Geoscience Reference
In-Depth Information
The works of Peric (1985), Majumdar (1988), and Ferziger and Peric (1995) fur-
ther popularized the non-staggered grid. The SIMPLE and SIMPLEC algorithms
based on the non-staggered grid with the momentum interpolation technique are
introduced below.
SIMPLE algorithm
The conservative form of the 2-D Navier-Stokes equations in the curvilinear grid
system reads
∂
∂τ
(ρ
)
+
∂
∂ξ
J
m
(ρ
J
u
m
ˆ
)
=
0
(4.185)
∂
∂τ
(ρ
)
+
∂
∂ξ
j
∂
u
i
∂ξ
=−
∂
∂ξ
m
j
m
m
i
p
Ju
i
ρ
J
u
m
u
i
ˆ
−
J
α
α
m
(
J
α
)
+
JS
(4.186)
m
m
where
S
includes the cross-derivative diffusion terms and the external forces.
Discretizating the momentum equation (4.186) in the control volume shown in
Fig. 4.21 yields the following equation for velocities
u
i
,
P
(
i
=
1, 2
)
:
⎛
⎝
⎞
⎠
+
1
a
P
u
n
+
1
i
,
P
a
l
u
n
+
1
D
i
(
p
n
+
1
w
p
n
+
1
e
D
i
(
p
n
+
1
s
p
n
+
1
n
=
+
S
ui
−
)
+
−
)
i
,
l
l
=
W
,
E
,
S
,
N
(4.187)
where
D
i
a
P
. Note that the values of
p
at faces
w
,
e
,
s
, and
n
are calculated by linear interpolation between two adjacent points, as
expressed in Eq. (4.151).
In analogy to Eq. (4.126), Eq. (4.187) can be used for both steady and unsteady
flows.
After an under-relaxation is introduced to stabilize the iterative solution process,
Eq. (4.187) is rewritten as (Majumdar, 1988)
1
i
a
P
, and
D
i
2
i
=
(
J
α
η)
/
=
(
J
α
ξ)
/
P
P
u
n
+
1
i
,
P
D
i
(
p
n
+
1
w
p
n
+
1
e
D
i
(
p
n
+
1
s
p
n
+
1
n
u
i
,
P
=
α
[
H
i
,
P
+
−
)
+
−
)
]+
(
1
−
α
)
u
u
(4.188)
where
H
i
,
P
=
(
l
=
W
,
E
,
S
,
N
a
l
u
n
+
1
a
P
, and
u
i
,
P
are the old values of
u
n
+
1
+
S
ui
)/
i
,
P
in the
i
,
l
previous iteration step.
Because the pressure is unknown, a pressure field
p
∗
is guessed, and then the
approximate values of the velocities are obtained by
u
i
,
P
=
α
H
i
,
P
+
D
i
(
p
w
−
p
e
)
+
D
i
(
p
s
−
p
n
)
]+
(
u
i
,
P
[
1
−
α
)
(4.189)
u
u
H
i
,
P
leads
Subtracting Eq. (4.189) from Eq. (4.188) and neglecting the terms
H
i
,
P
−
to the relation of velocity and pressure corrections at cell center:
u
n
+
1
i
,
P
u
i
,
P
+
α
D
i
(
p
w
−
p
e
)
+
D
i
(
p
s
−
p
n
)
]
=
[
(4.190)
u
