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346 Chapter 7 ■ Dimensional Analysis, Similitude, and Modeling 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 ß 1 , which in this case is m. We simply combine m with the other variables to make the combination dimensionless 1but do not use ¢p / in ß 2 , since we want the dependent variable to appear only in ß 1 2. For example, divide m by r 1to eliminate F2, then by V 1to eliminate T2, and finally by D 1to eliminate L2. Thus, and, therefore, a ¢p / rV 2b D a 1 b 1L2 L0 1cancels L ß 2 m rVD 1FL 2 T 2 1FL 4 T 2 21LT 1 21L2 F 0 L 0 T 0 ¢p / D rV 2 ß 1 ¢p /D rV 2 f a m rVD b L2 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, ß 2 contains m, which does not appear in ß 1 , 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 ß 2 can be formed by a combination of say ß 3 , ß 4 , and ß 5 such as ß 2 ß2 3 ß 4 ß 5 then ß 2 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. 7.6 Common Dimensionless Groups in Fluid Mechanics A useful physical interpretation can often be given to dimensionless groups. 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 F I along the streamline can be expressed as F I a s m, where a s 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 a s dV s dt V s dV s ds
7.6 Common Dimensionless Groups in Fluid Mechanics 347 TABLE 7.1 Some Common Variables and Dimensionless Groups in Fluid Mechanics Variables: Acceleration of gravity, g; Bulk modulus, E v ; Characteristic length, ; Density, r; Frequency of oscillating flow, v; Pressure, p (or ¢ p); Speed of sound, c; Surface tension, s; Velocity, V; Viscosity, m Dimensionless Interpretation (Index of Types of Groups Name Force Ratio Indicated) Applications rV/ m Reynolds number, Re inertia force viscous force Generally of importance in all types of fluid dynamics problems Special names along with physical interpretations are given to the most common dimensionless groups. V 1g/ p rV 2 rV 2 E v Froude number, Fr Euler number, Eu Cauchy number, a Ca inertia force gravitational force pressure force inertia force inertia force compressibility force Flow with a free surface Problems in which pressure, or pressure differences, are of interest Flows in which the compressibility of the fluid is important V c Mach number, a Ma inertia force Flows in which the compressibility force compressibility of the fluid is important v/ V Strouhal number, St inertia 1local2 force inertia 1convective2 force Unsteady flow with a characteristic frequency of oscillation rV 2 / s Weber number, We inertia force surface tension force 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. where s is measured along the streamline. If we write the velocity, V s , and length, s, in dimensionless form, that is, where V and / represent some characteristic velocity and length, respectively, then and V* s V s V a s V 2 F I V 2 s* s / / V * s dV* s ds* / V * s dV * s ds* m V s Streamline gm F I G U R E 7.3 The force of gravity acting on a fluid particle moving along a streamline.
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Achieve Positive Learning Outcomes
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WileyPLUS combines robust course ma
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Sixth Edition F undamentals of Flui
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About the Authors vii Bruce R. Muns
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P reface This book is intended for
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Preface xi Life Long Learning Probl
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Preface xiii WileyPLUS offers today
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Featured in this Book xv LEARNING O
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C ontents 1 INTRODUCTION 1 Learning
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Contents xix 6.4.3 Irrotational Flo
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12.6 Axial-Flow and Mixed-Flow Pump
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10 8 Jupiter red spot diameter 2 Ch
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4 Chapter 1 ■ Introduction and do
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6 Chapter 1 ■ Introduction up the
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8 Chapter 1 ■ Introduction the SI
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10 Chapter 1 ■ Introduction E XAM
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12 Chapter 1 ■ Introduction TABLE
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14 Chapter 1 ■ Introduction TABLE
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16 Chapter 1 ■ Introduction Crude
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18 Chapter 1 ■ Introduction Visco
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20 Chapter 1 ■ Introduction Kinem
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22 Chapter 1 ■ Introduction As th
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24 Chapter 1 ■ Introduction Boili
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26 Chapter 1 ■ Introduction E XAM
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28 Chapter 1 ■ Introduction The r
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30 Chapter 1 ■ Introduction fluid
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32 Chapter 1 ■ Introduction Secti
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34 Chapter 1 ■ Introduction avail
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36 Chapter 1 ■ Introduction compr
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2 Fluid Statics CHAPTER OPENING PHO
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40 Chapter 2 ■ Fluid Statics in a
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42 Chapter 2 ■ Fluid Statics p Fo
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44 Chapter 2 ■ Fluid Statics the
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46 Chapter 2 ■ Fluid Statics p 2
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48 Chapter 2 ■ Fluid Statics wher
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50 Chapter 2 ■ Fluid Statics F l
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52 Chapter 2 ■ Fluid Statics E XA
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54 Chapter 2 ■ Fluid Statics diff
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56 Chapter 2 ■ Fluid Statics Bour
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58 Chapter 2 ■ Fluid Statics Free
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60 Chapter 2 ■ Fluid Statics The
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62 Chapter 2 ■ Fluid Statics is t
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64 Chapter 2 ■ Fluid Statics h 1
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66 Chapter 2 ■ Fluid Statics SOLU
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68 Chapter 2 ■ Fluid Statics COMM
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70 Chapter 2 ■ Fluid Statics V2.8
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72 Chapter 2 ■ Fluid Statics CG
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74 Chapter 2 ■ Fluid Statics The
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80 Chapter 2 ■ Fluid Statics Sect
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88 Chapter 2 ■ Fluid Statics 2.81
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92 Chapter 2 ■ Fluid Statics 2.12
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94 Chapter 3 ■ Elementary Fluid D
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100 Chapter 3 ■ Elementary Fluid
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148 Chapter 4 ■ Fluid Kinematics
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188 Chapter 5 ■ Finite Control Vo
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264 Chapter 6 ■ Differential Anal
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) 266 Chapter 6 ■ Differential An
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396 Chapter 8 ■ Viscous Flow in P
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WARNING Stand clear of Hazard areas
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462 Chapter 9 ■ Flow over Immerse
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∋ 530 Chapter 9 ■ Flow over Imm
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10 Open-Channel Flow CHAPTER OPENIN
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536 Chapter 10 ■ Open-Channel Flo
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580 Chapter 11 ■ Compressible Flo
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646 Chapter 12 ■ Turbomachines Ex
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648 Chapter 12 ■ Turbomachines Q
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650 Chapter 12 ■ Turbomachines E
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656 Chapter 12 ■ Turbomachines Co
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702 Appendix A ■ Computational Fl
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A ppendix B Physical Properties of
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716 Appendix B ■ Physical Propert
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720 Appendix C ■ Properties of th
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722 Appendix D ■ Compressible Flo
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724 Appendix D ■ Compressible Flo
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Answers to Selected Even-Numbered H
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I ndex Absolute pressure, 13, 48 Ab
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Index I-3 guidelines for selection,
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Index I-5 hydrostatic: on curved su
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Index I-7 piezometer tube, 50, 105,
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Index I-9 Pressure force, 40 Pressu
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Index I-11 Stress field, 277 Strouh
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Index I-13 Weber, M., 29, 350 Weber
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V4.8 Jupiter red spot V4.9 Streamli
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V10.3 Water strider V10.13 Triangul
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■ TABLE 1.4 Conversion Factors fr
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■ TABLE 1.7 Approximate Physical
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