## Porous Convection Modeling

Here are some results from a porous convection model that uses one of Matlab's ODE solvers (ODE23) to solve the heat transport equation, and uses the backslash operator to solve the streamfunction (Laplace's) equation. This model shows us how heat is transported within a slab of saturated porous media when the slab is heated from below, and the top is open to a fluid bath. The sides and bottom have no fluid flow across them, and the sides have no heat transport across them. Heat is transported across the bottom by conduction, and is transported through the top by advection and conduction.

Shown are four different model runs, each with a different Rayleigh number (Ra). The Rayleigh number characterizes the vigor of convection within a system. Below a critical Rayleigh number, no convection will occur. In such a regime, enough heat is transported by conduction, and instabilities shrink rather than grow. Above the critical Rayleigh number, convection occurs because instabilities grow, and the solution becomes time-dependent. There are time-independent convecting solutions, but none are shown here.

Shown here is one model run below the critical Rayleigh number, and three at progressively larger Rayleigh numbers above the critical value.

Ra = 275: Ra = 825: Ra = 1375: Ra = 5500: 