Exact coherent states in plane Couette flow under spanwise wall oscillation

A set of several exact coherent states in plane Couette flow is computed under spanwise wall oscillation control, with a range of wall oscillation amplitudes and periods (A(w), T). It is found that the wall oscillation generally stabilises the upper branch of the equilibrium solutions and achieves the corresponding drag reduction, while it influences modestly the lower branch. The stabilisation effect is found to increase with the oscillation amplitude with an optimal time period around T+ approximate to 100. The exact coherent states reproduce some key dynamical behaviours of streaks observed in previous studies, while exhibiting the rich coherent structure dynamics that cannot be extracted from a phase average of turbulent states. Visualisation of state portraits shows that the size of the state space supporting turbulent solution is reduced by the spanwise wall oscillation, and the upper-branch equilibrium solutions become less repelling, with many of their unstable manifolds being stabilised. This change of the state space dynamics leads to a significant reduction in lifetime of turbulence. Finally, the main stabilisation mechanism of the exact coherent states is found to be the suppression of the lift-up effect of streaks, explaining why previous linear analyses have been so successful for turbulence stabilisation modelling and the resulting drag reduction.

Automated summary

A set of exact coherent states in plane Couette flow is computed under spanwise wall oscillation control, which stabilises the upper branch of equilibrium solutions and achieves corresponding drag reduction. The stabilisation effect increases with the oscillation amplitude, with an optimal time period around T+ approximately 100. The exact coherent states reproduce key dynamical behaviours of streaks observed in previous studies and exhibit rich coherent structure dynamics. The size of the state space supporting turbulent solution is reduced by the spanwise wall oscillation, leading to a significant reduction in turbulence lifetime. The main stabilisation mechanism is the suppression of the lift-up effect of streaks.