Biomedical Engineering Reference
In-Depth Information
to highlight that the pressure term and any other scalar value is stored in the scalar
control volume. The above equations can be rewritten as:
a
nb
u
nb
−
p
P
−
p
W
x
a
i
,
J
u
i
,
J
b
u
V
=
V
+
a
nb
v
nb
−
p
P
−
p
S
a
I
,
j
v
I
,
j
b
v
V
=
V
+
(7.57)
y
The coefficients
a
nb
are defined by the discretisation scheme that is chosed (Central
Differencing, Upwind Differencing, QUICK). The SIMPLE scheme provides a ro-
bust method of calculating the pressure and velocities for an incompressible flow.
When coupled with other governing variables such as temperature and turbulent
quantities, the calculation needs to be performed sequentially since it is an iterative
process. The details of the sequence of operations in a typical CFD iterative process
that embodies the SIMPLE scheme is discussed below.
Step 1
The iterative SIMPLE calculation process begins by first initialising the
pressure field, by giving a guessed value
p
* to solve the momentum equations. The
discretised Eq. (7.57) are solved based on
p*
to give
a
nb
u
nb
−
p
P
−
p
*
W
a
i
,
J
u
i
,
J
=
V
x
a
nb
v
nb
−
p
P
−
p
S
y
a
I
,
j
v
I
,
j
=
V
(7.58)
where the neighbouring coefficients and
u
nb
and
v
nb
denotes the calculated neigh-
bouring nodal velocity components based on
p
∗
. The asterisk notation represents the
value is the guessed value.
Step 2
The velocities
u
∗
, and
v
∗
obtained from the momentum equations, typically
will not satisfy the continuity equation. We need to apply corrections
u
and
v
at
each node so that the velocity terms satisfy the continuity equation. Thus we relate
the desired velocities
u
and
v
to the guessed velocities
u
∗
and
v
∗
and similarly the
correct pressure
p
with the guessed pressure
p
∗
by
p
=
p
*
+
p
u
*
u
u
=
+
v
v
*
v
=
+
(7.59)
Note that at this stage
u
∗
and
v
∗
are the velocities determined from the guessed
p
∗
, that
initially satisfied the momentum equation but don't satisfy the continuity equation.
The correction terms
u
and
v
are not known and these need to be determined. We
begin by subtracting Eqs. (7.58) from (7.57) we obtain
a
nb
(
u
nb
−
p
*
W
)
p
P
)
(
p
W
−
−
(
p
P
−
a
i
,
J
(
u
i
,
J
−
u
i
,
J
)
u
nb
)
=
+
V
x
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