Linear and Nonlinear Convection in Porous Media between Coaxial Cylinders

Abstract

In this thesis we develop a mathematical model for describing three-dimensional natural convection in porous media filling a vertical annular cylinder. We apply a linear stability analysis to determine the onset of convection and the preferred convective mode when the annular cylinder is subject to two different types of boundary conditions: heat insulated sidewalls and heat conducting sidewalls. The case of an annular cylinder with insulated sidewalls has been investigated earlier, but our results reveal more details than previously found. We also investigate the case of the radius of the inner cylinder approaching zero and the results are compared with previous work for non-annular cylinders. Using pseudospectral methods we have built a high-order numerical simulator to uncover the nonlinear regime of the convection cells. We study onset and geometry of convection modes, and look at the stability of the modes with respect to different types of perturbations. Also, we examine how variations in the Rayleigh number affects the convection modes and their stability regimes. We uncover an increased complexity regarding which modes that are obtained and we are able to identify stable secondary and mixed modes. We find the different convective modes to have overlapping stability regions depending on the Rayleigh number. The motivation for studying natural convection in porous media is related to geothermal energy extraction and we attempt to determine the effect of convection cells in a geothermal heat reservoir. However, limitations in the simulator do not allow us to make any conclusions on this matter.