Biomedical Engineering Reference
In-Depth Information
Fig. 5.29  A one-dimensional
mesh used to solve the
convection-diffusion equa-
tion. The mesh is uniform,
but the finite volume method
is not restricted to uniform
meshes
Central Differencing 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 on a uniform grid, we obtain
φφφ φφφ
=+ =+
(
)/2
(
)/2
(5.49)
e
PE
w W P
Putting the terms together, and rearranging, the discretised equation becomes
u
u
e
w
(5.50)
(
φφ φφ φφ φφ
+− += −− −
)
(
)
D
(
)
D
(
)
P
E
W
P
eE
P
wP
W
2
2
We rearrange the terms to identify the coefficients of ϕ E and ϕ W and ϕ P as
(5.51)
u
u
u
u
−+++− =− ++
e
w
e
w
D
D
(
uu
]
φ
D
φ
D
φ
e
w
e
wP
e
E
w
W
2
2
2
2
so that
(5.52)
a
φ
=
a
φ
+
a
φ
PP
EE
WW
); aD u
E
; aD u
Ww
e
w
where aaauu
PEWe
=+ 2 ;
The central differencing discretisation is second order accurate however it is
unable to exhibit any bias in the flow direction by the transportiveness property.
The scheme fails in strongly convective flows as it is unable to identify the flow
direction. To overcome this, variations to the interpolation scheme are made. Two
commonly used schemes are the Upwind Differencing and QUICK which are intro-
duced here (Fig. 5.30 ).
=++−
(
=− 2
w
e
Upwind Differencing In upwind differencing, the value of a cell face is equal to
the value at the upstream node. If we limit our example to a flow direction from left
to right, then ϕ e = ϕ P and ϕ w = ϕ W
 
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