7708d_c07_384* 7/23/01 10:00 AM Page 384 Flow past a circular cylinder with Re ϭ 2000: The pathlines of flow past any circular cylinder regardless of size, velocity, or fluid are as shown provided that the dimensionless1 parameter called the Reynolds2 number, Re, is equal to 2000. For other values of Re, the flow pattern will be different air bubbles in water . Photograph courtesy of ONERA, France. 1 2 1 2 7708d_c07_384* 7/23/01 10:00 AM Page 385 7 Similitude, Dimensional Analysis, and Modeling 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 Experimentation solved by analysis alone. The solution to many problems is achieved through the use of a and modeling are combination of analysis and experimental data. Thus, engineers working on fluid mechanics widely used tech- problems should be familiar with the experimental approach to these problems so that they niques in fluid can interpret and make use of data obtained by others, such as might appear in handbooks, mechanics. 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 for example, in the laboratory can be used to describe the behavior of other 1 2 similar systems outside the laboratory . The laboratory systems are usually thought of as 1 2 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. 7.1 Dimensional Analysis 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. An important characteristic of this system, which would be of interest 385 7708d_c07_384* 7/23/01 10:00 AM Page 386 386 I Chapter 7 / Similitude, Dimensional Analysis, and Modeling 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 even with the aid of large computers 1 2 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. Thus, we can express this relationship as ¢p/ ϭ f D, r, m, V (7.1) 1 2 which simply indicates mathematically that we expect the pressure drop per unit length to be some function of the factors contained within the parentheses. At this point the nature of the function is unknown and the objective of the experiments to be performed is to deter- mine the nature of this function. It is important to To perform the experiments in a meaningful and systematic manner, it would be nec- develop a meaning- essary to change one of the variables, such as the velocity, while holding all others constant, ful and systematic and measure the corresponding pressure drop. This series of tests would yield data that could way to perform an be represented graphically as is illustrated in Fig. 7.1a. It is to be noted that this plot would experiment. 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. We could repeat the process by varying each of the other variables in turn, as is illustrated in Figs. 7.1b, 7.1c, and 7.1d. This approach to determining the functional relationship between the pressure drop and the vari- ous factors that influence it, although logical in concept, is fraught with difficulties. Some of the experiments would be hard to carry out—for example, to obtain the data illustrated in Fig. 7.1c it would be necessary to vary fluid density while holding viscosity constant. How would you do this? Finally, once we obtained the various curves shown in Figs. 7.1a, 7.1b, 7.1c, and 7.1d, how could we combine these data to obtain the desired general functional re- lationship between ¢p/, D, r, m, and V which would be valid for any similar pipe system? ∆pᐉ ∆pᐉ D, ρµ , – constant V, ρµ , – constant V D (a) (b) ∆pᐉ ∆pᐉ D, ρ , V– constant D, V, µ – constant I FIGURE 7.1 Illus- trative plots showing how the pressure drop in a pipe may ρ µ be affected by several different (c) (d) factors. 7708d_c07_384* 7/23/01 10:00 AM Page 387 7.1 Dimensional Analysis I 387 ∆ _____D pᐉ ρV2 I ρ ____ VD FIGURE 7.2 An illustrative plot of pressure drop µ data using dimensionless parameters. Dimensionless Fortunately, there is a much simpler approach to this problem that will eliminate the products are impor- difficulties described above. In the following sections we will show that rather than working tant and useful with the original list of variables, as described in Eq. 7.1, we can collect these into two nondi- in the planning, mensional combinations of variables called dimensionless products or dimensionless groups 1 2 execution, and so that interpretation of D ¢p/ rVD experiments. ϭ f (7.2) rV 2 a m b Thus, instead of having to work with five variables, we now have only two. The necessary experiment would simply consist of varying the dimensionless product rVDրm and deter- 2 mining the corresponding value of D ¢p/րrV . The results of the experiment could then be represented by a single, universal curve as is illustrated in Fig. 7.2. This curve would be valid for any combination of smooth-walled pipe and incompressible Newtonian fluid. To obtain this curve we could choose a pipe of convenient size and a fluid that is easy to work with. Note that we wouldn’t have to use different pipe sizes or even different fluids. It is clear that the experiment would be much simpler, easier to do, and less expensive which would cer- 1 tainly make an impression on your boss . 2 The basis for this simplification lies in a consideration of the dimensions of the vari- ables involved. As was discussed in Chapter 1, a qualitative description of physical quantities can be given in terms of basic dimensions such as mass, M, length, L, and time,T.1 Alternatively, we could use force, F, L, and T as basic dimensions, since from Newton’s second law F Џ MLT Ϫ2 Recall from Chapter 1 that the notationЏ is used to indicate dimensional equality. The 1 Ϫ3 Ϫ2 4 2 dimensions of the variables in the pipe flow example are ¢p/ Џ FL , D Џ L, r Џ FL T , m Џ FLϪ2T, and V Џ LT Ϫ1. A quick check of the dimensions of the two groups that appear in Eq. 7.2 shows that they are in fact dimensionless products; that is, 3 D ¢p/ L FրL Џ 1 2 Џ F 0L0T 0 rV 2 FLϪ4T 2 LT Ϫ1 2 1 21 2 and rVD FLϪ4T 2 LTϪ1 L Џ 1 21 21 2 Џ F 0L0T 0 m FLϪ2T 1 2 Not only have we reduced the numbers of variables from five to two, but the new groups are dimensionless combinations of variables, which means that the results presented in the form 1 As noted in Chapter 1, we will use T to represent the basic dimension of time, although T is also used for temperature in ther- modynamic relationships such as the ideal gas law . 1 2 7708d_c07_384* 7/23/01 10:00 AM Page 388 388 I Chapter 7 / Similitude, Dimensional Analysis, and Modeling of Fig. 7.2 will be independent of the system of units we choose to use. 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. 7.2Buckingham Pi Theorem A fundamental question we must answer is how many dimensionless products are required to replace the original list of variables? The answer to this question is supplied by the basic theorem of dimensional analysis that states the following: If an equation involving k variables is dimensionally homogeneous, it can be reduced to a relationship among k Ϫ r independent dimensionless products, where r is the minimum number of reference dimensions required to describe the variables.