Graphics Reference
In-Depth Information
5.2 Buoyancy
We now turn to the effect that
T
and
s
have on velocity. In this section we
introduce a simplified buoyancy model commonly used in graphics.
We all know that hot air rises and cool air sinks; similarly it seems
plausible that air laden with heavier soot particles will be pulled downwards
by gravity. We can model this by replacing the acceleration
g
due to gravity
in the momentum equation with a buoyant acceleration
b
=[
αs
−
β
(
T
−
T
amb
)]
g,
where
α
and
β
are non-negative coecients, and
T
amb
is the ambient tem-
perature (say 273 K). Note that we take this proportional to the downward
gravity vector—indeed, buoyancy doesn't exist in a zero-G environment.
Also note that the formula reduces to zero wherever
s
=0and
T
=
T
amb
,
as might be expected.
Since
T
and
s
are generally stored at grid cell centers, we need to do
some averaging to add the acceleration to the MAC grid velocities, e.g.,
T
i,j
+1
/
2
,k
=
2
(
T
i,j,k
+
T
i,j
+1
,k
). Alternatively put, when we add buoyancy
to the velocity field prior to projection, the contribution of acceleration
evaluated at the grid cell center (
i, j, k
) is equally split between
v
i,j−
1
/
2
,k
and
v
i,j
+1
/
2
,k
.
5.3 Variable Density Solves
Underlying the buoyancy model is the fact that fluid density is a function
of temperature and—if we treat the soot as actually dissolved in the air—
smoke concentration. Let's begin with just the effect of temperature, for
now taking
s
= 0. From the ideal gas law, thermodynamics derives that
the density of the air should be
P
RT
,
ρ
air
=
(5.2)
10
5
Pa
in SI units),
R
is the
specific gas constant for air (approximately 287 J/kg K in SI units), and
T
is the temperature. It should be underscored here that the absolute pres-
sure
P
we propose using here is approximated as a constant—not coupled
with the pressure solve for incompressibility—as otherwise we end up in
a significantly more complicated compressible flow model; this has been
worked out by Bonner [Bonner 07] in a generalization of the MAC grid
incompressible simulation developed in this topic, if you are interested.
where
P
is the absolute pressure (say 1
.
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