Geoscience Reference
In-Depth Information
Substituting Eqs. (4.105) and (4.106) into Eq. (4.104) yields
1
2
(ρ
1
2
(ρ
)
=
e
(φ
E
−
φ
P
)
−
w
(φ
P
−
φ
W
)
u
)
(φ
+
φ
)
−
u
)
(φ
+
φ
e
E
P
w
P
W
x
e
x
w
(4.107)
at faces
w
and
e
can be obtained by interpolation of their
values at points
W
,
P
, and
E
. This is to be discussed in the end of Section 4.3.2 for
general situations. The discretized equation (4.107) is reformulated as
The values of
ρ
,
u
, and
a
P
φ
=
a
W
φ
+
a
E
φ
(4.108)
P
W
E
where
a
W
,
a
P
, and
a
E
are coefficients:
a
W
=
D
w
+
F
w
/
2
a
E
=
D
e
−
F
e
/
2
(4.109)
a
P
=
a
W
+
a
E
+
(
F
e
−
F
w
)
with
F
x
.
Integrating the 1-D continuity equation over the control volume shown in Fig. 4.14
leads to
F
e
=
ρ
u
and
D
=
/
=
F
w
, which is not introduced here in detail. Therefore,
F
e
−
F
w
can be
eliminated from the expression of
a
P
in Eq. (4.109).
Because the coefficients in Eq. (4.109) could become negative and
|
a
P
|
<
|
a
E
|+|
a
W
|
when
2), the central scheme may result in unrealistic solutions;
see also Section 4.2.1.2. Here,
P
is the Peclet number, defined in Eq. (4.20) or as
F
|
F
|
>
2
D
(or
|
P
|
>
D
. The numerical oscillations for the central scheme at large Peclet numbers are
due to the assumption that the convected property of
/
at a cell face is given the aver-
age of its values at two neighboring points. Schemes that overcome this problem are
upwind scheme (Courant
et al
., 1952), exponential scheme (Spalding, 1972), hybrid
upwind/central scheme (Spalding, 1972), QUICK scheme (Leonard, 1979), SOUCUP
(Zhu and Rodi, 1991), HLPA scheme (Zhu, 1991), etc., as discussed below.
φ
Upwind scheme
In the upwind scheme, the formulation of the diffusion flux remains unchanged. For
the convection flux the value of
at face
w
is set as its value at the upwind adjacent
grid point, as shown in Fig. 4.16, thus yielding
φ
φ
W
,f
F
w
≥
0
φ
=
(4.110)
w
φ
P
,
if
F
w
<
0
which can be rewritten as
F
w
φ
=
φ
W
max
(
F
w
,0
)
−
φ
P
max
(
−
F
w
,0
)
(4.111)
w
An expression similar to Eq. (4.111) can be derived for the convection flux at
face
e
. When Eq. (4.105) is replaced by this concept, the coefficients of the discretized
