Biomedical Engineering Reference
In-Depth Information
1
6
3
1
3
6
u
−++ −− ++
φ
φ
φ
u
φ
φ
φ
e
w
P
E
w
WW
P
W
8
8
8
8
8
8
(5.54)
= −− −
D
(
φφ φφ
)
D
(
)
eE
P
wP
W
and rearranging the terms to identify the coefficients of
ϕ
E
and
ϕ
W
and
ϕ
P
as before
(5.55)
a
φ
=
a
φ
+
a
φ
PP
EE
WW
=−
3
8
6
8
1
8
);
aDu
;
aDu
=+ +
u
where
aaaa uu
PEWWWe
=+++−
(
;
E
e
e
Ww
w
e
w
u
w
=−
8
Example: Comparisons of Discretisation Schemes
In central differencing for a
uniform mesh,
aD
u
E
b
e
=−
2
which means that the coefficient becomes negative if
2
D
e
< leading to a diverging and unstable solution. The Peclet number is used
as a guide to determine stability of a problem. If we consider the characteristic
length scale
L
c
as the mesh cell length ∆
x
, then the quantity is referred to as the cell
Peclet number. Across face
e,
Pe uxD
c
e
Pe
≤
then we
are guaranteed positive coefficients and the solution will converge. Similarly in the
QUICK scheme we obtain
aDu
=∆/
and that as long as
2
e
e
c
=−
3
8
which means the stability criterion is
E
e
e
Pe
≤
. The Peclet number is a function of the grid size, and this implies a mini-
mum spatial grid size to ensure that Pe is small enough to maintain the stability cri-
terion. In Upper Differencing, the value of
ϕ
on a cell face is determined by the flow
direction and the neighbouring coefficients are guaranteed to have positive values.
To demonstrate the dependence of Pe and the discretisation scheme, on the solu-
tion, we can compare the numerical solution with the exact analytical solution. The
analytical solution for a 1D steady convection-diffusion problem is
8/3
c
d u
(
φ
)
d
d
φ
φφ
−
exp(
ux
/
Γ−
)
1
− Γ =→ =
0
(5.56)
0
dx
dx
dx
φφ
−
exp(
uL
/
Γ−
)
1
L
0
In this example, the length of the domain is 1 m divided into five control volumes.
The boundary conditions are
ϕ
0
= 0 and
ϕ
L
= The velocity,
u
is 1.6 m/s and diffusion
coefficient, Γ is 0.1. The resulting Pe number is Pe = 3.2, producing an unstable
solution for the Central Differencing. Figure
5.32
shows the 'undershoots' and
'overshoots' occurring in the solution for Central Differencing. A similar oscillating
pattern is found for the QUICK scheme as well (although not shown in the figure)
since it has a restriction of Pe < 8/3. The Upper Diffrencing has no restrictions on
the Pe number and produces the trendline of the analytical solution. However the
accuracy of the Upper Difference solution drops off as
x
approaches 1. For both
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