💨 Fluid Lattice-Boltzmann

LBM discretises space into a lattice (here 160×80 cells) and tracks nine particle-distribution functions per cell instead of directly solving the Navier-Stokes equations for pressure and velocity. Each simulation step alternates a local collision (relaxing f_i toward its equilibrium value f_i^eq over relaxation time τ) with a streaming step that shifts each f_i to the neighbouring cell in its lattice direction. Density and velocity are simply computed as moments of the distribution functions. This local, explicit update rule makes LBM straightforward to parallelise, unlike traditional pressure-Poisson-based Navier-Stokes solvers.

D2Q9 means a 2-dimensional lattice with 9 discrete velocity directions per cell: one rest direction, four axis-aligned neighbours, and four diagonal neighbours, each with its own lattice weight (4/9, 1/9 or 1/36). This is the standard, minimal stencil for 2D incompressible-ish flow simulation and is exactly what this simulation implements with the CX/CY/W arrays in the code.

The simulation computes the Reynolds number as Re = U·L/ν, where U is the inlet velocity, L is the obstacle diameter (twice the Obstacle size slider), and ν is the Viscosity slider value, which also sets the LBM relaxation time τ = 3ν + 0.5. Lowering viscosity or raising inlet velocity increases Re and pushes the flow from smooth creeping motion toward vortex shedding and eventually turbulence; the four presets are just pre-tuned combinations of these two sliders.

Once Re rises above roughly 40, the symmetric wake behind the cylinder becomes unstable to small perturbations, and vortices begin shedding alternately from the top and bottom of the obstacle instead of staying attached. This periodic shedding is the Kármán vortex street, and its frequency f obeys the Strouhal relation St = f·D/U ≈ 0.21 near Re = 100, which is the constant value the simulation reports in the Flow Stats panel.

Speed colours each cell by the magnitude of its local velocity vector, computed from the first velocity moment of the distribution functions. Vorticity approximates the local curl of the velocity field using finite differences between neighbouring cells' momentum, highlighting the rotating cores of shed vortices in red/blue. Pressure/density mode colours cells by ρ = Σf_i, which in this near-incompressible LBM scheme is proportional to local pressure through an ideal-gas-like equation of state.