Graphics Reference
In-Depth Information
The right-hand side of this equation doesn't depend on
y
,so
∂v/∂y
must
be a constant along the
y
-direction too—which implies
v
has to be a linear
function of
y
. It's fully determined from its value at the bottom
y
=
b
(
x, z
)
and the gradient we just derived.
The bottom velocity comes from the boundary condition
u
n
=0,
remembering again that we're assuming the bottom is stationary. Re-
calling some basic calculus, the normal at the bottom is proportional to
(
·
−
∂b/∂x,
1
,
−
∂b/∂z
), so at the bottom
y
=
b
(
x, z
):
u
∂b
∂x
+
v − w
∂b
−
∂z
=0
v
=
u
∂b
∂x
+
w
∂b
∂z
.
Note that if the bottom is flat, so the partial derivatives of
b
are zero, this
reduces to
v
= 0 as expected. Combined with Equation (12.3) we get the
following vertical velocity at any point in the fluid:
⇔
∂u
(
y
v
(
x, y, z, t
)=
u
∂b
∂x
+
w
∂b
∂x
+
∂w
∂z
−
−
b
)
.
(12.5)
∂z
In other words, for shallow water we take
v
to be whatever it requires for
the flow to be incompressible and to satisfy the bottom solid boundary
condition.
12.1.3 Height
We can also evaluate the vertical velocity at the free surface, in a different
way. Note that the function
φ
(
x, y, z
)=
y
h
(
x, z
) implicit defines the free
surface as its zero isocontour—similar to how we tracked general liquid
surfaces back in Chapter 6. We know that the free surface, i.e., the zero
isocontour, moves with the velocity of the fluid, and so
φ
should satisfy an
advection equation
−
Dφ
Dt
=0
∂φ
∂t
+
u
∂φ
∂x
+
v
∂φ
∂y
+
w
∂φ
⇔
=0
∂z
∂t
+
u
+
v
(1) +
w
=0
∂h
∂h
∂x
∂h
∂z
⇔−
−
−
at least at the surface
y
=
h
itself. Plugging in what we derived for the
velocity in Equation (12.5) at
y
=
h
gives us an equation for the rate of
