Modeling the Motion of a Hot, Turbulent Gas
Nick Foster and Dimitris Metaxas
SIGGRAPH '97

review by Kenneth E. Hoff III

Typically gases are modeled and rendered as simple particle systems
that require significant effort by the animator in order to appear
realistic. This paper presents a physically-based volumetric
technique for modeling hot, turbulent gases that can interact
with arbitary environments. This paper emphasizes the use of a
low-resolution discretization of a very accurate system of gas
equations as an efficient strategy for modeling gases in a
physically realistic way with minimal user-intervention.

Gases exhibit complex inter-rotational motion as a result of 
complex interactions between areas of different temperature,
pressure, and velocity, and with objects in the scene. There model
accounts for these visual effects by using a reduced form of the
Navier-Stokes equations that considers only convection and drag,
and thermal buoyancy.

The convection and drag equation models the velocity
effects of a gas taken as a system of particles that obey Newtonian
laws of motion. External drag forces are computed to create
vortex effects (billowing). The compression of the gas is ignored
by assuming that the gases are not under extreme pressure. The
resulting equation models the change in velocity, the pressure
gradient, and the external drag forces.

The thermal buoyancy equation models the induced motion resulting
from combining of areas of different temperatures. Hot regions
surrounded by cooler areas rise against gravitational forces. In
order to calculate these "buoyant" forces, the changes in
temperature of regions of space must be accounted for.

The resulting equations that model velocity, pressure, and
temperature are solved by discretizing the space into voxels.
These parameters are solved for discrete regions rather than
for continuous regions of space. This paper shows that even
a fairly coarse grid can result in a very realistic appearing
simulation (depending, of course, on the volume rendering
strategy employed).

The equations are solved in the voxel space using finite
differences. Each of the equations are solved for a particular
voxel and are rewritten in terms of the neighboring voxels.
A method is presented to maintain the accuracy despite the
low resolution of the solution. Stability of the system is
maintained by restricting the stepsize such that the velocity
is sampled enough to be smaller than a voxel grid.

The resulting animations look stunningly realistic despite the
often obvious discretized look of the surrounding objects that
have to be approximated by the voxel grid. However, it is 
important to note that the interacting objects only need to be
voxelized for the gas simulation not for rendering. The small
errors will probably go unnoticed since the model will be
rendered cleanly and the gas does not show the discretization
so obviously. 

The computational complexity of the algorithm is O(n^3) where
the voxelization is a nxnxn cube (not necessarily). However,
n is kept very small (around 40-60) since they emphasize the
use of coarse voxel grids. This computation time may seem much
greater than a typical particle system, but the results are
more realistic with virtually no dependence on the skills of the
animator to get the proper gas effects.