ABSTRACT

INTRODUCTION
The flow separation and reattachment can be frequently found in our living environment (e.g., the flows around ve- hicles and buildings, etc.).  However, an energy loss occurs at the flow separation point due to the increased velocity and pressure fluctuations caused by the vibration of the separation bubble.  On the one hand, since the turbulence is promoted, these phenomena help to improves heat transfer and enhances mixing effects.  Therefore, understanding both phenomena is required in engineering design. The flow over the backward-facing step (BFS), in which there is only one separation region with reattachment, has been studied well due to its simplicity.  In particular, the reattach- ment length has been investigated extensively because of the occurrence of important engineering phenomena (i.e., the in- creased pressure resistance, the accelerated heat transfer, the noise generation, etc.). Ötügen (1991) investigated the effect of the expansion ratio ER on the reattachment length, and this experiment showed that the reattachment length increases with increasing ER.  Recently, Nadge & Govardhan (2014) con- ducted a parametric study to comprehensively investigate the effects of the expansion ratio and Reynolds number on the reat- tachment length. Several methods for controlling the turbulent flow have been proposed, and one of them, traveling wave control, has a high drag reduction effect.  Min et al. (2006) performed a direct numerical simulation to investigate the effect of travel- ing wave blowing and suction control in a channel flow. They achieved a drag reduction rate of about 30% by upstream trav- eling waves.  It is known that the upstream traveling wave in blowing and suction has the pumping effect (Hœpffner & Fuk- agata (2009)). However, since blowing and suction are diffi- cult to make practical, Mamori & Fukagata (2014) generated traveling waves by body force and investigated the effect of 1 this control in a turbulent channel flow.  They reported that the pumping effect of the traveling wave enhanced the vortical structures. It is known that the separation bubbles have a great in- fluence on the oscillation of the reattachment length (Schäfer et al. , 2009).  Therefore, we expect that the pumping effect of the traveling wave control will enhance the mixing of the separation bubbles, which will reduce the reattachment length. Morita et al. (2022) reported that a traveling wave-like body force control reduces the reattachment length in the BFS tur- bulent flow.  The traveling wave control had a uniform input in the spanwise direction. In order to reduce the reattachment length further, it is necessary to promote turbulence in the re- circulation region, and we employ a wave-machine-like trav- eling wave (Nabae & Fukagata, 2020). The control has spatial periodicity in the spanwise direction, which is expected to pro- mote turbulence and affect the reattachment length. In this study, we perform direct numerical simulations of the turbulent flows over the backward-facing step. To control the flow separation, the spanwise uniform traveling wave-like body force and the wave-machine-like traveling body force are applied to the top of the step.  The purpose of this study is to investigate the effect of the controls on the reattachment length and to clarify the mechanism.  Especially, the effect of the wavespeed is discussed in this paper.
Direct Numerical Simulation
Direct numerical simulations of turbulent flows over the backward-facing step are performed.  The governing equa- tions are incompressible continuity and Navier-Stokes equa- tions, and these equations are defined as follows,
where, x i u i , and f i (i = 1 ～ 3) are the coordinates, the veloc- ities, and the body force term in the streamwise, wall-normal and spanwise direction, respectively. In addition, t is time, and p is pressure. The coordinates are interchangeably used as x , y and z , the corresponding velicity components are used as u , v and w , respectively, The Reynolds number defined by the channel half-width at the inlet δ and the inflow bulk velocity 2ub is Reb = 2u g δ* /v* = 5600. Here, the asterisk denotes the dimensional variable. The flow condition is the constant flow rate. We employ the DNS code for the channel flows by Fuka- gata & Kasagi (2006). The governing equations are discretized using an energy-conserving second-order accurate central dif- ference method for space.  For time integration, the second- order accurate Crank- Nicolson scheme and the low-storage third-order accurate Runge–Kutta scheme are used. Figure l shows the computational domain of the backward-facing step flow and the driver.  Here, the driver is the fully developed turbulent channel flow and is used to define the inflow velocity of the turbulent flow over the backward-facing step.  The computational domain is set to (L x,L y,L z ) = (6π,2.4,π) and (L q,L q,L q) = (2π,2,π), and the number of grids is set to ( N x , N z ) = (768, 128, 128) and ( N đ , N đ , N đ ) = (128,96,128). The streamwise length in a step is L x = 2 π , the step height is h = 0.4. The expansion ratio is ER = L γ/(L γ − h) = 1.2. The superscript d denotes the physi- cal quantity at the driver. In the boundary condition of the backward-facing step flow, the periodic boundary condition is applied in z -direction. As the inflow velocity, we use the velocity distribution of the fully developed flow generated in the driver part. The convec- tion condition is imposed on the outlet.  The non-slip condi- tion is imposed on the wall.  The step is represented by the immersed boundary method proposed by Kim et al. (2001). They used the fractional-step method, while the authors used the SMAC method. In the driver, we impose periodic bound- ary conditions in x and z directions and, the no-slip condition on the wall. In order to control the flow separation, we apply the span- wise uniform traveling wave-like body force and the wave- machine-like traveling body force on the top of the step (Fig. 1). Since the body acts in the region near the wall, the body (b) force control is assumed to exponentially decay along the wall- normal direction. Therefore, each controls are defined as, (3) (4) The control parameters are the amplitude A , the penetration length Δρ , and the wavespeed c . The wavelengths in the main and spanwise directions are λ π and λ π , respectively.  In this study, we use a parameter set of A = 2, Δρ = 0.1, and λ x = 2π, which significantly reduced the reattachment length in Morita et al. (2022). The wavelength in the spanwise direction is fixed at λ ε = π .  We investigate the effect of wavespeed c on the reattachment length. The time averaged skin-friction coefficient C f on the lower wall of the uncontrolled flow is shown in Fig. 2. The definition of the skin-friction coefficient C f is,
Here, τ * is the lower wall shear stress, and ρ * is the density. The horizontal axis is normalized by the step height h , and the origin x /h = 0 is the separation point.  The reattachment length x r is the distance from the origin to the reattachment point of C f = 0 (except for the secondary recirculation region).
Results and discussion
Figure 3 shows the effect of wavespeed on the reattach- ment length in the Traveling wave and the WM-like wave. Here, "Traveling wave" means the spanwise uniform travel- ing wave-like body force, and "WM-like wave" means wave- machine-like traveling body force.  Both the Traveling wave and the WM-like wave reduce the reattachment length regard- less of the wavespeed. The minimum x r is obtained at c = 0.4 in both contols. The reattachment length of the WM-like wave was shorter than that of the Traveling wave at − 1 < c < 1.5. In the WM-like wave, the direction of wave propagration had little effect on the reattachment length.  On the other hand, the effect of wavespeed in the Traveling wave on the reattach- ment length was different among the three cases: downstream traveling wave ( c > 0), upstream traveling wave ( c < 0), and stationary wave ( c = 0). For the downstream traveling wave (c > 0), the reattachment length is decreased as the wavespeed decrease.  For the upstream traveling wave ( c < 0), the reat- tachment length is kept constant.  Compared with that in the traveling wave ( c̸= 0), the reattachment length in the station- ary wave (c = 0) increased. The effect of the Traveling wave is discussed by compar- ing the results of the case without the control and the following Case 1 ～ 3: Cases 1, stationary wave-like body force (c = 0) as shown in Fig. 1(a); Case 2, stationary wave with the oppo- site phase to Case 1; Case TW, traveling wave ( c = 0.6) in the downstream direction. Figure 4 shows the instantaneous streamwise velocity dis- tribution. In Case 1, the reattachment length and recirculation region decrease because the control generated backward flow on the step and suppressed the velocity near the separation point. In Case 2, the upward flow is induced near the sepa- ration point, and the recirculation region is enlarged. This en- hanced the fluid mixing caused by the shear between the main flow and the backward flow in the recirculation region. In Case TW, the strong vortical structure is generated in the recircula- tion region and is released as shown in Fig. 5, which causes a significant decrease in the reattachment length. Figure 6 shows the distribution of the Reynolds shear stress ( − u ν ). Here, the instantaneous velocity u ; is decom- posed into the velocity averaged in the time and spanwise di- rection īī , and the variation ι ′ .
In Case 1, the velocity fluctuation decreases as compared to the uncontrolled case; in Case 2 and Case TW, the velocity fluctuation increases. In order to investigate the relationship between the release period of the vortical structure and the period of the traveling wave (T = λ / | c |), we analyze the phase-averaged statistics. The phase average is defined as follows,
where, τ = t − nT (0 < τ < T,n ∈ Z ) is the time within one traveling wave cycle. N p is the number of data used for phase average. Figure 7 shows the phase-averaged reattachment length in Case TW. It shows the temporal variation of the reattach- ment length within one cycle of the traveling wave. The sep- aration bubble is released when the reattachment length de- creases rapidly.  The release period of the separation bubble corresponds with the period of the traveling wave. Therefore, it is considered that the release period of the vortical structure can be controlled by changing the wavelength and wavespeed in the traveling wave-like body force control. The mechanism of the reattachment length reduction in each case was different.  In Case TW, the pumping effect in the traveling wave generated a strong vortical structure in the recirculation region, which caused the release of the vortical structure, thus reducing the reattachment length. Phase anal- ysis showed that the release period of the vortical structure corresponds with the period of the traveling wave. The control effect was different in the stationary wave because the input phase distribution was different.  In Case 1, the reverse flow region generated on the step suppressed the velocity near the separation point and the velocity fluctuation in the recircula- tion region, which caused the reduction of the reattachment length. In Case 2, reattachment length was reduced because of the enhanced mixing of the separation vortical structure, but the recirculation region was enlarged because of the upward flow induced near the separation point. Therefore, the reduc- tion rate of the reattachment length was smaller than those in the other cases. Figure 8 compares the turbulent vortical structures.  The vortical structures are visualized by the isosurface of the sec- ond invariant of the velocity deformation tensor (i.e., so-called the Q value). The wavespeed is set at c = 0.4 for both cases. In the uncontrolled case, there are many vortical structures in the separation bubble compared with the upper surface of the step. In the Traveling wave, the large and spanwise uniform vortical structure is observed at the top of the step. In the case of WM- like wave, the vortical structures distribute non-uniformly in the spanwise direction, unlike the Traveling wave.  In addi- tion, the streamwise vortical structure is generated on the up- per surface of the step, which may work as vortex generators. Therefore, the WM-like wave control was more effective in reducing the reattachment length than the spanwise uniform traveling wave because the control generates the streamwise vortical structure on the top surface of the step.
Conclusion
Direct numerical simulations of the incompressible turbu- lent flow over the backward-facing step are performed. In or- der to control the separation flow, the spanwise uniform travel- ing wave-like body force control and wave-machine-like trav- eling body force are applied to the top surface of the step.  In this study, we investigate the effects of wavespeed c on the reattachment length. Compared with the uncontrolled case, the reattachment length decreased in all the control cases. How- ever, the mechanism was different and depended on the con- trol. In the spanwise uniform stationary wave, the control ef- fect was different because the input phase distribution was dif- ferent.  The velocity fluctuation in the recirculation affected the increase or decrease of the reattachment length.  In the spanwise uniform cases, the control generated a strong vortical structure in the recirculation region, which caused the release of the vortical structure, thus reducing the reattachment length. The release period of the vortical structure corresponded with the period of the traveling wave. In the WM-like wave control, the control generates the streamwise vortical structure on the top surface of the step. Therefore, mixing of separation bubble is more enhanced and the reattachment distance is shortened.
Acknowledgments
We would like to thank Mr.  Kimura for his significant contribution to this research.