Making fluid from velocity

Incompressible Navier–Stokes, decomposed

Fluid is not a texture. Each frame moves the field, disturbs it, measures compression, and removes that compression so ∇·u = 0. Stam’s stable-fluids decomposition is the method;^[113] the claim is perceptual rather than numerical — that an incompressible field is what makes the motion read as fluid rather than as a smear. Dijkstra’s point about abstraction holds here too: the field is exact at a level other than the numeric readout.^[036]^[Th.]

Momentum

Incompressible Navier–Stokes

\dfrac{\partial \htmlClass{fs fs--u}{\mathbf{u}}}{\partial t}+\bigl(\htmlClass{fs fs--u}{\mathbf{u}}\cdot\htmlClass{fs fs--grad}{\nabla}\bigr)\htmlClass{fs fs--u}{\mathbf{u}}=-\htmlClass{fs fs--grad}{\nabla}\htmlClass{fs fs--p}{p}+\htmlClass{fs fs--nu}{\nu}\htmlClass{fs fs--grad}{\nabla}^{2}\htmlClass{fs fs--u}{\mathbf{u}}+\htmlClass{fs fs--f}{\mathbf{f}}

Hover a symbol

Fig. 03.1

Momentum, divergence-free flow, and the pressure Poisson solve — the three relations the fluid chapter depends on.

Trace backward to move forward

Semi-Lagrangian advection is unconditionally stable. The price is blur; vorticity confinement buys the swirl back.^[043]^[116] Stam’s later GPU notes treat the same trade as a real-time constraint rather than a numerical luxury.^[114]

Velocity field

How a 2D flow looks on a grid.

A synthetic curl-noise field for teaching — not the Metal advection or vorticity kernels. Click a mode to focus on flow, swirl, or speed.

01 · FlowEach arrow is a velocity sample. Direction is where the fluid moves; longer means faster.

Fig. 03.2

Teaching velocity field (curl noise). Arrows show flow; tint shows swirl. Not the Metal vorticity kernel.

Pressure makes the field incompressible

Sixteen red/black pairs on Apple Silicon; twenty Jacobi sweeps on the fallback. Same constraint, two ownership models. Bridson’s fluids notes are the engineering reading of that constraint.^[022]

Pressure solve

Checkerboard updates, faster convergence.

Brightness is pressure on a 14×9 grid. Red and black cells take turns so neighbours stay fixed during each sweep — about 2× faster than Jacobi for the same sweep count.

01 · RedUpdate every red cell from its four black neighbours. Half the grid advances in one parallel pass.

Fig. 03.3

Checkerboard pressure solve on a 14×9 grid. Red then black sweeps; sixteen pairs on Apple Silicon.

A second field for wave memory

A damped wave rides the carrier and couples back into velocity. Roll and azimuth become rings, not badges.

Wave profile

A 1D slice of the ripple field.

Same damped wave update as WaveCompute.metal. Open ends, two pulse sites, defaults c=0.4 and d=0.02.

01 · HeightSolid curve is wave height h(x,t) right now — the surface the ripples ride on.

Fig. 03.4

1D damped wave matching WaveCompute.metal. Ghost = previous frame; pulses ≈ oar entry (c=0.4, d=0.02).