Reduced description of exact coherent states in parallel shear flows

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.

Exact coherent structures in a reduced model of parallel shear flow

A reduced description of shear flows consistent with the Reynolds number scaling of lower-branch exact coherent states in plane Couette flow [J. Wang et al., Phys. Rev. Lett. 98, 204501 (2007)] is constructed. Exact time-independent nonlinear solutions of the reduced equations corresponding to both lower and upper branch states are found for Waleffe flow [F. Waleffe, Phys. Fluids 9, 883--900 (1997)]. The lower branch solution is characterized by fluctuations that vary slowly along the critical layer while the upper branch solutions display a bimodal structure and are more strongly focused on the critical layer. The reduced model provides a rational framework for investigations of subcritical spatiotemporal patterns in parallel shear flows.