RIKEN Review No. 40 (October, 2001): Focused on High Performance Computing in RIKEN 2000

Direct numerical simulation of turbulent flow over acompliant surfaceTakahide Endo,∗,† and Ryutaro Himeno

∗Computer and Information Division, Advanced Computing Center, RIKEN

Direct numerical simulation of turbulent channel flow over a compliant surface has been performed to evaluatethe friction drag reduction effect. It is found that a slight drag reduction is observed over a compliant surface.This observation indicates that drag reduction is possible using a compliant surface that is deformed passivelyby wall pressure fluctuation. A small wall displacement and velocity are observed in the present study, however,a large-scale pressure field becomes dominant. The typical shape of displacement of the compliant surface isa wave, which is almost homogeneous in the spanwise direction.

Introduction

From the viewpoint of saving power and protecting the en-vironment, it is highly desired to develop efficient turbulencecontrol techniques for drag reduction and heat transfer aug-mentation. Organisms living in water have developed effi-cient turbulent control techniques through their unique evo-lutions. A typical example is the shark. The scales of ashark have grooves in the streamwise direction, and a ribletsurface is made after the shark’s scales as a drag reductiondevice. Walsh 1) performed a series of experiments on ribletsurfaces, and found that a maximum drag reduction rate of8% is obtained with a V-shaped riblet surface. Recently, thefriction drag mechanism on a riblet surface was clarified bya detailed experiment with three-dimensional particle track-ing velocimetry (3-D PTV) 2) and direct numerical simulation(DNS).3)

A dolphin swims as fast as 40 knots per hour at the maxi-mum. However, its muscle is not sufficiently strong for suchfast swimming;4) this is well known as “Gray’s paradox”.Therefore, it is expected that the flexible skin of a dolphin(compliant surface) plays a key role in controlling the sur-rounding fluid flow. Research studies of compliant surfaceswere started by Kramer,5) and it has been reported that acompliant surface causes transition delay and friction dragreduction.

Many experiments and numerical calculations using the Orr-Sommerfeld equation have been conducted to investigate thetransition delay on compliant surfaces (e.g., Refs. 6–8). Car-penter and Garrad 9) modeled a compliant surface to an elas-tic plate which is supported by an array of springs, and foundthat the transitional Reynolds number increased. They sug-gested that there exists an optimal combination of the mate-rial properties of a compliant surface for the turbulent tran-sition delay.

On the other hand, a friction drag reduction of more than20% was reported by Kramer 5) and Chu and Blick.10) How-ever, the reliability of these results is open to question withrespect to the accuracy of the measurement. No detailed data

† e-mail address: tendo@postman.riken.go.jp

has been obtained, because there are many difficulties in theconduct of experiments over compliant surfaces; i.e., the ma-terial properties are sensitive to changes in the environmentof the experimental facility, and there are the difficulties inmeasuring the flow field over a moving boundary.

The objective of the present study is to obtain detailed dataon the turbulent flow field over a deforming compliant surfacewith the aid of DNS. The effect of friction drag reductionusing a compliant surface is evaluated, and the mechanismof the drag reduction is investigated. It is also desired toobtain the optimal material properties of a compliant surfacefor drag reduction.

Numerical calculations

The governing equations are the incompressive Navier-Stokesequations and the continuity equation. The wall deforma-tion is described with a boundary-fitted coordinate systemfor a moving boundary. Periodic boundary conditions areemployed in the streamwise (x-) and the spanwise (z-) di-rections, while the nonslip boundary condition is imposed onthe top and bottom deformable walls.

A modified Crank-Nicolson-type fractional-step method 11) isused for the time advancement, while a second-order finitedifference scheme is employed for the spatial discretizationof both flow variables and metrics on a staggered mesh.12)

The pressure Poisson equation is solved with the multigridmethod.13) The successive overrelaxation (SOR) method isadopted for the finer and coarser meshes.

The sizes of the computational volume are 2.5πδ in thestreamwise direction and 0.75πδ in the spanwise direction(where δ is the channel half-width), which correspond toabout 1180 and 360 viscous length units, respectively. Thesizes are about 2.5 and 3.6 times larger than the so-calledminimal flow unit.14) Hereafter, all the parameters with a su-perscript + represent quantities nondimensionalized by thefriction velocity uτ in the plane channel flow and the kine-matic viscosity ν.

The numbers of grid points are 96, 97 and 96 in the x-, y- and

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z-directions, respectively. Uniform meshes are used in the x-and z-directions with spacings �x+ = 12 and �z+ = 3.7.A nonuniform mesh with a hyperbolic tangent distribution isused in the y- direction. The first mesh point located awayfrom the wall is given at y+ = 0.25.

The computational time step is 0.33ν/u2τ . The simulation is

performed under the constant flow rate condition throughoutthe present study. The Reynolds number based on the bulkmean velocity Ub and the channel width 2δ is 4600 (about 150based on uτ and δ for the plane turbulent channel flow). Aninstantaneous flow field of a fully developed turbulent channelflow was used as the initial condition.

Modeling of compliant surface

In the first stage of the present study, the material propertyof the compliant surface is assumed to be isotropic. Carpen-ter and Morris 15) found that the anisotropy of deformationof a compliant surface contributes to the stabilization of thetraveling-wave flutter (TWF). However, a simple model of acompliant surface is used in the present study to investigatethe influence of deformation of a compliant surface on theturbulent coherent structure. Each grid point on the compli-ant wall is assumed to move only in the y-direction, and thedeformation of the compliant surface is modeled by a spring,mass, and damper system (Fig. 1), The wall deformation isdetermined as follows:

m∂2yw

∂t2+ c

∂yw

∂t+ kyw

= ∆x∆z

�−p′

w + Tx∂2yw

∂x2+ Tz

∂2yw

∂z2

�, (1)

where m is the mass of a compliant surface for one calcula-tion grid volume, and is determined as m ≡ ρc · ∆x · ∆z · d(ρc and d are the density and the thickness of a compliantsurface, respectively). c and k are the damping parameterand the spring stiffness, respectively, p′

w denotes wall pres-sure fluctuation, and Tx and Tz are tensions employed in thestreamwise and spanwise directions, respectively.

It is shown that many parameters are necessary to be de-termined in Eq. (1). The material properties of the com-pliant surface used in the present study are δ = 0.04m,ν = 10−6m2/s, d = 28mm and fluid density ρf = 10

3 kg/m3,and are set to be constant. In the present study, the tensionsemployed in horizontal directions Tx and Tz are neglected forsimplicity. The remaining parameters c and k are under in-

Fig. 1. Schematic model for a compliant surface.

vestigation. Note that the natural circular frequency ω0 isrelated to Young’s modulus Y of the compliant surface bythe following equation:

ω0 =

rk

m=

sY∆x∆z/d

ρc d ∆x∆z=1

d

rY

ρc. (2)

Young’s modulus of elastic rubber is of the order of 1MPa,and that of human skin is 150MPa. In the present study,Young’s modulus (the spring stiffness) is determined to be rel-atively small, so that the deformation of the compliant surfaceis sufficiently large to investigate the influence of the defor-mation on turbulent coherent structures. Here, Y = 92.75Pa(k = 0.011 kg/s2) is adopted. Gad-el-Hak et al. 16) useda compliant surface that is a mixture of polyvinyl chlorideresin (PVC) and dioctyl phthalate with Young’s modulus of5 ∼ 12500Pa depending on the mixture rate. Therefore,the assumed material of the compliant surface used in thepresent study is not impractical. For the damping coeffi-cient, Case1: c = 0, Case2: c = 4.0× 10−4 kg/s, and Case3:c = 1.6× 10−3 kg/s are tested.

Results and discussion

The time trace of the volume-averaged streamwise pressuregradient, which is normalized by that in the plane channelflow, is shown in Fig. 2. Note that the form drag of the de-formable wall is found to be negligible, therefore, the frictionand total drags are employed synonymously in the presentstudy. In Case 1, both the friction drag and turbulent ki-netic energy increased from t+ = 300, and the calculationdiverged since the grid resolution becomes insufficient for an-alyzing the turbulence. On the other hand, in Cases 2 and 3,in which the appropriate damping coefficient is adopted, thefriction drag decreased slightly compared to that of the planechannel flow. The mean rates of reduction of the friction dragin Cases 2 and 3 during the period of t+ = 0 – 1000 are 1.7%and 2.7%, respectively.

When the compliant surface is applied to the turbulence con-trol device, the drag reduction rates obtained in Cases 2 and3 are not practical when the production cost is taken intoaccount. However, it is found that drag reduction is possibleusing the compliant surface that is deformed passively by thepressure fluctuation of the flow. Moreover, it is expected thata greater drag reduction rate is obtained by optimizing thematerial property of the compliant surface.

Fig. 2. Time trace of the normalized mean pressure gradient.

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Fig. 3. Time trace of the rms value of wall displacement.

Fig. 4. Time trace of the rms value of wall pressure fluctuation.

Figure 3 shows the time trace of the rms value of the walldisplacement y+

w . In Case 1, wall displacement becomes ex-cessively large by resonating with the pressure field. On theother hand, wall displacements in Cases 2 and 3 remain asdefinite values in the period of t+ > 200 due to the dampingof the compliant surface. The values are small and are ofthe order of y+

w rms ∼ 0.03. Although it is not shown here,the rms value of wall velocity v+

w rms is approximately 0.25,which is small.

The time traces of the rms value of wall pressure fluctuationsp+

w rms of the plane channel flow and Case 3 are shown inFig. 4. Although the wall velocity v+

w rms is small, the wallpressure fluctuation in Case 3 drastically increased comparedto that in the plane channel flow. Therefore, it is expectedthat by optimizing the material parameters, a small deforma-tion of the compliant surface can change turbulence coherentstructures drastically.

Figure 5 shows the profiles of the two-point correlation ofvelocity and pressure fluctuations in the x-direction for theplane channel flow and Case 3. Although there is small dif-ference in the velocity field, the pressure field differs signif-icantly. The two-point correlation of the pressure fluctua-tion in the spanwise direction is almost unity over the entirespanwise length of the calculation domain (not shown here).Therefore, it is expected that a large scale of the pressurefield becomes dominant over a deformable compliant surface.

The two-dimensional (x–z) spectrum of the wall displacementyw and velocity vw in Case 3 are shown in Fig. 6. The peak is

Fig. 5. Two-point correlations of velocity and pressure fluctuationsin streamwise direction at y+ = 15.

Fig. 6. Two-dimensional spectrum. (a) Wall displacement y+w and

(b) Wall velocity v+w .

observed at (κx, κz) = (1, 0) for both yw and vw, which showsthat the typical displacement and the velocity have scales ofapproximately 600 and 360 viscous lengths in the streamwiseand spanwise directions, respectively. And, the strength ofthe wall displacement and the velocity with a wave numberof more than (κx, κz) = (5, 5) are small. Thus, the deforma-tion of the compliant surface is relatively large. This fact isexpected to be closely related with the large-scale pressurefluctuation at y+ = 15 as shown in Fig. 5. However, it isyet unclear why the large-scale pressure fluctuation becomesdominant over the compliant surface; this is a subject of ourfuture study.

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(a)

(b)

Fig. 7. Instantaneous flow field of plane channel flow. (a) t+ = 4.86and (b) t+ = 602.1. Blue: u′+ = −3.5, Red: u′+ = 3.5, White:II′+ = −0.03.

Figure 7 shows the top views of instantaneous flow fields ofthe plane channel flow. The flow direction is from left toright. The red and blue contours are the high- and low-speedstreaks, respectively, while the white contour shows the vorti-cal structure (II ′+ = −0.03, II ′ is the second invariant of thedeformation tensor).17, 18) Vortical structures are mostly ob-served at the downstream edge of the meandering low-speedstreaks.19, 20) It is observed that regeneration of the vorticalstructure occurs continuously by activating the meanderingstreaks with its rotation.

Figure 8 shows the top view of the instantaneous flow fieldof Case 3. The wall shading describes the wall displacementyw. Since a marked drag reduction is not obtained in thepresent calculation, there is small qualitative difference invortical and streaky structures. However, the contours of thehigh- and low-speed streaks are found to be slightly small.This finding corresponds to the slight decrease in the two-point correlation of the streamwise velocity fluctuation in thestreamwise direction at y+ = 15 shown in Fig. 5. The typi-cal shape of wall deformation is a wave whose deformation ishomogeneous in the spanwise direction. This shape is qual-itatively the same as that visualized in the experiments ofHansen et al. 21) and Gad-el-Hak et al.16)

Figure 9 shows an instantaneous pressure field and the wallvelocity in Case 3. The flow is from the front of the figureto the back. Light blue and yellow contours show the low-and high- pressure fluctuations of p′+ = ±2.5, respectively.The wall color indicates the wall velocity in the y- direction,i.e., red is a positive wall velocity, and blue is a negative wallvelocity. It is observed that the wall velocity is positive nearthe low-pressure region on the wall, and vice versa. Therefore,it is expected that the wall velocity is in phase with the wallpressure in the present calculation.

Fig. 8. Instantaneous flow field of Case 3. For caption, see Fig. 7(b).

Fig. 9. Instantaneous wall velocity and pressure field in Case 3. Lightblue: p′+ = 2.5, Yellow: p′+ = −2.5.

In order to examine flow characteristics associated with thehigh- and low- pressure regions on the wall, a conditionalaverage of a plane channel flow is made, given the conditionof the signs of p′+

w on the wall. A threshold of |p′+w| = 2.0

is employed to extract strong events only. Figure 10 showscontours of the conditionally averaged streamwise vorticity <ω+

x > at y+ = 15 near the low-pressure region. The detectionpoint is at the center of the figure. The wall-normal velocitycomponents above the high- and low-pressure regions on thewall are negative and positive, respectively, indicating thatsweep (Q4) and ejection (Q2) events occur, respectively.

When there is no phase difference between the wall velocityand the wall pressure fluctuation, the compliant wall mayhave a positive wall-normal velocity near the low-pressure re-gion where the ejection event occurs, as schematically shownin Fig. 11. A negative wall-normal velocity occurs near thesweep event. It is expected that the wall velocity wouldstrengthen the rotation of vortices accompanied by the ejec-tion and sweep events, and contributes to the increase in theReynolds shear stress.

Jeong et al.19) proposed a schematic model of the regenera-tion cycle of quasi-streamwise vortices near the wall as shownin Fig. 12. A child quasi-streamwise vortex is continuouslyregenerated upstream of the parent vortex. The stream-wise vorticity of the child vortex has a sign opposite that ofits parent vortex. Therefore, when the wall movement thatstrengthens the rotation of the parent vortex occurs beneaththe child vortex, it should encounter the rotation of the childvortex. It is expected that the compliant surface suppressesthe regeneration cycle of the quasi-streamwise vortex, and asa result, drag reduction is possible using a compliant surface

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Fig. 10. Contours of the conditionally averaged streamwise vorticityat y+ = 15 given the conditions of (a) p′+ = −2.5 and (b)p′+ = 2.5.

Fig. 11. Schematic of flow field near low-pressure region.

Fig. 12. Schematic model of regeneration cycle proposed by Jeonget al.19)

whose material properties are optimized so that a phase delayoccurs between the wall velocity and the wall pressure.

Conclusions

DNS of turbulent channel flow over compliant walls, whichare deformed passively by the wall pressure fluctuation, wasperformed to evaluate the friction drag reduction effect. Inthe present study, the compliant surface is modeled by aspring, mass and damper system, for simplicity.

The typical shape of the deformation is a wave whose pe-riod in the streamwise direction is approximately 1200 vis-cous length units. Approximately 2–3% drag reduction isobserved in the present study.

Although there are small effects on the velocity field, thepressure field is drastically affected. Therefore, it is expectedthat a compliant wall that has an impact on the velocity fieldis possible, when the material properties of the compliantsurface are optimized.

Based on the conditionally averaged flow field, the ejec-tion and sweep events occur upstream of the high- and low-pressure regions on the wall, respectively. Therefore, it issupposed that a phase delay in the spring, mass, and dampersystem is necessary to suppress the rotation of the quasi-streamwise vortical structure accompanied by ejection andsweep events.

References1) M. J. Walsh: AIAA Paper 82-0169 (1982).2) Y. Suzuki and N. Kasagi: AIAA. J. 32, 1781 (1994).3) H. Choi, P. Moin, and J. Kim: J. Fluid Mech. 255, 503

(1993).4) J. Gray: J. Exp. Biol. 50, 233 (1936).5) M. O. Kramer: Nav. Eng. J. 72, 25 (1960).6) D. M. Bushnell, J. N. Hefner, and R. L. Ash: Phys. Fluids

20, S31 (1977).7) J. J. Riley, M. Gad-el-Hak, and R. W. Metcalfe: Annu. Rev.

Fluid Mech. 20, 393 (1988).8) M. Gad-el-Hak: Appl. Mech. Rev. 49, S147 (1996).9) P. W. Carpenter and A. D. Garrad: J. Fluid Mech. 155, 465

(1985).10) H. H. Chu and E. F. Blick: J. Spacecr. Rockets 6, 763 (1969).11) H. Choi and P. Moin: J. Comput. Phys. 113, 1 (1994).12) Y. Mito and N. Kasagi: Int. J. Heat Fluid Flow 19, 470

(1998).13) A. O. Demuren and S. O. Ibraheem: AIAA J. 36, 31 (1998).

14) J. Jimenez and P. Moin: J. Fluid Mech. 225, 213 (1991).15) P. W. Carpenter and P. J. Morris: J. Fluid Mech. 218, 171

(1990).16) M. Gad-el-Hak, R. F. Blackbwelder, and J. J. Riley: J. Fluid

Mech. 140, 257 (1984).17) M. S. Chong, A. E. Perry, and B. J. Cantwell: Phys. Fluids

A 2, 765 (1990).18) N. Kasagi, Y. Sumitani, Y. Suzuki, and O. Iida: Int. J. Heat

Fluid Flow 16, 2 (1995).19) J. Jeong, F. Hussain, W. Schoppa, and J. Kim: J. Fluid Mech.

332, 185 (1997).20) T. Endo, N. Kasagi, and Y. Suzuki: Int. J. Heat Fluid Flow

21, 568 (2000).21) R. J. Hansen, D. L. Hunston, C. C. Ni, N. M. Reischman,

and J. W. Hoyt: in Viscous Flow Drag Reduction, edited byG. R. Hough (AIAA, New York, 1980), p. 439.

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