Geoscience Reference
In-Depth Information
where
A
w
=
(
J
ξη)
=
(
A
W
+
A
P
)/
2, and
A
s
=
(
J
ξη)
=
(
A
S
+
w
s
2.
Note that only formulations for the quantities at faces
w
and
s
are given in Eqs.
(4.137)-(4.139). The reason is that the face
e
of each cell is the face
w
of the next cell
on the east side, and the face
n
of each cell is the face
s
of the next cell on the north
side. The quantities at each cell face need to be calculated only once. This ensures the
quantities across cell faces to be consistent.
A
P
)/
Discretization of 3-D transport equation
The 3-D transport equation in the fixed, curvilinear coordinate system is written in
conservative form as
∂
∂
φ)
+
∂
∂ξ
j
∂φ
∂ξ
+
∂
∂η
j
∂φ
∂η
j
j
t
(ρ
J
ρ
J
u
ξ
φ
−
ˆ
J
α
α
ρ
J
u
η
φ
−
ˆ
J
α
α
+
∂
∂ζ
j
∂φ
∂ζ
3
j
3
ρ
J
u
ˆ
ζ
φ
−
J
α
α
=
JS
(4.140)
where
-directions,
related to the velocity components
u
x
,
u
y
, and
u
z
in the Cartesian coordinate system
by
u
ξ
,
ˆ
u
η
, and
ˆ
u
ζ
are the components of flow velocity in the
ˆ
ξ
-,
η
-, and
ζ
1
1
1
2
2
2
3
3
3
3
u
z
.
Fig. 4.22 shows the 3-D control volume centered at point
P
, which is embraced by
six faces
w
,
e
,
s
,
n
,
b
, and
t
. The cell faces are identified by the grid lines, and the point
P
is placed at the geometric center of the cell. Compared to the 2-D case, point
P
is
connected to two more points
B
(bottom) and
T
(top) in the
u
ξ
=
α
ˆ
1
u
x
+
α
2
u
y
+
α
3
u
z
,
u
η
=
α
ˆ
1
u
x
+
α
2
u
y
+
α
3
u
z
, and
u
ζ
=
α
ˆ
1
u
x
+
α
2
u
y
+
α
ζ
direction. Integrating
Eq. (4.140) in this control volume leads to
n
+
1
e
η
n
+
1
n
+
1
P
P
ρ
φ
−
ρ
φ
j
∂φ
∂ξ
P
P
1
j
1
(
J
ξηζ)
+
ρ
J
u
ξ
φ
−
ˆ
J
α
α
ζ
P
e
e
t
n
+
1
w
η
n
+
1
n
ξ
j
∂φ
∂ξ
j
∂φ
∂η
1
j
1
2
j
2
−
ρ
J
u
ξ
φ
−
ˆ
J
α
α
ζ
+
ρ
J
u
η
φ
−
ˆ
J
α
α
ζ
w
w
n
n
n
+
1
s
ξ
s
ζ
s
+
n
+
1
t
ξ
t
η
t
j
∂φ
j
∂φ
2
j
2
3
j
3
−
ρ
J
u
η
φ
−
ˆ
J
α
α
ρ
J
u
ζ
φ
−
ˆ
J
α
α
∂η
∂ζ
n
+
1
b
ξ
b
η
b
=
j
∂φ
3
j
3
−
ρ
J
u
ζ
φ
−
ˆ
J
α
α
S
(
J
ξηζ)
P
(4.141)
∂ζ
The backward difference scheme (4.23) is used to discretize the time-derivative term,
and the numerical schemes introduced in Section 4.3.1 are employed for the convection
and diffusion fluxes, thus yielding
n
+
1
n
+
1
n
P
n
P
ρ
φ
−
ρ
φ
P
P
n
+
1
n
+
1
n
+
1
n
+
1
V
P
=
a
W
φ
+
a
E
φ
+
a
S
φ
+
a
N
φ
W
E
S
N
t
n
+
1
n
+
1
n
+
1
+
a
B
φ
+
a
T
φ
−
a
P
φ
+
b
(4.142)
B
T
P
