We note that in the definition used here we require geometric similarity in

We note that in the definition used here we require

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We note that in the definition used here, we require geometric similarity in order to even consider dynamic similarity; some treatments relax this requirement, but such approaches are typically non-intuitive and sometimes not even self consistent. We note that the specific requirements of the above definition are not easily checked, and we will subsequently demonstrate that all that is actually needed is equality of all dimensionless parameters associated with the flow in, or around, the two objects. There are two ways by means of which we can determine the dimensionless parameters, and thus requirements for dynamic similarity, in any given physical situation. In cases for which governing equations are known, straightforward scaling of these equations will lead to the requirements needed to satisfy the above definition. On the other hand, when the governing equations are not known, the standard procedure is to employ the Buckingham Π theorem. We will introduce each of these approaches in the following two subsections, but we note at the outset that in the case of fluid dynamics the governing equations are known —they are the Navier–Stokes equations derived in preceding sections. Thus, we would expect to usually make direct application of scaling procedures. Nevertheless, in the context of analyzing experimental data it is sometimes useful to apply the Buckingham Π theorem in order to obtain a better collapse of data from a range of experiments, so at least some familiarity with this approach can be useful. 3.6.2 Scaling the governing equations Although in most elementary fluid dynamics texts considerable emphasis is placed on use of the Buckingham Π theorem, as we have already noted, when the governing equations are known it is more straightforward to use them directly to determine the important dimensionless parameters for any particular physical situation. The approach for doing this will be demonstrated in the current section. We begin by observing that from the definition of dynamic similarity we see that it is the ratios of forces at various corresponding locations in two (or more) flow fields that are of interest. Now if we could somehow arrange the equations of motion (via scaling) so that their solutions would be the same in each of the flow fields of interest, then obviously the ratios of forces would be the same everywhere in the two flow fields—trivially. In light of this, our goal should be to attempt to cast the Navier–Stokes equations in a form that would yield exactly the same solution for two geometrically similar objects, via scaling. Then, although the “unscaled” solutions would be different (as would be their solutions), they would differ in a systematic way related to geometric similarity of the objects under consideration.
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