Input-output analysis of stochastic base flow uncertainty

TL;DR

This paper introduces an input-output framework to analyze how stochastic base flow uncertainties impact the stability and response of linearized Navier-Stokes equations, providing verifiable conditions and insights into flow destabilization.

Contribution

It presents a novel input-output approach for analyzing stochastic base flow effects without costly simulations, including stability conditions, frequency response analysis, and perturbation methods for small disturbances.

Findings

Reveals Reynolds number scaling of destabilizing perturbation variances.

Shows how base flow shape influences flow amplification.

Explains robust streak amplification due to streamwise base flow variations.

Abstract

We adopt an input-output approach to analyze the effect of persistent white-in-time structured stochastic base flow perturbations on the mean-square properties of the linearized Navier-Stokes equations. Such base flow variations enter the linearized dynamics as multiplicative sources of uncertainty that can alter the stability of the linearized dynamics and their receptivity to exogenous excitations. Our approach does not rely on costly stochastic simulations or adjoint-based sensitivity analysis. We provide verifiable conditions for mean-square stability and study the frequency response of the flow subject to additive and multiplicative sources of uncertainty using the solution to the generalized Lyapunov equation. For small-amplitude base flow perturbations, we bypass the need to solve large generalized Lyapunov equations by adopting a perturbation analysis. We use our framework to study the destabilizing effects of stochastic base flow variations in transitional parallel flows, and the reliability of numerically estimated mean velocity profiles in turbulent channel flows. We uncover the Reynolds number scaling of critically destabilizing perturbation variances and demonstrate how the wall-normal shape of base flow modulations can influence the amplification of various length scales. Furthermore, we explain the robust amplification of streamwise streaks in the presence of streamwise base flow variations by analyzing the dynamical structure of the governing equations as well as the Reynolds number dependence of the energy spectrum.