Simple 2D fluid — theory

What this demo solves, how viscosity and Reynolds number enter, and how to read the overlays. Educational lattice Boltzmann — not production CFD.

Lattice Boltzmann (D2Q9 BGK)

Instead of discretising Navier–Stokes on a mesh with pressure Poisson solves, LBM evolves discrete velocity distribution functions on a lattice. Macroscopic density ρ and velocity (u, v) are moments of those populations.

collide (relax toward feq) → stream along eᵢ → bounce-back on solids

Viscosity

ν = cₛ² (1/ω − 1/2)
cₛ² = 1/3
ω = 1 / (3ν + 1/2)

Higher ν → more damping, thicker boundary layers, less unsteadiness. Too low ν (or high inlet speed) can make the scheme unstable; the demo softly clamps velocity (Mach control).

Reynolds number

Re = U L / ν

• U — inlet speed u₀ (lattice units)
• L — characteristic length (cylinder diameter, step height, …)
• ν — kinematic viscosity (lattice units)

Cylinder wakes: low Re attached; moderate Re → von Kármán street. On-screen Re is an order-of-magnitude lattice estimate.

Boundary conditions in the demo

• Channel — left Dirichlet-like equilibrium inlet, right copy outlet, optional top/bottom walls
• Lid-driven cavity — closed box with moving top lid (classic CFD benchmark geometry)
• Solids — bounce-back reflection of populations

Scalars & overlays

• |u| — speed map
• Vorticity — discrete curl; vortex cores and shear layers
• Density / pressure — LBM is weakly compressible; ρ variations proxy pressure
• Dye — passive scalar, semi-Lagrangian advection + decay
• Tracers / stream ribbons / vectors — Lagrangian intuition aids

Limits

• Coarse lattice; not mesh-converged
• Simplified BCs; no wall model / turbulence model
• Lattice units — not calibrated SI CFD
• For teaching wakes, Re, viscosity — not certification