Transition Threshold for the 3D Couette Flow in Sobolev Space

Abstract

In this paper, we study the transition threshold of the 3D Couette flow in Sobolev space at high Reynolds number Re. It was proved that if the initial velocity v 0 satisfies kv 0 .y; 0; 0/k H 2 c 0 Re 1 for some c 0 > 0 independent of Re, then the solution of the 3D Navier-Stokes equations is global in time and does not transition away from the Couette flow. This result confirms the transition threshold conjecture proposed by Trefethen et al. in 1993. Moreover, we prove that the long-time dynamics of the solution behaves as: for t . Re, the solution will experience a transient growth from O.Re 1 / to O.c 0 / due to the 3D lift-up effect; for t & Re 1=3 , the solution will rapidly converge to a streak solution due to the mixing-enhanced dissipation effect.

Linear stability of the Couette flow in the 3D isentropic compressible Navier-Stokes equations

“…The main results. The aim of this paper is to rigorously study the stability properties of the Couette flow for the 3D linearized isentropic compressible Navier-Stokes equations on T × R × T. We confirm the enhanced dissipation and the lift-up phenomena which have been deeply studied for the 3D incompressible fluids [6,64].…”

“…It is a challenging task to study the transition threshold problem of the Couette flow in the three dimensional finite channel T×[−1, 1]×T. Despite the boundary layer effect, it was shown by Chen, Wei and the second author in a very rencent work [18] that the threshold is still not larger than 1 (just as in the case without boundary [64]), and thus the conjecture proposed by Trefethen et al [63] was confirmed.…”

“…Significant progress has been made by Bedrossian, Germain and Masmoudi [4,5,6] in this direction for the domain T × R × T. More precisely, in [6], the authors proved that if the initial perturbation satisfies u in H s ≤ δRe − 3 2 , with Re the Reynolds number, δ = δ(s) > 0, and s > 9 2 , then the solution remains within O(Re − 1 2 ) of the Couette flow (y, 0, 0) ⊤ in L 2 for all time, and converges to the streaks solution for t ≫ Re 1 3 . Recently, Wei and the second author [64] proved the nonlinear stability of the Couette flow for the initial perturbation satisfying u in H 2 ≤ δRe −1 . This result implies that the transition threshold may be insensitive to the regularity of the perturbation above H 2 .…”

Structured input-output analysis of transitional wall-bounded flows

“…We now explore scaling of H ∇ µ at higher Reynolds numbers for plane Couette flow, which is known to be linearly stable for Re → ∞ (Romanov 1973). The scaling of permissible perturbation amplitude versus Reynolds number in the asymptotic limit Re → ∞ is also widely studied (Kreiss et al 1994;Chapman 2002;Bedrossian et al 2015Bedrossian et al , 2019Wei & Zhang 2020). Here we analyze this behavior by investigating computing results at Re = 15000, 25000, 35000, 100000, 280000, with associated increased wall-normal resolution including N y = 60, 80, 100, 120, 140 wall-normal grid points, respectively.…”

Stability threshold of two-dimensional Couette flow in Sobolev spaces

“…• Three-dimensional Couette flow in T R T : If X is taken as a Sobolev space, then D 1 gives stability ( [3][4][5]40]). Also see the very recent paper [15] for threedimensional Couette flow in a finite channel.…”