From oblique-wave forcing to streak reinforcement: A perturbation-based frequency-response framework

TL;DR

This paper introduces a perturbation-based frequency-response framework that unifies linear and nonlinear mechanisms of flow transition, elucidating streak formation and amplification in shear flows.

Contribution

It develops a systematic expansion of flow dynamics that links resolvent analysis with nonlinear interactions, providing a comprehensive view of transition mechanisms.

Findings

Quadratic interactions of oblique waves generate streaks via lift-up.

Higher-order nonlinear coupling influences streak reinforcement or attenuation.

Breakdown of weakly nonlinear regime coincides with sustained unsteadiness and secondary instability.

Abstract

We develop a perturbation-based frequency-response framework for analyzing amplification mechanisms that are central to subcritical routes to transition in wall-bounded shear flows. By systematically expanding the input-output dynamics of fluctuations about the laminar base flow with respect to forcing amplitude, we establish a rigorous correspondence between linear resolvent analysis and higher-order nonlinear interactions. At second order, quadratic interactions of unsteady oblique waves generate steady streamwise streaks via the lift-up mechanism. We demonstrate that the spatial structure of these streaks is captured by the second output singular function of the streamwise-constant resolvent operator. At higher orders, nonlinear coupling between oblique waves and induced streaks acts as structured forcing of the laminar linearized dynamics, yielding additional streak components whose relative phase governs reinforcement or attenuation of the leading-order streak response. Our analysis identifies a critical forcing amplitude marking the breakdown of the weakly nonlinear regime, beyond which direct numerical simulations exhibit sustained unsteadiness. We show that this breakdown coincides with the onset of secondary instability, revealing that the nonlinear interactions responsible for streak formation also drive the modal growth central to classical transition theory. The resulting framework provides a mechanistically transparent and computationally efficient description of transition that unifies non-modal amplification, streak formation, and modal instability within a single formulation derived directly from the Navier-Stokes equations.