Page 236 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
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fully developed flow in a channel, as given in reference (Aung and Worku 1986a), the
non-dimensional scales require changing. The scales used by Aung and Worku are
x 2 u 1 L
∗ ∗
x = ; u = (7.191)
2 1
ReL ν
All other scales are the same as the forced convection scale discussed in Section 7.3. The
above scales lead to some changes in the non-dimensional form of the mixed convection
2
∗
equation. The source term GrT /Re in the mixed convection equation will be GrT /Re
and the Reynolds number at all other locations will disappear. The great advantage of
applying this scale is that the non-dimensional length of the channel can be considerably
reduced. The analytical solution for a fully developed mixed convection profile is given by
Aung and Worku as
Gr x 1 3 x 1 2 x 1 2
u 1 = (1 − r T ) − + − − 6x + 6x 1 (7.192)
1
Re 6 4 12
where
T c − T a
r T = (7.193)
T h − T a

Figure 7.30 Mixed convection in a vertical channel. Unstructured finite element mesh