PhysSandbox
Classical MechanicsWaves & SoundElectricity & MagnetismOptics & LightGravity & OrbitsLabs
🌙Astronomy & The Sky🌡️Thermodynamics🌍Biophysics, Fluids & Geoscience📐Math Visualization🔧Engineering🧪Chemistry

More from Math Visualization

Other simulators in this category — or see all 78.

View category →
NewUniversity / research

Finite-Volume Advection-Diffusion 2D

Conservative scalar transport on a 2D grid: face fluxes, upwind vs central interpolation, Peclet number, CFL, and numerical diffusion.

Launch Simulator
NewUniversity / research

Conjugate Gradient Solver

SPD system Ax=b as quadratic minimization: contour geometry, CG vs steepest descent path, residual norm, and condition number.

Launch Simulator
NewUniversity / research

Power Iteration Eigenvalue Convergence

Visualize dominant eigenvector convergence: spectral gap ratio, Rayleigh quotient, eigen residual, and normalized power iterates on the unit circle.

Launch Simulator
NewUniversity / research

Newton-Raphson Basins in 2D Systems

Map Newton basins for nonlinear F(x,y)=0 systems: initial-guess sensitivity, iteration counts, root attraction, and singular-Jacobian failures.

Launch Simulator
NewUniversity / research

Monte Carlo Integration & Variance Reduction

Compare plain Monte Carlo, importance sampling, and stratified sampling for ∫f(x)dx, with convergence curves, standard error, and the 1/√N rate.

Launch Simulator
NewUniversity / research

LMS / NLMS Adaptive Noise Cancellation

Primary p = s + v with v a fixed unknown FIR of Gaussian reference x[n]. Watch an L-tap FIR adapt by LMS or NLMS so error e = p − wᵀx → s; running MSE and ‖w − h‖.

Launch Simulator
PhysSandbox

Interactive physics, chemistry, and engineering simulators for students, teachers, and curious minds.

Physics

  • Classical Mechanics
  • Waves & Sound
  • Electricity & Magnetism

Science

  • Optics & Light
  • Gravity & Orbits
  • Astronomy & The Sky

More

  • Thermodynamics
  • Biophysics, Fluids & Geoscience
  • Math Visualization
  • Engineering
  • Chemistry

© 2026 PhysSandbox. Free interactive science simulators.

PrivacyTermsContact
Home/Math Visualization/Lattice Boltzmann D2Q9 Flow

Lattice Boltzmann D2Q9 Flow

The lattice Boltzmann method (LBM) solves fluid-like motion by evolving particle distribution functions on a lattice instead of directly discretizing the Navier-Stokes equations. This simulator uses the standard D2Q9 velocity set and a BGK relaxation step: distributions collide toward local equilibrium, stream to neighboring cells, and bounce back from solid boundaries. Two setups are included: a lid-driven cavity and flow past a circular cylinder. The canvas colors vorticity so recirculation, shear layers, and wake structures become visible as Reynolds number changes.

Who it's for: Scientific computing, numerical methods, computational fluid dynamics, applied math, and physics students learning mesoscopic solvers.

Key terms

  • Lattice Boltzmann method
  • D2Q9
  • BGK collision
  • Vorticity
  • Reynolds number
  • Bounce-back boundary

This is a compact teaching LBM: small grid, BGK collision, simple bounce-back walls, and qualitative inlet/outlet boundaries. It is meant for numerical-method intuition rather than production CFD.

Live graphs

D2Q9 flow setup

120
0.055
5

Measured values

Relaxation time τ0.5178
Lattice viscosity ν0.00594
Collision rate ω1.931
Grid96×54
Stability hintlow τ

How it works

Interactive D2Q9 lattice Boltzmann fluid simulation with lid-driven cavity and flow past a cylinder, vorticity colors, and Reynolds-number control.

Key equations

D2Q9: f_i(x+c_i,t+1) = f_i(x,t) − ω(f_i − f_i^eq)
ν = c_s²(τ−1/2), c_s²=1/3, Re = U L / ν

Frequently asked questions

Why does LBM use distribution functions instead of velocity directly?
LBM evolves small populations moving in discrete lattice directions. Macroscopic density and velocity are moments of those populations, which makes streaming local and boundary handling intuitive.
What does the relaxation time tau control?
For D2Q9 BGK, viscosity is nu = (tau - 1/2)/3 in lattice units. Tau too close to 0.5 means very low viscosity and can become numerically unstable.