Biomedical Engineering Reference
In-Depth Information
Fig. 7.7
Finite volume method showing
a
vertex centred and
b
cell-centred
false/numerical diffusion when simple numerics and poor choice of mesh design is
applied.
The first step of the FV method is to divide the computational domain into a finite
number of discrete contiguous control volumes. Within each control volume, there
is a centroid where the variable values are calculated and stored. These values can
be coincident with a grid node, and hence is a
vertex-centred
FV mesh or the control
volume can coincide with the grid boundaries and nodes, and hence the centroid is
cell-centred
(Fig.
7.7
). In either case, interpolation is used to express variable values
at the control volume surfaces in terms of the values at the control volume centre
and suitable quadrature formulae are applied to approximate the surface and volume
integrals.
In a control volume, the bounding surface areas of the element are the faces in
which the flux of a dependent variable moves across. Here, the surface areas in the
normal direction to the volume surfaces as indicated in Fig.
7.8
shows the projected
surface areas where each face is projected in the normal direction (
n
). The projected
areas are positive if their outward normal vectors from the volume surfaces are
directed in the same directions of the Cartesian coordinate system; otherwise they
are negative.
To begin the discretisation let us consider the first order differential term,
∂φ
/
∂x
integrated over a control volume so that the rate of change of the variable
φ
with
respect to
x
is defined by the total change that is occurring inside the control volume,
∂φ
∂x
=
1
V
∂φ
∂x
dV
V
By applying Gauss' divergence theorem to the volume integral, we relate the volume
integral to the net flux of
φ
that crosses the bounding surface of the control volume.
N
N
∂φ
∂x
dV
total change inside
control volume
1
V
1
V
1
V
∂φ
∂x
≈
1
V
φdA
x
φ
i
A
i
φ
i
A
i
=
≈
→
i
=
1
i
=
1
V
A
net flux crossing
control surface
(7.18)
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