Evolution of channel flow and Darcy law beyond the critical Reynolds number

Channel flow is usually described by Darcy law with the Poiseuille flow profile. However, for incompressible channel flow there is a critical state, characterized by a critical Reynolds number $Re_c$ and a critical wavevector mc, beyond which the channel flow becomes unstable in the linear regime. By obtaining the analytical eigenfunctions of the linearized, incompressible, three dimensional (3D) Navier-Stokes (NS) equation in the channel geometry, i.e., the hydrodynamic modes (HMs), we reduce the full NS equation to a system of coupled autonomous ordinary differential equations (ODEs) by expanding the velocity in terms of the HMs; time becomes the only independent variable. The nonlinear term of the NS equation is converted to a third-rank tensor that couples pairs of the expansion coefficients to effect the time variation on the third. In the linear regime, the value of $Re_c$ is obtained to five significant digit accuracy when compared to the Orszag result. We numerically time evolve the autonomous ODEs at $Re>Re_c$ with a finite set of thermally excited initial HMs to find a fluctuating equilibrium state with a reduced net flow rate, accompanied by vortices. Through the perspective of force balance, interesting features are uncovered in the counter-flow profiles at $Re>Re_c$.