Turbulent skin-friction drag reduction by spanwise wall oscillation with generic temporal waveform

To generalize the well-known spanwise-oscillatingwall technique for drag reduction, non-sinusoidal oscillations of a solid wall are considered as a means to alter the skin-friction drag in a turbulent channel flow. A series of Direct Numerical Simulations is conducted to evaluate the control performance of nine different waveforms, in addition to the usual sinusoid, systematically changing the maximum wave amplitude and the period for each waveform. The turbulent average spanwise motion is found to coincide with the laminar Stokes solution that can be constructed, for the generic waveform, through harmonic superposition. A newly defined penetration depth of the Stokes layer is then used to build a simple tool that allows predicting turbulent drag reduction and net energy saving rate for any waveform. Among all the cases considered, the sinusoid at optimal amplitude and period is found to yield the maximum net energy saving rate. However, when the wave amplitude and period deviate from the optimal values, other waves are found to perform better than the sinusoid. This is potentially interesting in view of applications, where a particular actuator limitations might preclude reaching the optimal operating conditions for the sinusoidal wall oscillation. It is demonstrated that the present model can predict the locally optimal waveform for given wave amplitude and period, as well as the globally optimal sinusoidal wave.