Geoscience Reference
In-Depth Information
Figure 4.24
Staggered grid in MAC method.
Substituting Eqs. (4.158) and (4.159) into (4.157) leads to the discretized Poisson
equation for the pressure:
p
n
+
1
i
2
p
n
+
1
i
,
j
p
n
+
1
i
p
n
+
1
i
,
j
2
p
n
+
1
i
,
j
p
n
+
1
i
,
j
−
+
−
+
+
1,
j
−
1,
j
+
1
−
1
+
x
2
y
2
F
i
+
1
/
2,
j
−
D
i
,
j
F
i
−
1
/
2,
j
G
i
,
j
+
1
/
2
−
G
i
,
j
−
1
/
2
ρ
+
ρ
=
+
(4.160)
t
t
x
y
In Eq. (4.160)
D
i
,
j
/
t
may be interpreted as a discretization of
−
(∂
D
/∂
t
)
i
,
j
with
D
n
+
1
j
,
k
0. Thus, the pressure solution resulting from Eq. (4.160) is such as to allow
the discretized continuity equation (4.157) to be satisfied at time level
n
=
1.
Eq. (4.160) can be solved by using an iterative or direct method. Once it is solved,
substituting the obtained
p
n
+
1
into Eqs. (4.158) and (4.159) permits
u
n
+
1
+
x
and
u
n
+
y
to
be calculated. Because Eqs. (4.158) and (4.159) are explicit algorithms, the maximum
time step for a stable solution is restricted by (Peyret and Taylor, 1983)
2
x
2
0.25
(
|
u
x
|+|
u
y
|
)
t
Re
≤
1 and
t
/(
Re
)
≤
0.25
(4.161)
with the assumption of
x
=
y
. Re is the Reynolds number.
4.4.2 Primitive variables: projection formulation
on staggered grid
The projection method first proposed by Chorin (1968) solves the transport equations
to predict intermediate velocities and then project these velocities onto a space of
