Finite amplitude analysis of non-isothermal parallel flow in a vertical channel filled with a highly permeable porous medium

Abstract

This paper addresses the finite amplitude instability of stably stratified non-isothermal parallel flow in a vertical channel filled with a highly permeable porous medium. A cubic Landau equation is derived to study the limiting value of growth of instabilities under nonlinear effects. The non-Darcy model is considered to describe the flow instabilities in a porous medium. The nonlinear results are presented for air as well as water. The analysis is carried out in the vicinity of as well as away from the critical point (bifurcation point). It is found that when the medium is saturated by water then supercritical bifurcation is the only type of bifurcation at and beyond the bifurcation point. However, for air, depending on the strength of the flow and permeability of the medium, both supercritical and subcritical bifurcations are observed. The influence of nonlinear interaction of different harmonics on the heat transfer rate, friction coefficient, nonlinear kinetic energy spectrum and disturbance flow is also studied in both supercritical as well as subcritical regimes. The variation of neutral stability curves of parallel mixed convection flow of air with wavenumber reveals that a bifurcation that is supercritical for some wavenumber may be subcritical orvice versaat other nearby wavenumbers. The analysis of the nonlinear kinetic energy spectrum of the fundamental disturbance also supports the existence of supercritical/subcritical bifurcation at and away from the critical point. The effect of different harmonics on the pattern of secondary flow, based on linear stability theory, is also studied and a significant influence is found, especially in the subcritical regime.