Geoscience Reference
In-Depth Information
n
+
1
u
n
+
1
ζ
n
+
1
u
n
+
1
ζ
F
b
=
ρ
(
J
ξη)
ˆ
,
F
t
=
ρ
(
J
ξη)
ˆ
,
(4.144)
t
b
t
,
t
b
,
b
and
1
j
1
j
1
j
1
j
D
w
=
(
J
α
α
ηζ)
/ξ
w
,
D
e
=
(
J
α
α
ηζ)
/ξ
e
,
w
e
2
j
2
j
2
j
2
j
D
s
=
(
J
α
α
ξζ)
/η
s
,
D
n
=
(
J
α
α
ξζ)
/η
n
,
(4.145)
s
n
3
j
3
j
3
j
3
j
D
b
=
(
J
α
α
ξη)
b
/ζ
b
,
D
t
=
(
J
α
α
ξη)
/ζ
t
.
t
V
P
and the quantities
F
and
D
at cell faces in Eqs. (4.144)
and (4.145) can be calculated using only the parameters in the Cartesian coordinate
system, and the final discretized equation does not involve the increments
Like the 2-D case,
ξ
,
η
,
and
(Peric, 1985; Zhu, 1992b). This is demonstrated below.
Kordulla and Vinokur (1983) suggested a method to calculate the volume of a
3-D cell. As shown in Fig. 4.23, the cell with points
A
,
B
,
ζ
, and
H
as its eight vertices
is decomposed into six tetrahedra, all containing the same diagonal joining points
A
and
H
. The volume of the tetrahedron with vertices
A
,
E
,
G
, and
H
can be calculated as
...
1
6
|
(
−
AE
−
AG
−
AH
V
1
=
×
)
·
|
(4.146)
Thus, the volume of the total cell is
−
AE
−
AG
−
AF
−
AE
−
AG
−
AC
1
6
V
=
×
+
×
+
×
−
AH
−
AB
−
AF
−
AC
−
AD
−
AD
−
AB
+
×
+
×
+
×
·
(4.147)
It is of interest to note that the above method of calculating cell volumes ensures
the conservation of space, i.e., the sum of all cell volumes gives exactly the total
volume of the solution domain. This is a necessary condition for guaranteeing the true
conservation of transported quantities (Zhu, 1992b).
The convection fluxes at faces
w
,
s
, and
b
are determined by
+
u
n
+
1
ξ
+
+
n
1
n
1
b
1
u
x
b
2
u
y
b
3
u
z
n
1
F
w
=
ρ
(
J
ηζ)
ˆ
=
ρ
(
+
+
)
w
w
w
w
,
w
n
+
1
u
n
+
1
n
+
1
b
1
u
x
+
b
2
u
y
+
b
3
u
z
)
n
+
1
F
s
=
ρ
(
ξζ)
s
ˆ
=
ρ
(
J
(4.148)
s
η
,
s
s
s
n
+
1
u
n
+
1
ζ
n
+
1
b
1
u
x
b
2
u
y
b
3
u
z
n
+
1
F
b
=
ρ
(
J
ξη)
ˆ
=
ρ
(
+
+
)
b
b
,
b
b
b
where
b
i
1
i
,
b
i
2
i
, and
b
i
3
i
=
J
α
ηζ
=
J
α
ξζ
=
J
α
ξη
(
i
=
1, 2, 3). The
m
i
and, in turn,
b
i
coefficients
α
can be calculated using Eq. (4.65) and the cofac-
i
m
in the Jacobian matrix
B
in Eq.
(4.64). For example, using
b
1
tors of
β
=
