Geoscience Reference
In-Depth Information
In a similar manner, the
u
y
-equation is derived by subtracting Eqs. (4.172) and
(4.174) as
u
y
,
n
=
p
P
−
p
N
)
d
n
(
(4.178)
a
n
.
The control volume for the pressure is shown in Fig. 4.25(d), over which the
continuity equation (4.168) can be integrated as
where
d
n
=
A
n
/
n
+
1
P
ρ
−
ρ
P
n
+
1
n
+
1
n
+
1
n
+
1
x
y
+[
(ρ
u
x
)
−
(ρ
u
x
)
]
y
+[
(ρ
u
y
)
−
(ρ
u
y
)
]
x
=
0
(4.179)
e
w
n
s
t
Inserting Eqs. (4.176)-(4.178) into Eq. (4.179) leads to the discrete equation for
p
:
a
p
a
p
a
p
a
p
a
p
P
p
P
=
W
p
W
+
E
p
E
+
S
p
S
+
N
p
N
+
b
p
(4.180)
where
a
p
W
y
,
a
p
E
y
,
a
p
S
x
,
a
p
N
x
,
a
p
P
w
d
w
n
+
n
+
1
n
+
1
n
+
1
=
ρ
=
ρ
d
e
=
ρ
d
s
=
ρ
d
n
=
e
s
n
a
p
W
+
a
E
+
a
S
+
a
N
, and
b
p
+
n
1
n
P
n
+
1
u
x
,
e
−
ρ
n
w
u
x
,
w
)
+
1
=−
(ρ
−
ρ
)
x
y
/
t
−
(ρ
y
−
e
P
n
+
1
u
y
,
n
−
ρ
n
+
1
u
y
,
s
)
(ρ
x
.
The computation is performed in the following sequence:
n
s
(1) Guess the pressure field
p
∗
;
(2) Solve the momentum equations (4.173) and (4.174) to obtain
u
x
and
u
y
;
(3) Calculate
p
using (4.180);
(4) Calculate
p
using Eq. (4.175);
(5) Calculate
u
n
+
1
x
and
u
n
+
y
using Eqs. (4.176)-(4.178);
(6) Treat the corrected pressure
p
as a new guessed pressure
p
∗
, and repeat the
procedure from step 2 to 6 until a converged solution is obtained, and
(7) Conduct the calculation of the next time step if unsteady flow is concerned.
SIMPLEC algorithm
Because the term
l
a
l
u
x
,
l
is neglected in the derivation of Eq. (4.177), the pressure
is not exactly solved in the aforementioned SIMPLE algorithm. Several algorithms,
such as SIMPLER (SIMPLE Revised, Patankar, 1980), PISO (Issa, 1982), and SIM-
PLEC (SIMPLE Consistence, van Doormaal and Raithby, 1984), have been proposed
to improve this. Van Doormaal and Raithby (1984) have reported that significant
savings on computation time can be achieved by the SIMPLEC algorithm in several
applications, as compared to the SIMPLE and SIMPLER algorithms. Therefore, the
SIMPLEC algorithm is introduced below.
The full velocity correction equation reads
a
e
u
x
,
e
=
a
l
u
x
,
l
+
p
P
−
p
E
)
A
e
(
(4.181)
l
