Computation of Effective Viscosities for Rarefied Gas Flows Using Ray-Tracing

The recovery of rarefied gas flow fields with the Boltzmann equation is too computationally expensive whenever solid–gas boundary conditions exhibit a complicated shape. In the literature, an extension of the Navier–Stokes equations has been proposed to model the stress–strain relationship through a proportionality relationship between the effective mean free path and the effective viscosity of a gas. In this article, it is proposed to compute effective mean free paths through a numerical integration procedure using a ray-tracing algorithm, which allows the determination of local values of mean free paths on any voxelization or mesh discretization of any computational domain. It is first shown that the use of a ray basis aligned with a lattice structure can create a large error in the evaluation of the effective mean free path, highlighting the importance of dealignment of the ray-basis. A second-order angle step convergence is found for a planar geometry, while a sublinear convergence is reached for an array of cylinders, as a consequence of the non-convexity of the gas domain. Thus, a larger number of rays than previously considered in the literature is required for an accurate computation of the effective mean free paths.