Biomedical Engineering Reference
In-Depth Information
Fig. 7.17
A one-dimensional
mesh used to solve Eq. (7.38).
The mesh is uniform, but the
finite volume method is not
restricted to uniform meshes
Central Differencing (CD)
The values of
φ
e
and
φ
w
in the convective flux needs to
be evaluated, since the values of the scalar are stored at the nodes
P, E
, and
W
. Using
central differencing and for a uniform grid, we obtain
φ
e
=
(
φ
P
+
φ
E
)
/
2
φ
w
=
(
φ
W
+
φ
P
)
/
2
Putting the terms together, and rearranging, the discretised equation becomes
u
e
2
u
w
2
(
φ
P
+
φ
E
)
−
(
φ
W
+
φ
P
)
=
D
e
(
φ
E
−
φ
P
)
−
D
w
(
φ
P
−
φ
W
)
D
e
−
D
w
+
u
w
)
φ
P
D
e
−
φ
E
+
D
w
+
φ
W
u
e
2
u
w
2
u
e
2
u
w
2
+
+
(
u
e
−
=
We can rearrange the terms to identify the coefficients of
φ
E
and
φ
W
and
φ
P
as
a
P
φ
P
=
a
E
φ
E
+
a
W
φ
W
(7.39)
Where
u
e
2
u
w
2
a
P
=
a
E
+
a
W
+
(
u
e
−
u
w
);
a
E
=
D
e
−
;
a
W
=
D
w
+
;
(7.40)
The central differencing discretisation is second order accurate, however it is unable
to exhibit any bias in the flow direction. It has been well documented (Patankar 1980;
Versteeg and Malalasekera 2007) that the scheme fails in strongly convective flows
as it is unable to identify the flow direction. To overcome the problem due to central
differencing, variations to the interpolation scheme are made. Two commonly used
schemes are the upwind differencing and QUICK which are introduced here. The
reader should be aware that other schemes exist, such as Total Varaible Diminishing
(TVD), which is not shown here. Through the central differencing approximation,
the face value at
e
is always assumed to be weighted by the influence of the available
variables at the neighbouring points; the downstream value at
E
is always required
during the evaluation of
e
which is usually not known
a priori
in the majority of flow
cases. However, by exerting an unequal weighting influence, a numerical solution can
be designed to recognize the direction of the flow in order to appropriately determine
the interface values. This is essentially the hallmark of the
upwind
or
donor-cell
concept.
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