Structured Input-Output Modeling and Robust Stability Analysis of Compressible Flows

TL;DR

This paper extends structured input-output analysis to compressible flows, enabling robust stability assessment and insight into flow instabilities across Mach numbers using a quadratic reformulation of the Navier-Stokes equations.

Contribution

It introduces a novel method for analyzing compressible flows by embedding nonlinearities into structured uncertainties and applying the structured singular value framework.

Findings

Identified spanwise elongated instability structures at subsonic Mach numbers.

Revealed oblique instability structures at sonic and supersonic Mach numbers.

Compared with unstructured analysis, showed the importance of nonlinearity structure in flow physics.

Abstract

The recently introduced structured input-output analysis is a powerful method for capturing nonlinear phenomena associated with incompressible flows, and this paper extends that method to the compressible regime. The proposed method relies upon a reformulation of the compressible Navier-Stokes equations, which allows for an exact quadratic formulation of the dynamics of perturbations about a steady base flow. To facilitate the structured input-output analysis, a pseudo-linear model for the quadratic nonlinearity is proposed and the structural information of the nonlinearity is embedded into a structured uncertainty comprising unknown `perturbations'. The structured singular value framework is employed to compute the input-output gain, which provides an estimate of the robust stability margin of the flow perturbations, as well as the forcing and response modes that are consistent with the nonlinearity structure. The analysis is then carried out on a plane, laminar compressible Couette flow over a range of Mach numbers. The structured input-output gains identify an instability mechanism, characterized by a spanwise elongated structure in the streamwise-spanwise wavenumber space at a subsonic Mach number, that takes the form of an oblique structure at sonic and supersonic Mach numbers. In addition, the structured input-output forcing and response modes provide insight into the thermodynamic and momentum characteristics associated with a source of instability. Comparisons with a resolvent/unstructured analysis reveal discrepancies in the distribution of input-output gains over the wavenumber space as well as in the modal behavior of an instability, thus highlighting the strong correlation between the structural information of the nonlinearity and the underlying flow physics.