Transition Threshold for the <scp>3D</scp> Couette Flow in Sobolev Space

Dongyi Wei, Zhifei Zhang

2020Communications on Pure and Applied Mathematics66 citationsDOI

Abstract

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 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. © 2020 Wiley Periodicals LLC

Topics & Concepts

Couette flowMathematicsLift (data mining)Reynolds numberFlow (mathematics)Space (punctuation)Sobolev spaceStreakConjectureMathematical analysisMechanicsPhysicsCombinatoricsGeometryTurbulenceComputer scienceOperating systemData miningOpticsFluid Dynamics and Turbulent FlowsNavier-Stokes equation solutionsHydrology and Sediment Transport Processes