Geoscience Reference
In-Depth Information
Non-hydrostatic pressure model
The governing equations in the non-hydrostatic pressure model are Eqs. (6.103)-
(6.105). Some approaches used for solving the 2-D Navier-Stokes equations in
Section 4.4 can be applied here. For example, Karpik and Raithby (1990) proposed the
SIMPLED algorithm, which is a modification of the SIMPLE algorithm on the stag-
gered grid, to solve this set of equations. In analogy to the depth-averaged 2-D model,
the width-averaged 2-D model can also be solved using the SIMPLE(C) algorithms on
the non-staggered grid. The details are not presented here, because the formulations
in both 2-D models are similar.
The stream function and vorticity method can be extended to the width-averaged
2-D model by defining the stream function
ψ
corresponding to the continuity
equation (6.103) as
1
b
∂ψ
1
b
∂ψ
U
x
=
z
,
U
z
=−
(6.119)
∂
∂
x
and the vorticity as
=
∂
U
z
∂
−
∂
U
x
∂
(6.120)
x
z
Therefore, the following equation for stream function is obtained by inserting
Eq. (6.119) into Eq. (6.120):
2
2
∂
ψ
+
∂
ψ
1
b
∂
b
x
∂ψ
1
b
∂
b
z
∂ψ
−
x
−
=−
b
(6.121)
x
2
z
2
∂
∂
∂
∂
∂
∂
z
Cross-differentiating Eqs. (6.104) and (6.105) with respect to
z
and
x
and
subtracting them yields the vorticity equation:
∂
∂
+
∂(
U
x
)
+
∂(
U
z
)
=
∂
∂
)
∂
∂
+
∂
∂
)
∂
∂
(ν
+
ν
(ν
+
ν
+
S
t
t
t
∂
x
∂
z
x
x
z
z
(6.122)
where
S
includes all the remaining terms.
Eqs. (6.121) and (6.122) constitute the governing equations of the width-averaged
2-D stream function and vorticity model. Unlike the depth-averaged 2-D model, the
width-averaged 2-D stream function and vorticity model is applicable to both steady
and unsteady flows.
A special task in the width-averaged 2-D model is handling the free surface. The
techniques introduced in Section 7.1 for the 3-D model can be used here.
