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 shock wave and analyzing vortex dynamics on the spherical surface.

Code repository: https://github.com/agabbana/vielbein_lbm

References in corpus (19)

• Topological Origin of Equatorial Waves
• Formation of Jets and Equatorial Superrotation on Jupiter
• Modelling fluid deformable surfaces with an emphasis on biological interfaces
• Lattice Boltzmann study on Kelvin-Helmholtz instability: the roles of velocity and density gradients
• Massively parallel lattice-Boltzmann codes on large GPU clusters
• A fully relativistic lattice Boltzmann algorithm
• Dynamical patterns in active nematics on a sphere
• Hydrodynamic Flows on Curved Surfaces: Spectral Numerical Methods for Radial Manifold Shapes
• Conservative 3+1 General Relativistic Boltzmann Equation
• Performance and Portability of Accelerated Lattice Boltzmann Applications with OpenACC
• Corner transport upwind lattice Boltzmann model for bubble cavitation
• Active Flows on Curved Surfaces
• Lattice Boltzmann models based on the vielbein formalism for the simulation of flows in curvilinear geometries
• Linearly forced fluid flow on a rotating sphere
• Fast kinetic simulator for relativistic matter
• Multicomponent Flow on Curved Surfaces: A Vielbein Lattice Boltzmann Approach
• Two-dimensional off-lattice Boltzmann model for van der Waals fluids with variable temperature
• Semi-Lagrangian implicit Bhatnagar-Gross-Krook collision model for the finite-volume discrete Boltzmann method
• Axisymmetric flows on the torus geometry