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

TL;DR

This paper develops a resolvent-based framework to analyze how compliant surfaces influence wall turbulence, enabling predictions of flow structures and guiding the design of surfaces to suppress turbulence effectively.

Contribution

It extends the resolvent formulation to include compliant surfaces, allowing for accurate prediction and rational design of wall coatings to control turbulence.

Findings

Negative damping walls can suppress near-wall cycle modes.

Positive damping walls are effective against large-scale motions.

Walls that suppress certain structures may amplify others.

Abstract

This paper extends the resolvent formulation proposed by McKeon & Sharma (2010) to consider turbulence-compliant wall interactions. Under this formulation, 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. With minimal computation, this framework accurately predicts the emergence of the quasi-2D propagating waves observed in recent direct numerical simulations. Further, the analysis also enables the rational design of compliant surfaces, with properties optimized to suppress flow structures energetic in wall turbulence. It is shown 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. Positive-damping walls are effective for modes resembling the so-called very large-scale motions (VLSMs), indicating that compliant surfaces may be better suited for application at higher Reynolds number. Unfortunately, walls that suppress structures energetic in natural turbulence are also predicted to have detrimental effects elsewhere in spectral space. Consistent with previous experiments and simulations, slow-moving spanwise-constant structures are particularly susceptible to further amplification. Mitigating these adverse effects will be central to the development of compliant coatings that have a net positive influence on the flow.