On the stability analysis in the transition to turbulence problem

TL;DR

This paper examines key stability assumptions in the transition to turbulence, highlighting the impact of domain choice and nonlinear energy conservation, and demonstrates spectral instability of Couette flow at finite Reynolds numbers.

Contribution

It clarifies how domain considerations and nonlinear effects influence stability results, challenging classical views on Couette flow stability.

Findings

Couette flow is spectrally unstable at finite Reynolds numbers.

Stability results depend critically on whether the domain is infinite or semi-infinite.

The nonlinear terms of Navier-Stokes conserve energy, affecting transition analysis.

Abstract

In this note, which is of general stability theory interest, we discuss some of the key assertions usually stated in the context of the transition to turbulence problem. In particular, the two main points made here in the setting of the transition problem are (i) the crucial dependence of the stability results on whether the problem is considered on infinite or semi-infinite domain, and (ii) the energy conservation by the nonlinear terms of the Navier-Stokes equations. As an application, we demonstrate that the Couette flow, when analyzed in the mathematical setting most correctly reflecting the way the experiments are usually done, is spectrally unstable for finite Reynolds numbers in apparent contradiction to the commonly accepted classical century-old results. Also, the interrelation of various stability notions, the effects of infinite dimensionality, the covariant nature of the transition phenomena and how non-normality of the linear operators and finite-amplitude instability fit into this picture are discussed as well.