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

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.