Improved uncertainty representation for reducing artificial energy production in structured input-output stability analysis

Abstract

This work employs a new form of fixed structured uncertainty within the structured small-gain theorem approach proposed by Frank-Shapir & Gluzman (J. Fluid Mech., vol. 1030, 2026, pp A8) for the stability analysis of incompressible shear flows subject to finite-magnitude disturbances. Within this framework, the nonlinear advection term in the Navier-Stokes equations is replaced by a structured feedback uncertainty interconnection with the linearized dynamics to account for the impact of nonlinear feedback. Herein, a new uncertainty representation is derived via linear transformations of the input and output channels, transforming the feedback loop such that the resulting structured uncertainty has a repeated-diagonal structure. This structure aims to preserve the component-wise pathways of the nonlinear advection term while keeping the structured singular value computation tractable. We apply the method to two canonical base flows: Couette and plane Poiseuille flows. The resulting thresholds on disturbance magnitude to preserve stability are less conservative and more accurate. We compare the novel methodology presented here with previously proposed repeated and non-repeated block approximations of the uncertainty structure, where our stability threshold provided the closest agreement with previous numerical and experimental studies. We show that repeated and non-repeated block structures that were proposed in past studies result in an artificial energy-production term arising from using constant structured uncertainty in the structured input-output formulation, violating the divergencefree assumption. This energy-production term is smallest when using the methodology presented in this work, providing a more faithful representation of the impact of nonlinear feedback interconnection with the linearized dynamics of the Navier-Stokes system.