Heat transfer regimes and hysteresis in porous media convection

Results of an investigation of different heat transfer regimes in porous me dia convection are presented by using a truncated Galerkin representation o f the governing equations that yields the familiar Lorenz equations for the variation of the amplitude in the time domain. The solution to this system is obtained analytically by using a weak non-linear analysis and computati onally by using Adomian's decomposition method. Expressions for the average d Nusselt number are derived for steady, periodic, as well as weak-turbulen t (temporal-chaotic) convection. The phenomenon of Hysteresis in the transi tion from steady to weak-turbulent convection, and backwards, is particular ly investigated, identifying analytically its mechanism, which is confirmed by the computational results. While the post-transient chaotic solution in terms of the dependent variables is very sensitive to the initial conditio ns, the affinity of the averaged values of these variables to initial condi tions is very weak. Therefore, long-term predictability of these averaged v ariables, and in particular the Nusselt number, becomes possible, a result of substantial practical significance. Actually, the only impact that the t ransition to chaos causes on the predicted results in terms of the averaged heat flux is a minor loss of accuracy. Therefore, the predictability of th e results in the sense of the averaged heat flux: is not significantly affe cted by the transition from steady to weak-turbulent convection. The transi tion point is shown to be very sensitive to a particular scaling of the equ ations, which leads the solution to an invariant value of steady-state for sub-transitional conditions, a result that affects the transition point in some cases.