Lattice Boltzmann Method

The Navier-Stokes equations describe macroscopic fluid behavior; they emerge from the Boltzmann equation through statistical averaging over molecular velocities. LBM discretizes Boltzmann's equation on a lattice, allowing efficient computation. The connection is rigorous: correct moments of the distribution function yield density, momentum, and energy conservation exactly.

The notation DxQy denotes x dimensions and y discrete particle velocities. D2Q9 is 2D with 9 velocities (including rest), sufficient for 2D Navier-Stokes. D3Q27 is 3D with 27 velocities, capturing full momentum and stress tensors. Higher Q numbers improve accuracy and stability but increase memory and compute cost.

Relaxation time τ controls viscosity: ν = c_s² (τ - 0.5). Choose τ > 0.5 for stability; τ close to 0.5 gives high Reynolds numbers but risks oscillations. Typical values are 0.6-1.0. For high Reynolds numbers, use multiple-relaxation-time (MRT) or turbulence models instead of simply increasing τ.

Lattice Mach number Ma_L = |u|/c_s compares flow velocity to lattice sound speed. LBM assumes Ma_L << 1 for accurate incompressible behavior. If Ma_L > 0.1, compressibility errors and acoustic waves appear. Keep Ma_L < 0.05 in most applications by adjusting resolution or flow speed accordingly.

Instead of specifying velocity or pressure like in Navier-Stokes, you specify the distribution function at boundaries. For no-slip walls, use bounce-back: incoming particles reflect. For pressure or velocity inlets, use equilibrium distributions matching the desired macroscopic values. Moving boundaries update particle distributions based on wall velocity, naturally incorporating drag.