Fluid Mechanics 7

1Recall from Chapter 1 that the notation is used to indicate dimensional equality.2 The dimensions of the variables in the pipe flow example are and 3Note that the pressure drop per unit length has the dimensions of 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, and Not only have we reduced the number of variables from five to two, but the new groups are dimensionless combinations of variables, which means that the results presented in the form of Fig. 7.2 will be independent of the system of units we choose to use. This type of analysis is called dimensional analysis, and the basis for its application to a wide variety of problems is found in the Buckingham pi theorem described in the following section

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 independent dimensionless products, where r is the minimum number of reference dimensions required to describe the variables. The dimensionless products are frequently referred to as "pi terms," and the theorem is called the Buckingham pi theorem. 2 Edgar Buckingham used the symbol to represent a dimensionless product, and this notation is commonly used. Although the pi theorem is a simple one, its proof is not so simple and we will not include it here. 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. 1-152. Students interested in pursuing the subject in more depth 1including the proof of the pi theorem2 can refer to one of these books. The pi theorem is based on the idea of dimensional homogeneity which was introduced in Chapter 1. Essentially we assume that for any physically meaningful equation involving k variables, such as the dimensions of the variable on the left side of the equal sign must be equal to the dimensions of any term that stands by itself on the right side of the equal sign. It then follows that we can rearrange the equation into a set of dimensionless products 1pi terms2 so that where is a function of through The required number of pi terms is fewer than the number of original variables by r, where r is determined by the minimum number of reference dimensions required to describe the original list of variables. Usually the reference dimensions required to describe the variables will be the basic dimensions M, L, and T or F, L, and T. However, in some instances perhaps only two dimensions, such as L and T, are required, or maybe just one, such as L. Also, in a few rare cases

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 independent dimensionless products, where r is the minimum number of reference dimensions required to describe the variables. The dimensionless products are frequently referred to as "pi terms," and the theorem is called the Buckingham pi theorem. 2 Edgar Buckingham used the symbol to represent a dimensionless product, and this notation is commonly used. Although the pi theorem is a simple one, its proof is not so simple and we will not include it here. 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. 1-152. Students interested in pursuing the subject in more depth 1including the proof of the pi theorem2 can refer to one of these books. The pi theorem is based on the idea of dimensional homogeneity which was introduced in Chapter 1. Essentially we assume that for any physically meaningful equation involving k variables, such as the dimensions of the variable on the left side of the equal sign must be equal to the dimensions of any term that stands by itself on the right side of the equal sign. It then follows that we can rearrange the equation into a set of dimensionless products 1pi terms2 so that where is a function of through The required number of pi terms is fewer than the number of original variables by r, where r is determined by the minimum number of reference dimensions required to describe the original list of variables. Usually the reference dimensions required to describe the variables will be the basic dimensions M, L, and T or F, L, and T. However, in some instances perhaps only two dimensions, such as L and T, are required, or maybe just one, such as L. Also, in a few rare cases the variables may be described by some combination of basic dimensions, such as and L, and in this case r would be equal to two rather than three. Although the use of the pi theorem may appear to be a little mysterious and complicated, we will actually develop a simple, systematic procedure for developing the pi terms for a given problem

A little reflection on the process used to determine pi terms by the method of repeating variables reveals that the specific pi terms obtained depend on the somewhat arbitrary selection of repeating variables. For example, in the problem of studying the pressure drop in a pipe, we selected D, V, and as repeating variables. This led to the formulation of the problem in terms of pi terms as (7.5) What if we had selected D, V, and as repeating variables? A quick check will reveal that the pi term involving becomes and the second pi term remains the same. Thus, we can express the final result as (7.6) Both results are correct, and both would lead to the same final equation for . Note, however, that the functions in Eqs. 7.5 and 7.6 will be different because the dependent pi terms are different for the two relationships. As shown by the figure in the margin, the resulting graph of dimensionless data will be different for the two formulations. However, when extracting the physical variable, , from the two results, the values will be the same. We can conclude from this illustration that there is not a unique set of pi terms which arises from a dimensional analysis. However, the required number of pi terms is fixed, and once a correct set is determined, all other possible sets can be developed from this set by combinations of products of powers of the original set. Thus, if we have a problem involving, say, three pi terms, we could always form a new set from this one by combining the pi terms. For example, we could form a new pi term, by letting where a and b are arbitrary exponents. Then the relationship could be expressed as or All of these would be correct. It should be emphasized, however, that the required number of pi terms cannot be reduced by this manipulation; only the form of the pi terms is altered. By using ß1 f21ß2, ß¿22 ß1 f11ß¿2, ß32 ß¿2 ßa 2 ßb 3 ß¿2, ß1 f1ß2, ß32 ¢p/ f and f1 ¢p/ ¢p/D2 Vm f1 a rVD m b ¢p/D2 Vm ¢p/ m ¢p/D rV2 f a rVD m b r 344 Chapter 7 ■ Dimensional Analysis, Similitude, and Modeling and, therefore, so that which is the same as obtained using the FLT system. The other pi terms would be the same, and the final result is the same; that is, (Ans) d D f a h D, d D, E Dg b ß4 ß4 E Dg a 1, b 1 This will always be true—you cannot affect the required number of pi terms by using M, L, and T instead of F, L, and T, or vice versa. Once a correct set of pi terms is obtained, any other set can be obtained by manipulation of the original set. D2Δp Vm VD ____ μ ρ VD ____ μ ρ V2 ρ DΔp JWCL068_ch07_332-382.qxd 9/23/08 10:46 AM Page 344 this technique we see that the pi terms in Eq. 7.6 could be obtained from those in Eq. 7.5; that is, we multiply in Eq. 7.5 by so that which is the of Eq. 7.6. There is no simple answer to the question: Which form for the pi terms is best? Usually our only guideline is to keep the pi terms as simple as possible. Also, it may be that certain pi terms will be easier to work with in actually performing experiments. The final choice remains an arbitrary one and generally will depend on the background and experience of the investigator. It should again be emphasized, however, that although there is no unique set of pi terms for a given problem, the number required is fixed in accordance with the pi theorem

Although many practical engineering problems involving fluid mechanics can be solved by using the equations and analytical procedures described in the preceding chapters, there remain a large number of problems that rely on experimentally obtained data for their solution. 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 theoretical and numerical 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 execution 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 possible. 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

Another group of typical pi terms 1such as the Reynolds number in the foregoing example2 involves force ratios as noted in Table 7.1. The equality of these pi terms requires the ratio of like forces in model and prototype to be the same. Thus, for flows in which the Reynolds numbers are equal, the ratio of viscous forces in model and prototype is equal to the ratio of inertia forces. If other pi terms are involved, such as the Froude number or Weber number, a similar conclusion can be drawn; that is, the equality of these pi terms requires the ratio of like forces in model and prototype to be the same. Thus, when these types of pi terms are equal in model and prototype, we have dynamic similarity between model and prototype. It follows that with both geometric and dynamic similarity the streamline patterns will be the same and corresponding velocity ratios and acceleration ratios are constant throughout the flow field. Thus, kinematic similarity exists between model and prototype. To have complete similarity between model and prototype, we must maintain geometric, kinematic, and dynamic similarity between the two systems. This will automatically follow if all the important variables are included in the dimensional analysis, and if all the similarity requirements based on the resulting pi terms are satisfied

Another interesting point to note from Fig. 7.7 is the rather abrupt drop in the drag coefficient near a Reynolds number of As is discussed in Section 9.3.3, this is due to a change in the flow conditions near the surface of the sphere. These changes are influenced by the surface roughness and, in fact, the drag coefficient for a sphere with a "rougher" surface will generally be less than that of the smooth sphere for high Reynolds number. For example, the dimples on a golf ball are used to reduce the drag over that which would occur for a smooth golf ball. Although this is undoubtedly of great interest to the avid golfer, it is also important to engineers responsible for fluid-flow models, since it does emphasize the potential importance of the surface roughness. However, for bodies that are sufficiently angular with sharp corners, the actual surface roughness is likely to play a secondary role compared with the main geometric features of the body.

Application of the pi theorem indicates that if the number of variables minus the number of reference dimensions is equal to unity, then only one pi term is required to describe the phenomenon. The functional relationship that must exist for one pi term is where C is a constant. This is one situation in which a dimensional analysis reveals the specific form of the relationship and, as is illustrated by the following example, shows how the individual variables are related. The value of the constant, however, must still be determined by experiment.

As an alternative, we see from Eq. 7.21 that could be reduced by using a different fluid in the model such that For example, the ratio of the kinematic viscosity of water to that of air is approximately so that if the prototype fluid were air, tests might be run on the model using water. This would reduce the required model velocity, but it still may be difficult to achieve the necessary velocity in a suitable test facility, such as a water tunnel. Another possibility for wind tunnel tests would be to increase the air pressure in the tunnel so that thus reducing the required model velocity as specified by Eq. 7.20. Fluid viscosity is not strongly influenced by pressure. Although pressurized tunnels have been used, they are obviously more complicated and expensive. The required model velocity can also be reduced if the length scale is modest; that is, the model is relatively large. For wind tunnel testing, this requires a large test section which greatly increases the cost of the facility. However, large wind tunnels suitable for testing very large models 1or prototypes2 are in use. One such tunnel, located at the NASA Ames Research Center, Moffett Field, California, has a test section that is 40 ft by 80 ft and can accommodate test speeds to 345 mph. Such a large and expensive test facility is obviously not feasible for university or industrial laboratories, so most model testing has to be accomplished with relatively small models.

As an example of the procedure, consider the problem of determining the drag, on a thin rectangular plate placed normal to a fluid with velocity, V, as shown by the figure in the margin. The dimensional analysis of this problem was performed in Example 7.1, where it was assumed that Application of the pi theorem yielded (7.11) We are now concerned with designing a model that could be used to predict the drag on a certain prototype 1which presumably has a different size than the model2. Since the relationship expressed by Eq. 7.11 applies to both prototype and model, Eq. 7.11 is assumed to govern the prototype, with a similar relationship (7.12) for the model. The model design conditions, or similarity requirements, are therefore The size of the model is obtained from the first requirement which indicates that (7.13) We are free to establish the height ratio but then the model plate width, is fixed in accordance with Eq. 7.13. The second similarity requirement indicates that the model and prototype must be operated at the same Reynolds number. Thus, the required velocity for the model is obtained from the relationship Vm (7.14) mm m r rm w wm V hmh, wm, wm hm h w wm hm w h rmVmwm mm rVw m dm w2 mrmV2 m f a wm hm , rmVmwm mm b d w2 rV2 f a w h , rVw m b d f 1w, h, m, r, V2 1w h in size2 d, ß1 ß1m ß1 ß1m f ßnm ßn o ß3m ß3 ß2m ß2 ß1 ß1m f1ß2m, ß3m, . . . , ßnm2 7.8 Modeling and Similitude 355 The similarity requirements for a model can be readily obtained with the aid of dimensional analysis. h w V ρ, μ JWCL068_ch07_332-382.qxd 9/23/08 10:46 AM Page 355 Note that this model design requires not only geometric scaling, as specified by Eq. 7.13, but also the correct scaling of the velocity in accordance with Eq. 7.14. This result is typical of most model designs—there is more to the design than simply scaling the geometry

As the number of required pi terms increases, it becomes more difficult to display the results in a convenient graphical form and to determine a specific empirical equation that describes the phenomenon. For problems involving three pi terms it is still possible to show data correlations on simple graphs by plotting families of curves as illustrated in Fig. 7.5. This is an informative and useful way of representing the data in a general ß1 f1ß2, ß32 7.7 Correlation of Experimental Data 353 density, fluid viscosity, and the velocity, V. Thus, and application of the pi theorem yields two pi terms Hence, To determine the form of the relationship, we need to vary the Reynolds number, and to measure the corresponding values of The Reynolds number could be varied by changing any one of the variables, V, D, or or any combination of them. However, the simplest way to do this is to vary the velocity, since this will allow us to use the same fluid and pipe. Based on the data given, values for the two pi terms can be computed, with the result: 0.0195 0.0175 0.0155 0.0132 0.0113 0.0101 0.00939 0.00893 These are dimensionless groups so that their values are independent of the system of units used so long as a consistent system is used. For example, if the velocity is in fts, then the diameter should be in feet, not inches or meters. Note that since the Reynolds numbers are all greater than 2100, the flow in the pipe is turbulent 1see Section 8.1.12. A plot of these two pi terms can now be made with the results shown in Fig. E7.4a. The correlation appears to be quite good, and if it was not, this would suggest that either we had large experimental measurement errors or that we had perhaps omitted an important variable. The curve shown in Fig. E7.4a represents the general relationship between the pressure drop and the other factors in the range of Reynolds numbers between and Thus, for this range of Reynolds numbers it is not necessary to repeat the tests for other pipe sizes or other fluids provided the assumed independent variables are the only important ones. Since the relationship between and is nonlinear, it is not immediately obvious what form of empirical equation might be used to describe the relationship. If, however, the same data are ß1 ß2 1D, r, m, V2 9.85 104 4.01 10 . 3 9.85 104 8.00 104 5.80 104 3.81 104 2.00 104 9.97 103 6.68 103 4.01 103 D¢pRV RVDM 2 r, m, D ¢p/rV2 . Re rVDm, D ¢p/ rV2 f a rVD m b ß1 D ¢p/ rV2 and ß2 rVD m ¢p/ f 1D, r, m, V2 r, m, plotted on logarithmic graph paper, as is shown in Fig. E7.4b, the data form a straight line, suggesting that a suitable equation is of the form where A and n are empirical constants to be determined from the data by using a suitable curve-fitting technique, such as a nonlinear regression program. For the data given in this example, a good fit of the data is obtained with the equation (Ans) COMMENT In 1911, H. Blasius 11883-19702, a German fluid mechanician, established a similar empirical equation that is used widely for predicting the pressure drop in smooth pipes in the range 1Ref. 162. This equation can be expressed in the form The so-called Blasius formula is based on numerous experimental results of the type used in this example. Flow in pipes is discussed in more detail in the next chapter, where it is shown how pipe roughness 1which introduces another variable2 may affect the results given in this example 1which is for smooth-walled pipes2. D ¢p/ rV2 0.1582a rVD m b 14 4 103 6 Re 6 105 ß1 0.150 ß 0.25 2 ß1 Aßn 2 For problems involving more than two or three pi terms, it is often necessary to use a model to predict specific characteristics. 0.022 0.020 0.018 0.016 0.014 0.012 0.010 0.008 0 20,000 40,000 60,000 80,000 100,000 (a) D Δp ______ V2 ρ 4 2 10-2 8 6 4 × 10-3 103 2 4 6 8 104 2 4 6 8 105 D Δp ______ V2 ρ (b) _____ VD μ ρ Re = _____ VD μ ρ Re = F I G U R E E7.4 JWCL068_ch07_332-382.qxd 9/23/08 10:46 AM Page 353 way. It may also be possible to determine a suitable empirical equation relating the three pi terms. However, as the number of pi terms continues to increase, corresponding to an increase in the general complexity of the problem of interest, both the graphical presentation and the determination of a suitable empirical equation become intractable. For these more complicated problems, it is often more feasible to use models to predict specific characteristics of the system rather than to try to develop general correlations

As was discussed in the previous section, one of the common difficulties with models is related to the Reynolds number similarity requirement which establishes the model velocity as (7.20) or (7.21) where is the ratio of kinematic viscosities. If the same fluid is used for model and prototype so that then and, therefore, the required model velocity will be higher than the prototype velocity for greater than 1. Since this ratio is often relatively large, the required value of may be large. For example, for a length scale, and a prototype velocity of 50 mph, the required model velocity is 500 mph. This is a value that is unreasonably high to achieve with liquids, and for gas flows this would be in the range where compressibility would be important in the model 1but not in the prototype

At the top of Table 7.1 is a list of variables that commonly arise in fluid mechanics problems. The list is obviously not exhaustive but does indicate a broad range of variables likely to be found in a typical problem. Fortunately, not all of these variables would be encountered in all problems. However, when combinations of these variables are present, it is standard practice to combine them into some of the common dimensionless groups 1pi terms2 given in Table 7.1. These combinations appear so frequently that special names are associated with them, as indicated in the table. It is also often possible to provide a physical interpretation to the dimensionless groups which can be helpful in assessing their influence in a particular application. For example, the Froude number is an index of the ratio of the force due to the acceleration of a fluid particle to the force due to gravity 1weight2. This can be demonstrated by considering a fluid particle moving along a streamline 1Fig. 7.32. The magnitude of the component of inertia force along the streamline can be expressed as where is the magnitude of the acceleration along the streamline for a particle having a mass m. From our study of particle motion along a curved path 1see Section 3.12 we know that

Cauchy Number and Mach Number. The Cauchy number and the Mach number are important dimensionless groups in problems in which fluid compressibility is a significant factor. Since the speed of sound, c, in a fluid is equal to 1see Section 1.7.32, it follows that and the square of the Mach number is equal to the Cauchy number. Thus, either number 1but not both2 may be used in problems in which fluid compressibility is important. Both numbers can be interpreted as representing an index of the ratio of inertial forces to compressibility forces. When the Mach number is relatively small 1say, less than 0.32, the inertial forces induced by the fluid motion are not sufficiently large to cause a significant change in the fluid density, and in this case the compressibility of the fluid can be neglected. The Mach number is the more commonly used parameter in compressible flow problems, particularly in the fields of gas dynamics and aerodynamics. The Cauchy number is named in honor of Augustin Louis de Cauchy 11789-18572, a French engineer, mathematician, and hydrodynamicist. The Mach number is named in honor of Ernst Mach 11838-19162, an Austrian physicist and philosopher.

Common examples of this type of flow include pipe flow and flow through valves, fittings, and metering devices. Although the conduit cross sections are often circular, they could have other shapes as well and may contain expansions or contractions. Since there are no fluid interfaces or free surfaces, the dominant forces are inertial and viscous so that the Reynolds number is an important similarity parameter. For low Mach numbers compressibility effects are usually negligible for both the flow of liquids or gases. For this class of problems, geometric similarity between model and prototype must be maintained. Generally the geometric characteristics can be described by a series of length terms, and where is some particular length dimension for the system. Such a series of length terms leads to a set of pi terms of the form where and so on. In addition to the basic geometry of the system, the roughness of the internal surface in contact with the fluid may be important. If the average height of surface roughness elements is defined as then the pi term representing roughness will be This parameter indicates that for complete geometric similarity, surface roughness would also have to be scaled. Note that this implies that for length scales less than 1, the model surfaces should be smoother than those in the prototype since To further complicate matters, the pattern of roughness elements in model and prototype would have to be similar. These are conditions that are virtually impossible to satisfy exactly. Fortunately, in some problems the surface roughness plays em l/e. e, e/. i 1, 2, . . . , ßi /i / /1, /2, /3, . . . , /i , , // 1Ma 6 0.32, 7.9 Some Typical Model Studies Geometric and Reynolds number similarity is usually required for models involving flow through closed conduits. JWCL068_ch07_332-382.qxd 9/23/08 10:46 AM Page 360 a minor role and can be neglected. However, in other problems 1such as turbulent flow through pipes2 roughness can be very important. It follows from this discussion that for flow in closed conduits at low Mach numbers, any dependent pi term 1the one that contains the particular variable of interest, such as pressure drop2 can be expressed as (7.16) This is a general formulation for this type of problem. The first two pi terms of the right side of Eq. 7.16 lead to the requirement of geometric similarity so that or This result indicates that the investigator is free to choose a length scale, but once this scale is selected, all other pertinent lengths must be scaled in the same ratio. The additional similarity requirement arises from the equality of Reynolds numbers From this condition the velocity scale is established so that (7.17) and the actual value of the velocity scale depends on the viscosity and density scales, as well as the length scale. Different fluids can be used in model and prototype. However, if the same fluid is used 1with and 2, then Thus, which indicates that the fluid velocity in the model will be larger than that in the prototype for any length scale less than 1. Since length scales are typically much less than unity, Reynolds number similarity may be difficult to achieve because of the large model velocities required. With these similarity requirements satisfied, it follows that the dependent pi term will be equal in model and prototype. For example, if the dependent variable of interest is the pressure differential,3 between two points along a closed conduit, then the dependent pi term could be expressed as The prototype pressure drop would then be obtained from the relationship so that from a measured pressure differential in the model, the corresponding pressure differential for the prototype could be predicted. Note that in general ¢p ¢p

Distorted Models. Although the general idea behind establishing similarity requirements for models is straightforward 1we simply equate pi terms2, it is not always possible to satisfy all the known requirements. If one or more of the similarity requirements are not met, for example, if then it follows that the prediction equation is not true; that is, Models for which one or more of the similarity requirements are not satisfied are called distorted models. Distorted models are rather commonplace, and they can arise for a variety of reasons. For example, perhaps a suitable fluid cannot be found for the model. The classic example of a distorted model occurs in the study of open channel or free-surface flows. Typically, in these problems both the Reynolds number, and the Froude number, are involved. Froude number similarity requires If the model and prototype are operated in the same gravitational field, then the required velocity scale is Reynolds number similarity requires and the velocity scale is Since the velocity scale must be equal to the square root of the length scale, it follows that (7.15) where the ratio is the kinematic viscosity, Although in principle it may be possible to satisfy this design condition, it may be quite difficult, if not impossible, to find a suitable model fluid, particularly for small length scales. For problems involving rivers, spillways, and harbors, for which the prototype fluid is water, the models are also relatively large so that the only practical model fluid is water. However, in this case 1with the kinematic viscosity scale equal to unity2 Eq. 7.15 will not be satisfied, and a distorted model will result. Generally, hydraulic models of this type are distorted and are designed on the basis of the Froude number, with the Reynolds number different in model and prototype

Distorted models can be successfully used, but the interpretation of results obtained with this type of model is obviously more difficult than the interpretation of results obtained with true models for which all similarity requirements are met. There are no general rules for handling distorted mr n. mmrm mr nm n 1l/2 32 Vm V mm m r rm / /m rmVm/m mm rV/ m Vm V B /m / 1l/ Vm 1gm/m V 1g/ rV/m, V1g/, ß ß1 Z ß1m. ß2m Z ß2, 1 ß1m 7.8 Modeling and Similitude 359 Models for which one or more similarity requirements are not satisfied are called distorted models. V7.12 Distorted river model V7.11 Model of fish hatchery pond JWCL068_ch07_332-382.qxd 9/23/08 10:46 AM Page 359 models, and essentially each problem must be considered on its own merits. The success of using distorted models depends to a large extent on the skill and experience of the investigator responsible for the design of the model and in the interpretation of experimental data obtained from the model. Distorted models are widely used, and additional information can be found in the references at the end of the chapter. References 14 and 15 contain detailed discussions of several practical examples of distorted fluid flow and hydraulic models

Euler Number. The Euler number can be interpreted as a measure of the ratio of pressure forces to inertial forces, where p is some characteristic pressure in the flow field. Very often the Euler number is written in terms of a pressure difference, so that Also, this combination expressed as is called the pressure coefficient. Some form of the Euler number would normally be used in problems in which pressure or the pressure difference between two points is an important variable. The Euler number is named in honor of Leonhard Euler 11707-17832, a famous Swiss mathematician who pioneered work on the relationship between pressure and flow. For problems in which cavitation is of concern, the dimensionless group is commonly used, where is the vapor pressure and is some reference pressure. Although this dimensionless group has the same form as the Euler number, it is generally referred to as the cavitation numb

Flows in canals, rivers, spillways, and stilling basins, as well as flow around ships, are all examples of flow phenomena involving a free surface. For this class of problems, both gravitational and inertial forces are important and, therefore, the Froude number becomes an important similarity parameter. Also, since there is a free surface with a liquid-air interface, forces due to surface tension may be significant, and the Weber number becomes another similarity parameter that needs to be considered along with the Reynolds number. Geometric variables will obviously still be important. Thus a general formulation for problems involving flow with a free surface can be expressed as (7.24) As discussed previously, is some characteristic length of the system, represents other pertinent lengths, and is the relative roughness of the various surfaces. Since gravity is the driving force in these problems, Froude number similarity is definitely required so that The model and prototype are expected to operate in the same gravitational field and therefore it follows that (7.25) Thus, when models are designed on the basis of Froude number similarity, the velocity scale is determined by the square root of the length scale. As is discussed in Section 7.8.3, to simultaneously have Reynolds and Froude number similarity it is necessary that the kinematic viscosity scale be related to the length scale as (7.26) The working fluid for the prototype is normally either freshwater or seawater and the length scale is small. Under these circumstances it is virtually impossible to satisfy Eq. 7.26, so models involving free-surface flows are usually distorted. The problem is further complicated if an attempt is made to model surface tension effects, since this requires the equality of Weber numbers, which leads to the condition (7.27) for the kinematic surface tension It is again evident that the same fluid cannot be used in model and prototype if we are to have similitude with respect to surface tension effects for Fortunately, in many problems involving free-surface flows, both surface tension and viscous effects are small and consequently strict adherence to Weber and Reynolds number similarity is not required. Certainly, surface tension is not important in large hydraulic structures and rivers. Our only concern would be if in a model the depths were reduced to the point where surface tension becomes an important factor, whereas it is not in the prototype. This is of particular importance in the design of river models, since the length scales are typically small 1so that the width of the model is reasonable2, but with a small length scale the required model depth may be very small. To overcome this problem, different horizontal and vertical length scales are often used for river l/ 1. 1sr2. smrm sr 1l/2 2 nm n 1l/2 32 Vm V B /m / 1l/ 1gm g2, Vm 2gm/m V 2g/ e/ / /i Dependent pi term f a /i / , e / , rV/ m , V 2g/ , rV2 / s b c cm n nm 7.9 Some Typical Model Studies 367 Froude number similarity is usually required for models involving freesurface flows. V7.17 River flow model V7.18 Boat model JWCL068_ch07_332-382.qxd 9/23/08 10:46 AM Page 367 models. Although this approach eliminates surface tension effects in the model, it introduces geometric distortion that must be accounted for empirically, usually by increasing the model surface roughness. It is important in these circumstances that verification tests with the model be performed 1if possible2 in which model data are compared with available prototype river flow data. Model roughness can be adjusted to give satisfactory agreement between model and prototype, and then the model subsequently used to predict the effect of proposed changes on river characteristics 1such as velocity patterns or surface elevations2

For any given problem it is obviously desirable to reduce the number of pi terms to a minimum and, therefore, we wish to reduce the number of variables to a minimum; that is, we certainly do not want to include extraneous variables. It is also important to know how many reference dimensions are required to describe the variables. As we have seen in the preceding examples, F, L, and T appear to be a convenient set of basic dimensions for characterizing fluid-mechanical quantities. There is, however, really nothing "fundamental" about this set, and as previously noted M, L, and T would also be suitable. Actually any set of measurable quantities could be used as basic dimensions provided that the selected combination can be used to describe all secondary quantities. However, the use of FLT or MLT as basic dimensions is the simplest, and these dimensions can be used to describe fluid-mechanical phenomena. Of course, in some problems only one or two of these are required. In addition, we occasionally find that the number of reference dimensions needed to describe all variables is smaller than the number of basic dimensions. This point is illustrated in Example 7.2. Interesting discussions, both practical and philosophical, relative to the concept of basic dimensions can be found in the books by Huntley 1Ref. 42 and by Isaacson and Isaacson 1Ref. 122.

For large hydraulic structures, such as dam spillways, the Reynolds numbers are large so that viscous forces are small in comparison to the forces due to gravity and inertia. In this case, Reynolds number similarity is not maintained and models are designed on the basis of Froude number similarity. Care must be taken to ensure that the model Reynolds numbers are also large, but they are not required to be equal to those of the prototype. This type of hydraulic model is usually made as large as possible so that the Reynolds number will be large. A spillway model is shown in Fig. 7.8. Also, for relatively large models the geometric features of the prototype can be accurately scaled, as well as surface roughness. Note that which indicates that the model surfaces must be smoother than the corresponding prototype surfaces for l/ 6 1.

Fortunately, in many situations the flow characteristics are not strongly influenced by the Reynolds number over the operating range of interest. In these cases we can avoid the rather stringent similarity requirement of matching Reynolds numbers. To illustrate this point, consider the variation in the drag coefficient with the Reynolds number for a smooth sphere of diameter d placed in a uniform stream with approach velocity, V. Some typical data are shown in Fig. 7.7. We observe that for Reynolds numbers between approximately and the drag coefficient is relatively constant and does not strongly depend on the specific value of the Reynolds number. Thus, exact Reynolds number similarity is not required in this range. For other geometric shapes we would typically find that for high Reynolds numbers, inertial forces are dominant 1rather than viscous forces2, and the drag is essentially independent of the Reynolds number.

Froude Number. The Froude number is distinguished from the other dimensionless groups in Table 7.1 in that it contains the acceleration of gravity, g. The acceleration of gravity becomes an important variable in a fluid dynamics problem in which the fluid weight is an important force. As discussed, the Froude number is a measure of the ratio of the inertia force on an element of fluid to the weight of the element. It will generally be important in problems involving flows with free surfaces since gravity principally affects this type of flow. Typical problems would include the study of the flow of water around ships 1with the resulting wave action2 or flow through rivers or open conduits. The Froude number is named in honor of William Froude 11810-18792, a British civil engineer, mathematician, and naval architect who pioneered the use of towing tanks for the study of ship design. It is to be noted that the Froude number is also commonly defined as the square of the Froude number listed in Table 7.1.

If a given phenomenon can be described with two pi terms such that the functional relationship among the variables can then be determined by varying and measuring the corresponding values of For this case the results can be conveniently presented in graphical form by plotting versus as is illustrated in Fig. 7.4. It should be emphasized that the curve shown in Fig. 7.4 would be a "universal" one for the particular phenomenon studied. This means that if the variables and the resulting dimensional analysis are correct, then there is only a single relationship between and as illustrated in Fig. 7.4. However, since this is an empirical relationship, we can only say that it is valid over the range of covered by the experiments. It would be unwise to extrapolate beyond this range, since as illustrated with the dashed lines in the figure, the nature of the phenomenon could dramatically change as the range of is extended. In addition to presenting the data graphically, it may be possible 1and desirable2 to obtain an empirical equation relating and by using a standard curve-fitting technique.

In the preceding sections of this chapter, dimensional analysis has been used to obtain similarity laws. This is a simple, straightforward approach to modeling, which is widely used. The use of dimensional analysis requires only a knowledge of the variables that influence the phenomenon of interest. Although the simplicity of this approach is attractive, it must be recognized that omission of one or more important variables may lead to serious errors in the model design. An alternative approach is available if the equations 1usually differential equations2 governing the phenomenon are known. In this situation similarity laws can be developed from the governing equations, even though it may not be possible to obtain analytic solutions to the equations. To illustrate the procedure, consider the flow of an incompressible Newtonian fluid. For simplicity we will restrict our attention to two-dimensional flow, although the results are applicable to the general three-dimensional case. From Chapter 6 we know that the governing equations are the continuity equation (7.28) and the Navier-Stokes equations (7.29) (7.30) where the y axis is vertical, so that the gravitational body force, only appears in the "y equation." To continue the mathematical description of the problem, boundary conditions are required. For example, velocities on all boundaries may be specified; that is, and at all boundary points and In some types of problems it may be necessary to specify the pressure over some part of the boundary. For time-dependent problems, initial conditions would also have to be provided, which means that the values of all dependent variables would be given at some time 1usually taken at 2. Once the governing equations, including boundary and initial conditions, are known, we are ready to proceed to develop similarity requirements. The next step is to define a new set of t 0 y yB x x . B u uB v vB rg, r a 0v 0t u 0v 0x v 0v 0y b 0p 0y rg m a 02 v 0x2 02 v 0y2 b r a 0u 0t u 0u 0x v 0u 0y b 0p 0x m a 02 u 0x2 02 u 02 y b 0u 0x 0v 0y 0 7.10 Similitude Based on Governing Differential Equations Similarity laws can be directly developed from the equations governing the phenomenon of interest. JWCL068_ch07_332-382.qxd 9/23/08 10:47 AM Page 370 variables that are dimensionless. To do this we select a reference quantity for each type of variable. In this problem the variables are and t so we will need a reference velocity, V, a reference pressure, a reference length, and a reference time, These reference quantities should be parameters that appear in the problem. For example, may be a characteristic length of a body immersed in a fluid or the width of a channel through which a fluid is flowing. The velocity, V, may be the free-stream velocity or the inlet velocity. The new dimensionless 1starred2 variables can be expressed as as shown in the figure in the margin. The governing equations can now be rewritten in terms of these new variables. For example, and The other terms that appear in the equations can be expressed in a similar fashion. Thus, in terms of the new variables the governing equations become (7.31) and (7.32) (7.33) The terms appearing in brackets contain the reference quantities and can be interpreted as indices of the various forces 1per unit volume2 that are involved. Thus, as is indicated in Eq. 7.33, inertia 1local2 force, inertia 1convective2 force, pressure force, gravitational force, and viscous force. As the final step in the nondimensionalization process, we will divide each term in Eqs. 7.32 and 7.33 by one of the bracketed quantities. Although any one of these quantities could be used, it is conventional to divide by the bracketed quantity which is the index of the convective inertia force. The final nondimensional form then becomes (7.34) (7.35) We see that bracketed terms are the standard dimensionless groups 1or their reciprocals2 which were developed from dimensional analysis; that is, is a form of the Strouhal number

It is clear from the preceding section that the ratio of like quantities for the model and prototype naturally arises from the similarity requirements. For example, if in a given problem there are two length variables and the resulting similarity requirement based on a pi term obtained from these two variables is so that We define the ratio or as the length scale. For true models there will be only one length scale, and all lengths are fixed in accordance with this scale. There are, however, other scales such as the velocity scale, density scale, viscosity scale, and so on. In fact, we can define a scale for each of the variables in the problem. Thus, it is actually meaningless to talk about a "scale" of a model without specifying which scale. We will designate the length scale as and other scales as and so on, where the subscript indicates the particular scale. Also, we will take the ratio of the model value to the prototype value as the scale 1rather than the inverse2. Length scales are often specified, for example, as 1 : 10 or as a scale model. The meaning of this specification is that the model is one-tenth the size of the prototype, and the tacit assumption is that all relevant lengths are scaled accordingly so the model is geometrically similar to the prototype

Models are widely used in fluid mechanics. Major engineering projects involving structures, aircraft, ships, rivers, harbors, dams, air and water pollution, and so on, frequently involve the use of models. Although the term "model" is used in many different contexts, the "engineering model" generally conforms to the following definition. 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 physical system for which the predictions are to be made is called the prototype. Although mathematical or computer models may also conform to this definition, our interest will be in physical models, that is, models that resemble the prototype but are generally of a different size, may involve different fluids, and often operate under different conditions 1pressures, velocities, etc.2. As shown by the figure in the margin, usually a model is smaller than the prototype. Therefore, it is more easily handled in the laboratory and less expensive to construct and operate than a large prototype (it should be noted that variables or pi terms without a subscript will refer to the prototype, whereas the subscript m will be used to designate the model variables or pi terms). Occasionally, if the prototype is very small, it may be advantageous to have a model that is larger than the prototype so that it can be more easily studied. For example, large models have been used to study the motion of red blood cells, which are approximately in diameter. With the successful development of a valid model, it is possible to predict the behavior of the prototype under a certain set of conditions. We may also wish to examine a priori the effect of possible design changes that are proposed for a hydraulic structure or fluid-flow system. There is, of course, an inherent danger in the use of models in that predictions can be made that are in error and the error not detected until the prototype is found not to perform as predicted. It is, therefore, imperative that the model be properly designed and tested and that the results be interpreted correctly. In the following sections we will develop the procedures for designing models so that the model and prototype will behave in a similar fashion.

Models have been widely used to study the flow characteristics associated with bodies that are completely immersed in a moving fluid. Examples include flow around aircraft, automobiles, golf balls, and buildings. 1These types of models are usually tested in wind tunnels as is illustrated in Fig. 7.6.2 Modeling laws for these problems are similar to those described in the preceding section; that is, geometric and Reynolds number similarity is required. Since there are no fluid interfaces, surface tension 1and therefore the Weber number2 is not important. Also, gravity will not affect the flow patterns, so the Froude number need not be considered. The Mach number will be important for high-speed flows in which compressibility becomes an important factor, but for incompressible fluids 1such as liquids or for gases at relatively low speeds2 the Mach number can be omitted as a similarity requirement. In this case, a general formulation for these problems is (7.18) where is some characteristic length of the system and represents other pertinent lengths, is the relative roughness of the surface 1or surfaces2, and is the Reynolds number. Frequently, the dependent variable of interest for this type of problem is the drag, developed on the body, and in this situation the dependent pi term would usually be expressed in the form of a drag coefficient, where The numerical factor, is arbitrary but commonly included, and is usually taken as some representative area of the object. Thus, drag studies can be undertaken with the formulation (7.19) d 1 2rV2 /2 CD f a /i / , e / , rV/ m b / 1 2 2, CD d 1 2rV2 /2 CD, d, e/ rV/m / /i Dependent pi term f a /i / , e / , rV/ m b pr 1pr pv2 1 2rV2 pv, 7.9 Some Typical Model Studies 363 Geometric and Reynolds number similarity is usually required for models involving flow around bodies. V7.13 Wind engineering models F I G U R E 7.6 Model of the National Bank of Commerce, San Antonio, Texas, for measurement of peak, rms, and mean pressure distributions. The model is located in a long-test-section, meteorological wind tunnel. (Photograph courtesy of Cermak Peterka Petersen, Inc.) JWCL068_ch07_332-382.qxd 9/23/08 10:46 AM Page 363 It is clear from Eq. 7.19 that geometric similarity as well as Reynolds number similarity must be maintained. If these conditions are met, then or Measurements of model drag, can then be used to predict the corresponding drag, on the prototype from this relationship.

Note that we end up with the correct number of pi terms as determined from Step 3. At this point stop and check to make sure the pi terms are actually dimensionless 1Step 72. We will check using both FLT and MLT dimensions. Thus, or alternatively, Finally 1Step 82, we can express the result of the dimensional analysis as This result indicates that this problem can be studied in terms of these two pi terms, rather than the original five variables we started with. The eight steps carried out to obtain this result are summarized by the figure in the margin. Dimensional analysis will not provide the form of the function This can only be obtained from a suitable set of experiments. If desired, the pi terms can be rearranged; that is, the reciprocal of could be used, and of course the order in which we write the variables can be changed. Thus, for example, could be expressed as and the relationship between and as as shown by the figure in the margin. This is the form we previously used in our initial discussion of this problem 1Eq. 7.22. The dimensionless product is a very famous one in fluid mechanics—the Reynolds number. This number has been briefly alluded to in Chapters 1 and 6 and will be further discussed in Section 7.6. To summarize, the steps to be followed in performing a dimensional analysis using the method of repeating variables are as follows: Step 1 List all the variables that are involved in the problem. Step 2 Express each of the variables in terms of basic dimensions. Step 3 Determine the required number of pi terms. Step 4 Select a number of repeating variables, where the number required is equal to the number of reference dimensions 1usually the same as the number of basic dimensions2. Step 5 Form a pi term by multiplying one of the nonrepeating variables by the product of repeating variables each raised to an exponent that will make the combination dimensionless. Step 6 Repeat Step 5 for each of the remaining nonrepeating variables. Step 7 Check all the resulting pi terms to make sure they are dimensionless and independent. Step 8 Express the final form as a relationship among the pi terms and think about what it means.

One final note with regard to Fig. 7.7 concerns the interpretation of experimental data when plotting pi terms. For example, if , , and d remain constant, then an increase in Re comes from an increase in V. Intuitively, it would seem in general that if V increases, the drag would increase. However, as shown in the figure, the drag coefficient generally decreases with increasing Re. When interpreting data, one needs to be aware if the variables are nondimensional. In this case, the physical drag force is proportional to the drag coefficient times the velocity squared. Thus, as shown by the figure in the margin, the drag force does, as expected, increase with increasing velocity. The exception occurs in the Reynolds number range where the drag coefficient decreases dramatically with increasing Reynolds number (see Fig. 7.7). This phenomena is discussed in Section 9.3. For problems involving high velocities in which the Mach number is greater than about 0.3, the influence of compressibility, and therefore the Mach number 1or Cauchy number2, becomes significant. In this case complete similarity requires not only geometric and Reynolds number similarity but also Mach number similarity so that (7.22) This similarity requirement, when combined with that for Reynolds number similarity 1Eq. 7.212, yields (7.23) c cm n nm /m / Vm cm V c 2 105 6 Re 6 4 105 mr 3 105 . 366 Chapter 7 ■ Dimensional Analysis, Similitude, and Modeling At high Reynolds numbers the drag is often essentially independent of the Reynolds number. Drag coefficient, CD 0.06 2 4 68 2 4 68 2 4 68 2 4 68 2 4 68 2 4 68 2 4 68 106 105 104 103 102 101 100 10-1 Reynolds number, Re = Vd ____ ρ μ 0.08 0.1 0.2 0.4 0.6 0.8 1 2 4 6 8 10 20 40 60 80 100 200 400 F I G U R E 7.7 The effect of Reynolds number on the drag coefficient, CD, for a smooth sphere with where A is the projected area of sphere, (Data from Ref. 16, used by permission.) Pd2 CD d 4. 1 2ARV2 , V JWCL068_ch07_332-382.qxd 9/23/08 10:46 AM Page 366 Clearly the same fluid with and cannot be used in model and prototype unless the length scale is unity 1which means that we are running tests on the prototype2. In high-speed aerodynamics the prototype fluid is usually air, and it is difficult to satisfy Eq. 7.23 for reasonable length scales. Thus, models involving high-speed flows are often distorted with respect to Reynolds number similarity, but Mach number similarity is maintained.

One of the most important uses of dimensional analysis is as an aid in the efficient handling, interpretation, and correlation of experimental data. Since the field of fluid mechanics relies heavily on empirical data, it is not surprising that dimensional analysis is such an important tool in this field. As noted previously, a dimensional analysis cannot provide a complete answer to any given problem, since the analysis only provides the dimensionless groups describing the phenomenon, and not the specific relationship among the groups. To determine this relationship, suitable experimental data must be obtained. The degree of difficulty involved in this process depends on the number of pi terms, and the nature of the experiments 1How hard is it to obtain 7.7 Correlation of Experimental Data JWCL068_ch07_332-382.qxd 9/23/08 10:46 AM Page 350 the measurements?2. The simplest problems are obviously those involving the fewest pi terms, and the following sections indicate how the complexity of the analysis increases with the increasing number of pi terms.

Reynolds Number. The Reynolds number is undoubtedly the most famous dimensionless parameter in fluid mechanics. It is named in honor of Osborne Reynolds 11842-19122, a British engineer who first demonstrated that this combination of variables could be used as a criterion to distinguish between laminar and turbulent flow. In most fluid flow problems there will be a characteristic length, and a velocity, V, as well as the fluid properties of density, and viscosity, which are relevant variables in the problem. Thus, with these variables the Reynolds number arises naturally from the dimensional analysis. The Reynolds number is a measure of the ratio of the inertia force on an element of fluid to the viscous force on an element. When these two types of forces are important in a given problem, the Reynolds number will play an important role. However, if the Reynolds number is very small this is an indication that the viscous forces are dominant in the problem, and it may be possible to neglect the inertial effects; that is, the density of the fluid will not be an important variable. Flows at very small Reynolds numbers are commonly referred to as "creeping flows" as discussed in Section 6.10. Conversely, for large Reynolds number flows, viscous effects are small relative to inertial effects and for these cases it may be possible to neglect the effect of viscosity and consider the problem as one involving a "nonviscous" fluid. This type of problem is considered in detail in Sections 6.4 through 6.7. An example of the importance of the Reynolds number in determining the flow physics is shown in the figure in the margin for flow past a circular cylinder at two different Re values. This flow is discussed further in Chapter 9.

Several methods can be used to form the dimensionless products, or pi terms, that arise in a dimensional analysis. Essentially we are looking for a method that will allow us to systematically form the pi terms so that we are sure that they are dimensionless and independent, and that we have the right number. The method we will describe in detail in this section is called the method of repeating variables. It will be helpful to break the repeating variable method down into a series of distinct steps that can be followed for any given problem. With a little practice you will be able to readily complete a dimensional analysis for your problem. Step 1 List all the variables that are involved in the problem. This step is the most difficult one and it is, of course, vitally important that all pertinent variables be included. Otherwise the dimensional analysis will not be correct! We are using the term "variable" to include any quantity, including dimensional and nondimensional constants, which play a role in the phenomenon under investigation. All such quantities should be included in the list of "variables" to be considered for the dimensional analysis. The determination of the variables must be accomplished by the experimenter's knowledge of the problem and the physical laws that govern the phenomenon. Typically the variables will include those that are necessary to describe the geometry of the system 1such as a pipe diameter2, to define any fluid properties 1such as a fluid viscosity2, and to indicate external effects that influence the system 1such as a driving pressure drop per unit length2. These general classes of variables are intended as broad categories that should be helpful in identifying variables. It is likely, however, that there will be variables that do not fit easily into one of these categories, and each problem needs to be carefully analyzed. Since we wish to keep the number of variables to a minimum, so that we can minimize the amount of laboratory work, it is important that all variables be independent. For example, if in a certain problem the cross-sectional area of a pipe is an important variable, either the area or the pipe diameter could be used, but not both, since they are obviously not independent. Similarly, if both fluid density, and specific weight, are important variables, we could list and or and g 1acceleration of gravity2, or and g. However, it would be incorrect to use all three since that is, and g are not independent. Note that although g would normally be constant in a given experiment, that fact is irrelevant as far as a dimensional analysis is concerned. Step 2 Express each of the variables in terms of basic dimensions. For the typical fluid mechanics problem the basic dimensions will be either M, L, and T or F, L, and T. Dimensionally these two sets are related through Newton's second law so that For example, Thus, either set can be used. The basic dimensions for typical variables found in fluid mechanics problems are listed in Table 1.1 in Chapter 1. Step 3 Determine the required number of pi terms. This can be accomplished by means of the Buckingham pi theorem, which indicates that the number of pi terms is equal to where k is the number of variables in the problem 1which is determined from Step 12 and r is the number of reference dimensions required to describe these variables 1which is determined from Step 22. The reference dimensions usually correspond to the basic dimensions and can be determined by an inspection of the dimensions of the variables obtained in Step 2. As previously noted, there may be occasions 1usually rare2 in which the basic dimensions appear in combinations so that the number of reference dimensions is less than the number of basic dimensions. This possibility is illustrated in Example 7.2. Step 4 Select a number of repeating variables, where the number required is equal to the number of reference dimensions. Essentially what we are doing here is selecting from the original list of variables several of which can be combined with each of the remaining k r, r ML3 or r FL4 T 2 . F MLT 2 1F ma2 . g rg; r, g, r , rg g r, g, 7.3 Determination of Pi Terms A dimensional analysis can be performed using a series of distinct steps. JWCL068_ch07_332-382.qxd 9/23/08 10:46 AM Page 336 variables to form a pi term. All of the required reference dimensions must be included within the group of repeating variables, and each repeating variable must be dimensionally independent of the others 1i.e., the dimensions of one repeating variable cannot be reproduced by some combination of products of powers of the remaining repeating variables2. This means that the repeating variables cannot themselves be combined to form a dimensionless product. For any given problem we usually are interested in determining how one particular variable is influenced by the other variables. We would consider this variable to be the dependent variable, and we would want this to appear in only one pi term. Thus, do not choose the dependent variable as one of the repeating variables, since the repeating variables will generally appear in more than one pi term. Step 5 Form a pi term by multiplying one of the nonrepeating variables by the product of the repeating variables, each raised to an exponent that will make the combination dimensionless. Essentially each pi term will be of the form where is one of the nonrepeating variables; and are the repeating variables; and the exponents and are determined so that the combination is dimensionless. Step 6 Repeat Step 5 for each of the remaining nonrepeating variables. The resulting set of pi terms will correspond to the required number obtained from Step 3. If not, check your work—you have made a mistake! Step 7 Check all the resulting pi terms to make sure they are dimensionless. It is easy to make a mistake in forming the pi terms. However, this can be checked by simply substituting the dimensions of the variables into the pi terms to confirm that they are all dimensionless. One good way to do this is to express the variables in terms of M, L, and T if the basic dimensions F, L, and T were used initially, or vice versa, and then check to make sure the pi terms are dimensionless. Step 8 Express the final form as a relationship among the pi terms, and think about what it means. Typically the final form can be written as where would contain the dependent variable in the numerator. It should be emphasized that if you started out with the correct list of variables 1and the other steps were completed correctly2, then the relationship in terms of the pi terms can be used to describe the problem. You need only work with the pi terms—not with the individual variables. However, it should be clearly noted that this is as far as we can go with the dimensional analysis; that is, the actual functional relationship among the pi terms must be determined by experiment. To illustrate these various steps we will again consider the problem discussed earlier in this chapter which was concerned with the steady flow of an incompressible Newtonian fluid through a long, smooth-walled, horizontal circular pipe. We are interested in the pressure drop per unit length, along the pipe as illustrated by the figure in the margin. First (Step 1) we must list all of the pertinent variables that are involved based on the experimenter's knowledge of the problem. In this problem we assume that where D is the pipe diameter, and are the fluid density and viscosity, respectively, and V is the mean velocity. Next 1Step 22 we express all the variables in terms of basic dimensions. Using F, L, and T as basic dimensions it follows tha

Strouhal Number. The Strouhal number is a dimensionless parameter that is likely to be important in unsteady, oscillating flow problems in which the frequency of the oscillation is It represents a measure of the ratio of inertial forces due to the unsteadiness of the flow 1local acceleration2 to the inertial forces due to changes in velocity from point to point in the flow field 1convective acceleration2. This type of unsteady flow may develop when a fluid flows past a solid body 1such as a wire or cable2 placed in the moving stream. For example, in a certain Reynolds number range, a periodic flow will develop downstream from a cylinder placed in a moving fluid due to a regular pattern of vortices that are shed from the body. 1See the photograph at the beginning of this chapter and Fig. 9.21.2 This system of vortices, called a Kármán vortex trail [named after Theodor von Kármán 11881-19632, a famous fluid v. St v/ V Ma2 rV2 Ev Ca Ma V A r Ev c 1Ev r Ma V c Ca rV2 Ev pr pv 1 pr pv2 1 2 rV2 ¢p 1 2 rV2 Eu ¢prV 2 ¢p, . Eu p rV2 7.6 Common Dimensionless Groups in Fluid Mechanics 349 The Mach number is a commonly used dimensionless parameter in compressible flow problems. V7.5 Strouhal number JWCL068_ch07_332-382.qxd 9/23/08 10:46 AM Page 349 mechanician], creates an oscillating flow at a discrete frequency, such that the Strouhal number can be closely correlated with the Reynolds number. When the frequency is in the audible range, a sound can be heard and the bodies appear to "sing." In fact, the Strouhal number is named in honor of Vincenz Strouhal 11850-19222, who used this parameter in his study of "singing wires." The most dramatic evidence of this phenomenon occurred in 1940 with the collapse of the Tacoma Narrows bridge. The shedding frequency of the vortices coincided with the natural frequency of the bridge, thereby setting up a resonant condition that eventually led to the collapse of the bridge. There are, of course, other types of oscillating flows. For example, blood flow in arteries is periodic and can be analyzed by breaking up the periodic motion into a series of harmonic components 1Fourier series analysis2, with each component having a frequency that is a multiple of the fundamental frequency, 1the pulse rate2. Rather than use the Strouhal number in this type of problem, a dimensionless group formed by the product of St and Re is used; that is The square root of this dimensionless group is often referred to as the frequency parameter

The method of repeating variables for forming pi terms has been presented in Section 7.3. This method provides a step-by-step procedure that if executed properly will provide a correct and complete set of pi terms. Although this method is simple and straightforward, it is rather tedious, particularly for problems in which large numbers of variables are involved. Since the only restrictions placed on the pi terms are that they be 112 correct in number, 122 dimensionless, and 132 independent, it is possible to simply form the pi terms by inspection, without resorting to the more formal procedure. To illustrate this approach, we again consider the pressure drop per unit length along a smooth pipe. Regardless of the technique to be used, the starting point remains the same—determine the variables, which in this case are Next, the dimensions of the variables are listed: and subsequently the number of reference dimensions determined. The application of the pi theorem then tells us how many pi terms are required. In this problem, since there are five variables and three reference dimensions, two pi terms are needed. Thus, the required number of pi terms can be easily obtained. The determination of this number should always be done at the beginning of the analysis. Once the number of pi terms is known, we can form each pi term by inspection, simply making use of the fact that each pi term must be dimensionless. We will always let contain the dependent variable, which in this example is Since this variable has the dimensions we need to combine it with other variables so that a nondimensional product will result. One possibility is to first divide by so that The dependence on F has been eliminated, but is obviously not dimensionless. To eliminate the dependence on T, we can divide by so that a ¢p/ r b 1 V2 a L T 2 b 1 1LT 1 2 2 1 L 1cancels T2 V2 ¢p/r ¢p/ r 1FL3 2 1FL4 T2 2 L T2 1cancels F2 ¢p/ r FL3 ¢p , /. ß1 V LT 1 m FL2 T r FL4 T 2 D L ¢p/ FL3 ¢p/ f 1D, r, m, V2 7.5 Determination of Pi Terms by Inspection Pi terms can be formed by inspection by simply making use of the fact that each pi term must be dimensionless. JWCL068_ch07_332-382.qxd 9/23/08 10:46 AM Page 345 Finally, to make the combination dimensionless we multiply by D so that Thus, Next, we will form the second pi term by selecting the variable that was not used in which in this case is We simply combine with the other variables to make the combination dimensionless 1but do not use in since we want the dependent variable to appear only in 2. For example, divide by 1to eliminate F2, then by V 1to eliminate T2, and finally by D 1to eliminate L2. Thus, and, therefore, which is, of course, the same result we obtained by using the method of repeating variables. An additional concern, when one is forming pi terms by inspection, is to make certain that they are all independent. In the pipe flow example, contains which does not appear in , and therefore these two pi terms are obviously independent. In a more general case a pi term would not be independent of the others in a given problem if it can be formed by some combination of the others. For example, if can be formed by a combination of say and such as then is not an independent pi term. We can ensure that each pi term is independent of those preceding it by incorporating a new variable in each pi term. Although forming pi terms by inspection is essentially equivalent to the repeating variable method, it is less structured. With a little practice the pi terms can be readily formed by inspection, and this method offers an alternative to more formal procedures.

The preceding section provides a systematic approach for performing a dimensional analysis. Other methods could be used, although we think the method of repeating variables is the easiest for the beginning student to use. Pi terms can also be formed by inspection, as is discussed in Section 7.5. Regardless of the specific method used for the dimensional analysis, there are certain aspects of this important engineering tool that must seem a little baffling and mysterious to the student 1and sometimes to the experienced investigator as well2. In this section we will attempt to elaborate on some of the more subtle points that, based on our experience, can prove to be puzzling to students. 7.4.1 Selection of Variables One of the most important, and difficult, steps in applying dimensional analysis to any given problem is the selection of the variables that are involved. As noted previously, for convenience we will use the term variable to indicate any quantity involved, including dimensional and nondimensional constants. There is no simple procedure whereby the variables can be easily identified. Generally, one must rely on a good understanding of the phenomenon involved and the governing physical laws. If extraneous variables are included, then too many pi terms appear in the final solution, and it may be difficult, time consuming, and expensive to eliminate these experimentally. If important variables are omitted, then an incorrect result will be obtained; and again, this may prove to be costly and difficult to ascertain. It is, therefore, imperative that sufficient time and attention be given to this first step in which the variables are determined. Most engineering problems involve certain simplifying assumptions that have an influence on the variables to be considered. Usually we wish to keep the problem as simple as possible, perhaps even if some accuracy is sacrificed. A suitable balance between simplicity and accuracy is a desirable goal. How "accurate" the solution must be depends on the objective of the study; that is, we may be only concerned with general trends and, therefore, some variables that are thought to have only a minor influence in the problem may be neglected for simplicity. For most engineering problems 1including areas outside of fluid mechanics2, pertinent variables can be classified into three general groups—geometry, material properties, and external effects. Geometry. The geometric characteristics can usually be described by a series of lengths and angles. In most problems the geometry of the system plays an important role, and a sufficient number of geometric variables must be included to describe the system. These variables can usually be readily identified. Material Properties. Since the response of a system to applied external effects such as forces, pressures, and changes in temperature is dependent on the nature of the materials involved in the system, the material properties that relate the external effects and the responses must be included as variables. For example, for Newtonian fluids the viscosity of the fluid is the property that relates the applied forces to the rates of deformation of the fluid. As the material behavior becomes more complex, such as would be true for non-Newtonian fluids, the determination of material properties becomes difficult, and this class of variables can be troublesome to identify. 7.4 Some Additional Comments about Dimensional Analysis It is often helpful to classify variables into three groups— geometry, material properties, and external effects. JWCL068_ch07_332-382.qxd 9/23/08 10:46 AM Page 341 External Effects. This terminology is used to denote any variable that produces, or tends to produce, a change in the system. For example, in structural mechanics, forces 1either concentrated or distributed2 applied to a system tend to change its geometry, and such forces would need to be considered as pertinent variables. For fluid mechanics, variables in this class would be related to pressures, velocities, or gravity. The above general classes of variables are intended as broad categories that should be helpful in identifying variables. It is likely, however, that there will be important variables that do not fit easily into one of the above categories and each problem needs to be carefully analyzed. Since we wish to keep the number of variables to a minimum, it is important that all variables are independent. For example, if in a given problem we know that the moment of inertia of the area of a circular plate is an important variable, we could list either the moment of inertia or the plate diameter as the pertinent variable. However, it would be unnecessary to include both moment of inertia and diameter, assuming that the diameter enters the problem only through the moment of inertia. In more general terms, if we have a problem in which the variables are (7.3) and it is known that there is an additional relationship among some of the variables, for example, (7.4) then q is not required and can be omitted. Conversely, if it is known that the only way the variables u, w, . . . enter the problem is through the relationship expressed by Eq. 7.4, then the variables u, w, . . . can be replaced by the single variable q, therefore reducing the number of variables. In summary, the following points should be considered in the selection of variables: 1. Clearly define the problem. What is the main variable of interest 1the dependent variable2? 2. Consider the basic laws that govern the phenomenon. Even a crude theory that describes the essential aspects of the system may be helpful. 3. Start the variable selection process by grouping the variables into three broad classes: geometry, material properties, and external effects. 4. Consider other variables that may not fall into one of the above categories. For example, time will be an important variable if any of the variables are time dependent. 5. Be sure to include all quantities that enter the problem even though some of them may be held constant 1e.g., the acceleration of gravity, g2. For a dimensional analysis it is the dimensions of the quantities that are important—not specific values! 6. Make sure that all variables are independent. Look for relationships among subsets of the variables.

The theory of models can be readily developed by using the principles of dimensional analysis. It has been shown that any given problem can be described in terms of a set of pi terms as (7.7) In formulating this relationship, only a knowledge of the general nature of the physical phenomenon, and the variables involved, is required. Specific values for variables 1size of components, fluid properties, and so on2 are not needed to perform the dimensional analysis. Thus, Eq. 7.7 applies ß1 f1ß2, ß3, . . . , ßn2 8 mm 7.8 Modeling and Similitude V7.8 Model airplane Prototype V Model m Vm JWCL068_ch07_332-382.qxd 9/25/08 6:56 PM Page 354 to any system that is governed by the same variables. If Eq. 7.7 describes the behavior of a particular prototype, a similar relationship can be written for a model of this prototype; that is, (7.8) where the form of the function will be the same as long as the same phenomenon is involved in both the prototype and the model. Variables, or pi terms, without a subscript will refer to the prototype, whereas the subscript m will be used to designate the model variables or pi terms. The pi terms can be developed so that contains the variable that is to be predicted from observations made on the model. Therefore, if the model is designed and operated under the following conditions, (7.9) then with the presumption that the form of is the same for model and prototype, it follows that (7.10) Equation 7.10 is the desired prediction equation and indicates that the measured value of obtained with the model will be equal to the corresponding for the prototype as long as the other pi terms are equal. The conditions specified by Eqs. 7.9 provide the model design conditions, also called similarity requirements or modeling laws.

There are, unfortunately, problems involving flow with a free surface in which viscous, inertial, and gravitational forces are all important. The drag on a ship as it moves through water is due to the viscous shearing stresses that develop along its hull, as well as a pressure-induced component of drag caused by both the shape of the hull and wave action. The shear drag is a function of the Reynolds number, whereas the pressure drag is a function of the Froude number. Since both Reynolds number and Froude number similarity cannot be simultaneously achieved by using water as the model fluid 1which is the only practical fluid for ship models2, some technique other than a straightforward model test must be employed. One common approach is to measure the total drag on a small, geometrically similar model as it is towed through a model basin at Froude numbers matching those of the prototype. The shear drag on the model is calculated using analytical techniques of the type described in Chapter 9. This calculated value is then subtracted from the total drag to obtain pressure drag, and using Froude number scaling the pressure drag on the prototype can then be predicted. The experimentally determined value can then be combined with a calculated value of the shear drag 1again using analytical techniques2 to provide the desired total drag

To illustrate a typical fluid mechanics problem in which experimentation is required, consider 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 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 decide on the factors, or variables, that will have an effect on the pressure drop per unit length, We expect the list to include the pipe diameter, D, the fluid density, , fluid viscosity, , and the mean velocity, V, at which the fluid is flowing through the pipe. Thus, we can express this relationship as (7.1) 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 determine the nature of this function. To perform the experiments in a meaningful and systematic manner, it would be necessary to change one of the variables, such as the velocity, while holding all others constant, and measure the corresponding pressure drop. This series of tests would yield data that could be represented graphically as is illustrated in Fig. 7.1a. 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. 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 various 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 relationship between and V which would be valid for any similar pipe system? Fortunately, there is a much simpler approach to this problem that will eliminate the difficulties described above. In the following sections we will show that rather than working with the

Two additional points should be made with regard to modeling flows in closed conduits. First, for large Reynolds numbers, inertial forces are much larger than viscous forces, and in this case it may be possible to neglect viscous effects. The important practical consequence of this is that it would not be necessary to maintain Reynolds number similarity between model and prototype. However, both model and prototype would have to operate at large Reynolds numbers. Since we do not know, a priori, what a "large Reynolds number" is, the effect of Reynolds numbers would In some problems Reynolds number similarity may be relaxed. JWCL068_ch07_332-382.qxd 9/23/08 10:46 AM Page 362 have to be determined from the model. This could be accomplished by varying the model Reynolds number to determine the range 1if any2 over which the dependent pi term ceases to be affected by changes in Reynolds number. The second point relates to the possibility of cavitation in flow through closed conduits. For example, flow through the complex passages that may exist in valves may lead to local regions of high velocity 1and thus low pressure2, which can cause the fluid to cavitate. If the model is to be used to study cavitation phenomena, then the vapor pressure, becomes an important variable and an additional similarity requirement such as equality of the cavitation number is required, where is some reference pressure. The use of models to study cavitation is complicated, since it is not fully understood how vapor bubbles form and grow. The initiation of bubbles seems to be influenced by the microscopic particles that exist in most liquids, and how this aspect of the problem influences model studies is not clear. Additional details can be found in Ref. 17.

Validation of Model Design. Most model studies involve simplifying assumptions with regard to the variables to be considered. Although the number of assumptions is frequently less stringent than that required for mathematical models, they nevertheless introduce some uncertainty in the model design. It is, therefore, desirable to check the design experimentally whenever possible. In some situations the purpose of the model is to predict the effects of certain proposed changes in a given prototype, and in this instance some actual prototype data may be available. The model can be designed, constructed, and tested, and the model prediction can be compared JWCL068_ch07_332-382.qxd 9/23/08 10:46 AM Page 358 with these data. If the agreement is satisfactory, then the model can be changed in the desired manner, and the corresponding effect on the prototype can be predicted with increased confidence. Another useful and informative procedure is to run tests with a series of models of different sizes, where one of the models can be thought of as the prototype and the others as "models" of this prototype. With the models designed and operated on the basis of the proposed design, a necessary condition for the validity of the model design is that an accurate prediction be made between any pair of models, since one can always be considered as a model of the other. Although suitable agreement in validation tests of this type does not unequivocally indicate a correct model design 1e.g., the length scales between laboratory models may be significantly different than required for actual prototype prediction2, it is certainly true that if agreement between models cannot be achieved in these tests, there is no reason to expect that the same model design can be used to predict prototype behavior correctly.

We could also use M, L, and T as basic dimensions if desired—the final result will be the same. Note that for density, which is a mass per unit volume we have used the relationship to express the density in terms of F, L, and T. Do not mix the basic dimensions; that is, use either F, L, and T or M, L, and T. We can now apply the pi theorem to determine the required number of pi terms 1Step 32. An inspection of the dimensions of the variables from Step 2 reveals that all three basic dimensions are required to describe the variables. Since there are five variables 1do not forget to count the dependent variable, 2 and three required reference dimensions then according to the pi theorem there will be or two pi terms required. The repeating variables to be used to form the pi terms 1Step 42 need to be selected from the list and V. Remember, we do not want to use the dependent variable as one of the repeating variables. Since three reference dimensions are required, we will need to select three repeating variables. Generally, we would try to select as repeating variables those that are the simplest, dimensionally. For example, if one of the variables has the dimension of a length, choose it as one of the repeating variables. In this example we will use D, V, and as repeating variables. Note that these are dimensionally independent, since D is a length, V involves both length and time, and involves force, length, and time. This means that we cannot form a dimensionless product from this set. We are now ready to form the two pi terms 1Step 52. Typically, we would start with the dependent variable and combine it with the repeating variables to form the first pi term; that is, Since this combination is to be dimensionless, it follows that The exponents, a, b, and c must be determined such that the resulting exponent for each of the basic dimensions—F, L, and T—must be zero 1so that the resulting combination is dimensionless2. Thus, we can write The solution of this system of algebraic equations gives the desired values for a, b, and c. It follows that and, therefore, The process is now repeated for the remaining nonrepeating variables 1Step 62. In this example there is only one additional variable so that or and, therefore, Solving these equations simultaneously it follows that so that

Weber Number. The Weber number may be important in problems in which there is an interface between two fluids. In this situation the surface tension may play an important role in the phenomenon of interest. The Weber number can be thought of as an index of the inertial force to the surface tension force acting on a fluid element. Common examples of problems in which this parameter may be important include the flow of thin films of liquid, or the formation of droplets or bubbles. Clearly, not all problems involving flows with an interface will require the inclusion of surface tension. The flow of water in a river is not affected significantly by surface tension, since inertial and gravitational effects are dominant However, as discussed in a later section, for river models 1which may have small depths2 caution is required so that surface tension does not become important in the model, whereas it is not important in the actual river. The Weber number is named after Moritz Weber 11871-19512, a German professor of naval mechanics who was instrumental in formalizing the general use of common dimensionless groups as a basis for similitude studies.

With the foregoing similarity requirements satisfied, the prediction equation for the drag is or Thus, a measured drag on the model, must be multiplied by the ratio of the square of the plate widths, the ratio of the fluid densities, and the ratio of the square of the velocities to obtain the predicted value of the prototype drag, Generally, as is illustrated in this example, to achieve similarity between model and prototype behavior, all the corresponding pi terms must be equated between model and prototype. Usually, one or more of these pi terms will involve ratios of important lengths 1such as in the foregoing example2; that is, they are purely geometrical. Thus, when we equate the pi terms involving length ratios, we are requiring that there be complete geometric similarity between the model and prototype. This means that the model must be a scaled version of the prototype. Geometric scaling may extend to the finest features of the system, such as surface roughness, or small protuberances on a structure, since these kinds of geometric features may significantly influence the flow. Any deviation from complete geometric similarity for a model must be carefully considered. Sometimes complete geometric scaling may be difficult to achieve, particularly when dealing with surface roughness, since roughness is difficult to characterize and control.

on the ship. Ship models are widely used to study new designs, but the tests require extensive facilities 1see Fig. 7.92. It is clear from this brief discussion of various types of models involving free-surface flows that the design and use of such models requires considerable ingenuity, as well as a good understanding of the physical phenomena involved. This is generally true for most model studies. Modeling is both an art and a science. Motion picture producers make extensive use of model ships, fires, explosions, and the like. It is interesting to attempt to observe the flow differences between these distorted model flows and the real thing

original list of variables, as described in Eq. 7.1, we can collect these into two nondimensional combinations of variables 1called dimensionless products or dimensionless groups2 so that (7.2) 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 and determining the corresponding value of 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 1which would certainly make an impression on your boss2. The basis for this simplification lies in a consideration of the dimensions of the variables 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, Alternatively, we could use force, F, L, and T as basic dimensions, since from Newton's second law

the Euler number, the reciprocal of the square of the Froude number, and the reciprocal of the Reynolds number. From this analysis it is now clear how each of the dimensionless groups can be interpreted as the ratio of two forces, and how these groups arise naturally from the governing equations. Although we really have not helped ourselves with regard to obtaining an analytical solution to these equations 1they are still complicated and not amenable to an analytical solution2, the dimensionless forms of the equations, Eqs. 7.31, 7.34, and 7.35, can be used to establish similarity requirements. From these equations it follows that if two systems are governed by these equations, then the solutions will be the same if the four parameters and are equal for the two systems. The two systems will be dynamically similar. Of course, boundary and initial conditions expressed in dimensionless form must also be equal for the two systems, and this will require complete geometric similarity. These are the same similarity requirements that would be determined by a dimensional analysis if the same variables were considered. However, the advantage of working with the governing equations is that the variables appear naturally in the equations, and we do not have to worry about omitting an important one, provided the governing equations are correctly specified. We can thus use this method to deduce the conditions under which two solutions will be similar even though one of the solutions will most likely be obtained experimentally. In the foregoing analysis we have considered a general case in which the flow may be unsteady, and both the actual pressure level, and the effect of gravity are important. A reduction in the number of similarity requirements can be achieved if one or more of these conditions is removed. For example, if the flow is steady the dimensionless group, can be eliminated. The actual pressure level will only be of importance if we are concerned with cavitation. If not, the flow patterns and the pressure differences will not depend on the pressure level. In this case, can be taken as and the Euler number can be eliminated as a similarity requirement. However, if we are concerned about cavitation 1which will occur in the flow field if the pressure at certain points reaches the vapor pressure, 2, then the actual pressure level is important. Usually, in this case, the characteristic pressure, is defined relative to the vapor pressure such that where is some reference pressure within the flow field. With defined in this manner, the similarity parameter becomes This parameter is frequently written as and in this form, as was noted previously in Section 7.6, is called the cavitation number. Thus we can conclude that if cavitation is not of concern we do not need a similarity parameter involving , but if cavitation is to be modeled, then the cavitation number becomes an important similarity parameter. The Froude number, which arises because of the inclusion of gravity, is important for problems in which there is a free surface. Examples of these types of problems include the study of rivers, flow through hydraulic structures such as spillways, and the drag on ships. In these situations the shape of the free surface is influenced by gravity, and therefore the Froude number becomes an important similarity parameter. However, if there are no free surfaces, the only effect of gravity is to superimpose a hydrostatic pressure distribution on the pressure distribution created by the fluid motion. The hydrostatic distribution can be eliminated from the governing equation 1Eq. 7.302 by defining a new pressure, and with this change the Froude number does not appear in the nondimensional governing equations. We conclude from this discussion that for the steady flow of an incompressible fluid without free surfaces, dynamic and kinematic similarity will be achieved if 1for geometrically similar systems2 Reynolds number similarity exists. If free surfaces are involved, Froude number similarity must also be maintained. For free-surface flows we have tacitly assumed that surface tension is not important. We would find, however, that if surface tension is included, its effect would appear in the free-surface boundary condition, and the Weber number, would become an additional similarity parameter. In addition, if the governing equations for compressible fluids are considered, the Mach number, would appear as an additional similarity parameter. It is clear that all the common dimensionless groups that we previously developed by using dimensional analysis appear in the governing equations that describe fluid motion when these equations are expressed in terms of dimensionless variables. Thus, the use of the governing equations to obtain similarity laws provides an alternative to dimensional analysis. This approach ha advantage that the variables are known and the assumptions involved are clearly identified. In addition, a physical interpretation of the various dimensionless groups can often be obtained.

to study the phenomenon of interest under carefully controlled conditions. From these model studies, empirical formulations can be developed, or specific predictions 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" system. In the following sections, we find out how this can be accomplished in a systematic manner.

Models are used to investigate many different types of fluid mechanics problems, and it is difficult to characterize in a general way all necessary similarity requirements, since each problem is unique. We can, however, broadly classify many of the problems on the basis of the general nature of the flow and subsequently develop some general characteristics of model designs in each of these classifications. In the following sections we will consider models for the study of 112 flow through closed conduits, 122 flow around immersed bodies, and 132 flow with a free surface. Turbomachine models are considered in Chapter 12.

vModels are used to investigate many different types of fluid mechanics problems, and it is difficult to characterize in a general way all necessary similarity requirements, since each problem is unique. We can, however, broadly classify many of the problems on the basis of the general nature of the flow and subsequently develop some general characteristics of model designs in each of these classifications. In the following sections we will consider models for the study of 112 flow through closed conduits, 122 flow around immersed bodies, and 132 flow with a free surface. Turbomachine models are considered in Chapter 12.

where s is measured along the streamline. If we write the velocity, and length, s, in dimensionless form, that is, where V and represent some characteristic velocity and length, respectively, then and FI V2 / V*s dV*s ds* m as V2 / V*s dV*s ds* / V*s Vs V s* s / Vs, 7.6 Common Dimensionless Groups in Fluid Mechanics 347 F I G U R E 7.3 The force of gravity acting on a fluid particle moving along a streamline. Streamline gm Vs Special names along with physical interpretations are given to the most common dimensionless groups. TABLE 7.1 Some Common Variables and Dimensionless Groups in Fluid Mechanics Variables: Acceleration of gravity, g; Bulk modulus, Characteristic length, Density, ; Frequency of oscillating flow, ; Pressure, p (or p); Speed of sound, c; Surface tension, ; Velocity, V; Viscosity, Dimensionless Interpretation (Index of Types of Groups Name Force Ratio Indicated) Applications Reynolds number, Re Generally of importance in all types of fluid dynamics problems Froude number, Fr Flow with a free surface Euler number, Eu Problems in which pressure, or pressure differences, are of interest Cauchy Flows in which the compressibility of the fluid is important Mach Ma Flows in which the compressibility of the fluid is important Strouhal number, St Unsteady flow with a characteristic frequency of oscillation Weber number, We Problems in which surface tension is important a The Cauchy number and the Mach number are related and either can be used as an index of the relative effects of inertia and compressibility. See accompanying discussion. number,a number,a Ca v ¢ s m Ev; ; r rV2 / s v/ V V c rV2 Ev p rV2 V 1g/ rV/ m inertia force surface tension force inertia 1local2 force inertia 1convective2 force inertia force compressibility force inertia force compressibility force pressure force inertia force inertia force gravitational force inertia force viscous force JWCL068_ch07_332-382.qxd 9/23/08 10:46 AM Page 347 The magnitude of the weight of the particle, so the ratio of the inertia to the gravitational force is Thus, the force ratio is proportional to and the square root of this ratio, is called the Froude number. We see that a physical interpretation of the Froude number is that it is a measure of, or an index of, the relative importance of inertial forces acting on fluid particles to the weight of the particle. Note that the Froude number is not really equal to this force ratio, but is simply some type of average measure of the influence of these two forces. In a problem in which gravity 1or weight2 is not important, the Froude number would not appear as an important pi term. A similar interpretation in terms of indices of force ratios can be given to the other dimensionless groups, as indicated in Table 7.1, and a further discussion of the basis for this type of interpretation is given in the last section in this chapter. Some additional details about these important dimensionless groups are given below, and the types of application or problem in which they arise are briefly noted in the last column of Table 7.1.

Fluid Mechanics 7

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