Graphics Reference
In-Depth Information
A basic theorem of vector calculus tells us that if a smooth vector field
has zero curl in a simply-connected region, it must be the gradient of some
scalar potential:
u
=
∇
φ.
Note that the
φ
used here has nothing to do with the signed distance func-
tion or any other implicit surface function we looked at earlier. Combining
this with the incompressibility condition,
∇·
u
= 0, indicates that the
potential
φ
must satisfy Laplace's equation:
∇·∇
φ
=0
.
This is the basis of
potential flow
: instead of solving the full non-linear
Navier-Stokes equations, once we know the fluid is irrotational and the
region is simply-connected, we only need solve a single linear PDE.
The boundary conditions for potential flow are where it gets interesting.
For solid walls the usual
u
·
n
=
u
solid
·
n
condition becomes a constraint on
∇
n
. Free surfaces, where before we just said
p
= 0, are a bit trickier:
pressure doesn't enter into the potential flow equation directly. However,
there is a striking resemblance between the PDE for the potential and the
PDE for pressure in the projection step: both involve the Laplacian
φ
·
∇·∇
.
We'll use this as a clue in a moment.
The equation that pressure does appear in is momentum: let's substi-
tute
u
=
∇
φ
into the inviscid momentum equation and see what happens:
∂
φ
∂t
+(
∇
φ
)+
1
∇
φ
)
·
(
∇∇
ρ
∇
p
=
g.
Exchanging the order of the space and time derivatives in the first term,
and assuming
ρ
is constant so it can be moved inside the gradient in the
pressure term, takes us to
∂φ
∂t
+(
p
ρ
=
g.
∇
∇
φ
)
·
∇∇
φ
)+
∇
(
Seeing a pattern start to form, we can also write the gravitational accelera-
tion as the gradient of the gravity potential,
g
gy
where
g
=9
.
81 m
/
s
2
and
y
is height (for concreteness, let's take
y
= 0 at the average sea level).
·
x
=
−
∂φ
∂t
+(
p
ρ
+
∇
∇
φ
)
·
(
∇∇
φ
)+
∇
∇
(
gy
)=0
.
