Variational multiscale element free Galerkin method for natural convection with porous medium flow problems

In this paper, the variational multiscale element free Galerkin (VMEFG) method is extended to simulate the two-dimensional natural convection with porous medium flow problems. The Darcy-Brinkman-Forchheimer extended model is used for the non-Darcy porous medium flow. The VMEFG method is a meshless method which had successfully applied to simulate incompressible flow problems with equal order basis approximation for the velocity and pressure fields, meanwhile, this method can easily deal with complex geometry problems. Three numerical examples, including the natural convection in a square cavity, a cavity with cosine vertical wavy wall and a triangular cavity with zig-zag shaped bottom are considered to investigate the validation and accuracy of the proposed method. The numerical results are presented in the form of streamlines, isotherms and average Nusselt numbers. A good agreement is observed between the present results and those available in the previous studies. Meanwhile, effects of Rayleigh number, porosity of the porous medium, amplitude of wavy surface and number of undulation on the flow structure and heat transfer are also investigated.