Direct numerical simulation of backward-facing step turbulent flow controlled by wave-machine-like traveling wave

A direct numerical simulation of turbulent flow over a backward-facing step is performed at Reynolds number of ${\mathrm{Re}}_{\mathrm{b}}=5600$ based on inflow bulk velocity and channel half-width at the inlet, with an expansion ratio of $\text{ER}=1.2$. To control the flow separation, a wall-normal body force in the form of a wave-machine-like traveling wave, which has streamwise and spanwise periodicity, is applied to the top surface of the step. To clarify the effect of spanwise periodicity on the reattachment length, spanwise-uniform traveling waves (SUTW) and wave-machine-like traveling waves (WMTW) are compared. The maximum reduction rates of the reattachment length in SUTW and WMTW are 51.4% and 58.9%, respectively. SUTW and WMTW exhibit a common flow behavior, in which the recirculation bubble is periodically released in time and in the streamwise direction, contributing significantly to the reduction of the reattachment length. However, spanwise periodicity produces differences in the flow structure. In SUTW, all flow structures become spanwise-uniform. In WMTW, a pair of longitudinal vortices is generated above the step and in the recirculation bubble and the released separation bubble. The pair of longitudinal vortices causes three effects that reduce the reattachment length—a decrease in the streamwise length of the secondary bubble, an enhancement of the negative wall-normal velocity, and an increase in Reynolds shear stress.

Figure 1

Schematic view of the computational domain and traveling wave control (control parameters are colored in red).

Figure 2

Flow statistics in the driver part: (a) mean streamwise velocity profile; (b) RMS values of streamwise, wall-normal, and spanwise velocity ($u_{i,\text{rms}}^{\prime +}=\sqrt{{\overline{u_i^{\prime +}{u}_i^{\prime +}}}^{txz}}$).

Figure 3

Flow statistics in the main part: (a) skin-friction coefficient and (b) pressure coefficient averaged in time and $z$ direction.

Figure 4

Skin-friction coefficient (left) and pressure coefficient (right) averaged in time and $z$ direction: (a) w/o control; (b) controlled.

Figure 5

Skin-friction coefficient and reattachment length: (a) definition of reattachment length determined from the skin-friction coefficient averaged in time and $z$ direction; effect of wavespeed (b) and spanwise wavelength (c), streamwise wavelength (d), amplitude (e), and penetration length (f) on reattachment length. In (b–f), the other control parameters are fixed at $A=2, \mathrm{\Delta }=0.1, {\lambda }_x=2\pi , {\lambda }_z=\pi$, and $c=0.4$.

Figure 6

Skin-friction coefficient (a) and pressure coefficient (b): black, w/o control; red, SUTW; blue, WMTW.

Figure 7

Wall-normal velocity averaged in time and $z$ direction: (a) w/o control; (b) SUTW; (c) WMTW.

Figure 8

Reynolds shear stress averaged in time and $z$ direction: (a) w/o control; (b) SUTW; (c) WMTW.

Figure 9

Instantaneous vortical structures visualized by the second invariant of the velocity gradient tensor. The threshold is $Q=4$, and the contour shown above the step is wall-normal body force $f_y$ at $y=0.405$. The time is $t=518.4$: (a) w/o control; (b) SUTW at $c=0.4$; (c) SUTW at $c=-0.4$; (d) WMTW at $c=0.4$; (e) WMTW at $c=-0.4$.

Figure 10

Coherent component of vortical structures around the separation point. The structures are visualized by the second invariant of the phase-averaged velocity gradient tensor, and the threshold is $\langle Q\rangle =0.1$. The contour shown above the step is wall-normal body force $f_y$ at $y=0.405$, and the surface of the vortical structures is colored with phase-averaged wall-normal velocities $\langle v\rangle$. Left column is SUTW at $c=0.4$ and right column is WMTW at $c=0.4$: (a) ${\varphi }_t=0$; (b) ${\varphi }_t=T/4$; (c) ${\varphi }_t=T/2$; (d) ${\varphi }_t=3T/4$.

Figure 11

Coherent component of vortical structures around the separation point. The structures are visualized by the second invariant of the phase-averaged velocity gradient tensor, and the threshold is $\langle Q\rangle =0.1$. The contour shown above the step is wall-normal body force $f_y$ at $y=0.405$, and the surface of the vortical structures is colored with phase-averaged wall-normal velocities $\langle v\rangle$. Left column is SUTW at $c=-0.4$ and right column is WMTW at $c=-0.4$: (a) ${\varphi }_t=0$; (b) ${\varphi }_t=T/4$; (c) ${\varphi }_t=T/2$; (d) ${\varphi }_t=3T/4$.

Figure 12

Coherent component of vortical structures around the separation point in WMTW at $c=0.4$. The structures are visualized by the second invariant of the phase-averaged velocity gradient tensor, and the threshold is $\langle Q\rangle =0.1$. The contour shown above the step is wall-normal body force $f_y$ at $y=0.405$, and the surface of the vortical structures is colored with phase-averaged wall-normal velocities $\langle v\rangle$. Left column is top view and right column is viewed from the lower wall: (a, b) ${\varphi }_t=0$; (c, d) ${\varphi }_t=T/4$.

Figure 13

Phase-averaged streamwise vorticity at ${\varphi }_t=0$ and $x=10.46$ in WMTW with $c=0.4$. Arrows are the phase-averaged velocity vector of $\langle w\rangle$ and $\langle v\rangle$.

Figure 14

Coherent component of vortical structures around the separation point in WMTW with $c=0.4$. The structures are visualized by the second invariant of the phase-averaged velocity gradient tensor, and the threshold is $\langle Q\rangle =0.1$. The contour shown above the step is wall-normal body force $f_y$, and the surface of the vortical structures is colored with phase-averaged wall-normal velocities $\langle v\rangle$. The structures are viewed from spanwise direction: (a) ${\varphi }_t=0$; (b) ${\varphi }_t=T/4$ (c) ${\varphi }_t=T/2$; $\left(d\right){\varphi }_t=3T/4$.

Figure 15

Time trace of the reattachment length in (a) w/o control, (b) SUTW at $c=0.4$, and (c) WMTW at $c=0.4$. Left column is $z=\pi /4$ and right column is $z=\pi /2$.

Figure 16

Phase-averaged reattachment length in (a) SUTW and (b) WMTW.

Figure 17

Velocity vector averaged in phase and $z$ direction $({\overline{\langle u\rangle }}^z,{\overline{\langle v\rangle }}^z)$ around the separation point in SUTW. The contour shown in the step is wall-normal body force $f_y$: (a) ${\varphi }_t=0$; (b) ${\varphi }_t=T/4$; (c) ${\varphi }_t=T/2$; (d) ${\varphi }_t=3T/4$.

Figure 18

Phase-averaged velocity vector $(\langle u\rangle ,\langle v\rangle )$ around the separation point at ${\varphi }_z={\lambda }_z/4$ in WMTW with $c=0.4$. The contour shown in the step is wall-normal body force $f_y$: (a) ${\varphi }_t=0$; (b) ${\varphi }_t=T/4$; (c) ${\varphi }_t=T/2$; (d) ${\varphi }_t=3T/4$.

Figure 19

Phase-averaged velocity vector $(\langle u\rangle ,\langle v\rangle )$ around the separation point at ${\varphi }_z={\lambda }_z/2$ in WMTW with $c=0.4$. The contour shown in the step is wall-normal body force $f_y$: (a) ${\varphi }_t=0$; (b) ${\varphi }_t=T/4$; (c) ${\varphi }_t=T/2$; (d) ${\varphi }_t=3T/4$.

Figure 20

Phase-averaged velocity vector $(\langle v\rangle ,\langle w\rangle )$ and phase-averaged wall-normal velocity $\langle v\rangle$ in WMTW with $c=0.4$. The contour shown in the step is wall-normal body force $f_y$. Left column is at the separation point $x=3\pi$ and right column is at the reattachment length $x=10.46$: (a, b) ${\varphi }_t=0$; (c, d) ${\varphi }_t=T/8$; (e, f) ${\varphi }_t=T/4$; (g, h) ${\varphi }_t=3T/8$.