: geometry of the golf ball
Figure 2: geometry of the golf ball dimples
Page 3
Figures. 1 and 2 show the geometry and boundary of a typical golf ball. The golf ball diameter is 42.6 mm
while the dimples diameter is 3.58681 mm. The golf ball has 389 dimples (See Figure 1).The domain size
is 600 mm × 400 mm × 400 mm in the x, y, and z-directions (See appendix-1- ). The fluid is air at
25°C and the relative pressure is 1 atm. The inlet velocity is changed to vary the Reynolds number. Air properties at 25°C and P= 1 atm:
𝜌 = 1.18 𝑘𝑔/𝑚3
𝜇 = 1.84 × 10−5 𝑘𝑔 𝑚 𝑠
𝜗 = 1.595 × 10−5𝑚2
𝑠
The root mean square of errors (RMS) was set to 10E-04, it is defined as: RMSerr = erri2n
i=1
Methods
Meshing
The mesh details for the smooth sphere and the golf ball are listed in appendix-2-. Both of the meshes use
Delaunay surface meshing and advancing front and inflation meshing strategy. Advancing front method is
used for the volume mesh as well.
Figure 3: Surface and volume mesh of a smooth
sphere
Figure 4: Volume mesh for the golf ball
Page 4
Under solver control, the advection scheme and the turbulence numerics are set to “High resolution”. The
automatic scaling was chosen for time and conservative for space. As for convergence criteria, the RMS
residuals were set to 10-5
for the sphere simulations and to 10-4
for the golf ball simulations.
The inlet boundary is defined by normal velocity whereas a zero gauge pressure condition was set at the
outlet. The domain walls are considered as free slip walls. The smooth sphere and the golf ball are
considered to have a smooth wall with a no slip condition to take into account the viscosity of the fluid,
which is air at 25 °C.
ANSYS CFX theory
Two Equation Turbulence Models
Two-equation turbulence models are very widely used, as they offer a good compromise between
numerical effort and computational accuracy. Two-equation models are much more sophisticated than the
zero equation models. Both the velocity and length scale are solved using separate transport equations
(hence the term „two equation').
The k-ε and k-ω two-equation models use the gradient diffusion hypothesis to relate the Reynolds stresses
to the mean velocity gradients and the turbulent viscosity. The turbulent viscosity is modeled as the
product of a turbulent velocity and turbulent length scale. In two-equation models, the turbulence velocity
scale is computed from the turbulent kinetic energy, which is provided from the solution of its transport
equation. The turbulent length scale is estimated from two properties of the turbulence field, usually the
turbulent kinetic energy and its dissipation rate. The dissipation rate of the turbulent kinetic energy is
provided from the solution of its transport equation.
The k-epsilon and SST Models in ANSYS CFX
The present numerical simulation of the airflow distribution around a golf ball requires the use of various
theoretical mathematical models based on fluid dynamics principles. The k-ω based Shear-Stress
Transport (SST) model was designed to give highly accurate predictions of the onset and the amount of
flow separation under adverse pressure gradients by the inclusion of transport effects into the formulation
of the eddy-viscosity. The SST model has a slight additional cost over other two equation models since a
wall scale equation is also solved.
The present model in CFX consists of the continuity equation, the momentum equation, and the energy
equation. k is the turbulence kinetic energy and is defined as the variance of the fluctuations in velocity. It
has dimensions of (L2 T
-2); for example, m
2/s
2. ε is the turbulence eddy dissipation (the rate at which the
Figure 5: Surface mesh of the golf ball
Page 5
velocity fluctuations dissipate), and has dimensions of k per unit time (L2 T
-3); for example, m
2/s
3.These
equations employed in the present numerical model are presented below.
Continuity equation:
𝜕𝜌
𝜕𝑡+ ∇. 𝜌𝑈 = 0
Momentum equation
𝜕(𝜌𝑈 )
𝜕𝑡+ ∇. 𝜌𝑈 𝑈 = −∇𝑃 + ∇. 𝜇𝑣∇𝑈 + 𝜌𝐹
Where 𝜇𝑣 = 𝜇 + 𝜇𝑡
In early research, turbulent model was applied in high Reynolds number incompressible flows. But it was
later experimentally proven that the air flow near the wall is associated with low Reynolds numbers.
Therefore, the development of turbulence model for low Reynolds numbers has been an intensive focus
for research activities. One remedy to this scenario is to introduce a wall function so that the low
Reynolds number air flow near the wall and the high Reynolds number flow far away from the wall can
be simulated at the same time. In this paper, the turbulent model used is the amended standard κ-ε model
because it has been proven to give good predictions for complex flows.
The k-ε model is given as
𝜕(𝜌𝑘)
𝜕𝑡+ ∇. 𝜌𝑘𝑈 = ∇.
𝜇𝑡
𝜎𝑘∇𝑘 + 𝐺 + 𝐵 − 𝜌𝜀
𝜕(𝜌𝜀)
𝜕𝑡+ ∇. 𝜌𝜀𝑈 = ∇.
𝜇𝑡
𝜎𝜀∇𝜀 + 𝐺1
𝜀
𝑘 𝐺 + 𝐵 (1 + 𝐶3𝑅𝑓) − 𝐶2𝜀𝜌
𝜀2
𝑘
Where
𝐺 = 2𝜇𝑡𝐸𝑖𝑗 . 𝐸𝑖𝑗
𝐵 = 𝛽𝑔𝑖
𝜇
𝜎𝑇
𝜕𝜌
𝜕𝑇
𝛽 = − 1
𝜌
𝜕𝜌
𝜕𝑇
𝜇𝑡 = 𝜌𝐶𝜇
𝑘2
𝜀
𝑅𝑓 =−𝐺1
2(𝐵+𝐺) , 𝐺1 = 2𝐵
Calculation of the drag coefficient [4]
The drag equation is a practical formula used to calculate the force of drag experienced by an object
moving through a fluid. The force on a moving object due to a fluid due to Lord Rayleigh is
Page 6
𝐹 =1
2 𝜌 𝑉2 𝐶𝐷 𝐴
F is the force of drag.
ρ is the density of the fluid.
V is the velocity of the object relative to the fluid.
A is the reference area.
𝐶𝐷 is the drag coefficient (a dimensionless constant).
The reference area A is the area of the projection of the object on a plane perpendicular to the direction of
motion (i.e. cross-sectional area).
Drag is the net force on the body in the direction of the flow. In the above diagram, the drag is the sum of
the forces on the wall in the horizontal direction, i.e. the sum of the pressure force and the viscous force
components in the x direction. It is apparent from this that viscous force is not a pure shear force since it
also has a small component in the normal direction, arising in part from a normal component in the
laminar flow shear stress.
The pressure and viscous moments are related to the pressure and viscous forces calculated at the Wall.
The pressure moment is the vector product of the pressure force vector and the position vector r. The
viscous moment is the vector product of the viscous force vector and the position vector r. i.e. where and
are the pressure and viscous moments respectively. These are summed over all the surface elements in the
Wall.
It is important to note that forces do not include reference pressure effects. The pressure force is
calculated as the integral of the relative pressure over the wall area and not as the integral of the sum of
the reference and relative pressures.
Results and discussion
Flow over a smooth sphere
Early aerodynamics researchers were quite puzzled by the theoretical result stating that there is no drag on
a sphere because it contradicted experimental measurements indicating that a sphere does generate drag.
The conflict between theory and experiment was one of the great mysteries of the late 19th century that
became known as d'Alembert's Paradox, named for famous French mathematician and physicist Jean le
Rond d'Alembert (1717-1783) who first discovered the discrepancy[5].
The reason d'Alembert's ideal theory failed to explain the true aerodynamic behavior of a sphere is that he
ignored the influence of friction in his calculations. The actual flowfield around a sphere looks much
different than his theory predicts because friction causes a phenomenon known as flow separation. We
The CFX-Solver calculates the pressure and viscous
components of forces on all boundaries specified as
Walls. The drag force on any wall can be calculated
from these values as follows:
Lift is the net force on the body in the direction
perpendicular to the direction of flow. In the above
diagram, the lift is the sum of the forces on the wall in
the vertical direction, i.e. the sum of the pressure
force and the viscous force components in the y
direction.
Figure 6: Drag forces on a body
Page 7
can better understand this effect by studying the following diagram of the actual flow around a smooth
sphere. Here we see that the flowfield around the sphere is no longer symmetrical. Whereas the flow
around the ideal sphere continued to follow the surface along the entire rear face, the actual flow no
longer does so. When the airflow follows along the surface, we say that the flow is attached. The point at
which the flow breaks away from the surface is called the separation point, and the flow downstream of
this point is referred to as separated. The region of separated flow is dominated by unsteady, recirculating
vortices that create a wake[5].
Although the values of critical Reynolds numbers are not exactly the same, the computational prediction
is acceptable as far as the overall trend is concerned. The drag coefficient plot as a function of Reynolds
numbers displays a slightly higher value for Re= 105 than the chart in appendix-3-.
0.45
0.46
0.47
0.48
0.49
0.5
0.51
0.52
0.53
0.54
0.55
0.56
0 50000 100000 150000 200000 250000 300000
Dra
g co
eff
icie
nt
Reynolds number
Figure 8: Drag coefficient variation as a function of increasing Reynolds number
Cd=f(Re)
Since the laminar boundary layer
around the smooth sphere separates so
rapidly, it creates a very large wake
over the entire rear face. Re-
circulating vortices in the wake are
pointing out from the main axis of the
flow. This large wake maximizes the
region of low pressure and, therefore,
results in the maximum difference in
pressure between the front and rear
faces (Figure 7). This difference
creates a large drag like that seen
below the transition Reynolds number.
Figure 7: Pressure streamlines around
the smooth sphere for V=1 m/s
Page 8
- The transition to a turbulent boundary layer, on the other hand, adds energy to the flow allowing it
to remain attached to the surface of the sphere. Since separation is delayed, the resulting wake is
much narrower. This thin wake reduces the low-pressure region on the rear face and reduces the
difference in pressure between the front and back of the sphere. This smaller difference in
pressure creates a smaller drag force comparable to that seen above the transition Reynolds
number.
- These results tell us that causing a turbulent boundary layer to form on the front surface
significantly reduces the sphere's drag. For a given sphere diameter, a designer has only two
options encourage this transition, either increase the speed of the flow over the sphere to increase
the Reynolds number beyond transition or make the surface rough in order to create turbulence.
The latter case is often referred to as "tripping" the boundary layer.
Figure 9: Velocity streamlines around
the smooth sphere for V=100 m/s
Figure 10: Eddy viscosity contour plot for
the flow over the sphere for V=100 m/s
Figure 10: Pressure vortex around the
smooth sphere for V=20 m/s
Page 9
Flow over a golf ball
In the case of a golf ball, increasing the speed is not an option since a golfer can only swing the club so
fast, and this velocity is insufficient to exceed the transition Reynolds number. That leaves tripping the
boundary layer as the only realistic alternative to reducing the drag on a golf ball. The purpose of the
dimples is to do just that--to create a rough surface that promotes an early transition to a turbulent
boundary layer. This turbulence helps the flow remain attached to the surface of the ball and reduces the
size of the separated wake so as to reduce the drag it generates in flight. When the drag is reduced, the
ball flies farther.
Figure 11: Velocity 3D streamlines around the golf ball for V=100 m/s
Air flows smoothly over the contours
of the front side and eventually
separates from the ball toward the
back side. The flying golf ball also
leaves behind a turbulent wake region
where the air flow is fluctuating or
agitated (Figure 11), resulting in lower
pressure behind it. The size of the
wake affects the amount of drag on
the object. Dimples on a golf ball
create a thin turbulent boundary layer
of air that clings to the ball's surface.
This allows the smoothly flowing air
to follow the ball's surface a little
farther around the back side of the
ball, thereby decreasing the size of the
wake. See figure 12. Figure 12: Pressure 3D streamlines around the golf
ball for V=100 m/s
Page 10
Moving in to about 45 degrees from the leading surface of the golf ball, we can see a number of trains of
vortices are developed (Figure 13). These lines of whirlpools follow the scallops of the dimples in the
direction of the airstream. In the behaviour of the air shown just around a pair of dimples (Figure 14), we
see the airflow evolve, from front to back within a single dimple, as the air detaches and shears away,
varying with the stream‟s direction. We see that the cumulative effect of the vortices cause air to come
down onto the ball to delaying energy-wasting separation. Hence, Golf balls with dimples turn out to be
more slippery than smooth spheres.
Figure 15: Velocity vectors in the vortex core
region around the golf ball for V=100m/s
Figure 13: Velocity contour plot in the vortex
core region around the golf ball for V=100 m/s
Figure 14: Detaching vertices from dimples in
the vortex core region for V=100 m/s
As figure 15 shows, the velocity
vectors field is pointing to the center
of the wake. This convergence pattern
of the vertices tends to reduce the size
of the wake. In fact, the vectors are
pointing against the flow main stream
which allows the wake to have a
counter effect the drag force on the
ball.
Page 11
Turbulence around the golf ball has a beneficial effect on reducing the wake of an object. If the surface
air in the boundary layer becomes turbulent, the higher kinetic energy in the turbulent region will help the
air stick to the surface longer before separating (Figure 16). The result is lower form drag. Figure 17
illustrates the high level of turbulence by a maximum eddy viscosity values just behind and very close to
the golf ball.
Drag coefficient
Figure 16: Turbulence kinetic energy contour
plot around the golf ball for V=100m/s
Figure 17: Eddy viscosity contour plot around
the golf ball for V=100m/s
It is the difference between the high and
low pressure values that account for drag
forces a body experiences. In the case of
separated flow around a sphere the drag
force and hence drag coefficient is
dominated by form drag which depends on
the separation point on the sphere. Hence
anything that effects the location of the
separation point has a large effect on the
drag coefficient. For example, the dimples
on a golf ball cause the laminar boundary
layer to become turbulent sooner and this
moves the separation point rearward
decreasing the from drag and the drag
coefficient as shown in figure.
Figure 18: Pressure contour plot around
the golf ball for V=100 m/s
Page 12
Validation of the simulations
For validation, this study used a 3-D sphere. The turbulence model being validated is the shear stress
transport model. Drag coefficient is the lowest at the critical Reynolds number of 4×104. After that, drag
coefficient will raise slowly with Reynolds number. The drag coefficient of the sphere starts to drop off at
a Reynolds number of 8×104, but stays fairly constant afterwards. This corresponds to the transition of
air flow from laminar to turbulent. Figure 19 below shows the comparison of drag coefficients at different
Reynolds numbers for the golf ball against the smooth ball.
A golf ball usually flies at a Reynolds number more than 105, which is near the critical Reynolds number.
Figure 19 shows that for Reynolds bigger than 105, the drag coefficient for the golf ball starts to decrease
with higher intensity than the smooth sphere. These results qualitatively agree well with each other.
Although the values of critical Reynolds number are not exactly the same, the computational prediction is
acceptable as far as the overall trend is concerned.
The dimples, paradoxically, do increase drag at low Reynolds numbers. But they also increase "Magnus
lift", that peculiar lifting force experienced by rotating bodies travelling through a medium. Magnus lift
is present because a driven golf ball has backspin[6].
Domain convergence analysis
An attempt has been made to simulate different flows around the golf ball with different Reynolds
numbers with a domain 1.5 bigger in each direction. These simulations (that came up with a fatal error at
CFX-post except three cases) have an RMS error of 0.017. Through this small error, we cannot judge the
error of the other flows. Still, it is a good indication of the accuracy of the chosen domain and confirming
this choice stated in the literature [7].
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
1000 10000 100000 1000000
Dra
g co
eff
icie
nt
Reynolds number
Figure 19: Drag coefficient variation with Reynolds number
Smooth sphere
Golf Ball
Page 13
Conclusions
- Drag on a golf ball comes mainly from air-pressure forces. This drag arises when the pressure in
front of the ball is significantly higher than that behind the ball. The only practical way of
reducing this differential is to design the ball so that the main stream of air flowing by it is as
close to the surface as possible. This situation is achieved by a golf ball's dimples, which augment
the turbulence very close to the surface, bringing the high-speed airstream closer and increasing
the pressure behind the ball. The effect is plotted in figure 19, which shows that for Reynolds
numbers achievable by hitting the ball with a club, the coefficient of drag becomes lower for the
dimpled ball.
- The critical Reynolds number (Recr) holds the explanation for the fact that golf balls have
dimples. Recr is the Reynolds number at which the flow transitions from a laminar to a turbulent
state. For a smooth sphere, Recr is much larger than the average Reynolds number experienced by
a gold ball. The dimpled ball has a lower Recr and the drag is fairly constant for Reynolds
numbers greater than Recr. Therefore, the dimples cause Recr to decrease, which implies that the
flow becomes turbulent at a lower velocity than on a smooth sphere. This causes the flow to
remain attached longer on a dimpled golf ball, which implies a reduction in drag. As the speed of
the dimpled golf ball is increased, the drag decreases. This is a good property in a sport like golf
where the main goal is to maintain the ball in this post-critical regime throughout its flight.
- Dimpled surface causes air to “grip” the ball for a longer period of time before passing, creating
turbulence and a thickened boundary layer. A smoother surface will allow the air to flow easier
over the ball creating what is called laminar flow. Unfortunately, laminar flow, while initially
having less drag, is also prone to separation, which produces an increased drag.
- On the golf ball, the pressure drag is much larger than the skin friction, so adding dimples is
beneficial. There was a lot of pressure drag to be reduced so the increase in skin friction is an
acceptable trade off.
- Computational Fluid Dynamics can be a powerful tool to investigate effects of dimple geometry
on the flow field around a golf ball and enable more efficient design process of dimple geometry
for less drag and longer flight distances.
Page 14
References
[1] Materials in sports equipment, volume 1, Mike Jenkins, 2003 ,Woodhead Publishing Ltd and CRC
Press LLC.
[2] "Flying Characteristics and Flow Pattern of a Sphere with Dimples", K. Aoki, A. Ohike, K.
Yamaguchi and Y. Nakayama, Journal Of Visualization, vol. 6, no. 1, pp. 67-76, 2003.
[3] http://www.aerospaceweb.org, 18 April 2010.
[4] ANSYS CFX 12.0 help, April 2010.
[5] Applied and Computational Fluid Mechanics, Scott Post, 2011 copyright (c) by Jones and Bartlett
publishers, LLC.
[6] A statistical study on reduction of drag force for golf balls, Takeyoshi Kimura and Mitsuru
Sumiyama, Memoirs of Fukui University of Technology, Vol.34, Part 1, 2004.
[7] Effects of golf ball dimple configuration on aerodynamics, trajectory, and acoustics, Chang-Hsien Tai
, Chih-Yeh Chao, Jik-Chang Leong, Qing-Shan Hong, Department of Mechanical engineering, National
Ping-Tung University of Science and Technology.
Page 1
Appendices
Appendix-1-: Geometry of the domain
Problem domain
Golf ball sketch
Page 2
Golf ball surface and volume mesh
Page 3
Appendix-2- Mesh Inputs
Smooth sphere
Volume mesh for the smooth sphere
Face spacing
Option Angular resolution
Angular resolution 18 degrees
Minimum edge length (mm) 1.5
Maximum edge length (mm) 30
Radius of influence (mm) 0
Expansion factor 1.2
Sizing
Used advanced sizing On: proximity
Relevance center Coarse
Smoothing Medium
Transition Slow
Span angle center coarse
Proximity accuracy 0.5
Min size 9e-003 m
Max face size 3e-002 m
Max tet size 3e-002 m
Growth rate 1.5
Minimum edge length 9.3934e-003 m
Inflation
Inflation option Smooth transition
Transition ratio 0.77
Maximum layers 5
Growth rate 1.2
Inflation algorithm Pre
Total thickness 7 mm
Mesh statistics
Total number of nodes 59340
Total number of tedrahedral 307214
Total number of prisms 980
Total number of elements 308194
Page 4
Golf ball meshing
Mesh spacing
Sphere spacing
Option Angular resolution
Angular resolution [Degrees] 18
Minimum edge length [mm] 0.5
Maximum edge length [mm] 3
Radius of influence [mm] 0
Expansion factor 1.2
Dimple spacing
Angular resolution [Degrees] 18
Minimum edge length [mm] 0.5
Maximum edge length [mm] 3
Radius of influence [mm] 0
Expansion factor 1.2
Edge spacing
Angular resolution [Degrees] 18
Minimum edge length [mm] 0.5
Maximum edge length [mm] 1
Radius of influence [mm] 0
Expansion factor 1.2
Inflation
Inflation option Smooth transition
Transition ratio 0.77
Maximum layers 5
Growth rate 1.2
Inflation algorithm Pre
Sizing
Used advanced sizing On: Curvature
Relevance center Coarse
Smoothing Medium
Transition Slow
Span angle center Fine
Curvature Default (18.0)
Min size Default(4.1102e-004 m)
Max face size Default(4.1102e-004 m)
Max tet size Default(8.2204e-004 m)
Growth rate 1.2
Minimum edge length 9.3934e-003 m
Mesh statistics
Total number of nodes 531932
Total number of tedrahedral 1988165
Total number of pyramids 22456
Total number of prisms 317746
Totalk number of elements 2328367
Page 5
Appendix-3- Drag coefficient variation with Reynolds numbers for different
blunt bodies
Page 6
Appendix-4-Simulations numerical results
V 100 90 80 70 60 50 40 30 20 10 7 5 2 1
Re 3E+05 2E+05 2E+05 2E+05 2E+05 1E+05 1E+05 80125 53417 26708 18696 13354 5342 2671
Sphere F
3.892 3.157 2.47 1.917 1.413 0.985 0.634 0.359 0.157 0.04 0.02 0.01 0.002 5E-04
Cd 0.463 0.463 0.459 0.465 0.467 0.468 0.471 0.475 0.465 0.472 0.476 0.483 0.502 0.548
Golf ball
F 3.818 3.09 2.461 - 1.403 0.984 0.639 0.368 0.157 0.041 0.021 0.011 0.002
6E-04
Cd 0.454 0.454 0.457 - 0.463 0.468 0.475 0.486 0.466 0.489 0.506 0.525 0.605 0.724
Simulation results of the drag coefficient over a smooth sphere and dimpled golf ball
V 100 90 80 70 60 50 40 30 20 10 7 5 2 1
Re 3E+05 2E+05 2E+05 2E+05 2E+05 1E+05 1E+05 80125 53417 26708 18696 13354 5342 2671
Golf ball
F 4.021 - - - - 1.029 - - - - - - -
6E-04
Cd 0.462 - - - - 0.48 - - - - - - - 0.732
Simulation results of the drag coefficient over a dimpled golf ball with a domain expansion factor of 1.5
( - ): Fatal error in CFX-post
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