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Influence of dimple design on
aerodynamic drag of Golf balls
Zaheer Abbas
CAE, National University of Science & Technology,
Pakistan
Zaheer.abbas@cae.nust.edu.pk
Syed Irtiza Ali Shah2, Ali Javed3, M. Jamil4
CAE, National University of Science & Technology2,3,4
Pakistan
irtiza_shah@gatech.com2, ali.javed@cae.nust.edu.pk3,
mjkhurshid01@gmail.com4
Abstract—-Golf ball aerodynamics is significantly multifaceted as
compared to other sports balls due to the presence of small
dimples, which serve as a source of surface roughness and drag
reduction. Overall aerodynamic performance of a golf ball is
influenced by flow behavior over these dimples. Consequently,
flow phenomena over the dimples is effected by various dimple
characteristics, of which dimples geometry, size, shape, depth,
and pattern are considered significant. Dimple characteristics
continue to be an active area of research because the results
thereof, affect the assertions, sales, and the performance of
different commercially available golf balls. Dimples on the other
hand also enhance the intricacy of flow over a golf ball; and due
to this fact aerodynamics of a golf ball is yet to be fully
recognized despite of considerable amount of available literature.
Previous studies focused on overall aerodynamic performance of
the golf ball through experimental analysis and Computational
Fluid Dynamics (CFD) simulations; however, available literature
still lacks detailed analysis of pertinent dimple characteristics. In
this paper, dimple depth effect on the drag performance of a golf
ball has been examined by employing numerical simulations,
followed by validation through bench mark wind tunnel results
of F. Alam (Procedia-2011). CFD simulations in ANSYS Fluent ®
have been carried out for 5 balls with varied dimple depth over
various velocities (wind). It has been established that golf ball
drag coefficient varies considerably as the dimple geometry is
varied. The results specify a direct relationship between dimple
depth ratio and transitioning Reynolds number. As the depth
ratio is increased the transitioning Reynolds number is lowered
and this also serves to increase the drag coefficient in
transcritical regime. A positive linear relationship was also
established between coefficient of drag and relative roughness.
Keywords—Aerodynamic Efficiency; Dimple
Characteristics;Dimples Depth;Wind Tunnel;CFD; Drag
Reduction.
Nomenclature
d = Golf Ball diameter
ε = Parameter for relative roughness
k= Golf Ball dimple depth
Re= Reynolds Number
ρ = Air density
ϵ = Dissipation
I. INTRODUCTION
Origin of golf is uncertain and much deliberated. Though,
it is by and large acknowledged that modern golf developed in
northern Scotland in the middle ages. Introduction of gutta-
percha ball in nineteenth century proved to be the major
advancement in golf. Gutta-percha could fly higher and farther
if irregularities were introduced on its surface. This finding led
to initiation of various cover designs, selected entirely by
personal cognition. Golf balls with elevated surface were not
favorable, since they had a tendency to collect mud and lacked
distance presumably due to higher drag force. Ball covers with
varying depressions were employed but round dimpled balls
were accepted as the standard choice for design. A standard
golf ball has regular arrangement of either 336 (American) or
330 (British) round dimples. In golf ball aerodynamics, the
key objective is to attain maximum distance and accuracy
resulting from different forces that act of the surface of the
golf ball.
Fig. 1. Sketch of aerodynamic forces acting on a golf ball. Lift force is
generated in the direction from the side moving against to the side moving
along the direction of the flow, commonly known as Magnus effect.
The dissimilar nature of aerodynamic forces that act on a
golf ball distinguishes it from a smooth sphere. The prime
dissimilarity is the presence of small dimples with varying
shape, size, depth, pattern and configuration. These tiny
dimples are a source of roughness and drag crises along with
reduction of critical Reynolds number. Therefore, study of
dragcrises over the dimples is of paramount interest. [1] On a
larger scale, forces that act on a golf ball, are dependent upon
the behavior of flow transversing over these dimples.
Therefore, the optimum performance of a golf ball can be
attained by investigating the effect of various factors
impacting the aerodynamic flow over the golf ball. Flow
phenomenon over the dimples has remained an active research
area for past many years. Boundary layer formulation and its
analysis over the individual dimples is an intricate process.
II. LITERATURE REVIEW
It has been revealed that Golf ball dimples efficiently promote
drag crises phenomenon [2-5]. At lower critical Reynolds
number the drag drops when compared to a same sized smooth
sphere.Davies [6] was probably the pioneer of endorsing this
fact by experimentally evaluating the aerodynamic forces that
act on a golf ball in flight and concluded that dimpled surface
gives greater distance and better control as compared to golf
balls with smooth surface. Further advancements led to the
development of wind tunnel technique by Bearman and
Harvey[7], and subsequent experimental measurements of
aerodynamic forces acting on a golf ball with varying dimple
geometries over different Reynolds numbers. It was concluded
that golf ball dimples cause reduction in drag force as
compared to a smooth sphere of identical size for low
Reynolds number.Based on the Bearman and Harvey
conclusions, Choi [3, 8] comprehensively explored the drag
reduction by the presence of dimples on a golf ball. He
focused on the how- part of dimples‟ contribution towards the
overall reduction of drag at low Reynolds number in
comparison with a sphere of identical size.Choi [3, 18]
highlighted the formation oflocal separation bubbles as the
flow moves past the individual dimples through direct
measurement of stream wise velocity and its variation over
each dimple. These separation bubbles contain an induced
unstable shear layer, which aids in energizing the boundary
layer flow and, subsequently, delays the complete detachment
of flow. Furthermore, Choi et al.[3, 8] investigated the cause
and proposed that with the rise in Reynolds number the
reduced drag coefficient barely increases in supercritical
regime. This points out that the instability in the shear layer
contained within local separation bubble, is shifted to further
upstream angular locations once the Reynolds number is
increased.
Over the years various numerical techniques have been
employed to capture the effect of dimples on a sphere [8-11].
Smith [10] carried out numerical demonstration of flow
behaviors revealed by Choi [3, 8] through Direct Numerical
Simulation (DNS) technique. The numerical simulations were
effective in capturing the decline drag force of the golf ball
and results were closely related to the experimental
measurement of drag force coefficients (CD). Additionally,
numerical demonstration of Choi‟s [3, 8] idea of local
separation bubble existing over individual dimples was also
carried out and related turbulent flow structures were captured
at small-scale level.
Feroz and Harun Chowdhury [12]carried out experimental
measurement of drag forces exerted on commercially available
golf balls at various speeds inside a wind tunnel.[13]Based on
the results, it was concluded that, drag coefficient (CD) of
tested golf balls differed considerably with different dimple
geometries. And configurations. Feroz and Harun [13] further
explored the dimple characteristics and their influence on golf
ball drag. Computational Fluid Dynamics (CFD) along with
wind tunnel testswere employed for measurement of drag
forces acting on 3D printed balls at different wind speeds with
varying dimple characteristics.To capture the effect of dimples
depth on aerodynamic performance, other geometrical
characteristics (size and diameter) of the dimples were kept
constant.The results revealed that the CD of golf balls differed
considerably due to varying dimples geometry and
configurations.
Takeshi Naruo [14] investigated the effect of dimples
depth on lift coefficient of the golf ball with varying launch
speeds. For investigation purposes, wind tunnel technique was
developed by installing rotating device inside the golf ball.
Flow was explored by PIV (Particle Image Velocimetry) and
images of flow were captured by high-speed video. Based on
the results it was concluded that golf balls with shallow
dimples has a high lift for higher flow velocities (35 m/s or
more). However, after a certain limit, the coefficient of lift
will decay rapidly if dimples are made too shallow for lower
speeds (30 m/s or less) and dependency on Reynolds number
will appear. Similar effect is experienced when taking shallow
dimples and expanding the size of dimple and reducing the
quantity of dimples.
Ting [15]investigated the effect of varying dimple size,
depth, number, coverage area, and distribution pattern on the
aerodynamic performance of a golf ball. Dimple depth effects
were computed for minimum drag by keeping same quantity,
distribution pattern, diameter and optimum depth of dimples.
Subsequently, with the optimum depth, the drag was found to
be decreasing with an increase in diameter until a limit beyond
which, there was no change in the drag. The effect of dimple
depth was examined in such a manner that dimple depressions
were varied until an optimal value of minimum drag loss was
achieved. Based on the analysis, it was deduced that by
employing the optimal dimple depth, drag results will continue
to decrease with the increase in the dimple diameter, till the
time a limit is reached after which, drag coefficient will not
exhibit any change with further increase of dimple diameter.
III. METHODOLOGY
A. Geometrical Model
For numerical analysis, 5 full scale golf balls are
modelled by employing Design Modeler (DM) ANSYS ® a
well-known computational Fluid Dynamics (CFD) software.
Average diameter and dimple size of commercially available
golf ballsare 42.67 mm and 3.5 mm, respectively. Geometry
and dimple characteristics of all models are kept same.
However, dimple depth was varied from 0.6 to 1.5 mm. Golf
ball dimple parameters. A smooth ball was also modelled for
comparison. Fig.2and 3, illustrates the geometry and
dimensions of dimple parameters of the golf balls.
Fig. 2. Geometry of Tested Golf ball. Modelling of Golf ball was carried
out in ANSYS ® Design Modeler
Fig. 3. Golf ball dimple parameters (All dimensions in mm). Dimple size
and golf ball diameter is kept constant for all golf ball models. Dimple depth is
varied for each model (0.5-1.5 mm).
B. Definition of Dimple Characteristics
Dimple characteristics can be defined in many ways.
Various researchers have developed different dimensionless
parameters to quantify the effect of dimple depth on overall
aerodynamic performance of a golf ball. To study the dimple
depth effects, Choi [3, 8] employed a dimensionless parameter
„dimple depth ratio‟. The formula for the same is as shown
below,
=
Parameter „k‟ defines the dimple depthand „d‟defines the
overall golf ball diameter.
Ting [15], Chowdhury [12] and Achenbach [11]
investigated the influence of dimple geometry in terms of
surface roughness on drag performance of a golf ball. They
investigated the effect of dimples depth with respect to ball
diameter on the coefficient of drag CD. For that matter they
introduced a surface roughness parameter (ϵ), a ratio of dimple
depth (k) and dimple diameter (d). The formula for the same is
as shown below,
= =
As it can be seen that both the parameters differ only
in nomenclature. In actuality, both parameters define and
annotates the dimensionless surface roughness introduced on
the golf ball by the dimples. Present study investigates the
effect of dimple depth on overall drag performance of the golf
ball, five golf balls with same dimple size and shape are
modelled with varying dimple depth. Table-1, illustrates the
dimple parametric characteristics of model golf balls,
TABLE I. VARIOUS TEST MODELS OF GOLF BALL WITH
DIMPLE PARAMETERS
C. Numerical Methodology
Use of computational fluid dynamics (CFD) is
always challenging while simulating sports projectiles for
aerodynamic performance analysis [1, 16-20]. Aerodynamics
of these balls are very similar to classical bluff bodies. Flow
transitioning is encountered during flight coupled with large
flow separations and intricate wake structures. Wake structure
analysis of such flows is primarilydetermined by employing
computationally demanding large eddy scale (LES) simulation
Model
A
B
C
D
E
Smooth
Depth of Dimples
(mm)
0.6
0.8
1
1.2
1.5
0
Golf ball diameter
(mm)
42.67
42.67
42.67
42.67
42.67
42.67
Relative
Roughness ( )
0.014
0.018
0.023
0.028
0.035
0
methods, and even direct numerical simulation (DNS). Use of
such approaches require careful application of the models, and
substantial computational means. The alternate is by
employing unsteady Reynolds-averaged Navier Stokes
(URANS) turbulence models, which are typically known to
struggle in such flow setups. URANS as compared to LESis
however economical from computational perspective. These
models have significant use in industrial and academia sector.
Therefore, development and refinement of these models is an
ongoing process. In the present research, major focus is
towards the drag force analysis around a golf ball and dimples
contribution in overall drag reduction. For the analysis, two
equation K-epsilon (k-) turbulence model is used. K-epsilon
(k-ε) turbulence model is the most common model used in
CFD to simulate mean flow characteristics for turbulent flow
conditions. This model employs two transport Partial
Differential Equations (PDEs) for analysis of turbulence. The
original motivation for the K-epsilon model is to provide
improvement in the mixing-length model, moreover to provide
a substitute to algebraically advocating turbulent length scales
in modest to high complexity flows [21]. Reduced standard k-
ε turbulence model (Launder and Spalding, 1974 [22]) set of
equations are,
For turbulent kinetic energy „k‟
()
+ ()
=
+ 2
For dissipation „
+
=
+1
2 22
Where defines the directional velocity magnitude,
definesrate of component deformation, is the eddy
viscosity. Moreover, ,,1 2are adjustable
constants for wide range of turbulent flows.
IV. RESULTS AND DISCUSSION
A. Solver Settings
In solver settings, time discretization of second-order
accurate is accomplished by employing the second-order-
accurate backward implicit scheme and the convective terms
are discretized using second-order upwind scheme. However,
the diffusive terms are discretized by employing second-order
central differencing.
B. Validation of Results
Domain and Grid independence study is carried out for
flow parameters on the surface of the model golf ball with
relative roughness =
= 0.035. The CD profiles were
compared with the wind tunnel test results [13]. Three
domains and grid sizes are chosen for the study.
Domain 1 (-6d ≤ x ≤ 12d,-3d ≤ y ≤ 3d,-4d ≤ z ≤ 4d),
Domain 2 (-10d ≤ x ≤ 20d, -5d ≤ y ≤ 5d, -6d ≤ z ≤ 6d),
Domain 3 (-12d ≤ x ≤ 24d, -7d ≤ y ≤ 7d,-8d ≤ z ≤ 8d).
Grid 1 (2.2.million elements, 0.45 million nodes)
Grid 2 (0.9 million elements, 0.2 million nodes)
Grid 3 (0.6 million elements, 0.1 million nodes)
For domain independence study, uniform grid is created for all
domains. Golf ball face sizing was kept at 0.1 mm with 25
inflation layers, as illustrated in Fig. 4 shown below. First
inflation layer thickness is calculated based on wall y+.
Complete domain was divided into 0.9 million hybrid mesh
containing tetrahedral and quadrilateral elements. The
variation of CD profile for the test case was compared with
wind tunnel results of bench mark solution [13]. The variation
of CD profile
for domain 2 and grid 2 matched well with the bench mark
solutions, same is illustrated in Fig. 5 shown below. Hence,
domain 2 and grid 2 was selected for the study, free from
domain dependency and grid dependency errors. Moreover,
the CD profile matching validates the numerical solution setup.
Fig. 4. Closer view of mesh used around Golf ball. Tetrahedral and quadrilateral hybrid elements are employedfor mesh generation. 25 inflation layers created
with first layer thickness of 0.59 mm calculated based on the wall y+. Inflation layers are created to capture the near surface flow variables inside the boundary layer.
Fig. 5. Domain Independence Study for three candidate domains versus benchmark solution. As depicted in the figure, Domain-2 CD profile matches well with
the wind tunnel results of bench mark solution. [13]
C. Variation of Drag coefficient for all Models
To investigate the effect on the drag coefficient
(CD)atdifferent Reynolds numbers (Re), numerical simulation
for each model and smooth sphereis carried out at various
velocities. Flow transitioning (from laminar to turbulent) is
observed in all test cases. It is observed that dimples trip the
boundary layer earlier, inducing shear layer instabilities, flow
separation for Model-E is illustrated in Fig. 6 shown
below.Flow transitioning varied for all the test cases due to
varied surface roughness in the form of different dimple
depths for each model. Moreover, CDvs. Re profile for each
case is also distinct, same is illustrated in Fig. 7 shown below.
It is concluded that transitioning effects varies distinctly bases
on the surface profile of test cases for a range of Re (2 x 104 -
4 x 104). It is found that golf balls with higher value of relative
surface roughness (more depth) initiate flow separation earlier
than golf balls with lower values of relative surface roughness.
Conversely, flow transition was not observed in the smooth
surfaced spherefor the testedReynolds numbers.
0
0.2
0.4
0.6
0.8
1
1.2
0246810
Coefficient of Drag (CD)
Reynolds Number, Re x 104
Grid Independence & Results Validation
Grid-1
Grid-2 (Optimized Grid)
Grid-3
Wind Tunnel Results
Linear (Wind Tunnel
Results)
Fig. 6. Flow separation of Model-E used in the study at Re = 3 x 104. Shear layer separation and curling of flow due to presence of dimples is evident in the
figure. These shear layer instabilities result into formation and detachment of Karman vortices. Two counter rotating Karman vortices are visible in the wake. Vortex
shedding considerably reduce the pressure drag and overall CD of the golf ball.
Fig. 7. Drag Coefficient of Model golf balls at various velocities (Reynolds number), calculated for all six test cases.
D. Variation of CDmin with respect to Relative Roughness
It is observed that minimum coefficient of drag
(CDmin) value at critical Reynolds number (Recrit) is lower for
models with less relative roughness. Model 1 with relative
roughness of 0.014 has CDmin value of around 0.17. However,
model-5 with relative roughness of 0.035 has CDmin value of
around 0.22.The minimum coefficient of drag increases by
increasing the relative roughness. It can be concluded that Recri
and CDminare related to relative roughness. The plot of relative
roughness against Recri and CDmin showed a nearly positive
linear correlation. Same is depicted in Fig. 8 and Fig. 9 shown
below.
Fig. 8. Minimum drag coefficient CDminfound as a function of Relative roughness (ε). The plot indicates a nearly positive linear relationship among CDmin
values of all test cases. Model „E‟ with the relative roughness of 0.035 has the highest value of CDmin of around 0.26; whereas, Model „A‟ with relative roughness
of 0.014 exhibited the lowest CDmin value of around 0.14. Therefore, a positive linear correlation was observed between the CDmin and relative roughness in the
form of dimple depth.
V. CONCLUSIONS
Overall aerodynamic performance of a golf ball is
influenced by flow behavior over tiny dimples. Consequently,
flow phenomena over the dimples is effected by various
dimple characteristics, of which dimples depth, size, shape,
and pattern are considered significant. Dimple characteristics
continue to be an active area of research because the results
thereof, affect the assertions, sales, and the performance of
different commercially available golf balls. Dimple depth
being the most significant parameter for aerodynamic
performance of a golf ball is investigated in this study. The
motivation factor was limited availability of experimental and
published data on the influence of depth parameter and its
overall effect on aerodynamic drag of a golf ball.Five golf ball
designs with varying dimple depth (1mm-1.5mm) were
modelled and analyzed in the study. To isolate the dimple
depth effects, all other dimple characteristics were kept same
based on average values of commercially available golf balls.
It is observed that the dimples trip the boundary layer and help
in early transitioning of flow from laminar to turbulent. This
creates a local separation bubble over individual dimples and
aids in reattachment of energized boundary layer significantly
reduces the pressure drag. However, this phenomenon differs
according to the dimple characteristics. Based on the results, it
is concluded that balls with higher value of relative roughness
in the form of dimple depth can lower the critical Reynolds
number. Moreover, at the same time the CD increases in the
transcritical regime. The variation of drag coefficient with
respect to relative roughness exhibits a positive linear
correlation.
VI. FUTURE WORK
This work may be extended to investigate impact of all
other dimple characteristics on overall aerodynamics of a golf
ball. That would result in an optimized golf ball for optimal
performance.
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