Dimensional Analysis: Examples of the Use of Symmetry
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After discussing several examples of method, the text examines pipe flow, material properties, gasdynamical examples, body in nonuniform flow, and turbulent flow. Additional topics include waves on a free liquid surface, examples with other fluid properties, and ideal gas equations of state. Figures appear throughout the text, which concludes with a bibliography.
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Dimensional Analysis - Hans G. Hornung
Bibliography
Preface
The contents of this little book were first compiled as part of a series of lectures in fluid mechanics presented at the Georg-August University of Gottingen, Germany, in the mid 1980’s and were published in German as an internal report (DFVLR Mitt. 88-27) of the Institute for Experimental Fluid Mechanics of the DFVLR (now DLR). The aim was not to treat dimensional analysis in a formal way but, instead, to emphasize the use of dimensional analysis for guiding more rigorous analysis. In particular, the power of using symmetry arguments to obtain the mathematical form of the functions that dimensional analysis yields is illustrated in all the examples. Readers should find it straightforward to apply such methods to their own fields of interest.
Because it was part of a course in fluid mechanics, the examples are nearly exclusively from this field. The methods used are, of course, equally applicable to any field of physics and, indeed, even more widely than that. However, the examples presented happened, at the time, to be the ones that appeared to be most useful for inspiring the fluid mechanics students to use such methods themselves. The presentation is aimed at students who have had some exposure to fluid mechanics and is best suited for students from junior to first graduate year.
Pasadena, September 2004.
Chapter 1
Introduction
In physics, a concept can attain meaning only if the properties that describe it can be measured. Since measurements are always subject to error, a quantity or a concept of physics can never be known precisely. The logical chain of an argument can, therefore, arrive at a meaningful conclusion (one for which the cumulative error does not dominate the result) only if the number of logical steps and the number of quantities used in the argument are not too large.
Measurement of a quantity, i. e., the association of a number with the quantity, consists of two essential steps:
Definition of a standard measure
Comparison of the quantity with the standard measure
The word comparison
implies a prescription that defines how the comparison is to be made. The result of a measurement, i. e., the association of a number to a quantity, consists of three elements:
physical quantity = number x (standard measure) ± error,
e. g.,
mass of my coffee cup = 0.150 x kg ± 0.001 kg.
We now restrict our interest to physical processes that involve only a small number of variables (because of the cumulative error). There is no need for a separate standard measure for each of the variables that arise in a problem. Many quantities can be measured by measures that are obtained by combination of standard measures, e. g.,
measure of viscosity = kg/(m x s).
For the description of any situation, there exists a minimum number of standard measures from which the measures of all variables describing the situation can be derived. For most of the processes of interest here, the standard measures for mass, length, and time suffice. We denote the type of a quantity by the word dimension
. For example, all quantities of the type mass
, i. e., those that can be measured by comparison with the standard measure for mass (kg), have the dimension mass
. We use the notation [M] for dimension mass
. Analogously, [L], [T] denote dimension length, time
respectively.