SCALES, SIMILITUDE, AND ASYMPTOTIC CONSIDERATIONS IN CONVECTIVE HEAT TRANSFER
Michael M. Chen
The application of asymptotic considerations in convective heat transfer, for the purposes of gaining insight on appropriate scales for non-dimensionalization, similitude and orders-of-magnitude estimates, and of deriving well-motivated computational schemes is discussed. The intent here is pedagogical rather than bibliographical. It is pointed out that many heat transfer problems, including especially the real world problems, are too complex for the basic non-dimensionalization schemes to yield unambiguous and clear physical understanding. With asymptotic considerations, a more systematic procedure exists for the testing and selection of the most meaningful scales for normalization, and for the accurate estimation of the relative importance of different effects. The interpretation of thermal boundary layers as the consequence of the singular asymptotic limit at high Peclet numbers, obvious to some but rarely utilized in heat transfer texts or research literature, is also demonstrated. Such singular limits are shown to require different non-dimensionalization for different regions of the flow and different coordinate directions. For conditions approximating either regular or singular asymptotic limits, much stronger and clearer statements on similitude can be obtained, as compared to those from basic dimensional analysis. Frequently the similarity parameters thus obtained can be useful even for conditions far from the limits, as long as they do not approach the opposite limits. As an example of the techniques discussed, the Prandtl number dependence of heat transfer is explored. Perturbation methods are not presented in detail, but their advantages in comparison to direct numerical solutions, even for problems of complex geometry, are listed. The potential of exploiting the asymptotic behavior of the equations in the development of numerical methods is also illustrated by a proposed scheme for solving exact Navier-Stokes and energy equations.
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