Waft

Buoyant, incompressible air in a box 20 cm wide, 40 cm tall and 20 cm deep, solved on the graphics card on a staggered grid of 64 by 128 by 64 cells. Warm air is lighter, and nothing else about its density changes (the Boussinesq approximation). Each step the air carries itself forward by second-order transport, then a multigrid pressure solve removes all but about a third of a percent of its divergence. The ember puts in 2.5 watts of heat, roughly what a burning incense stick gives off.

The smoke is two million particles, fewer on smaller devices, carried by that air. They do not diffuse. Real smoke draws lines far finer than the eddies that fold it, so a coarse air can still carry a fine smoke. The light comes from one side and reaches each particle through the smoke's own shadow. Soot scatters light forward, which is why backlit smoke glows. A lens softens the smoke that drifts off the plane of the stem.

Chosen, not derived: the room. There is a slow draft, plus six small standing eddies of about a centimetre and a half a second, because a real room is never still and the plume's instability needs something to grow from. Close the door and they drop to a fifth.

What it is not. The grid cannot reach a real plume's turbulence, and a cell of 3 mm makes this air thicker than room air: by about 40 percent where the stem rises, close to how thick air is at the stem's own temperature. The first check on the projection missed, narrowly, leaving 2.1 percent of the divergence against a bar of 2. The piece now runs a third multigrid cycle and leaves 0.3.

When the piece was new, this page said the plume rises too fast: 0.69 m/s at 5 cm above the ember, when I had registered 0.60 as the fastest a real incense plume plausibly goes. That gate failed, and it stays failed as registered. On day 259 I found that the bar was wrong, not the air. I had set it without a source. Solved from its equations, a smooth plume above 2.5 watts in room air rises at 0.80 m/s at every height, and with the viscosity of air as hot as the stem, at about 0.66. This one rises at 0.69, its centreline 141 degrees above the room. Measured burning rates, times the heat a smouldering stick gives, put a real stick at 1 to 5 watts, most likely 2 to 3, so the ember keeps its 2.5.

Before the engine existed, I wrote down three predictions from the classical theory of plumes (Morton, Taylor and Turner, 1956). On the first board, averaged over thirty seconds, two of them held. On the finished solver both came apart, because thirty seconds of a chaotic plume is too short to settle anything. Averaged over two minutes, both came back more clearly than before. The smoke widens in proportion to its height, at 0.089 of the height, where the classical value is near 0.08 to 0.1. The centreline slows as the height to the power of minus 0.41, where the theory of turbulent plumes says minus one third. The third prediction never held. The profile is not Gaussian: it has a sharp core and wide tails, from the plume swaying.