Vertically localised equilibrium solutions in large-eddy simulations of homogeneous shear flow

TL;DR

This paper discovers and analyzes vertically localized equilibrium solutions in large-eddy simulations of homogeneous shear flow, revealing bifurcation behavior and structural features akin to classical shear flows, with implications for understanding turbulence localization.

Contribution

It introduces the first numerical identification of vertically localized equilibrium solutions in LES of shear flow, showing their bifurcation, structure, and relation to turbulence.

Findings

Equilibrium solutions emerge via saddle-node bifurcation as R_S increases.

Localized vortical structures resemble those in plane Couette flow.

LES turbulence intermittently visits vertically localized states.

Abstract

Unstable equilibrium solutions in a homogeneous shear flow with sinuous symmetry are numerically found in large-eddy simulations (LES) with no kinetic viscosity. The small-scale properties are determined by the mixing length scale $l_S$ used to define eddy viscosity, and the large-scale motion is induced by the mean shear at the integral scale, which is limited by the spanwise box dimension $L_z$. The fraction $ R_S= L_z/l_S$, which plays the role of a Reynolds number, is used as a numerical continuation parameter. It is shown that equilibrium solutions appear by a saddle-node bifurcation as $R_S$ increases, and that they resemble those in plane Couette flow with the same symmetry. The vortical structures of both lower- and upper-branch solutions become spontaneously localised in the vertical direction. The lower-branch solution is an edge state at low $R_S$, and takes the form of a thin critical layer as $R_S$ increases, as in the asymptotic theory of generic shear flow at high-Reynolds numbers. On the other hand, the upper-branch solutions are characterised by a tall velocity streak with multi-scale multiple vortical structures. At the higher end of $R_S$, an incipient multiscale structure is found. The LES turbulence occasionally visits vertically localised states whose vortical structure resembles the present vertically localised LES equilibria.