Stability analysis of natural convection in porous cavities through integral transforms

The onset of convection and chaos related to natural convection inside a porous cavity heated from below is investigated using the generalized integral transform technique (GITT). This eigenfunction expansion approach generates an ordinary differential system that is adequately truncated in order to be handled by linear stability analysis (LSA) as well as in full nonlinear form through the Mathematica software system built-in solvers. Lorenzʹs system is generated from the transformed equations by using the steady-state solution to scale the potentials. Systems with higher truncation orders are solved in order to obtain more accurate results for the Rayleigh number at onset of convection, and the influence of aspect ratio and Rayleigh number on the cell pattern transition from n to n+2 cells (n=1,3,5,…) is analyzed from both local and average Nusselt number behaviors. The qualitative dependence of the Rayleigh number at onset of chaos on the transient behavior and aspect ratio is presented for a low dimensional system (Lorenz equations) and its convergence behavior for increasing expansion orders is investigated.