Graphics Reference
In-Depth Information
air, but we may not want the hassle of simulating the air as well. And since
air is 700 times lighter than water, it's not able to have that big of an effect
on the water anyhow. So instead we make the modeling simplification that
the air can be represented as a region with constant atmospheric pressure.
In actual fact, since only
differences
in pressure matter (in incompressible
flow), we can set the air pressure to be any arbitrary constant: zero is the
most convenient. Thus a free surface is one where
p
= 0, and we don't
control the velocity in any particular way.
The other case in which free surfaces arise is where we are trying to
simulate a bit of fluid that is part of a much larger domain: for example,
simulating smoke in the open air. We obviously can't afford to simulate
the entire atmosphere of the Earth, so we will just make a grid that covers
the region we expect to be “interesting.” (I'll preemptively point out here
that to simulate smoke, you
need
to simulate the smoke-free air nearby as
well, not just the smoky region itself—however, we can get away with not
simulating the air distant enough from all the action.) Past the boundaries
of the simulated region the fluid continues on, but we're not tracking it;
we allow fluid to enter and exit the region as it wants, so it's natural to
consider this a free surface,
p
= 0, even though there's not actually a visible
surface.
3
One final note on free surfaces: for smaller-scale liquids, surface tension
can be very important. At the underlying molecular level, surface tension
exists because of varying strengths of attraction between molecules of dif-
ferent types. For example, water molecules are more strongly attracted to
other water molecules than they are to air molecules: therefore, the water
molecules at the surface separating water and air try to move to be as
surrounded by water as much as possible. From a geometric perspective,
physical chemists have modeled this as a force that tries to minimize the
surface area or, equivalently, tries to reduce the mean curvature of the sur-
face. You can interpret the first idea (minimizing surface area) as a tension
that constantly tries to shrink the surface, hence the name surface ten-
sion; it can be a little more convenient to work with the second approach
using mean curvature. (Later, in Chapter 6 we'll talk about how to ac-
tually measure mean curvature and exactly what it means.) In short, the
model is that there is actually a jump in pressure between the two fluids,
3
Technically this assumes there is no gravitational acceleration
g
included in the
equations. If there is, we would take the hydrostatic pressure
p
=
ρg · x
as the open
boundary condition. To avoid having to do this, we can write the momentum equation
in terms of the pressure perturbation from hydrostatic rest:
p
=
ρg · x
+
p
. Substituting
this into the pressure gradient cancels out
g
on the other side, and we can use the simpler
open boundary condition
p
=0.
