Dynamics Near the Subcritical Transition of the 3D Couette Flow II: Above Threshold Case
Jacob Bedrossian, Pierre Germain, Nader Masmoudi
Abstract
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 <bold>Re</bold> . In this work, we show that there is constant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than c 0 much-less-than 1"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo> ≪ </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">0 > c_0 \ll 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , independent of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper R bold e"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> <mml:mi mathvariant="bold">e</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {Re}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , such that sufficiently regular disturbances of size <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon less-than-or-equivalent-to bold upper R bold e Superscript negative 2 slash 3 minus delta"> <mml:semantics> <mml:mrow> <mml:mi> ϵ </mml:mi> <mml:mo> ≲ </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> <mml:mi mathvariant="bold">e</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> <mml:mo> − </mml:mo> <mml:mi> δ </mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\epsilon \lesssim \mathbf {Re}^{-2/3-\delta }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi> δ </mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\delta > 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> exist at least until <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t equals c 0 epsilon Superscript negative 1"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:msup> <mml:mi> ϵ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">t = c_0\epsilon ^{-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and in general evolve to be <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis c 0 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(c_0)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> due to the lift-up effect. Further, after times <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t greater-than-or-equivalent-to bold upper R bold e Superscript 1 slash 3"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo> ≳ </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> <mml:mi mathvariant="bold">e</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">t \gtrsim \mathbf {Re}^{1/3}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t almost-equals epsilon Superscript negative 1"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo> ≈ </mml:mo> <mml:msup> <mml:mi> ϵ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn>