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

TL;DR

This paper introduces a lattice Boltzmann modeling approach using vielbein formalism to simulate flows in complex curvilinear geometries, enabling accurate handling of boundaries and diverse coordinate systems.

Contribution

It develops a novel lattice Boltzmann framework based on vielbein formalism, allowing flexible coordinate systems and improved boundary treatment in flow simulations.

Findings

Successfully simulated circular Couette flow with comprehensive benchmarking data.

Demonstrated capability to model flow in expanding channels with high accuracy.

Validated the formalism's effectiveness across different flow regimes.

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 time-stepping, respectively.