Dynamics near the subcritical transition of the 3D Couette flow I: Below threshold case

TL;DR

This paper proves that small initial disturbances in 3D Couette flow at high Reynolds number remain stable and return to the flow, highlighting the roles of mixing-enhanced dissipation and transient growth effects.

Contribution

It provides a rigorous analysis of stability for small perturbations in 3D Couette flow, incorporating the effects of transient growth, mixing, and nonlinear interactions at high Reynolds numbers.

Findings

Solutions remain close to Couette flow for small initial data.

Transient energy growth occurs due to lift-up effect.

Solutions are attracted to streamwise-independent streaks over time.

Abstract

We study small disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number $\textbf{Re}$. We prove that for sufficiently regular initial data of size $\epsilon \leq c_0\textbf{Re}^{-1}$ for some universal $c_0 > 0$, the solution is global, remains within $O(c_0)$ of the Couette flow in $L^2$, and returns to the Couette flow as $t \rightarrow \infty$. For times $t \gtrsim \textbf{Re}^{1/3}$, the streamwise dependence is damped by a mixing-enhanced dissipation effect and the solution is rapidly attracted to the class of "2.5 dimensional" streamwise-independent solutions referred to as streaks. Our analysis contains perturbations that experience a transient growth of kinetic energy from $O(\textbf{Re}^{-1})$ to $O(c_0)$ due to the algebraic linear instability known as the lift-up effect. Furthermore, solutions can exhibit a direct cascade of energy to small scales. The behavior is very different from the 2D Couette flow, in which stability is independent of $\textbf{Re}$, enstrophy experiences a direct cascade, and inviscid damping is dominant (resulting in a kind of inverse energy cascade). In 3D, inviscid damping will play a role on one component of the velocity, but the primary stability mechanism is the mixing-enhanced dissipation. Central to the proof is a detailed analysis of the interplay between the stabilizing effects of the mixing and enhanced dissipation and the destabilizing effects of the lift-up effect, vortex stretching, and weakly nonlinear instabilities connected to the non-normal nature of the linearization.