Statistical State Dynamics Eigenmodes and Equilibria support Couette Turbulence

Abstract

Wide-channel Couette (WCC) turbulence consists primarily of quasi-steady roll-streak structures (RSS) maintained by a self-sustaining process (SSP), yet the Navier-Stokes equations in velocity variables admit no linear RSS instability or stable equilibrium, leaving the WCC turbulent state's analytic basis obscure. We show that, in the statistical state dynamics (SSD) of a second-order closure of the Navier-Stokes equations, Couette turbulence arises from a modal RSS instability and equilibrates as a stable fixed point at spanwise wavenumber 3. This fixed point comprises a streamwise-mean flow and a pair of counterpropagating neutral eigenmodes, regularized by roll advection rather than viscosity. This stable equilibrium identifies both a canonical turbulence solution and its self-sustaining process (SSP), and characterizes their analytic structure. WCC turbulence arises as a spanwise tiling by this stable fixed-point RSS unit cell. Predictions of this SSD theory are corroborated by comparison with direct numerical simulation.