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value of T max is used. Let's make that clear: the actual temperature T in
the fuel side is kept constant at T ignition , but when advecting and diffusing
T on the burnt side, if reference is made to a T value in the fuel (e.g., when
doing interpolation in semi-Lagrangian advection) T max is used instead.
For the extreme temperatures apparent in fires, hot enough to make an
incandescent glow, a black-body radiation formula for the decay of T might
be used instead of the simple exponential decay mentioned earlier, where
the rate of cooling is proportional to the fourth power of the temperature
difference
c T
4
DT
Dt
T ambient
T max
=
T ambient
for some cooling constant c defined as a user parameter. After advecting
temperature around to get an intermediate T for a time step Δ t ,this
cooling equation can be solved analytically to give
T ambient ) 4 3 .
Similar to this treatment of temperature, we can also feed a smoke concen-
tration s max into the burnt region from the flame front, allowing it to be
advected and dissipated as usual. Temperature and smoke concentration
can feed into either a buoyancy force (if we make the Boussinesq approxi-
mation) or modify density, as discussed in Chapter 5.
The final issue is rendering, which mostly lies outside the scope of this
topic. The actual flame front itself is sometimes referred to as the “blue
core,” referring to the spectral emissions made when burning typical hy-
drocarbons (other fuels give rise to different spectra, giving different col-
ors): the level set itself is a light emitter. For a dirty flame, where soot
is produced, the bulk of the visual effect though is the black-body incan-
descence of the soot particles. That is, light is emitted from the burnt
region as well, proportional to the smoke concentration s and following the
black-body spectrum for the temperature T . Simultaneously, the soot is
somewhat opaque, so light emitted elsewhere should be absorbed at a rate
proportional to s . Further scattering effects can be included of course.
T n +1 = T ambient +
1
3 c Δ t
( T max
T ambient ) 3 +
( T
7.2 Volumetric Combustion
We now turn to an alternate model where combustion may take place
throughout the volume, loosely following Feldman et al. [Feldman et al. 03].
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