Nonmodal stability analysis of the plane Poiseuille flow in a multilayer porous-fluid channel

The stability of plane Poiseuille flow of a viscous Newtonian fluid in a multilayer channel with anisotropic porous walls is analyzed using the classical modal analysis, the energy method, and the non-modal analysis. The influence of porous wall parameters such as depth ratio (ratio of porous layer thickness to fluid layer thickness) and anisotropic permeability (in terms of meanpermeability and anisotropy parameter) on flow instability are investigated. The modal stability analysis and energy method show that the anisotropy parameter can stabilize the flow, whereas the depth ratio and mean permeability effects can cause destabilization. Furthermore, the energy budget analysis reveals that the energy production term transfers energy to the disturbance from the base flow through the Reynolds stress, amplifying the kinetic energies in all layers and, hence, enhancing the growth rates of the unstable modes. A significant disparity is observed between the critical Reynolds number obtained through modal analysis and the one determined by the energy method, which confirms the growth of transient perturbation kinetic energy. Specifically, transient growth and response functions are examined to understand the flow response to initial conditions and external excitation (receptivity analysis). It turns out that there is substantial transient growth at a sub-critical Reynolds number. These transient growths are greatly enhanced by increasing the mean permeability or the depth ratio and reducing the anisotropy parameter. The optimal perturbations leading to the maximum transient amplification are determined for various parameters, including counter-rotating vortices (rolls) and streaks.