36 Scaling and Dimensional Analysis The equations of fluid motion presented in

# 36 scaling and dimensional analysis the equations of

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3.6 Scaling and Dimensional Analysis The equations of fluid motion presented in the preceding section are, in general, difficult to solve. As noted there, they comprise a nonlinear system of partial differential equations, and in many situations it is not yet possible even to rigorously prove that solutions exist. (As mentioned in Chap. 1, the N.–S. equations, or more particularly the attempts to solve them, played a significant role in the development of modern mathematical analysis throughout the 20 th Century, and they continue to do so today.) At least in part because of these difficulties, until only very recently (with the advent of modern high-speed computers and CFD codes) much of the practical work in fluid dynamics required laboratory experiments. For example, flows about ships, trains and aircraft—and even tall buildings—were studied experimentally using wind tunnels and water tow tanks. Clearly, in any of these cases it could be prohibitively expensive to build a succession of full-scale models (often termed “prototypes”) for testing and subsequent modification until a proper configuration was found. This fact led to wind tunnel testing of much smaller scale models that were relatively inexpen- sive to build compared to the prototypes, and this immediately raises the question, “Under what circumstances will the flow field about a scale model be the same as that about the actual full-size object?” It is this question, along with some of its consequences from the standpoint of analysis, that will be addressed in the present section where we will show that the basic answer is: geometric and dynamic similarity must be maintained between scale model and prototype if data obtained from a model are to be applicable to the full-size object. We begin with a subsection in which we discuss these two key ingredients to scaling analysis, without which utilizing such analyses would be impossible, viz. , geometric and dynamic similarity. We then provide a subsection that details scaling analysis of the N.–S. equations, followed by a subsection describing the Buckingham Π theorem widely used in developing correlations of experi- mental data. Finally, we describe physical interpretations of various dimensionless parameters that result from scaling the governing equations and/or applying the Buckingham Π theorem. 3.6.1 Geometric and dynamic similarity Here we begin with a definition and discussion of a very intuitive idea, geometric similarity. We then introduce the important concept of dynamic similarity and indicate how it is used to interpret results from studies of small-scale models in order to apply these to analysis of corresponding full-scale objects. Geometric Similarity The requirement of geometric similarity is, intuitively, a rather obvious one; we would not expect to obtain very useful information regarding lift of an airfoil by studying flow around a scale model of the Empire State Building being pulled through a water tow tank. While this (counter)

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