Nonmodal stability analysis of Poiseuille flow through a porous medium

Abstract

We unravel the nonmodal stability of a three-dimensional nonstratified Poiseuille flow in a saturated hyperporous medium constrained by impermeable rigid parallel plates. The primary objective is to broaden the scope of previous studies that conducted modal stability analysis for two-dimensional disturbances. Here, we explore both temporal and spatial transient disturbance energy growths for three-dimensional disturbances when the Reynolds number and porosity of the material are high, based on evolution equations with respect to time and space, respectively. Modal stability analysis reveals that the critical Reynolds number for the onset of shear mode instability increases as porosity increases. Moreover, the Darcy viscous drag term stabilizes shear mode instability, resulting in a delay in the transition from laminar flow to turbulence. In addition, it demonstrates the suppression of three-dimensional shear mode instability as the spanwise wavenumber increases, thereby confirming the statement of Squire’s theorem. By contrast, nonmodal stability analysis discloses that both temporal and spatial transient disturbance energy growths curtail as the effect of the Darcy viscous drag force intensifies. But their maximum values behave like $O\left(R{e}^2\right)$ for a fixed porous material, where $Re$ is the Reynolds number. However, for different porous materials, the scalings for both temporal and spatial transient disturbance energy growths are different. Furthermore, increasing porosity also suppresses both temporal and spatial disturbance energy growths. Finally, we observe that temporal transient disturbance energy growth becomes larger for a spanwise perturbation, while spatial transient disturbance energy growth becomes larger for a steady perturbation when angular frequency vanishes. The initial disturbance that excites the largest temporal energy amplification generates two sets of alternating high-speed and low-speed elongated streaks in the streamwise direction.