Reduced description of exact coherent states in parallel shear flows

Abstract

Exact coherent states of a linearly stable, plane parallel shear flow confined between stationary stress-free walls and driven by a sinusoidal body force (a flow first introduced by F. Waleffe, Phys. Fluids 9, 883 (1997)) are computed using equations obtained from a large Reynolds-number asymptotic reduction of the Navier-Stokes equations. The reduced equations employ a decomposition into streamwise-averaged (mean) and streamwise-varying (fluctuation) components and are characterized by an effective order one Reynolds number in the mean equations along with a formally higher-order diffusive regularization of the fluctuation equations. A robust numerical algorithm for computing exact coherent states is introduced. Numerical continuation of the lower branch states to lower Reynolds numbers reveals the presence of a saddle-node; the saddle-node allows access to upper branch states that, like the lower branch states, appear to be self-consistently described by the reduced equations. Both lower and upper branch states are characterized in detail.

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• The Self-Sustaining Process in Taylor-Couette Flow
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• Marginally-Stable Thermal Equilibria of Rayleigh-Bénard Convection
• Three-dimensional coherent structures in a curved pipe flow
• Large-scale structures in high Reynolds number rotating Waleffe flow
• Eigenvalue bounds for compressible stratified magneto-shear flows varying in two transverse directions
• A self-sustaining process theory for uniform momentum zones and internal shear layers in high Reynolds number shear flows
• Closing the loop: nonlinear Taylor vortex flow through the lens of resolvent analysis