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Lower branch equilibria in Couette flow: the emergence of canonical states for arbitrary shear flows

Published online by Cambridge University Press:  06 June 2013

H. M. Blackburn ,
P. Hall  and
S. J. Sherwin
H. M. Blackburn*
Affiliation:
Department of Mechanical and Aerospace Engineering, Monash University, Victoria 3800, Australia
P. Hall
Affiliation:
Department of Mathematics, Imperial College London, South Kensington, London SW7 2AZ, UK
S. J. Sherwin
Affiliation:
Department of Aeronautics, Imperial College London, South Kensington, London SW7 2AZ, UK
*
†Email address for correspondence: Hugh.Blackburn@monash.edu
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Abstract

We consider the development of nonlinear three-dimensional vortex–wave interaction equilibria of laminar plane Couette flow for a range of spanwise wavenumbers. The results are computed using a hybrid approach that captures the required asymptotic structure while at the same time providing a direct link with full numerical calculations of equilibrium states. Each equilibrium state consists of a streak flow, a roll flow and a wave propagating on the streak. Direct numerical simulations at finite Reynolds numbers using initial conditions constructed from these parts confirm that the scheme generates equilibrium solutions of the Navier–Stokes equations. Consideration of the form of the vortex–wave interaction equations in the high-spanwise-wavenumber limit predicts that for small wavelengths the equilibria take on a self-similar structure confined near the centre of the flow. These states feel no influence from the walls, and lead to a class of canonical states relevant to arbitrary shear flows. These predictions are supported by an analysis of computational results at increasing values of the spanwise wavenumber. For the wave part of these new canonical states, it is shown that the mass-specific kinetic energy density per unit wavenumber scales with the $- 5/ 3$ power of the wavenumber.

JFM classification

Boundary Layers: Boundary layer stability Waves/Free-surface Flows: Critical layers Instability: Transition to turbulence
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 726 , 10 July 2013 , R2
Copyright
©2013 Cambridge University Press 

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Blackburn Supplementary Material

Animation of plane Couette flow DNS corresponding to the self-sustaining state shown in figure 4 (black line for k=1). Time runs from t=0 to t=1000. The grey translucent isosurface is drawn at (streamwise velocity component) u=0 and represents the critical layer. Red and blue isosurfaces are equal magnitude positive and negative values of spanwise velocity component (w), and serves to illustrate wave structure.

Download Blackburn Supplementary Material(Video)
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