Biomedical Engineering Reference
In-Depth Information
Fig. 5.30
Control volume
representation for Upwind
Differencing discretisation of
nodal point P
The resulting discretised equation becomes
u
φ φ φφ φφ
− = −− −
u
D
(
)
D
(
)
e P
wW
e
E
P
w
P
W
and rearranging the terms to identify the coefficients of
ϕ
E
and
ϕ
W
and
ϕ
P
as before
(5.53)
a
φ
=
a
φ
+
a
φ
PP
EE
WW
where
aaauu
PEWe
=+;
Effectively, the convection term is discretised by applying a backward difference
scheme which is first order accurate. If a flow direction is from right to left then a
forward difference scheme is used (Fig.
5.31
).
=++−
(
);
aD
E
=
;
aDu
Ww
w
e
w
Quick
The Quadratic Upstream Interpolation for Convective Kinetics, (QUICK)
scheme uses a quadratic approximation across two variable points at the upstream
and one at the downstream. The unequal weighting influence of this particular
scheme hinges on knowledge biased towards the upstream flow conditions. For a
flow direction from left to right the nodal points are:
1
3
6
1
6
3
φ
=−
φ
+
φ
+
φ
and
φ
=−
φ
+
φ
+
φ
w
WW
P
W
e
W
P
E
8
8
8
8
8
8
The resulting discretised equation becomes
Fig. 5.31
Control volume representation for the QUICK discretisation of nodal point P
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