Why smoke is a record of the air
Air is invisible. Smoke is a dye that the air picks up at the source and carries away. Each puff stays with the air it was put into, so the trail is a picture of everywhere that air went.
That picture is a streakline: all the air that passed one point, at every moment so far. It is not the same as where the air is heading right now (a streamline), or the path of one puff (a pathline). In a steady draught the three coincide. When the air changes, as with a waving hand or a gust, they come apart, and the smoke shows curls that the air at that moment no longer has.
Here the air is simulated and the smoke rides on it. Hot gas from the candle and the incense rises, pulls room air in behind it, and leaves through the ceiling.
Numbers for this simulation
The three lines
Equations
The air (two-dimensional, incompressible, with buoyancy in the Boussinesq form):
Heat and smoke are carried by the air. Smoke is a passive tracer:
β = 1/T₀ for an ideal gas, so each kelvin of excess temperature lifts the air at about 3.3 cm/s². f stands for the sources: the candle's outflow, the hand, the breeze and the room draft.
Plume theory used as the check (Morton, Taylor and Turner), with Q the convective heat output and z the height above the source:
Constants
| Gravity g | 9.81 m/s² |
| Room temperature T₀ | 293.15 K (20 °C) |
| Viscosity of air ν (20 °C) | 1.5 × 10⁻⁵ m²/s |
| Heat diffusivity of air α (20 °C) | 2.2 × 10⁻⁵ m²/s |
| Smoke settling speed (0.3 µm, Stokes) | ≈ 3 × 10⁻⁶ m/s, ignored |
| Smoke Brownian diffusion | ≈ 10⁻¹⁰ m²/s, ignored |
| Visible window | 60 cm tall |
| Candle ★ | ΔT 250 K, 0.5 m/s, 5 mm patch just above the flame |
| Blown-out wick ★ | ΔT 150 K fading in 1 s, smoke fading in 2.5 s |
| Incense ★ | ΔT 40 K at a 1.6 mm tip |
| Hand ★ | 3.5 cm radius; waving ±15 cm every 1.6 s |
| Room draft ★ | 2 cm/s nudges, about 3 per second |
| Settling of room-temperature air ★ | time constant 2 s |
| Breeze · gusts ★ | 12 cm/s · ±80 % every 4 s |
How it is computed
What is estimated
- The sources are imposed patches of heat, smoke and (for the candle) upward speed, chosen so the plumes land in published ranges. The grid cells are about 2 mm, too coarse to model a flame or a smouldering tip. Values marked ★ are estimates.
- A steady candle flame burns almost without soot, so its plume is nearly invisible. That part is real. Choose Heat to see it.
- The simulation is a two-dimensional slice. Each source acts like a long line rather than a point, and a hand acts like a long bar that pushes more air than a real hand. The slice also lets a plume sway from side to side more than in life.
- In a flat slice, swirls merge into ever larger, room-sized whirls, which real three-dimensional air breaks up into small eddies that die out. To stand in for that, air at room temperature settles back to rest with a 2 s time constant. The warm plumes are left alone. Without this, the room fills with a wind of its own making within a minute.
- Numerical smearing of motion is roughly 10–30 times larger than air's own viscosity, so plumes stay smooth longer and turbulence is coarser than in life. There is no turbulence model.
- Near the flame ΔT/T₀ is about 0.85, outside the small-ΔT range that the Boussinesq form assumes.
- Above the visible ceiling, smoke and heat are faded out on purpose so they leave the computation. The air itself flows on unchanged.
- The room draft is invented: gentle random nudges of about 2 cm/s, typical of still indoor air. It is what eventually breaks a laminar incense thread.
- Smoke amounts are in arbitrary units. Only where the smoke goes is computed; how dense it looks is a display choice.
- The flame drawing leans with the local air speed for illustration only.
Sources
- J. Stam, "Stable Fluids", SIGGRAPH 1999.
- A. Selle, R. Fedkiw, B. Kim, Y. Liu, J. Rossignac, "An Unconditionally Stable MacCormack Method", J. Sci. Comput. 35 (2008).
- W. Briggs, V. E. Henson, S. McCormick, A Multigrid Tutorial, 2nd ed., SIAM (2000).
- B. R. Morton, G. I. Taylor, J. S. Turner, "Turbulent gravitational convection from maintained and instantaneous sources", Proc. R. Soc. Lond. A 234 (1956).
- H. Rouse, C. S. Yih, H. W. Humphreys, "Gravitational convection from a boundary source", Tellus 4 (1952).