Coupled dynamics of wall pressure and transpiration, with implications for the modelling of tailored surfaces and turbulent drag reduction

Wall-based active and passive flow control for drag reduction in low-Reynolds-number ( ${\textit{Re}}$ ) turbulent flows can lead to three typical phenomena: (i) attenuation or (ii) amplification of the near-wall cycle, and (iii) generation of spanwise rollers. The present study conducts direct numerical simulations of a low ${\textit{Re}}$ turbulent channel flow and demonstrates that each flow response can be generated with a wall transpiration at two sets of spatial scales, termed streak and roller scales. The effect of the transpiration is controlled by its relative phase to the background flow, which can be parametrised by the wall pressure. Streak scales (i) attenuate the near-wall cycle if transpiration and wall pressure are approximately in-phase or (ii) amplify it otherwise, and (iii) roller scales energise spanwise rollers when transpiration and wall pressure are out-of-phase. Conditions for establishing these robust phase relations are derived from the analytical solution to the pressure Poisson equation and rely on splitting the pressure into its fast, slow and Stokes component. The importance of each condition depends on the relative magnitude of the pressure components, which is significantly altered by the transpiration. The analogy in flow response suggests that transpiration with the two scale families and their phase relations to the wall pressure represent fundamental building blocks for flows over tailored surfaces including riblets, porous and permeable walls.