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Concepts:

dimensionless, fluid, pipe, dimensional analysis, pressure, illustrate, dimensionless products, basic dimensions, laboratory2, ables, Buckingham, curve, flow, phenomenon, fluid mechanics.

Summary:

Although many practical engineering problems involving fluid mechanics can be solved by using the equations and analytical procedures described in the preceding chapters, there re- main a large number of problems that rely on experimentally obtained data for their solu- tion.

In fact, it is probably fair to say that very few problems involving real fluids can be solved by analysis alone.

The solution to many problems is achieved through the use of a combination of analysis and experimental data.

Thus, engineers working on fluid mechanics problems should be familiar with the experimental approach to these problems so that they can interpret and make use of data obtained by others, such as might appear in handbooks, or be able to plan and execute the necessary experiments in their own laboratories.

In this chapter we consider some techniques and ideas that are important in the planning and exe- cution of experiments, as well as in understanding and correlating data that may have been obtained by other experimenters.

An obvious goal of any experiment is to make the results as widely applicable as pos- sible.

To achieve this end, the concept of similitude is often used so that measurements made on one system 1for example, in the laboratory2 can be used to describe the behavior of other similar systems 1outside the laboratory2.

The laboratory systems are usually thought of as models and are used to study the phenomenon of interest under carefully controlled condi- tions.

From these model studies, empirical formulations can be developed, or specific pre- dictions of one or more characteristics of some other similar system can be made.

To do this, it is necessary to establish the relationship between the laboratory model and the "other" sys- tem.

In the following sections, we find out how this can be accomplished in a systematic manner.

To illustrate a typical fluid mechanics problem in which experimentation is required, con- sider the steady flow of an incompressible Newtonian fluid through a long, smooth-walled, horizontal, circular pipe.

to an engineer designing a pipeline, is the pressure drop per unit length that develops along the pipe as a result of friction.

Although this would appear to be a relatively simple flow problem, it cannot generally be solved analytically 1even with the aid of large computers2 without the use of experimental data.

The first step in the planning of an experiment to study this problem would be to de- cide on the factors, or variables, that will have an effect on the pressure drop per unit length, p/.

We expect the list to include the pipe diameter, D, the fluid density, r, fluid viscosity, m, and the mean velocity, V, at which the fluid is flowing through the pipe.

To perform the experiments in a meaningful and systematic manner, it would be nec- essary to change one of the variables, such as the velocity, while holding all others constant, and measure the corresponding pressure drop.

It is to be noted that this plot would only be valid for the specific pipe and for the specific fluid used in the tests; this certainly does not give us the general formulation we are looking for.

?FIGURE 7.2An illustrative plot of pressure drop data using dimensionless parameters.

The results of the experiment could then be represented by a single, universal curve as is illustrated in Fig. 7.2.

The basis for this simplification lies in a consideration of the dimensions of the vari- ables involved.

This type of analy- sis is called dimensional analysis, and the basis for its application to a wide variety of prob- lems is found in the Buckingham pi theorem described in the following section.

The dimensionless products are frequently referred to as "pi terms," and the theorem is called the Buckingham pi theorem.2 Buckingham used the symbol to represent a dimensionless product, and this notation is commonly used.

Many entire books have been devoted to the subject of similitude and dimensional analysis, and a number of these are listed at the end of this chapter 1Refs.

, uk2 the dimensions of the variable on the left side of the equal sign must be equal to the di- mensions of any term that stands by itself on the right side of the equal sign.

Essentially we are looking for a method that will allow us to system2 Although several early investigators, including Lord Rayleigh 11842-19192 in the nineteenth century, contributed to the develop- ment of dimensional analysis, Edgar Buckingham's 11867-19402 name is usually associated with the basic theorem.

The method we will describe in detail in this section is called the method of repeating variables.

With a little practice you will be able to readily complete a dimensional analysis for your problem.

We are using the term "variable" to include any quantity, including dimensional and nondi- mensional constants, which play a role in the phenomenon under investigation.

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