# Growth mechanisms of perturbations in boundary layers over a compliant wall

## TL;DR

This study investigates how perturbations grow in boundary layers over compliant walls, revealing unique instability mechanisms, the influence of viscosity, and the transition from two-way to one-way fluid-structure interaction.

## Contribution

It introduces a simplified spectral method for analyzing perturbation growth over compliant walls and explores the effects of fluid-structure interaction on flow stability.

## Key findings

- Global instability aligns with favorable flow forcing on the compliant wall.
- Viscosity significantly influences transient growth in compliant wall flows.
- Maximum growth rate decreases with Reynolds number due to changing fluid-structure coupling.

## Abstract

The temporal modal and nonmodal growth of three-dimensional perturbations in the boundary-layer flow over an infinite compliant flat wall is considered. Using a wall-normal velocity/wall-normal vorticity formalism, the dynamic boundary condition at the compliant wall admits a linear dependence on the eigenvalue parameter, as compared to a quadratic one in the canonical formulation of the problem. This greatly simplifies the accurate calculation of the continuous spectrum by means of a spectral method, thereby yielding a very effective filtering of the pseudospectra as well as a clear identification of instability regions. The regime of global instability is found to be matching the regime of the favorable phase of the forcing by the flow on the compliant wall so as to enhance the amplitude of the wall. An energy-budget analysis for the least-decaying hydroelastic (static-divergence, traveling-wave-flutter and near-stationary transitional) and Tollmien--Schlichting modes in the parameter space reveals the primary routes of energy flow. Moreover, the flow exhibits a slower transient growth for the maximum growth rate of a superposition of streamwise-independent modes due to a complex dependence of the wall-boundary condition with the Reynolds number. The initial and optimal perturbations are compared with the boundary-layer flow over a solid wall; differences and similarities are discussed. Unlike the solid-wall case, viscosity plays a pivotal role in the transient growth. A slowdown of the maximum growth rate with the Reynolds number is uncovered and found to originate in the transition of the fluid-solid interaction from a two-way to a one-way coupling. Finally, a term-by-term energy budget analysis is performed to identify the key contributors to the transient growth mechanism.