Turbulent flows over porous/rough walls: Response of velocity to stress on surfaces*

Porous/rough walls with finite grain sizes and depths affect the overlying turbulence via multiple mechanisms including slip, transpiration, roughness, and bottom reflection. Understanding these mechanisms is a fundamental prerequisite for a higher-level theory on the behavior changes of turbulence itself. In this study, we investigate the scale-local response of velocity to stress on porous/rough wall surfaces. First, the superficial velocity is correlated with the stress based on solutions to the linearized Navier-Stokes (LNS) equations, yielding a response matrix as a function of exciting wavevectors and frequencies. This matrix reveals a substrate's inherent attributes, particularly the bottom pressure reflection that principally distinguishes rough walls from deep porous walls. Second, we examine the superficial velocity-stress correlations for the data of direct numerical simulations that fully resolve turbulence over porous/rough walls. Strong correlations are observed for most scales and are highly consistent with the LNS results. This indicates that the LNS framework captures the superficial responding mechanisms, particularly the phase delay of transpiration responding to pressure, a crucial effect that cannot be captured by conventional Stokes-flow-based approaches.