Graphics Reference
In-Depth Information
With
S
= 0, note that the volume of the fuel region is
not
conserved, and
thus a lot of the worries for tracking level sets for water don't bother us
here—just the usual advection approaches can be used as if
φ
were any other
scalar, along with an addition of
S
Δ
t
each time step, and perhaps periodic
reinitialization to signed distance. Note also that if an object is supposed
to be on fire, a boundary condition forcing
φ
0 on the burning sections
of its surface should be included; otherwise
φ
should be extrapolated into
solids to avoid artifacts near the surface, as with liquids.
As an aside, Hong et al. [Hong et al. 07] have more recently added ad-
ditional detail to this technique by using higher-order non-linear equations
for the speed and acceleration of the flame front (rather than keeping it
as a constant burn speed
S
). These promote the formation of “cellular
patterns” characteristic of some fires.
The next problem to address is how the fire affects the velocity field. If
the density of the burnt products is less than the density of the fuel mix
(as is often the case, either due to differences in a specific gas constant
or temperature), then conservation of mass demands that the fluid should
instantaneously expand when going through the flame front from the fuel
region to the burnt-products region. The rate of mass leaving the fuel
region per unit area is
ρ
fuel
S
, which must match the rate of mass entering
the burnt region per unit area,
ρ
burnt
(
S
+Δ
V
), where Δ
V
is the jump in
the normal component of fluid velocity across the interface. Solving gives
Δ
V
=
ρ
fuel
≤
1
S.
ρ
burnt
−
Since the tangential component of velocity is continuous across the flame,
this means the full velocity satisfies the following jump at the flame front:
u
burnt
=
u
fuel
+Δ
V n
=
u
fuel
+
ρ
fuel
1
Sn,
ρ
burnt
−
(7.2)
where
n
is the normal pointing outward from the fuel region into the burnt
region. This naturally defines a
ghost velocity
for use in advection: if you
trace through the flame front and want to interpolate velocity on the other
side, you need to add or subtract this change in normal velocity for the
advection to make sense (where normal, in this case, might be evaluated
from the normalized gradient of the level set
φ
wherever you happen to be
interpolating). This is one example of the
ghost fluid method
developed by
Fedkiw and coauthors.
