9.1 Introduction

Models are used widely in all types of engineering, and especially in fluid mechanics. The term model has many uses, but in the engineering context, it usually involves a representation of a physical system, a prototype, that may be used to predict the behavior of the system in some desired respect. These models can include physical models that appear physically similar to the prototype or mathematical models, which help to predict the behavior of the system, but do not have a similar physical embodiment. In this chapter, we will develop procedures for designing models in a way to ensure that the model and prototype will behave similarly.

Consider the question, "Why can a 1.5-mm-long flea fall a meter without injury but a 1.5-m-tall man can't fall 1 km unhurt?" A model is a representation of a physical system that may be used to predict the behavior of the system in some desired respect. The implied question in the stated question above is, "Can the flea be used to model the man?"

It is true that a 1.5-mm-long flea can jump about 1 m. If we use geometric scaling, we might predict that a 1.5-m-long person would jump about 1 km. Since that is clearly not true, then what is wrong with this picture? A careful look at the theory of models will help answer our question.

In Sec. 9.7, we attempt to use a flea to model a man (or vice versa) to model the terminal velocity of the prototype using the model. We will learn, by using the theory of models and dimensional analysis, why this model does not work very well.

The theory of models is developed from the principles of dimensional analysis. These principles tell us that any prototype can be described by a series of dimensionless parameters, which I will call Pi terms in this chapter. For the prototype system, the parameter that we would like to measure, n^, can be represented as a function $ of a set of n dimen-sionless Pi terms.

If the relationship $ between Pi terms describes the behavior of the system, then it would also be possible to develop a similar set of Pi terms for a model that has the same dimensional relationships.

9.2.1 Dimensional analysis and the Buckingham Pi theorem

Edgar Buckingham (1867-1940) was an American physicist who first generated interest in the idea of dimensional analysis. In 1914, Buckingham published the article, "On Physically Similar Systems: Illustration of the Use of Dimensional Equations," Phys Rev. 1914;4:345-376. The Buckingham Pi theorem states:

The number of independent dimensionless quantities required to describe a phenomenon involving k variables is n, where n = k — r, and where r is the number of basic dimensions required to describe the variable.

To demonstrate the use of the Buckingham Pi theorem, imagine that we would like to perform a test that describes the pressure drop per length of pipe as a function of other variables that affect the pressure gradient (Fig. 9.1).

The first step in the process is to choose three fundamental dimensions, which describe mechanical properties. One might choose from a list of fundamental dimensions like force, length, time, mass, temperature, charge, voltage, and the list would be extensive. In fact, it is pretty clear that charge, voltage, and temperature are not important in this problem, so we will choose force, length, and time. We will designate those dimensions by the characters F, L, and T representing force, length, and time, respectively.

The second and arguably most difficult step in the modeling process will be to determine all important variables. In some models, it will not be obvious which variables are important. Occasionally, to simplify a

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