Response of a turbulent boundary layer to steady, square-wave-type transverse wall-forcing

Abstract

This study investigates the spatial evolution of a zero pressure gradient turbulent boundary layer (TBL) imposed by a square-wave (SqW) of steady spanwise wall-forcing, which varies along the streamwise direction (x). The SqW wall-forcing is imposed experimentally via a series of streamwise periodic belts running in opposite spanwise directions, following the methodology of Knoop et al. [Exp. Fluids 65, 65 (2024)]0723-486410.1007/s00348-024-03799-9, with the streamwise extent increased to beyond ∼11 times the boundary layer thickness (δo) in the present study. This unique setup is leveraged to investigate the influence of viscous-scaled wavelength of SqW wall-forcing on the turbulent drag reduction efficacy for λx+=471 (suboptimal), 942 (near-optimal), and 1884 (postoptimal conditions), at fixed viscous-scaled wall-forcing amplitude, A+=12, and friction Reynolds number, Reτ=960. The TBL's response to this wall-forcing is elucidated by drawing inspiration from established knowledge on traditionally studied sinusoidal forcing, based on analysis of the streamwise-phase variation of the Stokes strain rate (SSR). The analysis reveals the SqW forcing to be characterized by a combination of two markedly different SSR regimes whose influence on the overlying turbulence is found to depend on the forcing waveform: subphase I of local and strong impulses of SSR downstream of the half- (λx/2) and full-phase (λx) locations, associated with a reversal in spanwise forcing directions, leading to significant turbulence attenuation, and subphase II of near-zero SSR over the remainder of forcing phase that enables turbulence recovery (when wall-forcing magnitudes and direction remain constant). Upon the initial imposition of the SqW forcing, the Reynolds stresses are strongly attenuated over the short streamwise extent of x/δ0<0.5 for all wavelengths, whereas the skin-friction transient is more gradual. Thereafter, once the forcing is ultimately established, the suboptimum and optimum wavelength regimes display no distinctive responses to the individual SSR subphases; rather, the drag-reduced TBL response is quasi-streamwise homogeneous. In contrast, an SSR-related phenomenology establishes itself clearly for the postoptimal case, in which a local attenuation of near-wall turbulence characterizes subphase I, while the turbulent energy recovers in subphase II owing to the extended region of near-zero SSR.