A Boltzmann-based mesoscopic model for contaminant transport in flow systems

Abstract

The objective of this paper is to demonstrate the formulation of a numerical model for mass transport based on the Bhatnagar–Gross–Krook (BGK) Boltzmann equation. To this end, the classical chemical transport equation is derived as the zeroth moment of the BGK Boltzmann differential equation. The relationship between the mass transport equation and the BGK Boltzmann equation allows an alternative approach to numerical modeling of mass transport, wherein mass fluxes are formulated indirectly from the zeroth moment of a difference model for the BGK Boltzmann equation rather than directly from the transport equation. In particular, a second-order numerical solution for the transport equation based on the discrete BGK Boltzmann equation is developed. The numerical discretization of the first-order BGK Boltzmann differential equation is straightforward and leads to diffusion effects being accounted for algebraically rather than through a second-order Fickian term. The resultant model satisfies the entropy condition, thus preventing the emergence of non-physically realizable solutions including oscillations in the vicinity of the front. Integration of the BGK Boltzmann difference equation into the particle velocity space provides the mass fluxes from the control volume and thus the difference equation for mass concentration. The difference model is a local approximation and thus may be easily included in a parallel model or in accounting for complex geometry. Numerical tests for a range of advection–diffusion transport problems, including one- and two-dimensional pure advection transport and advection–diffusion transport show the accuracy of the proposed model in comparison to analytical solutions and solutions obtained by other schemes.