Page 228 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
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This solution is a steady state solution generated by an artificial compressibility form of
the CBS scheme. The momentum boundary layer develops as the flow travels downstream.
Figure 7.20 shows a comparison of the velocity profiles for non-dimensional distances
between 0 and 6. It may be seen that the parabolic profile is developed close to a distance
of 4.0. The analytical solution obtained from boundary layer theory (Schlichting 1968)
gives an approximate relation for the non-dimensional developing length as
l e = 0.04Re (7.189)
which gives a l e = 4.0 for a Reynolds number of 100. It should be noted that the veloc-
ity profile is continuously changing in the downstream direction. A completely unchanged
u 1 velocity profile can be obtained only by extending the length of the channel further
(Schlichting 1968). Also, more accurate velocity profiles can be obtained by either employ-
ing a structured mesh or using a finer unstructured mesh. The interested reader is advised
to carry out a mesh convergence study on this type of problem.

7.11 Laminar Non-isothermal Flow

In this section, some examples of non-isothermal problems are discussed. In the previous
section, the temperature effects are ignored, but they are included in this section in order
to study some heat convection problems. The categories of forced convection, buoyancy-
driven convection and mixed convection are discussed in the following subsections:

7.11.1 Forced convection heat transfer

Forced convection heat transfer is induced by forcing a liquid, or gas, over a hot body or
surface. Two forced convection problems will be studied in this section. The first problem
is the extension of flow through a two-dimensional channel as discussed in the previous
section and the second one is of forced convection over a sphere. The difference between
the first problem and the one in the previous section is that the top and bottom walls are
at a higher temperature than that of the air flowing into the channel. The non-dimensional
temperature scale employed is

T − T a
∗
T = (7.190)
T w − T a
Since the CBS flow code is based on non-dimensional governing equations, a non-
dimensional scaling factor needs to be employed. This scale will give a temperature value of
unity on the walls (T = T w ) and zero at the inlet (T = T a ). Dirichlet boundary conditions for
temperature are not necessary at the exit. However, the boundary integrals resulting from the
discretization of the second-order terms need to be evaluated and added to the equations. For
a steady state solution, all four steps of the CBS scheme can be solved simultaneously, or
firstly a steady flow solution is obtained and then using this result a temperature distribution
can be established independently. The Reynolds number is again assumed to be equal to
100, and the velocity distribution is the same as shown in Figure 7.19. The temperature