Fluid Mechanics: An Intermediate Approach [1 ed.] 1466598980, 9781466598980

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FLUID MECHANICS

An Intermediate Approach

“… addresses a series of topics in this book that are often left to more advanced courses or not covered at all at university. … this book is recommended to students undertaking advanced topics in engineering fluid mechanics as well as practicing engineers who find they need something that goes beyond the treatment found in introductory texts.” —Professor Peter R.N. Childs, Head of School, Dyson School of Design Engineering, Imperial College London

“… the topics here… cover the arguments of at least two courses at MSc level. … The arguments were treated in a proper way and clearly explained. … I particularly appreciated [the] presence of a chapter on flow network modelling, which is usually treated in MSc courses using very crude assumptions. I think that this book can be used in many universities throughout the world. I am thinking to include parts of the arguments in my course…” —Professor Domenico Borello, Dipartimento di Ingegneria Meccanica e Aerospaziale, Sapienza Università di Roma, Italy

“Bijay Sultanian has brought together in one well-written book the necessary material an engineer needs to know to be able to design and analyze fluid machinery.” —Dr. Ali Ameri, The Ohio State University, Columbus, USA

—Dr. Phil Ligrani, Eminent Scholar in Propulsion, University of Alabama in Huntsville, USA

Sultanian

“This new volume offers excellent intermediate course material in fluid mechanics. The coverage, example problems, and material are carefully designed to guide students through challenging concepts for a variety of subject areas. I highly recommend this text.”

FLUID MECHANICS

FLUID DYNAMICS

Spiral Vortex Stream Function:

=

2

ln r

m 2

FLUID MECHANICS

An Intermediate Approach Bijay K. Sultanian

K20929

an informa business

K20929_cover_revised.indd 1

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ISBN: 978-1-4665-9898-0

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w w w. c rc p r e s s . c o m

Taylor & Francis Group

7/10/15 10:52 AM

FLUID MECHANICS

An Intermediate Approach

FLUID MECHANICS

An Intermediate Approach Bijay K. Sultanian

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2016 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20150420 International Standard Book Number-13: 978-1-4665-9899-7 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

To my parents, teachers, and friends; wife, Bimla; daughter, Rachna; son, Dheeraj (Raj); son-in-law, Shahin; and grandchildren, Aarti, Soraya, and Shayan, who have immensely enriched my life.

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Contents Preface..............................................................................................................................................xv Acknowledgments....................................................................................................................... xix Author............................................................................................................................................ xxi

1. Kinematics of Fluid Flow.......................................................................................................1 1.1 Introduction.................................................................................................................... 1 1.2 What Is a Fluid?.............................................................................................................. 2 1.2.1 Continuum Assumption.................................................................................. 3 1.2.2 Fluid Mechanics versus Solid Mechanics..................................................... 3 1.3 Streamline, Pathline, and Streakline...........................................................................4 1.3.1 Streamline..........................................................................................................4 1.3.2 Pathline...............................................................................................................5 1.3.3 Streakline........................................................................................................... 5 1.4 Conservation Principles for a Material Region..........................................................7 1.5 Basic Analysis Techniques............................................................................................ 7 1.6 Some Interesting Flows................................................................................................. 8 1.6.1 Sudden Expansion Pipe Flow without Swirl................................................ 8 1.6.2 Sudden Expansion Pipe Flow with Swirl......................................................9 1.6.3 Secondary Flows............................................................................................. 10 1.6.4 Rotor Cavity Flow Field................................................................................. 11 1.7 Properties of Velocity Field......................................................................................... 12 1.7.1 Local Mass Conservation in Fluid Flow...................................................... 12 1.7.2 Convection in Fluid Flow.............................................................................. 15 1.7.3 Vorticity............................................................................................................ 19 1.7.4 Vortex................................................................................................................ 20 1.7.5 Circulation....................................................................................................... 24 1.8 Concluding Remarks................................................................................................... 25 Problems................................................................................................................................... 26 References................................................................................................................................ 27 Nomenclature.......................................................................................................................... 28 2. Key Concepts of Thermofluids........................................................................................... 31 2.1 Introduction.................................................................................................................. 31 2.2 Pressure......................................................................................................................... 32 2.2.1 Static Pressure and Total Pressure............................................................... 32 2.2.2 Calculation of Total Pressure in Incompressible and Compressible Flows........................................................................................ 33 2.3 Temperature.................................................................................................................. 40 2.3.1 Static Temperature and Total Temperature................................................. 40 2.3.2 Adiabatic Wall Temperature......................................................................... 41 2.4 Internal Energy, Enthalpy, and Entropy...................................................................43 2.5 Stream Thrust............................................................................................................... 48 vii

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2.6 Rothalpy........................................................................................................................ 51 2.7 Concluding Remarks................................................................................................... 53 Problems................................................................................................................................... 53 Bibliography............................................................................................................................ 55 Nomenclature.......................................................................................................................... 55 3. Control Volume Analysis..................................................................................................... 59 3.1 Introduction.................................................................................................................. 59 3.2 Lagrangian versus Eulerian Approach..................................................................... 59 3.3 Reynolds Transport Theorem.................................................................................... 60 3.3.1 Integral Control Volume Equations............................................................. 61 3.3.2 Control Volume Moving at Constant Velocity............................................63 3.4 Integral Mass Conservation Equation......................................................................64 3.5 Differential Mass Conservation Equation................................................................65 3.5.1 Using a Differential Control Volume...........................................................65 3.5.2 Using Gauss’s Divergence Theorem............................................................. 66 3.6 Linear Momentum Equation in Inertial Reference Frame..................................... 67 3.6.1 Control Volume with Multiple Inlets and Outlets..................................... 68 3.6.2 Thrust of a Rocket Engine............................................................................. 74 3.6.3 Nonuniform Velocity Profiles at Inlets and Outlets..................................77 3.6.4 Pressure Distribution over a Control Volume............................................ 81 3.6.5 Body and Surface Forces Acting on a Control Volume.............................84 3.7 Linear Momentum Equation in Non-Inertial Reference Frame............................ 92 3.8 Angular Momentum Equation in Inertial and Non-Inertial Reference Frames....................................................................................................... 100 3.8.1 Non-Inertial Control Volume with Constant Rotation............................ 102 3.9 Energy Conservation Equation................................................................................ 109 3.9.1 Forms of Steady-Flow Energy Equation.................................................... 111 3.10 Entropy Equation....................................................................................................... 118 3.11 Concluding Remarks................................................................................................. 119 Problems................................................................................................................................. 120 Bibliography.......................................................................................................................... 125 Nomenclature........................................................................................................................ 125 4. Bernoulli Equation.............................................................................................................. 129 4.1 Introduction................................................................................................................ 129 4.2 Original Bernoulli Equation..................................................................................... 129 4.2.1 Exchange between Kinetic Energy and Potential Energy...................... 129 4.2.2 Exchange between Flow Work and Kinetic Energy................................ 132 4.2.3 Exchange of Flow Energies in a Stream Tube: The Bernoulli Equation......................................................................................................... 134 4.2.4 Derivation of Bernoulli Equation from Euler Momentum Equations........................................................................................................ 137 4.2.5 Derivation of Bernoulli Equation for a Potential (Irrotational) Flow................................................................................................................ 139 4.3 Extended Bernoulli Equation................................................................................... 141 4.3.1 Minor and Major Head Losses.................................................................... 143 4.4 Concluding Remarks................................................................................................. 147

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Problems................................................................................................................................. 148 References.............................................................................................................................. 151 Bibliography.......................................................................................................................... 151 Nomenclature........................................................................................................................ 151 5. Compressible Flow.............................................................................................................. 155 5.1 Introduction................................................................................................................ 155 5.2 Classification of Compressible Flows...................................................................... 155 5.2.1 Speed of Sound............................................................................................. 155 5.2.2 Compressible Flow Mach Numbers........................................................... 158 5.3 Compressible Flow Functions.................................................................................. 161 5.3.1 Mass Flow Functions.................................................................................... 162 5.3.2 Impulse Functions........................................................................................ 168 5.3.3 Normal Shock Function............................................................................... 170 5.3.4 Some Practical Considerations.................................................................... 172 5.4 Variable-Area Duct Flow with Friction, Heat Transfer, and Rotation................ 175 5.4.1 Change in Relative Flow Velocity............................................................... 177 5.5 Isentropic Flow in a Variable-Area Duct................................................................ 180 5.5.1 Critical Throat Area...................................................................................... 182 5.5.2 Key Characteristics of Isentropic Flows in a Variable-Area Duct....................................................................................... 184 5.6 Isentropic Flow in a Constant-Area Duct with Rotation...................................... 185 5.7 Isentropic Flow in a Variable-Area Duct with Rotation....................................... 188 5.8 Fanno Flow.................................................................................................................. 188 5.8.1 Fanno Line..................................................................................................... 191 5.8.2 Quantitative Evaluation of Properties in a Fanno Flow.......................... 193 5.8.3 Fanno Flow Table with the Reference State at M = 1............................... 201 5.9 Rayleigh Flow............................................................................................................. 202 5.9.1 Quantitative Evaluation of Properties in a Rayleigh Flow................................................................................................ 205 5.9.2 Rayleigh Line................................................................................................. 207 5.9.3 Trends of Changes in Rayleigh Flow Properties...................................... 209 5.9.4 Loss in Total Pressure in a Rayleigh Flow with Heating........................ 210 5.9.5 Rayleigh Flow Table with the Reference State at M = 1.......................... 213 5.10 Isothermal Constant-Area Flow with Friction....................................................... 214 5.10.1 Table for Isothermal Constant-Area Flow with Friction Using the Reference State at M = 1/ k .................................................................. 216 5.11 Normal Shock............................................................................................................. 220 5.11.1 Governing Conservation and Auxiliary Equations................................. 221 5.11.2 Rankine–Hugoniot Equations.................................................................... 221 5.11.3 Constancy of Normal Shock Function....................................................... 224 5.11.4 Prandtl–Meyer Equation..............................................................................225 5.11.5 Changes in Properties with Upstream Mach Number........................... 226 5.11.6 Normal Shock Table.....................................................................................234 5.11.7 Intersections of Fanno and Rayleigh Lines...............................................234 5.12 Oblique Shock............................................................................................................. 236 5.12.1 Analysis of an Oblique Shock..................................................................... 237 5.12.2 Changes in Properties with Upstream Mach Number........................... 238

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5.12.3 Equations Relating Wave Angle, Deflection Angle, and Upstream Mach Number............................................................................. 239 5.12.4 Oblique Shock Polar Hodograph................................................................ 243 5.13 Prandtl–Meyer Flow.................................................................................................. 246 5.14 Operation of Nozzles and Diffusers....................................................................... 249 5.14.1 Operation of a Converging Nozzle............................................................ 249 5.14.2 Operation of a Converging-Diverging Nozzle......................................... 253 5.14.3 Operation of a Supersonic Wind Tunnel................................................... 257 5.15 Concluding Remarks................................................................................................. 260 Problems................................................................................................................................. 261 References.............................................................................................................................. 268 Bibliography........................................................................................................................... 268 Nomenclature........................................................................................................................ 268 6. Potential Flow....................................................................................................................... 271 6.1 Introduction................................................................................................................ 271 6.2 Basic Concepts............................................................................................................ 271 6.2.1 Velocity Potential.......................................................................................... 271 6.2.2 Stream Function............................................................................................ 272 6.2.3 Complex Potential, Complex Velocity, and Complex Circulation......... 276 6.2.4 Two Basic Types of Singularities................................................................ 278 6.3 Elementary Plane Potential Flows........................................................................... 278 6.3.1 Uniform Flows.............................................................................................. 278 6.3.2 Source and Sink............................................................................................ 279 6.3.3 Vortex.............................................................................................................. 280 6.3.4 Corner Flows................................................................................................. 282 6.4 Superposition of Two or More Plane Potential Flows...........................................284 6.4.1 Superposition of Uniform Flow and Source: Flow around a Rankine Half-Body....................................................................................... 285 6.4.2 Superposition of Source and Sink: Dipole................................................ 288 6.4.3 Superposition of Source and Sink: Doublet.............................................. 289 6.4.4 Superposition of Uniform Flow and Doublet: Flow around a Cylinder....................................................................................................... 292 6.4.5 Superposition of Uniform Flow, Doublet, and Vortex: Flow around a Cylinder with Circulation.......................................................... 293 6.4.6 Milne-Thomson Circle Theorem................................................................ 296 6.5 Force and Moment on a Body in Plane Potential Flows....................................... 297 6.5.1 Blasius Integral Theorems........................................................................... 297 6.5.2 Force and Moment on a Cylinder with Circulation................................. 299 6.5.3 D’Alembert’s Paradox...................................................................................300 6.5.4 Magnus Effect................................................................................................300 6.6 Conformal Transformation....................................................................................... 301 6.6.1 Joukowski Transformation.......................................................................... 302 6.6.2 Kutta Condition.............................................................................................306 6.7 Concluding Remarks.................................................................................................308 Problems.................................................................................................................................309 References.............................................................................................................................. 310 Bibliography.......................................................................................................................... 310 Nomenclature........................................................................................................................ 310

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7. Navier–Stokes Equations: Exact Solutions..................................................................... 313 7.1 Introduction................................................................................................................ 313 7.2 Forces on a Fluid Element......................................................................................... 313 7.2.1 Surface Forces due to Stresses..................................................................... 313 7.2.2 Body Force due to Gravity........................................................................... 315 7.3 Deformation Rate Tensor.......................................................................................... 316 7.4 Differential Forms of the Equations of Motion..................................................... 318 7.4.1 Continuity Equation..................................................................................... 318 7.4.2 Linear Momentum Equation....................................................................... 318 7.5 Navier–Stokes Equations.......................................................................................... 319 7.5.1 Constitutive Relationship between Stress and Rate of Strain................ 319 7.5.2 Navier–Stokes Equations and Their Simplifications............................... 320 7.6 Exact Solutions........................................................................................................... 322 7.6.1 Couette Flow.................................................................................................. 322 7.6.2 Hagen-Poiseuille Flow................................................................................. 331 7.7 Navier–Stokes Equations in Terms of Vorticity and Stream Function.............. 338 7.7.1 Vorticity Transport Equation...................................................................... 339 7.7.2 Stream Function Transport Equation........................................................ 341 7.8 Slow Flow.................................................................................................................... 341 7.9 Concluding Remarks.................................................................................................342 Problems.................................................................................................................................343 References..............................................................................................................................345 Bibliography..........................................................................................................................345 Nomenclature........................................................................................................................345 8. Boundary Layer Flow.......................................................................................................... 349 8.1 Introduction................................................................................................................ 349 8.2 Description of a Boundary Layer............................................................................ 350 8.2.1 External Flow................................................................................................ 350 8.2.2 Internal Flow................................................................................................. 350 8.3 Differential Boundary Layer Equations................................................................. 351 8.4 Von Karman Momentum Integral Equation.......................................................... 355 8.4.1 Boundary Layer Thickness.......................................................................... 355 8.4.2 Displacement Thickness.............................................................................. 355 8.4.3 Momentum Thickness................................................................................. 356 8.4.4 Derivation of Von Karman Momentum Integral Equation.................... 357 8.4.4.1 Method 1: Integration of Equation 8.9........................................ 357 8.4.4.2 Method 2: Using Control Volume Analysis............................... 358 8.5 Laminar Boundary Layer on a Flat Plate............................................................... 360 8.5.1 Exact Solution Using Similarity Variables................................................ 361 8.5.2 Approximate Solution Using the Momentum Integral Method............ 366 8.6 Laminar Boundary Layer in Wedge Flows............................................................ 369 8.6.1 Free-Stream Velocity in a Wedge Flow...................................................... 369 8.6.2 Falkner–Skan Equation................................................................................ 371 8.7 Boundary Layer Separation...................................................................................... 373 8.8 Concluding Remarks................................................................................................. 375 Problems................................................................................................................................. 376 References.............................................................................................................................. 377 Nomenclature........................................................................................................................ 377

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9. Flow Network Modeling.................................................................................................... 379 9.1 Introduction................................................................................................................ 379 9.2 Anatomy of a Flow Network.................................................................................... 380 9.2.1 Lumped Fluid Element................................................................................ 380 9.2.2 Internal Junction........................................................................................... 381 9.2.3 Boundary Junction........................................................................................ 382 9.3 Physics-Based Modeling........................................................................................... 382 9.3.1 Lumped Fluid Element................................................................................ 382 9.3.2 Distributed Fluid Element........................................................................... 383 9.3.3 Internal Junction........................................................................................... 383 9.3.4 Bridge Equations...........................................................................................384 9.4 Incompressible Flow Network................................................................................. 385 9.4.1 Total-Pressure-Based Formulation............................................................. 386 9.4.2 Static-Pressure-Based Formulation............................................................ 386 9.4.3 Incompressible Flow Orifice: Loss Coefficient and Discharge Coefficient................................................................................... 388 9.5 Compressible Flow Network.................................................................................... 389 9.5.1 LFE Modeling with Area Change, Friction, Heat Transfer, and Rotation.................................................................................................. 390 9.5.1.1 Internal Choking and Normal Shock Formation..................... 394 9.5.1.2 Determination of LFE Flow Direction....................................... 394 9.5.2 Compressible Flow Orifice: Discharge Coefficient.................................. 398 9.5.3 Internal Junction Modeling......................................................................... 398 9.6 Flow Network Solution............................................................................................. 399 9.6.1 Initial Solution Generation.......................................................................... 401 9.6.2 Newton–Raphson Method.......................................................................... 401 9.6.3 Modified Newton–Raphson Method......................................................... 402 9.7 Concluding Remarks................................................................................................. 403 References.............................................................................................................................. 403 Bibliography.......................................................................................................................... 403 Nomenclature........................................................................................................................404 10. Turbulent Flow Computational Fluid Dynamics: An Industrial Overview........... 407 10.1 Introduction................................................................................................................ 407 10.2 Industrial Analysis and Design Systems................................................................408 10.3 CFD Technology Used in Various Industries........................................................ 409 10.4 CFD Methodology..................................................................................................... 410 10.5 Common Form of Governing Conservation Equations....................................... 412 10.6 Physics of Turbulence................................................................................................ 414 10.7 Turbulence Modeling................................................................................................ 415 10.7.1 Reynolds Equations: The Closure Problem............................................... 415 10.7.2 Prandtl Mixing-Length Model.................................................................... 417 10.7.3 One-Equation k–L Model............................................................................. 418 10.7.4 Two-Equation k–ε Model.............................................................................. 418 10.7.5 Reynolds Stress Transport Model.............................................................. 420 10.7.6 Algebraic Stress Model................................................................................422

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10.8 Boundary Conditions................................................................................................423 10.8.1 Inlet and Outlet Boundary Conditions......................................................423 10.8.2 Wall Boundary Conditions: The Wall-Function Treatment....................423 10.8.3 Alternative Near-Wall Treatments............................................................. 426 10.9 Choice of a Turbulence Model.................................................................................. 427 10.9.1 Turbulent Flow in a Noncircular Duct...................................................... 427 10.9.2 Turbulent Flow in a Sudden Pipe Expansion........................................... 427 10.9.3 Swirling Turbulent Flow in a Sudden Pipe Expansion........................... 428 10.9.4 Turbulent Flow and Heat Transfer in a Rotor Cavity.............................. 428 10.10 Illustrative Design Applications of CFD Technology........................................... 428 10.10.1 Cooling Flow in a Compressor Rotor Drum Cavity................................ 429 10.10.2 Aerodynamic Design of a Gas Turbine Exhaust Diffuser......................430 10.10.3 High-Pressure Inlet Bleed Heat System Design....................................... 431 10.10.4 Gas Turbine Enclosure Ventilation Design: ATEX Certification............ 431 10.11 Physics-Based Post-Processing of CFD Results..................................................... 431 10.11.1 Large Control Volume Analysis of CFD Results...................................... 432 10.11.2 Entropy Map from CFD Results.................................................................434 10.12 Concluding Remarks................................................................................................. 435 References.............................................................................................................................. 435 Bibliography.......................................................................................................................... 437 Nomenclature........................................................................................................................ 438 Appendix A: Compressible Flow Equations and Tables.................................................... 441 Table A.1 Compressible Flow Functions..........................................................................442 Table A.2 Isentropic Compressible Flow.........................................................................454 Table A.3 Fanno Flow......................................................................................................... 466 Table A.4 Rayleigh Flow..................................................................................................... 478 Table A.5 Isothermal Constant-Area Flow with Friction.............................................. 490 Table A.6 Normal Shock.................................................................................................... 502 Table A.7 Prandtl–Meyer Function and Mach Angle.................................................... 510 Appendix B: Analytical Solution of Coupled Heat Transfer and Work Transfer in a Rotating Duct Flow............................................................................................................ 515 Appendix C: Temperature and Pressure Changes in Isentropic Free and Forced Vortices............................................................................................................................ 521 Appendix D: Converting Quantities between Stator and Rotor Reference Frames.........527 Appendix E: Vorticity and Circulation.................................................................................. 531 Appendix F: Review of Necessary Mathematics.................................................................. 537 Appendix G: Suggested Project Problems.............................................................................543 Index.............................................................................................................................................. 547

Preface This book is a unique blend of my passion in fluid mechanics, 40+ years of industry ­experience, and nearly a decade of teaching graduate-level courses in turbomachinery and fluid mechanics at the University of Central Florida (UCF). Fluid mechanics can be the most beautiful and most enjoyable area of engineering. Its true beauty lies not in its mathematical descriptions but in the understanding of its underlying physics—the deeper you go, the more beautiful it feels. This is one subject that is vividly displayed in all aspects of nature—a rich source of complex flow visualization. In this book, as a refreshing change, the mathematics follows the flow physics, not the other way around. After over 35 years of industry experience in rocket propulsion, thermal and fluid flow modeling of processes in steel plant engineering, development of physics-based methods and tools for the design of some of the most advanced gas turbines for aircraft propulsion and power generation, and extensive application of three-dimensional (3D) c­ omputational fluid dynamics (CFD) technology to the solutions of many design problems, I  moved to Orlando, Florida, in 2006 to work for Siemens Energy. UCF invited me that year to join the faculty as an adjunct professor and teach the graduate course “EML5402— Turbomachinery” in the fall semester. I immediately accepted the assignment. It allowed me to bring my industry experience into the classroom to train a new generation of talent for the practical, rather than the theoretical, problems facing today’s engineers. The following year, UCF invited me to teach another graduate course, “EML5713—Intermediate Fluid Mechanics,” in the spring semester. The class size grew each year, from around 15 in 2007 to around 40 by the fall of 2013. In teaching the second course, I struggled to find an adequate textbook that stressed the fundamental physical harmony of fluid flows as a prerequisite to formulas and problem solving. Faced with this challenge, I did what any great engineer would do: I designed one that did. Most undergraduate engineering programs include only a couple of courses on fluid mechanics. The students taking these courses are generally exposed to the concepts of statics and dynamics from prior courses on solid mechanics. As a result, many tend to find the new concepts of fluid mechanics overwhelming and nonintuitive. Due to a continued emphasis on mathematical treatment, rather than the physical introduction to the subject, many students lose sight of the beauty and simplicity of fluid mechanics in favor of simply committing to memory its mathematical equations, including the difficult Navier–Stokes equations (which are nonlinear partial differential equations in three coordinate directions). They soon start treating these undergraduate fluids courses as just another course in advanced mathematics, developing little understanding of the underlying physical concepts. The pressure of examinations and grades in these courses only reinforces this treatment. As a prerequisite to many advanced graduate courses in the thermofluids stream, many students are required to take some form of graduate-level intermediate fluid mechanics as one of their core courses. Students, weary of their undergraduate experiences, are often immediately frustrated by the prospect of dealing with even more complex mathematics, at times in tensor notation! Although many can recite complex formulas, when asked to articulate the difference between the static pressure and total pressure in plain English, and how to compute the latter in incompressible and compressible flows, they cannot. The difference between static temperature and total temperature or the concept of adiabatic wall xv

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temperature is surprisingly hard to articulate in plain English without a fundamental physical understanding. Adding stationary (inertial) and rotating reference frames only amplifies these hidden issues. Overreliance on the Bernoulli equation is a classic demonstration of where a mathematics-first approach fails many engineering students. The concept of choking in one-dimensional compressible flows, for example, can be exceedingly complicated using a mathematics-first approach, but asking students to take a step back to first visualize the phenomenon in nature and understand the physical concept of Mach number can lead to a world of difference. Although many students may be conversant with the equations of Fanno and Rayleigh flows, they find it hard to explain in layman’s terms how wall friction accelerates a subsonic airflow in a constant-area duct with or without heating. Many practicing engineers dealing with fluid flows in various industries are, as a result of the classic academic approach (coupled with dependence on one or more in-house or commercial codes to perform design calculations), surprisingly weak in their intuitive understanding of the physics of these flows. When questioned about the accuracy of their calculations, often an unshaken faith in design tools and their predictions is the answer. For many in the industry, if they, and their competition, have been doing it in a particular way for so many years, it must be right! With the availability of so many user-friendly commercial 3D CFD codes, today’s challenge in the application of 3D CFD technology to design lies not in high-quality and highfidelity grid generation or in obtaining a fully converged solution with the state-of-the-art numerical schemes and turbulence models but in properly interpreting their computed 3D CFD results for intended design applications. Only a strong foundation in fluid mechanics will help the CFD lovers to make good sense of their 3D CFD results. I strongly believe that this book will benefit not only the graduate and senior undergraduate students of fluid mechanics pursuing their degree programs in the thermofluids stream but also many practicing engineers dealing with thermofluids in the design of commercial and military planes; submarines and cruise ships; automobiles; jet and rocket propulsion; oil and gas pipelines; gas turbines, steam turbines, and generators; pumps and compressors; air-conditioning and refrigeration units; heat exchangers; artificial hearts and valves; dams and irrigation systems; and many other areas of thermofluids engineering. This book will come to many senior undergraduate and graduate-level students, and practicing engineers, as a breath of fresh air. Empowered by the knowledge gained from this book, these students and engineers will develop a unique insight into various fluid flow phenomena and will fall in love with the subject like never before. With focus on a clear understanding of the key fundamental concepts (which are amply illustrated by a number of worked-out real-world examples), this book will help readers develop a variety of problem-solving skills to handle practical fluids engineering problems. For example, they will master the techniques of control volume analysis of mass, linear momentum, angular momentum, and energy in both inertial and non-inertial reference frames. They will also develop an unprecedented intuitive understanding of one-dimensional compressible flows, including Fanno flows, Rayleigh flows, isothermal flows, normal shocks, oblique shocks, and isentropic Prandtl–Meyer expansion flows. By eliminating most of the key conceptual gaps for senior undergraduate and graduate-level students, the book will help them transition to learning new topics in advanced graduate courses in thermofluids. The book also includes two value-added chapters on special topics that reflect the stateof-the-art in design applications of fluid mechanics. Chapter 9 deals with the key details of physics-based modeling of both incompressible and compressible flow networks. These details, notwithstanding the current reality that a number of commercial flow network codes are used in many industries, are not found in any leading contemporary book on

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fluid mechanics. Chapter 10 focuses on the applications of CFD technology to practical industrial design problems, often in concert with related flow network solutions, rather than presenting the mathematical details of CFD numerical formulations and solution methods. Additionally, this chapter includes a physics-based methodology to post-process 3D CFD results to generate section-averaged values for their useful interpretations and design applications. The book is deemed to be an indispensable companion to all students and practicing engineers engaged in various designs, both new and upgrades, involving fluid flow and heat transfer. Chapters 1 through 4 form the core course in fluid mechanics. To develop a solid foundation in this subject, all engineers dealing with thermofluids, both in industries and in universities, must master the material presented in the first four chapters. At universities around the world, at least three distinct graduate or senior undergraduate courses in fluid mechanics can be taught using this textbook. The following syllabi are suggested for these courses; instructors, however, are free to fine-tune these syllabi and reinforce them with their notes and/or additional reference material to meet their specific instructional needs:

1. Intermediate fluid mechanics—a graduate-level core course in fluid mechanics for senior undergraduate and graduate students pursuing MS and PhD programs in the thermofluids stream (fluid mechanics, heat transfer, propulsion, turbomachinery aerodynamics, combustion, etc.): fluids core course (Chapters 1 through 4) and selected topics from Chapters 5 through 8.

2. Compressible flows or gas dynamics—a senior-level undergraduate course on compressible flows or gas dynamics: selected topics from fluids core course (Chapters 1 through 4) and Chapter 5. 3. Industrial fluid mechanics—an elective graduate-level course primarily targeted at practicing engineers in various industries dealing with fluids engineering: fluids core course (Chapters 1 through 4) and selected topics from Chapters 5, 9, and 10 and Appendix G. Overall, this textbook will serve as an effective prerequisite for advanced courses on turbulent shear flows, CFD, turbulence modeling, and so on. The major highlights of this textbook include the following: • Mathematical treatment follows flow physics, not the other way around. • Over 60 systematically worked-out real-world examples. • Over 100 chapter-end problems for which the Solutions Manual is available to all instructors who adopt this textbook for teaching their university courses in fluid mechanics. • Over 250 figures. • Clearly explained key thermofluids concepts: rothalpy, stream thrust, impulse pressure, forced/free vortices, windage, vorticity, and circulation. • Control volume analysis of linear momentum and angular momentum in both inertial and non-inertial reference frames. • Enhanced intuitive understanding of internal compressible flows through the use of various flow functions: total- and static-pressure mass flow functions, total- and static-pressure impulse functions, and normal shock function.

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Preface

• Physics-based modeling of compressible flow over all Mach numbers in a ­variable-area duct with friction, heat transfer, and rotation with internal choking and normal shock formation. • Compressible and incompressible flow network modeling (Chapter 9). • Strengths and weaknesses of state-of-the-art turbulent flow CFD predictions, including physics-based post-processing of 3D CFD results for design applications (Chapter 10). • Compressible flow tables with equations used to generate the tabular values (Appendix A). • Closed-form analytical solution of the coupled heat transfer and work transfer in a rotating duct flow (Appendix B). • Systematically derived equations to compute pressure and temperature changes in isentropic free and forced vortices (Appendix C). • Systematically derived equations to transfer stagnation (total) flow properties from a stationary reference frame to a rotating reference frame and vice versa (Appendix D). Dr. Bijay (BJ) Sultanian Founder & Managing Member, Takaniki Communications LLC Adjunct Professor, The University of Central Florida

Acknowledgments This is my dream book! A contribution of this magnitude would not have been possible without the perpetual love and support of my entire family, to whom I shall forever remain indebted. The inspiration for this dream book originated during my 12-year career at General Electric (GE). I am very fortunate to have participated in the design and development of two of the world’s largest and most efficient gas turbines: the GE 90 to propel planes and the steam-cooled 9H/7H to generate power. The challenges of heat transfer and cooling/ sealing flow designs in these machines were beyond anything I had experienced before. Among all my distinguished colleagues at GE, three individuals stand out: Mr. Ernest Elovic and Mr. Larry Plemmons at GE Aircraft Engines (GEAE) and Mr. Alan Walker at GE Power Generation. They are my true professional heroes. I owe my most sincere gratitude to Mr. Elovic and Mr. Plemmons, who introduced me to the concept of physics-based design predictions. Since it has become an integral part of my conviction, I have used the term “physics based” very often in this book. For the Managerial Award I received at GEAE in 1992, every word in the following citation, crafted by Mr. Elovic, continues to inspire me to this day: On behalf of Advanced Engineering Technologies Department, it gives me great pleasure to present to you this Managerial Award in recognition of your significant contributions to the development of improved physics-based heat transfer and fluid systems analysis methodologies of rotating engine components. These contributions have resulted in more accurate temperature and pressure predictions of critical engine parts permitting more reliable designs with more predictable life characteristics.

A gift of knowledge is the greatest gift one can give and receive. Mr. Walker gave me such a gift by sponsoring me to complete the 2-year executive MBA program at the Lally School of Management and Technology. While I remain greatly indebted to Mr. Walker for this unprecedented recognition, I also thank him for keeping my technical skills vibrant through my direct involvement in the redesign of the gas turbine enclosure ventilation system for the first full-speed no-load testing of the 9H machine; the robust design of a high-pressure inlet bleed heat system; CFD-based high-performance exhaust diffuser designs in conjunction with a joint technology development program with Toshiba, Japan; and other challenging design activities, including the applications of 3D CFD technology. I thank Prof. Ranganathan Kumar, who invited me to teach graduate courses at the University of Central Florida (UCF) in 2006 as an adjunct faculty. Without this teaching opportunity, my dream book would not have become a textbook. I remain sincerely thankful to Prof. J-C Han and Prof. Tom Shih, who wholeheartedly supported my book proposal with their rave reviews. I will continue to cherish a highly referenced book chapter on computations of internal and film cooling that Dr. Shih and I coauthored at the turn of this century. I owe many thanks to my friends Dr. Larry Wagner, who spent countless hours going through most of the chapters in this book and timely providing me with very helpful changes, and Dr. Kok-Mun Tham, who reviewed Chapter 10 and suggested several improvements. xix

xx

Acknowledgments

During the course of teaching “EML5713—Intermediate Fluid Mechanics” at UCF, the first four chapters of this book were class-tested in the fall semester of 2013 and the first eight chapters, covering the entire course syllabus, in the fall semester of 2014. I extend my heartfelt thanks to all my students who provided me with valuable feedback that significantly improved the final manuscript of the book. I offer my sincere gratitude to Mr. Jonathan Plant, acquiring executive editor at Taylor & Francis, who believed in my book proposal and, more importantly, believed in my passion to complete this book on time. I thoroughly enjoyed all my conversations with him. I thank Mrs. Laurie Oknowsky for her excellent initial project coordination, Mr. Ed Curtis and Ms Ramya Gangadharan for superbly managing the high-quality book editing and production process, and all the staff at Taylor & Francis for their exemplary support and professional communications. Last but not least, I remain eternally grateful to all readers who will take the following advice to heart: if you want to learn fluid mechanics in all its simplicity, beauty, and practicality, study this book in depth. If you want to learn even more, teach this subject to others. If you want to learn even more than that, and thoroughly enjoy this subject, I hope you will one day write an even better book than this. After all, since all fluid flows are governed by only the four fundamental laws of conservation of mass, momentum, energy, and entropy, how difficult can this subject really be?

Author Bijay Sultanian, PhD, PE, MBA, ASME Fellow, is a recognized international authority in gas turbine heat transfer, secondary air systems, and computational fluid dynamics (CFD). Dr. Sultanian is a founder and managing member of Takaniki Communications LLC (www.takaniki.com), a provider of high-impact, web-based, and live technical training programs for corporate engineering teams. Dr. Sultanian is also an adjunct professor at the University of Central Florida, where he has taught graduate-level courses in turbomachinery and fluid mechanics since 2006. As an active member of the IGTI’s Heat Transfer Committee since 1994, he has instructed a number of workshops at ASME/IGTI Turbo Expos. During his three decades in the gas turbine industry, Dr.  Sultanian has worked in and led technical teams at a number of organizations, including Allison Gas Turbines (now Rolls-Royce), General Electric (GE) Aircraft Engines (now GE Aviation), GE Power Generation (now GE Power & Water), and Siemens Energy (now Siemens Power & Gas). He has developed several physics-based improvements to legacy heat transfer and fluid systems design methods, including new tools to analyze critical high-temperature gas turbine components with and without rotation. He particularly enjoys training large engineering teams at prominent firms around the globe on cutting-edge technical concepts and engineering and project management best practices. During his initial 10-year professional career, Dr. Sultanian made several landmark contributions toward the design and development of India’s first liquid rocket engine for a surface-to-air missile (Prithvi). He also developed the first numerical heat transfer model of steel ingots for optimal operations of soaking pits in India’s steel plants. Dr. Sultanian is a Fellow of the American Society of Mechanical Engineers (1986); a registered professional engineer in the State of Ohio (1995); International WHO’S WHO of Professionals (1999); a GE-certified Six Sigma Green Belt (1998); Member, GE Aircraft Engines, Engineering Design Board, Heat Transfer (1997–1999); and an emeritus member of Sigma Xi, the Scientific Research Society (1984). Dr. Sultanian earned BS and MS degrees in mechanical engineering at the Indian Institute of Technology, Kanpur (1971), and the Indian Institute of Technology, Madras (1978), respectively. He earned a PhD in mechanical engineering at Arizona State University, Tempe (1984), and an MBA at the Lally School of Management and Technology at Rensselaer Polytechnic Institute (1999).

xxi

1 Kinematics of Fluid Flow

1.1 Introduction Fluid flows are ubiquitous in our daily life, abounding in both nature and various engineering applications. About two-thirds of our Earth is covered by water, and we all live within an approximately 17 km thick invisible blanket of air called the atmosphere. Both air and water are essential for the survival of all life on Earth. The flow of air and blood in our body carries oxygen and nutrients to all our vital body parts, keeping them alive and functioning. We remain passive spectators to the immeasurable devastation of many precious lives and property caused by tornadoes and hurricanes in nature. We can never wipe out the horrible televised images of the unimaginable destruction caused by the 2011 tsunami in Japan. Most of us can relate to the scare during a flight when our plane encounters pockets of atmospheric turbulence. In 2012, our weather forecast models predicted well in advance the unusual westward turn of Hurricane Sandy, saving many lives in the New York and New Jersey area. In spite of the two-faced “beauty and the beast” display of fluid mechanics, it remains one of the most fascinating subjects in engineering and science. The continued research in various flow phenomena allows for a deeper understanding and more accurate predictions using state-of-the-art computational fluid dynamics (CFD) technology. By using this technology, we have acquired the ability to develop advanced designs of many fluid flow–related applications including commercial and military planes; submarines and cruise ships; automobiles, jet and rocket propulsion; oil and gas pipelines; gas turbines, steam turbines, pumps, compressors, and air-conditioning and refrigeration units; heat exchangers, artificial hearts and valves; dams and irrigation systems, and others. Fortunately, all these fluid flows are governed by only a handful of laws pertaining to the conservation of mass, momentum, energy, and so forth. A good understanding of these laws of fluid mechanics and their correct applications will enable us to compute various flow properties such as velocity, pressure, temperature, and how they influence drag and lift forces on bodies. Such an understanding will also help us to design safe equipment involving fluid flows and to develop more accurate weather prediction models. In this chapter, we will focus on understanding and describing various attributes of a velocity field, common to all fluid flows, without going into the details of its underlying force field, which we will discuss in later chapters.

1

2

Fluid Mechanics

1.2  What Is a Fluid? Our surroundings typically comprise combinations of solids, liquids, and gases. A solid retains its shape regardless of the space it occupies, while a liquid assumes the shape of its container and may feature a free surface. A gas fills all available space within a closed container without generating a free surface. Liquids and gases are both called fluids. A fluid is precisely defined as a substance that deforms continuously when subjected to a shear stress, no matter how small that shear stress may be. In other words, a fluid, unlike a solid, cannot withstand a shear stress unless it is in motion (flowing). Figure 1.1 depicts a fluid flow between two parallel plates. As the top plate moves, the fluid in contact with the moving plate also moves with the same velocity as the plate (no-slip boundary condition for a real fluid). The fluid in contact with the stationary bottom plate remains at rest, creating a velocity gradient and nonzero shear stress (constant in this case, see Couette flow in Chapter 7) distribution in the flow. Besides density, fluids have other thermophysical properties that are used in their flow analysis. Most prominent of them is the viscosity, which is defined by Newton’s law of viscosity, given in its simplest form (for a more general definition, see Chapter 7 on NavierStokes equations) as τ=µ



du (1.1) dy

where μ is the dynamic viscosity. If we divide this viscosity by fluid density, we obtain the kinematic viscosity ν (i.e., ν = µ ρ) having the units of m 2/s—units of just length and time, hence the name kinematic. When we multiply the units of kinematic viscosity by the units of density, we easily obtain the units of dynamic viscosity as follows:  m 2   kg   kg  = [µ] = [ ν][ρ] =   s   m 3   m.s 



According to Equation 1.1, for the linear velocity profile shown in Figure 1.1, the shear stress τ is constant for a fluid with constant viscosity. The fluids that obey Newton’s law of viscosity are called Newtonian fluids. We will only discuss the mechanics of Newtonian fluids in this book. In passing, let us briefly discuss another important fluid property called compressibility, which determines the speed of pressure disturbance (sound wave) in a fluid. The compressibility of a fluid measured in terms of its bulk modulus K is defined as the following:  ∆Ps  dP K = lim  = ρ s (1.2) ∆Ps → 0  ∆ρ ρ  dρ 



y F

u d

y u=0

FIGURE 1.1 Flow between two parallel plates—one moving and one stationary.

x

3

Kinematics of Fluid Flow

where Ps is the static pressure and ρ is the fluid density. Consider a perfect gas with its equation of state given by Ps = ρRTs , where R is the gas constant and Ts is the fluid static temperature. Let us further assume that the gas is undergoing an isentropic compression governed by Ps = C1ρκ , where C1 is a constant and κ is the ratio of specific heat at constant pressure to that at constant volume (κ = c P/cV ). Under these assumptions, Equation 1.2 yields the bulk modulus of the gas as

K=ρ

d(C1ρκ ) = κC1ρκ = κPs (1.3) dρ

The speed of sound in a fluid can be shown to equal to higher speed of sound in water than in air.

K/ρ, which yields a much

1.2.1  Continuum Assumption Fluids, like any substance, consist of discrete molecules. Within the scope of the concepts and methods presented in this book, fluids are considered as a continuum. This assumption is very helpful in engineering and technology where the behavior of individual molecules is usually not of primary interest. The average properties of the molecules in a small parcel of fluid are used as the local property of the continuous fluid, allowing us to treat these properties as analytic (continuously differentiable) functions of space and time. For example, the mass of all molecules per unit volume of the parcel is called the density of the fluid. If we realize that one cubic millimeter of air under standard atmospheric conditions contains about 2.7 × 1013 molecules, the continuum assumption seems realistic, provided the size of the flow system we are dealing with is much larger than the mean free path (average distance a molecule travels before colliding with another molecule) of the molecules. Our continuum assumption will break down when we are analyzing, for example, a foot-long satellite in the upper rarified atmosphere. In Chapter 5 on compressible flows, we discuss supersonic flows with normal and oblique shocks, which represent regions of abrupt changes in the flow properties, except total temperature and impulse pressure (or stream thrust). The continuum assumption breaks down in a shock region. We deal with this region of discontinuity by satisfying the governing equations of mass, momentum, energy, and entropy over a thin control volume that encloses the region. 1.2.2  Fluid Mechanics versus Solid Mechanics In a Lagrangian approach, where we follow an identified parcel (system) of fluid particles (for example, liquid droplets in a combustor), the laws of solid mechanics become applicable for analyzing the dynamics of these fluid particles. As discussed later in Chapter 3, it is more practical to use an Eulerian approach to analyze a fluid flow. The Reynolds transport theorem allows us to make the conversion from a systems approach to the control volume approach. There are a few other key differences between solid mechanics and fluid mechanics. First, the inertia force in solid mechanics is simply calculated as a product of mass and acceleration. In fluid mechanics, it is evaluated as the net outflow rate of linear momentum from the control volume. Second, solid mechanics deals with normal stresses acting on a solid object, but lacks equivalence to fluid static pressure, which is locally isotropic (equal in all directions). In fluid mechanics, we do not have the counterpart of Mohr’s circle used in solid mechanics. The concept of convective acceleration,

4

Fluid Mechanics

at the heart of fluid mechanics, is nonexistent in solid mechanics. Chapter 7 presents a detailed derivation of the famous Navier-Stokes equations for two-dimensional incompressible laminar flows of Newtonian fluid. The chapter also includes the exact solutions of these equations in a few practical problems formulated in Cartesian and cylindrical polar coordinates.

1.3  Streamline, Pathline, and Streakline In a fluid flow, fluid particles (a tiny parcel of fluid molecules) display three basic modes of motion: translation, rotation, and deformation, as shown in Figure 1.2. In this section, we will only present ways of describing various flow patterns without going into the details of the body and surface forces that cause them. Flow visualization, unique to fluid flows, offers a powerful means of understanding the behavior of a complex flow field. Van Dyke (1982) presents black and white photos with concise but clear explanations of many interesting flow phenomena. Samimy et al. (2003) have compiled additional photos of fluid motions in color. Readers are encouraged to study various fluid flows in these monograms to develop a true appreciation for the inspiring beauty of fluid mechanics. In both nature and laboratory, flows are made visible with dust, ink, clusters of leaves, droplets of a liquid, smoke, cloud, and so on. These days, CFD technology offers an inexpensive means of computer-based visualization of a complex flow field. 1.3.1 Streamline Figure 1.3a shows a streamline that is characterized as a line to which the local velocity vectors are tangent everywhere at a given instant. Accordingly, no flow crosses any streamline. When streamlines are bundled together they form a stream tube. No flow crosses the surface of a stream tube, which is conveniently used in the analysis of many ideal internal flows with impermeable and frictionless walls. Although the magnitude of velocity may vary along a streamline, its direction always coincides with the local tangent to the streamline. We see in Figure 1.3b that the flow accelerates along the streamline on the suction surface, relative to the flow over the pressure surface of the airfoil. Similarly, the flow in a convergent–divergent nozzle, Figure 1.3c, shows significant acceleration as the streamlines pass through the throat area. As streamlines are instantaneous lines, they cannot be made easily visible in an unsteady flow.

Translation

Rotation

Deformation

(a)

(b)

(c)

FIGURE 1.2 Three basic modes of fluid motion.

5

Kinematics of Fluid Flow

V Streamline Streamline definition (a) Suction surface

Stagnation point Pressure surface Streamlines over an airfoil

Streamlines in a convergent–divergent nozzle

(b)

(c)

FIGURE 1.3 (a) Streamline definition, (b) streamlines over an airfoil, and (c) streamlines in a convergent–divergent nozzle.

1.3.2 Pathline Figure 1.4 shows both a streamline and a pathline, which is an actual path traversed by a given fluid particle. Note that, while a streamline is instantaneous, the pathline is generated by the passage of time. 1.3.3 Streakline A streakline is the locus of fluid particles that have earlier passed through a prescribed point. As shown in Figure 1.5, at t = 0, particle 1 (cross) has passed through the prescribed point A at the nozzle exit. At t = 1, particle 2 (solid circle) has passed through A. In the meantime, particle 1 has moved to a new location in the flow. At t = 2, particle 3 (solid triangle) has passed through A, while particles 1 and 2 have moved to their new locations. Again at t = 3, particle 4 (solid square) has passed through the prescribed point A at the nozzle exit, while particles 1, 2, and 3 have moved to their new locations. If we now connect the locations of all four particles, which have all passed through the prescribed point A at the nozzle exit, the resulting solid line shown in the figure is the streakline. Like pathlines, the streaklines are also generated by the passage of time. It may be noted here that streamlines, pathlines, and streaklines coincide in a steady flow. These lines reveal some important properties of the flow field. For example, if the streamlines are curved, we can deduce that the flow must have a gradient of static pressure perpendicular to these streamlines, causing curvature in them. In the absence of such a transverse pressure gradient, the streamlines will be straight. Similarly, in an elliptic flow, the streamlines clearly show the regions of flow reversal and recirculation.

6

Fluid Mechanics

V Instantaneous streamline Pathline Fluid particle FIGURE 1.4 Streamline and pathline.

1 A X

A

t=0

A t=2

1 X 3

1 X

2

t=1

2

1 X

2 A

4

3

t=3

FIGURE 1.5 Streakline.

Example 1.1 For a steady two-dimensional incompressible flow, the velocity field is represented by Vx = Ax , Vy = − Ay , and Vz = 0 . Derive an equation for the streamlines in this flow field. Solution: Since streamlines have the same slope as the velocity vector at all points, we can write



Vy − Ay y  dy  =− = =   Ax x dx streamline Vx Separating the variables and integrating, we obtain



ln y = − ln x + C  or xy = C

The equation xy = C represents the streamlines that are a family of rectangular hyperbolas shown in Figure 1.6. For each value of C, we obtain a new streamline. The streamlines with C = 0 correspond to the x and y coordinate axes. The figure shows them along with the one with C = C1 as solid boundaries of a corner flow.

7

Kinematics of Fluid Flow

y

xy = C1

x FIGURE 1.6 Streamlines (Example 1.1).

1.4  Conservation Principles for a Material Region We have only a few conservation principles that all fluid flows must obey as laws. Understanding these conservation laws as well as applying them to solve a variety of engineering problems is the main focus of this book. Note that in the following, the entropy inequality is not a conservation principle, but is included here to ensure that the second law of thermodynamics must always be satisfied for a physically feasible process. Mass conservation. The time rate of change of mass of a material region is zero. Linear momentum. The time rate of change of linear momentum of a material region in each coordinate direction is the sum of the forces acting on the region in that coordinate direction. Angular momentum. The time rate of change of angular momentum of a material region in each coordinate direction is the sum of the torques acting on the region in that coordinate direction—forms the basis of the Euler’s turbomachinery equation. Energy conservation. The time rate of change of energy within a material region is equal to the rate at which the energy is received by work and heat transfer—a statement of the first law of thermodynamics. Entropy inequality. The time rate of change of entropy within a material region is greater than (for an irreversible process) or equal to (for a reversible process) the ratio of the rate of heat transfer into the region and the absolute temperature of the region—a statement of the second law of thermodynamics.

1.5  Basic Analysis Techniques Fluid flows are broadly analyzed using both theoretical and experimental methods. Theoretical methods may be further divided into dimensional analysis, analytical methods, and numerical (computational) methods. With the modern high-speed computers, computational methods, popularly known as CFD, are gaining prominence in the understanding and prediction of complex turbulent flows over the time-consuming and costly experimental investigations, which are primarily used to validate numerical prediction methods. Advantages and disadvantages of basic analysis techniques used in fluid mechanics are summarized in Table 1.1.

8

Fluid Mechanics

TABLE 1.1 Comparison of Fluid Flow Analysis Techniques Technique

Advantages

Experimental

Capable of being most realistic Needs no further validation

Analytical

General solution usually in formula form Limited post-processing needed to understand flow physics Can handle nonlinear problems Can handle complicated turbulent flows Can provide unsteady flow solutions

Numerical

Disadvantages Long lead time Equipment costs Operating costs Scaling difficulties Measurement uncertainties Limited measurements Limited to simple geometry and physics Usually restricted to linear problems Modeling and numerical errors Computer costs Often needs validation against benchmark quality test data Needs accurate boundary conditions for comparison with measurements

To make the fluid flow problem more tractable for analysis by a theoretical method (analytical or numerical), additional assumptions are often made for both the fluid and the flow under investigation. In this context, it behooves us to heed Albert Einstein’s advice, “Make the problem as simple as possible, but not simpler.” For example, the flow of air, which is a compressible fluid, may be assumed incompressible for flow Mach numbers less than 0.3 and solved by using simpler equations governing an incompressible flow. Similarly, as viscosity effects are most dominant inside a wall boundary layer, the fluid may be considered to have zero viscosity outside the boundary layer, and the flow field may be analyzed as a potential (ideal) flow.

1.6  Some Interesting Flows In this section, we will discuss a few interesting flows of practical importance. These flows, or some of their features, are ubiquitous in many engineering applications. We will not attempt to fully understand what causes these flow features or begin their analysis here; we will study them later in Chapter 3 using control volume analysis. 1.6.1  Sudden Expansion Pipe Flow without Swirl The gross features of separation and redevelopment found in an axisymmetric sudden expansion flow are depicted in Figure 1.7. The separation consists of detachment (the flow leaving the inlet pipe wall), recirculation, growth of the free shear layer, and reattachment to the larger pipe wall. The redevelopment region has a pseudo-boundary-layer character where the flow eventually becomes fully developed if the pipe is long enough. The early part of the redevelopment region in this flow is found to be quite different from the usual entrance region in a developing pipe flow.

9

Kinematics of Fluid Flow

Free shear layer Inviscid core

Dividing streamline Locus of flow reversal New wall boundary layer Corner counterrecirculation

Primary recirculation

Reattachment

FIGURE 1.7 A sudden expansion pipe flow without swirl.

Following the sudden expansion, the flow enters the larger pipe in the form of a circular jet. The detachment of flow from the wall converts what had been the wall boundary layer into a complex free shear layer that emanates from the edge of the expansion and envelops the core flow in the jet. This shear layer grows both inward toward the pipe axis and outward toward the pipe wall. Although the inward growth of the shear layer occurs at the expense of the potential (inviscid) core flow, which decays downstream until the shear layers merge at the pipe axis, its outward growth happens through the process of entrainment until it finally reattaches to the pipe wall. The fluid entrained into this shear layer is continuously replenished from downstream through a favorable reverse pressure gradient. This gives rise to a primary recirculation region such that the flow in its forward branch is driven by the central jet momentum, and in its reverse branch, by an adverse pressure gradient. Engineering consideration of a sudden expansion flow geometry dates back to the original analysis of Borda (1766). The existence of a large wall–bounded recirculation region in this flow makes it predominantly elliptic in character. Drewry (1978) reports an improved flow-visualization study of the recirculation region using a surface oil-film technique. His results show that the reattachment length ranges between 7.9 and 9.2 step (difference between the radii of two pipes) heights for Reynolds numbers (based on the smaller pipe diameter) in the range of 1.3 × 106 to 2.2 × 106. Sultanian, Neitzel, and Metzger (1986) present accurate CFD predictions of this flow field using an advanced algebraic stress model of turbulence and also demonstrate that the standard highReynolds number κ−ε turbulence model significantly under-predicts the experimentally found reattachment lengths. 1.6.2  Sudden Expansion Pipe Flow with Swirl Figure 1.8 shows the main features of a swirling flow in a sudden pipe expansion. The wall-bounded primary recirculation zone in this flow field compared to the case without swirl, shown in Figure 1.7 and described above, shrinks with increasing swirl. When the swirl exceeds a critical value (greater than 45-degree swirl angle), an on-axis recirculation region appears. Sultanian (1984) expounds on the overwhelming complexity of this flow

10

Fluid Mechanics

Free shear layer On-axis recirculation Inviscid core CL Dividing streamline Locus of flow reversal

New wall boundary layer Corner counter recirculation

Primary recirculation

Reattachment

FIGURE 1.8 A sudden expansion pipe flow with swirl.

field from a computational viewpoint. The major difficulties are associated with predicting both the mean flow, which is dominated by all three velocity components, and the turbulence field, which is highly anisotropic. The most distinguishing feature of a swirling flow in a sudden pipe expansion shown in Figure 1.8 is the appearance of an on-axis recirculation zone caused by vortex breakdown. Such a feature is widely used in modern gas turbine combustors as an aerodynamic flame holder, which in early designs corresponded to the wake region of a bluff body, to ignite the fresh charge entering the combustor. Here is a simple explanation behind the formation of the central recirculation zone in a strong swirling flow in a sudden pipe expansion. A swirl in a free or forced vortex causes a radial pressure gradient; pressure always increases radially outward—the higher the swirl, the higher this pressure gradient becomes. Near the sudden expansion, due to high swirl, the pressure near the downstream pipe axis is much smaller than the average pressure. Further downstream, swirl decays due to wall shear stress and the pressure distribution tends to become more uniform at the pipe cross section. As a result, the static pressure on the pipe axis is higher downstream than upstream, causing on-axis flow reversal. 1.6.3  Secondary Flows Secondary flows are flows normal to the main flow direction. In a pipe flow, for example, the streamlines in the pipe cross section (normal to the main flow direction) indicate secondary flows. Although the velocities in secondary flows are generally an order of magnitude smaller than those along the main flow direction, they play a significant role in augmenting heat, mass, and momentum transport properties of the flow. We have two kinds of secondary flows: the secondary flow of the first kind and the secondary flow of the second kind. They are discussed as follows.

11

Kinematics of Fluid Flow

A B

A

B

FIGURE 1.9 Secondary flow of the first kind.

FIGURE 1.10 Secondary flow of the second kind.

Secondary flow of the first kind. This kind of secondary flow is generated when the primary flow negotiates a curved path with a finite radius of curvature as happens in a bent pipe. Figure 1.9 shows the flow entering a circular pipe with a uniform velocity and pressure at the lower end and coming out of a 180-degree bend at the upper end. As the main flow goes through the bend, the centrifugal force drives the fluid in the central region from the pipe inner wall (lower radius of curvature) to the outer wall (higher radius of curvature). The fluid then returns back to its original position along the side wall, as shown in Figure 1.9, due to a pressure gradient caused by the centrifugal force in the pipe cross section. This results in a pair of counter-rotating vortices in the pipe cross section at AB. Secondary flow of the second kind. A turbulent flow in straight noncircular duct features the second kind of secondary flow, which is not found in constant-viscosity laminar flow in circular or noncircular ducts or in turbulent flow in a circular duct. These secondary flows are driven primarily by the gradient in turbulence-pressure along the duct wall. For example, the secondary flows shown in Figure 1.10 in a duct of triangular cross section result from the variation of turbulence normal intensity around the triangle. It may be conjectured that these secondary flows cannot be predicted by using an isotropic turbulence model such as the standard high-Reynolds number κ−ε turbulence model. 1.6.4  Rotor Cavity Flow Field Figure 1.11 shows streamlines of a complex shear flow in a cavity between two rotating disks. The axial flow entering the cavity through the upstream disk undergoes a sudden geometric expansion. The growth of the outer shear layer of the annular jet occurs through entrainment of the pressure-gradient–driven backflow from the downstream stagnation region. This creates the primary recirculation region shown in the figure. The size and strength of this recirculation region are found to depend mainly on the flow rate and rotational speed as discussed in Sultanian and Nealy (1987). The entering axial flow turns 90 degrees over the concave corner and flows radially outward, aided in part by frictional

12

Fluid Mechanics

Air outlet

S

Outer shroud

Outflow zone

Pseudo boundary layer zone

b

Primary recirculation zone

Upstream disk

Downstream disk Air inlet

Central drive shaft Ω

FIGURE 1.11 Rotor cavity flow field.

pumping over the downstream disk induced by its rotation. A part of the flow (almost half in this case!) turns back toward the upstream disk and moves radially outward as a result of similar pumping action over that disk. The development of boundary layers on the two disks shares many of the features of a regular Ekman layer, see Greenspan (1968) and Lugt (1995), where the inertia force is negligible, and the viscous and Coriolis forces nearly balance each other in a reference frame rotating with the cavity. Unlike regular wall boundary layers, the Ekman boundary layers are nearly non-entraining, maintaining a constant mass flow rate within them.

1.7  Properties of Velocity Field 1.7.1  Local Mass Conservation in Fluid Flow The velocity field is a vector field where the velocity at a point may have components in all three coordinate directions. In a steady flow, the velocity components are functions of spatial coordinates (x, y, and z) only; in an unsteady flow, they depend on time also. Mathematically, we can express velocity components in unsteady and steady flows as follows.

13

Kinematics of Fluid Flow

Unsteady flow: Vx = f1 ( x , y , z , t), Vy = f2 ( x , y , z , t), and Vz = f3 ( x , y , z , t)

Steady flow:

Vx = f1 ( x , y , z), Vy = f2 ( x , y , z), and Vz = f3 ( x , y , z)



Note that a turbulent flow is inherently always unsteady and is considered “steady” if it is stationary in the mean (i.e., the ensemble-average value of each velocity component at a point is not changing with time). All vector fields do not represent a flow field; they need to additionally satisfy the local mass conservation (continuity equation), which could be considered “the first law of fluid mechanics.” To derive the continuity equation in a steady flow at a point, consider a cubic control volume of side length δ shown in Figure 1.12. Assuming uniform properties at each face of the control volume with ρ = ρ(x, y, z), we can write the total entering mass flow rate as follows:

∑ m in = ρVxδ 2 + ρVy δ 2 + ρVzδ 2



Similarly, total mass flow rate exiting the control volume can be written as follows:

 ∂(ρVx )

∑ m out = (ρVxδ 2 + ρVyδ 2 + ρVzδ 2 ) + 

 ∂x

+

∂(ρVy ) ∂y

+

∂(ρVz )  3 δ ∂z 

As the total mass within the control volume remains constant under steady state, from conservation of mass, we must have

∑ m out = ∑ m in

giving

 ∂(ρVx ) ∂(ρVy ) ∂(ρVz )  3 + +  δ = 0 ∂y ∂z   ∂x



∂(ρVy)

ρVy +

∂y

ρVz

δ

ρVx +

ρVx y

δ ρVz + o

z FIGURE 1.12 Local mass conservation in a flow field.

∂(ρVz) δ ∂z

δ

δ

ρVy

x

∂(ρVx) δ ∂x

14

Fluid Mechanics

As δ 3 ≠ 0, the above equation yields  ∂(ρVx ) ∂(ρVy ) ∂(ρVz ) + + = 0 or ∇ • (ρV ) = 0  or (ρui ), i = 0 (1.4) ∂x ∂z ∂y



where i = 1, 2, 3 correspond to three coordinate directions along which the velocity components are given by u1, u2, and u3, respectively. For an incompressible flow with constant density, Equation 1.4 reduces to  ∂Vx ∂Vy ∂Vz + + = 0 or ∇ • V = 0 or ui , i = 0 (1.5) ∂x ∂z ∂y



Note that Equation 1.5 is also true for an unsteady incompressible flow, because the unsteady term ∂ρ/∂t = 0 for constant density. Example 1.2 A team of graduate students carried out accurate measurements of the x-component of velocity and temperature in a two-dimensional steady flow on a flat plate, as shown in Figure 1.13. They correlated these measurements to Vx = V0 x 2, where V0 is a constant. Noting that the flow velocity (Vy ) is zero along the x axis, they calculated the distribution of the y-component of velocity from their understanding of incompressible fluid flows. Looking at the data and the calculations, the research adviser pointed out that the calculated y-component of velocity is 33% higher everywhere. She asked the students to recalculate it taking into account the temperature variation, which caused the density variation given by ρ = ρ0 xy (where ρ0 is a constant) in the flow field. Both x and y are dimensionless coordinates. Verify the error estimated by the research advisor in the y-component of velocity computed by the graduate students. Solution: Initially, the graduate students assumed an incompressible flow field with ­constant density to calculate the y-component of velocity (Vy ) using the following continuity equation, which is a two-dimensional simplification of Equation 1.5: ∂Vy ∂Vx ∂Vy ∂V ∂(V0 x 2 ) = 0  or  =− x =− + = −2V0 x ∂y ∂y ∂x ∂x ∂x As Vy = 0 for y = 0, we obtain Vy = −2V0 xy



Considering the variation of density in the flow field due to nonuniform temperature distribution, the continuity equation becomes:

∂(ρVx ) ∂(ρVy ) + = 0  or  ∂y ∂x

∂(ρVy ) ∂y

=−

∂(ρ0 xyV0 x 2 ) ∂(ρVx ) =− = −3 ρ0V0 x 2 y ∂x ∂x

y Vy Vx o FIGURE 1.13 Two-dimensional flow field (Example 1.2).

x

15

Kinematics of Fluid Flow

where Vy represents the y-component of velocity for the actual flow with density variation. As Vy = 0 for y = 0, we obtain 3 3 ρVy = − ρ0V0 x 2 y 2 or Vy = − V0 xy 2 2



The percentage error in the initial calculation of the y-component of velocity can now be calculated as follows:

Error =

(Vy − Vy ) Vy

× 100 = 33.3% (The research adviser is right!)

1.7.2  Convection in Fluid Flow In general, the flow field properties such as temperature, velocity, and so on, may vary in both space and time. If we follow a fluid particle (Lagrangian viewpoint) in the flow field with nonuniform property distribution, and assume that the particle instantly assumes the local property of the flow field (Eulerian viewpoint), its property will change with time not only due to the local time-evolution of the property but also due to its change in position (by velocity). Although it may be easy to visualize this change for the particle temperature, it is less intuitive to do so for the particle velocity—requiring interaction of the particle velocity with itself in the flow field! We will later see that the particle convective acceleration, which becomes nonlinear, is behind the beauty in fluid flows and also the source of significant challenge in predicting properties of complex shear flows in engineering applications. About the two viewpoints mentioned here, we can simply say, “Lagrangian follows the particle, Eulerian watches the particle go by.” To develop a good physical understanding of convection in fluid flow, let us first consider the convection of fluid particle temperature in a steady one-dimensional flow with uniform velocity Vx shown in Figure 1.14. The flow has a fixed linear variation in temperature, given by Equation 1.6, within the dotted square from x1 to x2.

T = (T1 − ax1 ) + ax (1.6)

where a=



y

T2 − T1 x2 − x1

Vx

Vx P

T

T2

T1 o

x1

x

FIGURE 1.14 Particle temperature change in a steady one-dimensional flow.

x2 x

16

Fluid Mechanics

In following the fluid particle P from x1 to x2 in the flow field shown in Figure 1.14, it is clear that in time dt, the particle will undergo displacement dx. Thus, we can express the time-rate of change of particle temperature as

dT dT dx d = = Vx ((T1 − ax1 ) + ax ) = aVx (1.7) dt dx dt dx

Equation 1.7 reveals that the time-rate of change of particle temperature, following the particle (convection), at any point in a flow field is the product of the flow velocity and the spatial gradient of temperature at that point. If we double the particle flow velocity at a point, the time-rate of change of its temperature for the given spatial temperature gradient will also double. Let us now try to understand the convective particle acceleration (time-rate of change of particle velocity) in a one-dimensional flow. Figure 1.15 shows a steady one-dimensional flow through a convergent passage (planer flow) between x1 and x2 with the linear variation in velocity Vx given by Equation 1.8: Vx = Vx1 + b( x − x1 ) (1.8)

where

b=



Vx2 − Vx1 x2 − x1

The convective acceleration of the particle P in the flow through the convergent passage can be written as follows:

ax =

dVx dVx dx dVx 1 dVx2 (1.9) = = Vx = dt dx dt dx 2 dx

ax = Vx



d Vx − bx1 + bx = bVx (1.10) dx 1

(

)

Unlike the Lagrangian change in particle temperature with time given by Equation 1.7, the Lagrangian acceleration given by Equation 1.9 is nonlinear in velocity Vx. The local value of this acceleration is again the product of velocity and its spatial gradient. Having developed a physical understanding of Lagrangian time-change in the properties (temperature and velocity) of a fluid particle in a steady one-dimensional flow, let us now use a more formal mathematical approach to develop these changes in an unsteady y Vx1

Vx1

o FIGURE 1.15 Particle acceleration in a steady one-dimensional flow.

Vx2

Vx

P

x1

x

x2

x

17

Kinematics of Fluid Flow

three-dimensional flow field for a general particle property Φ(x, y, z, t), which is a function of all three spatial coordinates (x, y, and z) and time t. The total Lagrangian change in Φ(x, y, z, t) in a flow can be written as follows: DΦ =



∂Φ ∂Φ ∂Φ ∂Φ dt + dx + dy + dz ∂t ∂x ∂y ∂z

Dividing the above equation by dt yields DΦ ∂Φ ∂Φ dx ∂Φ dy ∂Φ dz + + + = Dt ∂t ∂ x dt ∂ y dt ∂ z dt



DΦ ∂Φ ∂Φ ∂Φ ∂Φ (1.11) + Vx + Vy + Vz = Dt ∂t ∂x ∂y ∂z



The term on the left-hand side of Equation 1.11 is called the particle derivative, substantial derivative, or total derivative. This equation shows that the total change in Φ, which is a function of several variables, of a particle as we follow it in a flow field results from the superposition of two effects: (a) the local rate of change (the first term on the right-hand side) and (b) the convective terms (the last three terms on the right-hand side). The local rate of change is the partial derivative of Φ with t, holding x, y, and z constant. The convective terms represent the rate of change in Φ as the particle moves into a new position with a different value of Φ. Thus, the total change experienced by a moving particle in a flow field is the sum of the local and the convective changes. In compact tensor notations (see Appendix F), we can express Equation 1.11 as follows: DΦ ∂Φ ∂Φ (1.12) + ui = Dt ∂t ∂ xi



When Φ stands for velocity ui, the total acceleration can be written as follows: Dui ∂u ∂u ∂u = ai = i + u j i = i + u j ui , j (1.13) Dt ∂t ∂x j ∂t

Example 1.3

For the flow field of Example 1.1, find the components of local acceleration. Solution: As the flow field is steady, we only have the acceleration of transport (convective acceleration) in this flow with Vx = Ax , Vy = − Ay , and Vz = 0. Using Equation 1.13 we can write



ax = Vx ay = Vx

∂Vx ∂V ∂V + Vy x + Vz x = ( Ax)( A) + (− Ay )(0) + (0)(0) = A 2 x ∂x ∂y ∂z

∂Vy ∂x

+ Vy

∂Vy ∂y

+ Vz

∂Vy ∂z

= ( Ax)(0) + (− Ay )(− A) + (0)(0) = A 2 y az = 0

The results show that, for this two-dimensional flow, the acceleration in the x-direction varies linearly with x and that in the y-direction varies linearly with y.

18

Fluid Mechanics

Example 1.4

 The velocity distribution in an incompressible flow is given by V = 10 x 3 y iˆ − 15 x 2 y 2 ˆj + 10t kˆ m/s. First, verify that the given velocity distribution satisfies the continuity equation everywhere in the flow and then determine the velocity and acceleration of a particle at x = 1 m,  y = 2 m,   z = 1 m, and t = 1 s. Solution: As the flow is incompressible, we must first check that the given velocity satisfies mass conservation (continuity equation) everywhere. Given velocity distribution:  V = 10 x 3 y iˆ − 15 x 2 y 2 ˆj + 10t kˆ m/s Continuity equation:

∂Vx ∂Vy ∂Vz + + =0 ∂y ∂x ∂z



∂(10 x 3 y ) ∂(−15 x 2 y 2 ) ∂(10t) = 30 x 2 y − 30 x 2 y = 0 + + ∂x ∂y ∂z

which shows that the given velocity distribution is mass conserving. Particle velocity at the required position and time:  V (1, 2, 1, 1) = 10(1)3 (2) iˆ − 15(1)2 (2)2 ˆj + 10(1) kˆ m/s  V (1, 2, 1, 1) = 20 iˆ − 60 ˆj + 10 kˆ m/s Particle acceleration at the required position and time: ax =



ax =

∂Vx ∂V ∂V ∂V + Vx x + Vy x + Vz x ∂t ∂x ∂y ∂z

∂(10 x 3 y ) ∂(10 x 3 y ) ∂(10 x 3 y ) ∂(10 x 3 y ) + (10 x 3 y ) + (−15 x 2 y 2 ) + (10t) ∂t ∂x ∂y ∂z



ax = 0 + (10 x 3 y )(30 x 2 y ) + (−15 x 2 y 2 )(10 x 3 ) + 0



ax = 0 + 300 x 5 y 2 − 150 x 5 y 2 = 150 x 5 y 2



ax (1, 2, 1, 1) = 150 × (1)5 (2)2 = 600 m/s 2



ay =

ay =

∂Vy ∂t

+ Vx

∂Vy ∂x

+ Vy

∂Vy ∂y

+ Vz

∂Vy ∂z

∂(−15 x y ) ∂(−15 x y ) ∂(−15 x 2 y 2 ) ∂(−15 x 2 y 2 ) + (10 x 3 y ) + (−15 x 2 y 2 ) + (10t) ∂t ∂x ∂y ∂z 2

2

2

2

ay (1, 2, 1, 1) = 0 + (10 x 3 y )(−30 xy 2 ) + (−15 x 2 y 2 )(−30 x 2 y ) + 0



ay = −300 x 4 y 3 + 450 x 4 y 3 = 150 x 4 y 3



ay (1, 2, 1, 1) = 150 × (1)4 (2)3 = 1200 m/s 2



az = az (1, 2, 1, 1) =

∂Vz ∂V ∂V ∂V + Vx z + Vy z + Vz z ∂t ∂x ∂y ∂z

∂(10t) ∂(10t) ∂(10t) ∂(10t) + (10 x 3 y ) + (−15 x 2 y 2 ) + (10t) = 10 m/s 2 ∂t ∂x ∂y ∂z  a(1, 2, 1, 1) = 600 iˆ + 1200 ˆj + 10 kˆ m/s 2

19

Kinematics of Fluid Flow

The results for the acceleration components show that both ax and ay are functions of x and y only and not of z. The acceleration in the z-direction is constant throughout the flow field.

1.7.3 Vorticity Like velocity, the vorticity is a kinematic vector property of a flow field and equals the curl of the velocity vector at each point in the flow. The component of the vorticity vector along a coordinate direction represents twice the rate of local counterclockwise rotation of the fluid particles about the coordinate direction. As shown in Figure 1.16, let us consider a tiny fluid element ABCD in a two-dimensional flow at time t. In time interval dt, the element occupies a new position A ′ B ′ C′ D ′ , the side AB rotates counterclockwise by dα into its new position A ′ B ′ , and the side AD rotates clockwise by dβ into its new position A ′ D ′. The net counterclockwise rotation of the diagonal element A ′ C′ about the z-axis becomes (dα − dβ)/2. Let us determine the value of this rotation in terms of velocities and their spatial gradients. The difference between the y-coordinates of A′ and B′ can be written from Figure 1.16 as   ∂Vy   ∂Vy   dx  dt − Vy dt =  ∆y = Vy +  dx dt  ∂ x  ∂ x     



Similarly, the difference between the x-coordinates of A′ and B′ can be written as

  ∂V   ∂V   ∆x = Vx +  x  dx  dt + dx − Vx dt = dx +  x  dx dt  ∂x    ∂x    ∂V  As  x  dx dt