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In-Depth Information
change of height:
∂x
+
u
∂b
∂u
(
h
b
)
∂h
∂t
−
u
∂h
∂x
+
w
∂b
∂x
+
∂w
w
∂h
∂z
−
∂z
−
−
−
=0
,
∂z
b
)
∂u
.
(12.6)
∂h
∂t
+
u
∂
(
h
−
b
)
+
w
∂
(
h
−
b
)
∂x
+
∂w
=
−
(
h
−
∂x
∂z
∂z
Using the depth
d
=
h
−
b
, and remembering that
b
is stationary, this can
be simplified to
d
∂u
.
∂d
∂t
+
u
∂d
∂x
+
w
∂d
∂x
+
∂w
=
−
(12.7)
∂z
∂z
That is, the water depth is advected by the horizontal velocity and, in
addition, increased or decreased proportional to the depth and the two-
dimensional divergence.
We can simplify Equation (12.7) even further, putting it into what's
called
conservation law form
:
∂d
∂t
+
∂x
(
ud
)+
∂
∂
∂z
(
wd
)=0
.
(12.8)
This can in fact be directly derived from conservation of mass, similar to the
approach in Appendix B. It's significant here because it leads to numerical
methods that exactly conserve the total volume of water in the system—
avoiding the mass-loss problems we saw earlier with three-dimensional free
surface flow.
2
However, discretizing this accurately enough (to avoid nu-
merical dissipation) is a topic that lies outside the scope of this topic.
12.1.4 Boundary Conditions
Equations (12.2) and (12.7) or (12.8) also need boundary conditions, where
the water ends (in the
x
-
z
horizontal plane) or the simulation domain ends.
The case of a solid wall is simplest: if
n
is the two-dimensional normal to
the wall in the
x
-
z
plane, then we require
(
u, w
)
·
n
=0
.
2
This conservation law form can also be applied in three dimensions, leading to a
volume-of-fluid
or
VOF
simulation that exactly conserves volume as well. However,
in three dimensions, VOF techniques have their own share of problems in terms of
accurately localizing the surface of the fluid.
