A framework for studying the effect of compliant surfaces on wall turbulence

It has long been recognized that compliant surfaces can serve as passive controllers for turbulent flows. However, the lack of a physics-based, computationally cheap theoretical framework that predicts the effect of compliant surfaces on turbulence has restricted progress towards designing performance-improving walls. To address this gap, we extend the resolvent analysis of McKeon {\&} Sharma (2010, \textit{J. Fluid Mech.}). Under this analysis, the turbulent velocity field is expressed as a linear superposition of propagating modes, identified via a gain-based decomposition of the Navier-Stokes equations. Compliant surfaces, modeled as a complex wall-admittance linking pressure and velocity, affect the gain and structure of these modes. Using a pattern search, we show that walls with unphysical negative damping are required to interact favorably with modes resembling the energetic near-wall cycle, which could explain why previous studies have met with limited success. Our results suggest that positive-damping walls could be effective for modes resembling the so-called very large-scale motions (VLSMs). Since the VLSMs have an organizing influence on smaller-scale turbulence, they may serve as a pathway for compliant walls to affect the entire flow.