Geoscience Reference
In-Depth Information
Subtracting Eq. (6.36) from Eq. (6.34) yields
U
∗
−
t
ρ
∇
U
n
+
1
p
=
(6.37)
Substituting Eq. (6.37) into Eq. (6.1) yields
∂
h
h
U
∗
)
−
t
2
p
−
t
ρ
∇
p
=
+∇·
(
h
∇
h
·∇
0
(6.38)
∂
t
ρ
Neglecting the last term on the left-hand side of Eq. (6.38) and applying
∂
h
/∂
t
=
h
n
+
1
h
n
p
/(ρ
(
−
)/
t
=
g
t
)
yields the following Poisson equation for pressure
correction:
h
U
∗
)
t
2
2
p
=−
ρ
(
1
−
gh
∇
)
g
t
∇·
(
(6.39)
Eq. (6.39) is solved using the efficient element method for the Laplace operator on
the left-hand side and the finite volume method for the term on the right-hand side.
The resulting algebraic equations are solved using the SIP method.
The calculation procedure starts with calculating
U
∗
, using Eq. (6.36) with the
known pressure
p
n
. Eq. (6.39) is then solved to obtain the pressure correction. Pressure
and velocity at time level
n
+
1 are obtained using the correction equations (6.35) and
(6.37), respectively.
Because the pressure correction is defined by Eq. (6.35), the time step length is some-
what limited. This limitation can be relaxed by using
p
n
+
1
p
∗
+
p
, as demonstrated
=
in Section 6.1.3.1. This modification requires
p
n
in Eq. (6.36) to be replaced by
p
∗
and a term of
p
n
p
∗
to be added on the right-hand side of Eq. (6.39). Iteration is
needed in the solution of the modified equations, but the time step can be longer and
the mass balance is less affected by the omission of the last term on the left-hand side
of Eq. (6.38), as
p
→
−
0 through the iteration.
6.1.3.3 Stream function and vorticity method
Stream function and vorticity equations
As described in Section 4.4.5, the stream function and vorticity method is highly
convenient for solving the 2-DNavier-Stokes equations of incompressible flows. How-
ever, its application in the depth-averaged 2-D model is not straightforward, because
the definition of stream function in Eq. (4.207) is not valid for the depth-integrated
continuity equation (6.1). Wu
et al
. (1995) redefined the stream function and extended
the stream function and vorticity method to the depth-averaged 2-D model for steady
open-channel flows.
The governing equations for the depth-averaged 2-D steady shallow water flows are
written as
∂(
hU
x
)
+
∂(
hU
y
)
=
0
(6.40)
∂
x
∂
y
