Biomedical Engineering Reference
In-Depth Information
Fig. 5.25  Velocity profile in the one-dimensional domain produced from the convection-diffusion
equation over time. The time dependent solution displays a decaying travelling wave. Case A uses
a diffusion term with value D = 0.1, and Case B uses a diffusion three times as great, with D = 0.3
D = 0. for Case A). The relative strength of convection from Au x
⋅∂
/
and diffu-
2
2
sion from Dux
e = / ,
where L is a characteristic length, usually taken as the blood vessel diameter. This
number is analogous to the Reynolds number that is used for the non-linear form,
and is an important number used to define the stability of a solution which is dis-
cussed in later sections in this chapter.
By enforcing constant values for the coefficients A, and D, we obtain the linear
form of the convection-diffusion equation which differs to its non-linear form of
the momentum equation. This means that the flow and diffusion don't interact.
⋅∂
/
is related by the dimensionless Peclet number, P LD
5.4.2
Finite Volume Method
The finite volume approach is based on conservation principles we saw earlier
in developing the momentum equations. If we consider a single control volume
(Fig. 5.26 ) and apply the fundamental principle that mass is conserved, then
That is, ' the flux of a variable (i�e� net rate of a mass of a variable that crosses the
control surface) is equal to the net change in quantity of the variable inside the control
volume '. This conservation principle is the cornerstone of the finite volume method.
Since the control volumes can take arbitrary shapes, it allows more flexibility in rep-
resenting the grid by either structured or unstructured mesh, as it is not limited by the
cell shape. A disadvantage of finite volume is that it is susceptible to false/numerical
diffusion when simple numerics and poor choice of mesh design is applied.
The physical domain is divided into discrete control volumes and nodes are placed
between control volume boundaries. The analysed node is given the variable P , and
its neighbour nodes are E and W denoting its east and west neighbour (Fig. 5.27 ).
Upon discretisation, the resulting equation for nodal point P is generalised as
(5.45)
a
φ
=
a
φ
+
b
P
P
nb
nb
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