A Convex Approach to Hydrodynamic Analysis

TL;DR

This paper introduces a convex optimization-based method for analyzing the stability and input-output properties of 3D viscous flows with invariance, using Lyapunov functionals and matrix inequalities, demonstrated on rotating Couette flow.

Contribution

It develops a convex framework leveraging invariance properties to analyze stability and input-output behavior of 3D flows, including polynomial laminar profiles.

Findings

Matrix inequalities can be checked via convex optimization for polynomial laminar flows.

The method effectively analyzes stability of streamwise constant flows.

Application to rotating Couette flow demonstrates practical utility.

Abstract

We study stability and input-state analysis of three dimensional (3D) incompressible, viscous flows with invariance in one direction. By taking advantage of this invariance property, we propose a class of Lyapunov and storage functionals. We then consider exponential stability, induced L2-norms, and input-to-state stability (ISS). For streamwise constant flows, we formulate conditions based on matrix inequalities. We show that in the case of polynomial laminar flow profiles the matrix inequalities can be checked via convex optimization. The proposed method is illustrated by an example of rotating Couette flow.