Vielbein Lattice Boltzmann approach for fluid flows on spherical surfaces

Abstract: In this paper, we develop a lattice Boltzmann scheme based on the vielbein formalism for the study of fluid flows on spherical surfaces. The vielbein vector field encodes all details related to the geometry of the underlying spherical surface, allowing the velocity space to be treated as on the Cartesian space. The resulting Boltzmann equation exhibits inertial (geometric) forces, that ensure that fluid particles follow paths that remain on the spherical manifold, which we compute by projection onto the space of Hermite polynomials. Due to the point-dependent nature of the advection velocity in the polar coordinate $\theta$, exact streaming is not feasible, and we instead employ finite-difference schemes. We provide a detailed formulation of the lattice Boltzmann algorithm, with particular attention to boundary conditions at the north and south poles. We validate our numerical implementation against two analytical solutions of the Navier-Stokes equations derived in this work: the propagation of sound and shear waves. Additionally, we assess the robustness of the scheme by simulating the compressible flow of an axisymmetric shockwave and analyzing vortex dynamics on the spherical surface.