Biomedical Engineering Reference
In-Depth Information
Comparison of the FD and FV discretisations
Although the same algebraic form
of equation for the steady-state one-dimensional diffusion process is obtained, dif-
ferent expressions for the coefficients of
a
E
,
a
W
and
a
P
are derived as seen in Eq.
(7.28) for the FD method and Eq. (7.35) for the FV method. Nevertheless, let us
consider for a special case where the diffusion coefficient,
is spatially invariant
and the mesh is uniformly distributed. The coefficients in the algebraic equation for
the FD and FV methods, both reduce to
x
2
;
x
2
;
a
E
=
a
W
=
a
P
=
a
E
+
a
W
;
b
=
S
φ
where the control volume is
x
(uniform grid and
y, z
is unity). From the example
of the continuity equation discretisation, the resultant algebraic equations are exactly
the same whether adopting either the FD or FV discretisation. It should be noted that
the FD method generally requires a uniformly distributed mesh in order to apply the
first and second order derivative approximations to the governing equation. For an
arbitrary grid shape, some mathematical manipulation (e.g. transformation functions)
is required to transform Eq. (7.24), into a computational domain in generalized
coordinates before applying the FD approximations. This requirement is however
not a prerequisite for the FV method. Because of the availability of having different
control volume sizes, any non-arbitrary grid could therefore be easily accommodated.
The FD method is mathematically derived from the Taylor series, whereas the FV
method applies the conservation principle by integration of the variable over the con-
trol volume, and thus it retains this physical significance throughout the discretisation
process. Almost all commercial CFD codes adopt the finite volume discretisation of
the Navier-Stokes equation to obtain numerical solutions for complex fluid flow
problems as the mesh is not restricted to structured-type elements but can include a
variety of unstructured-type elements of different shapes and sizes.
Worked Example
Consider a steady heat conduction problem in a large plate with a
uniform heat generation,
q
= 500 kW/m
3
. The left and right walls have temperatures
T
L
=
100
◦
C and
T
R
=
400
◦
C, and the thickness of the plate is
L
2
.
40 cm. The
diffusion coefficient
governing the heat conduction problem becomes the thermal
conductivity
k
of the material which is constant at
k
=
6
W/m
2
.K. Assuming that
the dimensions in the
y
direction and
z
direction are so large that the temperature
gradients are only significant in the
x
direction, we can reduce the problem to a one-
dimensional analysis. We apply the FV method to obtain the solution of this simple
heat conduction problem (Fig.
7.14
).
Firstly, let us divide the domain into four control volumes giving
x
=
0
.
006 m.
After, we will investigate the influence of a larger number of control volumes. There
are four nodal points, each representing the central location for the four control
volumes. For illustration purposes a unit area is considered in the
y
=
−
z
plane, and
thus
V
x
. The general discretised form of the one-dimensional diffusion
equation for point P (node 3) is
=
a
P
T
P
=
a
E
T
E
+
a
W
T
W
+
b
Search WWH ::
Custom Search