Lattice Boltzmann Method for Solving Poiseuille Flow in Fluid Dynamics

The Lattice Boltzmann Method (LBM) serves as a widely adopted approach for solving Poiseuille flow problems in fluid dynamics. This technique is grounded in lattice gas kinetic theory, where continuous Navier-Stokes equations are discretized into lattice-based Boltzmann equations through spatial and temporal discretization. The method operates through two fundamental steps: collision (where particle distributions relax toward equilibrium) and streaming (where distributions propagate to neighboring lattice nodes). Key implementation aspects include defining D2Q9 lattice structures for 2D simulations, implementing bounce-back boundary conditions for no-slip walls, and calculating macroscopic velocity fields through moment summation of distribution functions. Given specified iteration counts as input parameters, LBM generates dynamic velocity distribution profiles that visualize parabolic flow development characteristic of Poiseuille flow. The algorithm's advantages include high numerical accuracy, computational efficiency due to local operations, inherent parallelization capabilities through distributed memory architectures, and straightforward handling of complex geometries. These attributes have established LBM as a prevalent computational fluid dynamics tool across diverse engineering applications.