A projection method to symmetrize the momentum flux in a vector lattice Boltzmann formulation of hydrodynamics

Journal article

The standard lattice Boltzmann approach to hydrodynamics uses scalar distribution functions to represent the number densities of particles. Each particle moves with one of a small set of constant velocities. The momentum flux is the second moment of these distribution functions with respect to the particle velocities. The momentum flux is thus a symmetric tensor by construction, so the conservation law for linear momentum implies a conservation law for angular momentum. A more flexible approach uses vector distribution functions instead. Each particle carries a momentum that may not be aligned with its direction of propagation. The momentum flux tensor is in general not symmetric, so the evolution equation for angular momentum contains a source term as well as a flux. The momentum flux can be symmetrized using a more general model for collisions between particles that applies separate relaxation times to the symmetric and antisymmetric parts of the momentum flux and then sets the relaxation time for the antisymmetric part to zero. The resulting algorithm is equivalent to a projection of the momentum flux onto the space of symmetric tensors, analogous to the Chorin–Temam projection scheme for the incompressible Navier–Stokes equations that projects the velocity field onto the space of divergence-free vector fields.

Authors: Dellar, PJ

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: American Institute of Physics
• Journal: Physics of Fluids
• Volume: 37
• Issue: 6
• Article number: 063606
• Publication date: 2025-06-06
• Acceptance date: 2025-04-19
• DOI: 10.1063/5.0268328
• EISSN: 1089-7666
• ISSN: 1070-6631
• Language: English
• Keywords: stress strain relations; cauchy stress tensor; angular momentum coupling; lattice Boltzmann methods; Navier Stokes equations