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

NUMERICAL STUDY OF TURBULENT RESISTANCE REDUCTION
EFFECT BY STAGGERED SUPERHYDROPHOBIC SURFACE

Daito Hirata, Hiroya Mamori,Takeshi Miyazaki
Department of Mechanical and Intelligent Systems Engineering
The University of Electro-Communications
1-5-1 Chofugaoka, Chofu, Tokyo, 182-8585, Japanmamori@uec.ac.jp

Table 1.

ABSTRACT
Direct numerical simulations of the turbulent channel
ﬂow with superhydrophobic surfaces (SHS) are performed.
We employed a staggered type of the SHS and conﬁrmed adrag reduction effect and an increase of a mean streamwise velocity, while the reduction rate is lower than that of the straighttype. In addition, we investigate the inﬂuence of particle adhesion on the drag reduction effect. The particle adhesion decreases the drag reduction effect by the SHS. The number ofadhered particles on the staggered SHS is smaller than that of
the streamwise straight SHS. Therefore, we expect the staggered SHS to maintain the drag reduction effect longer than
the streamwise straight SHS.

Numerical conditions

Domain(Lx , Ly , Lz ) = (2π , 2, 2π )Number of grids(Nx , Ny , Nz ) = (256, 96, 256)

Total slip area

50%

Bx

Lx /10, Lx /2, Lx

Bz

Lz /10, Lz /2, Lzof the SHS are performed. The inﬂuence of the particle adhesion is investigated by evaluating the drag reduction rate, theﬂow statistics, and the number of adhered particles.

INTRODUCTION
In turbulent ﬂows, the skin-friction drag accounts for a
substantial proportion of the total drag. The superhydrophobic
surface (referred to as the SHS, hereafter) decreases the skinfriction drag and has been widely studied (e.g. Daniello, etal., 2009; Choi and Kim, 2006). The friction drag reduction
is obtained due to the slip velocity. The slip velocity occurs
at the interface between the trapped air into the micro-groove
and the working ﬂuid.
To simulate the turbulent ﬂow over the SHS, the SHS isrepresented by combining the no-slip and slip area. Hasegawa
et al.(2011) performed direct numerical simulations (DNS) ofturbulent channel ﬂows with straight and sinuous types of theSHS and found the increase of the bulk mean velocity. Michaelet al.(2009) performed the DNS of turbulent channel ﬂow toinvestigate the effects of width and gap of the slip area of thestraight type of the SHS on the drag reduction effect. Watanabe et al.(2017) investigated the effect of the angle between
the ﬂow direction and the direction of the straight SHS on thebulk mean velocity using the DNS. When the direction of theSHS coincided with the ﬂow direction, the bulk mean velocity
increased up to 15%.
While the straight SHSs have been studied frequently, weinvestigate the effect of the staggered type of the SHS on theturbulent ﬂow. Moreover, we investigate the inﬂuence of particle adhesion on the skin-friction drag. Since the SHS is expected to be used on ships as a practical application, the particles ﬂoating in the sea may adhere to the SHS and degrade thedrag reduction rate. Therefore, the investigation of the effectof the particle adhesion on the skin-friction drag is required.
Accordingly, in this study, the numerical simulations ofthe turbulent channel ﬂow with the straight and staggered types

Direct Numerical Simulation
The DNS of the turbulent channel ﬂow with or without
the SHS is performed (Kim, et al., 1987). The governing equations are continuity and the Navier-Stokes equations of the incompressible ﬂow, as∂ ui=0∂ xi
1 ∂ 2 ui
∂ ui
∂ ui
∂p+uj
=−+∂t
∂xj
∂ xi Reτ ∂ x2j

(1)
(2)

Here, t is time and p is pressure. The streamwise, wallnormal, and spanwise coordinates are denoted by x, y, andz, and the corresponding velocities are u, v, and w, respectively. And xi and ui (n = 1 ∼ 3) are interchangeably used.
The driving force of the main ﬂow is a constant pressure gradient condition, and the friction Reynolds number deﬁned by
the channel half-width and the wall-friction velocity is Reτ =u∗τ δ ∗ /ν ∗ = 180. This corresponds to the bulk Reynolds number Reb = 2u∗b δ ∗ /ν ∗ = 5600, where u∗b is the bulk velocity andδ ∗ is the channel half-width. The asterisk means the dimensional variables.
The velocity and pressure are deﬁned on the staggering
grid system. The governing equations are discretized by using
an energy-conserving second-order central difference method(Ham, et al., 2002). As time advance, we employ the secondorder Crank-Nicolson method for the viscous term and thememory-saving third-order Runge-Kutta method for the other1

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

Flowslipno-slip(Case1)Figure 1.

(Case2)(Case3)(Case4)

Schematic in the channel ﬂow with the SHS.

(a)

Case 2
17.6

Case 438.8
Case 314.1(b)Case 145.6

Figure 2. Drag reduction ratio for Cases 1 ∼ 4.

(Spalarta, et al., 1991). Figure 1 shows a schematic of thechannel ﬂow with the SHS, and the numerical conditions arein Table 1.
The numerical condition is shown in Table 1. We imposed
the periodic boundary condition in the x and z directions. The
channel wall is the SHS represented by a combination of slip
and no-slip conditions area alternately. In Fig. 1, the slip andno-slip areas are colored by white and black, respectively. Theslip area is 50 percent of the entire wall surface, and a single
slip area is deﬁned by the streamwise length Bx and the spanwise length Bz . To maintain the symmetry of the ﬂow ﬁeld, thesame SHS is installed on the upper and lower walls.

Figure 3. Schematic of the turbulent channel ﬂow near thelower wall: (a) no-slip, (b) Case 3.

between the no-slip and slip surfaces. There is no signiﬁcantdifference in the vortical structure above the wall between noslip and Case 3.
Figure 4(a) shows the mean streamwise velocity. The
streamwise velocity increased in all the as compared with theno-slip case. The velocity increased the most in Case 1, thenin Case 4, Case 2, and Case 3. The result is consistent withFig. 2, since the velocity increase corresponds to the decreaseof the skin-friction drag under the constant pressure gradientcondition. Figure 4(b) shows the streamwise velocity excluding the slip velocity on the wall u|wall . A downward shift ofvelocity proﬁle is observed in the buffer layer.
Figure 5(b) shows the Reynolds shear stress (referred toas the RSS, hereafter) in the area near the wall. The RSS in
Case 2 is larger than that of the no-slip case. In contrast, inCases 1 and 3-4, the negative RSS is found at y+ ≤ 3.

DRAG REDUCTION EFFECTWe perform the DNS of the turbulent channel ﬂow with
the SHS without the particles. Figure 2 shows the pattern of
the SHS together with the drag reduction rate: Cases 1 and2 are the straight types in the streamwise and spanwise directions, respectively; Cases 3 and 4 are the ﬁne and coarse staggered type of the SHS, respectively. The drag reduction rate isdeﬁned asRD =C f 0 −C f
Cf0

PARTICLE ADHESION
Subsequently, the inﬂuence of particle adhesion is discussed. We start simulations of the particle from the fully
developed turbulent channel ﬂow with the SHS. No particles
are added during the simulation. The particles do not rotate, deform and interact with each other. The one-way coupling method is employed: the ﬂow affects the particles by theStokes resistance, but the particles do not affect the ﬂow.
The equation of motion with the Stokes resistance of theparticle is(3)Where C f is the skin-friction coefﬁcient. The subscript of zeromeans the no-slip case (without the particles). The drag reduction rate RD is largest in Case 1 and low in Case 2. Furthermore, RD in the ﬁner staggered SHS of Case 3 is smaller thanin the coarse SHS of Case 4. Figure 3 visualizes the turbulent
vortical structures in the no-slip case and Case 3. The vorticalstructure is shown in white. In Case 3, the small vortical structure in the z direction was generated near the wall boundarym∗2∂ v ∗pπ= − ρ ∗f CD r∗2 v ∗p − v ∗f (vv∗p − v ∗f )∂t2(4)

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

(a)

(a)

(b)

wall(b)Figure 4. Mean streamwise velocity (a) u and (b) u − u|wall .
Figure 5. The Reynolds shear stress for different cases: (a)0 < y+ < 180 and (b) zoom-up view of the near wall area.

Here, m is the mass of particles, r is the radius of theparticle, ρ p is the density ratio of the particles to the ﬂuid,v p is the particle velocity, v f is the ﬂuid velocity, and ρ p f
is the density of the ﬂuid. The asterisk means dimensional
quantities. The drag coefﬁcient CD is expected as,24CD = Rep


24 

CD =
1 + 0.15Re0.678p
Re p

adhere to the slip area.
In order to show the particle adhesion qualitatively, Fig. 8
shows the ratio of the local number of attachments to the total
number of attachments. For Cases 1∼3, the ratio is displayedin 2π /5 × 2π /5 since it is the smallest unit for the Case 3. We
found that the particles tend to adhere to the slip area or the rearregion of the slip area, as shown in Cases 1 and 2, respectively.
In a ﬁne staggered arrangement of Case 3, the particles adhereat the side area of the slip area, where the adhesion at the centerarea is little. In a coarse staggered arrangement of Case 4, theparticles adhere uniformly on the slip area.

(Re p ≤ 1)
(5)
(Re p ≥ 1)

here, Re p is a particle Reynolds number based on the particle diameter and the relative velocity between the ﬂow andparticles. In this simulation, the parameters are ρ p f = 3 andr = 2.5×10−3 (r+ = 0.45). The number of particles is 10, 000.
Therefore, the fraction of particles in the ﬂuid is 0.008%. At
the beginning of the simulation, the particles distribute randomly in the ﬂow, and the local ﬂuid velocity interpolates thesevelocities. To express the decrease of the SHS performance by
the particle adhesion, we employ a simple model: the slip cellon the wall becomes the no-slip cell if the particle contacts. We
compute the no-slip area by the adhesion based on the spherical surface of the particle.
Figure 6(a) shows the time trace of the bulk velocity considering the particle adhesion. The bulk velocity gradually decreased in Cases 1 and 4, while it is almost unchanged in Case2 ∼ 3. The bulk velocity corresponds to the drag reduction
rate. Figure 6(b) shows the time trace of the number of particles adhering to the SHS. The number of particles adheringto the SHS increased monotonically with time in all the cases.
It implies that the particles are more likely to adhere to thestreamwise straight type than the staggered type.
Figure 7 shows the accumulated adhered-particle on thewall at 0 < t < 80 for all the cases. The white dot corresponds
to the adhered particle. We found that the particles tend toCONCLUSION
We obtain the signiﬁcant drag reduction rate by thestraight SHS in the streamwise direction, while the drag reduction rate by the staggered SHS is lower than that of the straight
SHS. For the staggered SHS, the coarse SHS of Case 4 gives
a more signiﬁcant drag reduction rate than the ﬁner staggered
SHS of Case 3. Subsequently, we investigate the inﬂuence ofparticle adhesion on the SHS. The particle adhesion increases
monotonically with time, and the number of the adhered particles in the staggered SHS is lower than that in the streamwise
straight type. In the ﬁner staggered SHS of Case 3, the decrement of the drag reduction rate and the number of the adhered
particles are slight. Accordingly, the staggered SHS is robust
for the drag reduction rate against the adhesion of the particles.
The distribution of the particle adhesion is also investigated. In
all the cases, the particles adhere to the slip area, and the distribution depends on the pattern of the SHS. In particular, ina ﬁne staggered arrangement of Case 3, the particles tend toadhere to the side area of the slip area rather than the centralarea.

312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022(a)(a) 22(b)ubZ

202/5πZ

2/5π180020t60

80

2/5π

Case 1Case 2
Case 3
Case 43000
2000(d)ZParticle adhesion Number(b) 4000

40(c)0010002/5π

X000

20406080X2/5πX

2π2πZ

160002/5π

X500101518Figure 8. The ratio of the local number of attachments to thetotal number of attachments: (a) Case 1, (b) Case 2, (c) Case3, and (d) Case 4.

tFigure 6. Time trace of (a) the bulk mean velocity and (b) thenumber of the particle adhesion.

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(a)

(b)

2π(c)

Z

Z

2π00

X

002π(d)

X

2π

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2πZ

2πZ

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X

2π00Figure 7. The accumulated adhered-particle on the wall at0 < t < 80. The white dot means a single adhered particle: (a)Case 1, (b) Case 2, (c) Case 3, and (d) Case 4.

4