Viscous Fluid Flow - 3rd ed. - Frank M. White (2006), Manuais, Projetos, Pesquisas de Engenharia Aeronáutica

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VISCOUS FLUID FLOW McGraw-Hill Series in Mechanical Engineering Alciatore and Histand: Introduction to Mechatronics and Measurement Systems Anderson: Computational Fluid Dynamics: The Basics with Applications Anderson: Fundamentals of Aerodynamics Anderson: Introduction to Flight Anderson: Modern Compressible Flow Barber: Intermediate Mechanics of Materials Beer/ Johnston: Vector Mechanics for Engineers Beer/ Johnston/De Wolf: Mechanics of Materials Budynas: Advanced Strength and Applied Stress Analysis Cengel and Boles: Thermodynamics: An Engineering Approach Cengel and Turner: Fundamentals of Thermal-Fluid Sciences Cengel: Heat Transfer: A Practical Approach Cengel: Introduction to Thermodynamics & Heat Transfer Crespo da Silva: Intermediate Dynamics Dieter: Engineering Design: A Materials & Processing Approach Dieter: Mechanical Metallurgy Doebelin: Measurement Systems: Application & Design Dunn: Measurement & Data Analysis for Engineering & Science EDS, Inc.: I-DEAS Student Guide Hamrock/Schmid/Jacobson: Fundamentals of Machine Elements Heywood: Internal Combustion Engine Fundamentals Holman: Experimental Methods for Engineers Holman: Heat Transfer Hutton: Fundamentals of Finite Element Analysis Kays/ Crawford/Weigand: Convective Heat and Mass Transfer Meirovitch: Fundamentals of Vibrations Norton: Design of Machinery Palm: System Dynamics Reddy: An Introduction to Finite Element Method Schaffer et al.: The Science and Design of Engineering Materials Schey: Introduction to Manufacturing Processes Shames: Mechanics of Fluids Shigley /Mischke/Budynas: Mechanical Engineering Design Smith: Foundations of Materials Science and Engineering Suryanarayana and Arid: Design and Simulation of Thermal Systems Turns: An Introduction to Combustion: Concepts and Applications Ugural: Mechanical Design: An Integrated Approach Ullman: The Mechanical Design Process White: Fluid Mechanics White: Viscous Fluid Flow Zeid: Mastering CAD/CAM ABOUT THE AUTHOR l I Frankj M. White is Professor Emeritus of Mechanical and Ocean Engineering at the Utjiversity of Rhode Island. He is a native of Augusta, Georgia, and went to undergraduate school at Georgia Tech, receiving a B .M.E. degree in 1954. Then he attend~d the Massachusetts Institute of Technology for an S.M. degree in 1956, return~·l g to Georgia Tech to earn a Ph.D. degree in mechanical engineering in 1959. e began teaching aerospace engineering at Georgia Tech in 1957 and moved to the · niversity of Rhode Island in 1964. He retired in January 1998. t the University of Rhode Island, Frank became interested in oceanographic and co~stal flow problems and in 1966 helped found the first Department of Ocean Engin~ering in the United States. His research interests have mainly been in viscous flow a' d convection heat transfer. Known primarily as a teacher and writer, he receiv d the ASEE Westinghouse Teaching Excellence Award in addition to seven Unive sity of Rhode Island teaching awards. His modest research accomplishments includ some 80 technical papers and reports, the ASME Lewis F. Moody Research Award in 1973, and the ASME Fluids Engineering Award in 1991. He is a Fellow of the ~SME and for 12 years served as editor-in-chief of the ASME Journal of Fluids JEngineering. He received a Distinguished Alumnus award from Georgia Tech it 1990 and was elected to the Academy of Distinguished Georgia Tech Alumn· in 1994. I addition to the present text, he has written three undergraduate textbooks: Fluid echanics, Heat Transfer, and Heat and Mass Transfer. He continues to serve ~n the ASME Publications Committee and has been a consulting editor of the McGraJW-Hill Encyclopedia of Science and Technology since 1992. He lives with his wiff, Jeanne, in Narragansett, Rhode Island. l My wife, Jeanne Faucher White, is the key to this book. Without her love and encouragement, I can't even get started. x CONTENTS 3 Solutions of The Newtonian Viscous-Flow Equations 96 3-1 Introduction and Classification of Solutions .................................................................... 96 3-2 Couette Flows Due to Moving Surfaces .......................................................................... 98 3-3 Poiseuille Flow through Ducts ....................................................................................... 106 3-4 Unsteady Duct Flows ..................................................................................................... 125 3-5 Unsteady Flows with Moving Boundaries ..................................................................... 129 3-6 Asymptotic Suction Flows ............................................................................................. 135 3-7 Wind-Driven Flows: The Ekman Drift ........................................................................... 141 3-8 Similarity Solutions ........................................................................................................ 144 3-9 Low Reynolds Number: Linearized Creeping Motion ................................................... 165 3-10 Computational Fluid Dynamics .................................................................................... 183 Summary ......................................................................................................................... 205 Problems ......................................................................................................................... 205 4 La1ninar Boundary Layers 215 4-1 Introduction .................................................................................................................... 215 4-2 Laminar Boundary-Layer Equations .............................................................................. 225 4-3 Similarity Solutions for Steady Two-Dimensional Flow ............................................... 230 4-4 Free-Shear Flows ............................................................................................................ 251 4-5 Other Analytic Two-Dimensional Solutions .................................................................. 257 4-6 Approximate Integral l'v1ethods ...................................................................................... 261 4-7 Digital-Computer Solutions ........................................................................................... 271 4-8 Thermal-Boundary-Layer Calculations .......................................................................... 278 4-9 Flow in the Inlet of Ducts .............................................................................................. 287 4-10 Rotationally Symmetric Boundary Layers ..................................................................... 290 4-11 Asymptotic Expansions and Triple-Deck Theory ......................................................... .300 4-12 Three-Dimensional Laminar Boundary Layers ............................................................... 307 4-13 Unsteady Boundary Layers: Separation Anxiety ........................................................... 318 4-14 Free-Convection Boundary Layers ................................................................................. 321 Summary ......................................................................................................................... 328 Problems ......................................................................................................................... 328 5 The Stability of Laminar Flows 337 5-1 Introduction: The Concept of Small-Disturbance Stability .......................................... .337 5-2 Linearized Stability of Parallel Viscous Flows .............................................................. 344 5-3 Parametric Effects in the Linear Stability Theory ......................................................... .357 5-4 Transition to Turbulence ................................................................................................. 370 5-5 Engineering Prediction of Transition ............................................................................. 378 Summary .......................................................................................................................... 394 Problems ......................................................................................................................... 394 I CONTEJ1'ffS xi 6 6-1 6-2 6-3 6-4 6-5 6-6 6-7 6-8 6-9 6-10 7 7-1 7-2 7-3 7-4 7-5 7-6 7-7 7.8 7-9 A B c I ~ncompressible Turbulent Mean Flow 398 i Physical and Mathematical Description of Turbulence ................................................. 398 I rhe Reynolds Equations of Turbulent Motion .............................................................. .406 1fhe Two-Dimensional Turbulent-Boundary-Layer Equations ...................................... .411 Velocity Profiles: The Inner, Outer, and Overlap Layers .............................................. .414 furbulent Flow in Pipes and Channels ......................................................................... .425 the Turbulent Boundary Layer on a Flat Plate ............................................................. .433 turbulence Modeling ..................................................................................................... 440 I ,f\nalysis of Turbulent Boundary Layers with a Pressure Gradient .............................. .454 free Turbulence: Jets, Wakes, and Mixing Layers ........................................................ .473 rurbulent Convective Heat Transfer .............................................................................. .485 ummary ......................................................................................................................... 498 froblems ......................................................................................................................... 498 I Compressible-Boundary-Layer Flow 505 I fotroduction: The Compressible-Boundary-Layer Equations ........................................ 505 $imil~rity Solutio~s for Compressible Lamin.ar Flo~ ................................................... 511 ~olut10ns for Lammar Flat-Plate and Stagnat1on-Pomt Flow ........................................ 514 ¢ompressible Laminar Boundary Layers under Arbitrary Conditions .......................... 525 $pecial Topics in Compressible Laminar Flow .............................................................. 539 I The Compressible-Turbulent-Boundary-Layer Equations ............................................. 544 'fvall and Wake Laws for Turbulent Compressible Flow ............................................... 547 ¢ompressible Turbulent Flow Past a Flat Plate ............................................................. 553 ¢ompressible-Turbulent-Boundary-Layer Calculation with a Pressure Gradient ......... 561 ~ummary ......................................................................................................................... 566 Piroblems ......................................................................................................................... 566 I I • Jj\.ppend1ces 571 iransport Properties of Various Newtonian Fluids ......................................................... 571 ~quations of Motion of Incompressible Newtonian Fluids in dylindrical and Spherical Polar Coordinates .................................................................. 581 I A. Runge-Kutta Subroutine for N Simultaneous Differential Equations ........................ 585 I *ibliography .......................................................................... 590 I Ibdex ...................................................................................... xxx I I I PREFAqE xv I I SUPlfLEMENTS 1, The 1ew Instructor and Student Resource Web Site, http://www.mhhe.com/ whitele, will house general text information, the solutions to end-of-chapter prob- lems qunder password-protection), additional problems (with password-protected solutiqns), and helpful Web links. I : I ACKlNOWLEDGMENTS I There ]are many people to thank. Much appreciated comments, suggestions, photos, charts~ corrections, and encouragement were received from Leon van Dommelen of Florid~ State University; Gary Settles of Penn State University; Steven Schneider of Purdu¢ University; Kyle Squires of Arizona State University; Chihyung Wen of Da- Yeh Upiversity, Taiwan; Brooks Martner of the NOAA Environmental Technology Labonhory; Jay Khodadadi of Auburn University; Philipp Epple of Friedrich- ' Alexa1der-Universitat; Jurgen Thoenes of the University of Alabama at Huntsville; Luca cjl' Agostino of Universita Degli Studi di Pisa; Raul Machado of the Royal Institute of Technology (KTH), Sweden; Gordon Holloway of the University of New Brunsi.rick; Abdulaziz Almukbel of George Washington University; Dale Hart of Louisi~na Tech University; Debendra K. Das of the University of Alaska Fairbanks; Alexa~der Smits of Princeton University; Hans Fernholz of Technische Unive*itaet Berlin; Peter Bernard of the University of Maryland; John Borg of Marqu~tte University; Philip Drazin of the University of Bristol, UK; Ashok Rao of Rant;ho Santa Margarita, CA; Deborah Pence of Oregon State University; Joseph I Katz of Johns Hopkins University; Pierre Dogan of the Colorado School of Mines; Philip I Burgers of San Diego, CA; Beth Darchi of the American Society of Mech~nical Engineers; and Norma Brennan of the American Institute of Aeron4utics and Astronautics. I I have tried to incorporate almost all of the reviewer comments, criticisms, correcdions, and improvements. The third edition has greatly benefited from the review~rs of the second edition text, as well as the reviewers of the third edition i • manus<pnpt: i Malcoljm J. Andrews, Texas A&M University Mehdi tsheghi, Carnegie Mellon University Rober~LBreidenthal, University of Washington H. A. tjlassan, North Carolina State University Herma~ Krier, University of Illinois, Urbana-Champaign Daniel \Maynes, Brigham Young University SureshiMenon, Georgia Institute of Technology Mered*h Metzger, University of Utah Kamra* Mohseni, University of Colorado Ugo Pipmelli, University of Maryland Steven r· Schneider, Purdue University xvi Kendra Sharp, Pennsylvania State University Marc K. Smith, Georgia Institute of Technology Leon van Dommelen, FAMU-FSU Steve Wereley, Purdue University PREFACE The editors and staff at McGraw-Hill Higher Education, Amanda Green, Jonathan Plant, Peggy Lucas, Rory Stein, Mark Neitlich, and Linda Avenarius, were constantly helpful and informative. The University of Rhode Island continues to humor me, even in retirement. Frank M. White whitef@egr.uri.edu j I LIST OF SYMBOLS Englis .. Symbols a A b B C, Ci, Cr cP, cv c Ci D Dh Du e, E er f, F f, F, g g G(Pr) h h, ho H HI J k speed of sound; acceleration (Chap. 2), body radius (Chap. 4) area; amplitude, Eq. (5-40); damping parameter, Eq. (6-90) jet or wake width, Fig. 6-35 stagnation-point velocity gradient (Sec. 3-8.1); turbulent wall-law intercept constant, Eq. (6-38a) wall-law shift due to roughness, Eq. (6-60) wave phase speeds (Chap. 5) specific heats, Eq. (1-69) Chapman-Rubesin parameter, Eq. (7-20) species concentrations (Chap. 1) diameter; drag force (Chap. 4); diffusion coefficient (Chap. 1) duct hydraulic diameter, Eq. (3-55) turbulent transport or diffusion, Eq. ( 6-111) internal energy internal plus kinetic plus potential energy, Eq. (2-113) force similarity variables acceleration of gravity heat-transfer parameter, Eqs. (3-172) and ( 4-78) enthalpy; duct width; heat-transfer coefficient metric coefficients, Eqs. (2-58) and (4-229) stagnation enthalpy, h + V 2 /2 shape factor, B* /8; stagnation enthalpy, Eq. (7-3) alternate shape factor, (o - 8*)/8 jet momentum, Eqs. (4-97), (4-206), and (6-144) thermal conductivity; roughness height (Chaps. 5 and 6) xvii LIST OF SYMBOLS xix Greek Symbols a, f3, y a*, f3,? f3 y o, ou o* o, 017 OT 03 o .. If ~ E A YJ µ, v 7T 11 <P thermal diffusivity, k/ pep; wedge angle (Fig. 3-32); wave number, Eq. (5-12); angle of attack finite-difference mesh-size parameters, Eq. (4-146); also compressible wall-law parameters, Eqs. (7-111) compressible finite-difference mesh-size parameters, Eq. (7-67) thermal expansion coefficient, Eq. (1-86); Falkner-Skan parameter, Eq. ( 4-71); Clauser parameter, Eq. ( 6-4 2) specific-heat ratio, c/cv; finite-difference parameter Eq. (4-163) intermittency, Fig. 6-5; compressibility parameter, Eq. (7-111) velocity boundary-layer thickness displacement thickness, Eq. (4-4) conduction thickness, Eq. (4-156) enthalpy thickness, Eq. ( 4-22) temperature boundary-layer thickness dissipation thickness, Eq. ( 4-128) Kronecker delta defect thickness, Eq. (6-43) perturbation parameter (Sec. 4-11 ); turbulent dissipation [term V of Eq. (6-17)] strain-rate tensor; Reynolds stress dissipation, Eq. (6-111) Karman constant, ~o.41 second viscosity coefficient (Chap. 2); Darcy friction factor, Eq. (3-38); Thwaites' parameter, Eq. ( 4-132); (2/Cr) 112 (Chap. 6) Karman-Pohlhausen parameter,(/ (dp/dx )/ µ,U; pipe-friction factor, Eq. (6-54) Graetz function eigenvalues, Table 3-1 similarity variable; free-surface elevation (Chap. 5) viscosity kinematic viscosity µ,/ p 3.14159 ... Coles' wake parameter, Eq. (6-47) velocity potential (Chap. 2); latitude (Chap. 3); wave angle, Eq. (5-12); dimensionless disturbance, Eq. (5-23) dissipation function, Eq. (2-46) stream function polar coordinate angle; momentum thickness, Eq. ( 4-6) dimensionless temperature ratio, Eq. (3-167) or ( 4-56) density molecular collision diameter (Chap. 1 ); numerical mesh parameter, Eq. (3-247); turbulent jet growth parameter, Eq. (6-147) i i UST OF !SYMBOLS ! Subscripts aw co e 0 c, crit m n rms sep t tr r w x I adiabatic wall far field freestream, boundary-layer edge initial or reference value critical, at the point of instability mean normal root mean square separation point turbulent, tangential transition recovery or adiabatic wall wall at position x Super$cripts I I I * + /\ time-mean differentiation; turbulent fluctuation dimensionless variable (Chaps. 2, 3, and 4) law-of-the-wall variable small-disturbance variable (Chap. 5) xxi I i PRELIMINARY CONCEPTS 3 beginnibg of the twentieth century. Indeed, the separation of fluid mechanics theo- ry fro~ experiment is not extinct even today, as witness the divergent views of the subject 11now held among aeronautical, chemical, civil, and mechanical engineers. A~ter Euler and his colleagues, the next significant analytical advance was the additio~ of frictional-resistance terms to Euler's inviscid equations. This was done,· with varying degrees of elegance, by Navier in 1827, Cauchy in 1828, Poisson in 1829, sit. Venant in 1843, and Stokes in 1845. The first four wrote their equations in term$ of an unknown molecular function, whereas Stokes was the first to use the first co~fficient of viscosity µ. Today these equations, which are fundamental to the subject,i are called the Navier-Stokes relations, and this text can do little to improve upon Stpkes' analysis. T~e Navier-Stokes equations, though fundamental and rigorous, are non-lin- ear, notjunique, complex, and difficult to solve. To this day, only a relatively few particul~r solutions have been found, although mathematicians are now taking an I interest I in the general properties of these remarkable equations [Constantin and Foias (lj988)]. Meanwhile, the widespread use of digital computers has given birth to man~ numerical models and published computations of viscous flows. Certain of I these mbdels can be implemented, for simple geometries, on a small personal com- puter aJd are described here in Chaps. 3, 4, and 6. Experimentation remains a strong c\omponent of viscous-flow research because even the largest supercomput- ers are i~capable of resolving the fine details of a high-Reynolds-number flow. Fdr practical fluids engineering, the biggest breakthrough was the demonstra- tion, by !Ludwig Prandtl in 1904, of the existence of a thin boundary layer in fluid flow witih small viscosity. Viscous effects are confined to this boundary layer, which may the~ be patched onto the outer inviscid flow, where so many powerful mathe- matical !techniques obtain. Boundary-layer theory applies to many, but definitely I not all, fngineering flows. The concept makes it possible, as Leslie Howarth said, "to thin~ intelligently about almost any problem in real fluid flow." Ttje second most important breakthrough, also accomplished at the turn of the twe~tieth century, was to put fluid-flow experimentation on a solid basis, using d~mensional analysis. Leaders in this effort were Osborne Reynolds (1842-11912), Lord Rayleigh (1842-1919), and Ludwig Prandtl (1875-1953). Modern l engineering studies-and textbooks-routinely place their results in dimensipnless form, thus making them applicable to any newtonian fluid under the sam¢ flow conditions. W~th thousands of researchers now active in fluid mechanics, present progress is incrdnental and substantial. Instrumentation has advanced greatly with the inventio~s of the hot-wire, the hot-film, the laser-Doppler velocimeter, and minia- ture prdsure and temperature sensors. Visualization of flow-through bubbles, smoke, ~ye, oil-films, holography, and other methods-is now outstanding [see, e.g., Vatj Dyke (1982), Nakayama and Tanida (1996), Smits and Lim (2000), and Nakaya1ia ( 1988)]. Computational fluid dynamics (CFD) has grown from a special topic to mfiltrate the entire field: Many user-friendly CFD codes are now available I so that tjrdinary engineers can attempt to model a realistic two- or three-dimen- sional viscous flow. 4 VISCOUS FLUID FLOW . The literature in fluid mechanics is now out of control, too much to keep up with, at least for someone as dedicated as this writer. The first edition of this text had a figure to show the growth of viscous-flow papers during the twentieth century. Prandtl's 1904 breakthrough could be considered as "Paper l ,"and the research out- put rose at a 7 percent annual rate to 70 papers per year in 1970. Well, that annual increase has continued to this day, so that thousands of papers are now being pub- lished each year. A dozen new fluids-oriented journals have been introduced, plus a half-dozen serials related to computational fluid dynamics. There are dozens of con- ferences and symposia every year devoted to fluids-oriented topics. Consider the fol- lowing statistics: • In 2002, the Journal of Fluid Mechanics printed 450 papers covering 9000 jour- nal pages. • In 2002, the Physics of Fluids printed 410 papers covering 4500 journal pages. • In 2002, the Journal of Fluids Engineering printed 120 papers covering I 068 journal pages. • In 2002, the Fluids Engineering Division of the American Society of Mechanical Engineers sponsored 650 papers. • The May 2003 Meetings Calendar of the Journal of Fluids Engineering listed 38 fluids-related conferences for the coming year. • The 2001 International Symposium on Turbulent Shear Flow Phenomena pre- sented 300 papers. This situation is, of course, much the same in other pure and applied sciences. The net effect of the above incomplete statistics is that the present text can only be an introduction to some of the many topics of ongoing research into laminar and tur- bulent shear flows. Many specialized monographs will be cited throughout this text for further reading. The historical details in this present section were abstracted from the excel- lent history of hydraulics by Rouse and Ince ( 1957). 1-2 SOME EXAMPLES OF VISCOUS-FLOW PHENOMENA Before embarking upon the inevitable detailed studies of theoretical and experi- mental viscous flows, let us discuss four examples, chosen to illustrate both the strength and the limitations of the subject: (1) airfoil flow, (2) a cylinder in cross- flow, (3) pipe-entry flow, and ( 4) prolate spheroid flow. These examples remind one that a textbook tends to emphasize analytical power while deemphasizing practical difficulties. Viscous-flow theory does have limitations, especially in the high-Reynolds-number turbulent flow regime, where the flow undergoes random fluctuations and is only modeled on a semiempirical time-mean or statistical basis. PRELIMINARY CONCEPTS 5 i ~lthough geometry and fluid buoyancy and compressibility will be important, in all iviscous flows, the primary controlling parameter is the dimensionless Reynol~s number. I ReL = pUL/µ, (l-1) where Vis a velocity scale, L is a characteristic geometric size, and p and µ, are the fluid dtjnsity and viscosity, respectively. Fluid properties alone can cause dramatic differe1kes in the Reynolds number and, consequently, the flow pattern. For exam- ple, if[:; = Im/sand L = Im at 20°C, ReL = 9E3, 7E4, and 1E6 for SAE-10 oil, air, and\ water, respectively. By adding in changes in size and speed, the Reynolds numbe~ can vary from a small fraction (falling dust particles) to 5E9 (a cruising supertapker). For a given geometry, as ReL increases, the flow pattern changes from smooth! or laminar through a transitional region into the fluctuating or turbulent regime.i I i E:~mmple 1 Flow past a thin airfoil. Consider flow past a thin airfoil at small angle o~ incidence, a < 5°, as sketched in Fig. 1-1 a. In practical applications, the Reynolds nUmber, ReL, is large. For example, if L = 1 m, U = 100 m/s, and v ;::::; I.SE-5 m 2 /s I (afr at 20°C and 1 atm), ReL = 6.7E6. In these circumstances the flow creates a thin bqundary layer near the airfoil surface and a thin wake downstream. The measured surface pressure distribution on the foil can be predicted by inviscid-flow theory [e.g., Wlhite (2003), Sec. 8.7], and the wall shear stress can be computed with the boundary- 1 later theory of Chaps. 4 to 6. The sharp trailing edge establishes the flow pattern, for a viscous fluid cannot go around such a sharp edge but instead must leave smoothly I an~ tangentially, as shown in Fig. 1-Ia. , Streamlines (a) Boundary layer with no separation Thin wake (b) FIGURE 1-1 Flow past a thin airfoil: (a) low inci- dence angle, smooth flow, no separation; (b) high incidence angle, upper surface separates or "stalls," lift decreases. 8 VISCOUS FLUID FLOW FIGURE 1-3 Continued. 20° (c) Example 2 Flow past a circular cylinder. A very common geometry in fluids engi- neering is crossflow of a stream at velocity U co past a circular cylinder of radius R. For plane inviscid flow, the solution superimposes a uniform stream with a line dou- blet and is given in polar coordinates by [see, e.g., White (2003), p. 537] v r = u co ( 1 - ~:) cos e (1-3) The streamlines of this flow can then be plotted as in Fig. 1-4. At the surface of the cylin- der, r = R, we have v,. = 0 and v0 = -2V cosine, the latter velocity being finite and thus violating the no-slip condition imposed by intermolecular forces between the fluid and the solid. The pressure distribution at the cylinder surface can be found from Bernoulli's equation, p + !P V2 = const, where p is the fluid density. The result is or P.1· = Pco + ~pU~(l - 4 sin2 8) C = Ps - Poe = 1 - 4 sin2 () I' I U2 2.P co PRELIMINARY CONCEPTS 9 FIGURE 1-4 Perfect-fluid flow past a circular cylinder. i Tljiis distribution is shown as the dash-dot line in Fig. 1-5. Equations (1-3) illustrate a bharacteristic of inviscid flow without a free surface or "deadwater" region: There ar~ no parameters such as Reynolds number and no dependence upon physical prop- erpes. Also, the symmetry of Cp(8) in Fig. 1-5 indicates that the integrated surface- pr~ssure force in the streamwise direction-the cylinder drag-is zero. This is an exiample of the d' Alembert paradox for inviscid flow past immersed bodies. The experimental facts differ considerably from this inviscid symmetrical pic- tute and depend strongly upon Reynolds number. Figure 1-5 shows measured CP by Flpchsbart ( 1932) for two Reynolds numbers. The pressure on the rear or lee side of the cylinder is everywhere less than the freestream pressure. Consequently, unlike the d' Alembert paradox, the real fluid causes a large pressure-drag force on the body. . Nor are the real streamlines symmetrical. Figure 1-6 shows the measured flow pa\ttern in water moving past a cylinder at ReD = 170. The flow breaks away or "sep- ar~tes" from the rear surface, forming a broad, pulsating wake. The pattern is visual- iz¢d by releasing hydrogen bubbles at the left of the photograph, in streaklines paral- ' Id to the stream and timelines normal to the flow. Note that the wake consists of pairs of:vortices shed alternately from the upper and lower part of the rear surface. They are caped Karman vortex streets, after a paper by Karman ( 1911) explaining this alterna- tidn to be a stable configuration for vortex pairs. Beginning for ReD > 35, the vortex streets occur in almost any bluff-body flow and persist over a wide range of Reynolds 2 1 cP'! \ 1 -1 \ ' Subcritical ' -2 1---+--1-'..__-+--,,,.__~ Supercritical ---'<~:-+---.'-ft---+--; Theoretical FI<i;URE 1-5 Comparison of perfect-fluid theory and an experiment for the pressure distribution on a cylinder. [After Flachsbart (1932).] 10 VISCOUS FLUID FLOW FIGURE 1-6 Timelines and streaklines for flow past a cylinder at ReD = 170. [From Nakayama (1988), courtesy of Y. Nakayama, Tokai University.] numbers, as shown in Fig. l-7. As the Reynolds number increases, the wake becomes more complex-and turbulent-but the alternate shedding can still be detected at Re= 107 . As shown in Fig. 1-8, the dimensionless cylinder shedding frequency or Strauhal number, St = fD/U:::::::: 0.2 for Reynolds numbers from 100 to 105 . Thus the shedding cycle takes place during the time that the freestream moves approximately five cylin- der diameters. Vortex shedding is one of many viscous flows which, though posed with fixed and steady boundary conditions, evolve into unsteady motions because of flow instability. The pressure distributions in Fig. 1-5 are time averages for this reason. Re=32 Re=55 - - Re=65 Re= 71 - Re= 102 Re= 161 - - Re= 225 Re= 281 FIGURE 1-7 The effect of the Reynolds number on the flow past a cylinder. [After Honwnn (1936).j PRELIMINARY CONCEPTS 13 I ""~ ~ - ~ ~ - ~ --..,,,,---- ... ,,_,._ --....., ---~-~~ - v ? %0 ~- f!IGURE 1-11 J1ydrogen-bubble visualization of laminar flow in the entrance of a tube at Re = 1600. [From Nakayama (1988), courtesy of Y. Nakayama, Takai University.] Rlow __,._ ---x Subsonic diffuser Flow separation FIGURE 1-12 Flow separation in a diffuser. Example 4 Flow past a prolate spheroid. Flows involving complicated three- dijmensional effects, such as the cylinder flow of Example 2, are rightly termed com- plex viscous flows and cannot be analyzed by traditional boundary-layer methods. Cpmplex flows are studied either experimentally or, increasingly, by computational fluid dynamics (CFD). Figure 1-13 illustrates the computation of viscous flow past a 6: 1 prolate sphe- rqid (a slender football shape) at a high angle of attack of 20°. The angle of attack is dtifined as the angle between the oncoming flow and the central axis of the body. The R¢ynolds number is large, pUL/ µ, = 4.2E6, and the flow near the body is turbulent. Hbw might one analyze this flow with CFD? Laminar (smooth, nonfluctuating) flow can be computed accurately with a suitable fine mesh. However, the wide spectrum of raj1dom fluctuations of turbulence will yield to direct numerical simulation (DNS) only fot low Reynolds numbers of order 104 . Higher Reynolds numbers require modeling the small scale eddies with a time-averaging scheme (Chap. 6). The prolate spheroid flow in Fig. 1-13 was computed using large-eddy simula- tion (LES) by Constantinescu et al. (2002). The authors directly simulated the large eddies but used a turbulence model for the fine-scale motions. The three-dimensional grjd in Fig. l- I 3a uses 2.6 million nodes, yet it still cannot resolve the important small fluctuations. Figure 1-13b shows the computed surface streamlines when the body is pl<ll.ced at a 20° angle to the freestream. The computations are in reasonable agreement 14 VISCOUS FLUID FLOW (a) (b) FIGURE 1-13 Computational fluid dynamics model of flow past a prolate spheroid: (a) meridian slice showing the million- node grid system; (b) surface pattern at ReL = 4.2E6 and an angle of attack of 20°. [After Constantinescu et al. (2002).] PRELIMINARY CONCEPTS 15 Jwith the data of Chesnakas and Simpson ( 1997) for surface pressures and shear stress- ~s, separation lines, and surface turning angles. Prolate spheroid flow is a good example of fluids engineering research. Detailed benchmark experiments are followed by prediction methods that are gradually improved. ln addition to the LES computations of Constantinescu et al. (2002), the same flow has been modeled by a multivortex simulation, Dimas et al. ( 1998), and by traditional turbu- lence modeling without large-eddy simulation, Paneras (1998). The general subject of computational fluid dynamics is beyond the scope of the present text. We will briefly discuss CFD simulations in Chaps. 3, 4, and 6 but always refer the reader to advanced monographs for details of the subject. 1-3 PROPERTIES OF A FLUID It is c~mmon in introductory physics to divide materials into the three classes of solids,\ liquids, and gases, noting their different behavior when placed in a contain- er. Th~s is a handy classification in thermodynamics, for example, because of the strong 1 differences in state relations among the three. In fluid mechanics, however, there a,re only two classes of matter: fluids and nonfluids (solids). A solid can resist an applied shear force and remain at rest, while a fluid cannot. This distinction is not cornpletely clear-cut. Consider a barrel full of pitch at room temperature. The pitch lpoks hard as a rock and will easily support a brick placed on its surface. But if the ltlrick is left there for several days, one will have trouble retrieving the brick from tfue bottom of the barrel. Pitch, then, is usually classed as a fluid. Consider the metal ~luminum. At room temperature, aluminum is solid to all appearances and will re$ist any applied shear stress below its strength limit. However, at 400°F, well below :its l 200°F melting point, aluminum flows gently and continuously under appliecJ stress and has a measurable viscosity. Nor is high temperature the criterion for fluii.d behavior in metals, since lead exhibits this gentle viscous creep at room temperature. Note also that mercury is a fluid and has the lowest viscosity relative to its own density (kinematic viscosity) of any common substance. This text is primarily concerned, then, with easily recognizable fluids which flow re~dily under applied shear, although some slight attention will be paid to the borderl!ine substances which pai1ly flow and partly deform when sheared. All gases are true fluids, as are the common liquids, such as water, oil, gasoline, and alcohol. Some liquid substances which may not be true fluids are emulsions, colloids, high- polymcir solutions, and slurries. The general study of flow and deformation of mate- rials constitutes the subject of rheology, of which viscous flow is a special case [see, e.g., the texts by Reiner (1969), Bird et al. (2001), Hutton (1989), and Owens and Phillips (2002)]. Restricting ourselves to true fluids, we now define and illustrate their proper- ties. These properties are of at least four classes: 1. Kinematic properties (linear velocity, angular velocity, vort1c1ty, acceleration, and strain rate). Strictly speaking, these are properties of the flow field itself rather than of the fluid. 2. Transport properties (viscosity, thermal conductivity, mass diffusivity). 18 VISCOUS FLUID FLOW Note that the acceleration is concerned with u, v, and wand 12 scalar derivatives, i.e., the local changes au/at, av/at, and aw/at and the nine spatial derivatives of the form auif ax1, where i andj denote the three coordinate directions. Henceforth, we shall not use i, j, and k to denote unit vectors but instead use them as Cartesian indexes. The convective terms in D/Dt present unfortunate mathematical difficulty because they are non-linear products of variable terms. It follows that viscous flows with finite convective accelerations are non-linear in character and present such vex- ing analytical problems as failure of the superposition principle; nonunique solu- tions, even in steady laminar flow; and coupled oscillating motion in a continuous frequency spectrum, which is the chief feature of a high Reynolds number, or tur- bulent, flow. Note that these non-linear terms are accelerations, not viscous stresses. It is ironic that the main obstacle in viscous-flow analysis is an inviscid term; the viscous stresses themselves are linear if the viscosity and density are assumed constant. In frictionless flow, non-linear convective accelerations still exist but do not misbehave, as can be seen with reference to the valuable vector identity y2 (V · V)V = V 2 - V X (V x V) (1-10) As we shall see, the term V X V usually vanishes if the viscosity is zero (irrota- tionality), leaving the convective acceleration equal only to the familiar kinetic- energy term of Bernoulli's equation. Inviscid flow, then, is non-linear also, but the non-linearity is confined to the calculation of static pressure, not to the determina- tion of the velocity field, which is linear. If we agree from this brief discussion that viscous flow is mathematically formidable, we deduce that a very important problem is that of modeling a viscous- flow experiment. That is, when can a velocity distribution V 1 measured in a flow about or through a model shape B 1 be scaled by (say) a simple multiplier to yield the velocity distribution V 2 about or through a geometrically similar but larger (say) model shape B 2? This happy condition is called similarity, and the conditions for achieving it or the frustrations in not achieving it are discussed in Chap. 2. 1-3.3 Other Kinematic Properties In fluid mechanics, as in solid mechanics, we are interested in the general motion, deformation, and rate of deformation of particles. Like its solid counterpart, a fluid element can undergo four different types of motion or deformation: ( 1) translation; (2) rotation; (3) extensional strain, or dilatation; and (4) shear strain. The four types are easy to separate geometrically, which is of course why they are so defined. The reader familiar with, say, the theory of elasticity for solids will find the following analysis of fluid kinematic properties almost identical to that in solid mechanics. Consider an initially square fluid element at time t and then again later at time t + dt, as illustrated in Fig. 1-14 for motion in the xy plane. We can see by inspec- tion the four types of motion which acted on the element. There has been transla- tion of the reference corner B to its new position B'. There has been a counter- clockwise rotation of the diagonal BD to the new position B' D'. There has been y dV dy+ - dydt dy B' dU Time, t A i-------./ D / / / dy / / dx+ - dxdt dX v dt D' 45°/" / /45<1> 8"'-v _ _._ _ __.. c----~----------- x dx FIGURE 1-14 Distortioln of a moving fluid element. PRELIMINARY CONCEPTS 19 dilatation; the element looks a little bigger. There has been shear strain, i.e., the square has become rhombic. Now put this discussion on a quantitative basis. Note in each case that the final result will be a rate, i.e., a time derivative. The translation is defined by the displacements u dt and v dt of the point B. The rare of translation is u, v; in three-dimensional motion, the rate of translation is the velocity u, v, w. The angular rotation of the element about the z axis is defined as the average counterclockwise rotation of the two sides BC and BA. As shown in Fig. 1-14, BC has rotfited an amount da. Meanwhile, BA has rotated clockwise, thus its counter- clockwise turn is (-d{3). The average rotation is thus (1-11) where the subscript z denotes rotation about an axis parallel to the z axis. Thus we perceived the counterclockwise rotation in Fig. 1-14 because da was drawn larger than d/$. Referring to Fig. 1-14, we find that both da and df3 are directly related to velocity derivatives through the calculus limit: da l" -I ax ( av dx dt ) = im tan dr~o dx + ~~ dx dt =av dt ax ( au d dt ) (1-12) y Y =au dt df3 = lim tan- 1 dr~o dy + ~~ dy dt ay 20 VISCOUS FLUID FLOW Substituting Eq. (1-12) into (1-11), we find that the rate of rotation (angular velocity) about the z axis is given by (1-13) In exactly similar fashion, the rates of rotation about the x and y axes are (1-14) These are clearly the three components of the angular velocity vector dO/dt. The three factors of one-half are irritating, and it is customary to work instead with a quantity w equal to twice the angular velocity: w = 2d0 dt (1-15) The new quantity w, of vital interest in fluid mechanics, is called the vorticity of the fluid. By inspecting Eqs. (1-13) to (1-15), we see that vorticity and velocity are related by the vector calculus: w = curl V = V X V (1-16) and hence the divergence of vorticity vanishes identically: div w = V · w = div curl V = 0 (1-17) Mathematically speaking, we say the vorticity vector is solenoidal. Note also that vorticity is intimately connected with convective acceleration through Eq. ( 1-10). If w = 0, the flow is irrotational. Now consider the two-dimensional shear strain, which is commonly defined as the average decrease of the angle between two lines which are initially perpen- dicular in the unstrained state. Taking AB and BC in Fig. 1-14 as our initial lines, the shear-strain increment is obviously 4Cda + d{3). The shear-strain rate is E = l(da + df3) = l(av + au) xy 2 dt dt 2 ax ay (1-18) Similarly, the other two components of shear-strain rate are 1 (aw av) Eyz = 2 ay + az I (au aw) Ez.t = 2 az + ax (1-19) By analogy with solid mechanics, the shear-strain rates are symmetric, that is, Eij = Eji· The fourth and final particle motion is dilatation, or extensional strain. Again with reference to Fig. 1-14, the extensional strain in the x direction is defined as the fractional increase in length of the horizontal side of the element. This is given by dt = (dx + au/ax dx dt) - dx = au dt €_\_\ dx ax (1-20) PRELIMINARY CONCEPTS 23 No-slip v----c;:;:====:::::::;===:;====~:::::::=====.::;:J-~V :::::: t // v (~ :.·~:.T.~viscous / Apparent::::; h /"""-- ·.·.·. :}~ t1u·1d / velocity ::::: .. / ~ :=:::: / profile :·:·: FIGURE 1-15 No-slip A fluid sheared between two plates. liquids such as oil, but the time to flow is not viscosity, any more than the speed of sound is the time it takes an echo to return from a mountainside. It is an intriguing fact that the flow of a viscous liquid out of the bottom of a cup is a difficult problem for which no analytic solution exists at present. A more fundamental approach to viscosity shows that it is the property of a fluid which relates applied stress to the resulting strain rate. The general relations are considered in Sec. 2-4. Here we consider a simple and widely used example of a fluid sheared between two plates, as in Fig. 1-15. This geometry is such that the shear stress Txy must be constant throughout the fluid. The motion is in the x direc- tion only and varies with y, u = u(y) only. Thus there is only a single finite strain rate in this flow: E = l (au + av) = l au = l du xy 2 ay ax 2 ay 2 dy (1-28) If one performs this experiment, one finds that, for all the common fluids, the applied shear is a unique function of the strain rate: Txy = f(Exy) (1-29) Since, for a given motion V of the upper plate, T xv is constant, it follows that in these fluids Ex_n and hence du/dy, is constant, so that the resulting velocity profile is linear across the plate, as sketched in Fig. 1-15. This is true regardless of the actual form of the functional relationship in Eq. (1-29). If the no-slip condition holds, the veloc- ity profile varies from zero at the lower wall to Vat the upper wall (Prob. 1-16 con- siders the case of a slip boundary condition). Repeated experiments with various values of Txy will establish the functional relationship Eq. (1-29). For simple fluids such as water, oils, or gases, the relationship is linear or newtonian: or V du Txy = µ -h = 2µExv = µ -d . . y (1-30) The quantityµ, called the coefficient of viscosity of a newtonianfiuid, is what hand- books commonly quote when listing the viscosity of a fluid (see App. A). Actually, there is also a second coefficient, A, related to bulk fluid expansions, but it is rarely encountered in practice (see Sec. 2-4). Equation (1-30) shows that the dimensions 24 VISCOUS FLUID FLOW Shear stress, r Plastic ..... / / / / Ideal Bingham / plastic Shear stress, r I I I Pseudoplastic Yield stress i Time rate of deformation, E FIGURE 1-16 Viscous behavior of various materials. Rheopectic Thixotropic Constant strain rate, E Time ofµ, are stress-time: N·s/m2 [or kg/(m·s)] in metric units and lbf·s/ft2 [or slugs/ft· s)] in English units. The conversion factor is 1 N · s/m2 = 0.020886 lbf · s/ft2 (1-31) The coefficient µ, is a thermodynamic property and varies with temperature and pressure. Data for common fluids are given in App. A. If the functional relationship in Eq. ( 1-29) is nonlinear, the fluid is said to be nonnewtonian. Some examples of nonnewtonian behavior are sketched in Fig. 1-16. Curves for true fluids, which cannot resist shear, must pass through the origin on a plot of 7 vs. E. Other substances, called yielding fluids, show a finite stress at zero strain rate and are really part fluid and part solid. The curve labeled pseudoplastic in Fig. 1-16 is said to be shear-thinning, since its slope (local viscosity) decreases with increasing stress. If the thinning effect is very strong, the fluid may be termed plastic, as shown. The opposite case of a shear-thickening fluid is usually called a dilatant fluid, as shown. Also illustrated in Fig. 1-16 is a material with a finite yield stress, followed by a linear curve at finite strain rate. This idealized material, part solid and part fluid, is called a Bingham plastic and is commonly used in analytic investigations of yielding materials under flow conditions. Yielding substances need not be linear but may show either dilatant or pseudoplastic behavior. Still another complication of some nonnewtonian fluids is that their behavior may be time-dependent. If the strain rate is held constant, the shear stress may vary, and vice versa. If the shear stress decreases, the material is called thixotropic, while the opposite effect is termed rheopectic. Such curves are also sketched in Fig. 1-16. A simple but often effective analytic approach to nonnewtonian behavior is the power-law approximation of Ostwald and de Waele: (1-3 la) PRELIMINARY CONCEPTS 25 where K and n are material parameters which in general vary with pressure and temperature (and composition in the case of mixtures). The exponent n delineates three cases on the left-hand side of Fig. 1-16. n < pseudoplastic n 1 newtonian (K = µ) (l-3lb) n > 1 dilatant Note the Eq. (1-31 a) is unrealistic near the origin, where it would incorrectly pre- dict the pseudoplastic to have an infinite slope and the dilatant a zero slope. Hence many other formulas have been proposed for nonnewtonian fluids; see, e.g., Bird et al. (2001) or Hutton et al. (1989). The reader is advised to become familiar with such flows although the present text is confined to the study of newtonian flow. 1-3.6 Viscosity as a Function of Temperature and Pressure The coefficient of viscosity of a newtonian fluid is directly related to molecular interactions and thus may be considered a thermodynamic property in the macro- scopic sense, varying with temperature and pressure. The theory of the transport properties of gases and liquids is sti11 being developed, and a comprehensive review is given by Hirschfelder et al. ( 1954). Extensive data on properties of fluids are given by Reid et al. (1987). No single functional relation µ(T, p) really describes any large class of fluids, but reasonable accuracy (±20 percent) can be achieved by nondimensionalizing the data with respect to the critical point (Tc., pc). This procedure is the so-called princi- ple of corresponding states [Keenan (1941)], wherein the given property, hereµ/ µ 0 is found to be roughly a function of T/Tc and p/pc., the reduced temperature and pres- sure. This principle is not justified on thermodynamic grounds but arises simply from dimensional analysis and experimental observation. Since changes occur very rapidly near the critical point, Tc and Pc are known only approximately, and it is essentia11y impossible to measure /Le accurately. Appendix A contains a table of critical constants (Tc., Pc,, J-Lc, kc) for common fluids, which should be regarded as best-fit values. Figure 1-17 shows a recommended correlation of reduced viscosity µ//-Le vs. reduced temperature T/Tc and reduced pressure P/Pc· As stated, the accuracy for any given fluid is about ±20 percent. By examining this figure, we can make the fo11owing general statements: 1. The viscosity of liquids decreases rapidly with temperature. 2. The viscosity of low-pressure (dilute) gases increases with temperature. 3. The viscosity always increases with pressure. 4. Very poor accuracy obtains near the critical point. Since Pc for most fluids is greater than 10 atm (App. A), typical gas flow problems are at low reduced pressure and approximate the low-density-limit curve of Fig. 1-17. 28 VISCOUS FLUID FLOW TABLE 1-2 Power-law and Sutherland-law viscosity parameters for gases [Eqs. (1-35) and (1-36)]T Error,%, temperature Gas T0,K µ 0, N · s/m 2 n range, K Air 273 I.716E-5 0.666 ±4 210-1900 Argon 273 2.125E-5 0.72 ±3 200-1500 C02 273 l.370E-5 0.79 ±5 209-1700 co 273 I.657E-5 0.71 ±2 230-1500 N1 273 l.663E-5 0.67 ±3 220-1500 02 273 1.919E-5 0.69 ±2 230-2000 H1 273 8.411E-6 0.68 ±2 80-1100 Steam 350 l.12E-5 1.15 ±3 280-1500 Source: Data from Hilsenrath et al. (1955). Temperature range for 2% S,K error 1 11 170-1900 144 120-1500 222 190-1700 136 130-1500 107 100-1500 139 190-2000 97 220-1100 1064 360-1500 ,.No data given above maximum temperature listed. Formulas inaccurate below minimum temperature listed. where n is of the order of 0.7 and To and µ 0 are reference values. This formula was suggested by Maxwell also and later deduced on purely dimensional grounds by Rayleigh. Table 1-2 lists empirical values of n for various gases and the accuracy obtainable for a given temperature range. A second widely used approximation resulted from a kinetic theory by Sutherland (1893) using an idealized intermolecular-force potential. The final for- mula is µ :::::::: (L)3/2 To + S µo T0 T + S (1-36) where S is an effective temperature, called the Sutherland constant, which is char- acteristic of the gas. The accuracy is slightly better than that of Eq. (1-35) for the same temperature range. Values of S for common gases are also given in Table 1-2. Less common gases are tabulated in App. A. These dilute-gas formulas are strictly valid only for a single component sub- stance; air qualifies only because its two principal components, oxygen and nitrogen, are nearly identical diatomic molecules. For mixtures of gases of markedly different species, the mixture viscosity varies strongly with species concentration. A good dis- cussion of transport properties of gas mixtures is given in Bird et al. (2001). 1-3.8 The Coefficient of Thermal Conductivity It is well established in thermodynamics that heat flow is the result of temperature variation, i.e., a temperature gradient. This can be formally expressed as a propor- tionality between heat flux and temperature gradient, i.e., Fourier's law: q = -kVT (1-37) PRELIMINARY CONCEPTS 29 where q is the vector rate of heat flow per unit area. The quantity k is our second trans- port coefficient, the thermal conductivity. Solid substances often show anisotropy, or directional sensitivity: (1-38) Fortunately, a fluid is isotropic, i.e., has no directional characteristics, and thus k is a thermodynamic property and, like viscosity, varies with temperature and pressure. Also, like viscosity, the common fluids will correlate with their critical properties, as shown in Fig. 1-18. Note that the general remarks for Fig. 1-17 also apply to Fig. 1-18 but the actual numerical values are quite different. The low-density limit for k is quite practical for the problems in this text, and the kinetic theory of dilute gases applies once more. 10~~--~-~-------~-~~~~ 91----+---+--+--+-----t----t----+--+--t--t-t--t--f 81---+----+---+---+-----+-----+---+--+--t--t--t--+--i 7 1-----+-- 6 1----1-- 5 41---+-~~,,,,,___~~~~~-+------+---+---+--+--+-+--+----l ~ I ~<J 3 1----+---+-~~~~---"<c----~-~ II ~._ ~ ·:; ·..= u ::l TI c 0 u ctl E .... Q) £ -0 Q) u ::l TI Q) a: 1 t-----+--- 0. 9 1----+---+--+---+H-+'t-'< 0.8 1----4-----+--- o. 7 l---+----+----r------1--'-" 0.6 l---+---+-- 0.5 f----+---+---"---r-f"----?-s.~-f----+---+--+-+-+---+--+-t 0 .4 l---+---+--0.4~.r7~---+--+---+-+--t--t-+-l---1 o .3 1----4--- _ o.2-""!.<'~R------+----+----+--+---+--f---1f--+--l l> 0.1 0.2 (ff / .§ o' 0~ .:{~ Low-density limit 0.1~~--~~-~---~-~-~~~~~~ 0.3 0.4 0.6 0.8 1.0 2 3 4 5 6 7 8 910 T Reduced temperature, Tr= - Tc FIGURE 1-18 Reduced thermal conductivity vs. reduced temperature. [After Owens and Thodos (1957).] 30 VISCOUS FLUID FLOW By inspection of Eq. (1-37), we see that thermal conductivity k should have dimensions of heat per time per length per degree, the common engineering unit being Btu/ (h)(ft)( 0 R). The metric unit is W / (m · K), and the conversion factor is 1 w /(m. K) = 0.5778 Btu/(h)(ft)(0 R) (1-39) Also, k has the dimensions of viscosity times specific heat, so that the ratio of these is a fundamental parameter called the Prandtl number: J-LCp Prandtl number = Pr = k (1-40) This parameter involves fluid properties only, rather than length and velocity scales of the flow. As we shall see, the Prandtl number is important in heat-transfer calcu- lations but does not enter into friction computations. The kinetic theory for conductivity of dilute gases is very similar to the vis- cosity analysis and leads to an expression similar to Eq. (1-33): k Wj(m. K) = 0.08330 a.2Dv VAf (1-41) where u, Dv, and M have the same meanings and the same numerical values as in Eq. (1-33) and Tis in kelvins. The collision integral flv is again given by Table 1-1. If Eq. (1-41) is divided by Eq. (1-33), the result is J-LCp 4y Pr= -- ;=:::---- k 15y - 15 (1-42) which relates the Prandtl number to the specific-heat ratio of the gas. The accuracy is only fair, however, and this relation was modified by Eucken (1913) and now serves as a purely empirical correlation formula: 4y {0.690 Pr ;:::::: 7.08y - 1.80 = 0.667 if y = 1.40 (diatomic gas) if y = 1.67 (monatomic gas) (1-43) This is seen to be a fair approximation to the Prandtl numbers of the gases plotted in Fig. 1-26. For routine calculations with dilute gases, the power law and Sutherland for- mula, which correlated viscosity data, can also be used for thermal conductivity: Power law: k -;:::::: ko Sutherland: k -;:::::: ko ( L)3/2 To+ S T0 T + S (l-44a) (l-44b) The accuracy is from 2 to 4 percent, depending upon the gas. Values of n, S, k0, and To are given for the common gases in Table 1-3, along with their accuracy when compared with the data compiled by White ( 1988). PRELIMINARY CONCEPTS 33 For dilute gases, kinetic theory yields the following expression for the binary- diffusion coefficient D 12 between species 1 and 2 [Chapman and Cowling (1970)]: 0.001858T312 [(M 1 + M 2)/ M 1M 2] 1!2 D12 = ~~~~~~~~~~~~~~- P<TT2!1 D (1-51) where D 12 = binary-diffusion coefficient, cm 2 /s M 1, M 2 = molecular weight of two gases flD = diffusion collision integral u 12 = effective collision diameter, A T = absolute temperature, K p = pressure, atm The collision integral is about 10 percent smaller than Dv from Table 1-1 and can be approximated closely by modifying Eq. (1-34): DD ~ l.OT*-0.145 + (T* + 0.5)-2.0 (1-52) where T* = T/TE 12 • The effective temperatures and diameters are averages com- puted from the separate molecular properties of each species: I <T12 = 2(a-1 + <T2) (1-53) TE12 = (TE1TE2) 112 The molecular properties listed in App. A are valid for these diffusion calculations also. Note that, unlike viscosity and conductivity, diffusion coefficient D varies inversely with pressure in Eq. (1-51). Thus, at low pressures, product pD is a func- tion only of temperature. For dense gases, there is considerable pressure depend- ence also, but very few data are available. Figure 1-19 is a tentative plot of this pres- sure effect for self-diffusion D 11 (nearly identical molecules). Note that reduced a a -10 33 1.0 I I 0- == 3.Q- 0.9 0.8 0.7 0.6 0.5 0.4 --..;;::: ,,,_ --- :r--r-----::::--..:::::::: r-- --..._-i--- r--i--- :;.a::::: ..._ - 1.6,,_ ............. .......... -........ r----:: r---....._ - ----- 1 ~ .......... ......_ " ...... ~ ~ '----- ::r=== ' "" ~ ""'~-\ " ·<-~ ~ ......... _!., - 0.3 0.2 0.1 _,,..-~ 7.o ---~ 0 0.1 0.15 0.2 0.3 0.40.5 0.6 0.81.0 1.5 2.0 3.0 4.0 p p,=- Pc FIGURE 1-19 Tentative generalized chart for self-diffusivity of a dense gas. [After Slattery and Bird (1958).] 34 VISCOUS FLUID FLOW diffusivity D/Dc cannot be used. The plot involves instead the ratio of pD to its value (pD)o at the same temperature but very low pressures. Since binary-diffusion data are scarce, we shall not attempt to present a power law or a Sutherland formula for estimating D 12 of dilute gases. For further infor- mation on diffusion estimates, consult Reid et al. ( 1987). 1-3.10 Transport Properties of Dilute-Gas Mixtures The previous discussion of viscosity has been confined essentially to fluids of a single species. For gases, the kinetic theory has been extended, for the low-density limit, to multicomponent mixtures. The details are given by Hirschfelder et al. (1954 ). For rou- tine calculations, the semiempirical formula of Wilke ( 1950) is recommended: 11 XiJ.Li /.Lmix :=::::: L 2: 11 cf> i= I }= JXj iJ (l-54) where This formula is for a mixture of n gases, where M; are the molecular weights and Xi are the mole fractions. In terms of the mass fractions Ci = pif p, we have C-/M· I I X; = "ll (C/M) .L.-j=l j j (l-55) A similar formula is recommended by Wilke for the thermal conductivity k of a mixture of n gases except that µ,; are replaced by k;. 1-3.11 Transport Properties of Liquids The theoretical analysis of liquid transport properties is not nearly as well devel- oped as that for gases. The difficulty is that liquid molecules are very closely packed compared to gases and thus dominated by large intermolecular forces. Momentum transfer by collisions-so dominant in gases-is small in liquids. The kinetic theory of liquids is summarized in the texts by Bird et al. ( 1977) and Reid et al. ( 1987). However, the theory is not yet quantitative, in the sense that a given liquid's properties cannot be predicted from other thermodynamic data. If no data are available, we recommend the reduced temperature and pressure plots, Figs. 1-17 and 1-18, for ± 20 percent accuracy. If data are available for cali- bration, however, they may be fit accurately to the empirical formula: Liquid: µ, (T0 ) (T0 ) 2 ln /.Lo ~ a + b T + c T (l-56) where (µ,0 , T0) are reference values and (a, b, c) are dimensionless curve-fit con- stants. For nonpolar liquids, c ~ 0, i.e., the plot is linear. PRELIMINARY CONCEPTS 35 0.5 T0 = 273 K Freezing 0.0 -0.5 -1.0 -1.5 • Data, App. A Boiling -2.0 ~-~--~--~--~--......._ __ ....____~ 0.70 0.80 FIGURE l-20 To T 0.90 1.00 Empirical plot of the viscosity of water in the manner suggested by Eq. ( 1-56). For example, App. A tabulates the viscosity of water at atmospheric pressure. If these data are plotted in the manner suggested by Eq. (l-56), the results are shown in Fig. 1-20. The plot is nearly linear, and the curve-fit values are a= -2.10 b = -4.45 c = 6.55 (1-57) corresponding to T0 = 273 K and µ 0 = 0.00179 kg/m · s. The accuracy of the curve fit is ±I percent when compared to the data of App. Table A. I. For thermal conductivity of liquids, Eq. ( 1-56) may not be a good fit-see, e.g., data for water in App. Fig. A-5. Reid et al. (1987) recommend the simple lin- ear fit k1iq or Dtiq ~ a + bT (1-58) which will be accurate at least over a limited temperature range. 38 VISCOUS FLUID FLOW are quite sufficient thermodynamically, as discussed previously. However, in prepar- ing such a Mollier chart for publication, it is usual to include other lines (constant density, temperature, speed of sound). Data for water and steam are given by Parry et al. (2000). 1-3.13 Secondary Thermodynamic Properties Still other properties are often used in flow analyses, particularly with idealized equations of state such as the perfect-gas law. Two of these are the specific heats (so-called) at constant pressure and constant volume: (1-69) which of course are not heats at all but expressions of energy change. The ratio of specific heats (l-70) is an important dimensionless parameter in high-speed ( con1pressible) flow problems. This ratio lies between 1.0 and 1.7 for all fluids. For a liquid, which is nearly incom- pressible, cP ~ cv and y ~ 1.0. Figure 1-22 shows values of y at atmospheric c..f'-1 c.S II ?--.. 1.6 1.5 1.4 1.3 1.2 1 .1 Ar 1000 Atmospheric pressure 2000 3000 Temperature, 0 R 4000 5000 FIGURE 1-22 Specific-heat ratio for eight common gases. [Data from Hilsenrath et al. (1955).] PRELIMINARY CONCEPTS 39 pressure for various gases as a function of absolute temperature. The pressure depend- ence of y is very weak, and the figure shows that the approximation ( y = const) is accurate over fairly wide temperature variations. Another minor but important thermodynamic property is the speed of sound a. defined as the rate of propagation of infinitesimal pressure pulses: 2 ap a=-ap s (1-71) [see, e.g., White (2003), Sec. 9.2]. The partial derivative in Eq. (1-71) is often rather clumsy to handle, in which case an alternate relation can be used: a2 = Y(ap) ap T (1-72) The student may prove as an exercise that Eqs. ( 1-71) and ( 1-72) are thermody- namically identical. Since viscous flows are definitely not isentropic in general, the speed of sound is not a natural inhabitant of viscous analyses but enters instead through the assumption of perfect-gas relations, as does the Mach number. 1-3.14 The Perfect Gas All the common gases follow with reasonable accuracy, at least m some finite region, the so-called ideal or perfect-gas law: p = pRT (1-73) where R is called the gas constant. Equation (1-73) has a solid theoretical basis in the kinetic theory of dilute gases, e.g., Brush ( 1972), and should not be regarded as an empirical formula. The gas constant R is the ratio of Boltzmann's constant to the mass of a single molecule: (1-74) Alternately, R may be written in terms of the molecular weight M of the gas: Ro Rgas = ~ gas (1-75) where Ro is a universal constant similar to Boltzmann's constant. In metric units, Ro = 8314 J/(kg · mol · K) (1-76) which is too many significant figures, since no gas really fits the law that well. Equations (1-73) and (1-75) are also suitable for mixtures of gases if the equivalent molecular weight is properly defined in terms of the mass fractions Ci = pJ p: (1-77) 40 VISCOUS FLUID FLOW In terms of the mole fractions Xi (number of moles of species i per mole of mix- ture), we have (1-78) As a classic example, air at ordinary temperatures has mole fractions of approxi- mately 78 percent nitrogen, 21 percent oxygen, and 1 percent argon. Then, from Eq. (1-78), Mair = 0.78(28.016) + 0.21(32.000) + 0.01(39.944) = 28.97 from which 8314 Rair = 28 .97 = 287 J/(kg · K) These are the accepted values for room-temperature dry air. According to Eq. (1-73), the so-called compressibility factor p z = pRT (1-79) should be unity for gases. Actually, Z varies from zero to 4.0 or greater, depending upon temperature and pressure. To good accuracy, Z is a function only of the reduced variables p,. = P/Pcrit and T,. = T/Tcrit referred to the critical point. This is illustrated in Figs. 1-23 and 1-24, which are representative of all gases. Examining these figures, we see that the perfect-gas law (Z = 1) is accurate to± 10 percent in the range 1.8 < T,. < 15 and 0 < p,. < 10, which is the range of interest of the majority of viscous-flow problems. The higher temperatures should be viewed with 0.7 QI~ 0.6 0.5 II N 0.3 0.2 0.1 0 0 0.5 1 Liquid region 1 .5 2 2.5 3 3.5 4 4.5 5 Reduced pressure = Pr FIGURE 1-23 Compressibility factors for gases. [After Weber (1939).] PRELIMINARY CONCEPTS 43 that cp and cv must increase with temperature. Since R is known from Eq. ( 1-75), Fig. 1-22 may be used to calculate cp and cu to good accuracy. Finally, Fig. 1-26 shows the Prandtl number, Pr = µ,cp/ k, for various com- mon gases. Note by comparison with Fig. 1-22 that Eucken's formula, Eq. (1-43), is only a fair approximation. , 1-3.16 Bulk Modulus In flow problems which involve sound-wave propagation, it is useful to have a ther- modynamic property which expresses the change of density with increasing pres- sure. This property is the bulk modulus K: (1-84) From Eq. (1-72), then, we see that the speed of sound can be written in terms of K: 2 yK a=-p (1-85) where 1· may be taken as 1.0 for liquids and solids.";- For a perfect gas, the reader may verify that K = p itself. For liquids, however, K is fairly constant, varying slightly with pressure and temperature. For water, a good average value is K = 2.2E9 Pa. Taking y = 1.0 and an average water density of 998 kg/ m3 , we cal- culate the speed of sound in water to be approximately 1480 m/s. 1-3.17 Coefficient of Thermal Expansion There is a subset of flow problems, called natural convection, where the flow pat- tern is due to buoyant forces caused by temperature differences. Such buoyant forces are proportional to the coefficient of thermal expansion f3, defined as f3 = _l(dp) P iJT P (1-86) For a perfect gas, the reader may show that f3 = 1/T. For a liquid, f3 is usually smaller than 1/T and may even be negative (the celebrated inversion of water near the freezing point). For imperfect gases, f3 can be considerably larger than l/T near the saturation line, particularly at high pressures. This is illustrated in Fig. 1-27 for steam. We see that steam is well approximated by the perfect-gas result f3T = I at low pressures and high temperatures. <A solid has tn;o sound speeds: (K/ p) 1/ 2 is the dilatation, or longitudinal-wave, speed, and ( G / p) 1/ 2 is the rota- tional, or shear-wave, speed. Fluids have only one sound speed. 44 VISCOUS FLUID FLOW 2.0~~~~~~,~~~~..--~---, 1.8 1.6 f3T 1.4 1.2 , Tc= 1165 °R I Pc= 3208 psi : I I , , 1.ol--~~__.:___:~~--=:::::::=::;;;;;:;::;==----:i p = O (perfect gas) 500 1000 1500 2000 FIGURE 1-27 Temperature, 0 R Thermal-expansion coefficient for steam. The quantity f3 is also useful in estimating the dependence of enthalpy on pressure, from the thermodynamic relation ( 1-87) where we remark that of course T must be absolute temperature. For the perfect gas, the second term vanishes, so that h = h(T) only. Figure 1-27 shows that imperfect gases like steam also often fit this approximation dh ~ cp dT. Table 1-4 gives measured values of the bulk modulus K and the thermal- expansion coefficient f3 for water at saturation pressures. TABLE 1-4 Bulk modulus K and expansion coefficient f3 for water at saturation conditions T,K Psat• kPa f3T K,MPa 273 0.61 -0.019 2,062 293 2.34 0.057 2,230 313 7.38 0.119 2,304 333 19.92 0.176 2,301 353 47.35 0.230 2,235 373 101.3 0.281 2,120 423 461 0.447 1,692 473 1,580 0.637 1,190 523 3,970 0.985 716 573 8,560 1.80 342 623 16,500 4.8 82 64T 22,090-;· 00 0 'Critical point. 1-4 BOUNDARY CONDITIONS FOR VISCOUS-FLOW PROBLEMS PRELIMINARY CONCEPTS 45 The equations of motion to be discussed in Chap. 2 will require mathematically ten- able and physically realistic boundary conditions. It is of interest here to study the underlying physical mechanisms of the boundary approximations commonly used. For fluid flow, there are five types of boundaries considered: J. A solid surface (which may be porous) 2. A free liquid surface 3. A liquid-vapor interface 4. A liquid-liquid interface 5. An inlet or exit section Cases 2 and 3 are related in the sense that a free liquid surface is a special case of the liquid-vapor interface where the vapor causes a negligible interaction. Let us take up these cases in order. 1-4.1 Conditions at a Solid Surface: Microflow Slip in Liquids Wall boundary conditions depend upon whether the fluid is a liquid or a gas. For macroflows, system dimensions are large compared to molecular spacing, so that both liquid and gas particles contacting the wall must essentially be in equilibrium with the solid. At a solid surface, the fluid will assume the velocity of the wall (the no-slip condition) and the temperature of the wall (the no-temperature-jump condition): V fluid = V sol Tfluid = Tso! (l-88) We shall use Eqs. ( 1-88) throughout this text for newtonian liquids. However, cer- tain liquid/solid combinations are known to slip under small-scale microflow con- ditions. The subject is somewhat controversial and is reviewed in the monograph edited by Gad-el-Hak (2001). Some liquid microftows studies show slip and others do not. One way to characterize slip in liquids is the slip length Lslip relating slip velocity to the local velocity gradient, a model first suggested by Navier himself: Uwall = Lslip (~~) wall (l-89) The slip length depends upon the liquid, the geometry, and the shear rate. Tretheway and Meinhart (2002) test water flowing in a 30 ,um width microchannel coated with hydrophobic octadecyltrichlorosilane (OTS). They measure Lslip ~ 1 µm and a wall- slip velocity equal to 10 percent of the centerline velocity. Choi et al. (2003), studying 48 VISCOUS FLUID FLOW For further study of micro- and nanoflows and rarefied gases, see the monographs by Karniadakis and Bestok (2001) and Cercignani (2000). For most gas flow analy- ses in this text, we will use the standard no-slip condition, Eqs. ( 1-88). For a new method of computer calculation of molecular flows, formulated by following and averaging thousands of individual particles, see the text by Bird (1994 ). This type of calculation is now called the direct simulation Monte Carlo method (DSMC). 1-4.3 Kinetic Theory for Wall-Temperature Jump The temperature condition in Eq. (1-88) will also fail for a gas flow if the mean free path is large compared to flow dimensions. This effect is called temperature jump and is analogous to velocity slip. The kinetic-theory expression for the jump Tgas - Twin a gas flow was given by Smoluchowski ( 1898): Tgas - Tw ~ (~ - 1) 2: l :~ (~T) (1-96) 1' p y H' where a is the so-called thermal-accommodation coefficient, defined as the fraction of impinging molecules which becomes accommodated to the temperature of the wall. Clearly a is analogous to the diffuse-reflection coefficient fin Eq. (1-92). Experiments have shown that a is also fairly close to unity. With a = 1.0, we can again substitute for C from Eq. (1-32) and for dT/dy from Eq. (1-37) to obtain 3y µk qw Tgas - Tw ~ 1' + l paµcp k (l-97) Eliminating µ and k, we divide by the temperature difference T,. - Tw which con- trols the wall heat transfer (see Sec. 7-3) and rearrange: Tgas - Tw Tr - Tw ~ 3y u q\V ')' + 1 a pcPU(Tr - Tw) 3y 1' + 1 MaC1z (1-98) where C1i is the so-called Stanton number, or wall heat-transfer coefficient of the flow. It will be shown in Chap. 4 that in boundary-layer flows, the Stanton number is approximately one-half the friction coefficient, an effect which is called the Reynolds analogy: C1i ~ 1cf (1-99) If finally we combine Eqs. (1-98) and (1-99) and take y = 1.4 (air), we find that the temperature jump, expressed as a fraction of the driving temperature difference Tr - Tw, becomes Tgas - Tw T _ T ~ 0.87MaCf r H' (1-100) which is nearly identical to the slip relation, Eq. (1-94). Thus the same remarks as before apply here. Temperature jump in turbulent flow is entirely negligible, and the PRELIMINARY CONCEPTS 49 jump in laminar flow is also extremely small except at low Reynolds number and high Mach number. It is customary, then, to adopt the no-slip and no-temperature- jump conditions in routine analysis of viscous gas-flow problems: (I-IOI) ln many cases, of course, the coordinate system is such that the wall is stationary, so that the velocity condition is simply V fluid = 0. 1-4.4 Conditions at a Permeable Wall Jn the event that the wall is porous and can permit fluid to pass through, the no-slip condition is relaxed on the velocity component normal to the wall. The proper con- ditions are complicated by the type of porosity of the wall, but, in general, we assume V =O tangential vnormal * 0 no slip flow through wall (1-102) The temperature condition is also complicated by a porous wall. For wall suction, where the fluid leaves the main flow and passes into the wall, it is sufficiently accu- rate to assume Ttluid = TH. suction (1-103) For injection through a porous wall into the main stream (sometimes called tran- spiration), the injected fluid may be, say, a coolant at a temperature considerably different from the wall, and one needs to consider an energy balance at the wall. A good approximation for coolant injection is to use the boundary condition proposed by Roberts [in Truitt (1960, Chap. 11)]: Injection: (l-104) where pH.V11 is the mass flow of coolant per unit area through the wall. The actual numerical value of V11 depends largely upon the pressure drop across the porous wall. In general, although a few simple porous-wall solutions will be given in this text, the flow characteristics of porous media are a self-contained specialized sub- ject of fluid mechanics. For more information, consult such texts on porous media as Bear (2000) or Crolet (2000). 1-4.5 Conditions at a Free Liquid Surface There are many flow problems where the liquid fluid ends, not at a solid wall, but at an open or free surface exposed to an atmosphere of either gas or vapor. We dis- tinguish between two cases: ( 1) the ideal or classic free surface that exerts only a known pressure on the liquid boundary and (2) a more complicated case where the atmosphere exerts not only pressure but also shear, heat flux, and mass flux at the 50 VISCOUS FLUID FLOW Vapor Liquid/ x z Z= TJ(X, y) , , , y FIGURE 1-29 Conditions at an ideal free surface. surface. This latter, more involved surface is more properly termed a liquid-vapor interface, the ocean surface being a splendid example. The classic free surf ace is sketched in Fig. 1-29. Let us assume that the xy plane is more or less parallel to the free surface, so that the actual deflected shape of the surface can be denoted by z = YJ(X, y). The two required conditions for this surface are ( 1) the fluid particles at the surface must remain attached (kinematic condition) and (2) the liquid and the atmospheric pressure must balance except for surface-tension effects. The kine- matic condition is specified mathematically by requiring the particle's upward velocity to equal the motion of the free surface: DYJ aY/ aY/ a17 w(x, y, YJ) = Dt = at + u ax + v ay (1-105) The pressure equilibrium is expressed by p(x, y, YJ) = p" - ~U, + ~J (1-106) where Rx and Ry are the radii of curvature of the surface and where 2J is the coeffi- cient of surface tension of the interface. For a two-dimensional surface deflection 17 = YJ(x) only, Eq. (1-106) becomes p(x, YJ) = Pa - / / 3/2 [ 1 + (dYJ dx)-]- (1-107) We see from this relation that, when the interface smiles (concave upward, positive curvature), p < Pa, while a frowning interface (concave downward) results in p > p 0 • Equations (1-105) and (1-106) are complex non-linear conditions, but they can be evaluated by numerical computer modeling. In the range 0 to 100°C, a clean air-water interface has a nearly linear varia- tion of surface tension with temperature: ~ (;;:) "° 0.076 - 0.00017T (0 C) (1-108) with accuracy of ± 1 percent. Measured values of air-water surface tension are given in Table 1-5 for temperatures up to the critical point. These are ideal data for a very clean interface. Under field conditions, '3 can vary greatly due to the pres- ence of surface contaminants or slicks. 1-4. 7 Conditions at a Deformable Fluid-Solid Interface PRELIMINARY CONCEPTS 53 In most of the problems of this text, we shall take a solid boundary to be a rigid interface which merely imposes no-slip and no-temperature-jump conditions on the fluid. The dynamics and thermodynamics of the solid are neglected. However, there is a growing field of research in which the solid is coupled to the fluid through deformable and dynamic interactions. In this case, we must match the velocity, stress, temperature, and heat transfer across the interface, and the equations of motion of fluid and solid are to be solved simultaneously. Material-behavior rela- tions analogous to Eqs. ( 1-111) must be used-perhaps the fluid is newtonian and the solid satisfies Hooke's law. For details, see the texts by Blevins (1977) or Au-Yang (200 I). 1-4.8 Inlet and Exit Boundary Conditions In many viscous-flow problems, it is convenient to limit the analysis to a finite region through which the flow passes. In the pipe-flow problem of Fig. 1-11, for example, we would like to limit our analysis to the short length of pipe shown. Both finite and infinite regions require known property conditions at all boundaries. In pipe flow, Fig. 1-11, we specify no-slip and no-temperature-jump at the pipe walls. An alternate temperature condition would be known heat flux, qw = -k(aT/an)w, which specifies the normal temperature gradient. In this latter case, the wall temperature would be found from the solution. At the pipe inlet, we would specify the distributions of V, T, and p. Often the inlet pressure is assumed uniform as a simplification. At the pipe exit, we specify V and T. No exit condition is required upon p, which is then found from the solution. As we shall see, pressure occurs only as first derivatives in the equations of motion (Chap. 2); thus only one pressure condition is needed. We can disregard temperature conditions if the flow is assumed isothermal. For the infinite-region example of flow about a cylinder (Fig. 1-6), we specify no-slip and no-temperature-jump at the cylinder walls. At the "inlet," or approach flow, we would specify the distributions of V, T, and p. Far downstream, we speci- fy V and T, which is very difficult, because the wake of the body will cause nonuni- formity and possible unsteadiness in V and T. We would not know, in advance, the exact velocity and temperature far downstream, but there are several useful down- stream approximations that lead to excellent CFD simulations of flow past immersed bodies. Pressure downstream will be calculated and need not be specified. 1-4.9 Resume of Fluid-Flow Boundary Conditions We have discussed the physical conditions which must hold at three different types of fluid-flow boundaries: (I) a fluid-solid interface, (2) a fluid-fluid interface, and (3) the inlet and exit of a flow. These various types of conditions are sketched in 54 VISCOUS FLUID FLOW Liquid-vapor interface: equality of m, w, p, q, and r across the surface Inlet: -....i known V, p, Tl L.- Outlet: I known V, T ,------------1 I I I I ~ I I i---- 1 Solid contact: I : / (V, T) fluid= (V, T) wall : /dr/7/~T//T/T////T/77/T////T//77 Solid impermeable wall FIGURE 1-31 Various boundary conditions in fluid flow. Fig. 1-31. The general results are that, for the majority of viscous-flow analyses, we must have the following information at the boundaries: 1. No slip or temperature jump at a fluid-solid interface: V fluid = V sol T fluid = Tso! (1-lOla) Note that there is no condition on p unless the wall is permeable. 2. Equality of velocity, momentum flux, heat flux, and mass flux at a liquid-fluid interface (neglecting surface tension): (V, p, T, q, m)fluid above = (V, p, T, q, m)liq below (1-114) If the fluid above is a gas with negligible interaction, the conditions on T, q, and m may be ignored. 3. Known values of V, p, and Tat every point on an inlet section of the flow. Also, unless simplifying approximations are made, V, and T must be known on any exit section. SUMMARY This chapter has reviewed the introductory concepts of fluid motion with which the reader should already be conversant. A brief history and some .sample viscous-flow problems were outlined, followed by an extensive discussion of the different quan- tities which distinguish a fluid: (1) the kinematic properties, (2) the transport prop- erties, and (3) the thermodynamic properties. The chapter closes with a detailed look at the various boundary conditions relevant to viscous-fluid flow. PRELIMINARY CONCEPTS 55 PROBl,EMS 1-1. A sphere 1.4 cm in diameter is placed in a freestream of 18 m/s at 20°C and 1 atm. Compute the diameter Reynolds number of the sphere if the fluid is (a) air, (b) water, (c) hydrogen. 1-2. A telephone wire 8 mm in diameter is subjected to a crossflow wind and begins to shed vortices. From Fig. 1-8, what wind velocity (in m/s) will cause the wire to "sing" at middle C (256 Hz)? 1-3. If the wire in Prob. 1-2 is subjected to a crossftow wind of 12 m/s, use Fig. 1-9 to esti- mate its drag force (in N/m). 1-4. For oil flow in a pipe far downstream of the entrance (Figs. 1-10 and 1-11 ), the axial velocity profile is a function of r only and is given by u = (C/ µ)(R2 - r2), where C is a constant and R is the pipe radius. Suppose the pipe is 1 cm in diameter and Umax = 30 m/s. Compute the wall shear stress (in Pa) ifµ = 0.3 kg/(m · s). 1-5. A tornado may be simulated as a two-part circulating flow in cylindrical coordinates, with v,. = v: = 0, v0 = wr ifr < R and Determine (a) the vorticity and (b) the strain rates in each part of the flow. 1-6. A plane unsteady viscous flow is given in polar coordinates by v,. = 0 where C is a constant and v is the kinematic viscosity. Compute the vorticity w:(r, t) and sketch an array of representative velocity and vorticity profiles for various times. 1-7. A two-dimensional unsteady flow has the velocity components: y v = 1 + 2t Find the equation of the streamlines of this flow which pass through the point (x 0, y0 ) at time t = 0. 1-8. Using Eq. (1-2) for inviscid flow past a cylinder, consider the flow along the stream- line approaching the forward stagnation point (r, 8) = (R, 7T). Compute (a) the distri- bution of strain rates E,.,. and E,-o along this streamline and (b) the time required for a particle to move from the point (2R, 7T) to the stagnation point. 1-9. A commonly used equation of state for water is approximately independent of tem- perature: p ( p )11 - ~(A+ 1) - - A Po Po where A ;:o:;; 3000, n ;:o:;; 7, p0 = 1 atm, and p 0 = 998 kg/m3. From this formula, com- pute (a) the pressure (in atm) required to double the density of water, (b) the bulk modulus of water at 1 atm, and (c) the speed of sound in water at 1 atm. 1-10. As shown in Fig. Pl-10, a 3 x 3-ft plate slides down a long 30° incline on which there is a film of oil 0.005 in. thick with viscosityµ = 0.0005 slug/(ft · s). Assuming that 58 VISCOUS FLUID FLOW Q Viscous fluid µ, f'..r<< R FIGURE Pl-24 1-25. Consider 1 m3 of a fluid at 20°C and 1 atm. For an isothermal process, calculate the final density and the energy, in joules, required to compress the fluid until the pres- sure is 10 atm, for (a) air and (b) water. Discuss the difference in results. 1-26. Equal layers of two immiscible fluids are being sheared between a moving and a fixed plate, as in Fig. Pl-26. Assuming linear velocity profiles, find an expression for the interface velocity U as a function of V, µ, 1, and µ2. v h/2 - - -- - - i------... ~- - - - - - - - - - U? h/2 y Fixed FIGURE Pl-26 1-27. Use the inviscid-flow solution of flow past a cylinder, Eqs. (1-3), to (a) find the loca- tion and value of the maximum fluid acceleration along the cylinder surface. Is your result valid for gases and liquids? (b) Apply your formula for amax to airflow at 10 m/s past a cylinder of diameter 1 cm and express your result as a ratio compared to the acceleration of gravity. Discuss what your result implies about the ability of fluids to withstand acceleration. 2-1 INTRODUCTION CHAPTER 2 FUNDAMENTAL EQUATIONS OF COMPRESSIBLE VISCOUS FLOW The equations of viscous flow have been known for more than 100 years. In their complete form, these equations are very difficult to solve, even on modem digital computers. In fact, at high Reynolds numbers (turbulent flow), the equations are, in effect, impossible to solve with present mathematical techniques because the bound- ary conditions become randomly time-dependent. Nevertheless, it is very instructive to derive and discuss these fundamental equations because they give many insights, yield several particular solutions, and can be examined for modeling laws. Also, these exact equations can then be simplified, using Prandtl's boundary-layer approx- imations. The resulting simpler system is very practical and yields many fruitful engineering solutions. 2-2 CLASSIFICATION OF THE FUNDAMENTAL EQUATIONS The basic equations considered here are the three laws of conservation for physical systems: 1. Conservation of mass (continuity) 2. Conservation of momentum (Newton's second law) 3. Conservation of energy (first law of thermodynamics) 59 60 VISCOUS FLUID FLOW The three unknowns that must be obtained simultaneously from these three basic equations are the veiocity V, the thermodynamic pressure p, and the absolute tem- perature T. We consider p and T to be the two required independent thermodyna- mic variables. However, the final forms of the conservation equations also contain four other thermodynamic variables: the density p, the enthalpy h (or the internal energy e), and the two transport propertiesµ, and k. Using our tacit assumption of local thermodynamic equilibrium, the latter four properties are uniquely deter- mined by the values of p and T. Thus the system is completed by assuming knowl- edge of four state relations p = p(p, T) µ, = µ,(p, T) h = h(p, T) k = k(p, T) (2-1) which can be in the form of tables or charts or semitheoretical formulas from kinetic theory. Many useful analyses simply assume that p, µ,, and k are constant and that h is proportional to T (h = cPT). Finally, to specify a particular problem completely, we must have known con- ditions (of various types) for V, p, and Tat every point of the boundary of the flow regime. The preceding considerations apply to a fluid of assumed uniform, homoge- neous composition, i.e., diffusion and chemical reactions are not considered. Multicomponent reacting fluids must consider at least two extra basic relations: 4. Conservation of species 5. Laws of chemical reaction plus additional auxiliary relations such as knowledge of the diffusion coefficient D = D(p, T), chemical-equilibrium constants, reaction rates, and heats of forma- tion. This text does not consider reacting boundary-layer flows [see Kee et al. (2003)]. Finally, even more relations are necessary if one considers the flow to be influenced by electromagnetic effects. This is the subject of the field of magneto- hydrodynamics. Such effects are not considered in the present text. Let us now derive the three basic equations of a single-component fluid, bear- ing in mind that the results will also apply to uniform nonreacting mixtures, such as air or liquid solutions. 7 2-3 CONSERVATION OF MASS: THE EQUATION OF CONTINUITY As mentioned in the discussion of Eq. (1-5), all three of the conservation laws are Lagrangian in nature, i.e., they apply to fixed systems (particles). Thus, in the ;Note, however, that air fl.owing at very high temperatures will undergo spontaneous diffusion and chemical reactions, as will many other mixtures. Even single-component fluids, such as oxygen, will dissociate into atomic oxygen at high temperatures. FUNDAMENTAL EQUATIONS OF COMPRESSIBLE VISCOUS FLOW 63 z y Tyy Tyx Tyz Tzy hzx Txz Tzz rij= stress in the j direction on a face normal to the i axis Txy Txx x FIGURE 2-1 Notation for stresses. electromagnetic potential. We ignore magnetohydrodynamic effects here and con- sider only the gravitational body force, which on our unit volume basis is fbody = pg (2-13) where g is the vector acceleration of gravity. The surface forces are those applied by external stresses on the sides of the ele- ment. The quantity stress Tu is a tensor, just as the strain rate Eu was in Sec. 1-3. The sign convention for stress components on a Cartesian element is shown in Fig. 2-1, where all stresses are positive. The stress tensor can be written as C" Txy T") T·· - Tn- Tyy y: (2-14) I) T .:x T:y T__ by analogy with Eu in Eq. (1-22). Like strain rate, Tu forms a symmetric tensor; that is, Tu = Tji· This symmetry is required to satisfy equilibrium of moments about the three axes of the element.:· The positions of the T's in the array of Eq. (2-14) are not arbitrary; the rows correspond to applied force in each coordinate direction. Considering the ·:·Here we assume the absence of concentrated couple stresses. 64 VISCOUS FLUID FLOW front faces of the element in Fig. 2-1, the total force in each direction due to stress is dFx = T xx dy dz + Tyx dx dz + T zx dx dy dFY = Txy dy dz + TYY dx dz + T.:y dx dy dF.: = Tx.: dy dz + Ty.: dx dz + T.:.: dx dy (2-15) preserving the positions in the array. For an element in equilibrium, these forces would be balanced by equal and opposite forces on the back faces of the element. If the element is accelerating, however, the front- and back-face stresses will be dif- ferent by differential amounts. For example, OTXX TXX, front = TXX, back + ax dx (2-16) Hence the net force on the element in the x direction, for example, will be due to three derivative terms: ( OTxx ) (oTrx ) (oT~r: ) dFx,net = ax dx dy dz+ a; dy dxdz + 0; dz dxdy or, on a unit volume basis, dividing by dx dy dz, since Tij = TJi, oTxx aTxv aTX7 f --+-· +--x - ax ay oz (2-17) which we note is equivalent to taking the divergence of the vector ( T xx' T xv' r r), the upper row of the stress tensor. Similarly, f,, and f.: are the divergences of th~ sec- ond and third row of 'T iJ· Thus the total vector surface force is OT·· I) fsur = V • T ij = OX. J (2-18) where the divergence of Tij is to be interpreted in the tensor sense, so that the result is a vector. Newton's law, Eq. (2-12), now becomes (2-19) and it remains only to express Tu in terms of the velocity V. This is done by relat- ing Tu to Eu through the assumption of some viscous deformation-rate law, e.g., the newtonian fluid. 2-4.1 The Fluid at Rest: Hydrostatics From the definition of a fluid, Sec. 1-3, viscous stresses vanish if the fluid is at rest. The velocity and shear stresses are zero, and the normal stresses become equal to FUNDAMENTAL EQUATIONS OF COMPRESSIBLE VISCOUS FLOW 65 the hydrostatic pressure. Equation (2-19) reduces to the hydrostatic equation if v = 0: Txx = Tyy = T;::;:: = -p T.. = 0 for i * 1· I) (2-20) \lp =pg If we take the z coordinate as up and assume p and g are constant, the pressure varies linearly with z, op = -pg oz. Pressure increases downward, proportional to the specific weight of the fluid. Recall that hydrostatics is treated extensively in undergraduate texts, e.g., White (2003). We must ensure here that our dynamic momentum equation reduces to Eq. (2-20) when V = 0. 2-4.2 Deformation Law for a Newtonian Fluid By analogy with Hookean elasticity, the simplest assumption for the variation of viscous stress with strain rate is a linear law. These considerations were first made by Stokes ( 1845), and, as far as we know, the resulting deformation law is satisfied by all gases and most common fluids. Stokes' three postulates are 1. The fluid is continuous, and its stress tensor Tu is at most a linear function of the strain rates Eu· 2. The fluid is isotropic, i.e., its properties are independent of direction, and therefore the deformation law is independent of the coordinate axes in which it is expressed. 3. When the strain rates are zero, the deformation law must reduce to the hydro- static pressure condition, Tu = -poi}, where oij is the Kronecker delta function ( o iJ = 1 if i = j and o iJ = 0 if i -:F j). Note that the isotropic condition 2 requires that the principal stress axes be identi- cal with the principal strain-rate axes. This makes the principal axes a convenient place to begin the deformation-law derivation. Let x 1, y 1, and z1 be the principal axes, where the shear stresses and shear strain rates vanish [see Eq. (1-24)]. With these axes, the deformation law could involve at most three linear coefficients, C 1, C2' C3 . For example, (2-21) The term-pis added to satisfy the hydrostatic condition (condition 3 above). But the isotropic condition 2 requires that the crossflow effect of E 22 and E 33 be identical, i.e., that C2 = C3. Therefore, there are really only two independent linear coefficients in an anisotropic newtonian fluid. We can rewrite Eq. (2-21) in the simpler form (2-22) where K = C 1 - C2 , for convenience. Note also that E 11 + E 22 + E 33 equals div V from Eq .. (2-5). 68 VISCOUS FLUID FLOW [Eq. (2-19)]. The result is the famous equation of motion which bears the names of Navier (1823) and Stokes (1845). In scalar form, we obtain P Du = pg - - ap + ~ (2µ, au + 1'. div v) + ~ [µ,(au + av) l Dt .x ax ax ax ay ay ax + :z [JL(~: + ~~) l p fy~ = pg, - ~~ + :x [JL ( ~~ + ~~) ] + :y ( 2JL :~ + A div V) + :z [JL (~~ + ~;)] (2-29a) Dw = _ ap + ~ [ (aw + au) J + ~ [ (av + aw) J P Dt pg~ az ax µ, ax az ay µ, az ay + ; 2 ( 2JL ~; + A div V) These are the Navier-Stokes equations, fundamental to the subject of viscous-fluid flow. Considerable economy is achieved by rewriting them as a single vector equa- tion, using the indicial notation: P ~'; = pg - Vp + a:JJL (~~; + ~~) + •V div VJ (2-29b) 2-4.5 Incompressible Flow: Thermal Decoupling If the fluid is assumed to be of constant density, div V vanishes due to the continuity Eq. (2-7) and the vexing coefficient 1'. disappears from Newton's law. Equations (2-29) are not greatly simplified, though, if the first viscosity µ, is allowed to vary with temperature and pressure (and hence with position). If, however, we assume thatµ, is constant, many terms vanish, leaving us with a much simpler Navier-Stokes equation for constant viscosity and density: DV V V? p Dt = pg - p + µ, -v (2-30) Most of the problems and solutions in this text are for incompressible flow, Eqs. (2-7) and (2-30). Note that, if p andµ, are constant, these equations are entirely uncoupled from temperature. One may solve continuity and momentum for velocity and pres- sure and then later, if one desires, solve for temperature from the energy equation of Sec. 2-5. This approximation often divides textbooks into "fluid mechanics" and, later, "heat transfer." However, the present text will maintain some heat-transfer dis- cussions throughout. FUNDAMENTAL EQUATIONS OF COMPRESSIBLE VISCOUS FLOW 69 2-4.6 Inviscid Flow: The Euler and Bernoulli Equations If we further assume that viscous terms are negligible, Eq. (2-30) reduces to (2-30a) This is called Euler's equation (derived by Leonhard Euler in 1755) for inviscid flow. It is first order in V and p and thus rather simpler than the second-order Navier-Stokes Eq. (2-30). At a fixed wall, the no-slip condition must be dropped, and tangential velocity is allowed to slip. Much research has been reported for Euler's equation: analytical (Currie 1993), numerical (Tannehill et al. 1997), and mathematical theorems (Kreiss and Lorenz 1989). As covered in undergraduate texts, for example, White (2003), Euler's equation for steady, incompressible, friction1ess flow may be integrated along a streamline between any points 1 and 2 to yield (2-30b) where z is up, that is, opposite gravity. This is Bernoulli's equation for steady fric- tionless flow. Though approximate, since all fluids are viscous, it has many appli- cations in aeronautics and hydrodynamics and serves as an outer boundary match- ing condition in boundary-layer theory (Chap. 4). The unsteady form of Bernoulli's equation will be given in Sec. 2-10. 2-5 THE ENERGY EQUATION (FIRST LAW OF THERMODYNAMICS) The first law of thermodynamics for a system is a statement of the fact that the sum of the work and heat added to the system will result in an increase in the energy of the system: dE, = dQ + dW (2-31) where Q = heat added W = work done on system The quantity E, denotes the total energy of the system; hence, in a moving system, such as a flowing fluid particle, E, will include not only internal energy but also kinetic and potential energy. Thus, for a fluid particle, the energy per unit volume is E, = p( e + 4v2 - g · r) (2-32) where e = internal energy per unit mass r = displacement of particle Just as in conservation of mass and momentum, the energy equation for a fluid is conveniently written as a time rate of change, following the particle. Thus 70 VISCOUS FLUID FLOW Heat flow per unit area: q = _ kaT x ax Wx / ---------------, , , , , dx , I q + aqx x ax dx - - I aw Wx+ _xdx ax FIGURE 2-2 Work done per unit area: w x = - ( ur xx+ vr xy + wr xz) Heat and work exchange on the left and right sides of an element. Eq. (2-31) becomes DE1 DQ DW -=-+-- Dt Dt Dt (2-33) From Eq. (2-32), we have DE1 (De DV ) Dt = p Dt + V Dt - g • V (2-34) It remains to express Q and Win terms of fluid properties. It is assumed that the heat transfer Q to the element is given by Fourier's law. From Eq. (1-37), the vector heat flow per unit area is q = -kVT Referring to Fig. 2-2, the heat flow into the left face of the element is qx dy dz while the heat flow out of the right face is ( aqx ) qx + ax dx dy dz (1-37) A similar situation holds for the upper and lower faces, involving qy, and the front and rear faces, involving qz. In each case, the net heat flow is out of the element. Hence the net heat transferred to the element is ( aq'( aqy aqz) - -- +-+- dxdydz ax ay az Dividing by the element volume dx dy dz, we have the desired expression for the heat-transfer term neglecting internal heat generation: ~; = -div q = + div(k VT) (2-35) FUNDAMENTAL EQUATIONS OF COMPRESSIBLE VISCOUS FLOW 73 2-5.1 The Incompressible-Flow Approximation From the thermodynamic identity (1-87), we can rewrite Eq. (2-48) as DT Dp pep Dt = {3T Dt + div(k VT) + <l> (2-49) Now, if flow velocity scale U becomes smaller, while heat transfer remains impor- tant, the fluid kinetic energy U2 will eventually become much smaller than the enthalpy change cP D..T. Since both Dp/Dt and <l> are of order U2, the limit of low- velocity or incompressible flow will be pep~~ ~ div(k VT) (2-50) If we further assume constant thermal conductivity, we obtain the more familiar incompressible heat-convection equation: DT 2 pep Dt ~kV T (2-51) Note that the correct specific heat is cP, not cu, even in the incompressible-flow limit of near-zero Mach number. A very good discussion of this point is given in Panton (1996, Sec. I 0.9). 2-5.2 Summary of the Basic Equations Summarizing, the three basic laws of conservation of mass, momentum, and energy have been adapted for use in fluid motion. They are, respectively, ap at+ div pV = 0 (2-6) DV p Dt = pg + V · r;1 - Vp (2-l 9a) Dh Dp . , au; p Dt = Dt + d1v(k VT) + riJax. J (2-48a) where, for a linear (newtonian) fluid, the viscous stresses are (2-27) As mentioned in the beginning of this chapter, Eqs. (2-6), (2-19), and (2-48) involve seven variables, of which three are assumed to be primary: p, V, and T (say). The remaining four variables are assumed known from auxiliary relations and data of the form P = p(p, T) h = h(p, T) µ. = µ.(p, T) k = k(p, T) (2-1) 7 4 VISCOUS FLUID FLOW Finally, we note that these relations are fairly general and involve only a few restrictive assumptions: ( 1) the fluid forms a (mathematical) continuum, (2) the par- ticles are essentially in thermodynamic equilibrium, (3) the only effective body forces are due to gravity, (4) the heat conduction follows Fourier's law, and (5) there are no internal heat sources. 2-6 BOUNDARY CONDITIONS FOR VISCOUS HEAT-CONDUCTING FLOW The various types of boundary conditions have been discussed in detail in Sec. 1-4 for the three different basic boundaries: (I) a fluid-solid interface, (2) a fluid-fluid interface, and (3) an inlet or exit section. A sketch was given in Fig. 1-31. In their full generality, the conditions in Sec. 1-4 can be quite complex and non-linear, necessitating digital-computer treatment. Thus the bulk of our analytical solutions in this text will be confined to simple but realistic approximations: 1. At a fluid-solid interface, there must be no slip V fluid = V sol and either no temperature jump (when the wall temperature is known) T fluid = Tso! or equality of heat flux (when the solid heat flux is known) ( aT) k- - q an fluid from solid to fluid (2-52) (2-53) (2-54) 2. At the interface between a liquid and a gaseous atmosphere, there must be kine- matic equivalence (2-55) where YJ is the surface coordinate, and there must be equality of normal momen- tum flux P!iq :::::::; Patm (2-56) We must also have equality of tangential momentum flux and heat flux, which leads to the approximations av an liq aT ~o an liq (2-57) when the atmosphere has negligibly small transport coefficients. Also, in some problems, such as the analysis of confused, stormy seas, we need to know the moisture evaporation rate n1 at the interface. 3. At any inlet section of the flow, we need the three quantities V, p, and Tat every point on the boundary. At the exit section, we generally need to know V and T, FUNDAMENTAL EQUATIONS OF COMPRESSIBLE VISCOUS FLOW 75 but not the pressure p, as you recall from the discussion of Fig. 1-31. Exit con- ditions are difficult because of the formation of wakes and other a priori unknown outflow behavior. One approximation is to let the streamwise flow gra- dients vanish far downstream of the flow field of interest. Note: If the pressure can be eliminated, as, for example, with the vorticity/stream-function approach to be discussed in Sec. 2-11, then there is no need to impose a pressure boundary condition. The new computer-oriented vortex methods, explained in the monograph by Cottet and Koumoutsakos (1999), do not compute the pressure at all. 2-7 ORTHOGONAL COORDINATE SYSTEMS Most of the previous discussion has been illustrated by Cartesian coordinates, and only lip service has been paid to other important orthogonal systems. The basic equations of motion, Eqs. (2-6), (2-19), and (2-48) are, of course, valid for any coordinate system when written in tensor form; the problem for non-Cartesian sys- tems is to derive the correct formula for the gradient vector V plus the related expressions for divergence and curl. A straightforward procedure is to use the met- ric stretching factors hi to relate the new curvilinear coordinates to a Cartesian sys- tem, as discussed in standard mathematical references, e.g., Pipes ( 1958). Let the general curvilinear system (x 1, x 2 , x 3) be related to a Cartesian system (x, y, z) so that the element of arc length ds is given by (ds)2 = (dx) 2 + (dy) 2 + (dz)2 (2-58) which defines the factors hi. Note that hi in general will be functions of the new coordinates xi. Then, in the new system, the components of the gradient of a scalar cf> are 1 a<P 1 (kp (2-59) (2-60) and the components of the vector B = curl A are given by B, = h2~Ja~2 (h3A3) - il~3 (h2A2) l (2-61) with exactly similar expressions for B2 and By Equations (2-59) to (2-61) are suf- ficient to derive the equations of motion in the new coordinate system xi. We illus- trate with the two classic (and probably most important) systems, cylindrical and spherical.