TY - JOUR

T1 - Dynamics Near the Subcritical Transition of the 3D Couette Flow II

T2 - Above Threshold Case

AU - Bedrossian, Jacob

AU - Germain, Pierre

AU - Masmoudi, Nader

N1 - Publisher Copyright:
© 2022 American Mathematical Society.

PY - 2022/9

Y1 - 2022/9

N2 - This is the second in a pair of works which study small disturbances to the plane, periodic 3D Couette flow in the incompressible Navier-Stokes equations at high Reynolds number Re. In this work, we show that there is constant 0 < c0 ≪ 1, independent of Re, such that sufficiently regular disturbances of size ∊ ≲ Re−2/3−δ for any δ > 0 exist at least until t = c0∊−1 and in general evolve to be O(c0) due to the lift-up effect. Further, after times t ≳ Re1/3, the streamwise dependence of the solution is rapidly diminished by a mixing-enhanced dissipation effect and the solution is attracted back to the class of “2.5 dimensional” streamwise-independent solutions (sometimes referred to as “streaks”). The largest of these streaks are expected to eventually undergo a secondary instability at t ≈ ∊−1. Hence, our work strongly suggests, for all (sufficiently regular) initial data, the genericity of the “lift-up effect ⇒ streak growth ⇒ streak breakdown” scenario for turbulent transition of the 3D Couette flow near the threshold of stability forwarded in the applied mathematics and physics literature.

AB - This is the second in a pair of works which study small disturbances to the plane, periodic 3D Couette flow in the incompressible Navier-Stokes equations at high Reynolds number Re. In this work, we show that there is constant 0 < c0 ≪ 1, independent of Re, such that sufficiently regular disturbances of size ∊ ≲ Re−2/3−δ for any δ > 0 exist at least until t = c0∊−1 and in general evolve to be O(c0) due to the lift-up effect. Further, after times t ≳ Re1/3, the streamwise dependence of the solution is rapidly diminished by a mixing-enhanced dissipation effect and the solution is attracted back to the class of “2.5 dimensional” streamwise-independent solutions (sometimes referred to as “streaks”). The largest of these streaks are expected to eventually undergo a secondary instability at t ≈ ∊−1. Hence, our work strongly suggests, for all (sufficiently regular) initial data, the genericity of the “lift-up effect ⇒ streak growth ⇒ streak breakdown” scenario for turbulent transition of the 3D Couette flow near the threshold of stability forwarded in the applied mathematics and physics literature.

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U2 - 10.1090/MEMO/1377

DO - 10.1090/MEMO/1377

M3 - Article

AN - SCOPUS:85137074521

SN - 0065-9266

VL - 279

SP - 1

EP - 147

JO - Memoirs of the American Mathematical Society

JF - Memoirs of the American Mathematical Society

IS - 1377

ER -