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

NUMERICAL SIMULATION OF TURBULENT CHANNEL FLOW WITH PARTICLEADHERED RIBLET SURFACES

Chikara Shimizu, Junichi Morita, 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

sinusoidal riblets provided a maximum drag reduction of
11.7%. In addition, the drag reduction effect is confirmed bythe direct numerical simulation (Sasamori et al. 2017), and thevortical structure over the riblet surface is analyzed by using adual-plane stereoscopic PIV (Mamori et al. 2019).
Accordingly, the 2-D and 3-D riblet surfaces are adequate
for the drag reduction of the turbulent flow. However, frompractical application, the riblet surface is employed for theouter flow where many small particles are floating and canadhere to the wall. The adhered particles may decrease theskin-friction drag reduction effect by the riblet, and therefore
the robustness of the riblet against the particle adhering isessential.
In this study, we perform numerical simulations toinvestigate the effect of particles adhering to the riblets on theskin friction drag. The target of this simulation is a turbulent
channel flow with a rectangular 2D-riblet surface. We perform
one shot-simulation: first, we compute the turbulent flow withfloating particles and obtain the distribution of the adhering
particle; second, the riblet shape is changed due to the adheredparticle; finally, we make the flow simulation again toestimate the variation of the drag reduction effect.

ABSTRACT
Numerical simulations of fully developed turbulent
channel flows with riblet surfaces were carried out at thefriction Reynolds number of 180. The influence of particles
adhering to the triangular riblets on the drag reduction effectis investigated. The one-way coupling method is employed:
the particles are affected by the fluid, but the fluid is notaffected by the particles. The simulation consists of threesteps: first, we calculate the adhesion position of the particles
to riblets in the turbulent channel flow; second, the riblet
shapes are changed based on the particle adhesion; third, we
calculate the flow over the riblet with the adhered particles. Asa result, most of the particles adhered near the upper edge ofthe riblets. Due to the particle adhesion on the riblets, the dragincreases as compared with the case of the initial riblet of theriblet.

INTRODUCTION
With the development of industry, there is a growinginterest in reducing energy consumption. Therefore, researchon fluid drag reduction for energy reduction has been activelyconducted. Turbulence control methods can be divided into
active and passive control. Active control requires externalenergy, such as traveling wave control (Mamori et al. 2014).
Passive control requires only initial processing, such as
superhydrophobic surfaces (Fuaad and Arul Prakash 2019),
polymers (Fujimura et al. 2017), and riblet surface.
The riblet surface means fine grooves on the wall, which
is known to reduce turbulent friction drag, and many patternsof the riblet design have been studied. Choi et al. (1993)
performed direct numerical simulations of a fully developed
turbulent channel flow to investigate the two-dimensional
riblet surface (referred to as the 2-D riblet, hereafter). They
revealed that the lateral space between the riblet walls plays
an important role, i.e., the drag reduction effect appears when
the lateral spacing is smaller than the diameter of the vorticalstructure. Bechert et al. (1997) showed the blade-type ribletswith very thin walls, and a maximum drag reduction was about
10%, where the lateral space between the riblets is 𝑠 ! = 15.
In recent years, the three-dimensional riblet surface (the 3-Driblet) has been investigated. The 3-D riblet means that theriblet walls vary in the streamwise direction. Sasamori et al.
(2014) made an experimental study to investigate the dragreduction effect by the sinusoidal riblet surface: it means thatthe lateral spacing between the riblet walls also varies
periodically in the streamwise direction. They show that

NUMERICAL SIMULATION
Numerical simulations of the particle-laden turbulent
channel flow with the riblet are performed. The governing
equations are continuity and Navier-Stokes equations for theincompressible flow, as

𝜕𝑢!
=0
𝜕𝑥!
𝜕𝑢!
𝜕𝑢!
𝜕𝑝
1 𝜕𝑢!$+ 𝑢"
=−+𝜕𝑡
𝜕𝑥"
𝜕𝑥! Re# 𝜕 $ 𝑥"

(1)
(2)

Here, 𝑢" and 𝑥" (𝑖 = 1~3) are the velocity and thecoordinates in the streamwise, wall-normal, and spanwise
directions, respectively. In addition, 𝑡 is time and 𝑝 is thepressure. The coordinates 𝑥" are interchangeably used as 𝑥, 𝑦,and 𝑧, the corresponding velocity components are used as 𝑢,𝑣, and 𝑤, respectively. The friction Reynolds number definedby the wall friction velocity 𝑢# and the channel half-height 𝛿is set to Re# = 180. This corresponds to the bulk Reynolds
number of Re$ ≈ 5600 (based on the bulk velocity 𝑈$ andthe channel half-height 𝛿). The mean pressure gradient is kept

1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
asterisk means dimensional quantities. The drag coefficient
𝐶- is expected as,

Lz=3.2
Lx=ʌ

Meanflow𝐶( =w

Ly=2＋＋s𝐶( =h＋y,v x,u24
Re*

24:1 + 0.15Re+.-./=*Re*:Re* ≤ 1.0= (4):Re* > 1.0= (5)z,wwhere, Re. is the Reynolds number of particles based on thediameter and the relative velocity between the flow andparticles. Particles are introduced only at the beginning of thecalculation and no additional particles are introduced.
As shown in Fig. 1, the channel walls have a rectangularriblet surface. In the initial shape of the riblet surfaces(referred to as the initial riblet, hereafter), the height of theriblets, the width of the riblet, and the lateral spacing between
the riblets are set to ℎ! = 7.5 , 𝑤 ! = 3 , and 𝑠 ! = 16.2 ,respectively. Here, the superscript of the plus denotes the wall
unit. Under these conditions, we installed 30 riblet walls on
the upper and lower walls, respectively.
Next, we explain a numerical treatment of the particle
adhered riblet surfaces. Although the adhered particles interact
with the flow in the actual phenomenon, we employed singleshot calculation to reduce the computational cost. The
procedure is as follows.
1.
Particles are tracked as mass points and the adhesion
distribution on the riblet surfaces is obtained at 0 ≤ 𝑡 ≤80. Here, the time origin 𝑡 = 0 corresponds that thebeginning of the simulation: the number of total addedparticles is set to 𝑁/ = 5.0 × 100 , the particles are
randomly distributed in the fully developed turbulent
channel flow with the riblet walls, and then we start totrack the particles.
2.
The adhesion distribution on the riblet and channel wall
surface is computed. The height of the adhesion is
obtained by the volume of adhered particles divided by
the wetting area of the riblet and channel surface. Here,
we make twofold assumptions: the adhesion distribution
does not change as the number of adhesion particles 𝑁/ ′increases; the particles uniformly adhere in the xdirection. Accordingly, we predict the riblet shape at
𝑁/1 = 5.0 × 100 and 2.5 × 100 .
3.
The turbulent channel flow with the reshaped riblet
surfaces is calculated, and we obtain the drag reduction
rate and the flow statistics.

Figure 1. Schematic of the channel flow with the riblet
surfaces.

Figure 2. Schematic of the turbulent channel flow withthe riblet surface together with the floating particles in theregion near the lower wall.

Figure 3. Time trace of number of adhering particles.
at constant. The velocity and pressure are defined on astaggered grid. The governing equations are discretized by thefinite volume method.
Figure 1 shows the schematic of the channel flow with theriblet surface. The computational domain is set to 𝐿% × 𝐿& ×
𝐿' = 2𝜋 × 2 × 3.2, and the number of computational grids isset to 𝑁% × 𝑁& × 𝑁' = 128 × 192 × 512 . In boundary
condition, the periodic condition is imposed in the streamwise
and spanwise directions. The no-slip conditions are imposedon channel walls, and the riblets are represented by theimmersed boundary method proposed by Kim et al. (2001).
The particle radius is set to 0.27 and 0.45 in the wall unit,
and the density ratio of particles to fluid is set to 1.0 × 10( .
One-way coupling method is employed: the fluid affects themotion of the particles, but the particles do not affect the fluid.
The equation of motion of the particles considering theStokes resistance is𝑚∗∂𝒗∗𝒑π= − 𝜌'∗ 𝐶( 𝑟 ∗ $ |𝒗∗𝒓 |𝒗∗𝒓∗∂𝑡2

SAMPLE RESULTS
Figure 2 visualizes the turbulent flow in the region near
the lower wall with the floating particles with 𝑟 ! = 0.27 atthe beginning of the simulation. The riblet surface is clean.
The riblet surface and floating particles are shown in gray andblue. The adhered particles are shown in red. The vorticalstructure is shown in green. In general, the riblet surfaces
prevent the vortical structure from approaching the channel
wall and the behaviour is observed.
Figure 3 shows the time variation of the total number ofparticles adhered to the channel and riblet walls. The number
increases with the time advance. The number of adhered
particles at 𝑡 = 80 is approximately 𝑁/ ≈ 4.6 × 100 and4.9 × 100 for 𝑟 ! = 0.27 and 0.45, respectively.

(3)Here, 𝑚 is the mass of particles, 𝜌) is fluid density, and 𝑟 isparticle radius. In addition, 𝒗𝒓 = 𝒗𝒑 − 𝒗𝒇 is the relativevelocity, 𝒗𝒑 is particle velocity, and 𝒗𝒇 is fluid velocity. The2

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

(a)

(b)

Figure 4. Distribution of the adhered particles on the riblet surface at 𝑡 = 80: (a) 𝑟 ! = 0.27 and (b) 𝑟 ! = 0.45.

Figure 5. The particle adhered riblet for 𝑁/1 = 5.0 × 100 : (a) 𝑟 ! = 0.27 (Case 1) and (b) 𝑟 ! = 0.45 (Case 2).

Figure 6. The particle adhered riblet for 𝑁/1 = 2.5 × 102 : (a) 𝑟 ! = 0.27 (Case 3) and (b) 𝑟 ! = 0.45 (Case 4).
Table 1. Parameter set for particle-adhered riblet shape.

Case 1Case 2
Case 3
Case 4of the riblet sidewalls, particles sparsely adhere. As shown in
Fig. 4(b), particles adhere to the entire riblet and channel
surfaces.
Next, we predict four particle-adhered riblet shapes, as
shown in Table 1. Figure 5 shows particle-adhered riblet
surfaces for 𝑁.1 = 5.0 × 100 . The initial riblet shape iscolored by black and the predicted shape is colored by grey.
As shown in Fig. 5, the change of the riblet shape is barely
discernible, because the radius and number of adhered
particles are small. Figure 6 shows the particle-adhered riblet
surfaces for 𝑁.1 = 2.5 × 102 . As shown in Fig. 6(a), theadhesion layer covers the top edge of the riblet, results in theincrease of the width of the top edge of the riblet. The
adhesive layer covers the entire riblet and channel surfaces,
and the riblets become a trapezoidal shape in Fig. 6(b). Inaddition, the particles are deposited on the channel wall
surface. In all the cases, the amount of the adhesion increases
at the ridges of the riblets. We suggest that particles are more
likely to adhere there.
Finally, we perform numerical simulations of theturbulent flow over the predicted riblet surfaces. Since thepressure gradient is constant, an increase in the bulk velocity
corresponds to an increase in the drag reduction rate. The dragreduction rate is shown in Table 2. The drag reduction rate isdefined as𝑟!
0.27
0.45
0.27
0.45

𝑁/
5.0 × 100
5.0 × 100
2.5 × 102
2.5 × 102

Table 2. The drag reduction rate by each case.

initial riblet
Case 1Case 2
Case 3
Case 4

𝑅4.24
4.18
3.29
3.38
2.09

Figure 4 shows the distribution of the adhered particles on
the riblet surface at 𝑡 = 80. The riblets are shown in black andadhered particles are shown in red. The figure is a perspective
view i.e., all the adhered particle in the x-direction is shownon the y-z plane and the two riblet walls are shown as sample
results. The particle size is 𝑟 ! = 0.27 in Fig. 4(a) and 𝑟 ! =0.45 in Fig. 4(b). As shown in Fig. 4(a), the particles adhere
to the whole of the riblet and channel surfaces. At the bottom

𝑅- =3𝐶)! − 𝐶)× 100𝐶)!

(6)

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022of the particles, the RSS increased slightly and the dragreduction rate decreases.

CONCLUSION
We performed numerical simulations of the turbulent
channel flow with the riblet and the floating particles. We
investigated the effect of particle adhesion on the riblet shapeand the drag reduction effect.
The particles tend to adhere to the ridge of the riblet
surfaces. Based on it, we predict the riblet shape adhering to
the particle. The drag reduction rate decreases as increase of
the number of particles adhered and the Reynolds shear stressincreases slightly.

REFERENCES
Bechert, D. W., Bruse, M., Hage, W., Van Der Hoeven, J.
G. T., and Hoppe, G., 1997, “Experiments on drag-reducing
surfaces and their optimization with an adjustable geometry.”,
Journal of Fluid Mechanics, Vol. 338, pp. 59-87.
Choi, H., Moin, P., and Kim, J., 1993, “Direct numericalsimulation of turbulent flow over riblets,” Journal of Fluid
Mechanics, Vol. 255, pp. 503-539.
Fuaad P.A. and Arul Prakash, K., 2019, “Enhanced dragreduction over superhydrophobic surfaces with sinusoidal
textures: A DNS study.”, Computers and Fluids, Vol. 181, pp.
208-223.
Fujimura, M., Iwamoto, K., Murata, A., Masuda, M.,Ando, H., and Mamori, H., 2017, “Influence of length ofpolymer aggregation on turbulent friction drag reduction
effect”, Journal of Fluid Science and Technology, Vol. 12: 2.
Fukagata, K., Iwamoto, K., and Kasagi, N., 2002,“Contribution of Reynolds stress distribution to the skinfriction in wall-bounded flows”, Physics of Fluids, Vol. 14, pp.
L73-L76.
Kim, J., Kim, D., and Choi, H., 2001, “An immersedboundary finite-volume method for simulations of flow incomplex geometries”, Journal of Computational Physics, Vol.
171, pp. 132-150.
Mamori, H., Iwamoto, K., and Murata, A., 2014, “Effectof the parameters of traveling waves created by blowing andsuction on the relaminarization phenomena in fully developed
turbulent channel flow.” Physics of fluids, Vol. 26: 015101.
Mamori, H., Yamaguchi, K., Sasamori, M., Iwamoto, K.,
and Murata, A., 2019, “Dual-plane stereoscopic PIV
measurement of vortical structure in turbulent channel flow onsinusoidal riblet surface.” European Journal of Mechanics,
Vol. 74, pp. 99-110.
Sasamori, M., Iihama, O., Mamori, H. Iwamoto, K., andMurata, A., 2017, “Parametric study on a sinusoidal riblet fordrag reduction by direct numerical simulation.” Flow
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Figure 7. The Reynolds shear stress in Cases 1 and 2 (a)and Cases 3 and 4 (b).
Here, 𝐶) is the coefficient of skin-friction drag, and thesubscript of zero means the plane channel case (i.e., without
the riblet and particle). In all the cases, the drag reduction ratio
decreases due to the particle adhesion, the increase in thenumber of adhered particles and the volume of the particles. Itis because the total volume of the adhered particles on theriblet increases as the number and radius of the particles
increase.
The skin-friction coefficient 𝐶) is decomposed as(Fukagata et al., 2002)

𝐶' =

12
6 3
2 𝑣 2 )𝑑𝑦
FFFFFF+ $ C (1 − 𝑦)(−𝑢Re0 𝑈1 +(7)The first term is the laminar contribution. The second term isthe contribution from the turbulence, which is the weighted
integration of the Reynolds shear stress (referred to as the RSS,hereafter). Therefore, the skin-friction drag reduction
corresponds to the decrease of the RSS.
Figure 7 shows the RSS at the center of the lateral spacingof the riblet walls. As compared to the clean channel flow
(displayed by the w/o control), the RSS near the wall is
reduced and the distribution shifts toward the center in the ydirection in the case of the initial riblet. Due to the adhesion4