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Fluid Mechanics

McGraw-Hill Series in Mechanical Engineering CONSULTING EDITORS Jack P. Holman, Southern Methodist University John Lloyd, Michigan State University

Anderson Computational Fluid Dynamics: The Basics with Applications Anderson Modern Compressible Flow: With Historical Perspective Arora Introduction to Optimum Design Borman and Ragland Combustion Engineering Burton Introduction to Dynamic Systems Analysis Culp Principles of Energy Conversion Dieter Engineering Design: A Materials & Processing Approach Doebelin Engineering Experimentation: Planning, Execution, Reporting Driels Linear Control Systems Engineering Edwards and McKee Fundamentals of Mechanical Component Design Gebhart Heat Conduction and Mass Diffusion Gibson Principles of Composite Material Mechanics Hamrock Fundamentals of Fluid Film Lubrication Heywood Internal Combustion Engine Fundamentals Hinze Turbulence Histand and Alciatore Introduction to Mechatronics and Measurement Systems Holman Experimental Methods for Engineers Howell and Buckius Fundamentals of Engineering Thermodynamics Jaluria Design and Optimization of Thermal Systems Juvinall Engineering Considerations of Stress, Strain, and Strength Kays and Crawford Convective Heat and Mass Transfer Kelly Fundamentals of Mechanical Vibrations

Kimbrell Kinematics Analysis and Synthesis Kreider and Rabl Heating and Cooling of Buildings Martin Kinematics and Dynamics of Machines Mattingly Elements of Gas Turbine Propulsion Modest Radiative Heat Transfer Norton Design of Machinery Oosthuizen and Carscallen Compressible Fluid Flow Oosthuizen and Naylor Introduction to Convective Heat Transfer Analysis Phelan Fundamentals of Mechanical Design Reddy An Introduction to Finite Element Method Rosenberg and Karnopp Introduction to Physical Systems Dynamics Schlichting Boundary-Layer Theory Shames Mechanics of Fluids Shigley Kinematic Analysis of Mechanisms Shigley and Mischke Mechanical Engineering Design Shigley and Uicker Theory of Machines and Mechanisms Stiffler Design with Microprocessors for Mechanical Engineers Stoecker and Jones Refrigeration and Air Conditioning Turns An Introduction to Combustion: Concepts and Applications Ullman The Mechanical Design Process Wark Advanced Thermodynamics for Engineers Wark and Richards Thermodynamics White Viscous Fluid Flow Zeid CAD/CAM Theory and Practice

About the Author

Frank M. White is Professor of Mechanical and Ocean Engineering at the University

of Rhode Island. He studied at Georgia Tech and M.I.T. In 1966 he helped found, at

URI, the first department of ocean engineering in the country. Known primarily as a

teacher and writer, he has received eight teaching awards and has written four text-

books on fluid mechanics and heat transfer.

During 1979–1990 he was editor-in-chief of the ASME Journal of Fluids Engi-

neering and then served from 1991 to 1997 as chairman of the ASME Board of Edi-

tors and of the Publications Committee. He is a Fellow of ASME and in 1991 received

the ASME Fluids Engineering Award. He lives with his wife, Jeanne, in Narragansett,

Rhode Island.

v

To Jeanne

Content Changes

New to this book, and to any fluid mechanics textbook, is a special appendix, Ap-

pendix E, Introduction to the Engineering Equation Solver (EES), which is keyed to

many examples and problems throughout the book. The author finds EES to be an ex-

tremely attractive tool for applied engineering problems. Not only does it solve arbi-

trarily complex systems of equations, written in any order or form, but also it has built-

in property evaluations (density, viscosity, enthalpy, entropy, etc.), linear and nonlinear

regression, and easily formatted parameter studies and publication-quality plotting. The

author is indebted to Professors Sanford Klein and William Beckman, of the Univer-

sity of Wisconsin, for invaluable and continuous help in preparing this EES material.

The book is now available with or without an EES problems disk. The EES engine is

available to adopters of the text with the problems disk.

Another welcome addition, especially for students, is Answers to Selected Prob-

lems. Over 600 answers are provided, or about 43 percent of all the regular problem

assignments. Thus a compromise is struck between sometimes having a specific nu-

merical goal and sometimes directly applying yourself and hoping for the best result.

There are revisions in every chapter. Chapter 1—which is purely introductory and

could be assigned as reading—has been toned down from earlier editions. For ex-

ample, the discussion of the fluid acceleration vector has been moved entirely to Chap-

ter 4. Four brief new sections have been added: (1) the uncertainty of engineering

data, (2) the use of EES, (3) the FE Examination, and (4) recommended problem-

solving techniques.

Chapter 2 has an improved discussion of the stability of floating bodies, with a fully

derived formula for computing the metacentric height. Coverage is confined to static

fluids and rigid-body motions. An improved section on pressure measurement discusses

modern microsensors, such as the fused-quartz bourdon tube, micromachined silicon

capacitive and piezoelectric sensors, and tiny (2 mm long) silicon resonant-frequency

devices.

Chapter 3 tightens up the energy equation discussion and retains the plan that

Bernoulli’s equation comes last, after control-volume mass, linear momentum, angu-

lar momentum, and energy studies. Although some texts begin with an entire chapter

on the Bernoulli equation, this author tries to stress that it is a dangerously restricted

relation which is often misused by both students and graduate engineers.

In Chapter 4 a few inviscid and viscous flow examples have been added to the ba-

sic partial differential equations of fluid mechanics. More extensive discussion con-

tinues in Chapter 8.

Chapter 5 is more successful when one selects scaling variables before using the pi

theorem. Nevertheless, students still complain that the problems are too ambiguous and

lead to too many different parameter groups. Several problem assignments now con-

tain a few hints about selecting the repeating variables to arrive at traditional pi groups.

In Chapter 6, the “alternate forms of the Moody chart” have been resurrected as

problem assignments. Meanwhile, the three basic pipe-flow problems—pressure drop,

flow rate, and pipe sizing—can easily be handled by the EES software, and examples

are given. Some newer flowmeter descriptions have been added for further enrichment.

Chapter 7 has added some new data on drag and resistance of various bodies, notably

biological systems which adapt to the flow of wind and water.

xii Preface

Supplements

EES Software

Chapter 8 picks up from the sample plane potential flows of Section 4.10 and plunges

right into inviscid-flow analysis, especially aerodynamics. The discussion of numeri-

cal methods, or computational fluid dynamics (CFD), both inviscid and viscous, steady

and unsteady, has been greatly expanded. Chapter 9, with its myriad complex algebraic

equations, illustrates the type of examples and problem assignments which can be

solved more easily using EES. A new section has been added about the suborbital X-

33 and VentureStar vehicles.

In the discussion of open-channel flow, Chapter 10, we have further attempted to

make the material more attractive to civil engineers by adding real-world comprehen-

sive problems and design projects from the author’s experience with hydropower proj-

ects. More emphasis is placed on the use of friction factors rather than on the Man-

ning roughness parameter. Chapter 11, on turbomachinery, has added new material on

compressors and the delivery of gases. Some additional fluid properties and formulas

have been included in the appendices, which are otherwise much the same.

The all new Instructor’s Resource CD contains a PowerPoint presentation of key text

figures as well as additional helpful teaching tools. The list of films and videos, for-

merly App. C, is now omitted and relegated to the Instructor’s Resource CD.

The Solutions Manual provides complete and detailed solutions, including prob-

lem statements and artwork, to the end-of-chapter problems. It may be photocopied for

posting or preparing transparencies for the classroom.

The Engineering Equation Solver (EES) was developed by Sandy Klein and Bill Beck-

man, both of the University of Wisconsin—Madison. A combination of equation-solving

capability and engineering property data makes EES an extremely powerful tool for your

students. EES (pronounced “ease”) enables students to solve problems, especially design

problems, and to ask “what if” questions. EES can do optimization, parametric analysis,

linear and nonlinear regression, and provide publication-quality plotting capability. Sim-

ple to master, this software allows you to enter equations in any form and in any order. It

automatically rearranges the equations to solve them in the most efficient manner.

EES is particularly useful for fluid mechanics problems since much of the property

data needed for solving problems in these areas are provided in the program. Air ta-

bles are built-in, as are psychometric functions and Joint Army Navy Air Force (JANAF)

table data for many common gases. Transport properties are also provided for all sub-

stances. EES allows the user to enter property data or functional relationships written

in Pascal, C, C, or Fortran. The EES engine is available free to qualified adopters

via a password-protected website, to those who adopt the text with the problems disk.

The program is updated every semester.

The EES software problems disk provides examples of typical problems in this text.

Problems solved are denoted in the text with a disk symbol. Each fully documented

solution is actually an EES program that is run using the EES engine. Each program

provides detailed comments and on-line help. These programs illustrate the use of EES

and help the student master the important concepts without the calculational burden

that has been previously required.

Preface xiii

Preface xi

vii

Hurricane Elena in the Gulf of Mexico. Unlike most small-scale fluids engineering applications, hurricanes are strongly affected by the Coriolis acceleration due to the rotation of the earth, which causes them to swirl counterclockwise in the Northern Hemisphere. The physical properties and boundary conditions which govern such flows are discussed in the present chapter. (Courtesy of NASA/Color-Pic Inc./E.R. Degginger/Color-Pic Inc.)

1.1 Preliminary Remarks Fluid mechanics is the study of fluids either in motion (fluid dynamics ) or at rest (fluid

statics ) and the subsequent effects of the fluid upon the boundaries, which may be ei-

ther solid surfaces or interfaces with other fluids. Both gases and liquids are classified

as fluids, and the number of fluids engineering applications is enormous: breathing,

blood flow, swimming, pumps, fans, turbines, airplanes, ships, rivers, windmills, pipes,

missiles, icebergs, engines, filters, jets, and sprinklers, to name a few. When you think

about it, almost everything on this planet either is a fluid or moves within or near a

fluid.

The essence of the subject of fluid flow is a judicious compromise between theory

and experiment. Since fluid flow is a branch of mechanics, it satisfies a set of well-

documented basic laws, and thus a great deal of theoretical treatment is available. How-

ever, the theory is often frustrating, because it applies mainly to idealized situations

which may be invalid in practical problems. The two chief obstacles to a workable the-

ory are geometry and viscosity. The basic equations of fluid motion (Chap. 4) are too

difficult to enable the analyst to attack arbitrary geometric configurations. Thus most

textbooks concentrate on flat plates, circular pipes, and other easy geometries. It is pos-

sible to apply numerical computer techniques to complex geometries, and specialized

textbooks are now available to explain the new computational fluid dynamics (CFD)

approximations and methods [1, 2, 29]. 1 This book will present many theoretical re-

sults while keeping their limitations in mind.

The second obstacle to a workable theory is the action of viscosity, which can be

neglected only in certain idealized flows (Chap. 8). First, viscosity increases the diffi-

culty of the basic equations, although the boundary-layer approximation found by Lud-

wig Prandtl in 1904 (Chap. 7) has greatly simplified viscous-flow analyses. Second,

viscosity has a destabilizing effect on all fluids, giving rise, at frustratingly small ve-

locities, to a disorderly, random phenomenon called turbulence. The theory of turbu-

lent flow is crude and heavily backed up by experiment (Chap. 6), yet it can be quite

serviceable as an engineering estimate. Textbooks now present digital-computer tech-

niques for turbulent-flow analysis [32], but they are based strictly upon empirical as-

sumptions regarding the time mean of the turbulent stress field.

Chapter 1

Introduction

(^1) Numbered references appear at the end of each chapter.

tion, pouring out over the lip if necessary. Meanwhile, the gas is unrestrained and ex-

pands out of the container, filling all available space. Element A in the gas is also hy-

drostatic and exerts a compression stress  p on the walls.

In the above discussion, clear decisions could be made about solids, liquids, and

gases. Most engineering fluid-mechanics problems deal with these clear cases, i.e., the

common liquids, such as water, oil, mercury, gasoline, and alcohol, and the common

gases, such as air, helium, hydrogen, and steam, in their common temperature and pres-

sure ranges. There are many borderline cases, however, of which you should be aware.

Some apparently “solid” substances such as asphalt and lead resist shear stress for short

periods but actually deform slowly and exhibit definite fluid behavior over long peri-

ods. Other substances, notably colloid and slurry mixtures, resist small shear stresses

but “yield” at large stress and begin to flow as fluids do. Specialized textbooks are de-

voted to this study of more general deformation and flow, a field called rheology [6].

Also, liquids and gases can coexist in two-phase mixtures, such as steam-water mix-

tures or water with entrapped air bubbles. Specialized textbooks present the analysis

1.2 The Concept of a Fluid 5

Static deflection

Free surface

Hydrostatic condition

Solid Liquid

A A A

( a ) ( c )

( b ) ( d )

0

0 A A

Gas

(1)

• p – p

p

p

p

= 0

τ

θ θ

2 θ

1

• = p – = p

σ

σ

1

τ

σ

τ

σ

τ

σ

Fig. 1.1 A solid at rest can resist shear. ( a ) Static deflection of the solid; ( b ) equilibrium and Mohr’s circle for solid element A. A fluid cannot resist shear. ( c ) Containing walls are needed; ( d ) equilibrium and Mohr’s circle for fluid element A.

1.3 The Fluid as a Continuum

of such two-phase flows [7]. Finally, there are situations where the distinction between

a liquid and a gas blurs. This is the case at temperatures and pressures above the so-

called critical point of a substance, where only a single phase exists, primarily resem-

bling a gas. As pressure increases far above the critical point, the gaslike substance be-

comes so dense that there is some resemblance to a liquid and the usual thermodynamic

approximations like the perfect-gas law become inaccurate. The critical temperature

and pressure of water are Tc  647 K and pc  219 atm,^2 so that typical problems in-

volving water and steam are below the critical point. Air, being a mixture of gases, has

no distinct critical point, but its principal component, nitrogen, has Tc  126 K and

pc  34 atm. Thus typical problems involving air are in the range of high temperature

and low pressure where air is distinctly and definitely a gas. This text will be concerned

solely with clearly identifiable liquids and gases, and the borderline cases discussed

above will be beyond our scope.

We have already used technical terms such as fluid pressure and density without a rig-

orous discussion of their definition. As far as we know, fluids are aggregations of mol-

ecules, widely spaced for a gas, closely spaced for a liquid. The distance between mol-

ecules is very large compared with the molecular diameter. The molecules are not fixed

in a lattice but move about freely relative to each other. Thus fluid density, or mass per

unit volume, has no precise meaning because the number of molecules occupying a

given volume continually changes. This effect becomes unimportant if the unit volume

is large compared with, say, the cube of the molecular spacing, when the number of

molecules within the volume will remain nearly constant in spite of the enormous in-

terchange of particles across the boundaries. If, however, the chosen unit volume is too

large, there could be a noticeable variation in the bulk aggregation of the particles. This

situation is illustrated in Fig. 1.2, where the “density” as calculated from molecular

mass  m within a given volume  is plotted versus the size of the unit volume. There

is a limiting volume * below which molecular variations may be important and

6 Chapter 1 Introduction

Microscopic uncertainty

Macroscopic uncertainty

0

1200

δ

δ * ≈ 10 -^9 mm^3

Elemental volume

Region containing fluid

= 1000 kg/m^3 = 1100

= 1200

= 1300

( a ) ( b)

ρ

ρ ρ

ρ

ρ (^) δ

Fig. 1.2 The limit definition of con- tinuum fluid density: ( a ) an ele- mental volume in a fluid region of variable continuum density; ( b ) cal- culated density versus size of the elemental volume.

(^2) One atmosphere equals 2116 lbf/ft 2  101,300 Pa.

even the metric countries differed in their use of kiloponds instead of dynes or new-

tons, kilograms instead of grams, or calories instead of joules. To standardize the met-

ric system, a General Conference of Weights and Measures attended in 1960 by 40

countries proposed the International System of Units (SI). We are now undergoing a

painful period of transition to SI, an adjustment which may take many more years to

complete. The professional societies have led the way. Since July 1, 1974, SI units have

been required by all papers published by the American Society of Mechanical Engi-

neers, which prepared a useful booklet explaining the SI [9]. The present text will use

SI units together with British gravitational (BG) units.

In fluid mechanics there are only four primary dimensions from which all other dimen-

sions can be derived: mass, length, time, and temperature.^4 These dimensions and their units

in both systems are given in Table 1.1. Note that the kelvin unit uses no degree symbol.

The braces around a symbol like { M } mean “the dimension” of mass. All other variables

in fluid mechanics can be expressed in terms of { M }, { L }, { T }, and { }. For example, ac-

celeration has the dimensions { LT ^2 }. The most crucial of these secondary dimensions is

force, which is directly related to mass, length, and time by Newton’s second law

F  m a (1.2)

From this we see that, dimensionally, { F }  { MLT ^2 }. A constant of proportionality

is avoided by defining the force unit exactly in terms of the primary units. Thus we

define the newton and the pound of force

1 newton of force  1 N  1 kg m/s^2

1 pound of force  1 lbf  1 slug ft/s^2  4.4482 N

In this book the abbreviation lbf is used for pound-force and lb for pound-mass. If in-

stead one adopts other force units such as the dyne or the poundal or kilopond or adopts

other mass units such as the gram or pound-mass, a constant of proportionality called

gc must be included in Eq. (1.2). We shall not use gc in this book since it is not nec-

essary in the SI and BG systems.

A list of some important secondary variables in fluid mechanics, with dimensions

derived as combinations of the four primary dimensions, is given in Table 1.2. A more

complete list of conversion factors is given in App. C.

8 Chapter 1 Introduction

(^4) If electromagnetic effects are important, a fifth primary dimension must be included, electric current { I }, whose SI unit is the ampere (A).

Primary dimension SI unit BG unit Conversion factor Mass { M } Kilogram (kg) Slug 1 slug  14.5939 kg Length { L } Meter (m) Foot (ft) 1 ft  0.3048 m Time { T } Second (s) Second (s) 1 s  1 s Temperature { } Kelvin (K) Rankine (°R) 1 K  1.8°R

Table 1.1 Primary Dimensions in SI and BG Systems

Primary Dimensions

Part (a)

Part (b)

Part (c)

EXAMPLE 1.

A body weighs 1000 lbf when exposed to a standard earth gravity g  32.174 ft/s 2. ( a ) What is its mass in kg? ( b ) What will the weight of this body be in N if it is exposed to the moon’s stan- dard acceleration g moon  1.62 m/s 2? ( c ) How fast will the body accelerate if a net force of 400 lbf is applied to it on the moon or on the earth?

Solution

Equation (1.2) holds with F  weight and a  g earth : F  W  mg  1000 lbf  ( m slugs)(32.174 ft/s 2 ) or m   3

  (31.08 slugs)(14.5939 kg/slug)  453.6 kg Ans. (a) The change from 31.08 slugs to 453.6 kg illustrates the proper use of the conversion factor 14.5939 kg/slug. The mass of the body remains 453.6 kg regardless of its location. Equation (1.2) applies with a new value of a and hence a new force F  W moon  mg moon  (453.6 kg)(1.62 m/s 2 )  735 N Ans. (b) This problem does not involve weight or gravity or position and is simply a direct application of Newton’s law with an unbalanced force: F  400 lbf  ma  (31.08 slugs)( a ft/s^2 ) or

a   3410 .0 ^08  12.43 ft/s 2  3.79 m/s 2 Ans. (c)

This acceleration would be the same on the moon or earth or anywhere.

1.4 Dimensions and Units 9

Secondary dimension SI unit BG unit Conversion factor Area { L^2 } m^2 ft^2 1 m 2  10.764 ft^2 Volume { L^3 } m^3 ft^3 1 m 3  35.315 ft^3 Velocity { LT ^1 } m/s ft/s 1 ft/s  0.3048 m/s Acceleration { LT ^2 } m/s^2 ft/s^2 1 ft/s^2  0.3048 m/s^2 Pressure or stress { ML ^1 T ^2 } Pa  N/m^2 lbf/ft 2 1 lbf/ft^2  47.88 Pa Angular velocity { T ^1 } s^1 s^1 1 s^1  1 s^1 Energy, heat, work { ML^2 T ^2 } J  N m ft lbf 1 ft lbf  1.3558 J Power { ML^2 T ^3 } W  J/s ft lbf/s 1 ft lbf/s  1.3558 W Density { ML ^3 } kg/m 3 slugs/ft^3 1 slug/ft^3  515.4 kg/m^3 Viscosity { ML ^1 T ^1 } kg/(m s) slugs/(ft s) 1 slug/(ft s)  47.88 kg/(m s) Specific heat { L^2 T ^2 ^1 } m^2 /(s^2 K) ft 2 /(s^2 °R) 1 m 2 /(s^2 K)  5.980 ft 2 /(s^2 °R)

Table 1.2 Secondary Dimensions in Fluid Mechanics