Proceedings of the Fourth International Conference on Fluid Mechanics, July 28∼31, Dalian, China c 2003 Tsinghua University Press & Sringer-Verlag °

STUDY ON REYNOLDS STRESS TRANSPORTATION IN A TURBULENT CHANNEL FLOW WITH SPANWISE OSCILLATING WALL Wei-Xi Huang, Chun-Xiao Xu, Gui-Xiang Cui and Zhao-Shun Zhang Department of Engineering Mechanics, Tsinghua University, Beijing 100084, P.R. China Abstract: An investigation into a turbulent channel flow subjected to spanwise wall oscillation is carried out via direct numerical simulation. By wall oscillation, turbulence is suppressed and friction drag is reduced. The transportation of Reynolds stresses under the influence of the oscillating wall is analyzed at the initial and statistically stationary stages to further disclose the mechanism of turbulence suppression and drag reduction. It is found that the moving wall can cause the transient growth of spanwise velocity fluctuations only at the very beginning of wall oscillation and thereafter the pressure-strain terms play an important role. The attenuation of turbulence intensities and skin friction at the statistically stationary stage is mainly due to the sustained reduction of the pressure-strain terms. Keywords: wall oscillation, drag reduction, wall turbulence.

1.INTRODUCTION The control of turbulence to achieve drag reduction is becoming a hot spot in turbulence research for its great potential of application. It has been shown that the skin-friction can be effectively reduced when the wall is oscillated in spanwise direction[1][2][3] . This newly devised control scheme need additional energy to move the wall, but requires no feedback of flow information, and hence it has drawn much attention in recent years. To disclose the mechanism of turbulence suppression and skin-friction reduction from the point of view of Reynolds stress transportation, a turbulent channel flow subjected to spanwise wall oscillation is studied by direct numerical simulation in this paper. The Reynolds stress transport equation is analyzed at both initial and statistically stationary stages of wall oscillation. The response of each budget terms to the oscillating wall is investigated, and the interrelationship between Reynolds stress components is explored to elucidate the mechanism of the attenuation of turbulence intensities and skin-friction drag by spanwise wall oscillation. 2.NUMERICAL METHOD The governing equations are the incompressible Navier-Stokes equations, the oscillation of the wall is directly considered by boundary conditions. For comparison, the upper wall oscillates in spanwise direction and the lower wall is fixed. The Fourier-Chebyshev spectral method is adopted in spatial discretization, and the third-order time-splitting method is used in time

advancement. To validate the code, a turbulent channel flow with fixed walls (with 128 × 129 × 128 grids ) is simulated. The statistical results are compared with those of Kim et al.[4] and they show a good agreement. The simulation is carried out with constant mass flux. The Reynolds number based on mean velocity Um and half channel width H is Rem = Um H/ν = 2666. The size of computational domain is 4πH × 2H × 2πH, and the corresponding grid numbers are 64×65×64 respectively. The period of wall oscillation is T + = 90, and the amplitude is A+ = 15.4. 3.RESULTS The transport equations of Reynolds stresses are Pij + Dij + P Sij − εij = Cij +

∂ 0 0 hu u i, ∂t i j

(1)

where Pij , Dij , P Sij , εij and Cij stand for the terms of production, diffusion, pressure-strain, dissipation and convection respectively. They can be expressed as Pij = −hu0j u0k i

∂Ui ∂Uj − hu0i u0k i , ∂xk ∂xk

¸ · ∂hu0i u0j i 1 0 0 ∂ 1 0 0 0 0 0 Dij = −hui uj uk i + ν − hui p iδjk − huj p iδik , ∂xk ∂xk ρ ρ ¿ 0µ 0 ¶À ∂u0j p ∂ui P Sij = + , ρ ∂xj ∂xi ¿

(2) (3)

(4)

À

εij = 2ν

∂u0i ∂u0j · ∂xk ∂xk

Cij = Uk

∂hu0i u0j i = 0, ∂xk

,

(5)

(6)

From the above equations, it can be deduced that ∂W/∂y caused by spanwise wall oscillation acts directly on the production terms of hu0 w0 i, hv 0 w0 i and hw02 i. Since hu0 w0 i and hv 0 w0 i is small compared with other terms, and also they are not directly related with skin-friction, we will only concentrate on the transportation of hw02 i, hv 02 i, hu0 v 0 i and hu02 i in the analysis. The effect of spanwise wall oscillation on hw02 i transportation is shown in Fig. 1. It can be seen that all the budget terms at the oscillating wall side are globally suppressed compared with those at the fixed wall side. Because of P33 > 0 near the oscillating wall (y + < 30), the budget of hw02 i transport equation in this region differs from that near the fixed wall. Near the oscillating wall, P33 and pressure-strain term P S33 both act as the contributor to hw02 i, balancing with diffusion term D33 and dissipation term ε33 . Far from the wall, the balance is mainly between P S33 and ε33 . Compared with the reduction of P S33 , the increment of P33 is small, as a result, hw02 i is reduced in the whole region. Investigation at the initial stage of applying wall oscillation can further depict the influence of 0 oscillating wall on the transportation of hw02 i. The response to wall oscillation of wrms and its

production term P33 , pressure-strain term P S33 and dissipation term ε33 is shown in Fig. 2. For

P33c

0.1

D33c

Budget

ε33c PS33c

0

2

P33n D33n ε33n

-0.1

PS33n 0

10

20 y+

30

40

Figure 1: Budget of hw02 i transport equation at the statistically stationary stage.

1.2

(a)

0.06

0.04 P33

w’rms

0.8

[0,T/2] [T/2,T] [T,3T/2] [3T/2,2T] No Control

0.4

0 0

0.06

(b)

10

20 y+

30

0.02

0 0

40

0

(c)

10

20 y+

30

40

10

20 y+

30

40

(d)

-0.04 ε33

PS33

0.04

-0.08

0.02

0 0

10

20 y+

30

40

-0.12 0

0 Figure 2: Time evolution of (a) wrms , (b) P33 , (c) P S33 and (d) ε33 in the first two periods of wall

oscillation.

clarity, the first two oscillation periods are equally divided into four time intervals, and the above terms are averaged in each time interval, which equals to half oscillation period. As is shown in 0 Fig. 2(a), wrms increases slightly in the first half-period of oscillation, and reaches its maximum

in the second half-period, and then it decreases. Fig. 2(b) shows the changes of P33 , which reflects the direct influence of introducing ∂W/∂y by oscillating the wall. P33 increases to its maximum in the first half-period. In the second half-period, its peak value decreases, but it is obviously increased in the region of y + > 15. It continues decreasing in the third and forth half-period. Unlike production term P33 , the pressure-strain term P S33 is reduced monotonically from the very beginning of oscillation, while the dissipation term increases in the first two half-periods, as can be seen in Fig. 2(c) and (d). According to the above analysis, it is clear that the appearance 0 of P33 leads to the increase of wrms in the first two half-periods, in which time, the reduction

of P S33 is less than the increment of P33 , leading to the increase of hw02 i. In the following two half-periods, the contribution of P33 decreases, and P S33 is reduced more, causing the reduction of hw02 i. The analysis at the initial stage of wall oscillation illustrate that the production term P33 is only important at the very beginning of oscillation. With time increasing, its influence is weakening, and P S33 play a more important role, and becomes the main contributor to hw02 i. The transportation of hv 02 i, hu0 v 0 i and hu02 i are also studied at the initial and statistically steady stages of wall oscillation. It can be shown that in hv 02 i transportation, pressure-strain term P S22 is the main contributor. It is monotonically reduced from the beginning, and causes hv 02 i decrease all the time. The reduction of hv 02 i is amplified by ∂U/∂y, and causes a considerable reduction in the production term of hu0 v 0 i, P12 . The reduction of hu0 v 0 i is also amplified by ∂U/∂y in the production term of hu02 i, P11 , and causes the suppression of hu02 i. Through the analysis of Reynolds stress transportation at the initial and statistically stationary stage of wall oscillation, the key role that pressure-strain term plays in drag reduction and turbulence suppression by spanwise wall oscillation is disclosed. ACKNOWLEDGEMENT The support by National Science Foundation of China is acknowledged(Grants 10072032 and 10232020). REFERENCES 1. Jung W J, Mangiavacci N and Akhavan R. Suppression of turbulence in wall-bounded flows by high-frequency spanwise oscillations. Phys. Fluids 4, 1605 (1992). 2. Quadrio M and Sibilla M. Numerical simulation of turbulent flow in a pipe oscillating around its axis. J. Fluid Mech. 424 217 (2000). 3. Choi J-I, Xu C-X and Sung H-J. Drag Reduction by Spanwise Wall Oscillation in Wall-Bounded Turbulent Flows. AIAA J. 40(5) 842 (2002). 4. Kim J, Moin P and Moser R K. Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133 (1987).