Geoscience Reference
In-Depth Information
The final discretized transport equation is written as
n
+
1
n
+
1
n
P
n
P
ρ
φ
−
ρ
φ
n
+
1
n
+
1
n
+
1
n
+
1
n
+
1
P
P
A
P
=
a
W
φ
+
a
E
φ
+
a
S
φ
+
a
N
φ
−
a
P
φ
b
(4.134)
+
W
E
S
N
P
t
where
F
w
exp
(
F
w
/
D
w
)
F
e
a
W
=
1
,
a
E
=
1
,
exp
(
F
w
/
D
w
)
−
exp
(
F
e
/
D
e
)
−
F
s
exp
(
F
s
/
D
s
)
F
n
a
S
=
1
,
a
N
=
1
,
(4.135)
exp
(
F
s
/
D
s
)
−
exp
(
F
n
/
D
n
)
−
a
P
=
a
W
+
a
E
+
a
S
+
a
N
+
(
F
e
−
F
w
+
F
n
−
F
s
)
, and
b
=
S
A
P
.
F
s
in the coefficient
a
P
can be treated by using the discretized
continuity equation introduced in Section 4.4.
In fact,
The term
F
e
−
F
w
+
F
n
−
A
P
and the quantities
F
and
D
at cell faces in Eqs. (4.132) and (4.133)
can be evaluated using only the parameters in the Cartesian coordinate systemwithout
involving the increments
ξ
and
η
(Peric, 1985; Zhu, 1992a). The area of the control
volume is calculated by
1
2
|
(
A
P
=
x
ne
−
x
sw
)(
y
nw
−
y
se
)
−
(
x
nw
−
x
se
)(
y
ne
−
y
sw
)
|
(4.136)
The convection fluxes at faces
w
and
s
are determined by
n
+
1
u
n
+
1
ξ
n
+
1
b
1
u
x
b
2
u
y
n
+
1
F
w
=
ρ
(
J
η)
ˆ
=
ρ
(
+
)
w
w
,
w
w
w
n
+
1
u
n
+
1
η
n
+
1
b
1
u
x
b
2
u
y
n
+
1
F
s
=
ρ
(
J
ξ)
ˆ
=
ρ
(
+
)
(4.137)
s
s
,
s
s
s
where
b
1
1
1
,
b
2
1
2
,
b
1
2
1
=
J
α
η
≈
(∂
y
/∂η)η
=
J
α
η
≈−
(∂
x
/∂η)η
=
J
α
ξ
≈
, and
b
2
=
2
2
−
(∂
y
/∂ξ)ξ
J
α
ξ
≈
(∂
x
/∂ξ)ξ
according to Eq. (4.79). The difference
equations for
b
i
at center
P
and faces
w
and
s
of the control volume shown in Fig.
4.21 are:
b
1
P
=
b
1
w
=
y
sw
,
b
1
s
=
y
n
−
y
nw
−
y
P
−
y
s
,
y
S
,
b
2
P
=
b
2
w
=
x
nw
,
b
2
s
=
x
s
−
x
n
,
x
sw
−
x
S
−
x
P
,
b
1
P
=
b
1
w
=
y
P
,
b
1
s
=
y
w
−
y
e
,
y
W
−
y
sw
−
y
se
,
b
2
P
=
b
2
w
=
x
W
,
b
2
s
=
x
e
−
x
w
,
x
P
−
x
se
−
x
sw
.
(4.138)
The diffusion parameters at faces
w
and
s
are computed by
1
j
1
j
b
1
b
1
+
b
2
b
2
)
w
/
D
w
=
(
J
α
α
η)
w
/ξ
w
=
w
(
A
w
2
j
2
j
b
1
b
1
+
b
2
b
2
)
D
s
=
(
J
α
α
ξ)
/η
=
(
/
A
s
(4.139)
s
s
s
s
