Part 5 of 6. Nine velocities on a square lattice, weighted so the fluid they make looks the same in every direction and in every moving frame, to the order that matters.
Part 4 ended with HPP failing because four axis velocities cannot build an isotropic rank-4 tensor. Lattice Boltzmann keeps the square lattice with real-valued populations fi on nine velocities (McNamara and Zanetti 1988; Qian, d'Humières and Lallemand 1992), weighted so the fluid respects two continuum symmetries: rotations and Galilean boosts.
The nine velocities form three orbits of the point group D4, so there are three weights, one fixed by normalisation. The rank-4 moment Σwi(ci·n)4 must not depend on the direction n, and must equal the Maxwellian's 3cs4, where cs2 = Σwicix2.
Figure 1. On the purple line the rank-4 moment is isotropic; on the orange curve its average over directions equals 3cs4. The polar plot is the moment over 3cs4; the Maxwellian is the dashed circle. Drag the point or use arrow keys.
The axes-only set matches along the axes and has half the moment on the diagonals. The eight moving vectors are isotropic at r = 1/4 but too small; the rest weight supplies the size, and the two curves cross only at 4/9, 1/9, 1/36.
Each step relaxes every population toward a local equilibrium with one time τ (the BGK collision), then shifts it one cell along its velocity. A Chapman–Enskog expansion turns this into incompressible Navier–Stokes with the viscosity below.
Figure 2 sums the equilibrium's moments. Through rank 2 they match the Maxwellian. At rank 3 they cannot, because cx3 = cx when every component is −1, 0 or 1.
| moment | lattice | Maxwellian | diff |
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Figure 2. fieq(ρ = 1, u) for the rest, axis and diagonal channels; ticks mark the weights at u = 0. The rank-3 rows miss by ρux3 and ρuxuy2.
Figure 3 lays a small shear wave on a periodic 32 × 32 lattice at seven orientations, fits the exponential decay of each amplitude and divides the rate by k2. The curves are the Chapman–Enskog prediction for general weights, ν(θ) = (τ − ½)(wa sin22θ + 4wd cos22θ)/cs2.
Figure 3. Points measured, curves predicted; the second chart is the selected direction's decay with its fit. With axes only, a wave along an axis never decays, as in HPP.
The eight-moving set passes that test. Figure 4 adds a uniform flow U along x to the shear wave, on a 32-cell periodic row,: a Galilean-invariant fluid carries the pattern at exactly U.
Figure 4. Heatmaps: the wave over 300 steps, columns rescaled; dashed, a crest moving at U. Lines: drift speed over U and shear viscosity, τ = 0.8.
The eight-moving set carries the wave at 5/9 of U, the radius of its circle in Figure 1. The axes-only set has no channel that moves y-momentum along x, so the wave stays put. D2Q9 drifts at exactly U, but its viscosity in the moving frame falls as U grows: the cubic error of Figure 2, and the reason runs are kept at low Mach number.
Walls are bounce-back: a population entering a solid site returns the way it came, so the wall velocity is zero. Figure 5 runs a 300 × 90 channel with a cylinder of diameter 22. Everything is mirror-symmetric about the centreline except a seeded random perturbation of 1% of U in the initial velocity.
Figure 5. τ comes from Re = UD/ν, floored at 0.52 for stability. The inlet imposes (U, 0), the outlet fixes density. St = fD/U counts zero crossings of uy four radii downstream.
The cylinder fills a quarter of the channel width. The measured Strouhal number sits above the unconfined value of about 0.2, and shedding starts above the free-stream Re ≈ 47: at 60 the wake stays symmetric, at 70 it sheds.
Figure 6 tracks the wake's mirror defect, the RMS difference between the flow and its reflection through the centreline, divided by U. It grows exponentially, then saturates, a Hopf bifurcation seen from inside: a complex pair of eigenvalues of the symmetric wake has crossed into the unstable half-plane.
Figure 6. From the run above. Dashed: the growth fit before the defect passes 0.05. The second chart compares velocities on a vertical probe line with themselves a time shift earlier, directly (grey) or reflected (pink); unrelated fields score about 1.
No snapshot of the street is symmetric, but reflection plus a shift of half a period maps it onto itself, so the Z2 symmetry survives as a space-time symmetry. A new seed changes only the phase of the street.
Practitioners keep the Mach number U/cs near 0.1; the slider here reaches 0.21. Compressible and thermal flows need larger velocity sets such as D2Q17 or D2Q37. Part 6 turns to the Ising model, where a Z2 symmetry breaks with no flow at all.