Analysis on the onset of buoyancy-driven convection in a fluid-saturated porous medium heated uniformly from below

Abstract

The onset of convective motion in an initially quiescent, horizontal isotropic porous layer is analyzed theoretically. Using linear theory, the stability equations are obtained with and without the quasi-steady state approximation (QSSA). They are solved by expanding the disturbances as a series of orthogonal functions. Initial value problem analyses also are conducted, and their results are compared with the eigenanalysis and the QSSA. For the large Darcy–Rayleigh number RaD, the critical time of the onset of convection is found to be τc=11.74RaD−1 with the critical wavenumber kc=0.29RaD1/2. Based on linear analyses, numerical simulations are conducted to consider nonlinear effects. The nonlinear analysis using the Fourier spectral method and the finite volume method reproduces the linear stability limit obtained theoretically. From the present nonlinear analysis, it is shown that manifest convection is observed experimentally from τm=51.42RaD−1.