Lattice Boltzmann Method for Natural Convection in Anisotropic Porous Media

Abstract

A lattice Boltzmann method (LBM) is proposed for the simulation of natural convection in anisotropic porous media with the generalized non-Darcy model. The Brinkman-Forchheimer momentum equation is recovered from a kinetic equation for the distribution function, which has a forcing term to introduce anisotropy of permeability. The LBM can analytically calculate fluid velocity using the inverse matrix for the Brinkman equation. The simulation results for a Poiseuille flow of anisotropic porous media demonstrate that the proposed LBM for the Brinkman equation has second-order accuracy in space for any permeability ratio. Since the forcing term of the generalized equation contains a nonlinear (Forchheimer) term containing the quadratic form of the velocity and the permeability tensor, the LBM must use a free-derivative optimization method. In the numerical simulation of natural convection, the streamlines and isotherms obtained by the LBM agree well with those of the finite difference method (FDM), as well as with those of previous investigations of various fundamental parameters, such as the Rayleigh number, the inclination of the principal permeability direction, and the permeability ratio. The numerical results of the LBM show good agreement with the reference solutions for the value of the stream function at the center of the primary vortex and the average Nusselt number.

Keywords

• anisotropic porous media
• lattice boltzmann method
• computational fluid dynamics
• convection
• non-darcy model