Modal and non-modal linear stability of Poiseuille flow through a channel with a porous substrate

Abstract

We present modal and non-modal linear stability analyses of Poiseuille flow through a plane channel with a porous substrate modeled using the Volume Averaged Navier–Stokes (VANS) equations. Modal stability analysis shows the destabilization of the flow with increasing porosity of the layer. The instability mode originates from the homogeneous fluid region of the channel for all the values of porosity considered but the governing mechanism changes. Perturbation kinetic energy analysis reveals the importance of viscous dissipation at low porosity values while dissipation in the porous substrate becomes significant at higher porosity. Scaling analysis highlights the invariance of the critical wavenumber with changing porosity. On the other hand, the critical Reynolds number remains invariant at low porosity and scales as Rec ∼ (H/δ) 1.4 at high porosity where δ is the typical thickness of the vorticity layer at the fluid– porous interface. This reveals the existence of a Tollmien–Schlichting-like viscous instability mechanism at low porosity values, and Rayleigh analysis shows the presence of an inviscid instability mechanism at high porosity. For the whole range of porosities considered, the non-modal analysis shows that the optimal mechanism responsible for transient energy amplification is the lift-up effect, giving rise to streaky structure as in single-phase plane Poiseuille flow. The present results strongly suggest that the transition to turbulence follows the same path as that of classical Poiseuille flow at low porosity values, while it is dictated by the modal instability for high porosity values.