Graphics Reference
In-Depth Information
along one edge, verify how fast-moving air can induce a pressure drop which
sucks things towards it.
1
Bernoulli's equation gives us a relationship, admittedly non-linear, be-
tween pressure and the potential
φ
. In the interior of the fluid this can be
used to get pressure from our solution for
φ
. At a free surface where
p
=0
is known, we can instead use it as a boundary condition for
φ
:
∂t
+
2
∇
2
+
gy
=0
.
∂φ
φ
Well, almost—this is more of a boundary condition on
∂φ/∂t
,not
φ
itself.
But it's not hard to see that as soon as we discretize in time, this will end
up as a boundary condition on the new value of
φ
that happens to also
depend on old values of
φ
.
13.2 Simplifying Potential Flow for the Ocean
Unfortunately as it stands we still have to solve a three-dimensional PDE
for the potential
φ
, and though it's a much simpler linear problem in the
interior of the water, it now has a fairly nasty non-linear boundary condition
at the free surface. In this section we'll go through a series of simplifications
to make it solvable in an ecient way. The critical assumption underlying
all the simplifications is that we're only going to look at fairly calm oceans.
The first step is to rule out breaking waves so the geometry of the free
surface can be described by a height field, just like the previous chapter:
y
=
h
(
x, z
)
.
Of course
h
is also a function of time,
h
(
x, z, t
), but we'll omit the
t
to em-
phasize the dependence on just two of the spatial variables. For a perfectly
calm flat ocean we'll take
h
(
x, z
) = 0; our chief problem will be to solve
for
h
as a function of time. In fact, given the velocity field
u
we know that
the free surface should follow it—and in fact viewing the surface as implicit
defined as the zero level set of
h
(
x, z
)
−
y
, we already know the advection
1
It also unfortunately figures in a bogus explanation of the lift on an airplane wing,
namely that due to the curved shape of the airfoil the air has to go faster over the top
than over the bottom to meet at the other side; hence there is lower pressure on the
top surface, which gives lift. It's not hard to see this is almost completely wrong: angle
of attack is the biggest factor in determining lift, allowing airplanes to fly even when
upside-down, and allowing flat fan blades with no fancy curvature to effectively push air
around.
