Supercritical states of two-dimensional wall-bounded shear flow

The Behavior of high Reynolds number two-dimensional incompressible viscous Navier-Stokes fluids confined to a channel with periodically identified ends is investigated numerically. Two-dimensional steady states, quasi-steady states and persistently temporally evolving states of plane Poiseuille flow are found by following the evolution of the fluid for very long times. Using the primitive variable formulation of the Navier-Stokes equations, a splitting method is implemented to enforce incompressibility and no-slip boundary conditions. The resulting equations are solved using a pseudo-spectral method with a Fourier series as the expansion basis in the streamwise direction and a Chebyshev polynomial expansion basis in the cross-stream direction. Steady states for Reynolds numbers up to 15000 are presented and characterized in the frame of reference where the vorticity is accurately time independent. In this frame of reference the vorticity and stream function display an interesting correlation, showing that the flow is divided into three spatially distinct regions. The simulation is motivated by the previous success of the mean-field statistical mechanics of large numbers of discrete line vortices in describing vortical structures found at late times in doubly-periodic two-dimensional turbulent decay calculations. The steady-states of channel flow are not globally well described by the mean-field theory nor were we able to formulate a similar theory to predict the form of the steady-states.