Geoscience Reference
In-Depth Information
φ
φ
Fig. 4.20 shows the relation between
W
and
w
for the HLPA scheme. When
≤
φ
0
w
is approximated by the parabolic function through three points
(0, 0), (0.5, 0.75), and (1, 1). When
φ
W
<
≤
1,
φ
W
0or
φ
W
>
φ
w
is approximated by the
first-order upwind scheme. Note that the point (0.5, 0.75) is the intersect point of
the QUICK scheme, central scheme, and second-order upwind scheme in the (
φ
1,
W
,
φ
w
)
plane.
The SOUCUP andHLPA schemes have good performance for convection-dominated
problems.
4.3.1.2 Discretization of 1-D unsteady problems
Integrating the 1-D unsteady, heterogeneous convection-diffusion equation
∂(ρφ)
∂
+
∂
∂
φ)
=
∂
∂
∂φ
∂
x
(ρ
u
+
S
(4.122)
t
x
x
over the control volume centered at point
P
shown in Fig. 4.14 yields
∂φ
∂
∂(ρφ)
∂
∂φ
∂
t
x
P
+
(ρ
u
φ)
−
(ρ
u
φ)
=
e
−
w
+
S
x
P
(4.123)
e
w
x
x
where
x
P
is the length of the control volume.
Applying the backward difference scheme (4.23) for the time-derivative term, one
of the numerical schemes introduced in Section 4.3.1.1 for the convection fluxes, and
the central difference scheme for the diffusion fluxes in Eq. (4.123) yields
n
+
1
n
P
(ρφ)
−
(ρφ)
n
+
1
n
+
1
n
+
1
P
x
P
=
a
W
φ
+
a
E
φ
−
a
P
φ
+
S
x
P
(4.124)
W
E
P
t
The source term in Eq. (4.124) can be linearized as
n
+
1
S
x
P
=
S
U
+
S
P
φ
(4.125)
P
where
S
U
and
S
P
are coefficients. The linearization formulation (4.125) should be a
good representation of the
S
∼
φ
relationship, and
S
P
must be nonpositive (Patankar,
1980).
The final form of the discretized equation is
n
+
1
n
+
1
n
+
1
a
P
φ
S
U
=
a
W
φ
+
a
E
φ
+
(4.126)
P
W
E
where
a
P
=
n
+
1
S
U
=
n
P
n
P
a
P
+
ρ
x
P
/
t
−
S
P
,
S
U
+
ρ
φ
x
P
/
t
(4.127)
P
