Geoscience Reference
In-Depth Information
4.4.3 Primitive variables: SIMPLE(C) formulation on
staggered grid
SIMPLE algorithm
In the finite volume method, the 2-D Navier-Stokes equations are usually written in
conservative form as
∂ρ
∂
+
∂(ρ
u
x
)
+
∂(ρ
u
y
)
=
0
(4.168)
t
∂
x
∂
y
u
x
)
+
∂(ρ
u
y
u
x
)
x
+
∂τ
∂(ρ
u
x
)
+
∂(ρ
F
x
−
∂
p
x
+
∂τ
xx
xy
=
(4.169)
∂
t
∂
x
∂
y
∂
∂
∂
y
u
y
)
∂(ρ
∂(ρ
u
y
)
+
∂(ρ
u
x
u
y
)
−
∂
p
y
+
∂τ
+
∂τ
yx
yy
+
=
F
y
(4.170)
∂
t
∂
x
∂
y
∂
∂
x
∂
y
Note that the flow density
ρ
may vary with sediment concentration, temperature,
salinity, etc.
Fig. 4.25(a) shows the staggered grid used in the SIMPLE algorithm of Patankar and
Spalding (1972). For simplicity, a rectangular grid is used here. The control volume
for the
x
-momentum equation is shown in Fig. 4.25(b). Applying the finite volume
discretization introduced in Section 4.3.2 to Eq. (4.169) in this control volume leads
to the following discretized equation for
u
x
,
e
:
a
e
u
n
+
1
a
l
u
n
+
1
p
n
+
1
P
p
n
+
1
E
=
+
S
u
+
A
e
(
−
)
(4.171)
x
,
e
x
,
l
l
where
A
e
is the width of face
e
, i.e.,
y
e
. Note that the index
l
sweeps over all four
u
x
neighbors outside the control volume in Fig. 4.25(b).
As explained in Eq. (4.126), in the case of unsteady flow the discretized time-
derivative term is split and added to the source term
S
ui
and the coefficient
a
e
.
Therefore, Eq. (4.171) can be used for both steady and unsteady flows.
The control volume for the
y
-momentum equation is shown in Fig. 4.25(c). The
discretized equation for
u
y
,
n
can be written as
a
n
u
n
+
1
a
l
u
n
+
1
p
n
+
1
P
p
n
+
1
N
=
+
S
v
+
A
n
(
−
)
(4.172)
y
,
n
y
,
l
l
where
A
n
x
n
.
Once the pressure field is given, the discretized momentum equations (4.171) and
(4.172) can be solved. However, the pressure field is still to be determined. In an
iterative solution process, a pressure field
p
∗
is first guessed and then an approximate
velocity field is obtained using the following equations:
=
a
e
u
x
,
e
=
a
l
u
x
,
l
+
p
P
−
p
E
)
S
u
+
A
e
(
(4.173)
l
