Three-dimensional vorticity-velocity formulation in a lattice Boltzmann method

In recent decades, a paradigm shift in macroscopic methods has favored the use of non-primitive variables, such as velocity and vorticity (V-V), over traditional primitive variables. This shift eliminates the need for solving a Poisson equation for pressure, aligning numerical treatments more closely with physical reality. However, the lattice Boltzmann method (LBM), renowned for its efficacy in studying fluid flow phenomena, continues to rely on the conventional pressure-velocity (P-V) approach. This conventional approach necessitates a pressure-density relation, posing challenges in maintaining the incompressible condition. This study pioneers a novel application of the LBM to three-dimensional velocity-vorticity equations, expanding upon our suggested recent method for two-dimensional cases [Kefayati, Phys. Fluids. 36, 013128 (2024)]. To address the complexities introduced by the vortex stretching term in three dimensions, a new equilibrium distribution function is formulated and introduced to the three-dimensional nature of the vorticity vector. The paper details the derivation of the three-dimensional LBM and substantiates its effectiveness through numerical examples, showcasing its applicability in fluid dynamics. By bridging the gap between traditional P-V formulations and the benefits of non-primitive V-V variables, this work contributes to the ongoing exploration of LBM applications in fluid dynamics. The focus on three-dimensional scenarios involving velocity-vorticity equations marks a significant advancement, offering insights into the nuanced dynamics of fluid flow and paving the way for more accurate and realistic simulations in complex environments.