CFD model: lattice Boltzmann method (LBM)
- streaming (part 2/2)f0temp(x,t) = f0(x, t)
fitemp(x,t) = f(t%2 ? i : (i%2 ? i+1 : i-1))(i%2 ? x : x-ei, t) for i ∈ [1, q-1]
- collisionρ(x,t) = (Σi fitemp(x,t)) + 1
u(x,t) = 1∕ρ(x,t) Σi ci fitemp(x,t)
fieq-shifted(x,t) = wi ρ · ((u°ci)2∕(2c4) - (u°u)∕(2c2) + (u°ci)∕c2) + wi (ρ-1)
fitemp(x, t+Δt) = fitemp(x,t) + Ωi(fitemp(x,t), fieq-shifted(x,t), τ)
- streaming (part 1/2)f0(x, t+Δt) = f0temp(x, t+Δt)
f(t%2 ? (i%2 ? i+1 : i-1) : i)(i%2 ? x+ei : x, t+Δt) = fitemp(x, t+Δt) for i ∈ [1, q-1]
- variables and notation
| variable | SI units | defining equation | description |
| :------------------: | :---------------------------------: | :-------------------------------------------------: | :------------------------------------------------------------------------------ |
| | | | |
| x | m | x = (x,y,z)T | 3D position in Cartesian coordinates |
| t | s | - | time |
| ρ | kg∕m³ | ρ = (Σi fi)+1 | mass density of fluid |
| p | kg∕m s² | p = c² ρ | pressure of fluid |
| u | m∕s | u = 1∕ρ Σi ci fi | velocity of fluid |
| ν | m²∕s | ν = μ∕ρ | kinematic shear viscosity of fluid |
| μ | kg∕m s | μ = ρ ν | dynamic viscosity of fluid |
| | | | |
| fi | kg∕m³ | - | shifted density distribution functions (DDFs) |
| Δx | m | Δx = 1 | lattice constant (in LBM units) |
| Δt | s | Δt = 1 | simulation time step (in LBM units) |
| c | m∕s | c = 1∕√3 Δx∕Δt | lattice speed of sound (in LBM units) |
| i | 1 | 0 ≤ i < q | LBM streaming direction index |
| q | 1 | q ∈ { 9,15,19,27 } | number of LBM streaming directions |
| ei | m | D2Q9 / D3Q15/19/27 | LBM streaming directions |
| ci | m∕s | ci = ei∕Δt | LBM streaming velocities |
| wi | 1 | Σi wi = 1 | LBM velocity set weights |
| Ωi | kg∕m³ | SRT or TRT | LBM collision operator |
| τ | s | τ = ν∕c² + Δt∕2 | LBM relaxation time |
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