From Chapter 3: Method of Similitude and

Introduction to Fractional Analysis of Overall Equations.

In Similitude and Approximation Theory

by Stephen Jay Kline McGraw-Hill Book Company 1965.

( later reprinted by Springer-Verlag )

Section a. Use of Force Ratios (p 38-40)

“The number of different kinds of forces found in nature is extremely large, and it is consequently impractical to deal with them all at once. Not only would this require a treatise larger in magnitude than this volume, but also it is seldom necessary. Since the purpose of this volume is to develop and examine methodology, it is sufficient to make an example of one field of analysis. The field chosen is fluid mechanics, since the method is well developed in that area and since the author is reasonably familiar with the subject. A table of basic dimensionless parameters similar to that developed for fluid mechanics can be prepared for use in other fields. What is more, the preparation of such tables is very instructive both as an exercise and as a reference in any given area of science and engineering. The construction of such a table enforces a general but especially careful consideration of the basic effects to be found in the field of study; it increases the physical understanding of the physical parameters normally employed; it provides standardization of these parameters for ready reference, and, most important, it provides a firm basis for checking for possible improvement of these parameters as further data is accumulated.

“There are six very common forces in fluid mechanics. Fifteen independent nondimensional numbers can be formed as ratios of these six forces, taken two at a time. These forces are defined, their dimensions shown, and the fifteen independent forces systematically displayed in Table 3.1.

“Examination of Table 3.1 shows that nearly all of the commonly employed correlating groups of fluid mechanics are contained in the first six numbers. Among the common groups only Mach number, drag coefficient, and the ratio of specific heats appear to be missing. It is readily shown that Mach number is merely the square root of the Cauchy number, which does appear in the table, and that drag coefficient is the same type of number as pressure coefficient and Euler number; that is, the ratio of forces acting on the surface to the inertia forces. Specific heat ratio apparently cannot be found from force considerations alone. It is also interesting to note that even among the fifteen simple numbers in Table 3.1, only six are apparently widespread enough to have been given names.

“The appearance in Table 3.1 of all the most commonly correlating groups of at least incompressible flow is, of course, not a coincidence. These groups are widely used not only for historical reasons, but also because the direct ratios of the governing forces express the correlating groups in a particularly useful, simple, and readily interpreted form.

Steve Kline devoted a very productive life to theoretical and experimental fluid mechanics, but also to methodology. He was cofounder and cochair , with Professor Walter Vincenti, of the Program in Values, Technology, Science and Society at Stanford. Steve wrote CONCEPTUAL FOUNDATIONS OF MULTIDISCIPLINARY THINKING (Stanford University Press, 1995) and died in 1997. He was a close friend and long-time partner, of mine – and some might be interested in a Short Eulogy for Steve Kline given at Steve's memorial service at Stanford Chapel.