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• W1538041412 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 $epsilon leq c_0textbf{Re}^{-1}$ 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." @default.
• W1538041412 created "2016-06-24" @default.
• W1538041412 creator A5057307606 @default.
• W1538041412 creator A5078240002 @default.
• W1538041412 creator A5081815930 @default.
• W1538041412 date "2015-06-11" @default.
• W1538041412 modified "2023-09-27" @default.
• W1538041412 title "Dynamics near the subcritical transition of the 3D Couette flow I: Below threshold case" @default.
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