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

TL;DR

This paper analyzes the transition of channel flow beyond a critical Reynolds number, revealing how flow becomes unstable and develops vortices, with a new analytical approach to understanding the flow dynamics.

Contribution

It introduces an analytical eigenfunction approach to reduce the Navier-Stokes equations to a system of ODEs, enabling detailed study of flow instability beyond the critical Reynolds number.

Findings

Flow becomes unstable and fluctuates beyond Re_c.

Reduced net flow rate with vortex formation at Re > Re_c.

Accurate determination of Re_c matching previous results.

Abstract

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 $Re>Re_c$.