Integration of vorticity–velocity formulation in a lattice Boltzmann method for porous media

Abstract

A lattice Boltzmann method (LBM) is proposed to address the two-dimensional macroscopic equations of velocity–vorticity for porous media in incompressible flows. The macroscopic equations of two-dimensional porous media using the representative element volume scale approach, employing various models, are shown. The momentum equations are transformed and presented in the velocity and vorticity format. Additionally, the energy and concentration equations are thoroughly examined. Subsequently, the LBM is presented to restore the dimensional macroscopic equations of the velocity–vorticity format for various porous models, accounting for external forces. The paper provides proof and derivations of the equations for the LBM, which are then demonstrated and discussed. The suggested approach is evaluated across a variety of well-established benchmark examples within the realm of fluid flow, heat, and mass transfer in porous media. Importantly, the LBM approach significantly reduces computational time compared to previous methods in the field by eliminating pressure in the momentum equation, thereby reducing the number of unknown variables and transforming the equation into a convection–diffusion form. This modification leads to a linear equilibrium distribution function and a noteworthy decrease in computational costs.