Fresnel reflection boundary for radiative transport lattice Boltzmann methods in highly scattering volume

Abstract

With its roots in kinetic theory, the lattice Boltzmann method (LBM) cannot only be used to solve complex fluid flows but also radiative transport in volume. The present work derives a novel Fresnel boundary scheme for radiative transport LBM, based on Fresnel's equation, which depicts the partly reflected radiation on surfaces. Driven from a boundary modeling and discussion on the microscopic level, incorporating Fresnel's equation, it is developed a boundary model for the mesoscopic radiative transport LBM. At an intermediate step, the Fresnel's equation is related to well known partial differential (Robin) equations, based on a bottom-up approach where the P1-Approximation is deployed. To connect the novel boundary scheme to the so derived target equation, a Chapman-Enskog expansion is examined in addition. Both techniques together, point out how to interpret microscopic modeling by the means of macroscopic expressions and as a consequence how, to chose simulation parameters according to the specific boundary. The numerical tests suggest that the proposed boundary is first order convergent. The paper closes with a showcase, where the novel boundary method for radiative transport LBM is applied to a setup with multiple LED spots.