12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

AN EFFICIENT DRAG REDUCTION OF BLUFF BODIES BYTESSELLATION

N. Beratlis, K. Squires
School for Engineering of Matter, Transport and Energy
Arizona State University
Tempe, AZ
nberatlis@asu.edu
squires@asu.edu

E. Balaras
Department of Mechanical and Aerospace Engineering
The George Washington University
Washington, DC
balaras@gwu.edu

ABSTRACT

rapidly and hits a minimum, which is an order of magnitude
lower (CD ∼ 0.08). With further increase in the Reynolds number the flow enters into the post-critical regime characterizedby turbulent boundary layers on the surface of the sphere. In
this regime the drag coefficient rises slowly with increasing
Reynolds number.

In this work a new method of passive drag reduction on
bluff bodies by tessellation is presented. Wind-tunnel measurements on tessellated spheres reveal that the variation of thedrag coefficient is similar to dimpled spheres, manifested by asudden decrease in the drag coefficient at a critical Reynolds
number followed by a nearly constant drag value in the postcritical regime. However, tessellated spheres can achieve afurther 10%-15% drag reduction compared to dimples without
shifting the critical Reynolds number. To further investigate
the underlying physics leading to this reduction we also conducted Direct Numerical Simulations of both tessellated and5dimpled spheres at Re = 1.50 × 10 . The predicted values ofdrag coefficient agree very well with the experiments and confirm the drag reduction. Analysis of the flow reveals that thetessellated panels introduce a smaller pressure ”penalty” compared to dimples at the front part of the body. In addition,transition to turbulence occurs later and near the top of the
body. As a result the boundary layer grows thinner and global◦
separation is delayed by approximately 10 .

In dimpled spheres the drag crisis happens at a much4
lower critical Reynolds number (Re < 7 × 10 in Fig. 1). The
critical value of the Reynolds number, as well as the attained
minimum drag coefficient in the post-critical regime dependon the dimple geometry and arrangement. In general as thetotal dimple volume is increased the drag crisis is accelerated
and the drag coefficient in the post-critical regime increases.
Note however that the drag coefficient for dimpled spheresin the post-critical regime is considerably higher than that ofa smooth sphere. To investigate the origin of this behaviour
Beratlis et al. (2019) carried out direct numerical simulations(DNS) of the flow over a dimpled sphere in the post-criticalregime. By comparing the computed pressure and skin friction coefficients to that of a smooth sphere they showed thati) dimples decrease the overall skin friction drag, which rectifies the common assumption that dimples typically increase it;
and ii) dimples, despite their small size, incur a local pressure
penalty, which is significant and accounts for the most part ofthe difference in drag between a smooth and a dimpled spherein the post-critical regime. The above study as well as earlier
experiments in the literature point to the limitations of further
reducing the total drag on a dimpled sphere without lowering
the critical Reynolds number.

INTRODUCTION
Passive roughness elements such as trip wires, dimples,
protrusions of different shapes, etc. are effective in reducing
the drag force on bluff (i.e. sphere, cylinder) as well as streamlined (i.e. airfoils) bodies. Among the different types of elements dimples are very efficient in maintaining the low dragconfiguration for a large range of Reynolds numbers. An example is shown in Fig. 1, where the variation of the drag coef2
ficient, CD = 2FD /ρU∞ A, as a function of Reynolds number,
Re = U∞ D/ν, (FD is the drag force, U∞ is the freestream velocity, D the diameter, A the projected area, and ρ, ν the density and kinematic viscosity of the fluid respectively) is plotted for smooth and dimpled spheres. For the former case thedrag coefficient remains constant (CD ∼ 0.5) until the Reynolds
5
number approaches a critical value (Re ∼ 3 × 10 ). At thispoint, which is usually referred to as drag crisis, CD decreases

In this work we present a new topological configurationthat can achieve a drag reduction of up to 15% compared toa dimpled sphere without shifting the drag crisis. The modification is based on the tessellation of the sphere resulting in
spherical polyhedral containing mostly hexagonal and some
pentagonal flat panels. An example of such a sphere is shownin Fig. 1. The drag on the tessellated spheres is measured bywind tunnel experiments that show that the low drag configuration is maintained through the post-critical regime. To under-

1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
stand the flow physics a DNS of one spherical polyhedral and
one dimpled sphere are carried out in the post-critical regime.
Analysis showed that similar to dimples the sharp change inangle between the flat panels cause local flow separation leading to the formation of a detached shear layer that becomes unstable resulting in the generation of turbulence, which causes
transport of high speed fluid closer to the wall.

Table 1.

Summary of the computations

Name

V /Vs

A f /A fs

CD (DNS)

Dimpled

0.989

0.990

0.188

Poly192

0.979

0.986

0.155

Poly162

0.973

0.982

0.178

METHODOLOGIESBoth wind tunnel experiments and direct numerical simulations have been carried out. The experiments have been
carried out in an open return wind tunnel at the George Washington University with a test section of 30cm × 25cm × 70cm
in the spanwise, vertical and streamwise direction respectively
running at speeds between 18-63 m/s. The drag force wasmeasured using a single component piezoelectric force sensor mounted on a sting on one end and attached to the back of
the spheres on the other end as illustrated in Fig. 2. The sensor
had a diameter of 0.6cm and width of 0.2cm. The sting itself
was attached to a rigid test stand located approximately 20cm
behind the spheres. A very thin piano wire 0.05cm in diameter was wrapped around the sting and securely attached to thefloor of the test section to reduce the vibrations on the spheres.
The models were 3D printed on the Projet 3500 printer and had
a diameter of 6.85cm resulting in a blockage ratio of less than
5%. For validation the drag coefficient of a smooth sphere45in the sub-critical regime (8.0 × 10 < Re < 2.2 × 10 ) was
measured, and was in agreement to prior experiments in theliterature.
For the numerical simulations the Navier-Stokes equations for viscous incompressible flow are solved on a structured grid in cylindrical coordinates. In the following and unless otherwise stated, the letters r, φ , and z denote the radial,azimuthal and axial coordinates respectively, while θ denotes◦
the polar angle, going from θ = 0 at the stagnation point at the◦front of the sphere to θ = 180 at wake side. The governing
equations are advanced in time using a semi-implicit projecrd
tion method, treating the explicit part with a 3 order Rungend
Kutta scheme, and the implicit part with a 2 order CrankNicholson scheme. All spatial derivatives are discretized using second-order central-differences on a staggered grid. To
overcome the severe time step limitation imposed by the nature of the cylindrical coordinate grid, the viscous and convective terms are treated implicitly in the azimuthal direction, near
the centerline, and in the radial direction near the top of thesphere. The pressure Poisson equation, which enables the projection of the predicted velocity into a divergence-free field,
is solved using a direct solver based on a divide-and-concur
strategy. All geometrical configurations considered here arerepresented by a Lagrangian grid consisting of triangular elements. The requirement for the Eulerian grid to conform to thebody is relaxed, and the non-slip boundary conditions are imposed using the immersed-boundary formulation proposed byndYang & Balaras (2006). The resulting formulation is 2 -orderaccurate both in space and time. An extensive validation on
practical flow problems over a wide range of Reynolds numbers can be found in Balaras et al. (2015); Posa et al. (2011,2015); Posa & Balaras (2016); Rahromostaqim et al. (2016);
Pal et al. (2017); Posa & Balaras (2018). The code is parallelized using a domain decomposition approach in the streamwise direction, where all communication between processors
is handled utilizing Message Passing Interface (MPI) librarycalls.
The computational domain extends 10D upstream and

30D downstream of the sphere (the center of the sphere is located at r/D = 0, z/D = 0, where r and z are the radial and axial
coordinates respectively). The computational grid consists of1100 × 3002 × 3002 points in the radial, azimuthal and axial
directions respectively. The grid resolution is very similar tothe one used in DNS of the flow over a dimpled sphere by Beratlis et al. (2019), which is sufficient to resolve the dominant
flow structures near the wall as well as in the near wake. The
5
Reynolds number was set to Re = 1.5 × 10 . Approximately
1 flow-through time was required for the solution to become
independent of the initial conditions. After that statistics wereaccumulated over 2.5 flow-through times. In the following andfor a generic flow variable the lower case, β , represents an instantaneous quantity, the upper case, B, represents the variable′averaged over time only, β represent the instantaneous fluctuations for the time-averaged quantity, and B, represents thevariable averaged over both time and azimuthal direction.

DISCUSSION
Fig.1 shows the variation of the drag coefficient with
Reynolds number measured in the wind tunnel for one dimpled sphere and for two polyhedral spheres, one with 162 and192 polygonal panels (referred to from now on as poly162 andpoly192 respectively). Some basic geometrical properties arelisted in Table 1. The dimpled sphere contains 312 spherical
dimples, with an average diameter, d = 0.095D, and maximum
depth k = 0.003D. The resulting dimple coverage, which is
the ratio of the surface area occupied by the dimples to thetotal surface of a sphere, is approximately 70%. The geometry is representative of a commercial golf ball. The dimpled
sphere has the largest volume of the three, about 0.989 of thatof a sphere with D = 1, followed by poly192 and poly162.
The ratio of frontal area to that of a smooth sphere is also very
close to 1 for all spheres, with the dimpled sphere, poly192 andpoly162 being 0.990, 0.986 and 0.982 respectively. Therefore
the reduction in the drag coefficient exhibited by the above
polyhedral and shown next can not be attributed to changes in
the frontal area.
The drag curves for the polyhedral and dimpled spheresare very similar, with the polyhedral exhibiting a drag crisis
4around Re ∼ 8 × 10 and maintaining the low drag in the postcritical regime. In general, as the number of polygonal panels increases, the polyhedral approaches a smooth sphere andthe drag crisis shifts towards higher Reynolds numbers while
the drag coefficient in the post-critical regime decreases. It
is important to note that the drag crisis for poly192 appears
to be occurring at the same critical Reynolds number as the
dimpled sphere but the drag coefficient is consistently lower
by 10% − 15% throughout the post-critical regime. For thepoly162 the drag coefficient in the post-critical regime is almost identical to that of the dimpled sphere but the drag crisis
occurs earlier. The predicted drag coefficient from the present
DNS for poly192 and the dimpled sphere are also shown in the2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
0.5

0.4

0.3

0.2

`
```
```

0.1

0

100000200000`300000400000Figure 1. Drag coefficient vs Reynolds number for various spheres.
smooth sphere Achenbach (1972);
dimpled sphere with dimple depth k = 0.0035D (present experiment);
dimpled sphere with k = 0.003D (present
icosahedral sphere with 192 polygonal panels (present experiment);
icosahedral with 162
experiment);
polygonal panels (present experiment); • icosahedral with 192 polygonal panels (present DNS); • icosahedral with 162
polygonal panels (present DNS); • dimpled sphere with k = 0.003D (present DNS).

Figure 2.

ameter D. For the case of the dimpled sphere k/D exhibits
considerable variation while for poly192 it is more uniform
and also consistently higher. For the former, C f exhibits local
peaks and valleys consistent with the presence of the dimples,while for poly192 it is considerably smoother. Global separation, identified as the cross from positive to negative values◦◦of C f , occurs at θ = 116 and θ = 126 for the dimpled andpolyhedral spheres respectively. The integral of the skin friction as a function of θ is shown in Fig. 4b. When the integral isevaluated over the entire sphere it is equal to the skin friction◦
drag. In the front part of both spheres (θ < 45 ) the integral of◦
the skin friction is very similar. For θ > 45 the skin friction
drag for the tessellated sphere is slightly larger than that of thedimpled sphere. This is expected since for poly192 the flow
remains attached for longer than the dimple sphere. Overall
the skin friction drag for both spheres is less than 10% of thetotal drag.

Wind tunnel setup for measuring the drag force

plot. The agreement with the experimental results is very goodwith the DNS values are within 2-3% of the experiments. The
drag coefficient for the dimpled sphere is 0.185, while that forpoly192 is 0155 (16% lower).
It is clear from the behaviour of the force coefficients thatthe polyhedral designs are more efficient in reducing the dragthan the dimples. To better understand this behavior we willcompare the results from the corresponding DNS. Fig. 3 shows
contours of the time-averaged skin friction coefficient, C f , forthe dimpled sphere and poly192. The skin friction was cal2
culated as, C f = 2ν/U ⋅ dUt /dn, where Ut is the tangential
velocity minus the azimuthal component and n is the surface
normal. The separation line is indicated by a black line and thepolar angle θ measured from the stagnation point at the front
is denoted by vertical dashed lines. For the dimpled sphere the◦flow separates locally inside some dimples as early as 50 andremains attached at the next row of dimples. Small local sep◦aration bubbles are again observed at 70 and they appear tobe present more consistently in the azimuthal direction. Larger
local separation bubbles occupying the first half of the dimples◦
occur at the next row of dimples around θ = 80 . For poly192
the behavior of the skin friction is very different. A couple◦of small separation bubbles appear around θ = 80 and more◦consistently in the azimuthal direction around θ = 90
Fig. 4a shows the skin friction coefficient, C f , averaged
over time and the azimuthal direction. The average surface
depth distribution k/D is also plotted at the bottom. The depth
is measured relative to the surface of a smooth sphere of di-

The distribution of the average pressure coefficient, C p ,together with the average surface depth, k/D, as a function ofθ is shown in Fig. 5a. For poly192, C p is relatively smooth
while for the dimpled sphere it exhibits small oscillations thatare correlated with the average dimple depth. In particular, thelocal peaks in C p occur slight after local maxima in k/D and◦◦vice versa. At the back of the sphere (90 < θ < 125 ) C p forpoly192 is lower. However due to the delayed separation it recovers to a greater value and remains consistently higher thanthat of the dimpled sphere. As for C f above, the integral of◦◦
C p as a function of θ is shown in Fig. 5b. For 0 < θ < 45◦
the pressure integrals are very close. For θ > 45 the dimples
start to incur a small pressure penalty relative to the polygonal◦
panels. By θ = 90 the dimples contribute approximately 30%of the total drag penalty. At the back the remaining 70% contribution to the additional drag for the dimpled spheres comes
form the lower back pressure due to the earlier separation.
The difference in the separation point between the two
types of spheres can be explained by looking at the evolutionof the boundary layer. First of all, the approximate locationof transition to turbulence can be determined by looking at the
behavior of the velocity fluctuations. Fig. 6 shows the variationof the turbulent kinetic energy, q, averaged over time and in the3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
0.006 (a)azimuthal direction. The quantity is plotted along an arc thathas a radius of 0.505D, that is 0.005D above the surface of asmooth sphere. The points on that arc are shown with markers
in the subset. The levels of q are negligible at the front part for◦◦
both spheres and they start to rise around θ = 75 and θ = 87
for the dimpled sphere and poly192 respectively. It is therefore
reasonable to assume that transition to turbulence is delayed
for the latter case. The location of the peak in q also occurs alittle later but overall the maximum levels of q are similar.
Fig. 7 shows contours of the instantaneous azimuthal vorticity at a plane cutting through the middle of a dimple near
the top of poly192 sphere where transition occurs. Contoursof the instantaneous skin friction coefficient, C f , are plottedon the surface of the sphere along with the separation line denoted by a black line. It is clearly seen that a shear layer isformed as the flow separates over the leading edge of the flat
hexagonal panel. The flow separation is not uniform across thespan and occurs over a small portion near the first half of thepanel. The flow reattaches again near the center of the panel.
Shortly after the flow separates the shear layer becomes un-

60
50

◦

◦

70

◦

◦

80

90

Cf
0.002

0
50

100

150

50

100

150

50

100

150

k/D

0.0050
0

(b)
0.015

0.01

◦

∫ Cf dA

a)

0.004

0.5

C f Re0.005

0Figure 4. a) Distribution of the average skin friction coefficient, C f , (top part) and average surface depth (bottom part);b) Integral of C f .
dimpled sphere;
poly192;
difference between dimpled sphere and poly192.

b)stable and starts to roll-up into vortical structures, annotatedas rollers A and B in the figure. The evolution of these rollers,
which is essential in the transition process, can be better traced
in Fig. 7b, where a top view is shown and the vortical structures are colored by their streamwise vorticity. It can be seenthat roller A is not uniform in the azimuthal direction anddoes not extend across the entire span of panel. A similar behaviour can be observed for roller B. As these rollers undergo
instabilities in the azimuthal direction their vorticity is reoriented from the azimuthal to the streamwise direction. Also,in between the two rollers pairs of counter-rotating streamwise vortices are present. These vortices are reminiscent of
the braid-vortices containing mainly streamwise vorticity of
opposite sign, typically found in free-shear layers undergoing
Kevin-Helmhotz type instability. Towards the end sides of therollers thin elongated vortices aligned in the sreamwise direction and containing streamwise vorticity are observed. Farther
downstream and close to the trailing edge of the dimple various vortical structures resembling a Λ-type vortex are clearlyobserved. This pack of Λ-type vortices has evolved from thecontinuous bending of a previously shed roller and its legs areconnected to a pair of braid vortices. We should note that therollers aligned primarily in the azimuthal direction are transformed into Λ-type vortices always within the length of one
panel for a given Reynolds number. This mechanism is qualitatively similar to the one observed for flow over a dimpled

0.5

C f ReFigure 3. Contours of the time-average skin friction coeffi0.5
cient, C f , scaled by Re . a) dimpled sphere; b) poly192. The
separation line is shown with a solid black line while the polar
angle at various locations is indicated by vertical dashed lines.
The red rectangular outline corresponds to the location of thehexagonal dimple shown in Fig. 7.

4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
sphere Beratlis et al. (2019).
To further quantify this, profiles of the time averaged
streamwise velocity and turbulent kinetic energy around θ =◦83 are shown in Fig. 8. Three stations are selected: one before, one inside and one after the shear layer where transition
takes place. At the first station the velocity profile and the lackof turbulent kinetic energy confirms that the boundary layer isstill laminar. In the second station the negative velocity near
the wall indicates the flow and the velocity profile resembles
that of a shear layer. The turbulent kinetic energy has a sharp
peak at approximately r/D = 0.0035 indicating that the shear
layer become unstable. Farther downstream near the trailingedge of the hexagonal panel the peak in the turbulent kinetic
energy moves closer to the wall, at r/D = 0.017. However the
profile of q is much broader with significant levels of turbulent
kinetic energy extending up to r/D = 0.012. This is characteristic of momentum transport across the boundary layer.
Fig. 9 shows the average boundary layer thickness, δ , as afunction of the polar angle θ , up to the separation point. Since
the location of the wall varies in the azimuthal direction theboundary layer thickness is first calculated at each azimuthal
plane, as shown in the insert of Fig. 9. This gives a set of displacement thicknesses δ1 , δ2 , ... , δnφ with respect to the local
wall location, which are then averaged over the azimuthal direction φ to give δ . Note that δ is calculated utilizing the vor-

0.1

2

q/U∞

0.15

0.05

0

0

50

100

150

θ
Figure 6. Variation of the turbulent kinetic energy, q, alongpoly192;
an arc line with radius R = 0.505D.
dimpled sphere. The inset on the top left shows contours of q
for poly192 with black dots representing the arc line.

Cf

ωθ
a)
A
@

B
@
@

Λ
@
@

1

(a)
0.5

Cf

ωθ
a)

Cp

0

braid

A
B@
@ BB

-0.5
-1

0.0050

50

100

150

k/D

B

0

0.2

50

100

150
Figure 7. a) Isosurface of the Q-criterion visualizing vorticalstructures near the top of the tesselated sphere. Contours of
the instantaneous azimuthal vorticity, ωθ , are also shown at
an azimuthal plane going through the middle of a hexagonal
panel along with contours of the instantaneous skin friction,C f , plotted on the surface of poly192. The separation line isdenoted by solid black line. The location of the hexagonal
panel is indicated with a red box in Fig. 3b. b) Top view withvortical structures colored by contours of tangential vorticity
ωt .

(b)

∫ CpdA

0.1

0

50

100

ticity definition proposed by Spalart & Watmuff (1993), which
is more appropriate in the presence of curvature and pressure
gradients. For the dimpled sphere the boundary layer does not
grow in a monotonic fashion, and there are local peaks and valleys associated with the presence of the dimples (for reference
the average dimple depth is shown on the lower part of the figure). The peaks and valleys in δ correspond to the locationof the maximum and minimum dimple depth respectively, andoverall dimples promote a thickening of the boundary layer.

150

Figure 5. a) Distribution of the average pressure coefficientC p (top part) and average surface depth (bottom part); b) Integral of C p .
dimpled sphere;
poly192;
difference between dimpled sphere and poly192.

512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
78.5

◦

83

95

0.01

0 0

0

◦

0 0

U/U∞1

0.01

◦

U/U∞1

REFERENCES

0.01

r/D

r/D

0.04

2

q/U∞

0

0 0

U/U∞1

0.01

0

◦

by 10 at the same Reynolds number. Transition to turbulence on a dimpled sphere also occurs closer to the stagnation
point. As a result the boundary layer grows thicker and theflow separates earlier giving rise to lower back pressure andhigher form-drag.

0.01

r/D

◦

◦

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the origin of the drag force on dimpled spheres. Journal of
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Pal, A., Sarkar, S., Posa, A. & Balaras, E. 2017 Direct numerical simulation of stratified flow past a sphere at a subcritical reynolds number of 3700 and moderate froude number.
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Rahromostaqim, M., Posa, A. & Balaras, E. 2016 Numericalinvestigation of the performance of pitching airfoils at highamplitudes. AIAA Journal 54 (8), 2221–2232.
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r/D

0

r/D

r/D

0.01

◦

0

2

q/U∞

0.04

0

2

q/U∞

0.04

Figure 8. Top part: Profiles of the time averaged streamwise
velocity U; Bottom part: profiles of the mean turbulent kinetic
energy q. Statistics are shown at three stations along an azimuthal plane. The locations of the stations are also indicated◦◦◦in the figure: A is at θ = 78.5 , B is at θ = 83 , C is at θ = 95
0.02
δ3
δ2
δ1

δ /D

0.01
φ

20

40

60

80

20

40

60 θ 80

100

120

100

120

k/D

0.00500

Figure 9. Plot of the boundary layer thickness δ averaged
over time and azimuthal direction and average dimple depth
k/D. Lines represent: –; poly192, –; dimpled sphere

For poly192 the boundary layer growth is smoother.

SUMMARY
Dimples have been long considered as an efficient mechanism for tripping the boundary layer and reducing drag on
spheres. Previous work by the authors revealed though that
dimples impose a significant pressure penalty which can hinder drag reduction. The authors present a novel topologicalmodification, comprising the tessellation of a sphere, that can
reduce drag even more without changing the Reynolds number
at which the drag crisis occurs.
We report wind tunnel measurements of the drag coefficient over a range of Reynolds numbers, as well as DNS for
selected cases in the post-critical regime. Dimpled and tessellated spheres are considered. The wind tunnel measurements
demonstrate that the variation of the drag coefficient as a function of the Reynolds number for the two tessellated spherecases is very similar to that of dimpled spheres: drag crisis occurs and the drag coefficient remains relatively constant in the
post-critical regime. As the number of tessellations increases
the drag crisis shifts towards higher Reynolds number and thevalue of the drag coefficient in the post-critical regime is reduced.
Detailed analysis of the results from the numerical simulations showed that a tessellated sphere reduces the pressure
penalty compared to dimples while also delaying separation

6