Low-Speed Aerodynamics, Second Edition.pdf

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Low-Speed Aerodynamics, Second Edition

Low-speed aerodynamics is important in the design and operation of aircraft fly-ing at low Mach number and of ground and marine vehicles. This book offers amodern treatment of the subject, both the theory of inviscid, incompressible, andirrotational aerodynamics and the computational techniques now available to solvecomplex problems.

A unique feature of the text is that the computational approach (from a singlevortex element to a three-dimensional panel formulation) is interwoven throughout.Thus, the reader can learn about classical methods of the past, while also learninghow to use numerical methods to solve real-world aerodynamic problems. Thissecond edition, updates the first edition with a new chapter on the laminar boundarylayer, the latest versions of computational techniques, and additional coverageof interaction problems. It includes a systematic treatment of two-dimensionalpanel methods and a detailed presentation of computational techniques for three-dimensional and unsteady flows. With extensive illustrations and examples, thisbook will be useful for senior and beginning graduate-level courses, as well as ahelpful reference tool for practicing engineers.

Joseph Katz is Professor of Aerospace Engineering and Engineering Mechanicsat San Diego State University.

Allen Plotkin is Professor of Aerospace Engineering and Engineering Mechanicsat San Diego State University.

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Cambridge Aerospace Series

Editors:

MICHAEL J. RYCROFT AND WEI SHYY

1. J. M. Rolfe and K. J. Staples (eds.): Flight Simulation2. P. Berlin: The Geostationary Applications Satellite3. M. J. T. Smith: Aircraft Noise4. N. X. Vinh: Flight Mechanics of High-Performance Aircraft5. W. A. Mair and D. L. Birdsall: Aircraft Performance6. M. J. Abzug and E. E. Larrabee: Airplane Stability and Control7. M. J. Sidi: Spacecraft Dynamics and Control8. J. D. Anderson: A History of Aerodynamics9. A. M. Cruise, J. A. Bowles, C. V. Goodall, and T. J. Patrick: Principles of

Space Instrument Design10. G. A. Khoury and J. D. Gillett (eds.): Airship Technology11. J. Fielding: Introduction to Aircraft Design12. J. G. Leishman: Principles of Helicopter Aerodynamics13. J. Katz and A. Plotkin: Low Speed Aerodynamics, Second Edition

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Low-Speed AerodynamicsSecond Edition

JOSEPH KATZSan Diego State University

ALLEN PLOTKINSan Diego State University

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CAMBRIDGE UNIVERSITY PRESSCambridge, New York, Melbourne, Madrid, Cape Town, Singapore,

São Paulo, Delhi, Dubai, Tokyo, Mexico City

Cambridge University Press32 Avenue of the Americas, New York, NY 10013-2473, USA

www.cambridge.orgInformation on this title: www.cambridge.org/9780521665520

© Cambridge University Press 2001

This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the written

permission of Cambridge University Press.

First published 200110th printing 2010

A catalog record for this publication is available from the British Library.

Library of Congress Cataloging in Publication Data

Katz, Joseph, 1947–Low-speed aerodynamics / Joseph Katz, Allen Plotkin. – 2nd ed.

p. cm. – (Cambridge aerospace series : 13)ISBN 0-521-66219-2

1. Aerodynamics. I. Plotkin, Allen. II. Title. III. Series.TL570 .K34 2000

629.132'3 – dc21 00-031270

ISBN 978-0-521-66219-2 HardbackISBN 978-0-521-66552-0 Paperback

Cambridge University Press has no responsibility for the persistence oraccuracy of URLs for external or third-party Internet Web sites referred to inthis publication and does not guarantee that any content on such Web sites is,

or will remain, accurate or appropriate.


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Contents

Preface page xiiiPreface to the First Edition xv

1 Introduction and Background 1

1.1 Description of Fluid Motion 11.2 Choice of Coordinate System 21.3 Pathlines, Streak Lines, and Streamlines 31.4 Forces in a Fluid 41.5 Integral Form of the Fluid Dynamic Equations 61.6 Differential Form of the Fluid Dynamic Equations 81.7 Dimensional Analysis of the Fluid Dynamic Equations 141.8 Flow with High Reynolds Number 171.9 Similarity of Flows 19

2 Fundamentals of Inviscid, Incompressible Flow 21

2.1 Angular Velocity, Vorticity, and Circulation 212.2 Rate of Change of Vorticity 242.3 Rate of Change of Circulation: Kelvin’s Theorem 252.4 Irrotational Flow and the Velocity Potential 262.5 Boundary and Infinity Conditions 272.6 Bernoulli’s Equation for the Pressure 282.7 Simply and Multiply Connected Regions 292.8 Uniqueness of the Solution 302.9 Vortex Quantities 322.10 Two-Dimensional Vortex 342.11 The Biot–Savart Law 362.12 The Velocity Induced by a Straight Vortex Segment 382.13 The Stream Function 41

3 General Solution of the Incompressible, Potential Flow Equations 44

3.1 Statement of the Potential Flow Problem 443.2 The General Solution, Based on Green’s Identity 443.3 Summary: Methodology of Solution 483.4 Basic Solution: Point Source 493.5 Basic Solution: Point Doublet 513.6 Basic Solution: Polynomials 543.7 Two-Dimensional Version of the Basic Solutions 563.8 Basic Solution: Vortex 583.9 Principle of Superposition 60

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viii Contents

3.10 Superposition of Sources and Free Stream: Rankine’s Oval 603.11 Superposition of Doublet and Free Stream: Flow around a Cylinder 623.12 Superposition of a Three-Dimensional Doublet and Free Stream:

Flow around a Sphere 673.13 Some Remarks about the Flow over the Cylinder and the Sphere 693.14 Surface Distribution of the Basic Solutions 70

4 Small-Disturbance Flow over Three-Dimensional Wings:Formulation of the Problem 75

4.1 Definition of the Problem 754.2 The Boundary Condition on the Wing 764.3 Separation of the Thickness and the Lifting Problems 784.4 Symmetric Wing with Nonzero Thickness at Zero Angle of Attack 794.5 Zero-Thickness Cambered Wing at Angle of Attack–Lifting Surfaces 824.6 The Aerodynamic Loads 854.7 The Vortex Wake 884.8 Linearized Theory of Small-Disturbance Compressible Flow 90

5 Small-Disturbance Flow over Two-Dimensional Airfoils 94

5.1 Symmetric Airfoil with Nonzero Thickness at Zero Angle of Attack 945.2 Zero-Thickness Airfoil at Angle of Attack 1005.3 Classical Solution of the Lifting Problem 1045.4 Aerodynamic Forces and Moments on a Thin Airfoil 1065.5 The Lumped-Vortex Element 1145.6 Summary and Conclusions from Thin Airfoil Theory 120

6 Exact Solutions with Complex Variables 122

6.1 Summary of Complex Variable Theory 1226.2 The Complex Potential 1256.3 Simple Examples 1266.3.1 Uniform Stream and Singular Solutions 1266.3.2 Flow in a Corner 1276.4 Blasius Formula, Kutta–Joukowski Theorem 1286.5 Conformal Mapping and the Joukowski Transformation 1286.5.1 Flat Plate Airfoil 1306.5.2 Leading-Edge Suction 1316.5.3 Flow Normal to a Flat Plate 1336.5.4 Circular Arc Airfoil 1346.5.5 Symmetric Joukowski Airfoil 1356.6 Airfoil with Finite Trailing-Edge Angle 1376.7 Summary of Pressure Distributions for Exact Airfoil Solutions 1386.8 Method of Images 1416.9 Generalized Kutta–Joukowski Theorem 146

7 Perturbation Methods 151

7.1 Thin-Airfoil Problem 1517.2 Second-Order Solution 1547.3 Leading-Edge Solution 157


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Contents ix

7.4 Matched Asymptotic Expansions 1607.5 Thin Airfoil between Wind Tunnel Walls 163

8 Three-Dimensional Small-Disturbance Solutions 167

8.1 Finite Wing: The Lifting Line Model 1678.1.1 Definition of the Problem 1678.1.2 The Lifting-Line Model 1688.1.3 The Aerodynamic Loads 1728.1.4 The Elliptic Lift Distribution 1738.1.5 General Spanwise Circulation Distribution 1788.1.6 Twisted Elliptic Wing 1818.1.7 Conclusions from Lifting-Line Theory 1838.2 Slender Wing Theory 1848.2.1 Definition of the Problem 1848.2.2 Solution of the Flow over Slender Pointed Wings 1868.2.3 The Method of R. T. Jones 1928.2.4 Conclusions from Slender Wing Theory 1948.3 Slender Body Theory 1958.3.1 Axisymmetric Longitudinal Flow Past a Slender

Body of Revolution 1968.3.2 Transverse Flow Past a Slender Body of

Revolution 1988.3.3 Pressure and Force Information 1998.3.4 Conclusions from Slender Body Theory 2018.4 Far Field Calculation of Induced Drag 201

9 Numerical (Panel) Methods 206

9.1 Basic Formulation 2069.2 The Boundary Conditions 2079.3 Physical Considerations 2099.4 Reduction of the Problem to a Set of Linear Algebraic Equations 2139.5 Aerodynamic Loads 2169.6 Preliminary Considerations, Prior to Establishing Numerical Solutions 2179.7 Steps toward Constructing a Numerical Solution 2209.8 Example: Solution of Thin Airfoil with the Lumped-Vortex Element 2229.9 Accounting for Effects of Compressibility and Viscosity 226

10 Singularity Elements and Influence Coefficients 230

10.1 Two-Dimensional Point Singularity Elements 23010.1.1 Two-Dimensional Point Source 23010.1.2 Two-Dimensional Point Doublet 23110.1.3 Two-Dimensional Point Vortex 23110.2 Two-Dimensional Constant-Strength Singularity Elements 23210.2.1 Constant-Strength Source Distribution 23310.2.2 Constant-Strength Doublet Distribution 23510.2.3 Constant-Strength Vortex Distribution 23610.3 Two-Dimensional Linear-Strength Singularity Elements 23710.3.1 Linear Source Distribution 238


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x Contents

10.3.2 Linear Doublet Distribution 23910.3.3 Linear Vortex Distribution 24110.3.4 Quadratic Doublet Distribution 24210.4 Three-Dimensional Constant-Strength Singularity Elements 24410.4.1 Quadrilateral Source 24510.4.2 Quadrilateral Doublet 24710.4.3 Constant Doublet Panel Equivalence to Vortex

Ring 25010.4.4 Comparison of Near and Far Field Formulas 25110.4.5 Constant-Strength Vortex Line Segment 25110.4.6 Vortex Ring 25510.4.7 Horseshoe Vortex 25610.5 Three-Dimensional Higher Order Elements 258

11 Two-Dimensional Numerical Solutions 262

11.1 Point Singularity Solutions 26211.1.1 Discrete Vortex Method 26311.1.2 Discrete Source Method 27211.2 Constant-Strength Singularity Solutions (Using the Neumann B.C.) 27611.2.1 Constant Strength Source Method 27611.2.2 Constant-Strength Doublet Method 28011.2.3 Constant-Strength Vortex Method 28411.3 Constant-Potential (Dirichlet Boundary Condition) Methods 28811.3.1 Combined Source and Doublet Method 29011.3.2 Constant-Strength Doublet Method 29411.4 Linearly Varying Singularity Strength Methods

(Using the Neumann B.C.) 29811.4.1 Linear-Strength Source Method 29911.4.2 Linear-Strength Vortex Method 30311.5 Linearly Varying Singularity Strength Methods

(Using the Dirichlet B.C.) 30611.5.1 Linear Source/Doublet Method 30611.5.2 Linear Doublet Method 31211.6 Methods Based on Quadratic Doublet Distribution

(Using the Dirichlet B.C.) 31511.6.1 Linear Source/Quadratic Doublet Method 31511.6.2 Quadratic Doublet Method 32011.7 Some Conclusions about Panel Methods 323

12 Three-Dimensional Numerical Solutions 331

12.1 Lifting-Line Solution by Horseshoe Elements 33112.2 Modeling of Symmetry and Reflections from Solid Boundaries 33812.3 Lifting-Surface Solution by Vortex Ring Elements 34012.4 Introduction to Panel Codes: A Brief History 35112.5 First-Order Potential-Based Panel Methods 35312.6 Higher Order Panel Methods 35812.7 Sample Solutions with Panel Codes 360


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Contents xi

13 Unsteady Incompressible Potential Flow 369

13.1 Formulation of the Problem and Choice of Coordinates 36913.2 Method of Solution 37313.3 Additional Physical Considerations 37513.4 Computation of Pressures 37613.5 Examples for the Unsteady Boundary Condition 37713.6 Summary of Solution Methodology 38013.7 Sudden Acceleration of a Flat Plate 38113.7.1 The Added Mass 38513.8 Unsteady Motion of a Two-Dimensional Thin Airfoil 38713.8.1 Kinematics 38813.8.2 Wake Model 38913.8.3 Solution by the Time-Stepping Method 39113.8.4 Fluid Dynamic Loads 39413.9 Unsteady Motion of a Slender Wing 40013.9.1 Kinematics 40113.9.2 Solution of the Flow over the Unsteady Slender

Wing 40113.10 Algorithm for Unsteady Airfoil Using the Lumped-Vortex Element 40713.11 Some Remarks about the Unsteady Kutta Condition 41613.12 Unsteady Lifting-Surface Solution by Vortex Ring Elements 41913.13 Unsteady Panel Methods 433

14 The Laminar Boundary Layer 448

14.1 The Concept of the Boundary Layer 44814.2 Boundary Layer on a Curved Surface 45214.3 Similar Solutions to the Boundary Layer Equations 45714.4 The von Karman Integral Momentum Equation 46314.5 Solutions Using the von Karman Integral Equation 46714.5.1 Approximate Polynomial Solution 46814.5.2 The Correlation Method of Thwaites 46914.6 Weak Interactions, the Goldstein Singularity, and Wakes 47114.7 Two-Equation Integral Boundary Layer Method 47314.8 Viscous–Inviscid Interaction Method 47514.9 Concluding Example: The Flow over a Symmetric Airfoil 479

15 Enhancement of the Potential Flow Model 483

15.1 Wake Rollup 48315.2 Coupling between Potential Flow and Boundary Layer Solvers 48715.2.1 The Laminar/Turbulent Boundary Layer

and Transition 48715.2.2 Viscous–Inviscid Coupling, Including

Turbulent Boundary Layer 49115.3 Influence of Viscous Flow Effects on Airfoil Design 49515.3.1 Low Drag Considerations 49815.3.2 High Lift Considerations 49915.4 Flow over Wings at High Angles of Attack 505


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xii Contents

15.4.1 Flow Separation on Wings with UnsweptLeading Edge – Experimental Observations 508

15.4.2 Flow Separation on Wings with UnsweptLeading Edge – Modeling 510

15.4.3 Flow Separation on Wings with Highly SweptLeading Edge – Experimental Observations 516

15.4.4 Modeling of Highly Swept Leading-EdgeSeparation 523

15.5 Possible Additional Features of Panel Codes 528

A Airfoil Integrals 537

B Singularity Distribution Integrals 540

C Principal Value of the Lifting Surface Integral IL 545

D Sample Computer Programs 546

Index 611


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Preface

Our goal in writing this Second Edition of Low-Speed Aerodynamics remains thesame, to present a comprehensive and up-to-date treatment of the subject of inviscid, incom-pressible, and irrotational aerodynamics. It is still true that for most practical aerodynamicand hydrodynamic problems, the classical model of a thin viscous boundary layer alonga body’s surface, surrounded by a mainly inviscid flowfield, has produced important engi-neering results. This approach requires first the solution of the inviscid flow to obtain thepressure field and consequently the forces such as lift and induced drag. Then, a solutionof the viscous flow in the thin boundary layer allows for the calculation of the skin frictioneffects.

The First Edition provides the theory and related computational methods for the solutionof the inviscid flow problem. This material is complemented in the Second Edition witha new Chapter 14, “The Laminar Boundary Layer,” whose goal is to provide a moderndiscussion of the coupling of the inviscid outer flow with the viscous boundary layer. First,an introduction to the classical boundary-layer theory of Prandtl is presented. The need for aninteractive approach (to replace the classical sequential one) to the coupling is discussed anda viscous–inviscid interaction method is presented. Examples for extending this approach,which include transition to turbulence, are provided in the final Chapter 15.

In addition, updated versions of the computational methods are presented and severaltopics are improved and updated throughout the text. For example, more coverage is given ofaerodynamic interaction problems such as multiple wings, ground effect, wall corrections,and the presence of a free surface.

We would like to thank Turgut Sarpkaya of the Naval Postgraduate School and H. K.Cheng of USC for their input in Chapter 14 and particularly Mark Drela of MIT whoprovided a detailed description of his solution technique, which formed the basis for thematerial in Sections 14.7 and 14.8. Finally, we would like to acknowledge the continuinglove and support of our wives, Hilda Katz and Selena Plotkin.

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Preface to the First Edition

Our goal in writing this book is to present a comprehensive and up-to-date treat-ment of the subject of inviscid, incompressible, and irrotational aerodynamics. Over thelast several years there has been a widespread use of computational (surface singularity)methods for the solution of problems of concern to the low-speed aerodynamicist and aneed has developed for a text to provide the theoretical basis for these methods as well asto provide a smooth transition from the classical small-disturbance methods of the past tothe computational methods of the present. This book was written in response to this need.A unique feature of this book is that the computational approach (from a single vortex el-ement to a three-dimensional panel formulation) is interwoven throughout so that it servesas a teaching tool in the understanding of the classical methods as well as a vehicle for thereader to obtain solutions to complex problems that previously could not be dealt with inthe context of a textbook. The reader will be introduced to different levels of complexity inthe numerical modeling of an aerodynamic problem and will be able to assemble codes toimplement a solution.

We have purposely limited our scope to inviscid, incompressible, and irrotational aero-dynamics so that we can present a truly comprehensive coverage of the material. The bookbrings together topics currently scattered throughout the literature. It provides a detailed pre-sentation of computational techniques for three-dimensional and unsteady flows. It includesa systematic and detailed (including computer programs) treatment of two-dimensionalpanel methods with variations in singularity type, order of singularity, Neumann or Dirich-let boundary conditions, and velocity or potential-based approaches.

This book is divided into three main parts. In the first, Chapters 1–3, the basic theory isdeveloped. In the second part, Chapters 4–8, an analytical approach to the solution of theproblem is taken. Chapters 4, 5, and 8 deal with the small-disturbance version of the problemand the classical methods of thin-airfoil theory, lifting line theory, slender wing theory, andslender body theory. In this part exact solutions via complex variable theory and perturbationmethods for obtaining higher-order small disturbance approximations are also included.The third part, Chapters 9–14, presents a systematic treatment of the surface singularitydistribution technique for obtaining numerical solutions for incompressible potential flows.A general methodology for assembling a numerical solution is developed and applied to aseries of increasingly complex aerodynamic elements (two-dimensional, three-dimensional,and unsteady problems are treated).

The book is designed to be used as a textbook for a course in low-speed aerodynamics ateither the advanced senior or first-year graduate levels. The complete text can be covered ina one-year course and a one-quarter or one-semester course can be constructed by choosingthe topics that the instructor would like to emphasize. For example, a senior elective coursewhich concentrated on two-dimensional steady aerodynamics might include Chapters 1–3,4, 5, 9, 11, 8, 12, and 14. A traditional graduate course which emphasized an analyticaltreatment of the subject might include Chapters 1–3, 4, 5–7, 8, 9, and 13 and a course whichemphasized a numerical approach (panel methods) might include Chapters 1–3 and 9–14and a treatment of pre- and postprocessors. It has been assumed that the reader has taken

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xvi Preface to the First Edition

a first course in fluid mechanics and has a mathematical background which includes anexposure to vector calculus, partial differential equations, and complex variables.

We believe that the topics covered by this text are needed by the fluid dynamicist becauseof the complex nature of the fluid dynamic equations which has led to a mainly experimentalapproach for dealing with most engineering research and development programs. In a widersense, such an approach uses tools such as wind tunnels or large computer codes where theengineer/user is experimenting and testing ideas with some trial and error logic in mind.Therefore, even in the era of supercomputers and sophisticated experimental tools, there isa need for simplified models that allow for an easy grasp of the dominant physical effects(e.g., having a simple lifting vortex in mind, one can immediately tell that the first wing ina tandem formation has the larger lift).

For most practical aerodynamic and hydrodynamic problems, the classical model of a thinviscous boundary layer along a body’s surface, surrounded by a mainly inviscid flowfield,has produced important engineering results. This approach requires first the solution ofthe inviscid flow to obtain the pressure field and consequently the forces such as lift andinduced drag. Then, a solution of the viscous flow in the thin boundary layer allows forthe calculation of the skin friction effects. This methodology has been used successfullythroughout the twentieth century for most airplane and marine vessel designs. Recently, dueto developments in computer capacity and speed, the inviscid flowfield over complex anddetailed geometries (such as airplanes, cars, etc.) can be computed by this approach (panelmethods). Thus, for the near future, since these methods are the main tools of low-speedaerodynamicists all over the world, a need exists for a clear and systematic explanationof how and why (and for which cases) these methods work. This book is one attempt torespond to this need.

We would like to thank graduate students Lindsey Browne and especially Steven Yonwho developed the two-dimensional panel codes in Chapter 11 and checked the integrals inChapter 10. Allen Plotkin would like to thank his teachers Richard Skalak, KrishnamurthyKaramcheti, Milton Van Dyke, and Irmgard Flugge-Lotz, his parents Claire and Oscar fortheir love and support, and his children Jennifer Anne and Samantha Rose and especiallyhis wife Selena for their love, support, and patience. Joseph Katz would like to thank hisparents Janka and Jeno, his children Shirley, Ronny, and Danny, and his wife Hilda for theirlove, support, and patience. The support of the Low-Speed Aerodynamic Branch at NASAAmes is acknowledged by Joseph Katz for their inspiration that initiated this project andfor their help during past years in the various stages of developing the methods presentedin this book.


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CB329-01 CB329/Katz September 22, 2000 13:49 Char Count= 58027

CHAPTER 1

Introduction and Background

The differential equations that are generally used in the solution of problems rel-evant to low-speed aerodynamics are a simplified version of the governing equations offluid dynamics. Also, most engineers when faced with finding a solution to a practical aero-dynamic problem, find themselves operating large computer codes rather than developingsimple analytical models to guide them in their analysis. For this reason, it is important tostart with a brief development of the principles upon which the general fluid dynamic equa-tions are based. Then we will be in a position to consider the physical reasoning behind theassumptions introduced to generate simplified versions of the equations that still correctlymodel the aerodynamic phenomena being studied. It is hoped that this approach will givethe engineer the ability to appreciate both the power and the limitations of the techniquesthat will be presented in this text. In this chapter we will derive the conservation of mass andmomentum balance equations and show how they are reduced to obtain the equations thatwill be used in the rest of the text to model flows of interest to the low-speed aerodynamicist.

1.1 Description of Fluid Motion

The fluid being studied here is modeled as a continuum, and infinitesimally smallregions of the fluid (with a fixed mass) are called fluid elements or fluid particles. Themotion of the fluid can be described by two different methods. One adopts the particle pointof view and follows the motion of the individual particles. The other adopts the field pointof view and provides the flow variables as functions of position in space and time.

The particle point of view, which uses the approach of classical mechanics, is called theLagrangian method. To trace the motion of each fluid particle, it is convenient to introducea Cartesian coordinate system with the coordinates x, y, and z. The position of any fluidparticle P (see Fig. 1.1) is then given by

x = xP (x0, y0, z0, t)

y = yP (x0, y0, z0, t)

z = zP (x0, y0, z0, t)

(1.1)

where (x0, y0, z0) is the position of P at some initial time t = 0. (Note that the quantity(x0, y0, z0) represents the vector with components x0, y0, and z0.) The components of thevelocity of this particle are then given by

u = ∂x/∂t

v = ∂y/∂t

w = ∂z/∂t

(1.2)

and those of the acceleration by

ax = ∂2x/∂t2

ay = ∂2 y/∂t2

az = ∂2z/∂t2

(1.3)

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2 1 / Introduction and Background

Figure 1.1 Particle trajectory lines in a steady-state flow over an airfoil as viewed from a body-fixedcoordinate system.

The Lagrangian formulation requires the evaluation of the motion of each fluid particle.For most practical applications this abundance of information is neither necessary nor usefuland the analysis is cumbersome.

The field point of view, called the Eulerian method, provides the spatial distribution offlow variables at each instant during the motion. For example, if a Cartesian coordinatesystem is used, the components of the fluid velocity are given by

u = u(x, y, z, t)

v = v(x, y, z, t)

w = w(x, y, z, t)

(1.4)

The Eulerian approach provides information about the fluid variables that is consistentwith the information supplied by most experimental techniques and that is in a form ap-propriate for most practical applications. For these reasons the Eulerian description of fluidmotion is the most widely used.

1.2 Choice of Coordinate System

For the following chapters, when possible, primarily a Cartesian coordinate systemwill be used. Other coordinate systems such as curvilinear, cylindrical, spherical, etc. will beintroduced and used if necessary, mainly to simplify the treatment of certain problems. Also,from the kinematic point of view, a careful choice of a coordinate system can considerablysimplify the solution of a problem. As an example, consider the forward motion of an airfoil,with a constant speed U∞, in a fluid that is otherwise at rest – as shown in Fig. 1.1. Here, theorigin of the coordinate system is attached to the moving airfoil and the trajectory of a fluidparticle inserted at point P0 at t = 0 is shown in the figure. By following the trajectories ofseveral particles a more complete description of the flowfield is obtained in the figure. It isimportant to observe that for a constant-velocity forward motion of the airfoil, in this frameof reference, these trajectory lines become independent of time. That is, if various particlesare introduced at the same point in space, then they will follow the same trajectory.

Now let us examine the same flow, but from a coordinate system that is fixed relative tothe undisturbed fluid. At t = 0, the airfoil was at the right side of Fig. 1.2 and as a resultof its constant-velocity forward motion (with a speed U∞ toward the left side of the page),later at t = t1 it has moved to the new position indicated in the figure. A typical particle’s


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1.3 Pathlines, Streak Lines, and Streamlines 3

Figure 1.2 Particle trajectory line for the airfoil of Fig. 1.1 as viewed from a stationary inertial frame.

trajectory line between t = 0 and t = t1, for this case, is shown in Fig. 1.2. The particle’smotion now depends on time, and a new trajectory has to be established for each particle.

This simple example depicts the importance of good coordinate system selection. Formany problems where a constant velocity and a fixed geometry (with time) are present, theuse of a body-fixed frame of reference will result in a steady or time-independent flow.

1.3 Pathlines, Streak Lines, and Streamlines

Three sets of curves are normally associated with providing a pictorial descriptionof a fluid motion: pathlines, streak lines, and streamlines.

Pathlines: A curve describing the trajectory of a fluid element is called a pathline or aparticle path. Pathlines are obtained in the Lagrangian approach by an integration of theequations of dynamics for each fluid particle. If the velocity field of a fluid motion is givenin the Eulerian framework by Eq. (1.4) in a body-fixed frame, the pathline for a particle at P0

in Fig. 1.1 can be obtained by an integration of the velocity. For steady flows the pathlinesin the body-fixed frame become independent of time and can be drawn as in the case offlow over the airfoil shown in Fig. 1.1.

Streak Lines: In many cases of experimental flow visualization, particles (e.g., dye orsmoke) are introduced into the flow at a fixed point in space. The line connecting all of theseparticles is called a streak line. To construct streak lines using the Lagrangian approach,draw a series of pathlines for particles passing through a given point in space and, at aparticular instant in time, connect the ends of these pathlines.

Streamlines: Another set of curves can be obtained (at a given time) by lines that areparallel to the local velocity vector. To express analytically the equation of a streamline ata certain instant of time, at any point P in the fluid, the velocity1 q must be parallel to thestreamline element dl (Fig. 1.3). Therefore, on a streamline:

q × dl = 0 (1.5)

If the velocity vector is q = (u, v, w), then the vector equation (Eq. (1.5)) reduces to thefollowing scalar equations:

w dy − v dz = 0

u dz − w dx = 0 (1.6)

v dx − u dy = 0

or in a differential equation form:

dx

u= dy

v= dz

w(1.6a)

1 Bold letters in this book represent vectors.


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Figure 1.3 Description of a streamline.

In Eq. (1.6a), the velocity (u, v, w) is a function of the coordinates and of time. However,for steady flows the streamlines are independent of time and streamlines, pathlines, andstreak lines become identical, as shown in Fig. 1.1.

1.4 Forces in a Fluid

Prior to discussing the dynamics of fluid motion, the types of forces that act on afluid element should be identified. Here, we consider forces such as body forces per unitmass f and surface forces resulting from the stress vector t. The body forces are independentof any contact with the fluid, as in the case of gravitational or magnetic forces, and theirmagnitude is proportional to the local mass.

To define the stress vector t at a point, consider the force F acting on a planar area S(shown in Fig. 1.4) with n being an outward normal to S. Then

t = limS→0

(F

S

)

To obtain the components of the stress vector consider the force equilibrium on an infinites-imal tetrahedral fluid element, shown in Fig. 1.5. According to Batchelor1.1 (p. 10) thisequilibrium yields the components in the x1, x2, and x3 directions:

ti =3∑

j=1

τi j n j , i = 1, 2, 3 (1.7)

where the subscripts 1, 2, and 3 denote the three coordinate directions. A similar treatmentof the moment equilibrium results in the symmetry of the stress vector components so thatτi j = τ j i .

These stress components τi j are shown schematically on a cubical element in Fig. 1.6.Note that τi j acts in the xi direction on a surface whose outward normal points in thex j direction. This indicial notation allows a simpler presentation of the equations, and the

Figure 1.4 Force F acting on a surface S.


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1.4 Forces in a Fluid 5

Figure 1.5 Tetrahedral fluid element.

subscripts 1, 2, and 3 denote the coordinate directions x , y, and z, respectively. For example,

x1 = x, x2 = y, x3 = z

and

q1 = u, q2 = v, q3 = w

The stress components shown on the cubical fluid element of Fig. 1.6 can be summarizedin a matrix form or in an indicial form as follows:⎛

⎝ τxx τxy τxz

τyx τyy τyz

τzx τzy τzz

⎞⎠ =

⎛⎝ τ11 τ12 τ13

τ21 τ22 τ23

τ31 τ32 τ33

⎞⎠ = τi j (1.8)

Also, it is customary to sum over any index that is repeated such that

3∑j=1

τi j n j ≡ τi j n j for i = 1, 2, 3 (1.9)

Figure 1.6 Stress components on a cubical fluid element.


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Figure 1.7 Flow between a stationary (lower) and a moving (upper) plate.

and to interpret an equation with a free index (as i in Eq. (1.9)) as being valid for all valuesof that index.

For a Newtonian fluid (where the stress components τi j are linear in the derivatives∂qi/∂x j ), the stress components are related to the velocity field by (see, for example,Batchelor,1.1 p. 147)

τi j =(

−p − 2

3μ

∂qk

∂xk

)δi j + μ

(∂qi

∂x j+ ∂q j

∂xi

)(1.10)

where μ is the viscosity coefficient, p is the pressure, the dummy variable k is summedfrom 1 to 3, and δi j is the Kronecker delta function defined by

δi j ≡(

1, i = j0, i �= j

)

When the fluid is at rest, the tangential stresses vanish and the normal stress componentbecomes simply the pressure. Thus the stress components become

τi j =⎛⎝−p 0 0

0 −p 00 0 −p

⎞⎠ (1.11)

Another interesting case of Eq. (1.10) is the one-degree-of-freedom shear flow betweena stationary and a moving infinite plate with a velocity U∞ (shown in Fig. 1.7), withoutpressure gradients. This flow is called Couette flow (see, for example, Yuan,1.2 p. 260) andthe shear stress becomes

τxz = μ∂u

∂z= μU∞

h(1.12)

Since there is no pressure gradient in the flow, the fluid motion in the x direction is entirelydue to the action of the viscous forces. The force F on the plate can be found by integratingτxz on the upper moving surface.

1.5 Integral Form of the Fluid Dynamic Equations

To develop the governing integral and differential equations describing the fluidmotion, the various properties of the fluid are investigated in an arbitrary control volumethat is stationary and submerged in the fluid (Fig. 1.8). These properties can be density,momentum, energy, etc., and any change with time of one of them for the fluid flowingthrough the control volume is the sum of the accumulation of the property in the controlvolume and the transfer of this property out of the control volume through its boundaries.


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1.5 Integral Form of the Fluid Dynamic Equations 7

Figure 1.8 A control volume in the fluid.

For example, the conservation of mass can be analyzed by observing the changes in fluiddensity ρ for the control volume (c.v.). The mass mc.v. within the control volume is then

mc.v. =∫

c.v.ρ dV (1.13)

where dV is the volume element. The accumulation of mass within the control volume is∂mc.v.

∂t= ∂

∂t

∫c.v.

ρ dV (1.13a)

The change in the mass within the control volume, due to the mass leaving (mout) and tothe mass entering (m in) through the boundaries (c.s.), is

mout − m in =∫

c.s.ρ(q · n) dS (1.14)

where q is the velocity vector (u, v, w) and ρq · n is the rate of mass leaving across andnormal to the surface element dS (n is the outward normal), as shown in Fig. 1.8. Sincemass is conserved, and no new material is being produced, then the sum of Eq. (1.13a) andEq. (1.14) must be equal to zero:

dmc.v.

dt= ∂

∂t

∫c.v.

ρ dV +∫

c.s.ρ(q · n) dS = 0 (1.15)

Equation (1.15) is the integral representation of the conservation of mass. It simply statesthat any change in the mass of the fluid in the control volume is equal to the rate of massbeing transported across the control surface (c.s.) boundaries.

In a similar manner the rate of change in the momentum of the fluid flowing through thecontrol volume at any instant d(mq)c.v./dt is the sum of the accumulation of the momentumper unit volume ρq within the control volume and of the change of the momentum acrossthe control surface boundaries:

d(mq)c.v.

dt= ∂

∂t

∫c.v.

ρq dV +∫

c.s.ρq(q · n) dS (1.16)

This change in the momentum, as given in Eq. (1.16), according to Newton’s second lawmust be equal to the forces �F applied to the fluid inside the control volume:

d(mq)c.v.

dt= �F (1.17)

The forces acting on the fluid in the control volume in the xi direction are either bodyforces ρ fi per unit volume or surface forces n jτi j per unit area, as discussed in Section 1.4:

(�F)i =∫

c.v.ρ fi dV +

∫c.s.

n jτi j dS (1.18)

where n is the unit normal vector that points outward from the control volume.


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By substituting Eqs. (1.16) and (1.18) into Eq. (1.17), the integral form of the momentumequation in the i direction is obtained:

∂

∂t

∫c.v.

ρqi dV +∫

c.s.ρqi (q · n) dS =

∫c.v.

ρ fi dV +∫

c.s.n jτi j dS (1.19)

This approach can be used to develop additional governing equations, such as the energyequation. However, for the fluid dynamic cases that are being considered here, the mass andthe momentum equations are sufficient to describe the fluid motion.

1.6 Differential Form of the Fluid Dynamic Equations

Equations (1.15) and (1.19) are the integral forms of the conservation of massand momentum equations. In many cases, though, the differential representation is moreuseful. In order to derive the differential form of the conservation of mass equation, bothintegrals of Eq. (1.15) should be volume integrals. This can be accomplished by the use ofthe divergence theorem (see, Kellogg,1.3 p. 39), which states that for a vector q:∫

c.s.n · q dS =

∫c.v.

∇ · q dV (1.20)

If q is the flow velocity vector then this equation states that the fluid flux through theboundary of the control surface (left-hand side) is equal to the rate of expansion of the fluid(right-hand side) inside the control volume. In Eq. (1.20), ∇ is the gradient operator, which,in Cartesian coordinates, is

∇ = i∂

∂x+ j

∂

∂y+ k

∂

∂z

or in indicial form

∇ = e j∂

∂x j

where e j is the unit vector (i, j, k, for j = 1, 2, 3). Thus the indicial form of the divergencetheorem becomes∫

c.s.n j q j dS =

∫c.v.

∂q j

∂x jdV (1.20a)

An application of Eq. (1.20) to the surface integral term in Eq. (1.15) transforms it to avolume integral:∫

c.s.ρ(q · n) dS =

∫c.v.

(∇ · ρq) dV

This allows the two terms to be combined as one volume integral:∫c.v.

(∂ρ

∂t+ ∇ · ρq

)dV = 0

where the time derivative is taken inside the integral since the control volume is stationary.Because the equation must hold for an arbitrary control volume anywhere in the fluid, theintegrand is also equal to zero. Thus, the following differential form of the conservation ofmass or the continuity equation is obtained:

∂ρ

∂t+ ∇ · ρq = 0 (1.21)


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1.6 Differential Form of the Fluid Dynamic Equations 9

Expansion of the second term of Eq. (1.21) yields

∂ρ

∂t+ q · ∇ρ + ρ∇ · q = 0 (1.21a)

and in Cartesian coordinates

∂ρ

∂t+ u

∂ρ

∂x+ v

∂ρ

∂y+ w

∂ρ

∂z+ ρ

(∂u

∂x+ ∂v

∂y+ ∂w

∂z

)= 0 (1.21b)

Use of the material derivative

D

Dt≡ ∂

∂t+ q · ∇ = ∂

∂t+ u

∂

∂x+ v

∂

∂y+ w

∂

∂z

transforms Eq. (1.21) into

Dρ

Dt+ ρ∇ · q = 0 (1.21c)

The material derivative D/Dt represents the rate of change following a fluid particle. Forexample, the acceleration of a fluid particle is given by

a = Dq

Dt= ∂q

∂t+ q · ∇q (1.22)

An incompressible fluid is a fluid whose elements cannot experience volume change.Since by definition the mass of a fluid element is constant, the fluid elements of an incom-pressible fluid must have constant density. (A homogeneous incompressible fluid is thereforea constant-density fluid.) The continuity equation (Eq. (1.21)) for an incompressible fluidreduces to

∇ · q = ∂u

∂x+ ∂v

∂y+ ∂w

∂z= 0 (1.23)

Note that the incompressible continuity equation does not have time derivatives (but timedependency can be introduced via time-dependent boundary conditions).

To obtain the differential form of the momentum equation, the divergence theorem(Eq. (1.20a)) is applied to the surface integral terms of Eq. (1.19):∫

c.s.ρqi (q · n) dS =

∫c.v.

∇ · ρqi q dV

∫c.s.

n jτi j dS =∫

c.v.

∂τi j

∂x jdV

Substitution of these results into Eq. (1.19) yields∫c.v.

[∂

∂t(ρqi ) + ∇ · ρqi q − ρ fi − ∂τi j

∂x j

]dV = 0 (1.24)

Since this integral holds for an arbitrary control volume, the integrand must be zero andtherefore

∂

∂t(ρqi ) + ∇ · ρqi q = ρ fi + ∂τi j

∂x j(i = 1, 2, 3) (1.25)

Expanding the left-hand side of Eq. (1.25) first, and then using the continuity equation, wecan reduce the left-hand side to

∂

∂t(ρqi ) + ∇ · (ρqi q) = qi

[∂ρ

∂t+ ∇ · ρq

]+ ρ

[∂qi

∂t+ q · ∇qi

]= ρ

Dqi

Dt


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(Note that the fluid acceleration is

ai = Dqi

Dt

which, according to Newton’s second law, when multiplied by the mass per volume mustbe equal to �Fi .)

So, after substituting this form of the acceleration term into Eq. (1.25), the differentialform of the momentum equation becomes ρai = �Fi or

ρDqi

Dt= ρ fi + ∂τi j

∂x j(i = 1, 2, 3) (1.26)

and in Cartesian coordinates

ρ

(∂u

∂t+ u

∂u

∂x+ v

∂u

∂y+ w

∂u

∂z

)= �Fx = ρ fx + ∂τxx

∂x+ ∂τxy

∂y+ ∂τxz

∂z

(1.26a)

ρ

(∂v

∂t+ u

∂v

∂x+ v

∂v

∂y+ w

∂v

∂z

)= �Fy = ρ fy + ∂τxy

∂x+ ∂τyy

∂y+ ∂τyz

∂z

(1.26b)

ρ

(∂w

∂t+ u

∂w

∂x+ v

∂w

∂y+ w

∂w

∂z

)= �Fz = ρ fz + ∂τxz

∂x+ ∂τyz

∂y+ ∂τzz

∂z

(1.26c)

(Note that in Eqs. (1.26a–c) the symmetry of the stress vector has been enforced.) For aNewtonian fluid the stress components τi j are given by Eq. (1.10), and by substituting theminto Eqs. (1.26a–c), the Navier–Stokes equations are obtained:

ρ

(∂qi

∂t+ q · ∇qi

)= ρ fi − ∂

∂xi

(p + 2

3μ∇ · q

)+ ∂

∂x jμ

(∂qi

∂x j+ ∂q j

∂xi

)

(i = 1, 2, 3) (1.27)

which in Cartesian coordinates are

ρ

(∂u

∂t+ q · ∇u

)= ρ fx − ∂p

∂x+ ∂

∂x

{μ

[2∂u

∂x− 2

3(∇ · q)

]}

+ ∂

∂y

[μ

(∂u

∂y+ ∂v

∂x

)]+ ∂

∂z

[μ

(∂w

∂x+ ∂u

∂z

)](1.27a)

ρ

(∂v

∂t+ q · ∇v

)= ρ fy − ∂p

∂y+ ∂

∂y

{μ

[2∂v

∂y− 2

3(∇ · q)

]}

+ ∂

∂z

[μ

(∂v

∂z+ ∂w

∂y

)]+ ∂

∂x

[μ

(∂u

∂y+ ∂v

∂x

)](1.27b)

ρ

(∂w

∂t+ q · ∇w

)= ρ fz − ∂p

∂z+ ∂

∂z

{μ

[2∂w

∂z− 2

3(∇ · q)

]}

+ ∂

∂x

[μ

(∂w

∂x+ ∂u

∂z

)]+ ∂

∂y

[μ

(∂v

∂z+ ∂w

∂y

)](1.27c)


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Figure 1.9 Direction of tangential and normal velocity components near a solid boundary.

Typical boundary conditions for this problem require that on stationary solid boundaries(Fig. 1.9) both the normal and tangential velocity components will reduce to zero:

qn = 0 (on solid surface) (1.28a)

qt = 0 (on solid surface) (1.28b)

The number of exact solutions to the Navier–Stokes equations is small because of thenonlinearity of the differential equations. However, in many situations some terms can beneglected so that simpler equations can be obtained. For example, by assuming constantviscosity coefficient μ, Eq. (1.27) becomes

ρ

(∂q

∂t+ q · ∇q

)= ρf − ∇ p + μ∇2q + μ

3∇(∇ · q) (1.29)

Furthermore, by assuming an incompressible fluid (for which the continuity equation(Eq. (1.23)) becomes ∇ · q = 0), Eq. (1.27) reduces to

ρ

(∂q

∂t+ q · ∇q

)= ρf − ∇ p + μ∇2q (1.30)

For an inviscid compressible fluid

∂q

∂t+ q · ∇q = f − ∇ p

ρ(1.31)

This equation is called the Euler equation.In situations in which the problem has cylindrical or spherical symmetry, the use of ap-

propriate coordinates can simplify the solution. As an example, we present the fundamentalequations for an incompressible fluid with constant viscosity. The cylindrical coordinatesystem is described in Fig. 1.10, and for this example the r, θ coordinates are in a planenormal to the x coordinate. The operators ∇, ∇2, and D/Dt in the r, θ, x system are (seePai,1.4 p. 38 or Yuan,1.2 p. 132)

∇ =(

er∂

∂r, eθ

1

r

∂

∂θ, ex

∂

∂x

)(1.32)

∇2 = ∂2

∂r2+ 1

r

∂

∂r+ 1

r2

∂2

∂θ2+ ∂2

∂x2(1.33)

D

Dt= ∂

∂t+ qr

∂

∂r+ qθ

r

∂

∂θ+ qx

∂

∂x(1.34)


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Figure 1.10 Cylindrical coordinate system.

The continuity equation in cylindrical coordinates for an incompressible fluid then be-comes

∂qr

∂r+ 1

r

∂qθ

∂θ+ ∂qx

∂x+ qr

r= 0 (1.35)

The momentum equations for an incompressible fluid are

r direction:

ρ

(Dqr

Dt− q2

θ

r

)= ρ fr − ∂p

∂r+ μ

(∇2qr − qr

r2− 2

r2

∂qθ

∂θ

)(1.36)

θ direction:

ρ

(Dqθ

Dt+ qr qθ

r

)= ρ fθ − 1

r

∂p

∂θ+ μ

(∇2qθ + 2

r2

∂qr

∂θ− qθ

r2

)(1.37)

x direction:

ρDqx

Dt= ρ fx − ∂p

∂x+ μ∇2qx (1.38)

A spherical coordinate system with the coordinates r, θ, ϕ is described in Fig. 1.11. Theoperators ∇, ∇2, and D/Dt in the r, θ, ϕ system are (Karamcheti,1.5 Chapter 2 or Yuan,1.2

p. 132)

∇ =(

er∂

∂r, eθ

1

r

∂

∂θ, eϕ

1

r sin θ

∂

∂ϕ

)(1.39)

∇2 = 1

r2

∂

∂r

(r2 ∂

∂r

)+ 1

r2 sin θ

∂

∂θ

(sin θ

∂

∂θ

)+ 1

r2 sin2 θ

∂2

∂ϕ2(1.40)

D

Dt= ∂

∂t+ qr

∂

∂r+ qθ

r

∂

∂θ+ qϕ

r sin θ

∂

∂ϕ(1.41)

The continuity equation in spherical coordinates for an incompressible fluid becomes(Pai,1.4 p. 40)

1

r

∂(r2qr )

∂r+ 1

sin θ

∂(qθ sin θ)

∂θ+ 1

sin θ

∂qϕ

∂ϕ= 0 (1.42)


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Figure 1.11 Spherical coordinate system.

The momentum equations for an incompressible fluid are (Pai,1.4 p. 40)

r direction:

ρ

(Dqr

Dt− q2

ϕ + q2θ

r

)

= ρ fr − ∂p

∂r+ μ

(∇2qr − 2qr

r2− 2

r2

∂qθ

∂θ− 2qθ cot θ

r2− 2

r2 sin θ

∂qϕ

∂ϕ

)

(1.43)

θ direction:

ρ

(Dqθ

Dt+ qr qθ

r− q2

ϕ cot θ

r

)

= ρ fθ − 1

r

∂p

∂θ+ μ

(∇2qθ + 2

r2

∂qr

∂θ− qθ

r2 sin2 θ− 2 cos θ

r2 sin2 θ

∂qϕ

∂ϕ

)(1.44)

ϕ direction:

ρ

(Dqϕ

Dt+ qϕqr

r+ qθqϕ cot θ

r

)

= ρ fϕ − 1

r sin θ

∂p

∂ϕ+ μ

(∇2qϕ − qϕ

r2 sin2 θ+ 2

r2 sin θ

∂qr

∂ϕ+ 2 cos θ

r2 sin2 θ

∂qθ

∂ϕ

)

(1.45)

When a two-dimensional flowfield is treated in this text, it will be described in either aCartesian coordinate system with coordinates x and z or in a corresponding polar coordi-nate system with coordinates r and θ (see Fig. 1.12). In this polar coordinate system, thecontinuity equation for an incompressible fluid is obtained from Eq. (1.35) by eliminating∂qx/∂x , and the r - and θ -momentum equations for an incompressible fluid are identical toEqs. (1.36) and (1.37), respectively.


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