Reduced description of exact coherent states in parallel shear flows

TL;DR

This paper develops a reduced-order model to compute and analyze exact coherent states in parallel shear flows, revealing bifurcation structures and providing a robust numerical method for these states.

Contribution

It introduces a novel asymptotic reduction of Navier-Stokes equations and a numerical algorithm for computing exact coherent states in shear flows.

Findings

Identification of saddle-node bifurcation in lower branch states.

Access to upper branch states via continuation from lower branch.

Detailed characterization of both lower and upper branch states.

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.