Linear stability analysis of pressure-driven flows in channels with porous walls

We present the three-dimensional linear stability analysis of a pressure-driven, incompressible, fully developed, laminar flow in a channel delimited by rigid, homogeneous, isotropic, porous layers. We consider porous materials of small permeability in which the maximum fluid velocity is small compared to the mean velocity in the channel region and for which inertial effects may be neglected. We analyse the linear stability of symmetric laminar velocity profiles in channels with two identical porous walls as well as skewed laminar velocity profiles in channels with only one porous wall. We solve the fully coupled linear stability problem, arising from the adjacent channel and porous flows, using a spectral collocation technique. We validate our results by recovering the linear stability results of a flow in a channel with impermeable walls as the permeabilities of the porous layers tend to zero. We also verify that our results are consistent with the assumption of negligible inertial effects in the porous regions. We characterize the stability of pressure-driven flows by performing a parametric study in which we vary the permeability, porosity, and height of the porous layers as well as an interface coefficient, tau, associated with the momentum transfer process at the interfaces between the channel and porous regions. We find that very small amounts of wall permeability significantly affect the Orr-Sommerfeld spectrum and can dramatically decrease the stability of the channel flow. Within our assumptions, in channels with two porous walls, permeability destabilizes up to two Orr-Sommerfeld wall modes and introduces two new damped wall modes on the left branch of the spectrum. In channels with only one porous wall, permeability destabilizes up to one wall mode and introduces one new damped wall mode on the left branch of the spectrum. In both cases, permeability also introduces a new class of damped modes associated with the porous regions. The size of the unstable region delimited by the neutral curve grows substantially, and the critical Reynolds number can decrease to only 10 % of the corresponding value for a channel flow with impermeable walls. We conclude our study by considering two real materials: foametal and aloxite. We fit the porosity and interface coefficient tau to published data so that the porous materials we model behave like foametal and aloxite, and we compare our results with previously published numerical and experimental results.