Wall-bounded turbulent flow and equilibrium states in plane Couette flow under spanwise wall oscillation

This talk concerns skin-friction drag of turbulent and equilibrium states in plane Couette flow subjected to spanwise-wall oscillations. We set the configuration to the minimal flow unit at $Re=400$ ($\lambda_z^+\approx 110$ the spanwise wavelength in viscous units) to maintain only one integral length scale dynamics. We use HKW box (Hamilton 1995) as it sustains turbulence for a very long time at $Re=400$. The control is parametrized by the amplitude $A_w$ and the period $T$. The wall oscillation provokes a significant drag reduction for the fully developed turbulence. As a consequence, the control delays the transition to turbulence and reduces the turbulence lifetime. We have considered three equilibrium solutions EQ$1$, EQ$4$, and EQ$7$, each of which has a lower and upper branch that rises from a saddle-node bifurcation. The control over a wide range of $Re$ induces a significant drag reduction for the upper solution and has a minor effect on the lower branch. Near the saddle-node bifurcation, the lower branch skin-friction increases with control—that of the upper branch decreases to merge with the lower solution at the optimal $A_w$ and disappear. Therefore, the critical Reynolds number $Re_c$ of the saddle-node bifurcations moves to a higher $Re$.