Geoscience Reference
In-Depth Information
Similarly, one can apply the three-level implicit scheme (4.43) for the time-derivative
term in Eq. (4.123). The resulting discretized equation is Eq. (4.126) with
a
P
=
n
+
1
S
U
=
n
P
n
P
n
−
1
n
−
1
a
P
+
1.5
ρ
x
P
/
t
−
S
P
,
S
U
+
(
2
ρ
φ
−
0.5
ρ
φ
)
x
P
/
t
P
P
P
(4.128)
4.3.2 Finite volume method for multidimensional
problems on fixed grids
Discretization of 2-D transport equation
The 2-D transport equation in the fixed, curvilinear coordinate system is written in
conservative form as
∂
∂
φ)
+
∂
∂ξ
j
∂φ
∂ξ
+
∂
∂η
j
∂φ
∂η
1
j
1
2
j
2
t
(ρ
J
ρ
J
u
ξ
φ
−
ˆ
J
α
α
ρ
J
u
η
φ
−
ˆ
J
α
α
=
JS
(4.129)
where
-directions, which
are related to the velocity components
u
x
and
u
y
in the Cartesian coordinate system
by
u
ξ
and
ˆ
u
η
are the components of flow velocity in the
ˆ
ξ
- and
η
2
2
u
y
. Note that the source term
S
includes the
cross-derivative terms but excludes the second derivatives of coordinates (curvature
terms) that are very sensitive to grid smoothness.
The computational domain is discretized into a finite number of control volumes
(cells) by a computational grid. The grid may be the body-fitted grid generated by Eq.
(4.74) or another more arbitrary grid. One of the commonly used methods for the
control volume setup is shown in Fig. 4.21. The grid lines are identified as cell faces,
and the computational point is placed at the geometric center of each control volume.
The control volume centered at point
P
is embraced by four faces
w
,
s
,
e
, and
n
, which
are the linear segments between cell corners
nw
,
sw
,
se
, and
ne
. It is connected with
four adjacent control volumes centered at points
W
,
E
,
S
, and
N
. Here,
W
denotes
the west or the negative
1
1
2
u
y
and
2
u
ξ
=
α
ˆ
1
u
x
+
α
u
η
=
α
ˆ
1
u
x
+
α
ξ
direction,
E
the east or the positive
ξ
direction,
S
the south
or the negative
direction.
Integrating the transport equation (4.129) over the control volume shown in
Fig. 4.21 yields
η
direction, and
N
the north or the positive
η
n
+
1
e
η
n
+
1
n
+
1
n
P
n
P
ρ
φ
−
ρ
φ
j
∂φ
∂ξ
P
P
j
(
J
ξη)
+
ρ
J
u
ξ
φ
−
ˆ
J
α
α
P
e
t
ρ
n
+
1
w
η
n
+
1
n
ξ
j
∂φ
∂ξ
j
∂φ
∂η
1
j
1
2
j
2
−
J
u
ξ
φ
−
ˆ
J
α
α
+
ρ
J
u
η
φ
−
ˆ
J
α
α
w
n
ρ
n
+
1
s
ξ
j
∂φ
∂η
2
j
2
−
J
u
η
φ
−
ˆ
J
α
α
=
S
(
J
ξη)
(4.130)
s
P
where
η
w
,
η
e
,
ξ
s
, and
ξ
n
are the widths of faces
w
,
e
,
s
, and
n
in the (
ξ
,
η
)
coordinate system; and
ξ
P
and
η
P
are the lengths of the control volume centered
at
P
in the
ξ
- and
η
-directions.
