Ofek Frank-Shapir | Searcharxiv

arXiv2026

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

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.

physics.flu-dyn↗

arXiv2026

Stability analysis of transitional flows based on disturbance magnitude

We propose a novel stability criterion for incompressible shear flows by combining input-output analysis and the small-gain theorem. The criterion yields an explicit threshold on the magnitude of velocity perturbations about a given base flow that guarantees stability. If this threshold is crossed--either due to nonmodal growth, exponential growth, or a bypass transition scenario--our analysis predicts a loss of stability that may lead to transition to turbulence. We consider three approximated models for nonlinearity: unstructured, structured with non-repeated blocks, and structured with repeated blocks. We show that the imposed threshold obtained by these three methods complies with a hierarchical relationship, where the unstructured case is the most conservative, imposing the lowest bound on disturbance magnitude. We apply this approach to three canonical and well-studied base flows: Couette, plane Poiseuille, and Blasius. For these three base flows, we compare our results with experiments, direct numerical simulation results, nonmodal nonlinear stability results, and linear stability theory (LST). In the limit of infinitesimally small perturbation magnitude, our stability criterion for the unstructured case recovers the results of LST. For finite perturbations, the structured cases that account for nonlinear interactions provided stability thresholds that are consistent with experimental observations and simulation results of transition at both subcritical and post-critical Reynolds numbers for the considered base flows in our study. In particular, we utilize our stability criterion to demonstrate that Couette flow can become unstable and transition can be triggered at different Reynolds numbers, which is consistent with past experimental observations.

physics.flu-dyn↗