Mesh-Based CFD
Computational fluid dynamics divides into two broad families: methods that work on a fixed grid (Eulerian) and methods that follow fluid parcels as they move (Lagrangian). Most industrial solvers — ANSYS Fluent, OpenFOAM — are Eulerian finite-volume codes. The simulations here cover both families, letting you see how the same physical flow emerges from very different mathematical frameworks.
f_i(x + e_i·Δt, t + Δt) = f_i(x, t) − (1/τ)[f_i − f_i^eq]
Equilibrium distribution: f_i^eq = ρ·w_i·[1 + (e_i·u)/c_s² +
(e_i·u)²/(2c_s⁴) − u²/(2c_s²)]
9 velocity directions (D2Q9), weights w_i = 4/9, 1/9, 1/36
Relaxation time τ controls kinematic viscosity: ν = c_s²(τ −
0.5)Δt
Macroscopic density ρ = Σf_i, momentum ρu = Σf_i·e_i
Particle & Wave Methods
While grid methods excel at steady and mildly unsteady flows, particle and wave methods shine for free-surface problems — splashing droplets, breaking ocean waves, sloshing tanks. The key advantage: the free surface is implicit in the particle positions; you never need to track an interface separately.
Why choose LBM over traditional Navier-Stokes solvers? The Lattice-Boltzmann method never explicitly solves the incompressibility pressure equation — the costliest step in finite-volume CFD. Instead, simple streaming and collision rules on a regular lattice recover the Navier-Stokes equations in the macroscopic limit (Chapman-Enskog expansion). This makes LBM trivially parallelisable and ideal for GPU implementation.
Reynolds Number — The Universal Fluid Parameter
Every fluid dynamics simulation is ultimately characterised by the Reynolds number Re = ρUL/μ, the ratio of inertial to viscous forces. Below Re ≈ 40, flow past a cylinder is steady and symmetric. Between 40 and 200, Kármán vortex shedding begins. Above 1000, the wake becomes turbulent. The same sequence appears whether the fluid is air, water, or blood — only the characteristic length and velocity differ.
Algorithms at a Glance
Suggested Learning Paths
- Kármán Vortex — Reynolds number basics
- Bénard Convection — buoyancy-driven flow
- Shallow-Water Waves — wave propagation
- Blood Flow — non-Newtonian rheology
- Cyclone — geophysical fluid dynamics
- Lattice-Boltzmann — mesh-free CFD method
- SPH Fluid — particle hydrodynamics
- Kármán Vortex — SIMPLE solver internals
- Bénard Convection — Boussinesq approximation
- Ocean Waves — spectral wave modelling