A stabilized stochastic finite element second-order projection method for modeling natural convection in random porous media

We consider natural convection in flow saturated porous media with random porosity. The porosity is treated as a random field and a stochastic finite element method is developed. The stochastic projection method is considered for the solution of the high-dimensional stochastic Navier–Stokes equations since it leads to the uncoupling of the velocity and pressure degrees of freedom. Because of the porosity dependence of the pressure gradient term in the governing flow equations, one cannot use the first-order projection method. A stabilized stochastic finite element second-order projection method is presented based on a pressure gradient projection. A two-dimensional stochastic problem with moderate and large variation in the random porosity field is examined and the results are compared with Monte-Carlo and sparse grid (Smolyak) collocation approaches. Excellent agreement between these results indicates the effectiveness and accuracy of the proposed methodology.