Lattice Boltzmann models based on the vielbein formalism for the simulation of flows in curvilinear geometries

Abstract

In this paper, we consider the Boltzmann equation with respect to orthonormal vielbein fields in conservative form. This formalism allows the use of arbitrary coordinate systems to describe the space geometry, as well as of an adapted coordinate system in the momentum space, which is linked to the physical space through the use of vielbeins. Taking advantage of the conservative form, we derive the macroscopic equations in a covariant tensor notation, and show that the hydrodynamic limit can be obtained via the Chapman-Enskog expansion in the Bhatnaghar-Gross-Krook (BGK) approximation for the collision term. We highlight that in this formalism, the component of the momentum which is perpendicular to some curved boundary can be isolated as a separate momentum coordinate, for which the half-range Gauss-Hermite quadrature can be applied. We illustrate the capabilities of this formalism by considering two applications. The first one is the circular Couette flow between rotating coaxial cylinders, for which benchmarking data is available for all degrees of rarefaction, from the hydrodynamic to the ballistic regime. The second application concerns the flow in a gradually expanding channel. We employ finite-difference lattice Boltzmann models based on half-range Gauss-Hermite quadratures for the implementation of diffuse reflection, together with the fifth order WENO and third-order TVD Runge-Kutta numerical methods for the advection and

Quadrature-Based Lattice Boltzmann Models for Rarefied Gas Flow

“…More complex flows, where the application of half-range Gauss-Hermite quadrature is essential are investigated in Refs. [50,54,55].…”

“…Since this projection is not relevant for the further development of this chapter, we refer the reader to Refs. [49,50] for further details.…”

Lattice Boltzmann approach to rarefied gas flows using half-range Gauss-Hermite quadratures: Comparison to DSMC results based on ab initio potentials

“…While the discussion in this paper is limited to the Couette flow between parallel plates, we note that the LB models based on half-range quadratures can be employed also for flows in non-rectangular (curved) domains, e.g., when coupled with the vielbein approach, as discussed in Ref. [25].…”

“…In order to accurately capture the Knudsen layer in the vicinity of the wall at x = L/2, a coordinate stretching procedure is employed following Ref. [24], which is given through the following coordinate transformation [16,25]:…”

“…This allows the advection scheme to be implemented using a finite difference formulation, as described in Refs. [16,25]. For simplicity, we only consider the case when A = 0.98 in the simulations presented in the following sections.…”