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This expansion also has to be modeled in the pressure projection step,
enforcing a non-zero divergence at the flame front—indeed, this is one of the
critical visual qualities of fire. The simplest discrete approach is to build
Equation (7.2) into the evaluation of divergence that defines the right-hand
side of the pressure equation. For example, when evaluating the divergence
at a cell where φ
0 (i.e., in the fuel region) then any of the surrounding
u -, v -, or w -velocity samples, where the averaged φ> 0 (i.e., in the burnt
region) should be corrected by the x -, y -or z -component of
Δ V n , respec-
tively. Similarly if φ> 0 at the cell center, then, at surrounding velocity
samples where the averaged φ
0, φ should be corrected by the appro-
priate components of +Δ V n . This can be interpreted as yet another use
for divergence controls. When setting up the matrix in the pressure solve,
greater fidelity can be obtained by accounting for the different densities
of fuel and burnt products: this shows up as variable densities just as in
water or the advanced smoke model discussed earlier in the topic. We can
estimate the density in, say, a u -cell ( i +1 / 2 ,j,k ) as a weighted average of
ρ fuel and ρ burnt :
ρ i +1 / 2 ,j,k = αρ fuel +(1
α ) ρ burnt ,
where the weight α could be determined from the level set values φ i,j,k and
φ i +1 ,j,k :
1:
φ i,j,k
0and φ i +1 ,j,k
0 ,
φ i,j,k
φ i,j,k
:
φ i,j,k
0and φi +1 ,j,k > 0 ,
φ i +1 ,j,k
α =
φ i,j,k
φ i,j,k
1
:
φ i,j,k > 0and φi +1 ,j,k
0 ,
φ i +1 ,j,k
0: φ i,j,k > 0and φ i +1 ,j,k > 0 .
This of course blends very elegantly with the variable density smoke solve,
where density is a function of temperature T and the specific gas constant
R ; this constant R can be taken as different for the fuel region and the
burnt-products region.
Speaking of temperature, we don't yet have a model for it. The simplest
approach is to keep a constant T = T ignition , the temperature at which
combustion starts, inside the fuel region and establish a discontinuous jump
to T max on the burnt products side of the flame front. The temperature
on the burnt side is advected and dissipated as before for smoke, but using
the trick that—just as was done for velocity above—when crossing over
the flame front and referring to a temperature on the fuel side, the “ghost”
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