Smoke simulation 1: a flow field from noise

Real smoke and water move according to fluid equations. Solving them properly needs memory (the state of the fluid in the last frame), which a single-pass pixel shader does not have. But there is a famous shortcut that looks very convincing: invent a velocity field that behaves like a fluid, and let things be carried along by it.

What makes a velocity field fluid-like? Water cannot be squashed. In a field that flows without any spot where fluid appears or vanishes (no sources, no sinks) everything just swirls. There is a recipe for building such a field from any smooth noise: take its curl. In 2D, if n(x, y) is the noise, the velocity is (dn/dy, -dn/dx): the noise "rotated by 90 degrees". We measure the slopes with finite differences, exactly like the normals earlier:

float a = noise2((q + float2(0.0, e)) * 1.8 + ...) - noise2((q - float2(0.0, e)) * 1.8 + ...);   // dn/dy
float b = noise2((q + float2(e, 0.0)) * 1.8 + ...) - noise2((q - float2(e, 0.0)) * 1.8 + ...);   // dn/dx
float2 curl = float2(a, -b) / (2.0 * e);

Scrolling the noise slowly with time (+ float2(0.0, t)) makes the swirls evolve.

This lesson only looks at the field: the red channel is the sideways flow, the green channel the vertical flow. You will see big soft vortices, and next lesson uses them to move smoke.