Biomedical Engineering Reference
In-Depth Information
The term on the right hand side is the continuity equation, which equals zero for
mass conservation and the equation becomes
∂x
d
u
p
W
−
p
P
+
∂y
d
v
p
S
−
p
P
=
∂v
*
∂y
mass residual
∂u
*
∂x
+
∂
∂
(7.64)
Similarly, Eq. (7.64) can be discretised into the general form of
a
nb
p
nb
+
a
I
,
J
p
I
,
J
a
P
p
P
b
P
=
=
(7.65)
where
b
'
P
constains the
u
∗
and
v
∗
terms.
Step 3
To recap, we started with an initial guess of
p
∗
which produces
u
∗
and
v
∗
.
These guessed values need to be corrected and therefore their correction formulae
were derived to obtain
p
,
u
and
v
.
The pressure and velocity components are sub-
sequently updated through the correction formulae of Eq. (7.59). If the solution only
concerns a laminar CFD flow problem, the iteration process proceeds directly to
check the convergence of the solution. If the solution is not converged, the process
is repeated by returning to Step 1. The source term appearing in the pressure cor-
rection Eq. (7.64), commonly known as the
mass residual
, is normally used in CFD
computations as a criteria to terminate the iteration procedure. As the mass resid-
ual continues to diminish, the pressure correction
p
will be zero thereby yielding a
converged solution of
p
∗
=
p
,
u
∗
=
u
and
v
∗
=
v
.
Under some cases where the mesh contains highly skewed cells or the physical
problem is complex, the correction equations encounter difficulties in converging,
and in fact can diverge, and the solution is not achieved. In these cases, an under-
relaxation factor,
α
is applied during the iterative steps, whereby the correction
formulae is rewritten as
p
*
α
p
p
p
=
+
u
*
+
α
u
u
u
=
v
*
α
v
v
=
+
v
(7.66)
The under-relaxation factor is a value between 0 and 1 which aims to reduce large
changes in the variable which can cause divergence in the solution. However the
value should not be set too small as this will slow down the solution. Therefore there
this is an optimal value that is low enough to prevent divergence in the solution but
still high enough to provide rapid convergence. The selection of a suitbale under-
relaxation factor is flow dependent and different for each case, and so a single value
cannot be used.
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