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arXiv · 2207.06922

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

Xiaohui Deng·Ping Sheng
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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 $Re>Re_c$.

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Xiaohui Deng, Ping Sheng. 2022-07-14. Evolution of channel flow and Darcy law beyond the critical Reynolds number. https://arxiv.org/abs/2207.06922

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