On the optimal period of spanwise wall forcing for turbulent drag reduction

Turbulent channel flow controlled by spanwise wall oscillations is studied using direct numerical simulations to improve how spanwise forcing reduces skin-friction drag. Harmonic wall oscillations generate a periodic transverse Stokes layer whose thickness $\delta$ is determined by the forcing period $T$. Although an optimal $T$ that maximizes drag reduction is known to exist, its physical significance remains unclear. To elucidate it, we extend the spanwise Stokes layer by augmenting wall oscillation with an additional spanwise body force. In this formulation, $\delta$ and $T$ become decoupled and can be varied independently. The oscillating wall thus appears as a special and suboptimal case of spanwise forcing. Optimal performance is obtained for substantially smaller $T$ and larger $\delta$ than those of the classical Stokes layer. For the conditions examined, with Reynolds number and forcing amplitude held fixed, the maximum drag reduction increases by approximately one third, while the maximum net energy saving improves markedly from $-35\%$ to $+16\%$. These findings suggest that drag-reduction strategies based on spanwise forcing deserve renewed scrutiny: wall oscillation represents only one possible actuation method, and not necessarily the most effective one.