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E ¢?Th2og TOEO g UMA INN ny 1OHM/TEin TURBULENT FLOWS AND HEAT TRANSFER BOARD OF EDITORS THEODORE VON KARMAN, Chairman Hueu L. DrypEn Hueu 8. Tayior CoLEMAN DUP. Donatpson, General Editor, 1956- Associate Editor, 1955-1956 JosepH V. Cuaryk, General Editor, 1952-1956 Associate Editor, 1949-1952 MartTIN SUMMERFIELD, General Editor, 1949-1952 RicHaRD §. SNEDEKER, Associate Editor, 1955—- . Thermodynamics and Physics of Matter. Editor: F. D. Rossini . Combustion Processes. Editors: B. Lewis, R. N. Pease, H. 8. Taylor . Fundamentals of Gas Dynamics. Editor: H. W. Emmons . Theory of Laminar Flows. Editor: F. K. Moore . Turbulent Flows and Heat Transfer. Editor: C. C. Lin . General Theory of High Speed Aerodynamics. Editor: W. R. Sears . Aerodynamic Components of Aircraft at High Speeds. Editors: A. F Donovan, H. R. Lawrence . High Speed Problems of Aircraft and Experimental Methods. Editors: A. F. Donovan, H. R. Lawrence, F. Goddard, R. R. Gilruth . Physical Measurements in Gas Dynamics and Combustion. Editors: R. W. Ladenburg, B. Lewis, R. N. Pease, H. 8. Taylor . Aerodynamics of Turbines and Compressors. Editor: W. R. Hawthorne . Design and Performance of Gas Turbine Power Plants. Editors: W. R. Haw- thorne, W. T. Olson . Jet Propulsion Engines. Editor: O. E. Lancaster VOLUME V HIGH SPEED AERODYNAMICS AND JET PROPULSION TURBULENT FLOWS AND HEAT TRANSFER EDITOR: C. C. LIN MARINE BIOLOGICAL LABORATORY LIBRARY WOODS HOLE, MASS. Wikis Ont: PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS 1959 Copyricut, 1959, By PRINCETON UNIVERSITY PRESS London: OxFoRD UNIVERSITY PRESS L. C. CARD 58-5028 Reproduction, translation, publication, use, and dis- posal by and for the United States Government and its officers, agents, and employees acting within the scope of their official duties, for Government use only, is per- mitted. At the expiration of ten years from the date of publication, all rights in material contained herein first produced under contract Nonr-03201 shall be in the public domain. PRINTED IN THE UNITED States or AMERICA BY Toe Marie Press Company, Inc., YorK, PENNA. FOREWORD On behalf of the Editorial Board, I would like to make an acknowledgement to those branches of our military establishment whose interest and whose financial sup- port were instrumental in the initiation of this publi- cation program. It is noteworthy that this assistance has included all three branches of our Services. The Department of the Air Force through the Air Research and Development Command, the Department of the Army through the Office of the Chief of Ordnance, and the Department of the Navy through the Bureau of Aeronautics, Bureau of Ships, Bureau of Ordnance, and the Office of Naval Research made significant con- tributions. In particular, the Power Branch of the Office of Naval Research has carried the burden of responsibilities of the contractual administration and processing of all manuscripts from a security stand- point. The administration, operation, and editorial functions of the program have been centered at Prince- ton University. In addition, the University has con- tributed financially to the support of the undertaking. It is appropriate that special appreciation be expressed to Princeton University for its important over-all role in this effort. The Editorial Board is confident that the present series which this support has made possible will have far-reaching beneficial effects on the further develop- ment of the aeronautical sciences. Theodore von Karman ; ren f ealitit ini pit sdieapal 4g se Hi wil ee ne tt hel \ peat. jane ry vida aver fi ae wets hy leita AP a bh eS MANS ie Tk yee Hh ars Nes he cy inyp eae at ae esd deieineas : Aish re 3 | Lise levott| RR “ee cane A? Mie Welt MEA me oie? se catia in i i) Baerrih aM ‘Paavo iti eRe eek ti Wa: mae Ly Spain ak jen ine na AG SAAN OR 4 NNO ‘fl wmindvty ; hay: etytnanes ‘ae Mistevat hs arigaclt liter Wek allt hive (A Lana Cah ee oe mea bagel A, f 12 a er ey MEO CLARET UPC Gnas CTT iS ned lee Caer Ya e te vawekt Pu Ag aime are veer. ar) Niet watiht Coad on dak’ Mae yy wn as Lice (ae DU aa aon eg Nh eee sily’s ities 04 vt morir'g PP ENC Nh hin sy COTY PRE Uh a Tee Oe wer oth nian Puig Yh ey ahd “dit Cue tN Ai iba reat at a sti NOL We Rea Reicesciesial imeeey va ha cil ogi Lean ee eM yet Pit des FUR TR aD: otal iii i bh 4 Pidee ey Wd) us yi ee PE fsa enailaitel aps hs oe a) PAPA BR Muee 2A arte. wig bi Liked sate a a i yee it 5 a, ” uy ; ie iy woe it My Prd 1 asc Sod) lett ay hime = i aii tw i Ye ' De eiy, 4 Uf rahe 7} owen o ia Ath) j I iy : Wee : ti au ie ey uth uy! } ba i PREFACE Rapid advances made during the past decade on problems associated with high speed flight have brought into ever sharper focus the need for a comprehensive and competent treatment of the fundamental aspects of the aerodynamic and propulsion problems of high speed flight, together with a survey of those aspects of the underlying basic sciences cognate to such problems. The need for a treatment of this type has been long felt in research institutions, universities, and private industry and its poten- tial reflected importance in the advanced training of nascent aeronautical scientists has also been an important motivation in this undertaking. The entire program is the cumulative work of over one hundred scientists and engineers, representing many different branches of engineer- ing and fields of science both in this country and abroad. The work consists of twelve volumes treating in sequence elements of the properties of gases, liquids, and solids; combustion processes and chemical kinetics; fundamentals of gas dynamics; viscous phenomena; turbulence; heat transfer; theoretical methods in high speed aerodynam- ics; applications to wings, bodies and complete aircraft; nonsteady aerodynamics; principles of physical measurements; experimental meth- ods in high speed aerodynamics and combustion; aerodynamic problems of turbo machines; the combination of aerodynamic and combustion principles in combustor design; and finally, problems of complete power plants. The intent has been to emphasize the fundamental aspects of jet propulsion and high speed aerodynamics, to develop the theoretical tools for attack on these problems, and to seek to highlight the directions in which research may be potentially most fruitful. Preliminary discussions, which ultimately led to the foundation of the present program, were held in 1947 and 1948 and, in large measure, by virtue of the enthusiasm, inspiration, and encouragement of Dr. Theodore von Kaérm4n and later the invaluable assistance of Dr. Hugh L. Dryden and Dean Hugh Taylor as members of the Editorial Board, these discussions ultimately saw their fruition in the formal establishment of the Aeronautics Publication Program at Princeton University in the fall of 1949. The contributing authors and, in particular, the volume editors, have sacrificed generously of their spare time under present-day emergency conditions where continuing demands on their energies have been great. The program is also indebted to the work of Dr. Martin Summerfield who guided the planning work as General Editor from 1949-1952. The coop- eration and assistance of the personnel of Princeton University Press and of the staff of this office has been noteworthy. In particular, Mr. H.S. Eva) PREFACE TO VOLUME V Bailey, Jr., the Director of the Press, and Mr. R. S. Snedeker, who has supervised the project at the Press, have been of great help. The figures were prepared by Mr. Zane Anderson. Special mention is also due Mrs. E. W. Wetterau of this office who has handled the bulk of the detailed editorial work for the program. Coleman duP. Donaldson General Editor PREFACE TO VOLUME V This volume deals with the interrelated problems of turbulent flow and heat transfer. It begins with an article on transition from laminar to turbulent flow. This is followed by a discussion of the problem of shear flow from the experimental and semi-empirical points of view and of the statistical theory of turbulence from a deductive point of view. Future developments in our knowledge should result in the merging of these two approaches into a comprehensive and unified treatment. The three modes of heat transfer—conduction, convection, and radi- ation—are presented in the remaining part of this volume. These articles are especially oriented toward high speed flows with high temperature differences. Free convection due to gravitational forces is considered in this portion of the volume only in connection with boiling heat transfer. In the sections on the physical basis of radiation and on the method of engineering calculations in radiant heat exchange, it is an interesting reflection on the current status of our knowledge in this important field that these two phases of the problem are presented from somewhat different points of view. As originally planned, this volume was to be the second part of a larger volume comprising the present Volumes IV and V, under the joint editorship of Lester Lees and myself. The desirability of separating the material into two volumes soon became clear, on account of both the size of the articles and the nature of the material involved. Professor Lees had the major share of the editorial work at the early stages. Unfortunately, he was unable to continue with his editorship after he moved from Princeton University to the California Institute of Technology, with the consequent increase of pressure from his duties. I believe I may speak for Professor Lees as well as for myself as I express appreciation for the fine cooperation of all the authors, the General Editor, and the Princeton University Press. C. C. Lin Volume Editor ( viii ) A. Transition from Laminar to Turbulent Flow Bo hw re MARINE BIOLOGICAL LABORATORY ad fo 4 CONTENTS -————— WOODS HOLE, fi W. H. 0. Hugh L. Dryden, National Aeronautics and Space Administra- tion, Washington, D.C. . Introduction . Transition on a Flat Plate in a Stream of Constant Velocity . Effect of Pressure Gradient on Transition on a Flat Plate . Effect of Curvature of Surface on Transition of a Two-Di- mensional Boundary Layer . Effect of Surface Roughness and Waviness on Transition of a Two-Dimensional Boundary Layer . Application of Dimensional Analysis to Transition of a Two- Dimensional Boundary Layer . Transition of Shear Layers in the Free Fluid . Transition of Shear Layers with Reattachment Following Laminar Separation . Breakdown of Laminar Flow vs. Transition . Tentative Conceptual Picture of Transition . Theory of the Influence of Turbulence on Transition . Schlichting’s Procedure for Computing Transition on an Airfoil . Adequacy of Transition Theories Based on Local Parameters . Transition to Turbulent Flow in a Pipe of Circular Cross Section . Transition in Pipes of Noncircular Cross Section . Transition on an Elliptic Cylinder . Transition on Airfoils . Transition on Airplane Configurations and on Airplanes in Flight . Transition on Bodies of Revolution . Transition in Flow between Rotating Cylinders . Transition in Flow near Rotating Disks . Transition in Flow at Boundary of a Jet . Transition at Subsonic Speed as Affected by Heat Transfer . General Remarks on Transition at Supersonic Speed . Effect of Mach Number on Transition for Bodies without Heat Transfer at Supersonic Speeds . Effect of Heat Transfer on Transition at Supersonic Speeds . Present Status and Future Direction . Cited References ( ix ) CONTENTS B. Turbulent Flow wo Re Galen B. Schubauer, Fluid Mechanics Section, National Bureau of Standards, Washington, D. C. C. M. Tchen, Aerodynamics Section, National Bureau of Standards, Washington, D. C. Chapter 1. Introduction . Subject Treatment . Nature of Turbulent Flow . Diffusiveness of Turbulence Chapter 2. General Hydrodynamical Equations for the Turbu- lent Motion of a Compressible Fluid . Equations of Continuity and Momentum . Equation of Kinetic Energy . Equation of Energy and Enthalpy Chapter 3. Turbulent Boundary Layer of a Compressible Fluid . Introduction . Fundamental Equations of Motion of a Compressible Bound- ary Layer . Relationships between Velocity, Pressure, and Temperature Distributions . Phenomena of Transport of Properties in a Turbulent Fluid . Reynolds Analogy between Heat Transfer and Skin Friction . Basis of Skin Friction Theories . Empirical Laws of Skin Friction . Comparison between Experiments and Theories Chapter 4. General Treatment of Incompressible Mean Flow along Walls . Power Laws . Wall Law and Velocity-Defect Law . Logarithmic Formulas . Smooth Wall Incompressible Skin Friction Laws . Effect of Pressure Gradient . Equilibrium Boundary Layers According to Clauser . Law of the Wake According to Coles . Mixing Length and Eddy Viscosity in Boundary Layer Flows . Effect of Roughness . Integral Methods for Calculating Boundary Layer Develop- ment . Three-Dimensional Effects Chapter 6. Free Turbulent Flows . Types and General Features (x) 75 75 76 79 80 83 85 87 89 90 27 104 107 113 116 119 122 124 127 129 135 139 143 147 153 156 158 27. 28. 29. 30. ol. 32. 33. 4. 30. CONTENTS Laws of Mean Spreading and Decay General Form and Structure Transport Processes in Free Turbulent Flow Velocity Distribution Formulas for Jets and Wakes Effect of Density Differences and Compressibility on Jets with Surrounding Air Stationary Effect of Axial Motion of Surrounding Air on Jets Chapter 6. Turbulent Structure of Shear Flows The Nature of the Subject References on Structure of Shear Turbulence Cited References C. Statistical Theories of Turbulence oP OD Re C. C. Lin, Department of Mathematics, Massachusetts In- stitute of Technology, Cambridge, Massachusetts Chapter 1. Basic Concepts . Introduction . The Mean Flow and the Reynolds Stresses . Frequency Distributions and Statistical Averages . Homogeneous Fields of Turbulence . Conventional Approach to the Statistical Theory of Turbu- lence Chapter 2. Mathematical Formulation of the Theory of Homogeneous Turbulence . Kinematics of Homogeneous Isotropic Turbulence. Correla- tion Theory . Dynamics of Isotropic Turbulence . The Spectral Theory of Isotropic Turbulence . Spectral Analysis in One Dimension . Spectral Analysis in Three Dimensions . General Theory of Homogeneous Anisotropic Turbulence Chapter 3. Physical Aspects of the Theory of Homogeneous Turbulence . Large Scale Structure of Turbulence . Small Scale Structure of Turbulence. Kolmogoroff’s Theory . Considerations of Similarity . The Process of Decay . The Quasi-Gaussian Approximation . Hypotheses on Energy Transfer Chapter 4. Turbulent Diffusion and Transfer . Diffusion by Continuous Movements . Analysis Involving More Than One Particle ( xi } 159 163 168 172 176 179 184 185 190 196 196 197 198 200 201 202 208 210 214 216 218 219 221 225 230 236 238 240 243 20. 21. 22. 23. 24. 25. CONTENTS Temperature Fluctuations in Homogeneous Turbulence Statistical Theory of Shear Flow Chapter 5. Other Aspects of the Problem of Turbulence Turbulent Motion in a Compressible Fluid Magneto-Hydrodynamic Turbulence Some Aerodynamic Problems Cited References D. Conduction of Heat wo Re 4. 5 6. ue 8. 9: M. Yachter, Special Projects Department, The M. W. Kellogg Company, Jersey City, New Jersey EK. Mayer, Arde Associates, Newark, New Jersey. Now with Rocketdyne, division of North American Aviation, Inc. Chapter 1. Introduction . General Remarks . Mathematical Formulation . Thermal Property Data and Range of Heat Transfer Coeffi- cients Chapter 2. One-Dimensional Heat Conduction in a Homogeneous Medium Slab of Thickness d The Semi-Infinite Solid Applications Chapter 3. Transient Radial Heat Conduction in a Homoge- neous Hollow Cylinder Classical Results for Newtonian Heat Transfer Applications. Thermal Stresses Remarks on Thermal Shock Chapter 4. Transient Heat Conduction in a Unidimensional Composite Slab . General Results for Newtonian Heat Transfer . The “Thin” Shield . “Thick”? Thermal Shields . Design Criterion for Minimum Weight . Remarks on the Composite Hollow Cylinder Chapter 5. Some Special Problems . Variable Thermal Properties . Surface Melting and Erosion . Axial Heat Conduction in Nozzle Walls . Cited References ( xii ) 244 245 247 248 249 251 254 254 255 CONTENTS E. Convective Heat Transfer and Friction in Flow of Liquids R. G. Deissler, Lewis Flight Propulsion Laboratory, National Aeronautics and Space Administration, Cleveland, Ohio R. H. Sabersky, Division of Engineering, California Institute of Technology, Pasadena, California Chapter 1. Turbulent Heat Transfer and Friction in Smooth Passages . Introduction . Basic Equations Expressions for Eddy Diffusivity . Analysis for Constant Fluid Properties . Analysis for Variable Fluid Properties . Concluding Remarks Chapter 2. Survey of Problems in Boiling Heat Transfer . Introduction . General Results . Nucleate Boiling . Film Boiling . Closing Remarks . Cited References and Bibliography F. Convective Heat Transfer in Gases Ne EK. R. van Driest, North American Aviation, Incorporated, Downey, California . Introduction . The Mechanism of Convective Heat Transfer Chapter 1. Survey of Theoretical Results Applicable to Aerodynamic Heat Transfer. Status of Experimental Knowledge LAMINAR FLOW . Flat Plate Solution . Heat Transfer . Numerical Results for Zero Pressure and Temperature Gradients along the Flow . Cone Solution . Stagnation Point Solution . Effect of Variable Free Stream Pressure and Variable Wall Temperature . Status of Experimental Knowledge exiii)) 288 288 288 290 292 303 313 313 314 319 339 334 339 339 339 339 041 348 000 362 365 368 368 10. i: 12. 13. 14. 15. 16. Wee 18. 19; 20. 21. 22. 23. CONTENTS TURBULENT FLOW Flat Plate Solution Heat Transfer Cone Solution Stagnation Point Solution Effects of Variable Free Stream Pressure, Wall Temperature, etc. Rough Walls Status of Experimental Knowledge TRANSITION Stability of the Laminar Boundary Layer and Relation to Transition Effect of Supply Tunnel Turbulence Effect of Surface Roughness Chapter 2. Application of Theory to Engineering Problems at High Speeds Aerodynamic Heating of High Speed Vehicles Heat Transfer in Rocket Motors Dissociation Effects Cited References G. Cooling by Protective Fluid Films oR oo S. W. Yuan, Department of Aeronautical Engineering, The University of Texas, Austin, Texas . Introduction Flow through Porous Metal . Physical Nature of Transpiration-Cooling Process Heat Transfer in Transpiration-Cooled Boundary Layer Heat Transfer in Transpiration-Cooled Pipe Flow . Comparison with Experimental Results on Transpiration Cooling . Film Cooling and Its Comparison with Transpiration Cooling . Cited References and Bibliography H. Physical Basis of Thermal Radiation QOnarwne S. S. Penner, Division of Engineering, California Institute of Technology, Pasadena, California . Introduction . Black Body Radiation Laws Nonblack Radiators . Basic Laws for Distributed Radiators . Theoretical Calculation of Gas Emissivities . Cited References ( xiv ) 370 372 388 388 391 391 391 396 399 403 405 415 419 425 428 428 430 434 437 460 475 481 485 489 489 489 491 492 494 500 CONTENTS I. Engineering Calculations of Radiant Heat Exchange SOC WO Hoyt C. Hottel, Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massa- chusetts . Radiating Characteristics of Surfaces . The View Factor. Direct Interchange between Surfaces . Radiation from Flames and Gases . Radiant Exchange in an Enclosure of Source-Sink and No- Flux Surfaces Surrounding a Gray Gas . Enclosure of Gray Source-Sink Surfaces Containing a Real (Nongray) Gas . Application of Principles . Cited References ( xv ) 502 502 507 511 523 531 535 539 AAD Br Mm hig hal stoi He: Ciamienas ras, eh fem cio al ‘ay ibaa ne wy ie ay Re ORY aa a ui is ; abit hy anita th Ap ae hee ae Miia an nt agi at Wess 4 ‘ty ia ibe he “A DR: analy, eon LH A Lassi waves Vip! wtih ful ADR WAtae sat ar ay itite ' neaiavulaayy i eK widiiusbors Oe i Fe 4 . tia aL | Bh + yee j heels eerie tah ti ull ne ' Mh fa Raed LR L NC)! ae Siete yr ih) 7 ee mt SA RN GL arm ne Tt | ‘nso Gites): Sulehur heey a: Ma: meee Bae tay dena haeg a ae ” eet } Ae yl i MaDe " a i OR hath fai Ji y ff 3 Ney 4 Po ' nat 4 5 } i ley’ i 4 } ee , ed yay u F } / Ne ee eh habe bey i j avo T Diy ie i TURBULENT FLOWS AND HEAT TRANSFER i i i Ay i if Pet A wy nb in j “— om at Ve \ } Pi TMD) ie 4 oT ’ TEOVareRCe YC te \ / ory a Rani \ ) 7 1 i Ra i : ons 4 ‘ Ke oa We | SECTION A TRANSITION FROM LAMINAR TO TURBULENT FLOW Ee DR YSD IN| A,l. Introduction. The motions of real fluids exhibit phenomena of a bewildering degree of diversity. The pressure distribution and the re- sultant force and moment on a body moving through a fluid are but grosser manifestations of complex motions of the fluid in the surrounding space. The body is surrounded by a flow field in which the velocity and pressure vary. There is no single theory adequate to describe this flow field in its entirety. Progress can be made only by recognizing charac- teristic phenomena in limited regions of complex flow fields that can be reproduced and studied in a simpler experimental environment. Transition is one of these characteristic phenomena of wide occur- rence. Everyone has observed it in the rising column of smoke from a cigarette lying on an ash tray in a quiet room, and at the same time has observed the types of flow which we designate as laminar and turbulent. For some distance above the cigarette the smoke rises in smooth filaments characteristic of laminar flow, only to break up into the confused swirling turbulent motion at a height which is dependent on the quietness of the surrounding air. If the drafts or convective air currents are strong, transition occurs close to the cigarette. If the air is very quiet, the fila- ment may persist to a considerable height. This breakdown of the laminar flow is transition. Considerable progress has been made in the study of transition at low subsonic speeds where the motion may be regarded as that of an incom- pressible fluid of uniform temperature. While transition has also been ob- served in localized regions of flow at transonic and supersonic speed, the information available is less extensive. The effects of heat transfer at subsonic speed are discussed in Art. 23. Art. 24, 25, and 26 deal with transition at supersonic speed. Art. 2 to 23, inclusive, deal with low speed subsonic flow. A,2. Transition on a Flat Plate in a Stream of Constant Velocity. Transition is observed to occur in the boundary layer, that region of the flow field near the surface of the body where the steepest velocity gradi- (3) A - TRANSITION FROM LAMINAR TO TURBULENT FLOW ents are concentrated through the action of viscous forces. Hence the simplest situation for the study of transition is the simplest boundary layer that can be investigated both theoretically and experimentally. This is the flow along a smooth thin flat plate, parallel to the flow in a stream of uniform velocity and hence without longitudinal pressure gradi- ent, for which the theoretical laminar flow solution was given by Blasius 120 119 135 153 40 Re 0m Fig. A,2a. Velocity distribution in the boundary layer of a plate. Contours of equal local mean speed as function of x and y Reynolds number. Turbulence of free stream, 0.5 per cent. in 1908. Many measurements have been made on the velocity distribu- tion near such a plate and transition is easily recognized by a typical departure from the Blasius distribution of the type illustrated in Fig. A,2a. Here for measurements in air [/] contour lines of equal values of u/U are plotted on abscissas and ordinates of Re, and Rez, u being the local velocity at the point where distances from the leading edge and the surface of the plate are x and y, U the free stream speed, and Re, and Re, the Reynolds number formed from z and y, respectively. Transition accelerates the fluid close to the plate. Geass A,2- TRANSITION ON FLAT PLATE For the particular measurements illustrated in Fig. A,2a transition occurred at Re, = 1.1 X 10°. However, other measurements have yielded values varying from 9 X 104 to 2.8 X 10%. The principal variable con- trolling transition is the turbulence initially present in the air stream (V,B and IX,F). For present purposes we will use as the quantitative measure of the turbulence the quantity 100 ~/2(u”? + vo”? + w’?) U where wu’, v’, and w’ are the root-mean-square values of the three com- ponents of the turbulent velocity fluctuations. Fig. A,2b shows a plot of Schubauer and Skramstad Dryden Hall and Hislop Bailey and Wright (M.I.T.) 0 1 2 3 100 [2 (u’? + vy’2 + w’?) U2 Fig. A,2b. Effect of intensity of free stream turbulence on the transition Reynolds number of a plate. Re, against the intensity of turbulence for the available experimental data. While the scale of the turbulence is also known to be of importance, there is insufficient evidence for flat plates to demonstrate its effect. (See Art. 11.) The magnitude of the effect of small fluctuations of velocity amount- ing to less than one per cent of the average stream velocity is quite surprising. When the free stream turbulence is less than about 0.1 per cent, Schubauer and Skramstad [2] found that transition was preceded by the appearance of the Tollmien-Schlichting oscillations predicted by the theory of instability of laminar boundary layers (IV,F). These oscillations grew in amplitude until transition ensued. The oscillations are not ob- served at higher turbulence, the random turbulent fluctuations apparently imposing random fluctuations within the boundary layer which mask the Tollmien-Schlichting oscillations. (6) A+ TRANSITION FROM LAMINAR TO TURBULENT FLOW The exact mechanism by means of which the growing oscillations pro- duce transition is not yet completely understood. It appears, however, that the laminar instability is not a factor except when all sources of disturbance are made very small. When the transition region is investigated with a hot wire anemometer capable of indicating the instantaneous velocity, it is found that, whereas, from measurements of mean speed, transition appears to be a gradual process, it is in fact quite sudden [/]. As the test probe is moved down- stream, turbulent “‘bursts’”’ appear, at first infrequently, then more fre- quently and of longer duration, until finally the flow is continually and completely turbulent. These observations were interpreted as a wander- ing of transition back and forth about a mean position. In 1951 Emmons [3] suggested as a result of observations of the gravity flow of a thin sheet of water over an inclined flat plate that turbulence appears in more or less random fashion at localized spots which grow in size as they move downstream. Mitchner [4] developed an experimental method of generating local turbulent spots in air artificially by passing an electrical spark through the boundary layer. Recently Schubauer and Klebanoff [5] made extensive studies of the mechanism of transition from amplifying Tollmien-Schlichting waves. They showed that turbulence did in fact originate as localized spots in natural transition. They studied the growth of artificial spots by Mitchner’s technique. Since publication of the cited reference, Schubauer and Klebanoff have shown that the Tollmien-Schlichting waves exhibit variations in ampli- tude along a direction parallel to the leading edge of the plate, and that turbulent spots appear in the regions of maximum amplitude of the wave. A,3. Effect of Pressure Gradient on Transition on a Flat Plate. The variation of pressure along the outer edge of the boundary layer has a marked effect on the location of transition. Many years ago experi- ments were made by Wright and Bailey [6] on the effect of pressure gradient on transition in a tunnel in which the turbulence was about 0.2 per cent. Relatively small gradients produced large effects, changing Re, from 2 X 10° for zero pressure gradient to 0.7 X 108 and 2.5 X 108 for small positive and negative gradients. Schubauer and Skramstad [2] gave the striking demonstration shown in Fig. A,3 of the stabilizing effect of a pressure which decreases in the downstream direction and the de- stabilizing effect of a pressure which increases in the downstream direc- tion. In those cases where transition results from the instability of laminar flow at low turbulence levels, transition is hastened by a positive (adverse) pressure gradient and delayed by a negative (favorable) one. The behavior of the Tollmien-Schlichting oscillations in a pressure gradient has been computed [7] and the above-mentioned observations are in qualitative ( 6 ) A,3 - PRESSURE GRADIENT ON FLAT PLATE agreement with the theory. The computed effect is very large. A suitable nondimensional measure of the pressure gradient is \ = (62/v)(du./dz) where du,/dz is the velocity gradient and 6 is the boundary layer thick- ness defined by the Pohlhausen four-term approximation to the velocity distribution in the boundary layer. For —3 < < 3, stability theory yields a critical Reynolds number Re; based on displacement thickness oO D> +9 ; se oO S 5 : € c ae = ) a © 6 Fig. A,3. Effect of pressure gradient on laminar boundary layer oscillations. Oscil- lograms of u’ at y = 0.021 in., U = 95 ft/sec. Time interval between dots, 34 sec. (see Art. 6 and IV,C) approximately proportional to e°®. The actual values are 160, 575, and 4000 for \ equal to —3, 0, and 3, respectively, corresponding to values of Re, for zero pressure gradient of 8500, 110,000, and 5,300,000. The critical Reynolds number for transition is, however, considerably greater than that at which amplification of disturbances begins and by an unknown ratio which may vary with 2X. Liepmann made some measurements of the influence of pressure gradi- ent on transition [8] on the convex surface of a plate of 20-foot radius of ae) A: TRANSITION FROM LAMINAR TO TURBULENT FLOW curvature obtaining changes in transition Reynolds number from 2.2 X 10° to 0.9 and 3.2 X 10° for \ = 0, —5.7, and +1.9, respectively. Feindt [9] studied the influence of pressure gradient on smooth and roughened hollow cylinders with axes parallel to the air stream at a stream turbulence of approximately 1.2 per cent, the turbulence level being inferred from sphere measurements. (From the observed value of Re, for the smooth cylinder at zero pressure gradient, the turbulence level derived from Fig. A,2b is 1.0 per cent.) For pressure gradient parameter \ = 0, —4.4, and 3.7, the observed transition Reynolds numbers for the smooth cylinder were 0.66 X 10°, 0.36 X 10°, and 0.80 X 10°, respec- tively. The effects of pressure gradient on roughened cylinders were also large. Thus the influence of pressure gradient has been observed to be large and qualitatively the same over a wide range of roughness and free stream turbulence. A,4. Effect of Curvature of Surface on Transition of a Two- Dimensional Boundary Layer. Liepmann made a systematic study [10] of the effect of a uniform radius of curvature of the surface on the transition of a two-dimensional laminar boundary layer. On convex sur- faces up to values of displacement thickness 6* equal to 0.0026 times the radius of curvature 7, the same Tollmien-Schlichting instability occurs as for the flat plate and the effect of curvature is negligible. The effect of turbulence is large as in the case of the flat plate. On a concave surface, the behavior is the same as for a flat plate, provided the ratio 6*/r is less than 0.00013. If 6*/r exceeds 0.0013, the laminar flow is dynamically unstable because of centrifugal forces pro- ducing three-dimensional disturbances as studied theoretically by Gértler [11]. Gértler used as a measure of the stability boundary the parameter Rey ~/6/r, based on the momentum thickness @ which is approximately equal to 0.3866*. Liepmann found the Gortler parameter equal to 9.0 in an air stream of the lowest turbulence available to him (turbulence in- tensity 0.2 per cent as judged from his flat plate measurements, u’/u, = 0.0006, v’ and w’ not measured), whereas at a higher turbulence level (u’/u. = 0.003, v’/u, and w’/u, not measured) the value was about 6.0. For values of 6*/r between 0.00013 and 0.0013 there appears to be a more or less continuous change from the Tollmien-Schlichting instability to the Gortler instability. A,5. Effect of Surface Roughness and Waviness on Transition of a Two-Dimensional Boundary Layer. Surface roughness and wavi- ness are known to influence transition presumably because of the disturb- ances introduced by their presence. The general nature of the effect of a single roughness element has been studied in detail by Liepmann [7/2], the roughness element being a half cylinder with axis normal to the stream (8) A,5 - SURFACE ROUGHNESS AND WAVINESS in the boundary layer of a flat plate. In every case tried, the flow separated from the surface of the roughness element but did not in every case lead to immediate transition. The results of many experiments suggest that a single roughness element must be comparable in height with the displace- ment thickness in order to produce transition. In Liepmann’s experiments an element for which k = 0.766* did not produce immediate transition, whereas elements for which k > 0.926* did produce immediate transition. Certain experiments of Fage [13], when analyzed, show that the minimum height to cause transition on a plate at certain definite locations down- stream from the roughness element varies from 0.82 to 1.776* for smooth ‘O2 0.4 0.6 0.8 k/d; Fig. A,5a. Effect of single cylindrical roughness element on the transition Reynolds number of a plate. Original measurements of Tani, Hama, and Mituisi [15]. bulges, 1.44 to 1.626* for a smooth hollow, and 0.43 to 0.676* for a flat ridge. The higher values are required to cause transition to approach the roughness element; the values to cause transition at the element as deter- mined by extrapolation are 1.86*, 1.65* and 0.986*, respectively. Holstein [74] has also found that roughness heights comparable to the displacement thickness are required to move transition appreciably forward. In Fage’s experiments it was found that a roughness element can influence the position of transition far downstream, presumably because it imposes a disturbance akin to turbulent fluctuations which slowly grow until transition occurs. This effect was not observed in Liepmann’s experi- ments since observations were made only to a downstream distance of 6 inches. It is probable that, in the case in which Liepmann observed no Oo? A - TRANSITION FROM LAMINAR TO TURBULENT FLOW transition, transition did in fact occur earlier when the roughness element was present than when it was absent. Systematic experiments by Tani, Hama, and Mituisi [/4] have clarified the effect of a two-dimensional roughness element. Their experiments were made on a flat plate with zero pressure gradient in a wind tunnel in which Re, was about 1.7 X 10° with cylindrical wires as roughness ele- ments. In the original paper the data were analyzed according to the con- cept suggested by Schiller [/6, pp. 189-192] that a roughness element 300 260 Fig. A,5b. Relation between Reynolds number of roughness element and transition Reynolds number of a plate. Original measurements of Tani, Hama, and Mituisi [76]. induces transition when the Reynolds number of the element itself reaches a definite critical value at which vortices appear in its wake. Dryden [7/7] reanalyzed the data and showed much better agreement with the concept that Re, is a function of k/5* where k is the height of the roughness ele- ment and 6% is the displacement thickness of the boundary layer at the location of the roughness element. Fig. A,5a shows Re, as a function of k/6* whereas Fig. A,5b shows the relationship between Re, and Rex. Tani and Hama presented additional results in a later paper [18]. They are plotted in Fig. A,5c. As the roughness height is increased, the position of transition moves forward from the smooth plate position until it reaches the position of the roughness element. Re, is a function of k/éf ( 10 ) A,5 + SURFACE ROUGHNESS AND WAVINESS so long as transition is forward of the smooth plate position but down- stream from and not too close to the roughness element. The dotted up- ward branches in Fig. A,5c correspond to transition at the roughness element. The location of these branches is determined by the parameter 2./k, where x; is the position of the roughness element. The results of Fage [/3] analyzed in this way are shown in Fig. A,5d; 7 typical results of Stiiper [79], discussed at some length in [17], are shown © Tani [15] Tani [18] Pic xk.Gm AVXOUO0O>5-in. at 1 ft eins at 2 ft Peinvat 2 ht in. at 1 ft Spacing effect 2 diameters 8 diameters 0.8 0.4 2-dimensional roughness 6 (cylindrical rods) 0 0.4 0.8 2.0 2.4 Fig. A,5i. Effect of spherical roughness elements on the transition Reynolds num- ber of a plate. Measurements of Klebanoff, Schubauer, and Tidstrom [23]. As contrasted with this behavior of two-dimensional roughness ele- ments, the data on three-dimensional elements correlate best with the assumption of a critical Reynolds number of the roughness element. Ex- periments have been made by Klebanoff, Schubauer, and Tidstrom [23] on single rows of spherical elements of various spacings at various dis- tances from the leading edge of a flat plate. The results are shown in Fig. A,5i and A,5j, showing Re, vs. k/6* and Re, vs. uzk/v respectively, where wu; is the velocity in the boundary layer at a distance k from the wall. There is considerable scatter, but from the results for the smallest spheres the surface may be considered aerodynamically smooth, i.e. the CU) A - TRANSITION FROM LAMINAR TO TURBULENT FLOW roughness element has little effect on transition, if u.k/v is less than 300. If the free stream velocity U were used, the corresponding value of Uk/» would be about 800. The numerical values obviously depend on the shape of the elements, although systematic data are not available. There are only a few experiments on distributed roughness on a flat plate, and these are difficult of analysis because of the necessary experi- mental treatment of the leading edge. Thus Holstein [74], using a “ plate”’ consisting of a structure of wood about 11 feet long and 1.4 inches thick with an elliptical nose piece about 7.09 inches long, left the nose piece 3-dimensional roughness (spheres) Saye 3g cin. at 2 ft = -in. at 1 ft a -in. at 2 ft 4 -in. at 2 ft ¥-in. at 1 ft Rep <1 Ore Q 100 300 500 700 900 uk/v Fig. A,5j.. Relation between Reynolds number of spherical roughness elements and transition Reynolds number of a plate [23]. smooth and roughened the remainder of the surface by attaching emery paper of various degrees of roughness. The value of Re, for the smooth plate varied with the speed, perhaps because of variation of the turbu- lence with speed. Furthermore, the plate was not aerodynamically smooth since fine emery paper often increased the transition Reynolds number. Holstein believed that the paper covered up the surface waviness. The average value of Re, was about 650,000 corresponding to a turbulence of 1.0 per cent, or more probably to departures from aerodynamic smoothness. The emery paper had a thickness of about 0.8 mm and the sharp step at the leading edge of the paper was smoothed with plasticine over a ( 16 ) A,5 - SURFACE ROUGHNESS AND WAVINESS length of about 1 cm, according to a letter from Holstein. The roughness height given did not include the paper thickness. It seems probable that the front edge of the paper is the critical roughness rather than the dis- tributed roughness. Fig. A,5k shows Re, plotted against k/éy where k is k/8% Fig. A,5k. Effect of distributed roughness on the transition Reynolds number of a plate. Measurements of Holstein as reported. Fig. A,51. Effect of distributed roughness on the transition Reynolds number of a plate. Measurements of Holstein with suggested correction. the nominal roughness height and 6% is the displacement thickness at the beginning of the roughened portion. Fig. A,51 shows a plot of the same results on the assumption that the roughness height is the tabulated value plus 0.2 mm. If Holstein’s results are analyzed in terms of the Reynolds number ( 17 ) A: TRANSITION FROM LAMINAR TO TURBULENT FLOW of the roughness element based on free stream velocity and nominal roughness height, it appears that Uk/»v must be less than about 100 + 50 if the surface is to be considered aerodynamically smooth. Feindt [9] studied the effect of distributed roughness on a hollow cylinder, using sand paper. The sand paper cylinder was extended 12 mm beyond the sharpened leading edge of the supporting cylinder. The sand paper was smoothed with shellac for 3 mm from the leading edge, and the leading edge of the sand paper sharpened. Roughness for which Uk/» was less than 60 to 100 based on nominal sand grain size had little effect on transition, the exact value depending somewhat on the pressure gradient, the lower value applying to an adverse pressure gradient. This value is approximately the same as that found in Holstein’s measurements, al- though Feindt’s measurements show less scatter. The decrease in Re, with increasing Uk/»v is very rapid. For the case of zero pressure gradient an increase of Uk/v from 100 to 150 reduces Re, to one half its initial value. Feindt proposed the use of the Géttingen equivalent sand grain rough- ness, defined as that value of roughness in certain Géttingen measure- ments of friction coefficients in roughened tubes which gave the same friction coefficient as his samples. Based on this Gottingen equivalent roughness height, the limiting value of Uk/y is from 100 to 170. The applicability of these limiting values of Uk/v to smoother sur- faces for which k lies outside the range of the experimental values or to higher Reynolds numbers is somewhat uncertain. Since transition is be- lieved to arise as a result of eddy production by the roughness elements in the case of three-dimensional distributed roughness, 1t appears more logical to base any extrapolation on u;k/v where u; is the velocity in the boundary layer at the transition position at the height of the roughness element. A value of u,k/v of 30 to 50 was found for sand grain roughness in air streams of relatively high turbulence. For distributed small spheres in a flow of very low turbulence without pressure gradient, the observed value was about 300. Since critical Reynolds numbers are usually depend- ent on air stream turbulence as well as on shape, the difference between 30 to 50 and 300 may be due in large part to the different air stream turbulence levels (see Art. 17). In summary, single roughness elements should be limited to heights less than 0.2 times the displacement thickness of the boundary layer at the element under the prevailing conditions of speed, viscosity, and element location to avoid reduction of transition Reynolds number. Distributed roughness should be limited to heights for which the Reynolds number based on nominal roughness height and the velocity in the boundary layer at the transition position at the height of the element is less than a value of the order of 30 to 300. The former value applies to sand roughness in a stream of turbulence of 1.0 per cent; the latter to small spheres in a stream celery) A,6 - APPLICATION OF DIMENSIONAL ANALYSIS of turbulence of less than 0.1 per cent. Favorable pressure gradients in- crease the permissible roughness height considerably. A,6. Application of Dimensional Analysis to Transition of a Two-Dimensional Boundary Layer. So far we have considered the study of transition on a flat plate, first without pressure gradient, then with simple linear variation of the pressure, with uniform curvature, and with simple roughness, each variable considered singly. However, in the cases of technical interest, all of the variables may vary simultaneously along the surface of the body generating the boundary layer. Further attempts have been made to apply the knowledge gained in the simpler cases by means of dimensional analysis. For a series of geometrically similar two-dimensional bodies the po- sition at which transition occurs depends on the Reynolds number at which the measurement is made and on the intensity and scale of the turbulence of the main stream. Dimensional reasoning gives the result that where x, is the coordinate locating the transition point with relation to a selected system of axes, c is the reference dimension, for example, the chord of an airfoil, U is the free stream velocity, v the kinematic viscosity of the fluid, wu’ the intensity, and L the scale of the turbulence. Since, as previously noted, transition occurs within the boundary layer, one would like to relate its occurrence more directly to boundary layer parameters rather than to the shape of the body. Boundary layers on different two-dimensional bodies differ only in the variation of pres- sure to which they are exposed by the flow around the body, and in the curvature and roughness of the surface on which they are formed. Meas- uring x along the surface from the stagnation point, x, is dependent on the three functions of x describing the variation of the stream velocity just outside the boundary layer with x (which fixes the pressure vari- ation), the variation of the curvature of the boundary with xz, and the variation of the roughness with x, as well as on the free stream turbulence. Since the number of possible functions is indefinitely large, no simplifi- cation results from this approach. It was for the reason of gaining some insight into the problem that we considered in Art. 2 to 5 the simplest types of functional variation, namely constant values of the parameters independent of x, or linear variations. Since transition occurs suddenly, another approach is to relate tran- sition to the local boundary layer parameters, eliminating 2 as a variable. The local situation is usually described by the boundary layer thickness 6, the velocity wu. at the outer edge and its derivative du./dx which is a ( 19 ) A +: TRANSITION FROM LAMINAR TO TURBULENT FLOW measure of the pressure gradient, and the kinematic viscosity v. To these must be added the free stream turbulence, and the radius of curvature r and roughness k of the surface at the selected location. There is no obvi- ous reason for omitting higher derivatives of u, or the complete distribu- tion of velocity within the boundary layer. If, however, the quantities listed are sufficient, dimensional reasoning yields the result that a ra ne y dx r 6 6 U, In line with this approach to the problem, it has become customary to state the location of transition in terms of the local boundary layer AO Ory oUt oie oo) U2 Fig. A,6. Effect of intensity of free stream turbulence on the transition Reynolds num- ber of a plate expressed in terms of displacement thickness of the boundary layer. Reynolds number rather than in terms of x, in the hope that values for bodies of different shape would be more nearly comparable. There is con- siderable diversity of practice in selection of the particular boundary layer thickness to be used. Since the velocity in the boundary layer approaches asymptotically that outside, there is no accurately determinable value of the actual thickness. The displacement thickness 6* is frequently used and will be used hereafter. NACA writers have used the value of y for which w/u, = 0.707 as the thickness, because of the ease with which it can be read from experimental curves and because it is approximately proportional to 6*. Thus for the Blasius exact solution, the NACA value ( 20 ) A,7 - SHEAR LAYERS IN THE FREE FLUID of 6 is equal to 1.3846* and for the Pohlhausen 4-term approximations to flows with pressure gradient the NACA value of 6 is 1.28 and 1.326* for values of \ = 12 and —12. The value of Re,» for transition is often con- verted to an equivalent flat plate value of Re, by the approximate relation Re = +Re?s. The values of Res+ at transition on a smooth flat plate with zero pressure gradient vary from 515 to 2900 corresponding to the values of Re, of 9 X 104 to 2.8 X 108 previously quoted. A plot of Res» against intensity of turbulence is shown in Fig. A,6. A,7. Transition of Shear Layers in the Free Fluid. Another of the frequently observed characteristic phenomena in addition to tran- sition is that of separation of the flow from a solid boundary and we shall see that there is an interplay between the two phenomena. Separation of the flow is accompanied by a reversal in the direction of the flow very close to the boundary behind the separation line and by the formation of a wake in which the velocity is much reduced. Separation is a boundary layer phenomenon; it occurs when the pressure increases in the downstream direction, or in our previous terminology, when the boundary layer en- counters an adverse pressure gradient of sufficient magnitude. When the pressure rises, the flow in the boundary layer is retarded by the pressure as well as by the surface friction and the fluid near the surface is ulti- mately brought to rest. When separation occurs, the boundary layer be- comes a shear layer in the free fluid, sometimes called a vortex layer. Such shear regions in which the velocity gradient is much larger than elsewhere are found where discontinuities of velocity, or of the physical properties of the fluid are introduced, as for example, in the case of a jet of fluid issuing in a surrounding quiescent fluid. The flow in such a shear layer may be laminar or turbulent and tran- sition is observed to occur in shear layers as well as in boundary layers. An early rule of thumb was that if the Reynolds number formed from the velocity at the outer edge of a boundary layer and its thickness were less than about 2000, transition would not occur. However, shear layers are much more unstable (IV,F) and transition has been observed at Reynolds numbers less than 100. Thus a laminar boundary layer often exhibits transition immediately following separation. If the Reynolds number is sufficiently low a laminar boundary layer may separate and continue as a laminar shear layer for a considerable distance. Transition in such a laminar shear layer has been studied in some detail by Schiller and Linke [24,25]. The shear layer was found in the flow field around a circular cylinder at Reynolds numbers (based on cylinder diameter) of 2000 to 20,000. Pitot tube traverses of the wake showed the existence of a shear layer proceeding from the separating laminar boundary layer. Its thickness increased relatively slowly and then much more rapidly at a line which approached the line of separation as ( 21 ) A - TRANSITION FROM LAMINAR TO TURBULENT FLOW the Reynolds number was increased. This behavior is typical of tran- sition. Fig. A,7 shows the results obtained at three Reynolds numbers. The boundaries of the shear layer were determined as the point at which the Pitot pressure ‘‘ begins to decrease” and the point at which the Pitot pressure ‘‘reaches a constant minimum value.” The critical Reynolds number for transition based on this thickness varied systematically with the Reynolds number of the cylinder, increasing from 510 at a cylinder Reynolds number of 3540 to 900 at a cylinder Reynolds number of 8540. A most important observation was that a disturbance introduced by a wire of small diameter in the boundary layer of the cylinder moved the transition closer to the cylinder. Hence the transition of a laminar shear layer is a function of the initial turbulence as is transition in a boundary layer. Re = 14,480 x = distance from cylinder Fig. A,7. Free shear layers behind a circular cylinder exhibiting transition. The shear layer behind the cylinder is subjected to pressure gradients. However, no accurate static pressure measurements could be made in the wake. The total pressure along the plane of symmetry of the wake falls to a minimum at the point where the shear layers from the two sides meet. The shear layers are not very thin and probably the static pressure is not constant across them. However, the rising value of the critical Reynolds number of the shear layer and the more rapid fall of the total pressure along the plane of symmetry of the wake suggests that transition in a shear layer may be delayed by a favorable pressure gradient as in the case of a boundary layer. Schiller and Linke do not give their data in sufficient detail to permit an accurate calculation of the displacement thickness which in this case might be defined by the relation (u.e — Uo) 6* = / (w — wo)dy where wo is the velocity at the inner boundary. As a rough guess 6* is of the order of 0.3 the thickness defined by Schiller and Linke, and hence Res+ is of the order of 150 to 270 at the turbulence of Schiller and Linke’s ( 22 ) Ay]: SHEAR LAVERS IN THE FREE FEUID stream which was probably quite high. This compares with an estimated value of the order of 515 for a boundary layer under similar conditions (Fig. A,6). At cylinder Reynolds numbers from about 20,000 to about 200,000 the laminar boundary layer becomes turbulent immediately on separation. Since the value of Re;« at any fixed point on the cylinder varies as the square root of the cylinder Reynolds number, the critical Res+ for the boundary layer is of the order of +/10 or 3.2 times that for a shear layer. This is an independent experimental determination of the relative values of Re;« for transition. In similar layers with zero pressure gradient and low turbulence, the distance from stagnation point to transition would be ten times as great for a boundary layer as for a shear layer. It is well known that a surface of discontinuity in an incompressible frictionless fluid is unstable in the sense that any small disturbance in- creases exponentially in amplitude with time as originally discussed by Helmholtz and by Rayleigh. Rosenhead [26] attempted to follow the motion to a later stage and showed that the disturbance becomes un- symmetrical and tends to concentrate the vorticity at points spaced at equal intervals. Because of the nonlinear character of the equations, the final stages cannot be computed by superposition of the effects of separate wavelengths. Rosenhead believed that the determination of the wave- length which ultimately dominates cannot be determined except by con- sidering the effects of viscosity and diffusion. The rolling-up process was well advanced in the time required for the fluid to travel a distance equal to one third of the wavelength. The effect of viscosity is not only to convert a discontinuity into a shear layer of finite thickness but also to provide a damping effect. Lessen [27] attempted to compute the stability of a shear layer using the Tollmien-Schlichting theory. His computations are not complete but they indicate a critical Re;+ for the beginning of amplification of about 15 and a predominant wavelength of approximately 35 times 6*. Shear layers produced by separation of the flow from a flat plate normal to the wind have been studied experimentally by Fage and Johansen [28]. These were, however, turbulent from their origin at the edge of the plate, the value of Res« being of the order of 600 or more. Flachsbart [29] shows some smoke flow pictures of the transition of free vortex layers behind a flat plate. Reynolds numbers based on the observed width of the smoke trail at transition vary from 40 to 100, but it is unlikely that the width of the smoke filament has anything to do with the usual measures of shear layer width. The values of Re,_, based on distance from separation point to transition are better defined and from them we infer values of Re;+ of the order of 95 to 100. These values are somewhat less than those found by Schiller and Linke for a cylinder, perhaps due to different turbulence of the air streams. ( 23 ) A+ TRANSITION FROM LAMINAR TO TURBULENT FLOW A,8. Transition of Shear Layers with Reattachment Following Laminar Separation. It has been noted that the laminar boundary layer of a cylinder appears to undergo transition immediately following separation when the cylinder Reynolds number is greater than 20,000. When the curvature of the body is not large, it is observed that the turbulent shear layer may reattach to the surface, leaving a localized region of separation usually referred to as a separation “bubble.”’ The attached layer continues as a turbulent boundary layer along the surface to the trailing edge or until separation of the turbulent boundary layer pase i a namaearess AN a aw AAA ht bereeee aes \Netaae = ee of separation i eee O6o) Waren ois Le 0.68 er 0.72 (0.74)8 Votetiitee x/c Fig. A,8a. Velocity distribution in the boundary layer of an airfoil exhibiting a separation bubble. Airfoil NACA 663-018, Re. = 1.7 X 108. Contours of equal local mean speed. occurs. This reattachment is due to the much greater lateral diffusion of momentum in a turbulent as compared to a laminar layer. Separation bubbles on airfoils were noted by Jones [30] some years ago and there have been a number of studies of the detailed structure of separation bubbles. In addition, various attempts have been made to develop semiempirical theories. Fig. A,8a shows the velocity distribution in the localized region of laminar separation behind the position of minimum pressure on an NACA 66;-018 airfoil section at zero angle of attack taken from measurements by Bursnall and Loftin [37] in the NACA Langley low turbulence wind tunnel. The corresponding pressure distribution on the airfoil is given in ( 24 ) A,8 - SHEAR LAYERS WITH REATTACHMENT Fig. A,8b. Further studies on the same airfoil and on a modified NACA 0010 airfoil were made by Gault [32]. Similar separation bubbles have been observed in the vicinity of the leading edge of airfoils at relatively high angles of attack [33], and on an elliptic cylinder [34]. Attempts to correlate the observations into a unified scheme have been unsuccessful, the data for bubbles near the leading edge exhibiting different relationships than the data for bubbles near or behind the mid- chord position. Jacobs and von Doenhoff [35, p. 311] suggested that transition oc- curred when the Reynolds number Re,_, formed from the local free stream speed and the distance along the shear layer from the separation point attained the critical value of 50,000, according to their fragmentary 0 0.2 0.4 0.6 0.8 1.0 Chordwise station x/c Fig. A,8b. Pressure distribution over airfoil with separation bubble. measurements. Gault [32,33] obtained values for the shear layer in sepa- ration bubbles near the leading edge of 13,000 to 103,000 for the NACA 663-018, 0 to 171,000 for the modified NACA 0010, and 20,000 to 60,000 for the NACA 63-009 airfoil. Schiller and Linke’s values [24,25] of Re,_, for the shear layer from a circular cylinder vary from 4000 to 5000, and Re,_, was reduced to about 2000 by a turbulence-producing wire. Schu- bauer’s value [34] for the shear layer in a separation bubble on an elliptic cylinder is about 28,000 in an air stream of turbulence 0.85 per cent. Maekawa and Atsumi [36] obtained a value of 25,000 for the shear layer in a separation bubble from the ridge of a model consisting of two flat plates making an angle of nearly 180° at the ridge. These authors infer that this value is independent of air stream turbulence from indirect evidence based on observations of the point of reattachment and a theory which is not too well established. Bursnall and Loftin’s data [3/] give ( 25 ) A + TRANSITION FROM LAMINAR TO TURBULENT FLOW values of Re,_, from 150,000 to 260,000 for the shear layer in a separation bubble at 61 per cent of the chord on an NACA 663-018 airfoil, whereas Gault’s values [32] for the same airfoil vary from 160,000 to 380,000. Thus the values of Re,_, vary from 0 to 380,000, an even wider range than observed for the boundary layer on a plate without pressure gradient. One might assume that the large variation is a reflection of the influ- ences of free stream turbulence and pressure gradient on transition of the free shear layer entirely analogous to those demonstrated to exist for boundary layers. However; the available data do not establish this assumption conclusively. The difficulties may be illustrated by Bursnall and Loftin’s measurements. Their values of Re,, vary with the Reynolds number of the airfoil, yet the pressure distribution is substantially inde- pendent of the airfoil Reynolds number. The values of Re,_, are 148,000, 197,000, and 256,000 for airfoil Reynolds numbers of 1.2, 1.7, and 2.4 X 108, respectively. There is no reason to assume that the turbu- lence is greatest at the lowest Reynolds number as would seem to be required if turbulence is the controlling element. It appears plausible that the thickness of the boundary layer at the separation point should have some influence on transition, or that a Reynolds number based on the thickness at transition might be more suitable than one based on length of the free shear layer. However, no satisfactory correlation has been established on either basis. Thus in the experiments of Bursnall and Loftin, the values of Res+ at transition are 2070, 2560, and 2940 for airfoil Reynolds number of 1.2, 1.7, and 2.4 X 10°. Bursnall and Loftin attempted to correlate the results available to them by plotting the ratio of x, — x, to 6, against the value of Re; at separation. (In their paper 6 is the value of y at which u/u, = 0.707.) The available data fall approximately into two groups, one for regions of separation near the leading edge at high angles of attack, the other for regions of separation near the midchord position at 0° angle of attack. Hence this attempt does not yield an integrated picture. Gault [32] attempted further analysis of his measurements and those of Bursnall and Loftin, but the results are not satisfactory. Some work on the effect of turbulence was carried out by Gault on the NACA 66;-018 airfoil. Increased turbulence moved transition up- stream for all conditions and completely eliminated the separation bubble near midchord for all Reynolds numbers from 1.5 to 10 million. Separation bubbles near the leading edge were not eliminated but their size was greatly reduced. The effect of increasing the turbulence from 0.2 to 1.1 per cent at a Reynolds number of 2 million was approximately the same as increasing the Reynolds number from 2 to 4 million at a turbulence level of 0.2 per cent. Whereas the values of Res for a completely free layer are from 100 to 300 as noted in the last article, the values for the separated layers in ( 26 ) A,I - BREAKDOWN OF LAMINAR FLOW VS. TRANSITION bubbles vary from 600 to 2000 for the bubbles near the leading edge and are of the order of 2000 to 3000 for those near midchord. It is obvious that there is room for much additional research, but the turbulence of the air stream and the pressure gradient must be varied and measured if progress is to be made in understanding the phenomena. A,9. Breakdown of Laminar Flow vs. Transition. Transition is often regarded as synonomous with the breakdown of laminar flow but wider experimental experience shows that finer distinctions must be made. Breakdown of laminar flow may be followed by a flow varying periodically with time, exhibiting regular vortex patterns. Such flows can be described without the introduction of the random element characteristic of that type of flow for which the name ‘‘turbulent”’ should be reserved. The more familiar examples of flows of this type for which theoretical treat- ments are available are (1) the Karman vortex street behind a circular cylinder; (2) the Taylor three-dimensional vortex cells between two con- centric rotating cylinders; (3) the Gértler vortices near a concave surface; and (4) the vortex cell pattern in a thin layer of fluid heated from below. These periodic patterns are well defined and mainly laminar in their motion only at Reynolds numbers not too far above that for flow break- down. In the case of the cylinder, for example, the beautiful pictures of Karman vortices can be obtained only at Reynolds numbers based on a cylinder diameter of a few thousand. Turbulent flow is characterized by the presence of irregular and ran- dom velocity fluctuations of relatively high frequency, but the experi- mental detection and measurement of the velocity fluctuations require equipment not widely available because of its complexity and cost. Tur- bulent flow is most readily detected by the very high rate of diffusion of momentum, heat, vapor, and material particles as compared with the molecular diffusion present in laminar flow. Some of the very large num- ber of techniques of determining the occurrence of transition based on diffusion phenomena are described in IX,F. A familiar method used by Osborne Reynolds depends on the diffusion of dye particles in water, or of smoke particles in air. In laminar flow, filaments maintain their identity over great distances whereas in turbulent flow the dye or smoke is dif- fused laterally very rapidly, destroying the filament. In many flows, as for example that behind a circular cylinder at Reynolds numbers from a few thousand to a few hundred thousand, the flow is of mixed character. The flow in the wake shows a periodic character with definite frequency but the rapid diffusion of smoke in the wake shows that the flow there is turbulent. The laminar boundary layer is shed periodically and alternately from the two sides but immediately be- comes turbulent on separation. The vorticity in the layers which roll up into Kérmén vortices at lower Reynolds numbers is now rapidly diffused (27) A: TRANSITION FROM LAMINAR TO TURBULENT FLOW to give a typical error-law distribution of vorticity across the wake a short distance downstream from the cylinder [37]. Accordingly, transition should not be defined as breakdown of laminar flow but as the onset of the highly diffuse turbulent motion of random character. A,10. Tentative Conceptual Picture of Transition. The account that has been given of the several typical flow situations in which tran- sition is recognizable and of the influence of many of the controlling variables suggests that we are dealing with an effect which may have many causes. Each variable, initial turbulence, pressure gradient, rough- ness, in the absence of the influence of the other variables, fixes transition at a definite location for a given Reynolds number of the body. That variable which gives the most forward location of transition is the one which will be the controlling one under the given set of circumstances. Thus any of the variables may be controlling depending on the values of the other variables. If the initial turbulence and roughness are sufficiently small, transition will be preceded by regular Tollmien-Schlichting oscil- lations of increasing amplitude. There has been considerable progress in understanding the breakdown of laminar flow as a result of the theoretical and experimental work on the stability of laminar flow as described in IV,F. There is, however, no mathematical theory of the transition process itself. A satisfactory mathe- matical theory most certainly will have to take into account the nonlinear terms in the equations of motion of a viscous fluid. A great deal of experi- mental material is available. In most of the experiments essential measure- ments of the controlling variables were not made; especially lacking are measurements of the intensity and scale of the initial turbulence of the fluid stream which is now known to be one of the most important con- trolling factors. In other cases the surface roughness and waviness are not known. For the most part the experiments were made in the absence of any guiding theory. For all these reasons it is exceptionally difficult to systematize and analyze the data. During the course of the past 15 years as the experimental data accu- mulated, new physical aspects of the phenomena have unfolded and sug- gestions as to a descriptive physical mechanism have come to mind. It has been suggested that an immediate prerequisite in every case is separation with the resulting formation of a free shear layer within the boundary layer or shear layer under observation, the scale of the newly formed shear layer being an order of magnitude lower than that of the layer under study. This separation is presumed to occur even when the appar- ent dominating experimental variables are initial turbulence or surface roughness. Like the progression of eddies of successively decreasing size in the modern theory of turbulence bounded at the lower end by eddies ( 28 ) A,10 - TENTATIVE CONCEPTUAL PICTURE so small that viscosity quickly damps their motion, this theory of tran- sition requires a progression of separations with formation of free shear layers of successively smaller scale until the chain is broken by shear layers of such small Reynolds number that turbulence is not generated. This tentative conceptual picture of transition can hardly be said to be firmly demonstrated, but such a picture may serve, at least for a time, as a useful guide in the presentation of existing data and as a guide to future more systematic study of the basic phenomena. Let us examine the observed experimental fact that transition is greatly affected by exceptionally small disturbances in an otherwise steady flow. Fluctuations with time of amplitude as small as one tenth of one per cent of the mean velocity have measurable effects on the po- sition of transition. Dryden [/] calculated the effect of a small sinusoidal ee ee Na ; Ser oa ees SI REE, t RRL YN BABS S LS Fig. A,10. Streamlines for flow in the boundary layer of a plate subjected to peri- odically oscillating pressure variations beginning at distance Lp from the leading edge. Amplitude of free stream velocity variation—3 per cent of mean value. Wavelength Ly equal to 0.072 times initial length Lo. variation of the free stream velocity with distance along the outer edge of a boundary layer using the K4rmdén-Pohlhausen approximate method of solution of the Prandtl boundary layer equations. Separation of the flow was found to occur after three complete cycles of a sinusoidal vari- ation of amplitude two per cent of the mean velocity. A far more satis- factory computation was made in Germany during the war by Quick and Schréder [38] by a step-by-step procedure. A sinusoidal velocity variation of one-half per cent of the mean velocity and of wavelength of the order of the boundary layer thickness produced 15 to 20 per cent variations in displacement thickness with a separation bubble during the third cycle and complete separation at the fourth cycle. The streamlines for this case are shown in Fig. A,10. It is surmised from these considerations that small disturbances from any source will lead to intermittent separation and the formation of free shear layers in the fluid. If the Reynolds num- bers of these shear layers are sufficiently high, small scale turbulence will ( 29 ) A + TRANSITION FROM LAMINAR TO TURBULENT FLOW be generated and spread throughout the boundary layer. Even if the shear layer does not itself undergo transition, it will roll up into discrete vortices of a very small scale which diffuse through the boundary layer. The recent work of Schubauer and Klebanoff [5] shows clearly that the concept of two-dimensional separation is not applicable to transition in a boundary layer in an air stream of small turbulence, and raises serious doubts as to the utility of the separation concept in describing the phe- nomena involved. The periodic Tollmien-Schlichting waves rapidly lose their two-dimensional character, their amplitude varying along a direc- tion normal to the flow. These variations have been shown to be directly coupled with very small variations of mean velocity across the air stream on a line parallel to the leading edge of the plate. The boundary layer parameters, including the amplification ratio, are sensitive to these small changes, resulting in increasing variations of wave amplitude across the flow as the wave travels downstream. Apparently no dynamic instability of a three-dimensional character is involved, at least in the early stages. Turbulence originates locally in the regions of maximum amplitude as an essentially three-dimensional phenomenon. The first sign of turbulence in a hot wire record is a sharp and momentary large increase in wire temper- ature which is normally interpreted as a momentary large decrease in velocity. This appears first well out in the boundary layer rather than close to the wall as would be expected from separation. In the present state of the experiments it is difficult to believe that the results indicate a three-dimensional localized separation bubble, whose shear layer may roll up into a horseshoe vortex as described by Theodorsen [39]. The alter- nate theory is that vortices of the Gértler type with axes parallel to the flow develop at the wave amplitude maxima. A,ll. Theory of the Influence of Turbulence on Transition. Taylor [40] assumed that transition due to turbulence resulted from momentary separation of the boundary layer caused by the pressure gradients within the layer resulting from the fluctuating pressure gradi- ents of the turbulence. Separation is determined by the parameter \ = (62/v) (du./dx) which may also be written as — (62/v)(1/pu.)(0p/dx) where ép/dzx is the instantaneous pressure gradient. From the theory of isotropic turbulence the root-mean-square pressure gradient is proportional to (u’®/L)3(p/vt). Hence, the root-mean-square value of 2d is (wel/v)3(6?/1?) (w’/u.)i(L/l)-? where | is a reference dimension. Hence if separation occurs at a fixed value of \, we have, noting that 6// at the transition is some function of the Reynolds number Re, = (uel/v),, ae /UNe re = FLT) | According to this theory transition depends only on this combination of ( 30 ) A,11 - INFLUENCE OF TURBULENCE the intensity and scale of turbulence and not on the two properties separately. Fig. A,1la shows the critical Reynolds number of a sphere plotted as a function of the Taylor turbulence parameter [4/7] and Fig. A,11b shows the location of transition on an elliptic cylinder as a function of the same 0.12 0.8 =e 1.6 2.0 2.4 2.8 Ree SO Fig. A,lla. Effect of intensity and scale of free stream turbulence on the critical Reynolds number of spheres. L = scale of turbulence, D = diameter of sphere. 0.02 ] Fig. A,llb. Effect of intensity and scale of free stream turbulence on transition on an elliptic cylinder. L = scale of turbulence, D = length of minor axis of elliptical section. parameter [34]. This theory accounts for the influence of stream turbu- lence when the turbulence is greater than about two tenths of one per cent. Experiments on a flat plate with zero pressure gradient were made by Hall and Hislop [42] in turbulent air streams behind two square-mesh screens. In this case there was little overlap between the results for the ( 31 ) A: TRANSITION FROM LAMINAR TO TURBULENT FLOW two screens and the scatter of points is about the same whether Re, is plotted against u’/u. or the modified Taylor parameter (u’/u.)(c/M)* where M is the mesh of the screen. Nevertheless, the results are not inconsistent with Taylor’s theory. Experiments on airfoils were made by Drougge [43] and the results were analyzed to show that the effect of turbulence is related to inter- mittent separation. Another analysis of this type has been made by Dorodnitsein and Loitsianskii [44] and applied to published data. A,12. Schlichting’s Procedure for Computing Transition on an Airfoil. In 1940 Schlichting published a theoretical method [45] for the computation of the transition position on an airfoil based on compu- tations of the stability of boundary layers in accelerated and retarded Fig. A,12a. Velocity distribution around elliptic cylinder of fineness ratio 4. s = distance along surface, c’ = distance from leading to trailing edge measured along surface. flows. The method was further refined in 1942 [46]. On the basis of the experimental data available to him at the time, Schlichting assumed that transition always occurred ahead of the separation of the laminar bound- ary layer; in other words he did not consider separation bubbles with tran- sition in the separated layer and subsequent reattachment. Schlichting also observed that for Reynolds numbers up to 3 million, transition never occurred ahead of the position of minimum pressure on the airfoil. Experi- ments available to Schlichting showed values of Re, for a flat plate with zero pressure gradient lying within the range 350,000 to 500,000 whereas now values between 90,000 and 2,800,000 have been observed. As a preliminary to his airfoil computations, Schlichting computed — the stability characteristics of a boundary layer subjected to uniform pressure gradients (Art. 3 and IV,F). Schlichting then assumes that tran- sition depends only on the local boundary layer Reynolds number Reg; ( 32 ) A,12 - SCHLICHTING’S PROCEDURE (oe) [e) Ne) (©) yn W —~sS. O (e) On (oe) co (@) le fe | O10) C—O) eve = 0,5 Fig. A,12b. Velocity distribution around Schlichting’s Joukowsky I airfoil. s = distance along surface, c’ = distance from leading to trailing edge measured along surface. S/o! Fig. A,12c. Displacement thickness of boundary layer for elliptic cylinder and Joukowsky I airfoil. ( 33 ) A - TRANSITION FROM LAMINAR TO TURBULENT FLOW Scr Fig. A,12d. Pohlhausen pressure gradient parameter \ of boundary layer for elliptic cylinder and Joukowsky I airfoil. (Res. cr tf o fe i” _ S re) a S 2 _ ce} iS D O _ vn Fig. A,12e. Critical Reynolds number at which boundary layer oscillations are amplified as function of pressure gradient parameter X. ( 34 ) A,12 - SCHLICHTING’S PROCEDURE transition occurs when Re;+ reaches the critical value computed for the local velocity distribution which is in turn determined by the local pres- sure gradient. The location of transition is then computed by the following procedure: 1. Compute the theoretical pressure distribution over the body from po- tential flow theory. The result is expressed as a plot of u./U vs. s/c’ where wu, is the velocity at a distance s measured along the surface from the stagnation point, U is the stream velocity at a great distance, and Z| a ZZ Fig. A,12f. Variation of displacement-thickness Reynolds number for elliptic cylinder as function of cylinder Reynolds number and local position. c’ is the distance from leading to trailing edge measured along the surface. Fig. A,12a and A,12b show this curve for an elliptic cylinder of fineness ratio 4 and a Joukowski airfoil respectively. 2. Compute by Pohlhausen’s method (4-term polynominal approximation to the velocity distribution) the displacement thickness 6* of the boundary layer. Compute also the Pohlhausen parameter \. There result curves of (6*/c) ~/Uc/y and \ vs. s/c’ as shown in Fig. A,12c and A,12d. 3. From the stability calculations and the values of 2, plot the critical Reynolds numbers (Re;:),, corresponding to each ) vs. s/c’. The rela- tion of (Re+)., to \ as used by Schlichting is shown in Fig. A,12e. 4. From the values of (6*/c) ~/Uc/» = (U/u-)(Res«/Re*), where Re is ( 35 ) A - TRANSITION FROM LAMINAR TO TURBULENT FLOW the body Reynolds number Uc/», plot curves of Res« vs. s/c’ for several values of Re as in Fig. A,12f and A,12g. On the same diagram plot (Re). vs. s/c’ as computed in 3. 5. For each value of Re determine the intersection of the Ress curve with the (Re5+)., curve and read the value of s/c’ which is the location of transition for this Reynolds number. Plot the values of (s/c’).. vs. Re to give the transition position as a function of Re as shown in Fig. A,12h and A,12i. 104 D2 Ree 3 10°. aera | Free ZA TN Vy cr 8 } = Ses. 7 : AS ies Fig. A,12g. Variation of displacement-thickness Reynolds number for Joukowsky I airfoil as function of airfoil Reynolds number and local position. Schlichting had no reliable data in air streams of low turbulence to compare with his theoretical computations. The method as outlined is capable of considerable improvement, since the approximate methods used both for the stability calculations and the boundary layer thickness calculations are relatively crude. Furthermore it is desirable to use the experimental pressure distribution, if available, rather than that com- puted on potential flow theory. The most serious criticism, however, is that we know definitely that transition does not occur at the critical Reynolds number for instability of the laminar boundary layer but at a considerably higher Reynolds number, and no method is known for com- puting the ratio of the two. Hence the method as described cannot be expected to give valid results. ( 36 ) A,13 -THEORIES BASED ON LOCAL PARAMETERS Schlichting’s approach suggests the investigation of a procedure in which the critical Reynolds number is taken not from stability calcu- lations but from experimental values determined under simplified con- ditions. Let us suppose that (fe;+),. could be determined from available experimental data as a function of air stream turbulence, local pressure gradient, and local surface roughness. We could then carry out a modified Toon ooom Soil lA Na aan oe 2 ss FC Seen A TT QO 1 ST 0.2 CONSE eee ec a i NOs 2: 10° Sy Pall S APOZ 2: 5 OH Fig. A,12h. Location on elliptic cylinder at which boundary layer oscillations are amplified as function of cylinder Reynolds number. ed oe ig ee | then it SCO ) 0431 A cer 5 eS. CI oT od SR oo PREETI TM EA i 9 COL CCIE COCCI CE Cart 104 105 106 107 108 Uc/v Fig. A,12i. Location on Joukowsky I airfoil at which boundary layer oscillations are amplified as function of airfoil Reynolds number. Schlichting computation with some hope of better agreement with experi- mentally observed transition points. Unfortunately, sufficient experi- mental data are not available, and a complete test of this type of theory must await further research. A,13. Adequacy of Transition Theories Based on Local Pa- rameters. The key assumption in Schlichting’s approach is that transition is determined by the local values of mean speed distribution, ( 37 ) A: TRANSITION FROM LAMINAR TO TURBULENT FLOW pressure gradient, curvature, and surface roughness; that the critical Reynolds number is the same as if all of these quantities were constant and independent of the distance s along the surface. Attempts have been made to check the adequacy of this assumption by Stephens and Haslam [47] in actual flight tests. In the flight tests values of (Re;+).. were ob- served ranging from 1460 to 3240, values of \ at the point of transition between +0.3 and —7.2, values of the curvature parameter 6*/r from 0.06 X 10-3 to 0.86 XK 10-%. No correlation whatever of (Res«)., with d and 6/r was observed. The values of \ seem sufficiently far from the Pohlhausen value —12 to preclude the presence of separation bubbles, although in view of the shortcomings of the Pohlhausen method it may not be safe to conclude that this phenomenon was absent in all cases. No quantitative measures of surface roughness and waviness were given, and it is now believed that the results obtained by Stephens and Haslam were controlled by this parameter, whose effect overshadowed the influ- ence of the other parameters. This conclusion is drawn since in later flight tests where exceptional attention was given to the smoothness of the sur- face and more particularly freedom from waviness, values of (Re>+)., in the range of 6000 to 6500 were obtained by both British and United States investigators as described in Art. 17. We conclude that the question of the adequacy of theories based on local parameters is not settled by these measurements. There is one further bit of evidence resulting from tests on a smooth low drag airfoil in the NACA low turbulence wind tunnel over a wide range of Reynolds numbers (from 14 to 58 million). In these tests [48] the position of transition on both upper and lower surfaces varied from the 25 per cent to the 50 per cent chord location. The observed values of (Re;*)., were between 5150 and 6150. The assumption of a fixed value of 6150 gives computed transition positions agreeing with the observed po- sitions with a maximum difference of 7 per cent of the chord. We have already seen that a roughness element may introduce a dis- turbance which produces transition at some distance downstream from the element. Hence it is clear that the disturbances originating upstream, which are not indicated by instruments measuring average values, must be considered in addition to the local values of 6*, \, k, and r. Whether a theory based on local parameters is adequate or not depends on whether all the important local parameters are included. So far as disturbances from upstream roughness elements are concerned, we may consider in- stead the direct influence of all upstream roughness elements on the critical Reynolds number for transition. If any upstream value of k/é* produces a value of (Re;«),, greater than the value of Res at the point under study, its effect is negligible at that point. If, however, the local Res» equals the (Fe;*),, of any upstream roughness element, transition will occur at that point due to roughness. By this device it may be possible to ( 38 ) A,14 - PIPE OF CIRCULAR CROSS SECTION retain the framework of the modified Schlichting method. Much careful experimental work needs to be done, however, before the adequacy of this method of predicting transition can be evaluated. At the Ninth International Congress of Applied Mechanics held in Brussels in September, 1956, A. M. O. Smith, in a paper entitled ‘‘ Tran- sition, Pressure Gradient, and Stability Theory,’”’ advanced the hypothe- sis that transition occurs when the amplification ratio of the initial dis- turbances as computed from the Tollmien-Schlichting theory reaches e? or about 8100, and showed a comparison of experimental data with com- puted results which indicated reasonably good agreement. It is difficult to understand how the magnitude of the initial disturbance can be omitted; certainly the theory cannot deal with the effects of free stream turbulence on a smooth plate in a flow with zero pressure gradient. How- ever, in many cases the amplification ratio varies so rapidly with increas- ing distance along the surface because of the effects of pressure gradient that the computed transition position varies slowly with changes in the selected value of the initial disturbance amplitude or amplification ratio. A,14. Transition te Turbulent Flow in a Pipe of Circular Cross Section. The nature of the flow in a pipe depends on the value of the Reynolds number Re = u,,d/v where u,, is the mean velocity, d the diam- eter, and y the kinematic viscosity of the fluid. Since the velocity dis- tribution in laminar flow is parabolic, the Reynolds number may be written aS Unax?/y Where Unax is the velocity at the center and r is the radius. For comparison with transition in boundary layers we note that the Reynolds number Re; based on the displacement thickness, which in this case is the thickness of an annulus bounded by the pipe wall which would pass all of the fluid at the maximum velocity, is equal to 0.303 Re. Transition from laminar to turbulent flow depends greatly on the initial disturbances which in turn depend on the shape of the entrance to the pipe and the disturbances in the flow in the tank or reservoir ahead of the pipe entry. The lowest critical Reynolds number for large initial disturbances has been measured by many investigators [49, p. 319] with results lying between 1900 and 2320 for Re or 576 and 703 for Re>.. At lower values of Re, initial disturbances die out far downstream. Reynolds was able to increase Re to 13,000 by reducing the initial dis- turbances. Other experimenters have had greater success, obtaining values of Re of 20,000 (Barnes and Coker [50] and Schiller [57]), 32,000 (Taylor [49, p. 321]), and 50,000 (Ekman [52]). Ekman’s value corresponds to Re; of 15,150. The values vary by a factor of 26, dependent on initial disturbances which were not quantitatively measured. It is probable that the initial turbulence in these experiments varied from several per cent to less than one hundredth of one per cent. For comparison with transition measurements in boundary layers it ( 39 ) A+ TRANSITION FROM LAMINAR TO TURBULENT FLOW should be noted that transition in a pipe actually begins near the entrance to the pipe. If the entrance is bell-mouthed or rounded such that sepa- ration does not occur, a thin boundary layer develops on the pipe wall and grows in thickness until it equals the pipe radius. Because of the continuity relation the flow near the axis must accelerate and hence the static pressure falls more rapidly than in the finally developed flow where the drop in pressure is due only to friction. Transition thus occurs in the wall boundary layer which is subjected to a favorable stabilizing pressure gradient and to whatever turbulence is present in the entering flow. For the same turbulence the critical Reynolds number would be expected to be somewhat higher than for the boundary layer on a plate with zero pressure gradient. With sharp-edged entrances, separation occurs at the entry with the formation under some conditions of regular vortex patterns [53,54]. These vortices give very large disturbances and hence low values of the critical Reynolds number. For very rough pipes the critical Reynolds number appears to be the same as for a smooth pipe with very disturbed entry conditions. How- ever, if the initial turbulence is small, roughness may produce larger dis- turbances than those already present and reduce the critical Reynolds number. Depending on the shape of the roughness elements, the maxi- mum permissible height to avoid disturbance in a smooth pipe has been estimated to be of the order of 4/Re? times the radius of the pipe [49, p. 311]. The fact that any roughness is permissible is thought to be associ- ated with the fact that roughness elements also possess a critical Reynolds number below which they set up no disturbance. For example for a flat plate roughness element, the critical Reynolds number is about 30. The above relation corresponds to a critical Reynolds number of 32 and the assumption that the critical height is small compared to the radius. See, however, the discussion in Art. 5. The effect on the critical Reynolds number of curving the axis of cylindrical pipes of circular cross section has been studied by several investigators [55,56]. The breakdown of the laminar flow does not in this case lead immediately to turbulent motion but to a regular type of second- ary motion under the influence of the centrifugal forces. Turbulence sets in at a critical Reynolds number which depends on L/d where L is the radius of curvature of the axis of the pipe and d is the diameter of the pipe. The values obtained increase as L/d is reduced, from about 2300 at L/d = 1025 to 7600 at L/d = 7.6. A,15. Transition in Pipes of Noncircular Cross Section. Transi- tion has been studied in pipes of rectangular, square, and annular cross section. Basing the Reynolds number on the hydraulic radius, values of 2100 for the square cross section, 1600 for a rectangular section with ratio ( 40 ) A,17 - TRANSITION ON AIRFOILS of sides of 2.43 to 1, 2800 for rectangular sections with ratio of sides between 104 and 165, and 2400 for an annular section with ratio of radii of 0.818 to 1, were obtained [49, p. 319]. The disturbances were relatively large in these measurements, so that the values are comparable with 1900-2320 for a circular cross section. A,16. Transition on an Elliptic Cylinder. Studies of transition on an elliptic cylinder of fineness ratio 2.96 have been made by Schubauer [34] over a Reynolds number range of 21,000 to 160,000, the Reynolds number being based on the minor axis of the elliptical section. The turbu- lence of the stream was relatively high, from 0.85 to 4.0 per cent. At a turbulence level of 0.85 per cent, and Reynolds numbers from 21,000 to 30,000, the boundary layer flow is laminar until separation occurs, and transition occurs in the shear layer so far downstream that the flow around the cylinder is not affected by it. From the observed pressure distribu- tions, transition at higher Reynolds numbers occurs in the free layer and the boundary layer reattaches. The pressure distribution, position of sepa- ration, and reattachment change with Reynolds number until a value of about 120,000 is reached. For still higher Reynolds numbers the flow and pressure distribution were independent of Reynolds number up to the maximum value reached of about 160,000. Here transition occurred soon after separation, as described in Art. 8. The air stream turbulence was increased by the use of turbulence- producing screens and the location of transition was measured as afunc- tion of the intensity and scale of the turbulence. The results have already been given in Fig. A,11b for comparison with-Taylor’s theory. Unfortunately the gap between a turbulence of 0.85 per cent and the lower end of the curve in Fig. A,11b was not covered, but the curve must turn sharply to the right at a smaller value of the turbulence parameter to reach values of 2/D of about 2.7 as experimentally observed at a turbulence of 0.85 per cent. The relatively stationary position of tran- sition at z/D of about 1.53 corresponds approximately to the point of minimum pressure where the pressure gradient changes from favorable to adverse. A,17. Transition on Airfoils. The measurements of transition on airfoils as reported in the literature constitute a record of the develop- ment of wind tunnels of lower and lower turbulence and of improved techniques of producing smooth surfaces free from waviness. At first no detailed boundary layer measurements were made and the only data given were drag curves vs. Reynolds number based on the air stream velocity and airfoil chord. The critical Reynolds number of the airfoil was taken as that at which the minimum drag coefficient was reached or at which a perceptible rise occurred. Later the location of transition Gay A :- TRANSITION FROM LAMINAR TO TURBULENT FLOW was determined by the surface-Pitot technique, but boundary layer data were not taken and a small separation bubble might possibly be over- looked. During the same period the laminar flow airfoils were being de- veloped, permitting much more extensive runs of laminar flow because of the extent of the favorable pressure gradient. Both wind tunnel and flight data give estimated equivalent flat plate Reynolds numbers cover- ing a range from about 600,000 to about 14,000,000, the high values being obtained much more recently on laminar flow sections in wind tunnels of turbulence less than 0.1 per cent and in flight on models in which extreme care had been taken to remove surface roughness and waviness. In any particular measurement it is almost impossible to separate the effects of the many variables. Fig. A,17a illustrates wind tunnel data on the chordwise position of transition on airfoils at approximately zero angle of attack [43,48 ,57,58, Fig. A,17a. Transition position on airfoils as a function of airfoil Reynolds number. 59,60,61|. In every case transition moves forward with increasing Reyn- olds number but the rate and the value of the Reynolds number at which the most rapid change occurs are dependent on the type of airfoil section, the smoothness and fairness of the surface of the airfoil, and on the turbu- lence of the wind tunnel in which the measurements are made. The results for the 65215-114 airfoil [48], showing transition as far back as 25 to 30 per cent of the chord at Reynolds number of 40 to 55 million, were obtained in the low turbulence wind tunnel of the Langley Aero- nautical Laboratory in which the turbulence intensity is a few hundredths of 1 per cent. The results for the Tani-Mituisi airfoil at the extreme left of Fig. A,17a were obtained in the FFA wind tunnel at Stockholm [43] behind a turbulence grid giving a turbulence level of approximately 1 per cent. Although there is some influence of airfoil shape and surface wavi- ness in this comparison, the principal differences between these two curves are believed to be due to effects of wind tunnel turbulence. ( 42 ) A,17 - TRANSITION ON AIRFOILS The effect of airfoil shape may be seen by comparing the curves for the 0012 airfoil [58] and that for the 6521;-114 low drag airfoil, although data at the same Reynolds number are not available. The results for the 0012 airfoil [58] in the NACA 8-ft wind tunnel as compared with the results on the same airfoil in the NACA low turbulence wind tunnel show the effect of an increase in wind tunnel turbulence from a few hundredths to several tenths per cent on a conventional airfoil. From the totality of information available on airfoils and other bodies, we may reconstruct the qualitative picture of the influence of Reynolds number on the location of transition. At extremely low Reynolds num- bers the boundary layer separates from the surface as a laminar layer and the separated shear layer remains laminar far downstream. As the Reynolds number increases, transition occurs in the shear layer nearer and nearer the point of separation. At some Reynolds number the flow reattaches to form a separation bubble which decreases in size as the Reynolds number increases. Over a certain range of Reynolds numbers, transition remains fixed just beyond laminar separation. With further in- crease in Reynolds number, transition moves forward, more rapidly while in the region of adverse pressure gradient and more slowly as the pressure minimum is passed to reach the region of favorable pressure gradient. At very large Reynolds numbers, transition approaches closer and closer to the forward stagnation point. Most of the available data are for the Reynolds number range in which the transition lies between the laminar separation point and the leading edge. When the airfoil is placed at a different angle of attack the pressure distribution changes and aerodynamically we have to do with a different body. A low drag airfoil experiences adverse pressure gradients at suf- ficiently high angles of attack and leaves the low drag region. In general terms transition moves forward on the upper surface and backward on the lower surface as the angle of attack is increased. Typical experimental data are found in the references previously cited. Fig. A,17b shows the data plotted in Fig. A,17a in a slightly different form, the ordinate now being (x,/c) Re which is a rough approximation to the equivalent flat plate Reynolds number at transition. Exact data from boundary layer computations are shown for the 65215-114 airfoil for com- parison; in this case the approximate values are too high by from 2 to 14 per cent. For most of the data plotted the equivalent flat plate Reyn- olds number of transition increases with the airfoil Reynolds number. It is believed that this increase is associated with the increased stability of the boundary layer at the more forward positions of transition corre- sponding to the higher airfoil Reynolds numbers, where the pressure gradient is increasingly more favorable. The available wind tunnel data on roughness effects [13,15,62,63] are plotted in Fig. A,17c as transition Reynolds numbers based on airfoil ( 43 ) A: TRANSITION FROM LAMINAR TO TURBULENT FLOW Approximate Re, = (x./c Fig. A,17b. Approximate transition Reynolds number (equivalent flat plate value) for airfoil boundary layers as function of airfoil Reynolds number. Loftin, rivets spaced 3 in., single row Airfoil 1 0.35c0) Rivet 0.50c.0| diameter 0.035 in. k/5* Fig. A,17c. Effect of roughness on transition Reynolds number of airfoils. (xa A,18 - TRANSITION ON AIRPLANE CONFIGURATIONS chord as a function of k/6*. Varying techniques were used for measuring the transition Reynolds number, principally that of rise in total drag coefficient and surface tubes. The detector tubes were located in various positions and correction to a fixed position does not seem practicable. The effect of shape of the roughness elements is obvious as is the large influence of air stream turbulence. Loftin’s results [62] refer to three-dimensional roughness elements in the Langley two-dimensional low turbulence wind tunnel and were ana- lyzed by him in terms of the Reynolds number of the roughness element uxk/v. The observed value was a function of the ratio of the diameter d of the cylindrical projections to the height k, varying from a value of about 1000 for d/k = 0.5 to 200 for d/k = 7.0. For d/k = 1.0, the value of uzk/v was 750. This compares with Klebanoff, Schubauer, and Tid- strom’s value of 300 for spherical elements [23]. Von Doenhoff [63] studied the effect of sand grain roughness elements on airfoils in considerable detail. His results, also obtained in a low turbu- lence wind tunnel, give a critical Reynolds number of 250 based on nomi- nal particle size, or 600 based on maximum particle size. The available data suggest that the critical Reynolds number of a roughness element is affected as much or more by the turbulence of the air stream in which the measurements are made as by the shape of the element (see Art. 5). In the absence of sufficient data one can only conclude that the tran- sition Reynolds number on airfoils is a function of air stream turbulence, pressure gradient, and surface roughness and that all variables have im- portant effects. The flight data on transition on airfoils presents as confusing a picture as the wind tunnel data. While the effects of air stream turbulence are presumably absent, the angle of attack changes with speed so that the Reynolds number cannot be systematically varied for a fixed angle of attack. The largest influence, however, appears to be that of surface wavi- ness, for increased care to secure smooth and fair surfaces has given higher and higher values. In the last eighteen years the values of (x,/c) Re have increased from 3.5 X 10® for conventional airfoil sections in 1938 to 11.4 X 10° and 17.0 X 10° for low drag sections in recent years. Some of the more recent work is still classified but the highest observed values are those given in [64]. It seems impractical to secure the required free- dom from surface waviness and roughness in normal construction and operation of aircraft to realize these high values. A,18. Transition on Airplane Configurations and on Airplanes in Flight. Additional variables influence transition on three-dimensional airplane configurations in wind tunnels and on airplanes in flight. The pressure distribution at wing-body junctures and nacelle-wing fairings is often such as to bring transition close to the leading edge of the wing. ( 45 ) A: TRANSITION FROM LAMINAR TO TURBULENT FLOW Moreover actual airplanes have unavoidable local roughness at access doors and elsewhere which generates local turbulence. Turbulence so gen- erated spreads laterally, and this process has been studied by several authors [65,66]. In practical testing, using surface films for detecting transition [67], dust particles produce the typical wedges of turbulence behind them. The observed angle of spread (one half the vertex angle of the wedge) is 8.8°, the various determinations scattering over a range from about 8.5° to 11°. If there are a sufficient number of sources of disturbance and the wing chord is sufficiently large, the turbulence will cover the entire wing span. A propeller generates turbulence in its wake and hence transition on that part of the wing lying within the slipstream will occur at a tran- sition Reynolds number corresponding to a stream of large turbulence [68]. A tractor propeller produces a large effect; in the case studied its operation moved transition from midchord to less than 10 per cent of the chord from the leading edge. A pusher propeller was observed to have no measurable effect on transition on the wing ahead of it. Likewise the vibration due to an operating power plant appears to have little effect; wind tunnel measurements for vibrations of frequency of 27 cycles per second and amplitude of 0.1 inches gave no measurable change in the transition point. The boundary layer on an airplane in flight is subjected to the noise emanating from its power plant. Wind tunnel measurements in a low turbulence wind tunnel [2] show that noise may affect transition under certain circumstances. A,19. Transition on Bodies of Revolution. The simplest body of revolution is a sphere and the effect of the occurrence of transition before laminar separation in greatly reducing the drag coefficient has been known for 37 years. That transition on a sphere is greatly dependent on the tur- bulence of the air stream has been known for the same period and for many years the critical Reynolds number of a sphere was used as a meas- ure of wind tunnel turbulence. The relationship is plotted in Fig. A,11la. For reasons not fully understood the sphere is not a good indicator of turbulence when the turbulence level is less than a few tenths per cent. One hypothesis is that, because of the blunt shape, disturbances are set up at the forward stagnation point which mask the effects of low turbu- lence levels. Gortler [69] has suggested that the concavity of the streamlines in the neighborhood of a stagnation point in two-dimensional flow leads to an instability of the type discussed in Art. 4, resulting in vortices with axes along the flow lines. Calculations for the two-dimensional case have been made by Himmerlin [70]. Presumably a similar instability would be found near a stagnation point in three-dimensional flow. ( 46 ) A,19 - TRANSITION ON BODIES OF REVOLUTION Little work has been done in correlating the sphere data with flat plate data in terms of Re». Fage [71] gives Res = 945 at transition occurring well ahead of separation on a sphere in an air stream of 0.85 per cent turbulence. The equivalent flat plate Reynolds number of 298,000 is considerably below the curve of Fig. A,2b, but it must be noted that transition on the sphere occurred in a region of large adverse pressure gradient (A = —5 to —7) in Fage’s experiments. Tomotika [72] computed the growth of the laminar boundary layer on the surface of a sphere in a uniform stream for an experimental pres- sure distribution obtained at a Reynolds number of 165,000. Separation occurred at an azimuth angle of 81° with Re;:/+1/Re = 2.72, hence at an equivalent flat plate Reynolds number of 2.5Re. Thus if the Reynolds number at which transition occurs just ahead of the separation is known, the value of Re, may be computed. The critical Reynolds number of a sphere as usually defined corresponds to a considerably more forward position of transition and to a considerably modified pressure distribu- tion. Examination of sphere drag coefficient and pressure coefficient curves shows a departure from an approximately constant value of the coefficient beginning at about 0.4Re,,. Hence the values of Re, are proba- bly of the same order as Re,,. The maximum value observed for a sphere is about 4 X 10° as compared with 28 X 10° for the flat plate in air streams of low turbulence. For a turbulence of 1.0 per cent the value of Re, from a sphere is about 200,000 as compared with 630,000 for a flat plate. For very high turbulence the values agree; for a turbulence of 3 per cent both sphere and plate give a value of Re, of about 100,000. The much lower values derived from the sphere at low turbulence are presumably due to the large adverse pressure gradient at the transition point on the sphere. In this discussion the well-known but much smaller effect of the scale of turbulence has been omitted. The most suitable turbulence parameter for the generalized case is (w’/u.)(6*/L)* where 6* is the displacement thickness of the boundary layer at transition. The drag of streamline bodies of revolution as a function of wind tunnel turbulence was studied in 1929 [73] and the observed results were interpreted as due to the effect of turbulence on transition in the bound- ary layer. Computations were made on the crude assumption that the velocity distribution in the boundary layer was linear. It was assumed that transition occurred at values of Re; of 1250, 2000, and 2750 (6 being the thickness based on a linear distribution) for turbulence levels of 2.3, 1.6, and 1.2 per cent. The corresponding values of Re;+ are 625, 1000, and 1375, and of Re,, 193,000, 333,000, and 630,000. In view of the crude ap- proximations in the theoretical computations, and the experimental errors involved in early hot wire measurements of turbulence, these values are in satisfactory agreement with Fig. A,2b. ( 47 ) A+: TRANSITION FROM LAMINAR TO TURBULENT FLOW More recent studies have been made by Fage and Preston [74] in a water stream using the fluid motion microscope, but these do not agree so well with the two-dimensional values of Fig. A,2b. One of the bodies used was a long cylinder 3 inches in diameter with a semi-ellipsoidal nosepiece 6 inches long, and the water stream was 7 inches in diameter. Most of the observations were taken with turbulence screens about a foot ahead of the nose of the body. The turbulence varied considerably along the length of the body, and if the local value at the transition point is used as abscissa in Fig. A,2b, the values of Re, are of the order of 4 to + those observed for the flat plate. It is obvious that the higher turbulence levels upstream are influencing the location of transition. Even if the average turbulence level from the nose to the transition point is used, the values still fall below the flat plate values, although the pres- sure distribution is mildly favorable and should yield higher values. As in many early experiments on transition it is possible that effects of surface waviness may have been present. At any rate the values observed for Re, for a turbulence level of approximately one per cent were of the order of 300,000 to 400,000 as compared with the 600,000 to 700,000 for the flat plate shown in Fig. A,2b. Fage and Preston also studied transition on a second body with the same semi-ellipsoidal nose shape, a 4-inch cylindrical mid-body section, and a tail tapering in diameter from 3 to 2 inches over a length of 16 inches. For this body, transition occurred following laminar separation and the phenomena observed were similar to those described in Art. 16. The boundary layer on a body of revolution is of course not compa- rable with that on a plate at the same distance from the stagnation point because of the three-dimensional character of the flow. Mangler [75] has obtained a general relationship between two-dimensional and axially sym- metrical boundary layers. When applied to compute the relation between the distance x; along the axis of the body of revolution and the distance Z along a flat plate at which the boundary layer thickness is identical for the two bodies, we find ti att r?(a)dx where r(x) is the radius of the body of revolution at axial distance x. Thus & is less than x; over the forward part of the body, equal at some point beyond the maximum cross section, and exceeds it near the rear end where r(x) is diminishing rapidly, causing a rapid thickening of the boundary layer from continuity considerations. The equivalent flat plate Reynolds number of transition differs from Re,, in the same way. Measurements of transition on a prolate spheroid of fineness ratio 9 and on a modified prolate spheroid of fineness ratio 7.5, modified to give more favorable pressure gradients over the nose, were made by Boltz, ( 48 ) A,20 - FLOW BETWEEN ROTATING CYLINDERS Kenyon, and Allen [76] in a wind tunnel of low turbulence (u’ = 0.02 per cent). The observed values of Re, ranged from 3.2 to 3.8 million for the body of fineness ratio 9 and from 3.6 to 4.3 million for the body of fineness ratio 7.5, the exact value being dependent on the axial location of transition. The effect of local surface roughness in the form of a wire ring on the surface in a plane normal to the body axis was studied by Fage and Preston [74]. The results were similar to those already described for wires on a flat plate. As the speed was increased, transition moved forward from its original position in the absence of the wire until at some speed it reached the position of the wire where it remained at higher speeds. The Reynolds number formed from the wire diameter and the velocity in meer © Model | Raittierleeclaca O Model II Pf as a ae Ass (pE) + cee Di (p’us) + = Di ae ui (2 + Uj us) ) = — aes (75 ujH® + Uipuu; + Cpp Tuy — k 27) fe mi: oe AE Vek Po — Yo (6-3a) where the operator D/ Dt on any function f denotes Die © jo a te aa; (fU;) The left-hand side of Eq. 6-3a expresses the rate of change of quanti- ties pH°, p’u;, pT’, and p. These rates are the result of diffusion and tur- bulent energy production expressed by the terms on the right-hand side. Here we find the production term —pu,u;0U;/dx; which decreases the energy of mean motion, and diffusion terms within the brackets which include Reynolds stress terms of transport of mass and temperature along with the better-known term pu,u;. Here also is the molecular contribution expressed by 9 — ¢o, which we have already noted in Eq. 5-3 as a spatial transfer and not an energy dissipation. No molecular dissipation appears in Eq. 6-3a because the kinetic energy dissipated appears in the form of heat. The corresponding equation for H may be obtained by adding Eq. 5-4 and 6-2. Thus < (6B) alee =~ (0B Us) + 3, < (VB) =- — Z = + pujE + p'ujh’ + U;p'E’ — po) +6é—@6 (6-4a) As would be expected, Eq. 6-4a, which accounts for both the mean and the turbulent energy, is of simpler form and does not contain the pro- duction term pu,u;0U;/02;. The diffusion terms in the energy equations (Eq. 6-3a and 6-4a) con- tain the thermal diffusion with flux kd7'/dz;. This may be expressed as an ( 86 ) Bai IN ERODUCTION energy diffusion 0#°/dz,; or 0H/dz; if we introduce xo according to Eq. 5-3c and x according to Eq. 5-3d. In this way, Eq. 6-3a and 6-4a become respectively Z Pe (oe een. & (BBY) + Us (p'ui) + — P oo) - (2 +, ®') aah an; (5 ujE® + Uipuij + cppT’uj; — — a Cron; EGE @))+= os < (6B) _ z, (PBU)) 1s “(p p'E’) ) =a ATLA ETRS ai GRAF Tit k ob —— a ae Czy + pu; + p u;H’ =. U;p KK’ —— =) ] Cp OX; 1 Le iW |e: (5 i ae (U2 + | + % (6-4b) where Pr = uc,/k is the Prandtl number. The specific heat cp is taken as constant. Eq. 6-3b and 6-4b become much simpler if Pr = 1, and if xo and x are negligible. Since in the incompressible case » = const, Pr = 1 and xo = x = 0, we can reduce Eq. 6-3b and 6-4b respectively to D (yoy — (9? = aie qv, — ob” L Di (pE°) e + u; aglminag Upu; + cpp Tu; a Oe, (6-3¢) iors Oui eklab he) aimers zz es 7) ey Eq. 6-4¢ is the well-known equation of turbulent heat transfer in an incompressible flow. Here pu,’ is the flux of energy transported by tur- bulent diffusion. CHAPTER 3. TURBULENT BOUNDARY LAYER OF A COMPRESSIBLE FLUID B,7. Introduction. When a fluid flows past the solid boundary of a body, a shear flow results. The condition of no-slip requires that the fluid immediately in contact with the wall be brought to rest. Next to it the fluid is retarded by the internal shear stresses. The retardation decreases with increasing distance from the wall and becomes vanishingly small in ( 87 ) B - TURBULENT FLOW a relatively short distance. The layer in which this occurs is called the boundary layer. A knowledge of the flow behavior within this layer is of prime importance, especially when effects associated with compressibility and aerodynamic heating come into play. The turbulent boundary layer occurs more generally than the laminar boundary layer, but is less well understood theoretically. The exceedingly complex character of turbulent flow and the inadequacy of theories of turbulence make an exact mathematical treatment of the flow impossible at present. Therefore a great number of approximations are necessary, and it is to be expected that the various proposed theories may turn out different results which are not always reconcilable. In order to clarify many obscure points in the theories, and to display in a simple manner the essential physical features governing boundary layer flow, it seems worthwhile to outline the main approaches of the analytical treatments, and especially to elucidate the bases and assumptions underlying the theories. Where possible the theoretical results will be compared with existing experimental results. First the fundamental hydrodynamic equations, as developed earlier, will be simplified in Art. 8 under the special conditions of the boundary layer. Consequently some simple relations between pressure, temperature, and velocity can be derived in Art. 9. These will at once show some features of heat transfer in the boundary layer, and especially of the recovery factor, without going into the turbulent transport processes. For a deeper understanding of the problems, some statistical methods of transport phenomena become necessary. Existing theories make extensive use of the concept of mixing length as a parameter of the turbulent ex- change of properties. Since several fundamental questions arise in con- nection with the application of mixing length to various types of transport (mass, momentum, and heat) governing the boundary layer, and in the analogy theories between heat transfer and skin friction (the so-called Reynolds analogy), the statistical foundation of the transport processes will be studied in Art. 10. As an immediate application, the Reynolds analogy can be better understood and will be treated in Art. 11. Theories relating to velocity profiles in a compressible turbulent boundary layer do not seem to differ much from the corresponding theories for the incompressible boundary layer, especially concerning their basis and method of attack. Therefore we shall reserve these for Chap. 4 where incompressibility is assumed, and be content here to give only some experimental data on the velocity distribution. The skin friction in a compressible boundary layer deserves special attention, because of its important compressibility effect and its practi- cal significance. The basis of the theories will be described in Art. 12; the empirical formulas illustrating the essential behavior of skin friction will be given in Art. 13; and finally the comparison between theories and ( 88 ) B,8 - FUNDAMENTAL EQUATIONS OF MOTION experiments will be given in Art. 14. Since no unique theory has evolved, the emphasis will be placed on the description and discussion of the bases and assumptions underlying the theoretical treatments rather than their detailed analysis. Experimental data will be compared with theories. This method of approach seems best to show the present state of the subject and to serve as a guide to future theoretical and experimental investigations. B,8. Fundamental Equations of Motion of a Compressible Boundary Layer. When applying the hydrodynamic equations of Art. 4, 5, and 6 to the boundary layer developed on a flat plate with steady free stream velocity, certain simplifying approximations may be made. First of all, the mean flow is assumed two-dimensional with mean veloci- ties denoted by U and V in the z and y directions respectively, where z is the coordinate parallel to the plate, measured from the leading edge, and y is normal to the wall. The turbulence is still three-dimensional, with components u, v, and w in the z, y, and z directions. We now consider the order of magnitude of terms involved in the hydrodynamic equations. If U is taken as a magnitude of standard order 0(1), and the thickness of the boundary layer 6 is small compared to the distance x, it follows that 0/dt, 0/dx, 0?/dx? ~ O0(1), and d/dy ~ 0(6—}), 0?/dy? ~ 0(6-?). Also we assume that V ~ 0(6), the mean density # is 0(1), and the total energy content per unit mass £ is 0(1). If the viscous term of Eq. 4-5 is to be at most of the same order as the remaining terms, then it follows that u is at most of the order of 6°. By the same 1e reasoning, the correlations involving u, v, p’, T’, such as uv, uT"; vT’: pu, p y, p'T’, are at most of the order of 6, while the triple conelation p'uv will be at most of the order of 6?. Retaining the predominant terms of the same order of magnitude, we can easily reduce the dynamic equations (Eq. 4-5, 4-6, and 6-4a) respec- tively to the ates forms: — 5B 0 oU = Ee < (BU) +2 = (BUD) ar 5 UV) = a = aii = = = Oe s) (8-1) OD 6 Opie ia <(pU) +2 5 OY) +2 |) = 0 (8-3) 5, (6B) +2 ; (PU) +3 — 7 EV). 0 mang ies Oe CY) -2(2 me FB) +5 [(pe— 1) eG] ( 89 ) B+ TURBULENT FLOW Similarly from Eq. 6-3a with the same order of approximations we obtain 0 pes ee 5. (GB) + 3. (pB'U) + 5. GEV) = 5 (. a = po! — 7) +2 ie = 1) Ms P| + pu 5 (8-5) Except for the production term puvdU/dy, Eq. 8-4 and 8-5 have the same form. In the following we shall be concerned with Eq. 8-4 rather than Kq. 8-5. ? Kq. 8-1, 8-2, 8-3, 8-4, and 8-5 form a system of basic equations of the compressible boundary layer. The effects of the density fluctuation are to contribute an additional Reynolds stress, an apparent source, and an addi- tional eddy conductivity respectively in the equations of momentum, con- tinuity, and energy. B,9. Relationships between Velocity, Pressure, and Tempera- ture Distributions. Some simple relations are now derived for velocity, pressure, and temperature by integrating the momentum and energy equations. This is done here without entering into the mechanism of tur- bulence in the boundary layer. We consider a steady boundary layer, with strictly parallel flow (all average quantities depend only on the coordinate y). First by integrating the momentum equation (Eq. 8-2) from y to 4, where 6 is the thickness of the boundary layer, we obtain p= pe + pr i pv? =n(1 +m ) (9-1) since M2 = p.U2/yp.. Here quantities without subscript are taken at the coordinate y, while subscript . denotes the quantity at the edge of the boundary layer. For future reference, superscript ° denotes the total or stagnation value, and subscript ~ denotes the value at the wall (y = 0). The assumption of a constant pressure within the boundary layer is valid if the free stream Mach number is of the order of 0(1), and if the previous assumption of small turbulence level (v?/U? « 1) is made. For the derivation of energy relations, the following conventional boundary conditions are used: Aty =0:. U=0,V =0,u4=0,7=0,w=0, 7 —T, (9-2a) At ye (VO NO, Ts (9-2b) ( 90 ) B,9 - VELOCITY, PRESSURE, AND TEMPERATURE DISTRIBUTIONS If Pr = 1 is assumed, the energy equation (Hq. 8-4) takes the form Oo OR ye ay (. a pH’v — Ep i) = 0 (9-32) This is similar to the momentum equation (Kq. 8-1), rewritten as follows: ) 0 — —\ _ ay oo — puv — Up s) = (i) (9-3b) Further from the equation of continuity, Eq. 8-3, we have p’v = const. Therefore a comparison between Eq. 9-3a and 9-3b leads to the following linear relationship between EF and U: BaoM thu +3 @ teste =or fan Uta] ow where the constant 7° and 7, as determined by the boundary conditions (Eq. 9-2a and 9-2b), are 9 = T/T? Cpl! = 650. U? T® is the stagnation temperature at y = 6. If we neglect as usual the turbulent intensity in Eq. 9-4, we obtain the approximate relation eT +5 U? = tnT? ja -) qt n| (9-5) Eq. 9-5 gives a relation between T and U on the basis that the laminar Prandtl number is unity. Some authors have derived the same relation requiring that the turbulent Prandtl number should also be unity (see e.g. [8]), but the latter condition is superfluous according to the above considerations. Eq. 9-5 gives the temperature-velocity relationship including heat transfer. If the wall is insulated, we must have oT (F).-9 ie but, according to Eq. 9-5, DM OU ee aU yey Ua ay a and since in general (0U/dy)~ # 0, the condition (Eq. 9-2c) imposed upon Eq. 9-6 requires that n= 1 hence Eq. 9-5 simplifies to the following form: Cpl + 2U? = epT'~ (= const) (9-7) ( 91 ) B= TURBULENT: PLOW. with an insulated wall. We conclude that for Pr = 1, the relationships between temperature and velocity in the turbulent boundary layer are the same as those in the laminar boundary layer, which were first ob- tained by Crocco [9]. In an insulated boundary layer at low speeds, Squire [/0] and Acker- man [//] have independently deduced the formula Cpl — ene > he (9-8) for Pr #1 or Pr = 1. Here T,, is the temperature at the wall. When there is no heat transfer, 7’, is sometimes called equilibrium temperature. This formula may be expected to be not seriously in error at high speeds, and includes Eq. 9-7 as a special case with Pr = 1. As Pr < 1 in general, the temperature at the wall is accordingly smaller than the total free stream temperature. In the light of Eq. 9-4, a more general formula for the case of Pr ¥ 1 can be written as follows: Cpe TC plore (9-9a) by introducing a factor r., called the recovery factor. The recovery factor can then be considered as defined by Eq. 9-9a, and it then becomes oa = Il, ie aa 7 = WE, (9-9b) Using the adiabatic relation T°/T. = 1 + (y — 1)M?2/2, Heke (r:) (ie tale! ig = (9-9c) Here M. is the Mach number at the edge of the boundary layer, and y is the ratio of specific heats. According to Eq. 9-8 and 9-9, the recovery factor should not differ very much from the value Fale (9-10) The turbulent recovery factor, which shows a close agreement with Kq. 9-10, has been measured by Mack [1/2] over the surface of a cone, in the range of free stream Mach number from 1.33 to 4.50, to be 0.88 + 0.01, as compared with the calculated value of Pr? = 0.89, based on the recovery temperature. Experiments for a flat plate have been made by Stalder, Rubesin, and Tendeland [13] (r. = 0.89 + 0.01) at Mach num- ber 2.4. Also the measurements of the laminar recovery factor show a close agreement with the theoretical value of Pr}. The experimental results of various investigations are summarized in Fig. B,9a and B,9b. In general the recovery factor depends on the Reynolds number. In ( 92 ) VELOCITY, PRESSURE, AND TEMPERATURE DISTRIBUTIONS B,9 - ‘roAB] ATVpunog oy} Jo oSpo oy} YB SUOT}IPUOD OY} UO puv dr} oUOd oY} WOIF OOUBISIP Of} uodn poseq S81 °ay Jaquinu spjousay ous, [27] YORI 10978 ‘Aoquinu spjousdy oy} YIM Souod Jo 10yOVF ATBAODOL oinyeiedui9} 94} JO UOTYVUIBA “Eq “SIT 9-O| X< 4equunu spjoucsay S v 080 anuj] >| oa c bal oT ‘S. 1") 3 = 06°0 aon] 000’S€9 : “Ul ZTLXCL df 227 o€L 000'€6r "Ul ZLXTL df aH9nN7 oEL 000 €6l 0G “Ul OCXBL df aon] EL 000 022 : “Ul OC@XBL df sp|B4aq!4 000'rvE "ul O@X8L df 1999S MO}JOY (OZ 000'06S 1294S MO}JOY OT 000‘76z DIWUDIBD OT 000’0€z Ppoom .OL J223S MO}|JOU .O| 000‘00Z J994S MO}JOY OL 000'98 J203S MO}JOY OL 000‘8EZ 9293S MO}[OY oO] 000'€61 2231S MO}[OY .O| 000'Z81 J293S MO}JOY OL 000'8Sz J223S MO}JOY OL 000'86€ J294S MO}JOY .OL 000'8ZS jepow sauo> “ul /?ay | (OU Ex, sowy "Ul GX LIDTV9 uaapiaqy’ ZOU EX] sowy 44 TXT SIME VW ~XZ sowy "Ul yLXO| sew 9x9 sow "Ul BLXBL SIM2T 9X SIMPT [ (OU J} Ex] Sau Joyo04 AlaA0D04 aN Osadwa | o-—-MOO, 00°L AAAALTNAOCTCO @OdIPVAA4VAXO AVVO 8 Rot = jauun | ( 93 ) B- TURBULENT FLOW the transition region, the recovery factor varies from the lower laminar value to the higher turbulent value. An increase of the turbulent recovery factor with the Reynolds number in the fully turbulent region predicted by the theoretical formula of Seban [74], re = 1 — (4.71 — 4.11B — 0.601Pr) Re~°? pabribr+? DY Heri and the theoretical formula of Shirokow [15], re = 1 — 4.55(1 — Pr) Re~°-? is not systematically detectable from the experimental results of Fig. B,9a and B,9b. Here Re is the Reynolds number based on the distance © Boundary layer undisturbed @ Boundary layer tripped near leading edge Temperature recovery factor 84 0.2 0.4 0.6 0.81.0 2 4) {162 C8 10 Reynolds number & 1078 Fig. B,9b. Variation of the temperature recovery factor with the Reynolds number in the case of a flat plate, after Stalder, Rubesin, and Tendeland [13], M. = 2.4. The Reynolds number xU./v is based on the conditions at the edge of the boundary layer and on the distance from the leading edge. from a leading edge. There is probably also some slight variation of the recovery factor with the Mach number. The measurements of Mack [1/2] show a slight increase of the recovery factor with the Mach number, while those of Stine and Scherrer [1/6] show no variation. The Mach num- ber effect predicted by the theoretical formula of Tucker and Maslen [17], (iy SS EL m = N21 + 0.528? 8N 41-2 N = 2.6Ret is not yet verified by experiments. According to Fig. B,9a and B,9b, the turbulent recovery factors on cones and flat plates are of the same order, ( 94 ) B,9 - VELOCITY, PRESSURE, AND TEMPERATURE DISTRIBUTIONS while the laminar recovery factors on plates are higher than those on cones and other models. These high values of the recovery factor can be attributed to heat conduction effects in the leading edge region of the flat plate. The foregoing relations between temperature and velocity will be referred to in Art. 12 in connection with the relation between the tem- perature profile and the velocity profile. If Pr = 1, the relations become especially simple, as shown by Eq. 9-5 with heat transfer and by Eq. 9-7 without heat transfer. If Pr 4 1, the viscous dissipation and the heat conduction render such a general relationship between temperature and 1.0 0.8 y/6 Fig. B,9c. Velocity profile across turbulent boundary layer. The free stream Mach number is 4.93 and there is no heat transfer. The data are drawn after Lobb, Winkler, and Persh [18] and private communications. velocity very difficult, and a basic approach to the problem must involve the detailed mechanism of turbulence. However, without turning to this approach a relation between the temperature at the wall and the velocity and temperature at y = 6 was made possible by introducing a recovery factor r.. For no heat transfer such a relation was found to be that of Eq. 9-9a, and this agreed rather well with measurements. The question may be asked as to what form the relation might take if it were general- ized to include the heat transfer and to cover all positions in the boundary layer. To this end and by similar reasoning, we could introduce a variable recovery factor r(y) which satisfies Eq. 9-9a at the limit and becomes r(y) = 1 for Pr = 1. It is expected that a relation between 7° and T,, ( 95 ) B - TURBULENT FLOW 1.00 0.99 TO 0.98 Tg 0.96 Fig. B,9d. Total temperature profile across turbulent boundary layer. The free stream Mach number is 4.93 and there is no heat transfer. The experimental data in circles are drawn after Lobb, Winkler, and Persh [18] and private communications. The curve is drawn according to Eq. 9-12, by assuming equal thickness of boundary layers for U and T. could then be formulated predicting that T°/T° increases as (U/U.)? in- creases. With this in view we shall introduce a variable recovery factor r(y) and write by analogy to Eq. 9-7, cp T + 4r(y)U? = cpl. + 3r.U? = cpl (9-11) Here it is assumed that the heat conduction through the wall is absent, i.e. (OT /dy)~ = 0, but Pr may be arbitrary. Eq. 9-11 gives T°(y) in terms of U(y) as follows: IBOLT be YR 1p WIN? Ca ad am with ory) -1 Rint Te — l ( 96 ) B,10 - TRANSPORT OF PROPERTIES IN A TURBULENT FLUID > For a velocity distribution given by Fig. B,9c, the curve in Fig. B,9d shows the distribution of T°/T°® given by Eq. 9-12 when it is assumed as a rough approximation that a = 1. On the same figure are shown the measurements of Lobb, Winkler, and Persh [/8]. Fig. B,9e shows the measurements of van Driest [19] and Spivack [20]. Both figures suggest that T°/T® passes through a maximum value in excess of unity, the indi- cation of this being most pronounced in Fig. B,9e. This phenomenon cannot be explained from the above considerations unless a proper dis- tribution of r(y) is taken into account. y, mm Fig. B,9e. Total temperature profile across turbulent boundary layer at free stream Mach number 2.8. The data are from Spivack [20] with an axial distance from throat of 12.96 inches. B,10. Phenomena of Transport of Properties in a Turbulent Fluid. Up to the present we have dealt with the mean turbulent flow and certain simple relationships between mean quantities. For the latter, analogies were employed rather than procedures based on a mechanism of turbulence. Since the superficial nature of this approach is apparent, it becomes advisable to look into the physical transport processes of tur- bulence which are embodied in transport terms like wv, p’v, Tv, etc., of the fundamental equations obtained in Art. 8. They represent the mean rate of transfer of wu, p’, and JT’ respectively, across a unit area perpen- dicular to y. One of the major aims of turbulent theory is to find a method of calculating these transport terms directly from the hydrodynamic equa- tions governing the turbulent motion (Eq. 4-3, 4-4, and 6-1). At present the difficulties of the theories make such a program not yet possible. ( 97 ) B - TURBULENT FLOW The other approach is to regard the statistical effect of turbulence on the mean flow as being similar to that of molecular viscosity or heat conduction, so that the turbulent transport terms can be treated by the same statistical methods as those applied to transport processes in non- turbulent motions. To this end, and as a basis of the statistical theory of transport processes by molecular motions in a gas, we use the Boltzmann equation = of Olea Of ai Si Ox; We AG Og; Feat (10-1) where 6f/6t is a symbol representing the collision integral, f(t, x; £)dxdé is the number of particles in the space and velocity elements dxdé at the instant ¢; x and = are the vectors of position and velocity, X is the external force per unit mass. The left-hand side of Eq. 10-1 represents the rate of increase in time of the number of particles in the phase element dxdé when we move together with the particles in the phase space x, —. The right-hand side represents the effect of restoring and direct collisions which throw the particles respectively in and out of the phase element. A consequence of the Boltzmann equation is the equation of evolution of a transferable property ®(¢, x) defined by [ ass j = (10-2) ®(i, x) = where ¢(é) is a function of the random velocity ~. As special cases it is interesting to put ¢ = 1, &;, 3, thus obtaining from Eq. 10-1 and 10-2 the general hydrodynamic equations Sy RADA Ee ak tt a OF ar gg OUD) =10) 3 DO a OU; OU; oie oan Vs po (Ui + u, 208) = ax, + oe) Die Sapp ga eo Oe Here the density p, the speed U;, and the internal thermal energy per unit volume J are defined by the mean values p=m { fdt = mn 1 Us=4 | estar - gj i a ( 98 ) B,10 - TRANSPORT OF PROPERTIES IN A TURBULENT FLUID where C; = &; — U; is the thermal velocity and m is mass. The mean values of high powers of C; are oi; = —pC,C; = stress tensor gq: = —3pC,C? = thermal flux I is the internal energy, which is equal to C,T for an ideal gas, and finally Ll (OWs | OGs cE = sy =) Eq. 10-3 express the conservation of mass, momentum, and energy re- spectively. In order to express the quantities o;; and q; in terms of the macroscopic quantities s, U;, and T (or J) we have to investigate further the Boltzmann equation (Eq. 10-1). Because it is nonlinear in character, it can only be solved by approximations. For the detailed calculations, reference may be made to the textbook of Chapman and Cowling [2/]. As a first approximation it is found that Oig = — Dbz + 2ulez — Fenndis) oT : Ox; (10-4) ies where » and k are found as functions of 7 and depend on the collision cross section. With this approximation, the second equation of Eq. 10-3 becomes the Navier-Stokes equation of motion. In particular, for a transport in the y direction of a property which either is a scalar, or has a component in the z direction, Eq. 10-4 reduce to 0U Sz = —pé.Cy = Moy 1 k al Ge) or 2 —— | q P92 C2Cy Cp OY or, in general, the laminar flux of the transport J of a transferable property ®, which is the momentum or temperature in Eq. 10-5, can be written in the following form: Sita = Dies 7 ' (10-6) where D,, is a laminar phenomenological coefficient equal to the coef- ficient of viscosity in the case of transport of momentum, and to the coefficient of heat conduction in the case of the transport of heat. When we deal with turbulent transport, it is necessary to replace the concept of the molecular collisions by the turbulent exchanges between fluid elements. Similarly the thermal velocity C; is replaced by the ve- locity u; of turbulent motions. If the property & is to be transferred by (199) )) B:- TURBULENT FLOW these motions, then it is to be expected, by analogy with Eq. 10-5 and 10-6, that a turbulent flux will result in the form vee WO gad Bas a ais where J is turbulent flux, v is the turbulent velocity in the direction y, and D is turbulent coefficient of transport. In general the transport coef- ficient D may depend upon a number of unknown factors among which are the property to be transferred, and the intensity and scale of the tur- bulent motion. For example, its value may vary according to whether we have a transport of momentum, heat, or matter. A knowledge of its structure necessitates a detailed investigation of the basis of the tur- bulent exchange term by a procedure analogous to that which yielded the exchange coefficients » and k of Eq. 10-4 from the solution of the complete Boltzmann equation including the collision term. It is hoped that some insight into the essential structure of the turbulent transport can be gained by proceeding in this way on a somewhat simplified basis made possible by adopting an approximate form of the Boltzmann equa- tion. When applied to turbulent motion, the right-hand side of Eq. 10-1 represents the effects of the turbulent exchanges on the distribution func- tion. It can be regarded as a forcing term which distorts the distribution from its equilibrium. Therefore we can write Eq. 10-1 approximately as follows: Die iy —K(f ais Tea) (10-7) where f is the nonequilibrium distribution, f., is the equilibrium distribu- tion, and « depends on the efficiency of the turbulent mixing. The idea of writing such a simple relaxation type of exchange term in Eq. 10-7 in the place of the complete collision integral in Eq. 10-1 is not new. Lorentz [22], Van Vleck and Weisskoff [23] had initiated such a simplification in their study of microwave line shapes. Later Bhatnagar, Gross, and Krook [24] applied an essentially similar simplification for studying the collision proc- esses in gases. According to Eq. 10-2 and 10-7, we can write ot = —K(@ — 8) (10-8) Here @ — @ is the fluctuation of the transferable property. Eq. 10-8 can be used to find the evolution of the property & carried by a lump of fluid when the latter moves and mixes with its surroundings. Being given ®, the value of ® at any instant ¢ is given by the integral of Eq. 10-8 as follows: B(t) = x if ° dre-B(t — 1) (10-9a) ( 100 ) B,10 - TRANSPORT OF PROPERTIES IN A TURBULENT FLUID where (tf — 7) is the mean value of the property © when the lump of fluid carrying ® found itself at the instant ¢ — 7. It can be expanded into series as follows: &(t — r) = O(¢) — mee iP dt'v(t’) (10-9b) where the integral term is the displacement of the lump of fluid in the interval of time r. The expansion is valid when the lump of fluid makes only small displacements and when it is assumed that © is stationary, but nonuniform. After substitution for &(¢ — 7), Eq. 10-9a becomes a ss , &(t) = 6 — = ef dre—*"* tes dt’v(t’) The double integral is : a dre" es dt'o(t!) = « i dre-™ i dt'’v(t — t’) =« fo doe — 0) [dre ik ° dtl'e—*"v(t — 8!) Thus ab(t) [| ~* H(t) = Bt) — ah - dt'’e—’v(t — t’’) Hence the flux for the transport of & is: J = —pbv = p—— . eT ae (10-10a) Consequently the turbulent coefficient of transport D is found as follows: D = [> deat = PO) (10-10b) The transport coefficient D in Eq. 10-10 depends on the autocorrelation function of velocities v(¢ — ¢’’)v(¢) and on x. In its turn « depends on a number of factors among which is the property to be transferred. This entails that D may differ according to the nature of properties to be transferred, i.e. heat, momentum, particles, etc. Now we shall compare this result with the concept of mixing length, so often used in the study of the turbulent motion. By analogy with the kinetic theory of gases one may suppose that there is a length J, which represents the distance traveled by the lump of fluid between the instant when it was freed from its surroundings carrying with it the mean property of these surroundings, and the instant when it arrives in a ( 101 ) B - TURBULENT FLOW new layer where it is supposed to mix with the new surrounding fluid. In this case, the transport coefficient is v/, which can be written in the integral form Me if ° atvE — yo) (10-11) if-the correlation of velocities converges. Eq. 10-11, based on the mixing length, does not distinguish between the transport of heat, momentum, and matter, because the same length is intrinsically implied in all cases. As an illustration, Eq. 10-10 may be applied to the special case of transport of momentum and heat along the y direction. We then obtain aU EES, Dus (10-12) oe oT q = —pe,l"v = Bop Di a where 7 is the turbulent shear stress, g is the rate of turbulent transport of heat, and D, and D, are respectively the transport coefficients of momentum and heat defined by Eq. 10-10b. The coefficients are com- monly termed ‘‘turbulent exchange coefficients.”’ The results (Eq. 10-12) can be compared with the Boussinesq formulas of turbulent transport of momentum and heat, written usually in the following form: pepe” = oS i 10-13 ee as aay where ¢, is the eddy viscosity and «, is the eddy heat conductivity, intro- duced by formal analogy to the corresponding laminar viscosity and heat conductivity of the Navier-Stokes and Fourier equations. Eq. 10-13 give neither the structure of the exchange coefficients nor the basis of the transfer. However, they can be made completely identical in form with Kq. 10-12, if the following expressions are assigned to e, and ¢; é, = pD.z ee — pCpDr The ratio Du _ &lp _ Pr, D, €k is called the turbulent Prandtl number by analogy to the laminar Prandtl number introduced in Eq. 6-4b. We see that the mixing length theory { 102 ) B,10 - TRANSPORT OF PROPERTIES IN A TURBULENT FLUID (Eq. 10-11), which implies D, = Dy, predicts a turbulent Prandtl num- ber of unity. However, experiments show that Pr, is about 0.7, a value very close, incidently, to the laminar Prandtl number for air. The fact that the turbulent Prandtl number, as given by the ratio D,/Dn, is different from unity is interesting and indicates that Eq. 10-10b, rather than the mixing length formula (Eq. 10-11), should be more correct. However, due to the simplification introduced in the trans- port equation (Eq. 10-7), the parameter « is not determined in terms of the transferable property, so that the numerical value of the ratio of the two exchange coefficients cannot be computed from Eq. 10-10b alone. An auxiliary equation is needed to determine the transfer of property, for example, heat or particles, under the action of a turbulent fluid. In the case of the transfer of particles, such an equation may govern the motion of a small spherical particle suspended in a turbulent fluid. On the basis of it, the velocity correlation for the particles can be computed in terms of the velocity correlation of the ambient fluid or vice versa, and hence the ratio of the two exchange coefficients can be obtained. This has been done by Tchen [25], and, for the case of « = 0, it has been found that the exchange coefficient of particles is equal to that of the fluid (Eq. 10-11). This case is not surprising, since consistently the relax- ation is neither involved in the motion of the fluid nor in the motion of the particles, and no difference in exchange coefficients should exist, as already revealed by the simple theory of Eq. 10-10b. The ratio of the two exchange coefficients for the case of «x ~ 0 has not yet been studied on this basis. Several authors are concerned with such difficulties of dif- fusion phenomena, see e.g. [25] and the Burgers lecture on the turbulent fluid motion [26]. In the integrand of Eq. 10-10b, the exponential term can be con- sidered as a retarding effect of the relaxation between the equilibrium and nonequilibrium distribution in the transport phenomena (Eq. 10-8). Hence the complete integrand of Eq. 10-10b can be considered as a corre- lation corrected for the relaxation by means of the exponential factor. In the diffusion problem based on the model of a random walk, such an effect has been considered by Tchen [27] in the form of a more general memory, which could be either negative or positive, so that the corrected correlation will contain a factor respectively smaller or larger than unity. Before leaving the discussion, it is important to remark that the dif- fusion phenomena, described by the above transport phenomena, are only valid for irregular movements of small scales, since we have used in Eq. 10-9b a series expansion in terms of a length and some gradient of the transferable property. Such a diffusion can be called diffusion of the gradient type. On the other hand, when the irregular movements are of coarse scales, the bulk property rather than its local gradient must be the governing factor. The latter diffusion can be called diffusion of the ( 103 ) B- TURBULENT FLOW bulk convective type, and will be discussed in Art. 29 in connection with the coarse eddies of free turbulent flow. The structure of the transport coefficients can be determined by means of kinetic equations more general than the Boltzmann equation. This attempt has been made in an article by TFchen published in the Proceedings of the International Symposium on Atmospheric Diffusion (1958). B,11. Reynolds Analogy between Heat Transfer and Skin Fric- tion. As an application of the transport processes treated in Art. 10, let us study the Reynolds analogy between heat transfer and momentum transfer. Let ¢ = —pc,vT” be the rate of turbulent heat transfer in the y direction across the unit area normal to this axis, and 7 = —puwv be the rate of momentum transfer or turbulent shear stress. According to Kq. 10-6 we have a) | — oT = —pc vl’ = PepDn au oy The following expressions written in nondimensional form may be compared: pipet i tl OS aE Rn pee ih ples A pe, (One) ea) pr = Us) Here U. and T. are the velocity and temperature at a reference plane, which, in the present discussion, is taken at the edge of the boundary layer. The Reynolds analogy is a statement of equality of the two ex- pressions of Eq. 11-1. Let us investigate this analogy in some detail. Of special interest are the heat and momentum transfers at the wall, where the two expressions of Eq. 11-1 become (11-1) ts Iw f 20 Eley hea tail : Teh a Bue ratte where the subscript , denotes the value at the wall, St is the coefficient of heat transfer or Stanton number, and c; the coefficient of skin friction. Then the Reynolds analogy leads to St = $c;z (11-3) This result was first obtained by Reynolds [28] and is also given by Squire [7, p. 819] and Goldstein [6, p. 654] in their study of heat transfer. It is easy to see that Eq. 11-3 cannot be valid in general because, in the compressible case of an insulated boundary layer, we must have St = 0 and c; # 0, which obviously violate Eq. 11-3. Therefore it is worthwhile to derive a more general relationship between the heat trans- fer and skin friction. For this purpose we make use of the relation (Eq. St ( 104 ) B,11 - REYNOLDS ANALOGY 9-5) between the temperature and velocity for the case of Pr = 1. By differentiating with respect to y, we obtain ORV ene, ook LOU: (F) oo ag (3). Thus in terms of (@U/dy)~, we can write St and c; as defined by Eq. 11-2 in the following form: Tee aU See a = Pe UZ (2). 1 1 /aU De D te (32), so that Vee 1 2 Sg Sed er = Dy, i, ; o(1+%>4ae) -1 where 7 = T.,/T°, M. is the Mach number of the stream at y = 6, and y is the ratio of specific heats. Eq. 11-4 can be considered as a generali- zation of the Reynolds analogy to include the effects of the heat transfer, compressibility, and the difference in the transport of heat and mo- mentum. As M. > 0, 7 #1, and D, + D, according to Art. 10, the right-hand side of Eq. 11-4 is in general not unity, and the Reynolds relation (Eq. 11-3) is not obtained. However, if we neglect the effect of compressibility, for example at low speeds, and if the transports of heat and momentum are similar (D;, = D.,,), the right-hand side of Eq. 11-4 is approximately unity, and the Reynolds analogy (Eq. 11-3) is then found to be valid. In general those restrictive conditions are not present, and the Reynolds analogy will not hold. For example, in the case of a heated plate (n > 1), the factor between brackets in Eq. 11-4 is smaller than unity, so that in many circumstances we have St < ic, (11-5a) This inequality is verified by experiments. On the other hand, with in- tense cooling of the plate (7 < 1) the term between brackets in Eq. 11-4 may become larger than unity so that we may get St > dey (11-5b) In spite of its defects, the Reynolds analogy (Eq. 11-3) is often used in theories of boundary layers with heat transfer because of its simplicity, and sometimes the experiments show that the analogy is a surprisingly good approximation. Instead of defining the shear stress and heat transfer on the basis of ( 105 ) B - TURBULENT FLOW 10 Common liquids Normal gases Reynolds Fig. B,lla. Analogy theories of turbulent heat transfer; M, = 0, R. = 107 after Chapman and Kester [29]. ® Liepmann and Dhawan ? Cf; @ Chapman and Kester 1.0 oO [33] AIS QO Wilson Cre oa cee © Rubesin, Maydew, and | Cf, YN | 0.9 Varga j pana A Brinich and Diaconis AIA 0.8 OO 07 [33] pee eae Ste Oe. +> Slack sti 4s Monaghan } (Stave and Cooke ) (Stay); Cr, ce; based on Karman-Schoenherr incompressible skin friction (Stav)i, Sti based on Colburn incompressible heat transfer . OW sO Ol8nc RZ a 1h Gay 220 Ae Ore S22 1 St Gren Mach number M, Fig. B,11b. Variation of skin friction and heat transfer coefficients with Mach number, after Pappas [33]. 0.4 { 106 ) B,12 - BASIS OF SKIN FRICTION THEORIES the elementary and fundamental transport phenomena by turbulence, as was done in Eq. 10-12, one could consider a more general shear stress and heat transfer, by defining them by the right-hand side of the boundary layer equations, Eq. 8-1 and 8-4 respectively. Then a new coefficient of skin friction and a new Stanton number are obtained with a rather com- plicated ratio between them. Such an analogy between shear stress and heat transfer is called in the literature ‘‘ Modified Reynolds Analogy”? [8]. While it is already hard to express a satisfactory analogy between two elementary transfers, one wonders sometimes whether an analogy between a complex transfer of different nature (turbulent and laminar combined) could be expected in a reasonably simple form. The results of some of those analogy theories and the effects of the Prandtl number are shown in Fig. B,lla, after Chapman and Kester [29]. The data mentioned in Fig. B,11a are found in [15,30,31,32]. Fig. B,11b, after Pappas [33] shows how the skin friction coefficient and the heat transfer coefficient vary with the Mach number. The data of Fig. B,11b refer to [29,32,33,34,35,36,37,38, 39,40,41,42,43). B,12. Basis of Skin Friction Theories. For the analysis of the stresses acting on a body moving at high speeds, a study of the skin friction in a compressible fluid becomes important. Not only is it neces- sary for drag calculations, but it is useful for estimating heat transfer by means of the Reynolds analogy discussed in Art. 11. Before discussing experimental results and their comparison with theories, it appears desirable to review in a simple and general way the main steps, concepts, and approximations underlying the theories which have been proposed. According to Eq. 8-1, the turbulent shear stress for a compressible flow is 7 = —puv — Up (12-1a) The first term of the right-hand side of Eq. 12-la represents a momentum transfer, and the second a mass transfer. The ratio of the second term to the first is estimated to be proportional to the square of the local Mach number. Now the theories of skin friction assume, as a first approxi- mation, a value of 7 = 7, for Eq. 12-la, where r,, is the total shear stress at the wall. Any variation of 7 through the boundary layer is taken into account only in the higher order of approximations. Since the local Mach number is small near the wall, the second term of the right-hand side of Eq. 12-1a can be neglected, and we obtain T = —puv (12-1b) The next step is to express the fluctuating quantities in terms of the Clog) B- TURBULENT FLOW mean quantities in accordance with the relations een 10-11 and 10-12) as follows: —0U = = pul Oy (12-2a) where, following Prandtl, vl is expressed by = aU pes pps im w= 2 By (12-2b) The mixing length / is now to be expressed in terms of the local mean flow parameters. This can be done [6, Chap. 8] either by means of the Karman similarity hypothesis aU /dy l —S 11S aU Jay? (12-3) or by means of the Prandtl hypothesis l —s KoY (12-4) where x; and x2 are numerical constants. With the Karman hypothesis, Eq. 12-2a can be written as follows: ae (aU /dy)* * (8?U/ay?)? In boundary layer theories the equations are usually rendered di- mensionless by introducing a reference velocity Tw a i 12-6a Pw ( ) pe we Vine (12-6b) where the subscript . denotes the value at the wall. Using the reference velocity and length, as defined by Eq. 12-6, we can write the following dimensionless quantities: pe a pol eel i (12-72) (12-5) and a reference length U ee OS A Ose yee s U Te AaUcNien (12-7b) Se 2s ee 12-7 Meret (12-7c) and rewrite Eq. 12-5 in the dimensionless form as follows: 277 * ap *2)\2 Ky pS (8U* /ay*)4 ( 108 ) B,12 - BASIS OF SKIN FRICTION THEORIES where cy, is the skin friction coefficient at the wall, defined by — Tw in F000? The expression (Eq. 12-7c) is based on the equation of state of a perfect gas and the constancy of pressure across a boundary layer (compare the assumptions underlying Eq. 9-1). In order to formulate the differential equation for U*, it is necessary to express p* in terms of U*. This is possible by using relations between the temperature and velocity, such as those discussed in Art. 9. However, it is more proper to regard the boundary layer as a composite layer, con- sisting of a laminar sublayer very close to the wall with a superposed fully developed turbulent layer. Obviously the computation for such a condition becomes more elaborate, requiring matching of flow conditions at the interface and consideration of the heat transfer through it. But the final result of the temperature-velocity relation turns out to be rather simple and is of the form *-1 = 2 p => T = Ao + ARO al. A,U*? (12-10) w (12-9a) as could be expected from the elementary considerations of Art. 9, al- though the coefficients Ao, A1, and A» are more complicated functions of Pr, cs, T/T», and M.. For the details of the analysis by which these are found the reader is referred to [44,45]. When p* in Eq. 12-8 is replaced by Eq. 12-10 there is obtained an ordinary nonlinear differential equation of second order for U*(y*), with Pr, c;,, T/T, M. as parameters. The integration gives two constants to be determined by two boundary conditions taken at the interface be- tween the laminar sublayer and the turbulent layer. According to experi- mental data for incompressible flow [46], these are Ue = of? = iS * _ = 0.218 (12-11) In principle, Eq. 12-8 and 12-10 with the boundary conditions (Kq. 12-11) can be solved, with the solutions of the following general form: 0 U* = U* (y+ Pr Cras = M.) (12-12) In practice the solution is very elaborate and various numerical and approximate methods must be used. Now we assume that all the parameters in Eq. 12-12 are constant, except cy, which depends on x. Thus after integration of U*(y*) given by { 109 ) B - TURBULENT FLOW Kq. 12-12, according to the formula of momentum thickness, 6 = i aa (1 oA i dy (12-13) the momentum thickness 6(c,,) must be a function of x. Since the solution (Eq. 12-12) is valid only in the turbulent boundary layer and not in the laminar sublayer, some error will be introduced in the integration of Eq. 12-13 by using Eq. 12-12. However, since the laminar sublayer is thin, the error must be very small. We recall that the coefficient of skin friction c,, is defined by Eq. 12-9a in terms of density p,, and that the momentum equation in the integral form, for a flat plate with zero pressure gradient, is T dé Ch, = I.U2 = 2 dae (12-9b) where cy, is the skin friction coefficient referred to p., which will be fre- quently used later on. It is to be noted that c,,/c,, = T./Tw. After inte- gration with respect to z, Eq. 12-9b can be rewritten as follows: T, [°t~ Ab(c;,) 1 r= can a on de; (i+? vat) 7 [. ee Ae, O28 ro dey, Cr, The value of the integrand of Eq. 12-14 is given by the differentiation of Eq. 12-13. In Eq. 12-14, the limits of integration are (~, c,,) for cy, corresponding to (0, x) for xz, because at x = O the boundary is so thin that the velocity gradient and the skin friction become infinite. If we write x in terms of the Reynolds number p,U.2/p., the integration of Eq. 12-14 gives a relation between the skin friction coefficient and the Reynolds number of the following form: / be saeie — Pr ee M.) (12-15) a Further the heat transfer coefficient may be found on the basis of the skin friction coefficient by means of the Reynolds analogy as examined in Art. 11. Instead of using the K4rman similarity hypothesis (Eq. 12-3), which serves as the foundation of the differential equation (Eq. 12-5), we can use the Prandtl hypothesis (Eq. 12-4) so that Eq. 12-2a now becomes GU aa ( 110 ) B,12 - BASIS OF SKIN FRICTION THEORIES Again by introducing the dimensionless quantities (Eq. 12-7), we can re- write Eq. 12-16a in the following dimensionless form: ae ma (12-16b) In Eq. 12-16 the assumption is again made that 7 is constant across the boundary layer with the value 7,. For p*, the expression given by Eq. 12-10 is again used, and all further steps to compute U* and the skin friction coefficient are similar to the treatment given above for the Karman hypothesis. It is remarked that the differential equation (Kq. Re. Fig. B,12a. Comparison of the mixing length and similarity hypotheses of skin friction. Curve 1 illustrates the Falkner law according to Eq. 13-15, curve 2 illus- trates the Karman law (Eq. 12-17), based on the similarity hypothesis, and curve 3 is the Prandtl law (Eq. 12-18) drawn with a coefficient 0.472, based on the mixing length hypothesis. 12-16b) from the Prandtl hypothesis corresponds to the differential equa- tion (Eq. 12-8) from the K4érmdn hypothesis. However, Eq. 12-16b is of the first order and needs only one boundary condition, namely the inter- face condition Of = WLS ere oR Ss Ts It is interesting to compare the effect of the two hypotheses (Eq. 12-3 and 12-4) on the skin friction. For the sake of simplicity and in order to avoid as much as possible other assumptions which may obscure the issue, the comparison is made for skin friction coefficients of incompressible flow. Fig. B,12a shows that the Ka4rmd4n and Prandtl hypotheses do not lead to an appreciable difference in skin friction coefficients. The curves are drawn according to the following formulas: Ci ) B - TURBULENT FLOW Karman hypothesis [47], 0.242 = lore (Casi 12-17 VG £ ( fi é) ( ) Prandtl hypothesis [48], C,, = 0.472(log Re)—?-58 (12-18) As an additional comparison, the power law of Falkner according to Eq. 13-5 with A = 0.0262, n = + is also plotted, and agrees well with the Fig. B,12b. Compressibility effect on skin friction (theories), after Chapman and Kester [29]. Karman and Prandtl hypotheses. The Prandtl law has originally the coef- ficient 0.472 but Schlichting adopts a coefficient 0.455. Eq. 12-17 is some- times called the Kaérmdn-Schoenherr formula, and Eq. 12-18 the Prandtl- Schlichting formula. Although the skin friction coefficient at low speeds does not depend very much on the Kdérmd4n or Prandtl hypotheses, with the application of those hypotheses to high speeds there arise many uncertainties. Now consideration must be given to new exchanges, such as density mixing and heat transfer, and to the variation of fluid properties across the boundary layer. An investigation of the theoretical basis of skin friction as given above will reveal many passages which are uncertain and arbi- ( 112 ) B,13 - EMPIRICAL LAWS OF SKIN FRICTION trary. Therefore it is not surprising that they yield a great number of different predictions of skin friction at high speeds. Fig. B,12b shows that the various theories differ appreciably at large M.. They are based on the recovery factor r. = 1, with the exception of the theory of Wilson which is based on r, = 0.89. The viscosity-temperature exponent a covers the range of a = 0.75 to 1, and the Reynolds number covers the range of Re = 7to 10 X 10°. The discrepancies between the values of skin friction predicted by the various theories increase as the Mach number increases. At a Mach number of 5, the theoretical values of the skin friction differ by a factor greater than 3. Only a small portion of the discrepancies can be attributed to the different values of Re, r., and a used in the various theories. Because of their uncertainties, we shall not enter into the detail of the theories, and the readers who are interested in such details are referred to [19,38,49,50,61 ,52,53,54,55,66,57,58,59,60]. In view of the diffi- culties of such theories of the skin friction coefficient, some empirical formula of skin friction coefficient may often be more useful in practice. These will be treated in Art. 13. B,13. Empirical Laws of Skin Friction. Let us define again the various skin friction coefficients used in the theories and experiments. The local skin friction coefficient is defined in its general form by Bale Qrw 7 ale The wall and free stream values are obtained by writing in Eq. 13-la, respectively, p = pw and p = p.: (13-1a) QT w Cin p U2 (13-1b) cr, = ae (13-1¢) which have already been introduced in Kq. 12-13. The local skin friction coefficient c; is a function of x. Its average value is called the ‘‘average skin friction coefficient” C; 1 Ho Ce = 2 i, C;(a)dx (13-2) 0 or inversely aC Ox Corresponding to c;, and c;,, we can write their average values C,, and C;,. In the dimensionless form x can be replaced by the Reynolds numbers c(x) = Cy(z) + (13-3) ie tt, Ol ee) — ne (13-4) een Baa ee Vey in respective cases. { 113) B - TURBULENT FLOW The empirical laws of the skin friction coefficient for a compressible fluid start from incompressible laws, and the compressibility effects are incorporated by comparison with experiments. The power law cy, = ARe™” (13-5) is an example. Here c;, is the skin friction coefficient for the incompressi- ble boundary layer, Re is defined by Eq. 13-4, A and n are numbers (A = 0.0262, n = 4, according to Falkner [61]). In order to estimate the compressible skin friction coefficient (for example c;,), we assume that a reference temperature 7’, can be found so that the compressible skin fric- tion coefficient c,,, defined by putting p, = p(T;) into Eq. 18-1, satisfies the incompressible formula (Eq. 13-5). Then we can write Cr, = Cy,(Rer) = 0 (Re ie 2 Mr Pe cy, | Re a a (13-6) with p,/pe = T/T, and pe/pr = (T./T;)%. Since cs, follows the power law (Eq. 13-5), Eq. 13-6 can be rewritten in the following form: —(1+a)n Cf, = cy,( Ree) e T Further, cs, can be expressed in terms of c;, by means of the definitions (Eq. 13-1) which can be rewritten as follows: so that Eq. 13-6 becomes cy,( Ree) a cy,(Ree) \T? FARES (13-7a) The right-hand side of Eq. 13-7a gives the effect of compressibility (or M.). In a compressible boundary layer T varies between T. and Ty. It can be assumed that the compressibility effect is covered on the average, if the average temperature fie = a(T. ap Te) (13-8) is taken as the reference temperature. Then 7/27, or T./(T. + Tw) can be computed in terms of M. on the basis of Eq. 9-9, so that finally Eq. 13-7a becomes cy,( Ree) CS B1-(Lta)n x Cr,( Ree) Bias (13-7b) ( 114 ) B,13 - EMPIRICAL LAWS OF SKIN FRICTION where aT ae ( a m2) (13-9) r. is the recovery factor defined by Eq. 9-9b, M. is the free stream Mach number, and y is the ratio of specific heats. 1.0 0.8 0.6} Cfe “4 a 0.2 Fig. B,13. Compressibility effect on skin friction (empirical laws). cs./cy; is the ratio of average skin friction coefficients respectively at free stream Mach numbers M, ¥ 0 and M. = 0. Curve 1 represents the theory of Frankl-Voishel [44,45]. Curves 2 and 3 represent Eq. 13-7b and 13-11, based respectively on the power law and the loga- rithmic law of incompressible skin friction coefficient. The experimental results of Coles [34] are shown in circles for comparison. A viscosity-temperature exponent a = Lisused in plotting curves 1, 2, and 3. Curve 4 is plotted with a = 0.75, according to Eq. 13-11. Instead of selecting the power law (Eq. 13-5) on which to base com- pressibility effect, we may take as an alternative example a logarithmic law of the form (decimal basis): C,,(r) = A(log Re)™ (13-10) Then by the procedures of Eq. 13-6, 13-7, and 13-8 we find the following compressibility effect: C;,( Ree) bid (1 + a) log a | 1 log Re. 6 C';,( Ree) (13-11) where 6 = T./T, is given by Eq. 13-9, when 7, assumes the value given by Eq. 13-8. It is interesting to note that the compressibility effect as given by Kq. 13-7b, on the basis of the power law (Eq. 13-5), is separated from the ( 115 ) B - TURBULENT FLOW Reynolds number effect, while the compressibility effect (Eq. 13-11), on the basis of logarithmic law (Eq. 13-10), includes a Reynolds number effect. The two formulas (Eq. 13-7b and 13-11) are illustrated in Fig. B,13, by taking a = 1, r, = 1. The Falkner constants [67] A = 0.0262, n= have been used in Eq. 13-5 and 13-7b, and the Prandtl constants [48] A = 0.455, = 2198 in Eq. 13-10 and 13-11. It is seen that they are in quite good agreement. Also plotted are the theoretical results of Frankl and Voishel [44,45], originally in tabulated form, and the experimental results of Coles and Goddard [34]. It seems that the empitical formulas (Eq. 13-7b and 13-11) agree rather well with the theory of Frankl-Voishel and with the experi- ments of Coles. Although the experiments of Coles are run with a slightly different Reynolds number (Re. = 8 X 10°) than the Reynolds number of the theoretical curves (Re. = 7 X 10°), the correction for such a dis- crepancy is not significant. The viscosity-temperature coefficient a has the value between 0.75 and 1. Eq. 13-11 is also plotted in Fig. B,14b with C;, vs. Re, to be com- pared with experiments, by taking a = 0.75, n = 2.58. Cy, is based on Eq. 13-10 with A = 0.455. It is seen that the theoretical formula (Eq. 13-11) is in good agreement with experiments. B,14. Comparison between Experiments and Theories. There exists an extensive history of experiments on skin friction. Because of the importance of skin friction to naval architecture, experiments on skin friction were started as early as 1793 by Beaufoy. Schoenherr [47] gives a good review of experiments prior to 1932. In Fig. B,14a are plotted the experimental values of skin friction in compressible flow. The ratio C,,/C;, or c;,/c,, is illustrated. Except for the measurements of Chapman-Kester and Liepmann-Dhawan, wherein the incompressible skin friction values are deduced experimentally, all data points shown are based on the incompressible skin friction formula (Eq. 12-17) of Kirmadn-Schoenherr. There are two methods of determining the skin friction coefficient. Liepmann-Dhawan, Coles, and Chapman-Kester determine the skin friction coefficient by direct force measurements. Others determine it by surveying the boundary layer and then calcu- lating the friction coefficient by the usual momentum method. At the Mach number of 5, the two methods yield a discrepancy of about 5 per cent. Since the empirical theories, as given in Art. 13, do not differ very much according to Fig. B,13, we have plotted the theoretical formula ( 116 ) B,14 - COMPARISON BETWEEN EXPERIMENTS AND THEORIES Brinich and Diaconis Chapman and Kester Coles Hakkinen Hilf Korgeki Lobb, Winkler, and Persh Monaghan and Cooke Rubesin, Maydew, and Varga Wilson HoHomnDe®@es OO Fig. B,14a. Compressibility effect on turbulent skin friction (experimental). The experimental data are plotted according to [18,29,32,34,38,39,42,62,63]. The curve is drawn according to Eq. 13-9, with a viscosity-temperature coefficient of 0.75. 0.81 Chapman and Kester 1.69 Pappas 2.27 Pappas 2.41 O'Donnell 20a Me =:2.5 Rubesin, Maydew, » and Varga 4 2.5 Chapman and Kester A 3.6 Chapman and Kester Oe 106 107 108 Re. Fig. B,14b. Variation of skin friction with Reynolds number. Cy; = average skin friction coefficient. Ree = peU./ue, Reynolds number based on the free stream. The experimental data are drawn from [29,32,33,66] for the free stream Mach number M, = 0.81 — 3.6. The theoretical curves (VM. = 0 — 3.6) are plotted according to Eq. 13-11, for n = 2.58, A = 0.455, a = 0.75. M7 » B - TURBULENT FLOW Schultz-Grunow Fig. B,14c. Variation of skin friction with Reynolds number. c; = local skin friction coefficient, Re. = Reynolds number based on the free stream conditions. The data are drawn from [66,67]. © Brinich and Diaconis @ Coles: ® Hakkinen @ Hill O Korgeki j Lobb, Winkler, and Persh Fig. B,14d. Variation of skin friction with Reynolds number. c; = local skin friction coefficient, Ree = Reynolds number based on the momentum thickness. The experi- mental data are drawn from [18,39,63,64,66] for the free stream Mach number M, = 1.5 eM 9.0. Cates) B,15 - POWER LAWS (Eq. 13-9) with a = 0.75, and n = } based on the power law of incom- pressible skin friction to compare with experiments. It is seen that more experiments at higher Mach numbers are needed in order to understand skin friction better and to formulate better theories. The data of Fig. B,14a are plotted according to [18,29,32,34,38,39,42,62,63,64]. Fig. B,14b, B,14c, and B,14d illustrate the variations of the experi- mental skin friction coefficients c; and Cy; with the Reynolds numbers, based on the free stream conditions and the momentum thickness. It is interesting to see whether the experiments would follow some semi- empirical laws of skin friction. For this purpose, Eq. 13-11 is plotted for n = 2.58, A = 0.455, and a = 0.75 in Fig. B,14b, and we see that Eq. 13-11 is in quite good agreement with experiments. CHAPTER 4. GENERAL TREATMENT OF INCOMPRESSIBLE MEAN FLOW ALONG WALLS B,15. Power Laws. In attempting to deal with turbulent flows con- fined within pipes and channels or bounded on one side by a wall, much attention was given in the older literature to power laws. These were found to be very useful in that they could be made to approximate ob- served mean velocity distributions and to yield resistance laws that were reasonably correct over a limited Reynolds number range. These laws are, of course, purely empirical, but they have not lost their usefulness when one wishes to express the general character of a velocity profile in a pipe or boundary layer, or wishes to make an estimate of skin friction. We should, however, be mindful of their limitations. The power laws stem from the Blasius resistance formula for smooth straight pipes of circular cross section [68]. They were found, however, to be transferable to two-dimensional channels with parallel walls and two-dimensional boundary layers, if the radius of a pipe, the half-width of a channel, and the thickness of a boundary layer were regarded as equivalent dimensions, and if velocities were referred to those at the center or free stream. In all cases the walls are assumed to be smooth. Since the detailed development is available elsewhere [69], only the main steps are given here. Because the condition of incompressibility has been imposed, the physical properties of the fluid are independent of the flow and constant for any set of conditions. Hence we may simply denote the density by p, the viscosity by u, and the kinematic viscosity by ». Since it is not necessary to distinguish among the flows in pipes, channels, and boundary layers in bringing out the elemental aspects of power formulas, the distance from the wall is expressed by y, the velocity at the center or in the free stream by U., and the value of y where the velocity is U. by 6. The only constraint on the flow considered is the ( 119 ) B+ TURBULENT FLOW shear stress at the wall 7,. This means, of course, that the effect of a pressure gradient is neglected, and we must limit ourselves to cases where the pressure is constant or changing so slowly in the stream direction that its effect is minor compared to the effect of 7,. The coefficients in- volved are accordingly: Tw The local friction coefficient age Cy (15-1) Py e The friction velocity a = (Ue (15-2) Assuming that the local friction coefficient depends on the Reynolds number and may be expressed in powers of the Reynolds number, the relation may be written i const (15-3) LUTON Wig where U.6/v is a Reynolds number based on the maximum velocity and the distance from the wall to the point of maximum velocity. It follows from Eq. 15-3 and the definitions (Eq. 15-1 and 15-2) that Usriy U,8\-— U, = const Fl (15-4) where U,6/y is a Reynolds number based on the friction velocity and 6. Kq. 15-3 and 15-4 are both expressions for conditions near the wall. How- ever, it may be argued that a formula similar to Eq. 15-4 may be used _ to express the velocity at any distance from the wall without appreciable error because the main increase in velocity takes place near the wall. Assuming this, the velocity distribution is written oe const (Ue) (15-5) where U is the mean velocity at the distance y from the wall. We assume now that all mean velocity profiles are similar, and accord- ingly that U/U. is a function of y/6. While this assumption is exactly true for laminar flow, it is only an approximation for turbulent flow. The appropriate power-law form of the function is indicated by Eq. 15-5 and is written Ol | Dee Wie “4 (15-6) 1 By taking m = ; it is found that Eq. 15-3 expresses the variation of friction coefficients in pipes over the range 3000 < U.6/» < 70,000. With ( 120 ) B,15 - POWER LAWS the constants also determined from pipe tests, Eq. 15-8, 15-4, 15-5, and 15-6 become c, = 0.0466 ce) (15-3a) CA oe (15-4a) ig 74 Sl (15-5a) Sle Sig =: (4) (15-6a) The foregoing formulas for pipes would not be expected to apply to other cases. However, they do apply to two-dimensional channels and flat plates over a limited range of Reynolds number for particular coef- ficients and exponents. Power-law velocity distributions fit the observed distributions in an over-all way but not in all detail. When applied to the flat plate, Eq. 15-3a and 15-6a may be used to calculate 6 and c; as functions of x and of a Reynolds number based on z, provided the boundary layer begins as a turbulent layer at the leading edge. The loss of momentum flux through any section of the boundary layer is given by i pU(U. — U)dy The momentum thickness 0, which when multiplied by pU? gives this quantity, is accordingly defined by ieee bo U0! U i a || pU(U. —— U)dy = i on (1 ary | dy (15-7) Since 7, alone accounts for the loss of momentum, it follows that dé Ton Pee ie = pU2 = 9 Cf (15-8) With the velocity distribution given by Eq. 15-6a, 6 = 76/72. By substi- tuting this and Eq. 15-3a into Eq. 15-8, and integrating with the bound- ary condition 6 = 0 when x = QO, the result is gals i (15-9) where x is the distance from the leading edge and U.x/v is a Reynolds number based on zx and the velocity of the free stream. From Hq. 15-3a and 15-9 it follows that —% c, = 0.0592 & (15-10) (ei) B - TURBULENT FLOW Again taking 06 = 76/72 and using Eq. 15-9, —} Cy = 0.074 i (15-11) where C; is the mean friction coefficient from the leading edge to the point x. Eq. 15-11 checks the tests on smooth plates for U.x/v up to about 3 X 10%. Power formulas should be regarded as interpolation formulas, useful over a limited range of Reynolds number. For U.6/» over 100,000, Eq. 15-6a agrees better with measurements when the exponent 4 is replaced by 4, and even § when the Reynolds number is sufficiently high. Skin friction formulas may likewise be improved for agreement with measure- ment over a greater range of Reynolds number by adjusting the exponent. For example, as we have seen in Art. 13, Falkner [6/] uses an exponent of —7 instead of —# and gives e; = 0.0262 SS : (15-12) Vv C; = 0.0306 Se (15-13) It must be remembered that the foregoing considerations apply only to smooth walls. Except for Art. 23, where the effect of roughness is con- sidered, and elsewhere where roughness is mentioned, the smooth-wall condition is implied throughout this chapter. B,16. Wall Law and Velocity-Defect Law. Two laws that have gone far toward giving order and meaning to the seemingly confusing and conflicting data on flows bounded or partially bounded by walls are the ‘‘law of the wall”’ attributed to Prandtl (for example [70]) and the “‘velocity-defect law’ introduced by von Kaérm4n [71]. The first pertains to the region close to the wall where the effect of viscosity is directly felt and the second pertains to the bulk of the shear layer where viscous forces become negligible. The law of the wall is based on the logical premise that the tangential stress at the wall 7,, must depend on the velocity U at the distance y from the wall and on the viscosity » and density p. Assuming that the stress at the wall is the only constraint on the flow, we may write F(t, U, y, #, p) = 0 This may be expressed in dimensionless form by UT ee Cry i= (22) vo Vv ( 122 ) B,16 - WALL LAW AND VELOCITY-DEFECT LAW in terms of the characteristic friction velocity U, and the characteristic length v/U,. The functional equation (Eq. 16-1) is the law of the wall. In the laminar sublayer it takes the special form U zs U,y Ole v (16-2) which arises from the circumstance that the sublayer is so thin that r therein is constant and equal to 7,. In Eq. 16-2 the density included in the terms automatically cancels out. The range of y over which Eq. 16-1 is valid must be established by experiment. It might be supposed that the range would be severely limited by pressure gradient effects when these are present, since, as we have seen, the pressure acting across an area of unit width and height y has been neglected. Recent data, to be discussed in Art. 19, show that there re- mains a considerable range over which the law is valid for both rising and falling pressures and that the law is not so restricted as to be useless until conditions of near-separated flow are reached. Thus there is a range, even though possibly short, beyond the laminar sublayer, where the func- tional relation (Eq. 16-1) is universally of the same form. This is true only when there is a laminar sublayer, and therefore true only when the wall is aerodynamically smooth. The argument leading to the velocity-defect law is that the reduction in velocity (U. — U) at distance y is the result of a tangential stress at the wall, independent of how this stress arises but dependent on the dis- tance 6 to which the effect has diffused from the wall. We may then write U. — U = G(U,, y, 8) and in terms of dimensionless ratios We aie U ae y 7 74) Fe This is the velocity-defect law. The law (Hq. 16-3), unlike Eq. 16-1, holds true for rough as well as smooth walls, provided the roughness elements are not so large that y becomes indeterminate. Data for boundary layers with constant pressure are found to fall on a single curve within the precision fixed by the experimental scatter. Thistis shown by Fig. B,16 which presents various data collected by Clauser [72] for different Reynolds numbers and for smooth and rough walls. Aside from the fact that the law cannot apply in the vicinity of the laminar sublayer nor at distances comparable to the height of roughness elements, it appears to exhibit a universality for constant pressure boundary layers. Clauser has shown, however, by a formal argument that the law is fundamentally not universal when U, varies from one set of data to another, but that the dispersion will gener- (1 B - TURBULENT FLOW ally be small and scarcely outside the usual random scattering due to observational errors. The function g is affected to degrees that are far from negligible by conditions imposed on the flow from without. The effect of the pressure gradient, which will be considered in Art. 19, is of most importance. It is also affected by free stream turbulence and is therefore different in pipes and channels than in boundary layers. This sensitivity of the velocity- defect law to outer conditions stands in sharp contrast to the law of the wall which is remarkably insensitive in this respect. y/6 | Fig. B,16. Data for smooth and rough walls plotted on basis of velocity-defect law. (Taken from Clauser [72] omitting data source and designation.) B,17. Logarithmic Formulas. From time to time it has been inferred in the literature that the two laws, Eq. 16-1 and 16-3, are em- pirical laws, and in the sense that their adoption has depended on experi- mental confirmation, they are empirical. Certainly they draw but little on any knowledge of turbulent structure. About their only connection with the behavior of turbulence is the justification of the assumption that transfer processes are affected by viscosity very near a wall, but are inde- pendent of viscosity and dependent on the scale of the shear layer in the bulk of the flow. The principal empirical fact about these laws is that their regions of validity overlap one another. There is nothing in their makeup that requires an overlap, and the only apparent reason for it is a gradual change from wall conditions to outer-flow conditions. Millikan [73] has shown that if there is any region of overlap, no matter how limited, in which both laws are valid, then the functions f and g must be logarithms. Since this is the same form which results from mixing length considerations, but which is arrived at without re- ( 124 ) B,17 - LOGARITHMIC FORMULAS course to a physical model, the authors feel that this deduction must be ranked among the major contributions to the subject. A simple way of arriving at this result is to reexamine Eq. 16-1 and 16-3, written in the following forms: Oe y\( U,6 U; =i Y ) mie OEE ty, es UF (1) oe Since these are two expressions for the same quantity, and since a multi- plying factor inside a function must have the same effect as an additive factor outside a function, the functions f and g must be logarithms. The two formulas are usually written in the form Wan 225 Uy ee = Fie log & ) ae Gi (17-3) Um Cee eae ey Tone = C2 Te log 5 (17-4) where K, ci, and cz are experimentally determined constants. It follows from Eq. 17-1 and 17-2, when f and g are expressed as logarithms, that K must be common to both Eq. 17-3 and 17-4. The constant K is uni- versal and the logarithmic form of the functions do fit the observations, but only over a limited range of the variables. More specifically they have the logarithmic form where they overlap, but not necessarily much beyond this region. This may be taken as evidence that the empirically established overlap is not a basic condition and therefore not a sufficiently strong one to impel a long range validity for the laws deduced from it. The extent to which these laws fit the data and are influenced by various conditions will be taken up in Art. 19, 20, and 21. For the present we direct our attention to Eq. 17-4 in order to call attention to the fact that the constant c. is found to be the same for pipes and channels, but that it has a different value for boundary layers of flat plates. This is shown in Fig. B,17a in which pipe data have been omitted. The data are taken from [67,74,75,76,77]. The Reynolds number Re; is in all cases U,6/v. The constant 5.75, corresponding to K = 0.40, is common to both, provided the curves are fitted near the wall. It is seen that the log law does not fit well for the full range of y/6. This means only that the logarithmic form of the defect law is at fault, not the functional form of the law itself. More significant is the fact that the function g in Eq. 16-3 is different in boundary layers from that in chan- nels. This difference is evidently due mostly to a sensitivity to conditions at the outer limit y = 6 rather than to the presence of a small falling ( 125 ) B - TURBULENT FLOW Res = 27,000, Schultz-Grunow [74] Res = 101,000, Freeman [75] Res = 48,000, Klebanoff-Diehl [76] Res = 152,000, Klebanoff-Diehl [76] Res = 61,600, Laufer [77] Res = 95,000, Donch [67] 0.01 0.02 0.04 0.06 0.10 0.2 0.40 0.60 a y/6 Fig. B,17a. Logarithmic law, comparison for channel and boundary layer of flat plate. Boundary between Thickness & x layer and free stream Fig. B,17b. Schematic diagram of boundary layer. ( 126 ) B,18 - INCOMPRESSIBLE SKIN FRICTION LAWS pressure in the case of a channel, and a constant pressure in the case of a boundary layer. The boundary layer bounded by a free stream of negligible turbulence is known to have a sharp but very irregular outer limit. This is illustrated schematically in Fig. B,17b. The phenomenon is common to all turbu- lent shear flows which are limited only by the extent to which they have diffused into nonturbulent fluid. There is no such limit for fully developed turbulent flow in pipes and channels where turbulent motions may freely cross the center. To a limited extent a similar condition can be produced in boundary layer flows by introducing turbulence into the free stream by means of a grid. It has been noted that the profiles then deviate toward those for the pipe and channel. B,18. Smooth Wall Incompressible Skin Friction Laws. So far our laws have been so general that pipes and channels on the one hand and boundary layers on the other could be treated as one subject. We may continue in this vein in expressing the general form of the skin fric- tion law, but shortly it will be necessary to make a distinction. Since skin friction depends on conditions near the wall, Eq. 17-3 and 17-4 are used to derive a formula for skin friction, as was first done by von Karman [71]. If these equations are added, the result is Wo BB U,6 Wee = 3 los »(4 ) + const (18-1) By using Eq. 15-1 and 15-2 and introducing Re; = U.6/», Eq. 18-1 becomes 2 = 2.3 log (Res +/ cy) + const (18-2) Cy K Eq. 18-2 has been verified by a number of reliable measurements in pipes. With the constants for pipe flow as given by von Karmdén [78], Eq. 18-2 becomes 1 Vy where fe; is based on the velocity at the center U. and the radius of the pipe. The constant 4.15 corresponds to K = 0.39, this value having been chosen to give the best all-around agreement. The Karman skin friction formula for flat plates [78,79] results from conversion of Eq. 18-3 into terms involving x, where zx is the distance from the leading edge and the assumed beginning of the turbulent bound- ary layer. It is expressed as = 4.15 log (Res ~/c;) + 3.60 (18-3) Pa a8) te me log (Rec;) + const (18-4) (127 ) B+: TURBULENT FLOW where Re = U.x/v and c; is again the local friction coefficient defined by Kq. 15-1. With the constants evaluated from Kempf’s measurements on a flat plate [80], Eq. 18-4 becomes 1 —— = 4,15 log (Rec;) + 1.7 18-5 Ve; sakes Schoenherr [47] found the coefficient of mean friction over the dis- tance x to be given by Dees a iRee)) (12-17) VC; and the relation between the local and the mean friction coefficients to be uf 0.558C; 0.558 + 2+/C; Eq. 12-17 is one of the most widely used formulas for incompressible flow and, as previously mentioned in Art. 12 and 14, is sometimes called the Karman-Schoenherr formula. As reported by Prandtl [48], Schlichting proposed an interpolation formula of the form Cy (18-6) C; = 0.455 (log Re)—?-58 (12-18) The comparison between Eq. 12-17 and 12-18 is shown in Fig. B,12a. The corresponding interpolation formula for cs, also given by Schlichting [80], is c; = (2 log Re — 0.65)-*:3 (18-7) Schultz-Grunow [74] adopted the Prandtl law with constants as follows: c; = 0.370 (log Re)—?-584 (18-8) While the foregoing formulas are expressed in the form usually de- sired for engineering purposes, they suffer from the drawback that the boundary layer is often laminar for a significant distance before transition occurs. In such cases formulas based on Re cannot be applied without assuming some fictitious origin for x. A formula like Eq. 18-3, based on the local parameter Re;, does not involve this difficulty. Because of the indefiniteness of the outer limit of the boundary layer, the momentum thickness @ is commonly used in place of 6, and Res = U.6/» takes the place of Res. Squire and Young [8/] obtained from Eq. 18-4 the approxi- mate relation —— = Alog Rey + B (18-9) Vcr with the constants A and B chosen to give the best agreement with Eq. ( 128 ) B,19 - EFFECT OF PRESSURE GRADIENT 12-18. Their final expression, written in the form most commonly used, is 2 = [5.890 log (4.075Rea)| (18-10) f On the experimental side, measurements using the floating-element technique, wherein the shear stress on an element of the wall is deter- mined from a direct force measurement, are now believed to be the most reliable. The best known examples of results employing this technique Fig. B,18. Local skin friction coefficient for smooth wall, zero pressure gradient. Experimental values represented by points. Curve 1: Eq. 18-5; curve 2: Eq. 18-8; curve 3: Eq. 15-10. are those of Kempf [82] and the more recent results of Schultz-Grunow [74] and Dhawan [37]. These are given in Fig. B,18. Represented for comparison are the curves corresponding to the power formula (Eq. 15-10) and the logarithmic formulas (Eq. 18-5 and 18-8). B,19. Effect of Pressure Gradient. When a body moves through a fluid, the pressure in the neighborhood of the body is different from that in the undisturbed fluid in ways that are too well known to be recounted here. It suffices merely to point out that pressure gradients are the rule rather than the exception. The present discussion will be limited to two- ( 129 ) B - TURBULENT FLOW dimensional flow where pressure gradients in the x and y directions are encountered. Boundary layers are usually so thin compared to the relatively large distances over which pressure changes occur that the changes across the layer are so small that they have insignificant effects. The pressure may change even more gradually in the z direction, but here the boundary layer extends over the full range of the pressure changes, and cumulative effects become important. The thickness of the layer is always affected, and the mean velocity profile will change form as the flow progresses unless conditions are so arranged that it is held in equilibrium by the balance between inertial, pressure, and friction forces. Pipe and channel flows are examples of equilibrium flows in which the pressure drop is exactly balanced by wall friction. As shown by Clauser [83], a balance is possible in boundary layer flows under certain conditions, and his con- tributions to this subject will be taken up in Art. 20. In general, mean velocity distributions undergo progressive changes when subjected to pressure gradients—the less so when the flow proceeds toward lower pressures, and the more so when the flow proceeds toward higher pres- sures. The latter therefore deserves, and usually receives, the greater attention. The importance of flow to higher pressures is emphasized by the possibility, and often the occurrence, of flow separation. Separation is the result of flow reversal and an accumulation of stagnant fluid over which the moving fluid passes without having to follow the contour of a body. An adverse pressure gradient opposes motion in the direction of the main flow and can set up motion in the reverse direction when the fluid has lost sufficient momentum through friction with a wall. Since the momentum approaches zero at a wall, only the shear stresses between the faster- and slower-moving fluids can prevent flow reversal. Whether or not reversal will occur depends on an interplay between the shear stresses and the pressure gradient. In any case the fluid movement is retarded, and shear stresses are expended against internal forces on the fluid arising from the pressure gradient. The maximum shear stress is no longer at the wall, as it is for constant pressure, but now occurs some fraction of the boundary layer thickness away from the wall depending on the state of retardation of the layer. These effects reduce skin friction and the momentum losses from this source, but only in exchange for even greater internal momentum losses resulting from shear stresses applied to pressure-retarded flow. A classic example of the typical evolution of velocity profiles occurring when a boundary layer is subjected to a monotonically increasing pres- sure sufficient to bring about eventual flow separation is the set of curves compiled by von Doenhoff and Tetervin [84] shown in Fig. B,19a. Here U, is the local free stream velocity just outside the boundary layer, and { 130 ) B,19 - EFFECT OF PRESSURE GRADIENT 6 is the momentum thickness. Each curve of the set is characterized by 4 constant value of the form parameter H, where H = 6*/@ and in accord- ance with the usual definitions 6 6* = displacement thickness = if (1 — v) dy 0 e é t 6 = momentum thickness = ih UF (1 _ ) dy o U. U,. On the grounds that all suitable boundary layer data available up to 1943 could be made to fit one or another of these curves, von Doenhoff and [= i i a 2 = ag oe) On CODMNUAMAN bodIvyv4arn NO Ol AN 98 Se > le} ©) [Ea Ww NO i y/0 Fig. B,19a. Velocity profiles corresponding to various values of H, after von Doenhoff and Tetervin [84]. Tetervin concluded that H was a suitable form parameter. When the pressure increases with increasing x and the gradient is sufficient to bring about eventual separation, H steadily increases, and each successive pro- file takes a shape similar to one of those in the figure. Separation is imminent when H is above 2 and is likely to occur when H is 2.6 or 2.7. Apparently this family characteristic is only true of nonequilibrium pro- files, for Clauser has recently shown that equilibrium profiles, for which H remains nearly constant with increasing xz, do not align themselves with the typical forms of this family (see Art. 20). However, the general features are preserved. There has been a great deal of speculation about the abrupt rise of the curves of Fig. B,19a near the origin. It will be noted that even when the flow is about to separate the steep initial rise is present. Thanks to the recent contributions of Coles [85], to be considered in Art. 21, and to facts (181) B - TURBULENT FLOW pointed out earlier by Clauser about the law of the wall, this feature now has a simple explanation. If we regard the phenomenon as a sharp drop in velocity to zero at the wall instead of a rise from the wall outward, we see that this is simply the region where wall friction becomes predominant over the pressure effect. In other words, this is the region governed by the law of the wall. Typical of the agreement with the law of the wall and of the manner of departing from it are the examples shown in Fig. B,19b taken from Coles’ paper [85]. When the Reynolds number is high and the pressure is either constant or the adverse gradients are not exces- sive, the agreement is more as shown in Fig. B,19c given by Clauser [83]. 50 40 30 U U, 20 10 0 1 10 100 1000 10,000 100,000 U.y/v Fig. B,19b. Agreement and departures from the law of the wall, after Coles [84]. The region of the wall is a region for which we have a unique relation- ship between the velocity and the shear stress at the wall. Sometimes, slightly different working formulas evolve from the fitting to experimental data. We find, for example: According to Clauser = 5.6 log i + 4.9 (19-1) Uy Vv According to Coles ela Sle = 5.75 log ( ) +. 5.10) sWyGlgea) It is difficult to specify where departures from the law occur, because this depends both on the Reynolds number and the pressure gradient. Departures occur at lower values of U,y/v and are greater as the effect of ( 132 ) B19: EFFECT OF PRESSURE GRADIENT the adverse pressure gradient on the profile becomes more marked, i.e. as H is greater. They also occur at lower values of U,y/v as the Reynolds number decreases. Landweber [86] has shown that the logarithmic part no longer exists if U.6*/v is less than 725. At the wall side the law begins to merge into Eq. 16-2 somewhere around U,y/v = 50. The outer limit of the laminar sublayer is usually taken as 11.5, representing the point where the curve of Eq. 16-2 and the logarithmic law intersect. For a number of years there was considerable uncertainty about the effect of the adverse pressure gradient on the skin friction, and most methods of treating turbulent boundary layers assumed that the gradient had no effect. Estimates by means of the momentum equation were un- reliable and in some cases showed an apparent increase in the skin friction 40 O Ludwieg and Tillman O Klebanoff-Diehl w Freeman O Schultz-Grunow 500 2000 10,000 Fig. B,19c. Test of the law of the wall, after Clauser [83]. coefficient in regions of strongly rising pressure. When data based on more direct methods became available, such as those of Schubauer and Klebanoff [87] and Newman [88], based on hot wire measurements of shear stress, and those of Ludwieg and Tillmann [89], based on the heated- element method, it became clearly evident that c; was decreased by an adverse pressure gradient, and was steadily reduced toward zero as sepa- ration was approached, as logic dictates that it should be. The whole question has been considerably clarified by the universal character of the law of the wall which establishes a unique relation between velocity near the wall and skin friction without explicitly involving the pressure gradi- ent. The effect of the pressure gradient on the skin friction is thereby seen to result from its reduction of velocity near the wall. The relation between the integral characteristics of a two-dimensional boundary layer and the pressure gradient is obtained by integrating the equation of motion from y = 0 to y = 6. The commonly used form, (183) ) B - TURBULENT FLOW obtained from the equation of motion with only first order terms in the boundary layer approximation, is known as the Ké4rman momentum equa- tion, and is expressed as dé dz LGpranelldel 2 2) 6dp piubora ne age (19-3) where q is the dynamic pressure in the free stream where the pressure is p. This equation gives a synoptic description of boundary layer development and is independent of detailed processes. The relation between the various quantities in the equation does, however, depend on the mechanics of the turbulent diffusion process. When the pressure gradient is positive (adverse) and large, the second term on the right-hand side of Eq. 19-3 may, and usually does, become large compared to c;/2. For this condition the growth of @ with x depends primarily on internal momentum losses resulting from the expenditure of tangential forces against those portions of the stream which are retarded by pressure gradient and which, by the action of the force, progress to higher pressures but do not gain momentum equivalent to the forces ex- pended. When a boundary layer exists, a pressure rise can be negotiated only by the loss of momentum. A reduction of c; by pressure gradient is not an indication that drag is reduced. When d6/dzx in Kq. 19-3 is due largely to the pressure gradient term, it is obvious that cy cannot be accurately determined from measurements of dé/dzx. It is now generally recognized that Eq. 19-3 is unsuited for this purpose when pressure gradients assume appreciable values. Not only is the accuracy poor but totally unrealistic values of c; have been indicated. Several explanations have been offered having to do with the neglected terms in the equation of motion, but it now appears in the light of Clauser’s experience [83] that departures of the flow from two-dimen- sionality are largely responsible. The universal character of the law of the wall has suggested itself as a useful and reliable means of obtaining local skin friction coefficients from measured velocity distributions. It seems that the first published recognition of this occurs in the paper by Clauser [83], who devised the following procedure and used it in the analysis of his experimental results. Using U, = U. ~/c;/2, the following expressions are written: OMRON Pr Uy ay AZ UNG ae a oN With these and Eq. 19-1 he obtained the family of curves shown in Fig. B,19d having cy; as the parameter. Application of the figure to a determi- nation of cs merely requires the placing of a measured velocity distribu- tion thereon and reading off the value of c;, interpolating where necessary. It is still necessary to measure velocities within a short distance of a, ( 134 ) B,20 - EQUILIBRIUM BOUNDARY LAYERS wall, but the requirement of nearness is considerably relaxed over that required to derive c; from the initial slope of a velocity distribution. Ludwieg and Tillmann [89], who first confirmed the validity of the law of the wall in a region of adverse pressure gradient by means of their heated-element measurements of c;, deduced the following formula for Cf: 0.246 a 109-8787 Pe ,0-268 Cf (19-4) where H is the form parameter and Rep = U.6/v. This formula gives c; reasonably well where the velocity profiles conform to the H-parameter family of Fig. B,19a. 0) O Typical set of experimental points 0 == 10 102 103 104 105 Fig. B,19d. Chart for experimental determination of turbulent skin friction coefficient, after Clauser [83]. B,20. Equilibrium Boundary Layers According to Clauser. Since the velocity profile beyond the immediate region of the wall is affected by the pressure gradient, a universal representation on the basis of the velocity-defect law, as shown by Fig. B,16, is not in general ob- tained. However, by means of an experiment in which long lengths of two-dimensional turbulent boundary layer could be subjected to various adverse pressure gradients, Clauser [83] showed that the pressure dis- tribution could be adjusted to give similar boundary layer profiles when plotted on the basis of the defect law. The form of the function was different from that for constant pressure flow and also different for each separate pressure distribution, but the significant fact was that the same functional relation applied over an essentially arbitrary number of cross sections for any one pressure distribution. He termed the resulting bound- ary layer an ‘“‘equilibrium boundary layer” on the grounds that the same- ness of the function g in the case of a pressure gradient implied the same similarity of major flow characteristics as was maintained in constant ( 135 ) B - TURBULENT FLOW pressure flow. Constant pressure flows are then just one member of a family of flows developed under specific kinds of pressure distributions. With regard to the kind of pressure distribution required to produce an equilibrium flow, Clauser points out that a gradient parameter like (5’/t,.)dp/dx, where 6’ represents some effective face area over which the pressure acts, represents the ratio of forces acting on the layer; and if this is held constant, the flow should have a constant history and therefore be in equilibrium. The choice of the proper quantity, 6’, was not known when the experiments were performed, and the attainment of equilibrium Fig. B,20a. Equilibrium boundary layer profiles on the basis of the velocity-defect law, after Clauser [72]. conditions proceeded on a cut-and-try basis. In a later article [72] Clauser concluded that the proper parameter was (6*/7,,)dp/dx. Studies were con- ducted for two pressure distributions, designated as pressure distribu- tion 1, corresponding to a mild adverse gradient, and pressure distribu- tion 2, corresponding to a considerably stronger adverse gradient but not sufficient to cause separation. The resulting mean velocity profiles are shown in Fig. B,20a compared to a constant pressure profile. Due to the uncertainty in defining 6, Clauser sought a more suitable thickness parameter. Obviously it was required that this be proportional to 6, since equilibrium profiles correlate on the basis of y/6. The customary 5* and @ were not suitable because their ratio to 6 could be shown to ( 136 ) B,20 - EQUILIBRIUM BOUNDARY LAYERS depend on c;. Similarly the customary shape parameter H was found to be unsuited to equilibrium profiles. He therefore adopted as the thickness parameter “U. -—U and as integral shape parameter ea UU \ y ae We Caray enuf tWiay aU Non fey oo | ea, oa | real (20-2) Their relations 6*, 0, and H are s* = ff A (20-3) 6 = ne (1 NG WZ ny) (20-4) 1 H (20-5) nacre) The reader is referred to the original paper for a more detailed dis- cussion of these parameters. The logarithmic plot of the data in terms of 0 — a0) 0.001 0.01 0.10 y/A Fig. B,20b. Logarithmic plot of equilibrium velocity profiles using the Clauser thickness parameter A, after Clauser [83]. y/A is given here in Fig. B,20b. It is seen that near the wall the defect law conforms to the logarithmic law, as it must according to the arguments of Art. 17 if it overlaps the region in which the law of the wall is valid. This would be true, however, whether equilibrium existed or not, but (387) B - TURBULENT FLOW without equilibrium a family of curves instead of a single curve would be obtained for any one pressure distribution. The parameters for these curves are: G A/é Constant pressure 6.1 36 Pressure distribution 1 Oa! 6.4 Pressure distribution 2 19.3 12.0 We shall return to Fig. B,20b in Art. 23 in connection with the universal skin friction law proposed by Clauser for equilibrium flows. Another interesting fact brought out by Clauser’s investigation is that equilibrium profiles do not conform to the H-parameter family of profiles shown in Fig. B,19a. Comparisons at two values of H are shown in Fig. B,20c. It will be seen that nonequilibrium profiles are considerably more von Doenhoff and Tetervin Fig. B,20c. Comparison of equilibrium profiles and von Doenhoff- Tetervin one-parameter profiles, after Clauser [83]. rounded than equilibrium profiles. Furthermore H remained nearly con- stant with downstream distance for equilibrium profiles, whereas H in- creases progressively for nonequilibrium profiles. This suggests that the increase in 6* is slower and therefore that mixing is more thorough when equilibrium exists. This may merely mean that the imposed changes from section to section are now slow enough for the turbulent mixing to better keep pace. It may also mean that the mixing rates are higher for equi- librium than for nonequilibrium flow. In this connection information on turbulent structure is needed. A start in this direction was made by Ruetenik and Corrsin [90] who investigated equilibrium turbulent flow in a channel with a 1-degree half angle of divergence. Even for this small divergence, the average turbulent energy was found to be greater than that for a parallel channel by a factor of about 3. However, what is still needed is information of this sort to compare equilibrium and non- equilibrium flows. ( 138 ) B,21 - LAW OF THE WAKE ACCORDING TO COLES The reader is referred to Clauser’s paper [83] for a number of signifi- cant facts brought to light in his investigation. One of these concerned the downstream instability of a turbulent boundary layer with a large adverse pressure gradient. When the pressure gradient was small, no diffi- culty was experienced in adjusting the pressure distribution to obtain a desired equilibrium profile; but when it was large, great difficulty was experienced. He attributes the condition for large pressure gradients to a downstream instability, meaning that a change, say in the local gradient or in 0, made at one point would produce further changes downstream as the layer developed, rather than become damped out. This is an insta- bility in x, not in time. B,21. Law of the Wake According to Coles. In the short space of this article it is impossible to cover adequately the careful and extensive study which led Coles [85] to propose the law of the wake as an extension to the law of the wall. After having examined practically all available experimental data on turbulent boundary layers in terms of the loga- rithmic form of the law of the wall, expressed by Eq. 19-2, and noting the universal agreement with the law near the wall and the characteristic departure from it away from the wall, he concluded that the flow had a wakelike character, modified in various degrees by wall constraints. He concluded further that the wakelike form could be reduced to a second universal similarity law which he called the ‘‘law of the wake.”’ A linear combination with the law of the wall was then proposed as an over-all similarity law representing the complete profile for equilibrium and non- equilibrium flows alike. Attempts to generalize the law of the wall and the defect law so as to fit experimental results are not new. Millikan [73], for example, proposed forms to fit the distribution in pipes and channels. Others have expressed and employed ideas bearing certain similarities to the present one, those known being Lees and Crocco [91], Ross and Robertson [93], and Rotta [93]. Coles, however, appears to have been the first to show evidence of a universal wake law and to give it a rational physical explanation. In general form the mean velocity profile in turbulent shear flow may be expressed as a =f a + h(z, y) (21-1) fe Vv For equilibrium flows it is found experimentally that Eq. 21-1 may be written U Uy) y 2p CF J ( v ) ! (« ) & : where m is a parameter which is independent of x and y for a specific ( 139 ) B - TURBULENT FLOW situation and pressure distribution. The defect law is correspondingly expressed as OG Se y aaiCpaaiE F (« “) (21-3) T Coles concluded from his survey of existing data that the central problem was not so much a study of the defect function F as a study of the original function g(z, y/6) which gives the departure of the mean ve- locity profile from the logarithmic law of the wall. Since the characteristic departure was obviously not confined to equilibrium flows, the mean- velocity profile was expressed in the form E-()eP() as where K is a constant, r(x) denotes that 7 is now in general a function of x, and w(y/6) is a universal wake function common to all two-dimen- sional turbulent boundary layer flows. The term mx) fy ORG (“4 in Eq. 21-4 gives the departure from the logarithmic law of the wall, i.e. from Ue Uy Tk = K in( ) + Cc where, according to Coles, K = 0.4 and c = 5.10 (Kq. 19-2). From an analysis of experimental data, Coles found the form of w(y/6) as given in Fig. B,21a, in which w(y/6) has been subjected to the normal- izing conditions w(0) = 0, w(1) = 2, and f2(y/ 5)dw = 1. When plotted against y/6 these curves have a nearly symmetrical S shape; and, due to the normalization, have the maximum value of 2 at y/6 = 1. The curves obtained from Clauser’s equilibrium profiles and the one obtained from Wieghardt’s data, which Coles finds to be also an equilibrium flow, are plotted against the parameter yU,/(6*U.), which is equal to y/A in Clauser’s notation. Included in this set are data from nonequilibrium profiles and the data of Liepmann and Laufer [94] for a region of turbu- lent mixing between a uniform flow and a fluid at rest. The general working form of Eq. 21-4 may be written U 1 U, 4 U = hin( H) +0 +74 (4) (21-5) where the constants K and c have the numerical values as given above. In order to use this formula, r(x) must be known. It follows from Eq. ( 140 ) B,21 - LAW OF THE WAKE ACCORDING TO COLES 21-5, using the normalizing condition w(1) = 2, that OA Wohi yee Ua Ke 27 (x) K (21-6) )tet Vv Bassa 2 OF i Cf Thus Eq. 21-6 is an expression for z(x) in terms of the skin friction coefficient c;. For other relationships and a tabulation of w(y/5) and related functions the reader is referred to Coles’ original paper [85]. where 2 ® 1 y/6 Fig. B,21a. The law of the wake, after Coles [85]. Coles found that in most cases (Eq. 21-5) fitted available experimental data on velocity distributions well and concluded, for unseparated flows at least, that the wake hypothesis appeared to be a useful concept. The analytic character of the method enabled him to express also the distribu- tion of shear stress across the boundary layer. Computed distributions represented observations, except where the adverse pressure gradients were large. Here there were large discrepancies, reminiscent of those ob- tained by using the momentum equation. The general success of the method led Coles to suggest that yawed or three-dimensional flows might be usefully represented by universal func- tions considered as vector rather than scalar quantities. For further dis- cussions along these lines the reader is again referred to the original paper. (141 ) B - TURBULENT FLOW It is, of course, not uncommon to find empirical formulas with enough adjustable constants to fit experimental results. In the present case, how- ever, the formula, with a specified function w(y/6) and constants previ- ously specified in the law of the wall, stands the test of a wide variety of conditions. In addition the present similarity law appears to be based on meaningful physical concepts, which may be described as follows. It is easily seen that a wake is a natural consequence of earlier fric- tional constraints no matter how they may have arisen. There is therefore coming from upstream a flow of wakelike character, modified obviously by the remnant of upstream effects which caused it and by the local effects which distort the profile so that the velocity approaches zero at the wall. The remarkable thing was that Coles could extract S-shaped profiles typical of the pure wake component. Fig. B,21b. Mean velocity profiles of hypothetical boundary layer, after Coles [85]. No claim is made that the turbulent structure is the same as that of a real wake. From the limited information available it appears that wake structure is coarser (has larger eddies) than boundary layer structure. However, the law of the wake may be interpreted as a manifestation of a large scale mixing process in which stress-controlling motions are inde- pendent of viscous effects. The wall effect, as we already know, superposes a viscous effect which increases in magnitude as the wall is approached. The concept is best illustrated by the diagram of a hypothetical boundary layer used by Coles, reproduced here in Fig. B,21b. The figure shows velocity profiles for various values of x in a flow proceeding from separation to separation through a region of attached flow. The dashed lines denote the wakelike component represented by the function w(y/6). At points of separation or reattachment we find the wake component only. In regions of attachment we see the effect of the wall friction, and the requirement of vanishing velocity at the wall being met by a sharp drop to zero at the wall. ( 142 ) B,22 - MIXING LENGTH AND EDDY VISCOSITY B,22. Mixing Length and Eddy Viscosity in Boundary Layer Flows. As mentioned in Art. 3 and 10, the transport of momentum by turbulent motions may be regarded as involving an eddy viscosity. We shall briefly reexamine the associated concepts in the light of certain known facts about the flow in various parts of the boundary layer. In turbulent boundary layers three fairly distinct regions are easily recognized. First there is the laminar sublayer which is typically 0.01 to 0.001 of the total thickness of the layer. Beyond this is a turbulent region which extends to 0.1 to 0.2 of 6 and comprises the inner part of the layer where the logarithmic law is valid and the mean flow is virtually un- affected by pressure gradient. A short time response and rapid adjust- ment to local conditions are also characteristic of this region (see dis- cussion by Clauser [72]). Finally, there is the outer 0.8 to 0.9 of the layer where the eddies are limited in lateral extent only by the confines of the layer and mixing is relatively free. In the laminar sublayer molecular diffusion predominates, being exclusively this at the wall. Turbulent dif- fusion progressively increases as we enter the logarithmic region from the wall side and soon predominates over molecular diffusion. For virtually everything except the laminar sublayer the transfer processes should be governed by a property of the motion. We wish to see whether this property may be legitimately and usefully expressed in terms of an eddy viscosity, €,. Dimensionally, ¢«, is a product of density, velocity, and length. Ac- cording to the mixing length theory ey = wll (22-1) where v is the y component of turbulent velocity and / is the reach of a turbulent motion while it has the velocity v and is called the mixing length. Prandtl’s assumption is that v = IdU/dy and 1 = coy (see Art. 10). It is implied in this assumption that the correlation between v and / is absorbed into the value of J. Using these assumptions and assuming further that 7 is independent of y and equal to 7,, the value at the wall, we find the well-known expression or using U? = +,,/p U, = coy —— (22-2) This expression may be integrated to give the velocity distribution if we know the lower limits of y and U. These are their values at the edge of the sublayer, which may be found from Eq. 16-2 and written in terms of ( 143 ) B- TURBULENT FLOW a free constant: y, = cv/U, and U; = cU,. If we now integrate Eq. 22-2 as follows: y U | -@ Alyn Cody, Y Orin: we obtain exactly the law of the wall Webs In Uy + const U, C2 Vv where cz has the same numerical value as K. If we assume at the outset, as Prandtl did also, that v = U, and again take 1 = coy, we again obtain dU we = (6 yane 2y dy This is identical to Eq. 22-2 and again yields exactly the law of the wall. It is time to examine the consequences of these results. Since the law of the wall is well founded and is one of the most universal features of turbulent flow, we cannot escape the conclusion that the above assump- tions are valid for the region in which the logarithmic law is obeyed. We know, of course, that we must stay near the wall, if for no other reason than that 7 changes with y. More specifically we may express the eddy viscosity = = cyU, (22-3) in the region where the logarithmic law of the wall is valid. Turning our attention to the outer 80 to 90 per cent of the layer, we find that both Townsend [/] and Clauser [72] have explored the possi- bility that «, is constant in this region. Townsend employed the rather straightforward procedure of solving the boundary layer approximation of the equation of mean motion, considering both constant pressure flow and equilibrium flow with pressure gradient. We call attention here only to his treatment of the constant pressure case. When the constants in- volving ¢, were chosen for the best fit of experimental results, fair agree- ment was found for y/é > 0.05. The principal defect was the usual one, namely that a constant e, yielded too slow an approach to the free stream velocity. Evidently «, effectively decreases near the outer edge, due no doubt to intermittency of turbulent flow. The extent and quality of the over-all agreement was, however, sufficiently good to show that an essen- tially constant and valid e, is a physical reality in the turbulent parts of the flow beyond the logarithmic region. Clauser employed the novel approach of making laminar profiles re- semble the outer portion of the constant pressure turbulent profile when the laminar profiles were reduced to the basis of (U — U.)/U, vs. y/6. ( 144 ) B,22 - MIXING LENGTH AND EDDY VISCOSITY He noted that the principal difference in appearance between constant pressure laminar profiles and turbulent profiles was that the turbulent profiles dropped so abruptly at the wall as to appear to extrapolate to a nonzero velocity at the wall, whereas laminar profiles went to zero much more gradually and did not give this impression. The characteristic shape of the turbulent profile arises from the circumstance that the laminar sublayer next to the wall and the flow adjacent to it has a lower viscosity than the eddy viscosity prevailing in the main body of the turbulent flow. Consequently a large part of the velocity change from the wall to the _ free stream occurs in this low viscosity region. If the same situation were made to prevail in a laminar layer, say by placing a layer of fluid of lower viscosity next to the wall, a laminar profile could be made to re- semble a turbulent profile. Clauser therefore proceeded to simulate this condition in a family of laminar profiles obtained by solving the Blasius equation for slip velocities U, at the wall, U./U. amounting to 0, 0.2, 0.4, 0.5, 0.6, 0.7, and 0.8. He then attempted to collapse the family to a single curve by dividing (U — U.)/U. and y/6 by suitable factors. Leaving details to the original paper [72], we merely point out the sig- nificant fact that exact coincidence proved to be impossible, but that two procedures each resulted in a narrow band of curves. Clauser concluded that the same basic dissimilarity would prevent turbulent profiles, which pertain to different values of U,, from collapsing to a single curve on the basis of the velocity-defect law. Accordingly there is an almost-but-not- quite universal curve. The next step was to relate the laminar profiles to turbulent profiles on a velocity-defect-law basis by an appropriate eddy viscosity, ¢,. The appropriate velocity and length were chosen by the same reasoning proc- ess that leads to a reference velocity U, and a reference length 6 in the velocity-defect law, and e, was expressed by €u —=alU,A p where a@ is a constant of proportionality to be determined. Since A is equal to U.6*/U, (see Art. 20) = = aU.6* (22-4) which is an expression for e, in readily available quantities. The original article must be consulted for the details of the fitting process and the curves showing comparisons with data of Fig. B,16. Best agreement was obtained with a = 0.018. Considering that a narrow band of laminar curves is obtained rather than a single curve and that experi- mental data are expected to show a similar dispersion, the agreement is excellent for the outer 80 to 90 per cent of the layer. The method pro- ( 145 ) B - TURBULENT FLOW posed for connecting the outer and inner portions is left to the original article. A treatment of the same character was applied to equilibrium flows involving adverse pressure gradients. Again a good fit was obtained by assuming a constant eddy viscosity given by Eq. 22-4 even for near-sepa- ration profiles. Some of the more significant results of this work were: (1) that (6*/7,,)dp/dx proved to be the proper pressure gradient param- eter which must be constant throughout an equilibrium layer, (2) that a turned out to be practically independent of pressure gradient (inde- pendent of the parameter (6*/7,,.)dp/dzx) and to have the value of approxi- mately 0.018 in all cases tested. An interesting outcome of a constant a is a constant eddy Reynolds number. If such a Reynolds number is defined by Oi: inert | we find from Eq. 22-4 that Re. = 1/a. Taking a = 0.018, Re. = 56..A constant eddy Reynolds number is just another way of expressing the behavior trend of all turbulent shear flows, namely a tendency for the transferring agents to be proportional to the length and velocity scales of the flow. Most important of all is the evidence from these sources that e, be- haves in equilibrium flows toward mean-velocity distributions beyond the range of the logarithmic law as though it were constant. This cannot be taken as a sweeping generalization, but it furnishes good evidence that e, is likely to have a strong leaning in this direction generally and there- fore will have only a weak dependence on local conditions. This being so, there is little foundation for a mixing length theory in such regions, and it renders of little significance the various arguments about how mixing length should be expressed. The degree to which «, is constant and the exactness with which a gradient type of diffusion is obeyed for coarse mixing are probably not sufficient to represent more sensitive quantities like shear stress distributions. Near the wall the mixing length theory may be applied, and we see that a valid procedure starts with an expression for e, that has a striking resemblance to that for the outer flow. The comparison is: Re, Inner flow Fe = cyU,; c2= 0.4 Outer flow a =aU,A; a= 0.018 In the first case the mixing scale is proportional to the distance from the wall; in the second case it is proportional to the thickness of the shear layer. ( 146 ) B,23 - EFFECT OF ROUGHNESS The foregoing considerations regarding a constant eddy viscosity are given more for the physical ideas that they embody than for any possible expediency in methods of computation. B,23. Effect of Roughness. The treatment of roughness and its effects is rendered difficult and somewhat inexact by the varied geometri- cal forms of roughness and the variety of ways in which it may be dis- tributed. Again we are confronted with a subject that cannot be treated adequately in a short space, and the reader can profit by consulting addi- tional sources of information, such as [95,78,6,96,97]. The pattern of roughness studies was set largely by the extensive work of Nikuradse [95] on sand-grain roughness in tubes. Sand-grain roughness has been adopted as a standard in skin friction studies, and is taken to mean roughness elements consisting of grains, either being sand or like grains of sand, of nearly uniform size but generally of irregular shape spread with maximum density on a plain surface. The significant dimen- sions then reduce to one, this one being the mean height of the roughness element, denoted by k. It is customary to express the effect of an arbi- trary type of roughness in terms of an equivalent sand-grain roughness. For example, the effect of a given distribution of rivets of height k, is re- duced to the effect of equivalent sand roughness of height k. A number of such equivalents are given by Schlichting [96]. It has been found that the onset of an effect of sand-grain roughness on skin friction and on the flow near the wall depends on k relative to the thickness of the laminar sublayer. A more precise length, avoiding the arbitrariness of the sublayer thickness, is v/U,. Using this, the criterion becomes a roughness Reynolds number Usk Vv It has been found that below some value of this number roughness has no effect. The surface is then said to be aerodynamically smooth. Above this value an effect sets in, at first as a mixture of smooth-wall and rough- wall behaviors, involving both the roughness and viscous effects. When U,k/v reaches a sufficiently large value, the behavior is characteristic of the roughness only, becoming independent of viscosity. The final con- dition is termed “fully rough.”” When the final condition is reached, the laminar sublayer no longer exists since the particles themselves induce turbulent mixing by the flow about them. Broadly speaking, the foregoing is true of all types of roughness but the limits are different for different types. We shall shortly return to these limits and the importance of the parameter U,k/v, but first we turn our attention to the fully rough con- dition where viscosity no longer enters explicitly into the picture. Here Cie B - TURBULENT FLOW tT» depends on the velocity U at some small distance y from the wall and on k& and p. By dimensional reasoning similar to that leading to Eq. 16-1 we find U Yy As we have already noted, the velocity-defect law is unaffected by rough- ness. Since it again develops that there exists a region of overlap where both laws are valid, a logarithmic function is indicated in Eq. 23-1, and the law may be written re = zn @ + const (23-2) where K is the same as that appearing in the smooth wall law and in the velocity-defect law. Just as in the case of the smooth wall law there is a linear relation- ship between U/U, and In (y/k) only for the region of the wall, not throughout the whole boundary layer. Obviously there is some question about a suitable reference point from which to measure y. If y is not expressed correctly, the region that should be linear becomes curved. Experimentally this is used to find the origin of y. No cases are known where the origin did not lie somewhere between the top and bottom of the roughness elements. The well-known skin friction law for fully rough walls is obtained by adding Eq. 23-2 and the defect law (Eq. 17-4) and using the relationship U./U, = V/2/c;. The result is NE = Zin (7) + const (23-3) Since the defect law is affected by the pressure gradient, Eq. 23-3 applies only to cases where the effect of the pressure gradient is negligible. The effect of the free stream conditions is also present, but this effect is small and may be absorbed in the constant. The effect of roughness is seen to depend on its height compared to the boundary layer thickness. The effect is independent of Reynolds number. These two circumstances illustrate in a very direct way an in- herent characteristic of turbulent diffusion in shear flow, namely that the length scale in eddy diffusion processes tends to remain proportional to the thickness of the shear layer. In other words, mixing tends to take place on a scale of coarseness proportional to the boundary layer thick- ness, or the radius of a pipe. Ordinarily this rule cannot hold true in the immediate neighborhood of a wall where the turbulent motions are influ- enced by the presence of the wall; but if flow about roughness elements introduces a scale of mixing proportional to the scale of the shear layer, ( 148 ) B,23 - EFFECT OF ROUGHNESS then the rule does hold true for the entire layer. This is true when k is proportional to 6, and at the same time U,k/v is sufficiently large to make viscous effects negligible. If we fully grasp the foregoing facts, it does not seem so strange that a small quantity like k should be associated with a much larger quantity like 6 and furthermore occupy a position of equal importance. An important characteristic of the roughness effect, first pointed out by Nikuradse [95], is a downward shift of the velocity near the wall from that corresponding to the smooth wall condition at a given value of U,. This is understandable in view of the fact that the mixing action of the roughness elements increases the rate of momentum transfer, and a lower velocity near the wall is required to keep U, the same. In connection with this downward shift it is necessary to recall that we now have two wall laws: In ee) + const ns K howl y Hag In @) + const Both are dependent on conditions near the wall and both are independent of stream conditions, such as boundary layer thickness and pressure gradi- ent. If we subtract the second equation from the first and call the differ- ence AU/U,, the downward shift in velocity is found to be Smooth wall Fully rough wall We ak Ae In & + const (23-4) This equation applied only for values of U,k/v for which the surface is fully rough. The behavior of AU/U, over a wide range of values of U,k/v has been determined by a number of investigators. A representative summary of results given by Clauser [72] is reproduced in Fig. B,23a. This figure is very instructive. It shows the behavior of different kinds of roughness through the range smooth, partially rough, and fully rough conditions. The limits of such ranges can be judged from this figure. Where the roughness elements are of uniform size, as for example uniform sand, the limit below which the wall is smooth is reasonably definite. It appears to be U,k/v = 4. However, when the roughness consists of a mixture of sizes or is not densely packed and a fictitious k is chosen to bring the curves into coincidence in the fully rough regime, then the lower limit cannot be specified. The lower limit for the fully rough condition is seen to be somewhere between 50 and 100. It is interesting to interpret these limits in terms of k/ 61am, Where diam 18 the thickness of the laminar sublayer on a smooth wall. The sublayer is ( 149 ) B - TURBULENT FLOW inherently an indistinctly defined region, but taking the conventionally defined sharp limit given by Olam = 11.5 we the effect of roughness begins when k/6,,., = % and the fully rough regime sets in when k/6,m is between 4 and 8. These figures tell us little that could not be inferred, namely that the roughness elements must be well Colebrook-White © 48% smooth, 47% fine grains, 5% large grains O 95% uniform sand, 5% large grains @ 97.5% uniform sand, 2.5% large grains A 95% smooth, 5% large grains A Uniform sand 20 ----- Prandtl-Schlichting sand-grain roughness 102 U,k/v Fig. B,23a. Effect of roughness on universal turbulent velocity profile, after Clauser [72]. 108 104 buried in the laminar sublayer to have no effect and must extend well above it to completely eradicate viscosity effects. It may be shown rather simply that in order for a surface to remain aerodynamically smooth the roughness must decrease almost inversely with the free stream velocity. If the critical value is designated as k,, and the limit is taken as U,k,,/v = 4, then v v 2 i 4% =4(7) 2 where cy is the smooth wall coefficient which varies with U, but only slowly. It is also apparent from the slow variation of cy that the require- ments on k,, are nearly as stringent on a large body as on a small one. Returning to Fig. B,23a it is significant that the data conform to the ( 150 ) B,23 - EFFECT OF ROUGHNESS law (Eq. 23-4) for the fully rough condition. This means that the linear portion of the velocity distribution curve for a rough wall parallels that for a smooth wall but is stepped down by an amount AU/U,. With experi- mentally determined values of AU/U,, the velocity distribution for a fully rough wall may be expressed by the aid of the smooth wall formula. For this we use Eq. 19-1 containing the constants given by Clauser. The rough wall formula is then Was Ua (NG Ue = 56 tog (2) (2) +49 (23-5) A skin friction formula results at once by subtracting Eq. 23-5 from the logarithmic form of the velocity-defect law. Clauser [83] has obtained a universal law applicable to equilibrium flows including the effect of the pressure gradient by noting, on the basis of Fig. B,20b, that a pressure gradient also has the effect of producing a step-down in the velocity, AU,2/U,. Accordingly he writes the generalized defect law for equilibrium flows ‘) + 0.6 (23-6) Since Eq. 23-5 is unaffected by the pressure gradient, and Eq. 23-6 takes the effect of the pressure gradient into account, a universal skin friction law results by subtraction of Eq. 23-6 from Eq. 23-5. The end result may be written Ae = 5.6 log Res — Se (Re JA) +45 ets = a) + 4.3 (23-7) if T where +/¢;/2 = U,/U., 6* = ~V/c;/2 A, Re, = U.k/v, Ress = U.6*/v, and (AU/U,) (Rez ~/c;/2) and (AU2/U,)(G) denote functions of the argu- ments. The integral shape parameter G is defined in Art. 20. In order to put Eq. 23-7 into a more convenient form for engineering applications, Clauser proposes the introduction of two auxiliary factors Fy = 1040/5.607, Fe = 10402/5.6U; which permit Eq. 23-7 to be written ae = 5.6 log (Re =) + 4.3 (23-8) f Factors F; and F. have been determined by Clauser using Prandtl- Schlichting data for sand-grain roughness for the calculation of Ff; and his own data for equilibrium profiles for the calculation of F». These are presented in Fig. B,23b and B,23c. A plot of Eq. 23-8 for F; and F2 equal to unity is given in Fig. B,23d. If a fictitious Reynolds number, Re;"F2/F,, is first obtained, c; may be found from this figure. Since values of F2 are based on only two equilibrium pressure distributions, more data are to ( 151 ) B - TURBULENT FLOW 1000 100 Fy 10 1 U,k/v Fig. B,23b. Factor for effect of sand-grain roughness on local skin friction coefficient, after Clauser [83]. 1000 100 Fo 10 Fig. B,23c. Factor for effect of pressure gradient on local skin friction coefficient, after Clauser [83]. be desired in order to test the universality of the method. The term ‘“universal’’ is here used in the restricted sense of applying only to equi- librium boundary layers. The effect of roughness on velocity distribution is reflected in a raising of the shape parameter H. This effect has been shown by Hama [97] for a wide range of conditions. Since Clauser’s integral shape parameter G is ( 152) ) B,24 - INTEGRAL METHODS not affected, the variation of H may be expressed as a function of c; by Eq. 20-5 for both smooth and rough walls. It is worth noting before we leave the subject that experimental de- terminations of roughness effect in terms of AU/U, vs. U,k/v may be made optionally in boundary layers, pipes, or channels. Application of Res* Fig. B,23d. Local skin friction coefficient for smooth plates with constant pressure, after Clauser [83]. the results then merely requires the introduction of AU/U, into the appropriate smooth wall formula. B,24. Integral Methods for Calculating Boundary Layer Develop- ment. A number of methods have been proposed for calculating boundary layer parameters and separation as functions of x for boundary layers developed on a smooth wall in the presence of pressure gradients, Most of the attention has been given to cases involving adverse pressure gradients, and the methods are mostly restricted to two-dimensional flow, although sometimes the problem is set up so as to include axially sym- metric flow for the conditions where the boundary layer is thin compared to the radius of the body about its axis. It is generally assumed that the boundary layer is so thin that pres- sure changes across it may be neglected. Then the equations of motion and continuity for two-dimensional flow reduce to Eq. 8-1, 8-2, and 8-3. For incompressible flow, and by neglecting viscous stress and turbulent normal stresses, these become aU aU _ _lép , lar Uae Fae han (24-1) aU , aU we Alay Te (24-2) B - TURBULENT FLOW By integrating Eq. 24-1 from y = 0 toy = 6, using Eq. 24-2 to elimi- nate V, and Bernoulli’s equation to express p in terms of the local free stream velocity U., the Karman integral relation is obtained. The inte- grals turn out to be the well-known expressions for 6* and 6. By intro- ducing these and their ratio H, the Karman momentum equation is ob- tained. It may be written as follows by again using Bernoulli’s equation to restore p: oe NE Sea ee (24-3) 2q where g = pU?. Kq. 24-3 is the starting point for most known methods. These pro- ceed on the basis of some empirically determined form parameter for the velocity profile. The earlier methods such as those of Buri [98] and Gruschwitz [99] seem now to be mainly of historical interest. Gruschwitz’s method and his shape parameter, U 2 Seve q (a). found considerable use, but both have now been largely replaced by the method of von Doenhoff and Tetervin [84], or variations of it, employing HT only. Von Doenhoff and Tetervin [84] made what appears to be the most thorough search for a suitable form parameter. This resulted in the adoption of the parameter H and the single parameter family of profiles shown in Fig. B,19a. It is now clear from evidence previously cited that all profiles do not fit this pattern, and that any method based on such an assumption cannot be expected to give correct results under all con- ditions. Nevertheless the method of von Doenhoff and Tetervin has had certain successes and has appeared sufficiently promising to lead others to attempt to improve upon it. The method is based on the assumption that it is only necessary to determine @ and H in order to establish the boundary layer character- istics. Since the momentum equation (Eq. 24-3) alone is not sufficient for this purpose, an auxiliary expression for H was set up. Recognizing that a sudden change in pressure should not produce a discontinuity in the velocity profile, it was assumed that the rate of change of H rather than H itself would depend on local forces, 7, and dp/dz. When the ratio of these forces was expressed by 0 dq 29 q dx Ty it was found that 6dH/dx was a function of this ratio and also, to some extent, of H itself, but it was independent of Reynolds number. Using the Squire and Young formula (Eq. 18-10) for 7,, thereby ignoring any { 154 ) B,24 - INTEGRAL METHODS effect of pressure gradient on skin friction, von Doenhoff and Tetervin arrived at the following expression for ¢dH /dz: pu GI) | - EAC 2.035(H — 1.286) | (24-4) dx LT Given dq/dz, the two equations (Eq. 24-4 and 24-3) were solved by a step-by-step procedure for @ and H as a function of x. Starting with some initial value, 0). and Ho, d6/dz and dH /dzx were found. Each when multiplied by an increment of x and added to the initial values gave the next value of 6 and H to repeat the process. Garner [100] undertook to improve on the method of von Doenhoff and Tetervin by using different auxiliary expressions for skin friction and H, again disregarding the effect of pressure gradient on skin friction. The method, however, remains basically the same. Tests of this general method have shown a closeness of agreement with observations sufficient to make it worthy of consideration when con- ditions are not out of the ordinary; that is, when profiles can be expected to have the form of Fig. B,19a. Since adverse pressure gradient domi- nates the development of the layer, the use of an incorrect expression for the skin friction apparently has minor consequences. Tetervin and Lin [/0/] initiated a fresh attack on the problem, again built around the H-parameter family. They set up integral expressions for momentum, moment of momentum, and kinetic energy in a form suf- ficiently general to include axially symmetric flow as well as two-dimen- sional flow, subject to the restriction that 6 is small compared to the radius of curvature about the axis of symmetry. Their principal objective was to avoid an empirical expression for H if possible. The moment of momentum equation was found to be best suited for this purpose, but it required auxiliary expressions for velocity and shear stress distributions across the layer. A power-law fitting of the H-parameter profiles was adopted as an approximate but reasonable procedure. More serious was insufficient information about the value and distribution of shear stress. While the work of Tetervin and Lin fell short of immediate success, it pointed the way to future progress. It must be remembered that while 7,, may be reduced to small values by an adverse pressure gradient, 7 may rise to large values away from the wall before falling to zero at the outer edge of the layer. Fediaevsky [102] proposed a method for calculating the distribution of 7/7, with y/6 employing a polynomial expression that would satisfy boundary condi- tions at the wall and the outer edge of the layer. Certain large discrepan- cies were observed between shear stress distributions calculated by this method and those directly measured by the hot wire method by Schu- bauer and Klebanoff [87]. Ross and Robertson [103] modified the Fediaev- sky method and obtained some improvement in accuracy. ( 155 ) B - TURBULENT FLOW Two contributions following the general method proposed by Tetervin and Lin are those of Granville [104] and Rubert and Persh [740]. Gran- ville’s work suggested that the difficulty in using the moment of momen- tum equation for H might be overcome. By examining a limited amount of experimental data he showed that the integral of the shearing stress across the layer in terms of y/6* was the same in adverse pressure gradi- ents as in constant pressure flow. Rubert and Persh chose the kinetic energy equation for the determination of H and hence had to evaluate the integral of the dissipation across the layer. This they did empirically using experimental data for a variety of conditions. They alsoincluded the Reynolds normal stress in the momentum equation. Values of 6 and H calculated by Rubert and Persh showed reasonably close agreement with experiment for two-dimensional boundary layers and flow in dif- fusers. Both of these methods draw on the work of Ludwieg and Tillmann [89] for the shearing stress at the wall and the existence of the law of the wall in an adverse pressure gradient. Two methods based on dividing the treatment between the inner part of the boundary layer and the outer part are those of Ross [141] and Spence [142]. Each uses a separate similarity for the inner and outer parts. Both use the law of the wall for the region next to the wall. Ross adopts a 3-power velocity-deficiency expression for the outer region with a new parameter D, thus avoiding the use of the shape parameter H. Spence retains the H-parameter for the outer region, but evaluates it by means of an expression for the velocity at the distance 6 from the wall, obtained from the equation of motion formulated for the distance y = @. The several methods here mentioned show that progress is being made on this difficult problem. In some cases more tests are needed to judge the amount of progress. There is general agreement that more information is needed on the behavior patterns of turbulent flow before a universally valid method can come within reach. B,25. Three-Dimensional Effects. It may seem that undue atten- tion is given to two-dimensional mean flows when in their totality all flows are three-dimensional. The justification for the convenience of avoiding the complications introduced by a third dimension is that motion in the third dimension is in many cases locally absent or so insignificant that two-dimensionality is an acceptable assumption. This fortunate circumstance comes about because boundary layers are usually thin compared to the expanse and radius of curvature of a wall. Obviously there are many cases where the edges are too close to the region in question or the boundary layers are too thick for three-dimen- sional effects to be ignored even under local inspection. Common exam- ples are flow in noncircular pipes, flow near wing tips, and flow near the ( 156 ) B,25 - THREE-DIMENSIONAL EFFECTS juncture between a wing and a body. Attention has already been called to the fact that three-dimensional effects are hard to avoid in regions of adverse pressure gradient. They become very pronounced in regions of flow separation. On low aspect ratio wings at large angles of attack, separation often manifests itself as a curving of the flow in a continuous fashion to form the large scale trailing vortices. Important as these cases are, we shall regard them as special problems beyond the scope of the present treatise. , Some mention will be made of a particular three-dimensionality known | as yawed flow. This is the condition where the leading edge of a two- dimensional body is at an angle other than normal to the mean flow, such as might be represented by an infinitely long swept wing. In such cases deviations from the mean flow direction occur in the boundary layer. Among the first quantitative measurements to show the effects on swept wings are those of Kuethe, McKee, and Curry [108]. In the case of laminar yawed flow it is well known, and readily shown by the equations of motion, that the boundary layer development with distance normal to the leading edge and the velocity components associ- ated with this direction are independent of yaw. In other words, bound- ary layer thickness and velocity profiles, based on the stream component normal to the leading edge are independent of the flow parallel to the leading edge. This is known as the “‘independence principle.” According to the best evidence at hand, the independence principle does not apply in turbulent flow. The experiments of Ashkenas and Rid- dell [106] conducted on yawed flat plates show that the thickness of the turbulent boundary layer at a given streamwise distance from the lead- ing edge increases with the angle of yaw. A 1-inch strip of sandpaper glued to the surface near the leading edge made turbulent flow a cer- tainty from that point on and gave an essentially fixed virtual origin for the boundary layer. In terms of distance é from the virtual origin parallel to the free stream direction, the displacement thickness 6* was found to be given by wo, DUE ae ~ (cos #)& \ pv where @ is the yaw angle. Except for the factor (cos @)*, this is the ordi- nary expression for 6* in terms of wall length traversed by the flow. According to Ashkenas and Riddell, yawing would have the effect of decreasing 6* at a given streamwise distance if the independence princi- ple were to apply. The arguments leading to this conclusion are left to the original paper. The above result is in disagreement with that of Young and Booth [107] who concluded that the independence principle does apply in the Cols 7) B - TURBULENT FLOW turbulent boundary layer. Ashkenas and Riddell have noted this dis- agreement and have pointed out possible causes of error in the experi- ments of Young and Booth. Even without putting this case to actual test, it may be seen that the independence principle would not be expected to apply in turbulent flow. Let us imagine a wind tunnel experiment in which we have a flat belt passing through slots in the tunnel walls and running diametrically across the stream with the stream crossing it edgewise. If the boundary layers on the two sides of the belt are laminar, running the belt has no effect on the boundary layer associated with the action of the stream, unless of course the belt is running so fast that heating effects change the viscosity and density of the air. If, on the other hand, the boundary layers are turbulent, then running the belt increases the turbulence be- cause of the greater velocity relative to the surface. The eddy viscosity is thereby increased, and this increase affects all motions. To the flow component normal to the leading edge, the boundary layer now exhib- its greater eddy viscosity. The friction to air flowing over the belt is thereby increased and the thickness of the boundary layer is increased correspondingly. CHAPTER 5. FREE TURBULENT FLOWS B,26. Types and General Features. The term “free turbulent flows”’ refers to flows which are free of confining walls and exist in shear motion relative to a surrounding fluid with which they mix freely. The flows of common technical interest are jets, wakes, and mixing zones between two uniform streams moving with different relative velocity. Problems of technical interest are the rate of spreading with distance from a source of the flow, velocity distributions, and the manner in which other transported quantities such as heat and matter are distributed and mixed with a surrounding medium. A characteristic common to this class of flows is a lack of viscous con- straints on the mean motion in all parts of the field when the Reynolds number is sufficiently high. This condition is practically always fulfilled unless the Reynolds number is so low that the turbulent regime cannot exist at all. In the case of mixing zones, jets, and two-dimensional wakes this condition never degenerates; for no matter how feeble the relative mo- tion may become with increasing distance from the source, the Reynolds number either remains constant or increases due to the increase in size. More specifically the Reynolds number increases with distance for mix- ing zones and two-dimensional jets, and remains constant for axially symmetric jets and two-dimensional wakes. The axially symmetric wake is the one exception, for here the rate of decay of mean motion (and ( 158 ) B,27 -: LAWS OF MEAN SPREADING AND DECAY turbulence) exceeds the growth in diameter, and the Reynolds number tends toward an eventual zero value. We are therefore dealing with a class of flows in which the effects of viscosity are removed from those turbulent motions which control the mean motion and are relegated to the small scale eddies which take part in the final decay and the production of heat. In this respect the flow fields are subject to a controlling mechanism similar to that found in the outer regions of a turbulent boundary layer, but lacking the influence of a wall such as prevails to varying degrees in the boundary layer. Once the flows have attained a fully developed state, they remain similar through- out upon subsequent development, merely changing scales of length and intensive properties. The fully developed state is an asymptotic condition reached only at some distance from a body in the case of a wake and from a nozzle in the case of a jet. Since the initial conditions in these two cases are vastly different, the distance for their effect to disappear is also different. Behind a body the flow is highly agitated by a succession of eddies comparable to the diameter of the body, and this coarse scale motion persists for a long distance. Townsend [1/08], in his investigation of the plane wake behind a cylinder, finds that the mean wake flow reaches similarity only after 100 cylinder diameters downstream, and that complete statistical equi- librium in the turbulent motions is not reached short of 1000 diameters. At a nozzle the initial jet consists of a potential core of relatively smooth flow, or a flow characteristic of the internal flow, bounded by a layer in which free mixing begins. Kuethe [109] finds that the potential core of a round jet is consumed between 4 and 5 nozzle diameters downstream of the plane of the nozzle, and that fully developed jet flow is established at 8 nozzle diameters. We shall here be concerned mainly with fully developed character- istics and shall attempt to describe the principal ones, paying most atten- tion to the plane wake (two-dimensional) and the round jet (axially symmetric) since these have been investigated in the most detail. Since little information is available on the wake of a self-propelled body, this case will not be considered. A discussion of its laws of spreading and decay may be found in [110]. B,27. Laws of Mean Spreading and Decay. A certain amount of useful information can be gathered from the equations of mean motion without requiring their actual solution. Using the condition that momen- tum, heat, and matter are conserved and that the flow when fully devel- oped preserves similarity among mean motions and those turbulent mo- tions which influence the mean motion, it is possible to obtain the laws of spreading and decay of mean properties. The conventional procedure, which will be followed here, is to assume ( 159 ) B - TURBULENT FLOW constant density. If heat is added or is generated by friction, or if another gas is added, it is assumed that the amounts are too small to affect the dynamical problem. The type of problem considered is that of fully devel- oped rectilinear flow such as applies to the jet in a stationary surrounding medium and a wake at sufficient distance from a body. The Reynolds equations in simple form become acceptable approxi- mations under the conditions that (1) the viscous stresses may be neg- lected compared to the turbulent stresses, and (2) the mean pressure is so nearly constant that the gradients have a negligible effect on the axial motion and momentum. With regard to condition 2, it should be pointed out that the pressure in jets is slightly different from the ambient pres- sure [1/1], but this may be disregarded as far as our present interests are concerned. Let x be measured along the axis of mean flow from some suitable origin, and U denote the mean velocity in the x direction. Let y be the lateral coordinate for two-dimensional flow and r be the radial coordinate for flow symmetrical about the z axis, and let V represent the lateral or radial component of mean velocity in each case. Then for steady mean flow the equations of motion and continuity are respectively: U—+V —— (27-1) Plane jet and Ox oy poy mixing zone aU. av ao (27-2) peasy oe oie eee (27-3) . Ox or rp or Round jet a(rU) . alrV) r r ae + a 0 (27-4) Here 7 is the shear stress. For wakes, equations corresponding to Eq. 27-1 and 27-3 may be further reduced because of conditions which apply at the great distances from the object necessary for similarity to exist. These are that V has become negligible, and U is nowhere much less than the free stream ve- locity U.. If we express the velocity reduction by AU =U, —U and substitute in Eq. 27-1, at the same time dropping the term VOU /dy, we obtain OAUS Weer: —(U, Tr AU) ax p dy To a sufficient degree of approximation this may be written ONGE aly On; 124) BY fy papa RES i et a ane wake We as oa (27-5) ¥A Cs % —* ee "is ied Plate B,28. Turbulent wake of bullet. (Courtesy Ballistic Research Laboratories, Aberdeen Proving Ground.) After Corrsin and Kistler (117). B,27 - LAWS OF MEAN SPREADING AND DECAY Following a similar procedure for the round wake we obtain @AU _ 1 A(rr) Round wake —U,. A Ea aEe (27-6) The equations for the conservation of momentum are: Plane jet p oe: U?dy = const | Round jet 2p ik ” U*rdr = const Plane wake (27-7) p fan U(U. — U)dy = const = pU. fe AUdy Round wake 2rp iki U(U. — U)rdr const & 2rpU. a AUrdr The method of employing the foregoing relations to find the laws of spreading and decay will be illustrated by carrying through the steps for the plane jet. Then the end results for all cases will be stated. If U, is the velocity at the center of the jet and 6 is any convenient measure of the width (6 may be the distance from the center to where U is zero or some fraction of U,), then similarity means OT he a7 ol) as where f and g are any function whose form may remain unknown. We now set b ~ x” and U, ~ 2”. Then the terms in the equation of motion (Eq. 27-1) become of the following order in z: and In order that the equation shall be independent of x, we must have 2n + 1 = 2n + m, or m = 1. The momentum relation p il is W2dy:— const because of order nm and since this must be independent of z, —2n + m = 0. Since m = 1, n = 4. Thus it is found that the plane jet spreads linearly with x and the CGil)) B - TURBULENT FLOW velocity at the center decreases as x}. A similar procedure may be used for the other cases, and the results are summarized for all in Table B, 27. Table B,27 Mixing Plane Round Plane Round zone jet jet wake wake Width parameter x”. Velocity at center x~”. The same results may be obtained by setting up integral relations for the energy and using these with the momentum relations to determine mand n. This procedure is illustrated in [94]. The diffusion of heat and other scalar quantities is also of practical and theoretical interest. The equations of heat transfer, written for assumptions consistent with those made for the equations of motion, are as follows: Plane jet and oT Ol ih oa mixing zone uh Ox tae OY ply OY rel) ; oT Olan ella) Se jet U ae +V BR my tae NOR (27-11) Olt leg Plane wake Gs aE EOF (27-12) Rengpal onal U. = = peti) (27-13) where T is the mean temperature, qg is the rate of heat transfer in the y or r directions per unit area (see Art. 10), and c, is the specific heat at constant pressure. In proper terms the same equations hold for the trans- fer of matter. Molecular diffusion is so slow compared to turbulent dif- fusion that the transfer can be regarded as due entirely to turbulent motions. Again assuming similarity, and expressing it in analogous terms, Eq. 27-10, 27-11, 27-12, and 27-13, together with the fact that the same amount of heat and matter must flow through each cross section, serve to determine the form of spreading and the decrease of center temper- ature or concentration as a function of x. These are the same as for the velocity, but the absolute magnitudes are different. In all cases the origin of x is that point from which the flow appears to originate with the same law from the beginning. The point is usually found by extrapolating the experimental curves to a virtual origin. For ( 162 ) B,28 - GENERAL FORM AND STRUCTURE the round jet this is usually between 0.5 and 1.5 orifice diameters down- stream from the orifice. The virtual origin appears to be less well defined for the plane wake and is different for the extrapolated center velocity than for the extrapolated width (Townsend [/]). The foregoing relations apply as long as the Reynolds number remains sufficiently high for similarity to exist. Since the Reynolds number is pro- portional to x”-”, it is seen from Table B,27 that, if the condition is initially satisfied, it will continue to be satisfied with ever-increasing x in all cases, with the exception of the round wake. For the latter the Reynolds number will eventually decrease to the point where the turbu- lent laws of spreading and decay merge into laminar laws with a new virtual origin. The distances for this change to occur can be expected to be very great, and in most cases any practical interest in the wake will have already been lost. As already indicated, it is required in the foregoing analyses that similarity extend to the turbulent motions responsible for diffusion. The same rules must therefore apply to the scales of length and velocity enter- ing into the diffusion process. If we adopt the concept of eddy viscosity, we may compare the behavior of a turbulent flow to that of a laminar flow in terms of the behavior of a viscosity. Denoting the mean eddy vis- cosity applicable to the flow by e,, we have, since e, is proportional to a length times a velocity, Referring to Table B,27, we find that e, is constant for the round jet and the plane wake. These flows should then behave as laminar flows with respect to their form of spreading and decay, as in fact they do. It must be borne in mind that we are here concerned only with the proportionality rule, not with absolute magnitudes. In the case of the plane jet, e, in- creases as x?, and we find, as we should, that the spreading and decay follow faster laws than those governing laminar flow. In this case the laminar exponents are eae, Sayed! Mam Par) Mam = By In the case of the round wake, e, decreases as x, and we find, again as we should, that the spreading and decay follow slower laws than those governing laminar flow. Here the laminar exponents are erin | =s Mam rae ed) Mam ae 1 B,28. General Form and Structure. The boundary which separates the turbulent fluid, of say a jet or a wake, from the nonturbulent sur- rounding fluid is determined only by how far the motions have pene- trated the surroundings. While it is self-evident that the boundary must be irregular, it was not until comparatively recent hot wire studies were (163 5 B - TURBULENT FLOW made that the highly irregular and sharply defined character of the bound- ary was revealed. An intermittency in the turbulence recorded from a hot wire probe in the outer regions of a round jet was first observed and studied by Corrsin [//2]. It soon became apparent that this effect was due to a sharp and irregular boundary convected past the hot wire. The phenomenon was studied in considerable detail by Townsend [1/13,114, 115,116] in connection with his studies of the plane wake. Corrsin and Kistler [117] later made an exhaustive study of free stream boundaries, and this together with studies in the boundary layer by Klebanoff [118] has resulted in a reasonably clear understanding of the character and meaning of the free boundary. In Art. 17 attention has already been directed to the outer boundary of a turbulent boundary layer, and the situation has been depicted sche- matically in Fig. B,17b. The character of the free boundary and the sharp separation between turbulent and nonturbulent fluid is shown in actual reality by the photograph of the turbulent wake of a bullet, displayed by Corrsin and Kistler, and shown here as Plate B,28. No turbulence and no other property transported by the shear flow, except some energy as- sociated with potential motions, has penetrated the surrounding medium beyond the boundary. Moreover, the boundary is a connected surface; there are no disconnected parcels of fluid. The billows and hollows are, of course, three-dimensional. All motion in the nonturbulent fluid outside the boundary is irrotational, and the velocity there is that accompanying the potential motion of a free stream. These phenomena are reproduced at all free boundaries, differing only in degree. An “intermittency factor’? has been adopted as one of the criteria of the irregularity of the boundary. If a hot wire probe, capable of following the fluctuations, is placed so that, as the flow passes by, it is alternately in and out of the turbulent fluid, a record of the signal will show intermittently turbulent and nonturbulent sections. From such a record, or by other instrumental means, the fraction of the time that the flow is found to be turbulent may be determined. This is defined as the intermittency factor. As the probe is moved from the center of the flow outward, the intermittency factor goes from unity to zero. The custom- ary symbol for the intermittency factor is y. This symbol when used here is not to be confused with the same symbol for the ratio of specific heats used earlier. It is instructive to compare y distributions for several types of flow along with their mean velocity distributions. These are given for the boundary layer, the round jet, and the plane wake in Fig. B,28a and B,28b. In Fig. B,28a Klebanoff’s data for a smooth wall and Corrsin’s and Kistler’s data for a very rough wall are compared. While there is considerable dispersion in the observations of intermittency, the differ- ence between the curves for smooth and rough walls is believed to be real. ( 164 ) B,28 - GENERAL FORM AND STRUCTURE Fig. B,28a. Intermittency factor compared with velocity distribution in boundary layers for smooth and rough walls. Fig. B,28b. Intermittency factor compared with velocity distribution for jet and wake. r:, b: are distances from axis to where velocity ratios are 3. ( 165 ) B - TURBULENT FLOW Comparing the boundary layer and free flow, it is seen that the region of intermittency occurs where the velocity is not much different from that of the free stream in the case of the boundary layer, whereas it pene- trates more deeply into the jet and wake flows. The range of mean ve- locities occurring in the region covered by the various instantaneous po- sitions of the boundary is therefore much less for the boundary layer than for jets and wakes; and while the boundaries may appear superficially similar in all cases, the bulges and hollows involve the greater portion of the mean velocity field in free flows. This applies particularly to the wake. According to Townsend [/] free flows contain large eddies which have a relatively small amount of energy, but which nevertheless serve to con- vect the fluid about in large bulks. He postulates a double structure con- sisting of the large eddies containing little turbulent energy and a smaller scale of eddies containing most of the turbulent energy. This would seem to be a reasonable picture in view of the freedom of motion in the absence of a wall, but, to the degree that the outer boundary of a wall flow is also free, the same picture might also apply to the outer region of a boundary layer. A statistical measure of the width of the intermittent zone is the standard deviation of the instantaneous boundary from its mean position given by [(Y — Y)?]?, where Y is the instantaneous position and Y is the average position. From Townsend’s point of view the standard deviation is determined primarily by the large eddies. Corrsin and Kistler [117] were able to pre- dict the observed behavior (not the absolute magnitude) of the standard deviation in the boundary layer, jet, and wake on the basis of Lagrangian diffusion by continuous movements (Taylor [119]). However, this required only the assumption of similarity of velocity and length scales to one another and to the main flow, and therefore does not rule out a possibly predominant part played by the large eddies. It seems evident that the contour of a marked surface completely within the turbulent region would be qualitatively like that of the free boundary, but that its coarseness would depend on the scale of the eddies in the neighborhood and on the presence of turbulence on both sides. The boundary is therefore a marker which gives us a picture of the eddy diffusion at the extreme limits. The next question of considerable interest has to do with the mecha- nism by which the turbulence spreads into fluid which was originally non- turbulent. This spreading and enveloping of new fluid is the only means by which the average position of the boundary can migrate laterally. Given that the outer flow is irrotational, it must become rotational when it crosses the boundary into the turbulent region. Corrsin and Kistler have concluded that the change takes place suddenly and wholly within a very thin laminar superlayer “plastered” over the boundary. Vorticity ( 166 ) B,28 - GENERAL FORM AND STRUCTURE can be transmitted to an irrotational flow only by tangential forces due to viscosity. The layer in which this takes place is the laminar superlayer. Corrsin and Kistler have shown that this layer must be very thin, partly on the grounds that stability considerations would not permit it to be otherwise, and partly on the grounds that the turbulent stretching of vortex lines increases the vorticity and therefore sharpens up the velocity gradient. The thickness has been estimated to be less than the dissipation length dX. The presence of the laminar superlayer cannot be detected ex- perimentally, but the observed sharp demarcation between turbulent and nonturbulent regimes tends to confirm the thinness of the layer. The spreading of the turbulent region therefore takes place by viscous action at the immediate boundary, and the rate of encroachment depends on the steepness of the laminar gradient and on the surface area, both of which are increased by the larger-scale, eddy-diffusion process acting from within. Viscosity is the vorticity-propagating agent, but it plays no con- trolling role in the spread of the turbulent region. Corrsin and Kistler point out that heat and matter are transported across the boundary in exactly the same way; and if the Prandtl and Schmidt numbers are not much smaller than unity, these scalar quantities should be transported at the same rate as momentum. The processes at the immediate boundary therefore do not explain why heat and matter spread faster than mo- mentum. We shall return to this question in Art. 29. The phenomena just described require that the fluid everywhere be- yond the boundary cannot have received any quantity by diffusion. If a jet is hot, all of the heat is confined within the sharp boundary. The same is true of all of the axial momentum. The only effect on the outer fluid is a pressure-induced flow toward the jet and pressure-induced fluctu- ations. Both are irrotational. The term, turbulence, cannot be applied to these fluctuations. Relatively slow, potential-type velocity fluctuations are in fact observed in the outer fluid. Jumps in mean velocity are also observed in passing from turbulent to nonturbulent regions. Apparently in some cases these are smaller than would be expected if free stream velocity prevailed in the nonturbulent regions. Townsend proposes that the fluid between two turbulent bulges is partially carried along as the bulges move downstream, but there is some disagreement on this point. Corrsin and Kistler find jumps in the intermittent region of a boundary layer of about the order to be expected if the outer fluid is not carried along. The sharp boundary is not to be confused with the limits as usually expressed in terms of mean velocity distribution. It will be noted from the y curves that the fluctuations in the sharp boundary generally ex- tend beyond the mean velocity boundary. A bulge protruding far out apparently carries so little mean velocity increment or defect that its effect cannot be detected by the usual methods. aCulGT >) B - TURBULENT FLOW B.29. Transport Processes in Free Turbulent Flow. In order to solve the equations of motion and heat transfer given in Art. 27 and thus obtain velocity and temperature distributions in y or r, it is necessary to express the quantities on the right-hand side of the equations in terms that can be related to the derivatives of velocity and temperature with respect to y or r. The auxiliary expressions for this purpose have been discussed in Art. 10. Specifically, Eq. 10-12 or 10-13 are used with the coefficients D., Dix, «., oF & specified either by general conditions of the problem or expressed in terms of local conditions. The former usually takes the form of an assumption that the coef- ficients are constant over a given cross section of the fiow but vary from one section to the next. In recent years the following expression proposed by Prandil [7/20] has been extensively used: DD. or D; — K(U—. — 0.)b (29-1) where 6 is the width of the region at a given cross section, U,.. and U.:. are the extremes of mean velocity across the section, and K is an experimentally determined constant of proportionality whose value de- pends on the quantity D, or Ds. Specification of transport in terms of local conditions takes the form of mixing length theory. This theory has already been discussed in Art. 12. Its application to free turbulent flows has been so widely discussed in the iterature, for example [6,111], that only a few remarks are called for here. Much of the discussion has had to do with the relative merits of mo- mentum transfer theory on the one hand and Taylor’s vorticity transfer theory on the other. Vorticity transfer theory is generally favored on the grounds that it is consistent with a wider distribution of temperature than of velocity, but which of the two theories agrees the better with observed velocity distributions depends on cases. We shall here concern ourselves with the broader question regarding the foundation of the foregoing procedures rather than with the details of their application. The basis for judgment rests largely on the work of Townsend with the plane wake and that of Corrsin with the round jet. As mentioned in Art. 28, there is evidence that large eddies operate in free turbulent flows to contort the whole flow field and thus transport fluid with smaller scales of turbulence over much of the width occupied by the flow. The next idea to be introduced is that mixing of all proper- ties by large and small scale motions has gone on for a considerable time over the previous course of the flow. In this connection it is advisable to restrict the discussion to jets and wakes, for in these cases all of the properties in question have been put in at the beginning and through mixing have covered much of the cross section during their previous his- tory. Eddies of any scale significant in diffusion will have existed for a considerable time, and their size and intensity found at a particular lo- ( 168 ) B,29 - TRANSPORT PROCESSES IN FREE TURBULENT FLOW cation will depend mainly on their past environment and will reflect the character of the flow as a whole rather than that of any particular locality. The large-eddy part of the structure helps greatly to promote this general averaging. The central idea here is that the lengths and velocities enter- ing into a turbulent transport coefficient are not primarily determined by local conditions. What has been stated here is true to a degree of all turbulent flow, but the greater preponderance of large eddies and the exposure to mixing from the beginning enhance the effects in jets and wakes. We have the picture, then, that any property that has been in the flow for a considerable length of time should be mixed to a fair degree of uniformity when it has arrived at a particular cross section. Dilution occurs at the sharp boundaries, and also new fluid has recently become turbulent there. Therefore we would not expect complete uniformity everywhere within the sharp boundaries. Experiments show that turbu- lent energy, temperature in the case of a heated jet or wake, and concen- tration of a tracer gas in a jet are nearly uniform over the fully turbulent core and decrease gradually in the turbulent bulges as the boundary is approached. The over-ail average decrease toward the boundaries is faster than that in the turbulent parts alone due to the absence of any contribu- tion from the nonturbulent paris. The foregoing behavior does not apply in the same degree to the axial momentum. The mean velocity difference decreases considerably across the core and continues to decrease in the protruding turbulent bulges. This is obviously why the mean velocity distribution is less broad than the mean-temperature distribution, but it is only a superficial explanation since it leaves unexplained why the momentum should have been given preferential treatment in the mixing process. We must now be concerned with the question of how to express the transfer processes. Mixing length theory and Eq. 29-1 both assume a gradient type of transfer in which the rate can be expressed in terms of the local gradient. This requires that the diffusing movements shall be small compared to the distance over which the gradient changes. This condition may be satisfied as far as the smaller eddies are concerned, but it is obviously not satisfied for eddies comparable in size to the width of the jet or wake. Townsend proposes that the total rate of transport is a combination gradient diffusion by the smaller eddies, which contain most of the turbulent energy, and bulk convection by the larger eddies. Since the gradients in scalar quantities, like heat, matter, and turbulent energy have been reduced due to the long continued mixing, it would appear that these quantities have been transported laterally more by the bulk convection than by gradient diffusion. On the other hand, since mo- mentum has not been so thoroughly mixed, the prospects for gradient diffusion are better. ( 169 ) B - TURBULENT FLOW With regard to the theories in question, three main facts stand out: (1) only the smaller eddies of this double-structure picture can take part in the gradient diffusion on which the theories are based, (2) the smaller eddies are mixed to a state of near uniformity, and (3) the scale and intensity of all eddies responsible for transfer are determined by general conditions rather than local conditions. Fact 1 means that we cannot predict to what extent the theories will apply. Fact 2 means that we can make a good case for Eq. 29-1 for that part of the transport which is of the gradient type. Fact 3 means that we must be skeptical of the kind of local dependence on which mixing length theory rests. This refers specifically to Eq. 12-2b and 12-13 of Art. 12 which expresses » and J in terms of local mean flow parameters. Some lessening of local dependence is achieved when / is taken to be constant over a section of the flow and proportional to the width. This is commonly done in free turbulent flows. We see that even with this compromise, mixing length theory is scarcely tenable in free turbulent flows. . Turning to comparisons with measured distributions, we find that mixing length theory cannot be shown to be definitely wrong, although the agreement with observations is rather casual, with vorticity transfer turning out to be better in some cases and momentum transfer being better in others. The vorticity transfer version of the theory when com- bined with the heat transfer version does at least yield a broader tem- perature distribution than velocity distribution [12/]. Hinze and van der Hegge Zijnen [/22] conducted an exhaustive series of experiments in which they measured distributions of velocity, tem- perature, and concentration of small amounts of added gas in a round air jet. After comparing their results with mixing length theories they concluded that these theories were unsatisfactory, and so set out to ex- plore the possibilities of constant turbulent exchange coefficient. From their measured velocity distributions and the equations of motion and continuity, D, was determined as a function of radius and axial distance. It was found to remain nearly constant with increasing r from the center outward, and then to decrease in the intermittent zone. They concluded, however, that a constant D, was a sufficiently good assumption to justify the adoption of the well-known laminar solution. The resulting velocity distribution formula and the expression for D,, are given in Art. 30. Hinze and van der Hegge Zijnen found that temperature and concen- tration profiles indicated practically identical exchange coefficients. We shall denote these by the common symbol D, and refer to the ratio D./D; = Pr, as the turbulent Prandtl number. (This ratio is known as the Schmidt number when referring to matter in place of temperature.) The value of Pr, on the axis of the jet was found to be 0.685. How- ever, Pr, increased steadily with r and became greater than unity for Car70) B,29 - TRANSPORT PROCESSES IN FREE TURBULENT FLOW U/U, < 0.2. This means that D; decreased where D,, remained con- stant and decreased more rapidly than D, in the outer regions. Corrsin and Uberoi [123] calculated values of Pr, from their measure- ments in a heated round jet. Their mean values over the cross section of the jet were very close to the value 0.7. They also obtained an indicated increase from the center outward, but did not regard their accuracy as sufficient to be certain of a definite trend. They noted the striking agree- ment with the laminar Prandtl number for air at the mean temperature of the jet. Forstall and Shapiro [/24] point out, however, that turbulent Prandtl numbers for jets are about 0.7 for various kinds of fluids irrespec- tive of their laminar Prandtl number. Townsend found in his investigations of the plane wake [/13] that both D, and / remained nearly constant in the central portion of the wake, but fell off rapidly in the outer part. However, D,, divided by the intermittency factor y was not far from constant over the greater part of the wake. These pieces of evidence tend to confirm what was conjectured earlier in this article, namely that insofar as theories based on gradient transfer can be applied at all, they should apply better to momentum than to temperature or concentration. The laminar-type solutions of the equa- tions of motion based on some appropriate constant value of D, over the section have consistently given accurate descriptions of the velocity dis- tribution. Discrepancies occur in the outer part of the flow due to the fact that D, decreases. Townsend has shown that improvement results for the plane wake if the eddy viscosity e, is allowed to decrease with the intermittency factor, 1.e. as yey. The situation with regard to the diffusion of heat and matter is not so favorable, and transfer based on local gradients is little better than a crude approximation at best. For the round jet, and presumably for the plane jet also, D; is nowhere constant, but the assumption of constant Pr, is believed to be acceptable for practical purposes. When Pr, is con- stant, the relation between temperature distribution and velocity dis- tribution for the round jet is Gre uk Fo Be heh (29-2) ratio ratio While there is some question about the appropriate value of Pr,, a reason- able value is Pr, & 0.7. According to Reichardt [125] Eq. 29-2 should be more generally appli- cable in free turbulent flow. For a review of Reichardt’s inductive theory of turbulence, reference is made to [96]. For plane wakes it does not seem possible to calculate temperature distribution on the basis of an exchange coefficient for heat. Paradoxi- cally, mixing length theory gives reasonably good agreement with ob- (aig B+: TURBULENT FLOW served temperature and velocity distributions when / is assumed constant over the cross section. Momentum transfer theory and vorticity transfer theory give the same results for velocity distribution, but vorticity trans- fer must be used in connection with the heat transfer equation to get the proper result for temperature distribution. The results are where U. and 7’. are respectively the velocity and temperature of the free stream, U, and T’, are respectively the velocity and temperature at the center, and y. is the extreme limit in each case, y. being the greater for temperature distribution. We may conclude this discussion by noting that recent findings have given us a clearer physical picture but little by way of a fundamental theory. It has not been possible to clarify the question as to why turbu- lent motions act differently toward heat and matter than toward mo- mentum. Some discussion of this question is given by Townsend; and since this cannot readily be taken out of context, the reader is referred to [/, pp. 164, 165]. B,30. Velocity Distribution Formulas for Jets and Wakes. The advantage of a constant exchange coefficient is not so much in any marked improvement in accuracy over mixing length theory, but rather that it permits the adoption of laminar-type solutions. When similarity exists, the form of the dependence of the exchange coefficient on x is known, but the absolute magnitude must be found from experiment. The purpose here is to give examples of final results based on this method. For the purpose of comparison a mixing length formula will be shown for one case. It is assumed that mixing length theory and the resulting formulas have been given sufficient attention in other literature, notably in [6,/1/]. A comparison of formulas for the plane wake, made by Townsend [126], is shown in Fig. B,380. Compared with an observed velocity dis- tribution curve are 1. Mixing length theory, / constant over the width: iss pki Nai it SSB E ee | (30-1) 2. Constant exchange coefficient: e 2 fi = 1.835 exp | - Gas) | (30-2) (i> B,30 - VELOCITY DISTRIBUTIONS IN JETS AND WAKES 3. Modified theory: e, = (e,)vy, where (e,), = constant eddy viscosity in the turbulent region, y = intermittency factor: 4 ii —eleSaolexp |-144e E Le - ae || (380-3) In 1, 2, and 3, fi and é are Wau EL o\ is y fi U. ( d i; = @ =u Zo = virtual origin (xo/d = +90) d diameter of cylinder producing the wake In these cases = = D,, = 0.0173U.d (30-4) It is seen that mixing length theory makes the distribution too narrow near the axis. The constant exchange coefficient fits in this region but 3 + Mixing length 1.835 [1 — (€/0.48)7]? @ Constant shear coefficient 1.835 e — (&/0-253) © Modified theory 1.835 e ~ 14.4711 + 46/035) Mean of observations Fig. B,30. Comparison of velocity distribution formulas for plane wake, after Townsend [126]. makes the velocity difference approach zero too slowly in the outer region. In reality e, is not constant, and an all-over fit is obtained only by adjust- ing «,, aS in Kq. 30-3. The distribution of axial velocity across the round wake may also be represented by a Gaussian error function. Such representations are char- acteristically faulty near the outer edges. The round wake has not been investigated so thoroughly as the plane wake. VED, B: TURBULENT FLOW The mixing zone between two uniform streams of velocities Ui; and U2 and the plane jet were treated on the basis of constant exchange coefficient by Gértler [127]. Since this work has been well reviewed by Schlichting [96], only selected results are repeated here. The calculated velocity profile of the mixing zone is in very good agreement with experiment. The calcu- lated distribution of axial velocity in the plane jet is also in good agree- ment with experiment except in the outer regions where the calculated profile approaches zero too slowly. The formula for the plane jet is w= we = (1 — tanh? 7) (30-5) where a = a free constant to be determined by experiment, y gee Os K = strength of jet = es Udy The velocity at the center U, is given by 8 ES 5 2 (30-6) It must be assumed in these formulas that x is the distance from the point where the jet appears to originate. Schlichting quotes Reichardt’s experimental value of a as equal to 7.67 and D, as given by D, = 0.037y;,U, (30-7) where y; is the value of y where U/U, = 3. It follows from Eq. 30-5, 30-6, and 30-7 that V8 Te D, = 0.0102 VW Kx (30-7a) As mentioned in Art. 29, the adaptation of the laminar-type solution for the round jet was investigated by Hinze and van der Hegge Zijnen. Their expression is UF aia (30-8) U. nr \* (1 ay z) where yn, = r/(x + 20) xo = virtual origin which turned out to be 0.6 times the orifice diameter (x + xo) = over-all distance from point where jet appears to originate const = 0.00196 ao (174 ) B,30 - VELOCITY DISTRIBUTIONS IN JETS AND WAKES In this case D, = 0.00196(@ + 2) U, (30-9) According to Schlichting [96] the velocity at the center may be ex- pressed by 3 KG Us 8r Dil + 20) (30-10) Here K, = strength of jet = on | U2rdira— 7 OPE (30-11) 0 where D is the diameter of the nozzle and U, is the jet exit velocity. By means of Eq. 30-9 and 30-11, Eq. 30-10 may be written i ae (80-12) According to Hinze and van der Hegge Zijnen the numerical constant in Eq. 30-12 turns out to be 6.39 on the basis of their observed axial dis- tribution of velocity. When U, given by Eq. 30-10 is substituted into Eq. 30-9, D, = 0.0153 VK, (80-13) In jets, as in wakes, the constant exchange coefficient makes the calculated velocity approach zero too slowly in the outer regions. This discrepancy is tolerated partly because it is in the region where the ve- locity is low and partly because the reason for it is S MCS Y in terms of intermittency. Since the exchange coefficient D, is the turbulent kinematic viscosity, it is interesting to compare it to ordinary kinematic viscosity v. For the plane wake from a cylinder the ratio D,,/v is found from Eq. 30-4 to be os = = 0.0173 — Ue (30-14) where U.d/v is the Reynolds number of the cylinder. A similar expression may be found for the round jet by replacing K, in Eq. 30-13 by Eq. 30-11. The result is =— OO) <= (30-15) Vv Du _ 9.9153 we a? = & Lp If, in these two examples, d and D are both one inch and JU, in both cases is 100 ft/sec, the Reynolds number for air at ordinary temperature and pressure is about 4.9 X 104. The two values of D,,/v are then found to be 850 and 660 for the wake and jet respectively. These figures serve to (175) B - TURBULENT FLOW convey an idea of the order of magnitude of the ratio of turbulent vis- cosity to ordinary viscosity. When the spreading is linear, as it is for jets, the angle of spreading affords a convenient means of visualizing the size. This angle may be found from Eq. 30-5 and 30-8 in terms of some suitably defined width. If we take this to be the line along which U/U, = i, we find that the plane jet is a wedge with a half angle of approximately 63} degrees and the round jet is a cone with a half angle of approximately 5 degrees. These angles are independent of the strength of the jet. The spreading of laminar jets, on the other hand, depends on the strength, becoming narrower as the strength increases. The plane laminar jet is not wedge- shaped; the width increases with x*. These differences between turbulent flow and laminar flow are mentioned as additional illustrations of the effect of an eddy viscosity which is regulated by the flow itself. B,31. Effect of Density Differences and Compressibility on Jets with Surrounding Air Stationary. In jet propulsion the jet is much hotter than the surrounding air and it issues at a high relative velocity. Density differences and compressibility are therefore expected to be of some importance. When we examine the situation realistically, however, we find that both temperature difference and relative velocity diminish rapidly with distance, and the jet soon behaves much like the constant- density, incompressible jet previously treated. When the jet issues rear- ward from a moving vehicle, it does not emerge into a surrounding me- dium at rest but rather into a medium with an axial velocity in the same direction. Under this condition the jet spreads more slowly, and the tem- perature and relative velocity diminish more slowly with distance from the orifice. The extent over which density and compressibility effects are possibly important therefore depends on the velocity of the outer stream. The effect of an outer velocity will be considered in Art. 32; here the problem is considered for the surrounding medium stationary with respect to the nozzle. The work of Corrsin and Uberoi on the heated round jet [123] has contributed substantially to what is known about the hot jet issuing into still surrounding air. They studied the jet issuing from a 1-inch orifice with velocities ranging from 65 to 115 ft/sec. The initial temperature rise was made slight when it was desired to study the spread of momentum and heat without introducing significant effects of density difference, and was raised to about 300°C when the effect of density was to be studied. In the latter case the density ratio was pi/po0 = 2, where po is the density of the jet at the nozzle and p; is the density of the surrounding air. The principal effects are illustrated in Fig. B,3la and B,31b taken from the report of Corrsin and Uberoi. Fig. B,31a shows the velocity and temperature profiles 16 nozzle diameters downstream. From these it is (176 ) B,31 - DENSITY DIFFERENCES AND COMPRESSIBILITY clear that the reduced density corresponding to the higher temperature causes a more rapid spreading of both the velocity and temperature pro- files. Again the temperature profile is wider than the velocity profile. Fig. B,31b shows the decrease of velocity and temperature along the axis. aia: Fig. B,3la. Radial velocity and temperature distribution in round jet showing effect of density, after Corrsin and Uberoi [123]. Section 16 nozzle diameters from orifice. 7’) = initial temperature of jet, 7. = temperature at center, 7, = temper- ature of surrounding air. x/d Fig. B,31b. Axial velocity and temperature distribution at center of round jet, after Corrsin and Uberoi [123]. x = distance from orifice, d = diameter of nozzle, To = initial temperature of jet, T. = temperature at center, 7; = temperature of surrounding air. The decreased density causes both velocity and temperature to fall more rapidly than for constant density. No appreciable change was noted in the shape of the profile in the fully developed jet for pi/po = 2. Corrsin and Uberoi showed definitely that there was no measurable change in the shape of the total head pro- SEZ) B- TURBULENT FLOW files. However, the density difference was not large after the distance of 7 or 8 nozzle diameters required for the flow to become fully developed. For example, at 15 diameters pi/p, = 1.3 when pi/po = 2. For a density ratio of this order there is no essential departure from similarity in any of the profiles, so that the spread is found to be proportional to x and the decrease of velocity and temperature along the axis is found to be inversely proportional to z. Cleeves and Boelter [1/28] made measurements of velocity and tem- perature in a round jet with an initial temperature difference of 650°C. The jet issued vertically from a pipe 14 inches in diameter at velocities ranging from 13 to 56 ft/sec. They did not detect any difference in the radial velocity distribution between the isothermal jet and the hot jet. In short, they found no effect of the decreased density on the rate of spreading of the jet. This disagrees with the results of Corrsin and Uberoi. They did, however, find the velocity on the axis decreased more for the hot jet than for the isothermal jet. The decrease was greater than that found by Corrsin and Uberoi, as would be expected from the higher tem- perature, but their isothermal results for the velocity on the axis do not agree with those of Corrsin and Uberoi. The Corrsin and Uberoi results should perhaps be given the greater weight in view of the accurate con- trol over experimental conditions. Since high relative velocities, and the compressibility and heating effects associated with them, are generally found close to a nozzle, the magnitude of these effects is of most interest in connection with the mixing-zone problem. Abramovich [/29] investigated the effects by apply- ing the vorticity transfer version of mixing length theory to the plane mixing zone between a stream of uniform velocity and a medium at rest, restricting the treatment to air speeds up to Mach number unity and temperature differences up to 120°C. He found that cooling the stream increased the width of the mixing zone, with the boundary on the stream side showing practically all of the effect. The effect of increasing the ve- locity was to decrease the width of the mixing zone, again with only the boundary on the stream side being affected. However, the predicted effects were such that practically identical velocity distributions were indicated if on the one hand a low speed stream is cooled to AT = —60°C and on the other hand a stream of Mach number unity has a stagnation temperature equal to that of the stationary medium (static temperature, AT = —60°C). Thus a jet cooled either by extraction of heat or by adi- abatic expansion will have a more rapidly diverging mixing region than a jet having the same static temperature as the surrounding medium. This would not be consistent with the findings of Corrsin and Uberoi. Gooderum, Wood, and Brevoort [/30] measured the density distribu- tion with an interferometer in the mixing zone of a jet issuing from a 3 by 38-inch nozzle at a Mach number of 1.6. The stagnation temperature (178 ) B,32 - EFFECT OF SURROUNDING AIR ON JETS of the jet was about the same as that of the surrounding air. The jet was therefore cold, the initial density being 1.5 times that of the surrounding air. The density and velocity were examined across the mixing zone from 2 inches to 74 inches from the nozzle. The distributions were similar at each cross section, and the velocity distribution could be represented by Tollmien’s theoretical curve for incompressible flow [131] in the subsonic portion of the mixing region. Such distributions have the typical s-shape of the Gaussian integral curve, and they reduce to a common curve for different values of x when plotted against cy/x, where g is a scale factor. The width of the mixing region is thus inversely proportional to o. The value for incompressible flow is generally around 12. Gooderum, Wood, and Brevoort found o = 15. The rate of spreading into the jet core and into the surrounding air was therefore less than that for incompressible flow. This would appear to disagree with the trends found by Abramovich, which of course apply to subsonic flow, but is what would be expected from the density effect found by Corrsin and Uberoi in the round jet. Similar results were reported by Bershader and Pai [132] from meas- urements on the discharge from a rectangular nozzle 1 by 2 cm at a Mach number of 1.7. Density measurements were made with an interferometer at several closely spaced stations within one nozzle width from the orifice. The density distributions were found to be similar, and o was found to be 17. The mixing zone was thus narrower than that for incompressible flow. The profile of density ratio is in reasonable agreement with a curve based on Pai’s theory [133] which employs the concept of a constant coefficient of eddy kinematic viscosity of the form of Eq. 29-1. These experimental results on supersonic jets do not distinguish be- tween the effect of a denser jet and the heating effect resulting from internal dissipation. We might assume, however, as pointed out by Pai in relation to laminar flow, that the greater momentum associated with higher density causes the stream to carry farther and thus decrease the divergence of the mixing zone. Evidence to substantiate this assumption is afforded by the work of Keagy and Weller [134] who found wider ve- locity profiles for helium jetting into air and narrower profiles for carbon dioxide. It may be concluded from this that the observed effects are pri- marily density effects. Taken as a whole, the observations disagree with the theoretical predictions of Abramovich. B,32. Effect of Axial Motion of Surrounding Air on Jets. When a jet is projected rearward from a vehicle moving through the air, it effec- tively emerges into a surrounding medium in motion in the same direction as the jet. Some attention is now given to the effect of this motion on the characteristics of the jet. We do not consider other cases, likewise of im- portance, where the jet is projected forward or at an angle to a moving stream. ( 179 ) B - TURBULENT FLOW It appears that no basic investigations have been carried out which would give us information on the turbulent structure and the boundary configuration when the surrounding medium is moving. Corrsin and Kistler [177] call attention to the limiting case where a turbulent and a nonturbulent stream are in contact with no mean relative velocity, and infer that the diffusing mechanism will be much the same as when a rela- tive velocity exists. If this is so, it follows that mixing again depends on the velocity and scale of the mixing motions as determined by a relation of the type of Eq. 29-1. We know that travel of the surrounding stream along with the jet lessens the divergence and decay of jet velocity. In a very real sense the jet fluid rides along with the outer stream and reaches a distance x from the nozzle in a shorter time. Fluid has had less time to diffuse and as a consequence has traveled a shorter distance laterally. Correspondingly, it has had less time to mix, and it would be expected that a greater distance is now required for the similarity regime to pre- vail. In the absence of any firm knowledge of the turbulent structure, the usual concepts are applied by investigators in this field, namely that either mixing length or turbulent exchange coefficient are constant over a cross section. Using mixing length theory for momentum transfer, Kuethe [109] in- vestigated the plane mixing region between streams moving in the same direction with different relative velocities and also treated the mixing zone of the round jet from the nozzle to the end of the potential core for the case where the outside medium is at rest. Gértler [127] later de- veloped the relations for the plane mixing region between two streams on the basis of a turbulent exchange coefficient given in the form of Eq. 29-1. Szablewski [135] then extended this method to the core-containing region of the round jet for the case where the surrounding stream has different velocities. Squire and Trouncer [/36], using mixing length theory, applied to momentum transfer developed relations for the characteristics of the round jet for various velocities of the surrounding stream, including both the initial core-containing region and the fully developed region. In addition they calculated the inflow velocity in the region surrounding the ~ jet. All of the methods apply to incompressible, isothermal flow. All of the methods agree, at least qualitatively, in showing a marked effect of velocity of the outer stream on the rate of jet spreading and decay of velocity differences. The effect depends on the ratio U;/Us, where U;, is the outer stream velocity and Up is the jet exit velocity. As the ratio increases, the divergence decreases, the core region extends farther from the nozzle, and the velocity increments decrease more slowly with xz. When U;/Uo = 0, the core region extends only to about 5 orifice diameters from the nozzle. When U,/U » = 0.5, the distance is increased to 11 diameters according to Szablewski and to 8.1 diameters according to Squire and Trouncer. Each of these two methods requires that a single ( 180 ) B,32 - EFFECT OF SURROUNDING AIR ON JETS constant be evaluated by experiment, and for this purpose existing data for still surrounding air were used. Szablewski’s method indicates some- what greater effects, but it is difficult to judge the reliability of these methods due to the basic assumptions and approximations made in the solutions. No attempt will be made here to reproduce the developments and final formulas, all of which tend to be cumbersome. Squire and Trouncer achieved some simplification by arbitrarily adopting a cosine velocity profile which for the fully developed jet takes the form Wo Wa ool if CR =e = (2 + cos x") (32-1) where U; is the velocity of the surrounding stream, U, the velocity on the jet axis, r the radial distance from the axis, and r, the radius of the jet boundary. We turn next to experiment, and here we find a comprehensive in- vestigation conducted by Forstall and Shapiro [/24] aimed at testing the analytical formulation of Squire and Trouncer and additionally com- paring mass transfer and momentum transfer. For obtaining the mass transfer 10 per cent by volume of helium was added to the jet as a tracer. Values of Uo up to 225 ft/sec and values of U, up to 90 {ft/sec were used. Velocity ratios Ui/Uo ranged from 0.2 to 0.75. Velocity and concentration profiles downstream from the end of the potential region could be closely represented by a formula of the type of Eq. 32-1. The assumption of this formula by Squire and Trouncer was therefore well justified. The profiles remained substantially similar at all values of x and were independent of the velocity ratio U:/Uo. In order to avoid the uncertainty in specifying the extremes of the jet, the size parameters rm, and rng were adopted, where rm, is the radius where the velocity is the mean of its value on the axis and in the outside stream, and rm: is the radius where the concentration is $ the concen- tration on the axis. Expressing these in terms of the diameter of the nozzle D the rate of spreading with x/D was found to be greater for concentration than for velocity. A turbulent Schmidt number of about 0.7 was indicated (compare Art. 29). The experiments checked the law of jet divergence derived by Squire and Trouncer. Both concentration and velocity were found to decay inversely with z/D. In general, concentration showed more of a drop than did the ve- locity, but the difference in behavior was small. The inverse law amounts to a faster decrease with x/D than that predicted by the Squire and Trouncer theory, although the theory gives the general order of magni- tude of the center line properties. Forstall and Shapiro give the following empirical formulas for the round jet in a surrounding stream of equal density to serve as rough rules for the velocity field: ( 181 ) B - TURBULENT FLOW Formulas Symbols ap U; = velocity of surrounding stream (i) Dir vittaas Uo = exit velocity of jet at nae (ii) U, - U1 _ % A= U1/Uo Yo SUE Vee x = axial distance from end of 5 RY pat aNin nozzle &) Di a = z, = distance to end of potential Gv) U-Ui eal (1 a eo ur ) core U, = Uy 2 21 map U, = center line velocity of jet for 49 S> Ur D = diameter of nozzle r = radial distance from axis U = velocity at r radius where U = U+Us 2 my It is noted that formula (i) for the case where \) = 0 does not agree with the one given by Hinze and van der Hegge Zijnen, which is OC, Gae Ty x Dt 0:6 Reference should be made to Pai’s book [111, p. 120] for another form of (ii) and further discussion of the effect of a surrounding stream. Turning next to the heated jet in a surrounding stream, we have the problem of the combined effects of a stream velocity and density differ- ences on the velocity and temperature fields. Some experimental infor- mation on the temperature field of the round jet in a supersonic stream was obtained by Rousso and Baughman [1/37] in connection with an NACA program on jets aimed primarily at answering certain engineer- ing problems. The only known account of work attempting to solve the transfer problem is the paper by Szablewski [138], in which a theoretical development is given, and the experimental work of Pabst [139] is dis- played as a test of the theory. The analysis applies specifically to the round jet and includes large density differences. It does not include the case where the surrounding air is stationary. It is left to the reader to consult [138] for the lengthy analytical development and the complete results. In brief, Szablewski bases his development on turbulent exchange coefficients given by the Prandtl ex- pression (Eq. 29-1). These are introduced into the usual equations ex- pressed in the form of continuity equations for mass, momentum, and heat. The ratio of the exchange coefficient for momentum to that for heat and mass (the turbulent Prandtl number, Pr,) was taken to be 0.5 on ( 182 ) B,32 - EFFECT OF SURROUNDING AIR ON JETS the basis of Pabst’s results. The computed examples cover the range Ue Ue 0.5, 0.25, 0.05 fey Le Th I Mechs), Ae where U,, 71 = outer stream velocity and absolute temperature respec- tively Uo, To = jet exit velocity and absolute temperature respectively. Pabst’s measurements of velocity and temperature distributions in a round jet were made with Uy & 400 m/sec, U1 = 18, 101, and 188 m/sec, and 7) = 300°C (T/T: & 2). Since Szablewski’s account of this work appeared to cover the significant points, the original work of Pabst was not consulted. The following are the major conclusions: 1. The theoretical predictions for small density differences agree with other work regarding the direction of the effect of an outside stream, namely to decrease the rate of spreading of both velocity and temper- ature. The predicted asymptotic boundary (jet so far from nozzle that nozzle size has no effect) varies as z' for any value of U; except zero. 2. An outside stream reduces the rate of velocity and temperature fall along the axis. This is qualitatively confirmed by experiment, but there is some question about the accuracy of Pabst’s temperature measure- ments. The asymptotic variation predicted by theory is x? for any value of U, except zero. 3. When U; + 0 the asymptotic velocity profile is given by where U, is the velocity at the center, 71 is the radius of the jet bound- ary, and c is a shape factor. The temperature profile is (ea g — =) T,-T, \U,-— Ui where Pr, is taken as 0.5. These distribution functions when fitted to Pabst’s measurements at 16, 20, and 24 nozzle diameters downstream show reasonably good agreement. 4. Reduced jet density, due to elevated temperature, increases the rate of velocity and temperature fall along the axis. Apparently Pabst’s work does not provide any test of this effect. However, this is con- sistent with the findings of Corrsin and Uberoi for U; = 0. 5. Reduced jet density, due to elevated temperature, decreases the rate ( 183 ) B - TURBULENT FLOW of spreading of both velocity and temperature. The effect of the tem- perature on the width is given as proportional to [1 + (To — T1)/Ti}"*. Pabst’s work, as quoted by Szablewski, does not provide any test of this effect. The effect is opposite to that found by Corrsin and Uberoi for U; = (), Conclusion 5 appears to be inconsistent with conclusion 4. Conclusion 5 can, however, be made to appear reasonable by following a suggestion by Squire and Trouncer to the effect that compressibility or heating might be dealt with in terms of an ‘‘equivalent jet.’’ Since the momentum 4 DU (Uo = E) is maintained at all sections, an equivalent incompressible jet should be- have like a compressible jet when the momentum is the same. Heating a jet decreases pUo(U» — U1) by decreasing p. We should obtain the same effect without heating by keeping p the same and decreasing Uo(U» — U1). This can be done either by decreasing Uo or by increasing U1. By either means U,/U> is increased, and this clearly has the effect of decreasing the divergence of the velocity field. Presumably the divergence of the temperature field would follow that of the velocity field. The tentative conclusion drawn from the present information is that the effect of density on spreading characteristics reverses in going from the case where U; = 0 to the case where U; > 0. More experimental results covering a greater range of conditions are needed to clarify the situation. Not all of the information on jets has been covered in this brief survey. Pai, for example, gives a mathematical procedure for dealing with turbulent jets by employing methods analogous to those for laminar flow. For this, the reader is referred to his book [1/1]. An extensive bibliography given by Forstall and Shapiro [124] will be helpful to readers wishing to pursue the subject of jets further. CHAPTER 6. TURBULENT STRUCTURE OF SHEAR FLOWS B,33. The Nature of the Subject. Dating from about 1925 many investigators have applied the hot wire anemometer in aerodynamic ex- periments in an effort to learn something about turbulence through meas- urement. Over the years these efforts have borne fruit; consequently there are many separate pieces of information contributing to our present knowledge of turbulence and the turbulent structure of various flow fields. ( 184 ) B,34 - STRUCTURE OF SHEAR TURBULENCE For the most part the measured quantities are the velocity fluctu- ations, their mean square values, their time derivatives, space and time correlations, probability distributions, and energy spectra. In a few cases measurements have been made of temperature fluctuations, including correlations, spectra, and velocity-temperature correlations. Due to limi- tations inherent in the hot wire technique, very few measurements have yet been made in the compressible flow range. Low speed boundary layers, jets, and wakes have been the principal objects of investigation: Theoretical studies on the turbulent structure of shear flows are not abundant because of the difficult nature of the statistical theories of turbulence and the additional complications arising from anisotropy and inhomogeneity associated with shear flows. Nevertheless, it becomes a lengthy task to present and discuss both the experimental and theoretical sides of the subject. We therefore restrict the present coverage to a listing of references. Qualitative aspects of structure have been discussed in previous articles, and the references there cited will be repeated here only by number. The list covers mainly relatively recent works which are available to us, and it can by no means pretend to be complete. Classification is by subject, each being headed by a brief discussion. Since many works cover topics belonging in different classes, the group- ing must not be considered as rigid. B,34. References on Structure of Shear Turbulence. General considerations on vorticity and structure of turbulence. The stretching of vorticity and the formation of vortex sheets play an im- portant role in shear turbulence. Studies of these phenomena are gener- ally theoretical, and often have to be based on a simple and isotropic model. Agostini, L., and Bass, J. Les théories de la turbulence. Publs. sci. et tech. Ministére air France 237, 1950. Betchov, R. An inequality concerning the production of vorticity in isotropic turbulence. J. Fluid Mech. 1, 497-504 (1956). Burgers, J. M. The formation of vortex sheets in a simplified type of turbulent motion. Proc. Acad. Sci. Amsterdam 58, 122-133 (1950). Corrsin, 8. Hypothesis for skewness of the probability of the lateral velocity fluctua- tions in turbulent flow. J. Aeronaut. Sci. 17, 396-398 (1950). Djang, F. G. A kinetic theory of turbulence. Chinese J. Phys. 7, 176 (1948). Liepmann, H. W. Aspects of the turbulence problem. J. Math. and Phys. 3, 321-342, 407-426 (1952). Lin, C. C. On Taylor’s hypothesis in wind tunnel turbulence. Mem. Nav. Ord. Lab. 10775, 1950. Lin, C. C. On Taylor’s hypothesis and the acceleration terms in the Navier-Stokes equations. Nav. Ord. Rept. 2306, 1952. Munk, M. M. A Simplified Theory of Turbulent Fluid Motion. Catholic Univ. of America, 1955. Pai, 8. I. Viscous Flow Theory. II: Turbulent Flow. Van Nostrand, 1957. Theodorsen, Th. Mechanism of turbulence. Proc. Second Midwestern Conf. Fluid Mech., The Ohio State Univ., 1952. ( 185 ) B - TURBULENT FLOW Statistical theories of shear and inhomogeneous turbulence. In statisti- cal theories of turbulence, it is important to study the structure of corre- lations and spectral functions on the basis of hydrodynamical equations of motion. The spectral tensor in anisotropic turbulence has a much more complicated form than in isotropic turbulence. Exact mathemati- cal theories are not yet possible. Dimensional arguments and simplifying reasoning are necessary. If a reasonably simple equation of motion of one-dimension is used, such as in Burgers’ model, the solution can be obtained exactly, and many characteristics of turbulence can be studied without introducing simplifying assumptions at an early stage. Burgers, J. M. Some considerations on turbulent flow with shear. Proc. Acad. Sct. Amsterdam B56, 125-136, 137-147 (1953). Burgers, J. M., and Mitchner, M. On homogeneous non-isotropic turbulence con- nected with a mean motion having a constant velocity gradient. Proc. Acad. Sci. Amsterdam B56, 228-235, 343-354 (1953). Kampé de Fériet, J. Le tenseur spectral de la turbulence homogéne non isotrope dans un fluide incompressible. Proc. Seventh Intern. Congress Appl. Mech., London, 6-26 (1948). von Kdérmdn, Th. The fundamentals of the statistical theory of turbulence. J. Aeronaut. Sci. 4, 131 (1937). Monin, A. 8. Characteristics of anisotropic turbulence. Doklady Akad. Nauk. S.S.S.R. 75, 621-624 (1950). Parker, E. N. The concept of physical subsets and application to hydrodynamic theory. Naval Ord. Test Station Tech. Mem. 988, China Lake, Calif., 1953. Rotta, J. Statische Theorie nichthomogener Turbulenz I, II. Z. Physik 129, 547 (1951); 131, 51 (1951). Tchen, C. M. On the spectrum of energy in turbulent shear flow. J. Research Nail. Bur. Standards 50, 51 (1953). Tchen, C. M. Transport processes as foundations of the Heisenberg and Obukhoff theories of turbulence. Phys. Rev. 93, 4 (1954). Structure of turbulence in wall-bounded flow (boundary layer, channel and pipe). Turbulent measurements are made on energy, shear stresses, correlation, spectral functions of energy, and shear stress. In the case of the boundary layer, the flow is complicated by the fact that there exist a laminar sublayer near the wall and an irregular outer limit producing a region of intermittent turbulence near the free edge of the boundary layer. The intermittencies and the probability of their occurrence are important for the understanding of the boundary layer, and for the formu- lation of a realistic theory of the boundary layer structure. Phenomeno- logical theories are based on transport concepts (such as mixing length) to express nonlinear turbulent terms. Other theories assume some definite relation between the fourth and second orders of correlations, and a third group of theories make some assumption involving physical and dimen- sional reasoning on the role of the turbulent pressure. Chou, P. Y. On velocity correlation and the solutions of the equations of turbulent fluctuation. Quart. Appl. Math. 3, 38-54 (1945). Chou P. Y. Pressure flow of a turbulent fluid between two infinite parallel plates. Quart. Appl. Math. 3, 198-209 (1945). ( 186 ) B,34 - STRUCTURE OF SHEAR TURBULENCE Chou, P. Y. On velocity correlations and the equations of turbulent vorticity fluctuation. Natl. Tsing-Hua Univ. Sci. Rept. 5, 1-18 (1948). Chou, P. Y. The turbulent flow along a semi-infinite flat plate. Quart. Appl. Math. 6, 346-353 (1947). Dryden, H. L. Recent advances in the mechanics of boundary layer flow. Advances in Applied Mechanics, pp. 1-40. Academic Press, 1948. Eskinazi, S., and Yeh, H. An investigation on fully developed turbulent flows in a curved channel. J. Aeronaut. Sci. 23, 23-35 (1956). Fage, A., and Townend, H.C. H. An examination of turbulent flow with an ultra- microscope. Proc. Roy. Soc. London A135, 656-677 (1932). Favre, A., Gaviglio, J..and Dumas, R. Nouvelles mesure dans la couche limite d’une plaque plane, des intensités de turbulence, et des correlations dans le temps; spectres. Recherche aéronaut. Paris 38, 7-12 (1954). Favre, A., Gaviglio, J.,and Dumas, R. Couche limite turbulente: Corrélations spatio- temporelles doubles; spectres. Recherche aéronaut. Paris 48, 3-14 (1955). Johnson, D. 8. Turbulent heat transfer in a boundary layer with discontinuous wall temperature. The Johns Hopkins Univ. Dept. of Aeronaut. Rept., 1955. Klebanoff, P.S. Characteristics of turbulence in a boundary layer with zero pressure gradient. NACA Tech. Rept. 1247, 1955. Laufer, J. Some recent measurements in a two-dimensional turbulent channel. J. Aeronaut. Sci. 17, 277-287 (1950). Laufer, J. The structure of turbulence in fully developed pipe flow. NACA Rept. 1174, 1955. Ludwig, H., and Tillman, W. Untersuchungen iiber die Wandschubspannung in turbulenten Reibungsschichten. Ing.-Arch. 17, 288-299 (1949). Malkus, W. V. R. Outline of a theory of turbulent shear flow. J. Fluid Mech. 1, 521 (1956). Mattioli, E. Una formula universale per lo spettro nella turbolenza di parete. Atti accad. sci. Torino 90, 1956. Mattioli, E. Richerche teoriche e sperimentali sulla turbolenza di parete. Aero- tecnica 86, (2), 1956. Michel, R. Contribution 4 l’etude des couches limites turbulentes avec gradient de pression. Publs. sci. et tech. Ministére air France 252, 1951. Newman, B.G. Skin friction in a retarded turbulent boundary layer near separation. Dept. of Supply, Australia, Aeronaut. Research Lab. Rept. A73, 1950. Rotta, J. Beitrag zur Berechnung der turbulent Grenzschiehten. Ing.-Arch. 19, 31 (1953). Rotta, J. Schubspannungsverteilungen und Energiedissipation bei turbulenten Grenzschichten. Ing.-Arch. 20, 195-207 (1952). Sandborn, V. A., and Braun, W.H. Turbulent shear spectra and local isotropy in the low-speed boundary layer. NACA Tech. Note 3761, 1956. Schubauer, G. B. Turbulent processes as observed in boundary layer and pipe. J. Appl. Phys. 25, 188-196 (1954). Steketee, J. A. Some problems in boundary layer transition. Univ. Toronto Inst. Aerophys. Rept. 38, 1956. Szablewski, W. Berechnung des turbulenten Str6émung in Rohr auf der Grundlage der Mischungsweg-hypothese. Z. angew. Math. u. Mech. 31, 13-142 (1951). Szablewski, W. Berechnung des turbulenten Str6émung Langs einer ebenen Platte. Z. angew. Math. u. Mech. 31, 309 (1951). Taylor, G. I. Correlation measurements in a turbulent flow through a pipe. Proc. Roy. Soc. London A157, 537-546 (1936). Townsend, A. A. The structure of the turbulent boundary layer. Proc. Cambridge Phil. Soc. 47, 375-395 (1951). Walz, A. Naherungstheorie fur kompressible turbulente Grenzschichten. Z. angew. Math. u. Mech., 50-56 (1956). Walz, A. Nouvelle méthode approchée de calcul des couches limites laminaire et turbulente en ecoulement compressible. Publ. sci. et tech. Ministére air France 309, 1956. ( 187 ) B - TURBULENT FLOW Yeh, H., Rose, W. G., and Lien, H. Further investigation on fully developed turbu- lent flows in a curved channel. The Johns Hopkins Univ. Dept. Mech. Eng. kept., 1956. Cited references [73,76,77,87,89,93]. Structure of turbulence in a free flow (jet, wake). In a free flow, the intermittencies produced near the boundary of the flow play an important role in the characteristics of the flow and the structure of turbulence. The shear flow in the present case has a weak mean velocity gradient, so that the spectrum of energy is not far from the spectrum of an isotropic turbu- lence. However, here a spectrum of shear exists, in contrast to its absence in isotropic flow. Chou, P. Y. On an extension of Reynolds’ method of finding apparent stress and the nature of turbulence. Chinese J. Phys. 4, 1-33 (1940). Corrsin, S., and Uberoi, M.S. Spectra and diffusion in a round turbulent jet. NACA Rept. 1040, 1951. Hinze, J., and van der Hegge Zijnen, B.G. Heat and mass transfer in the turbulent mixing zone of an axially symmetrical jet. Proc. Seventh Intern. Congress Appl. Mech., London, 1948. Kalinske, A. A., and Pien, C.C. Eddy diffusion. Ind. Eng. Chem. 36, 220-223 (1944). Kovasznay, L. 8. G. Hot-wire investigation of the wake behind cylinders at low Reynolds numbers. Proc. Roy. Soc. London A198, 174-190 (1949). Laurence J. C. Intensity, scale, and spectra of turbulence in mixing region of free subsonic jet. NACA Rept. 1292, 1956. Laurence, J. C., and Stickney, T. M. Further measurements of intensity, scale, and spectra of turbulence in a subsonic jet. NACA Tech. Note 3576, 1956. Squire, H. B. Reconsideration of the theory of free turbulence. Phil. Mag. 39, 1-20 (1948). Swain, L. M. On the turbulent wake behind a body of revolution. Proc. Roy. Soc. London A125, 647-659 (1929). Tamaki, H., and Oshima, K. Experimental studies on the wake behind a row of heated parallel rods. Proc. First Japan. Natl. Congress. Appl. Mech., 459-464 (1951). Cited references [94,112,113,114,115,116,121,123,124,126]. Structure of turbulence connected with turbulent diffusion and heat transfer. In both the statistical and phenomenological theories of the structure of shear turbulence, of bounded or free flows, turbulent dif- fusion plays an important role. A thorough coverage of turbulent dif- fusion is not intended, and only those works dealing with the mechanism of diffusion which help in better understanding the structure of turbu- lence are listed below, leaving aside works mainly connected with appli- cations of diffusion. Batchelor, G. K., Binnie., A. M., and Phillips, O. M. The mean velocity of discrete particles in turbulent flow in a pipe. Proc. Phys. Soc. London B68, 1095-1104 (1955). Batchelor, G. K., and Townsend, A. A. Turbulent diffusion. Surveys in Mechanics, pp. 353-399. Cambridge Univ. Press, 1956. Beckers, H. L. Heat transfer in turbulent tube flow. Appl. Sci. Research A6, 147 (1956). Brier, G. W. The statistical theory of turbulence and the problem of diffusion in the atmosphere. J. Meteorol. 7, 283-290 (1950). ( 188 ) B,34 - STRUCTURE OF SHEAR TURBULENCE Davies, D. R. The problem of diffusion into a turbulent boundary layer from a plane area source, bounded by two straight perpendicular edges. Quart. J. Mech. and Appl. Math. 7, 467-471 (1954). Dryden, H. L. Turbulence and diffusion. Ind. Eng. Chem. 31, 416 (1939). Ellison, T..H. Atmospheric turbulence. Surveys in Mechanics, pp. 400-430. Cam- bridge Univ. Press, 1956. Frenkiel, N. F. On the statistical theory of turbulent diffusion. Proc. Natl. Acad. _ Set. 88, 509-515 (1952). Frenkiel, F. N. Application of the statistical theory of turbulent diffusion to micro- meteorology. J. Meteorol. 9, 252-259 (1952). Frenkiel, N. F. Sur la mesure de la diffusion de la chaleur. Groupement frang. dévelop. recherches aéronaut., 1946. Hinze, J. O. Turbulent diffusion from a source in turbulent shear flow. J. Aeronaut. Sci. 18, 565 (1951). Inoue, E. On the temperature fluctuations in a heated turbulent field. Geophys. Notes, Geophys. Inst., Tokyo Univ., 3, 1950; Geophys. Mag. 23, (1), 1951. Inoue, E. Some remarks on the dynamical and thermal structure of a heated fluid. J. Phys. Soc. Japan 6, 392 (1951). Kitojima, K. On the mixing length of turbulence. Kysyu Univ. Research Inst. Fluid Eng, Rept. 4, 43-54 (1948). Lee, T. D. Note on the coefficient of eddy viscosity in isotropic turbulence. Phys. Rev. 77, 842 (1950). Levich, V. G. Diffusion. Doklady Akad. Nauk. S.S.S.R. 78, 1105-1108 (1951). Liu, V.C. Turbulent dispersion of dynamic particles. J. Meteorol. 13, 399-405 (1956). Monin, A. S. Equations of turbulent diffusion. Doklady Akad. Nauk. S.S.S.R. 105, 256-259 (1955). Ribaud, G. Some remarks on the subject of heat and momentum transfer in the boundary layer. C. R. Acad. Sci. Paris 240, 1, 25-28 (1955). Taylor, G.I. The dispersion of matter in turbulent flow through a pipe. Proc. Roy. Soc. London A223, 446 (1954). Tchen, C. M. Enige Wiskundige Betrekkingen Welke een Rol Spelen in Diffusie- problemen. Verhandl. Acad. voor Wet. 53, 400-410 (1944). Tchen, C. M. Stochastic processes and the dispersion of the configurations of linked events. J. Research Natl. Bur. Standards 46, 480-488 (1951). Wieghardt, K. On diffusion phenomena in turbulent boundary layer. Z. angew. Math. u. Mech. 28, 346-355 (1948). Cited references [25,26,27,116,119,122]. Instrumentation for the measurement of turbulence. What has been learned about the structure of turbulence from experiment has depended largely on the instruments available for making observations and meas- urements. Here the hot wire anemometer has played a predominantly important role. The number and accuracy of quantities measured have gone hand in hand with the development of hot wire probes and the adaptation of electronic circuits to amplify the signal, compensate for thermal lag of the hot wire, and perform a variety of operations such as adding, multiplying, and differentiating signals. The following refer- ences therefore pertain mainly to the hot wire and its auxiliary equipment. Dryden, H. L., and Kuethe, A. M. The measurement of fluctuations of air speed by the hot-wire anemometer. NACA Rept. 320, 1929. Fage, A. Studies of boundary-layer flow with a fluid-motion microscope. 50 Jahre Grenezschichtforschung, pp. 132-146. (Ed: H. Gértler and W. Tollmien.) Vieweg, Braunschweig, 1955. ( 189 ) B - TURBULENT FLOW Kovasznay, L. 8. G. Turbulence in supersonic flow. J. Aeronaut. Sci. 20, 657-675 (1953). Kovasznay, L. S. G. Development of turbulence-measuring equipment. NACA Rept. 1209, 1954. Laufer, J., and McClellan, R. Measurements of heat transfer from fine wires in supersonic flow. J. Fluid Phys. 1, 276-289 (1956). Laurence, J. C., and Landes, L.G. Auxiliary equipment and techniques for adapting the constant-temperature hot-wire anemometer to specific problems in air-flow measurements. NACA Tech. Note 28438, 1952. Mock, W.C., Jr. Alternating-current equipment for the measurement of fluctuations of air speed in turbulent flow. NACA Rept. 598, 1937. Newman, B. G., and Leary, B. G. The measurement of the Reynolds stresses in a circular pipe as means of testing a hot wire anemometer. Dept. of Supply, Australia, Aeronaut. Research Lab. Rept. A72, 1950. Ossofsky, E. Constant temperature operation of the hot-wire anemometer at high frequency. Rev. Sci. Instr. 19, 881-889 (1948). Sansborn, V. A. Heat loss from yawed hot wires at subsonic Mach numbers. NACA Tech. Note 3568, 1955. Schubauer, G. B. A turbulence indicator utilizing the diffusion of heat. NACA Rept. 524, 1935. Spangenberg, W. G. Heat-loss characteristics of hot-wire anemometers at various densities in transonic and supersonic flow. NACA Tech. Note 3381, 1955. Tchen, C. M. Heat delivery in a compressible flow at subsonic and supersonic speeds. NACA Tech. Note 2436, 1951. Uberoi, M. 8., and Kovdsznay, L. 8. G. Analysis of turbulent density fluctuations by the shadow method. J. Appl. Phys. 26, 1955. Weske, J. R. A hot-wire circuit with very small time lag. NACA Tech. Note 881, 1943. Willis, J. B. Review of hot-wire anemometry. Council for Sci. and Ind. Research, Div. of Aeronautics, Australia, Rept. A34, 1945. Wise, B., and Schultz, D. L. Turbulent measurements in supersonic flow with the hot-wire anemometer. Brit. Aeronaut. Research Council Rept. FM 2390, 1955. B,35. Cited References. 1. Townsend, A. A. The Structure of Turbulent Shear Flow. Cambridge Univ. Press, 1956. 2. Batchelor, G. K. The Theory of Homogeneous Turbulence. Cambridge Univ. Press, 1953. . Reynolds, O. Phil. Trans. A186, 123 (1894), or Papers 2, 535. . Lorentz, H. A. Abhandl. theoret. Physik 1, 43 (1907). . Lamb, H. Hydrodynamics, 6th ed. Dover, 1945. . Goldstein, S. Modern Developments in Fluid Dynamics, 1st ed., Vol. 2. Clarendon Press, Oxford, 1938. 7. Howarth, L. Modern Developments in Fluid Dynamics, High Speed Flow, Ist ed., Vol. 2. Clarendon Press, Oxford, 1953. 8. Rubesin, M. W. A modified Reynolds analogy for the compressible turbulent boundary layer on a flat plate. NACA Tech. Note 2917, 1953. 9. Crocco, L. Sulla Transmissione del Calore da una Lamina Piana a un Fluido Scorrente ad alta Velocita. Aerotecnica 12, 181-197 (1932). 10. Squire, H. B. Heat transfer calculation for aerofoils. Brit. Aeronaut. Research Council Repts. and Mem. 1986, 1942. 11. Ackerman, G. Forsch. Gebiete Ingenieurw. 18, 226-234 (1942). 12. Mack, L. M. An experimental investigation of the temperature recovery factor. Calif. Inst. Technol. Jet Propul. Lab. Rept. 20-80, 1954. 13. Stalder, J. R., Rubesin, M. W., and Tendeland, T. A determination of the laminar-transitional, and turbulent-boundary-layer temperature-recovery fac- tors on a flat plate in supersonic flow. NACA Tech. Note 2077, 1950. D> Ore CO ( 190 ) 41. 42. 43. B,35 - CITED REFERENCES . Seban, R. A. Analysis for the Heat Transfer to Turbulent Boundary Layers in High Velocity Flow. Ph.D. Thesis, Univ. Calif., Berkeley, 1948. . Shirokow, M. Tech. Phys. USSR 3, 1020-1027 (1936). . Stine, H. A., and Scherrer, R. Experimental investigation of the turbulent- boundary layer temperature-recovery factor on bodies of revolution at Mach numbers from 2.0 to 3.8. NACA Tech. Note 2664, 1952. . Tucker, M., and Maslen,S8.H. Turbulent boundary layer temperature recovery factors in two-dimensional supersonic flow. NACA Tech. Note 2296, 1951. . Lobb, K. R., Winkler, E. M., and Persh, J. J. Aeronaut. Sci. 22, 1-9 (1955). . van Driest, F. R. J. Aeronaut. Sci. 18, 145-160 (1951). . Spivack, H. M. Experiments in the turbulent boundary layer of a supersonic flow. North Amer. Aviation Rept. AL-1052, APL/JHU CM-615, 1950. . Chapman, S., and Cowling, T. G. The Mathematical Theory of Non-Uniform Gases. Cambridge Univ. Press, 1953. . Lorentz, H. A. The Theory of Electrons. Dover, 1952. . Van Vleck, J. H., and Weisskoff, V.G. Rev. Mod. Phys. 17, 227 (1945). . Bhatnagar, P. L., Gross, E. P., and Krook, M. Phys. Rev. 94, 511 (1954). . Tchen, C. M. Mean Value and Correlation Problems Connected with the Motion of Small Particles Suspended in a Turbulent Fluid. Thesis, Delft, 1947. Mededeel- ing No. 51 uit het Laboratorium voor Aero-en Hydrodynamica der Technische Hogeschool te Delft. . Burgers, J. M. On turbulent fluid motion. Calif. Inst. Technol. Hydrodynam. Lab. Rept. E-34.1, Chap. 5, 1951. . Tchen, C. M. J. Chem. Phys. 20, 214-217 (1952). . Reynolds, O. Proc. Manchester Lit. Phil. Soc. 14, 7-12 (1874) ; Collected Papers 1, 81-85. . Chapman, D. R., and Kester, R. H. J. Aeronaut. Sci. 20, 441-448 (1953). . Reichardt, E. Heat transfer through turbulent friction layers. NACA Tech. Mem. 1047, 1943. . Colburn, A. P. Trans. Am. Inst. Chem. Engr. 29, 174-210 (1933). . Rubesin, M. W., Maydew, R. C., and Varga, 8. A. An analytical and experi- mental investigation of the skin friction of the turbulent boundary layer on a flat plate at supersonic speeds. NACA Tech. Note 2305, 1951. . Pappas, C. C. Measurement of heat transfer in the turbulent boundary layer on a flat plate in supersonic flow and comparison with skin-friction results. NACA Tech. Note 3222, 1954. : . Coles, D. J. Aeronaut. Sci. Readers’ Forum 19, 717 (1952). . Coles, D., and Goddard, F. E. Direct measurement of skin friction on a smooth flat plate at supersonic speeds. Paper presented at 8th Intern. Congr. Theoret. and Appl. Mech., Istanbul, 1952. . Liepmann, H. W., and Dhawan, 8. Proc. First U.S. Natl. Congr. Appl. Mech., Chicago, 869-874 (1951). . Dhawan, S. Direct measurements of skin friction. NACA Tech. Note 2567, 1952. . Wilson, R. E. J. Aeronaut. Sci. 17, 585-594 (1950). . Brinich, P. F., and Diacomis, N. S. Boundary-layer development and skin friction at Mach number 3.05. NACA Tech. Note 2742, 1952. . Fallis, W. B. Heat transfer in the transitional and turbulent boundary layers of a flat plate at supersonic speeds. Univ. Toronto Inst. Aerophys., UTIA Rept. 19, 1952. Slack, E. G. Experimental investigation of heat transfer through laminar and turbulent boundary layers on a cooled flat plate at a Mach number of 2.4. NACA Tech. Note 2686, 1952. Monaghan, R. J., and Cooke, J. R. The measurement of heat transfer and skin friction at supersonic speeds. Part III: Measurements of overall heat transfer and of the associated boundary layers on a flat plate at Mi = 2.43. Roy. Air Establishment Tech. Note Aero. 2129, 1951. Monaghan, R. J., and Cooke, J. R. The measurement of heat transfer and skin ion \ 44, 45. B- TURBULENT FLOW friction at supersonic speeds. Part IV: Tests on a flat plate at M1 = 2.82. Roy. Air Establishment Tech. Note Aero. 2171, 1952. Frankl, F. Heat transfer in the turbulent boundary layer of a compressible - gas at high speeds. Also Frankl, F., and Voishel, V. Friction in the turbulent boundary layer of a compressible gas at high speeds. NACA Tech. Mem. 1082, 1942. Frankl, F., and Voishel, V. Turbulent friction in the boundary layer of a flat plate in a two-dimensional compressible flow at high speeds. NACA Tech. Mem. 1058, 1943. . von Kdérmdn, Th. Mechanical similitude and turbulence. NACA Tech. Mem. 611, 1931. . Schoenherr, K. E. Trans. Soc. Nav. Arch. and Marine Eng. 40, 279-313 (1932). . Prandtl, L. Géttingen Ergebnisse 4, 27 (1932). . Clemmow, D. M. The turbulent boundary layer flow of a compressible fluid along a flat plate. Brit. Directorate of Guided Weapons Research and Develop. Rept. 50/6, 1950. . van Driest, E. R. Proceedings of the Bureau of Ordnance symposium on aeroballistics. Comments on paper by R. E. Wilson, NAVORD Rept. 1961, 264-267 (1950). . Li, T.-Y., and Nagamatsu, H. T. Effects of density fluctuations on the tur- bulent skin friction of an insulated flat plate at high supersonic speeds. Calif. Inst. Technol. Guggenheim Aeronaut. Lab. Mem. 6, 1951. . Ferrari, C. Quart. Appl. Math. 8, 33-57 (1950). . Smith, F., and Harrop, R. The turbulent boundary layer with heat transfer and compressible flow. Roy. Aircraft Establishment Tech. Note Aero. 1759, 1946. . Eckert, H. U. J. Aeronaut. Sci. 16, 573-584 (1950). . Monaghan, R. J. Comparison between experimental measurements and a suggested formula for the variation of turbulent skin friction in compressible flow. Brit. Aeronaut. Research Council C.P. 45, 18260, 1951. . Cope, W. F. The turbulent boundary layer in compressible flow. NPL Eng. Dept., Brit. Aeronaut. Research Council 76384, 1943. . Tucker, M. Approximate turbulent boundary-layer development in plane compressible flow along thermally insulated surfaces with application to super- sonic-tunnel contour correction. NACA Tech. Note 2045, 1950. . Tucker, M. Approximate calculation of turbulent boundary-layer development in compressible flow. NACA Tech. Note 2337, 1951. . Young, G. B. W., and Janssen, E. J. Aeronaut. Sci. 19, 229-236 (1952). . von Kaérmdén, Th. The problems of resistance in compressible fluids. Mem. Reale Acad. D’Italia, Rome, 1936. . Falkner, V.N. Aircraft Eng. 15, 65 (1948). . Hill, F. K. J. Aeronaut. Sci. 23, 35 (1956). . Korkegi, R. H. J. Aeronaut. Sci. 23, 97 (1956). . Hakkinen, R. J. Measurements of Skin Friction in Turbulent Boundary Layer at Transonic Speeds. Ph. D. Thesis, Calif. Inst. Technol., 1953. . O'Donnel, R. M. Experimental investigation at a Mach number of 2.41 of average skin-friction coefficients and velocity profiles for laminar and turbulent boundary-layers and an assessment of probe effects. NACA Tech. Note 8122, 1954. . Coles, D. J. Aeronaut. Sci. 21, 433-448 (1954). . Doénch, F. Forsch.-Aro. Gebiete Ingenieurw-Wes. 282, 1926. . Blasius, H. Mitt. Forschung. Ver. deut. Ing. 181, 1-34 (1913). . Prandtl, L. Aerodynamic Theory, Vol. 3, Durand, W. F. ed. Durand Reprinting Committee, Calif. Inst. Technol., 1943. . Prandtl, L. Recent results of turbulence research. NACA Tech. Mem. 720, 1933. (Transl. Z. Ver. deut. Ing. 7, (6), 1933.) . von Karman, Th. WNachr. Ges. Wiss. Gottingen, 58-76 (1930). 2. Clauser, F. H. The turbulent boundary layer. Advances in Appl. Mech. 4, 1-51. Academic Press, 1956. ( 192 ) B,35 - CITED REFERENCES . Millikan, C. B. A critical discussion of the turbulent flows in channels and circular tubes. Proc. Fifth Intern. Congress Appl. Mech., Cambridge, Mass., 386-392 (1938). . Schultz-Grunow, F. New frictional resistance law for smooth plates. NACA Tech. Mem. 986, 1941. (Transl. Luftfahrtforschung 17, 239-246, 1940.) . Freeman, H. B. Force measurements on a 7y-scale model of the U.S. airship “Akron.” NACA Rept. 432, 1932. . Klebanoff, P. S., and Diehl, Z. W. Some features of artificially thickened fully developed turbulent boundary layers with zero pressure gradient. NACA Rept. 1110, 1952. . Laufer, J. Investigation of turbulent flow in a two-dimensional channel. NACA Rept. 1033, 1951. . von Karman, Th. J. Aeronaut. Sct. 1, 1-20 (1934). . von Kaérman, Th. Mechanische Ahnlichkeit und Turbulenz. Proc. Third Intern. Congress Appl. Mech., Stockholm, 1, 85-93 (1930). . Schlichting, 8. Ing.-Arch. 7, 1-34 (1936). . Squire, H. B., and Young, A. D. The calculation of the profile drag of airfoils. Brit. Aeronaut. Research Council Repts. and Mem. 1838, 1938. . Kempf, G. Werft, Reederei, Hafen, 10, (11), 234-239 (1929); (12), 247-253 (1929). . Clauser, F. H. J. Aeronaut. Sci. 21, 91-108 (1954). . von Doenhoff, A. E., and Tetervin, N. Determination of general relations for the behavior of turbulent boundary layers. NACA Rept. 772, 1943. . Coles, D. J. Fluid Mech. 1, Part 2, 191-226 (1956). . Landweber, L. Trans. S.N.A.M.E. 61, 5 (1953). . Schubauer, G. B., and Klebanoff, P. S. Investigation of separation of the turbulent boundary layer. NACA Rept. 1030, 1951. . Newman, B. G. Some contributions to the study of the turbulent boundary layer near separation. Dept. Supply, Australia, Rept. ACA-53, 1951. . Ludwieg, H., and Tillmann, W. Investigation of the wall-shearing stress in turbulent boundary layers. NACA Tech. Mem. 1285, 1950. Transl. from Z. angew. Math. u. Mech. 29, 15-16, 1949. . Ruetenik, J. R., and Corrsin, S. Equilibrium turbulent flow in a slightly divergent channel. 50 Jahre Grenzschichtforschung. (Ed: H. Gértler and W. Toll- mien), 446-459. Vieweg, Braunschweig, 1955. . Lees, L., and Crocco, L. J. Aeronaut. Sct. 19, 649-676 (1952). . Ross, D., and Robertson, J. J. Appl. Mech. 18, 95-100 (1951). . Rotta, J. On the theory of the turbulent boundary layer. NACA Tech. Mem. 1344, 1953. Transl. Uber die Theorie der turbulenten Grenzschichten. Mitt. Maz-Planck-Inst., Gottingen, 1, 1950. . Liepmann, H. W., and Laufer, J. Investigation of free turbulent mixing. NACA Tech. Note 1257, 1947. . Nikuradse, J. Laws of flow in rough pipes. NACA Tech. Mem. 1292, 1950. Transl. Stromungsgesetze in rauhen Rohren. Ver. deut. Ing. Forschungsheft 361, 1933. . Schlichting, H. Boundary Layer Theory. McGraw-Hill, 1955. . Hama, F.R. Trans. Soc. Nav. Arch. and Marine Engrs. 62, 333-358 (1954). . Buri, A. A method of calculation for the turbulent boundary layer with accelerated and retarded basic flow. Brit. Ministry Aircraft Production R. T. P. Transl. 2073. From Thesis 652, Federal Tech. College, Zurich, 1931. (Also available from CADO, Wright-Patterson Air Force Base, as AT143493.) . Gruschwitz, E. Ing.-Arch. 2, 321-346 (1931). . Garner, H. C. The development of turbulent boundary layers. Brit. Aeronaut. Research Council Repts. and Mem. 2133, 1944. . Tetervin, N., and Lin, C. C. A general integral form of the boundary-layer equation for incompressible flow with an application to the calculation of the separation point of turbulent boundary layers. NACA Tech. Note 2158, 1950. ( 193 ) 102. 130. 131. 132. 133. 134. 135. B - TURBULENT FLOW Fediaevsky, K. Turbulent boundary layer of an airfoil. NACA Tech. Mem. 822, 1937. Transl. Central Aero-Hydrodynam. Inst., Moscow, Rept. 282, 1936. . Ross, D., and Robertson, J. M. J. Appl. Phys. 21, 557-561 (1950). . Granville, P.S. A method for the calculation of the turbulent boundary layer in a pressure gradient. The David W. Taylor Model Basin Rept. 762, 1951. . Kuethe, A. M., McKee, P. B., and Curry, W.H. Measurements in the boundary layer of a yawed wing. NACA Tech. Note 1946, 1949. . Ashkenas, H., and Riddell, F. R. Investigation of the turbulent boundary layer on a yawed flat plate. NACA Tech. Note 3383, 1955. . Young, A. D., and Booth, T. B. The profile drag of yawed wings of infinite span. College of Aeronautics, Cranfield, Rept. 88, May 1950. . Townsend, A. A. Proc. Roy. Soc. London A190, 551-561 (1947). . Kuethe, A. M. J. Appl. Mech. 2, (8), 1935. In Trans. Am. Soc. Mech. Eng. 57, A-87, A-95 (1935). . Birkhoff, G., and Zarantonello, E. H. Jets, wakes, and cavities. Applied Math. and Mech., Vol. 2. Academic Press, 1957. . Pai, S.-I. Fluid Dynamics of Jets. Van Nostrand, 1954. . Corrsin, S. Investigation of flow in an axially symmetrical heated jet of air. NACA Wartime Rept. ACR 8L23, 19438. . Townsend, A. A. Proc. Roy. Soc. London A190, 551-561 (1947). . Townsend, A. A. Australian J. Sci. Research, Series A 1, 161-174 (1948). . Townsend, A. A. Proc. Roy. Soc. London A197, 124-140 (1949). . Townsend, A. A. Phil. Mag. 41, 890-906 (1950). . Corrsin, 8., and Kistler, A. L. The free-stream boundaries of turbulent flows. NACA Rept. 1244, 1955. . Klebanoff, P. S. Characteristics of turbulence in a boundary layer with zero pressure gradient. NACA Tech. Note 3178, 1954. . Taylor, G.I. Proc. London Math. Soc. 20, 196-212 (1921). . Prandtl, L. Z. angew. Math. u. Mech. 22, 241-243 (1942). . Taylor, G.I. Proc. Roy. Soc. London A186, 685-702 (1932). . Hinze, J. O., and van der Hegge Zijnen, B. G. Appl. Sci. Research A1, 435-461 (1949). . Corrsin, §., and Uberoi, M. 8. Further experiments on the flow and heat transfer in a heated turbulent air jet. NACA Rept. 998, 1950. . Forstall, W., and Shapiro, A. H. J. Appl. Mech. 17, 399-408 (1950). In Trans. Am. Soc. Mech. Eng. 72, 1950. . Reichardt, H. Z. angew. Math. u. Mech. 24, 268-272 (1944). . Townsend, A. A. Australian J. Sci. Research, Series A, 2, 451-468 (1949). . Gortler, H. Berechnung von Aufgaben der freien Turbulenz auf Grund eines neuen Naherungsansatzes. Z. angew. Math. u. Mech. 22, 244-254 (1942). . Cleeves, V., and Boelter, L. M. K. Chem. Eng. Progr. 48, 123-134 (1947). . Abramovich, G. N. The theory of a free jet of a compressible gas. NACA Tech. Mem. 1058, 1944. Transl. Central Aero-Hydrodynam. Inst., Moscow, Rept. 377, 1939. Gooderum, P. B., Wood, §G. C., and Brevoort, M., J. Investigation with an interferometer of the turbulent mixing of a free supersonic jet. NACA Rept. 963, 1949. Tollmien, W. Calculation of turbulent expansion processes. NACA Tech. Mem. 1085, 1945. Transl. from Z. angew. Math. u. Mech. 6, 1926. Bershader, D., and Pai, 8. I. J. Appl. Phys. 21, 616 (1950). Pai, S. I. J. Aeronaut. Sci. 16, 463-469 (1949). Keagy, W. R., and Weller, A. E. A study of freely expanding inhomogeneous jets. Heat Transfer and Fluid Mech. Inst., Univ. Calif., 89-98 (1949). Am. Soc. Mech. Eng., New York. Szablewski, W. Contributions to the study of the spreading of a free jet issuing from a nozzle. NACA Tech. Mem. 1811, 1951. Transl. Zur Theorie der Aus- breitung eines aus einer Diise austretenden freien Strahls. Untersuch. u. Mitt. Nr. 8003, Sept. 1944. ( 194 ) 136. 37 138. 139. 140. 141. 142. B,35 - CITED REFERENCES Squire, H. B., and Trouncer, J. Round jets in a general stream. Brit. Aeronaut. Research Council Repts. and Mem. 1974, 1944. Rousso, M. D., and Baughman, E.L. Spreading characteristics of a jet expand- ing from choked nozzles at Mach 1.91. NACA Tech. Note 3836, 1956. Szablewski, W. The diffusion of a hot air jet in air in motion. NACA Tech. Mem. 1288, 1950. Transl. Die Ausbreitung eines Heissluftstrahles in Bewegter Luft. GDC/2460, Sept. 1946. Pabst, O. Die ausbreitung heisser Gasstrahlen in bewegter Luft. Untersuch. u. Mitt., Part II, UM 8007, 1944. Rubert, K. F., and Persh, J. A procedure for calculating the development of turbulent boundary layers under the influence of adverse pressure gradients. NACA Tech. Note 2478, 1951. Ross, D. Trans. Am. Soc. Civil Eng. Paper 2838, 121, 1219-1254 (1956). Spence, D. A. J. Aeronaut. Sci. 23, 3-15 (1956). ( 195 ) SECTION C STATISTICAL THEORIES OF TURBULENCE Cc. C. LIN CHAPTER 1. BASIC CONCEPTS C,1l. Introduction. The general concepts of turbulent motion have been discussed in the previous section. It is recognized that the details of turbulent flow are so complicated that statistical description must be used. Indeed, only statistical properties of turbulent motion are experi- mentally reproducible. The purpose of the present section is to give a more comprehensive treatment of the statistical theory. Current literature on the statistical theory of turbulence is mainly limited to the treatment of the case of homogeneous turbulence! without any essential mean motion. Superficially, one might think that there is little to be known about such fluid motions. Actually, the very absence of mean motion allows one to go more deeply into the inherent nature of the turbulent flow itself. Many basic concepts have been developed in the study of homogeneous turbulence, and these concepts now gradually find their way into the study of shear flow. Since there is available an account of the theory of homogeneous tur- bulence [/] with a complete discussion of the mathematical background, a somewhat different presentation is adopted in the present section. Following the historical order, the isotropic case is taken up first. It is hoped that this will be helpful to those readers who wish to get an idea of the essentials without going through all the preliminaries required in a complete mathematical treatment. In the later parts of this section, other aspects of the statistical theory and their applications will be treated.” We have, however, omitted several other approaches to the problem of turbulence. Among these, the work of Burgers [3,4] and Hopf [5] should especially be mentioned; nor is any attempt made to include a discussion of related mathematical studies, such as that of Hopf [6] and Kampé de Fériet [7]. 1 These concepts are explained more precisely in the following pages. 2 For a brief survey of some aspects of the problem of turbulent motion, see [2]. ( 196 ) C,2 - THE MEAN FLOW AND REYNOLDS STRESSES C,2. The Mean Flow and the Reynolds Stresses. It is generally assumed that the motion can be separated into a mean flow whose com- ponents are U,, Us, and U; and a superposed turbulent flow whose com- ponents are 1, U2, and ws, the mean values of which are zero. In taking average values, the following principles will be adopted. If A and B are dependent variables which are being averaged, and S is any one of the space variables 2, y, z, or the time t, then 0A/dS = 0A/dS and AB = AB, where the bar denotes a mean value. When the mean flow is not varying, that is, when the average value defined by t+r AG, y, 2, #) = lim te i Al Ge ycat. at. Tmo 2r t is independent of the time ¢t, the time average is the natural mean value to use. Difficulties arise when the flow is variable, and other types of averages have to be introduced. For instance, in the problem of turbu- lent flow near an infinite plate moving with variable velocity, the mean values could be taken over planes parallel to the plate. In more general cases, neither the time nor the space mean values can be conveniently defined to possess all the desired properties. We then consider the sta- tistical average over a large (infinite) number of identical systems (en- semble average). The equation of continuity of an incompressible fluid, when averaged, becomes dU; = -1 OX; u (2 ) The Navier-Stokes equations of motion are’ Dye = an; (6%; pvid;), y3=U;+u (2 2) where o;; is the stress tensor due to pressure and viscous forces. If the mean value is taken, Eq. 2-2 becomes BU: Led) Dino Ee (6; — pU:U; — puss) (2-3) This equation has the same form as Eq. 2-2, if v; is replaced by U;, and the stress o;; is replaced by ¢;; — puiu;. Thus, the equations of mean flow are the same as the ordinary equations of motion except that there are the additional virtual stresses Tij = — pur; (2-4) which represent the mean rate of transfer of momentum across a surface due to the velocity fluctuations. These virtual stresses were first intro- duced by Reynolds [8], and are known by his name. Calo) C - STATISTICAL THEORIES OF TURBULENCE In the case of a turbulent flow with steady mean motion, the time average is taken at every point, and the above physical interpretation of the Reynolds stress is clear. In the case of variable mean motion, such as the case of the infinite plate mentioned above, where mean values are defined as the averages over parallel planes, the interpretation of —pu,u; as a local stress is not as direct. In the case of general variable - motion, where the averaging process is the arithmetical mean taken over a large (infinite) number of unrelated identical systems (ensemble average), the physical interpretation of an average quantity as an apparent stress requires even more careful examination, since the average momentum transfer is not directly associated with any one particular system. More- over, the time average is usually measured in the case of steady mean flow. Thus, if the general theory is developed on the basis of statistical averages, an ergodic hypothesis must be introduced to identify these two in that case. In this section, the statistical average shall be adopted, and the validity of such a hypothesis shall be implied. Further investigations of such basic problems are beyond the scope of the present treatment. C,3. Frequency Distributions and Statistical Averages. One basic concept in the discussion of statistical averages is the frequency of occurrence, or the distribution function. For example, in the classical kinetic theory of gases, one considers a distribution function f(u, v, w, z, y, 2, t) such that flu, v, w, x, y, 2, t)dudvdwdxdydz gives the fraction of molecules at time ¢, having velocities in the range u, u+ du; v, v + dv; w, w + dw and lying in the element of volume z,x+dz;y,y + dy; 2, z+ dz. The kinetic theory of gases may then be based on the law governing the change of this function f(w, v, w, 2, y, 2, t). For a homogeneous gas at rest, it is the well-known Maxwellian function. In the case of turbulent motion, a similar (but different) function F(u, v, w, 2, y, 2, £) can be introduced giving, for each point (2, y, z) and each instant ¢, the probability that the turbulent velocity shall lie in the range u, u + du;v, v + dv; w, w + du, or for shortness, u;, u; + du;. If this function is known, then the Reynolds shear is given by formulas of the kind —puv = ~o fff Ft v, W, x, Y, 2, thuvdudvdw (3-1) for each point x, y, 2 at each instant ¢. To analyze the structure of turbulence one also needs to know the joint probability distribution for quantities observed at several points. For example, if we are interested in the correlation of velocities at two points P’ and P’’, then we must know a distribution function of the form ( 198 ) C,3 - FREQUENCY DISTRIBUTIONS. STATISTICAL AVERAGES F(ui, x1; ui’, x\’; t). The correlation of the x components of the velocities at these points is then given by OTs y 1 @ sae) faa // F(w’, vw’ ; ie Ts Che ie, wo, ils Lis ts ca t) ulul’d(u’, v’, w')d(ul’, yo” w’’) (3-2) Further generalization of joint probability distributions involves quanti- ties observed at more than two points, and quantities other than velocity fluctuations. Some experimental information is available regarding the distribution function F(u;, z:, t). The Gaussian distribution has been found, in many cases, to be a fairly good approximation for each component (Fig. C,3a).* In the isotropic case, i.e. where the statistical properties of the motion x Measurements (x/M = 16, UM/v = 9600) x NS ——=—— Normal distribution u; fluctuation Fig. C,3a. Probability density function of the velocity component w; in the direction of the stream for the turbulence generated by a square-mesh grid in a wind tunnel (after [7]). are essentially independent of direction, the Maxwellian distribution of velocity holds approximately [9,10]. There is also some indication that the joint probability distribution at two points in an isotropic field is approximately jointly Gaussian. How- ever, this is known to be not accurate. To get a quantitative assessment of the departure from joint Gaussian distribution, one may introduce the quantities CHRD: SN pie Ned eS Gar and (3-3) CHEN, EG ee omit 3In this and the following figures, M denotes the width of the mesh of grid, 2 denotes the distance of the observation point from the grid, U denotes the velocity of air, and » denotes the kinematic viscosity coefficient. ( 199 ) CC: STATISTICAL THEORIES OF TURBULENCE called respectively the skewness factor and the flatness factor. Fig. C,3b and C,3c show the experimental values of these factors as obtained by Stewart ({/1], also as quoted in [/2]). For exact joint-Gaussian distribu- tion, the former should be zero, and the latter should have the value 3. Thus, the hypothesis of a joint-Gaussian distribution is not exact, but it may still be used for certain approximations (cf. Art. 16). One should r/M Fig. C,3b. Skewness factor for turbulent fluctuations behind a grid (after Stewart [11]). 0) 0.08 mG 0.24 0.32 0.40 AOGINA)) an Xap =x ee SIV) Fig. C,3c. Flatness factor for turbulent fluctuations behind a grid across a uniform air stream (after Stewart, as quoted in [/2]). especially note the departure of the experimental value of the flatness factor from the value 3 for a small value of r. C,4. Homogeneous Fields of Turbulence. As noted in Art. 1, theoretical investigations of turbulent flow are often limited to the ideal- ized case of an infinite field of turbulence which is statistically homogene- ous or even isotropic, and devoid of mean motion. Homogeneity means that the statistical properties of the field are independent of the particular ( 200 ) C,5 - CONVENTIONAL APPROACH position in the field, and isotropy means that they are independent of direction.t For example, if we consider two points P and P’ in such a field, the velocity components u, at P and uj at P’, both in the direction of PP’, has a statistical correlation u,u). dependent only on the distance r between P and P’, but independent of the coordinates of the point P and the direction PP’. In a homogeneous anisotropic field, this correlation would be unaltered by a translation of the vector PP’ but would be altered by a rotation. In particular, in a homogeneous isotropic field, the mean square value of the three components of the velocity are equal to each other and are the same throughout the field. Thus we have Wh = 1B = = we (4-1) where 11, w%2, Uz are the velocity components along the coordinate axes 0x1, Ore, and Oz;, and wu is the root mean square value. The statistical properties under consideration may be the spectrum, the joint probability distributions of velocity and pressure, etc. In many cases, we shall, however, be concerned with velocity correlations which are the easiest to measure with hot wire instruments. The turbulent motion behind a grid in a wind tunnel has been found to be approximately homogeneous and isotropic in the above sense. An- isotropy is, however, generally found in the large scale eddies, and becomes prominent when the Reynolds number is relatively low. C,5. Conventional Approach to the Statistical Theory of Turbu- lence. Since a basic theoretical treatment of the frequency distribution function has not yet been developed to an applicable stage (cf. [5]), cur- rent statistical theories of turbulence are usually concerned with readily measurable quantities. This has the advantage that experimental infor- mation can be easily resorted to when purely theoretical considerations become uncertain. Instead of dealing with the distribution functions, we consider correlation functions, which can be more readily measured by the hot wire technique. These are indeed the moments of the distribution functions, as one can readily see from the formulas (Hq. 3-1 and 3-2) and similar ones for correlations of higher orders. As higher and higher corre- lations are known the over-all properties of the distribution functions are known with increasing detail. Mathematically, the correlation represen- tation can be shown to be equivalent to a spectral representation, con- sidering energy distribution among various wave numbers or scales. The two types of descriptions, however, exhibit different aspects of the same physical phenomena. Both of them will therefore be used in the following developments of the theory. 4 Indeed, general isotropy implies homogeneity, but the phrase “homogeneous isotropic turbulence”’ is usually preferred as more descriptive. ( 201 ) C - STATISTICAL THEORIES OF TURBULENCE CHAPTER 2. MATHEMATICAL FORMULATION OF THE THEORY OF HOMOGENEOUS TURBULENCE C,6. Kinematics of Homogeneous Isotropic Turbulence. Cor- relation Theory. In this chapter, we shall develop the theoretical concepts used for the description of homogeneous turbulence. The main body of the discussion will be limited to the isotropic case. The general case of anisotropic turbulence will be taken up in Art. 11. The statistical correlation of velocity fluctuations at two points is the most commonly used quantity for describing the structure of an isotropic field of turbulence. Clearly, the larger the size of the eddies, the further the correlation extends. Velocity correlations are used not only because they are the easiest to measure, but also because correlations involving pressure fluctuations are theoretically representable in terms of them. Experimentally, it is as yet difficult to determine correlations involving pressure fluctuations. In general, we shall be dealing with the correlation of quantities at several points and at different instants of time. For ex- ample, for three points P, P’, and P” in a field of turbulent motion, we may wish to consider the correlation w;(P)u2(P’)p(P”), where wi(P) and u2(P’) are respectively the components of velocity in the direction of the x, axis at the point P and in the direction of the x2 axis at the point P’, and p(P’’) is the pressure at the point P’’. Statistical correlations of the velocity components at one point are exemplified by the Reynolds stresses. In the order of increasing complexity, we next consider correlations at two points. As explained above, we now deal with the special case of isotropic turbulence. Double velocity correlations. Since there is no preferred choice of the coordinate system in the isotropic case, it is clear that the correlations must be basically characterized by the directions of the velocity com- ponents relative to the vector PP’ joining the two points at which the velocities are considered. It is therefore convenient to consider a longi- tudinal correlation coefficient f(r) defined by (see Fig. C,6a)® Uru, = wf(r) (6-1) Similarly, one may define a transverse correlation coefficient g(r) by uu, = wg(r) (6-2) for two parallel velocity components perpendicular to PP’. It is obvious from isotropy that this correlation is independent of the particular pair 5 Here the line PP’ lies in the direction of the x; axis, and the three mutually per- pendicular components U,, Wt, Up are U1, U2, Us. ( 202 ) C,6 - KINEMATICS OF TURBULENCE of parallel components taken. Now the velocity at a given point may be expressed as a linear combination of three mutually perpendicular com- ponents, taken along and perpendicular to PP’. The general velocity correlation between P and P’ can therefore be expressed in terms of the nine correlations between u,, uw, u, and u,, ui, ui, where u, and wu’, are components perpendicular to both u, and u,. By isotropy, correlations like uru, and u,u, are zero, and we see that an arbitrary velocity correlation can be expressed in terms of the two basic correlations f(r) and g(r). In fact, if ui(2 = 1, 2, 3) are the components of velocity at P(x,), and a eR ee ee TEC) | te | ul, g(r) u2 ; h(r) u, SSS u2 u; | u2 | us q(r) Fig. C,6a. Diagram illustrating the definition of the principal correlation functions in isotropic turbulence. u;(7 = 1, 2, 3) are those at P’(x’), von Kd4rmé4n [1/3] has shown by direct calculation that Ry = uu, = wv eee rj + g(r) 5. | (6-3) where r; = x; — 2, and 6,;is the Kronecker delta (6,; = Lifz = 7, 6; = 0 if ¢ #7). The correlation coefficient uwv;,/u? will be denoted by Ri;, and is equal to the expression in the brackets in Eq. 6-3. A derivation of Eq. 6-3, following the method of Robertson [1/4] will be given at the end of this article. By using the correlation tensor (Eq. 6-3), correlations involving ve- locity derivatives can be conveniently calculated. For example, if one ( 203 ) C: STATISTICAL THEORIES OF TURBULENCE wants to calculate the correlation between wu; at P and the derivatives du/dx;, at P’, one has only to use the following identity: 0 Ne zk! Ui ele CV (uu) = ax! Uy = The above transformations are made by using the general rule for aver- aging a derivative and the definition of r;. Similarly du; du; _ = «9 9 On Oe DD Gham © 2 0 a Ui; (6-4) If we now make use of the equation of continuity du;/dz; = 0, we may obtain from Eq. 6-4 — — (6-5) By using Eq. 6-3, this eventually gives rise to the single relation g=f+52 (6-6) connecting the two correlation functions f and g. Thus, in homogeneous isotropic turbulence, all the correlation functions of the second order can be expressed in terms of a single correlation function, say f(r). We shall now show that the correlation tensor (Eq. 6-3) is an even function of 7;. To do this, it is only necessary to show that f(r) is an even function of r. It then follows from Eq. 5-6 that g(r) is also an even func- tion, and the desired result becomes obvious from the formula (Eq. 6-3). Consider two points P and P’ along the z axis at a distance r apart. Then u*f(r) = ulxz)u(@ + 1) Expanding u(x + r) into a Taylor series, we obtain UU ii fQ) = 14+ r+ ert The coefficients of this power series can be simplified as follows by using the condition of homogeneity: UU; = 5(U")» = io (Oe) — Ori It can be easily seen that all coefficients of the odd powers of r are zero. Thus, fo) =1-5Mr tee (6-7) is an even function of r. C,6 - KINEMATICS OF TURBULENCE The idea of using the correlation function for isotropic turbulence was first introduced by Taylor [1/5], who also gave the above proof that they are even functions. The correlation tensor was first introduced by von Karman [73], who also deduced Eq. 6-6. Detailed experimental veri- fication of this relation (Eq. 6-6) seems to have first been made by Macphail [16], and reconfirmed by later experimenters (see Fig. C,6b). Fig. C,6b. Experimental verification of von Kdérmén’s relation for isotropic turbu- lence, after Macphail [16]. Ri = f(r), Re = g(r). M denotes mesh width, y and z are distances parallel to the grid. Triple velocity correlations. Continuing the study of velocity corre- lation, one would naturally be led to correlations for velocity components at three points P, P’, and P”: Te = wT 5h = AUP OH UP aie) (6-8) This triple correlation tensor is a function of the two vectors PP’ and PP”, say, and shows clearly that we are dealing with multiple-point tensors. Often one needs only the correlation tensor 7';;,, for two points JE pyavel J2Me (yn = Whe, == VAP AOA Ue) (6-9) It then becomes a function of the vector PP’. Such a two-point triple correlation tensor was first studied by von Kérmdén and Howarth [17], who showed that, because of isotropy, it can be expressed in terms of ( 205 ) C : STATISTICAL THEORIES OF TURBULENCE three scalar triple correlation coefficients h, k, and g in the following manner :° = k—-h-—2 h Ine = Soe Trite + Site — + Ox; i =F Suri 2 (6-10) r r r r The definitions of h, k, and q are shown in Fig. C,6a. Again, the equation of continuity leads to , OT 5k se ale (6-11) From this, the following two relations between the three quantities h, k, and g may be deduced: k = —2h r oh (6-12) aaa rasta 2 or expressing all triple correlation functions in terms of a single scalar function. t can be shown, by the method of power-series expansion used above for the study of the double correlations, that h, k, and q are odd functions of r, and that their series expansions begin with the third powers of r. Triple correlations seem to have been first measured directly by Townsend [1/0]. The more recent results of Kistler, O’Brien, and Corrsin [18] are shown in Fig. C,6c. Higher velocity correlations. Correlation tensors involving one ve- locity component each from n different points are multiple-point tensors involving n — 1 positional vectors. Correlations involving pressure. Correlations involving pressure are exemplified by (1) pP)pP") and (2) p(P)u(P’) The first one is obviously a scalar quantity, which, from kinematical considerations alone, is not connected with velocity correlations. How- ever, by making use of dynamical relations, it can be connected with velocity correlations of the fourth order. This will therefore be taken up in later sections (Art. 7 and 16). On the other hand, von K4rmdn and Howarth [17] showed that pu; = 0 (6-13) from the requirements of isotropy and incompressibility alone. Robertson’s invariant theory. As the need for developing more com- plicated correlations arises, one must employ some systematic methods 6 Derivation of Eq. 6-10 can be carried out by direct transformation of coordinates and application of the condition of continuity. That is the original method used by von Ka4rmd4n and Howarth. The method of Robertson described below yields the result more readily. ( 206 ) /C,6 - KINEMATICS OF TURBULENCE ee ere ce onl © eee aie ND eee ee r,cm Fig. C,6c. Top, double correlation functions involving velocity and temperature fluctuations u and 6 behind a heated grid set perpendicular to a uniform air stream. O(x) O(a + 1) 6’(x)0'(x + 1) fr) = u(x)u(x + 7) ee uw’ (x)u' (a + 1) where x and r denote distance in the direction of the stream and a dash indicates the root mean square value. M is the mesh width of the grid (after Kistler, O’Brien, and Corrsin [18]). Bottom, triple correlation functions obtained by the same authors under the same conditions as those in Fig. C,6c, top. mr) = w?(x)u(x + r) u’2(z)u' (a + 1) u(x) 0(x)O(x + 1) u(x) 6’ (x2)6'(x + 1) k(r) = p(r) = ( 207 ) C - STATISTICAL THEORIES OF TURBULENCE for their deduction. Robertson [1/4] gave such a method based on a con- sideration of invariants. It proved very useful in later developments, par- ticularly in the study of homogeneous anisotropic turbulence. We shall give below the development for the double velocity correlation tensor as an illustration of the method. Consider two arbitrary unit vectors a; and b;. Then the correlation between the velocity components u,a; at P and uib; at P’ is a scalar quantity Q, independent of rotation of the coordinate system: Q = UiaiUzb; = wR aid; (6-14) It must therefore be a scalar function of all the scalar quantities involved in the problems, namely, all the scalar quantities formed with the vectors a:, b;, and r;. These invariants are the following: (1) a,a; = I b,b; = i TK = r2 (ii) Qibs, airs, Ort: In addition, the determinant formed of these vectors is an invariant under rotation. This may be written in the form (iii) Eijk QiD iT k where ¢;;, is the alternating symbol: e:;, = 1if (7,7, k) is C, 2, 3) or its cyclic permutation, ej. = —1,if (2,7, k) is (1, 2, 3) or its cyclic permutation, and €ijx = 0 otherwise. We note that Q is a bilinear expression in the vectors a; and b;. Hence, Q = Qi(7)ab; + Qo(r)aribjr; + Qs(r)ejnaidsre since this is the most general bilinear form in a; and b; that can be formed from the invariants cited above. If one now imposes the further condition that Q must also be invariant under a reflection, which changes 7; into —r;, it is clear that Q; = 0, and hence Q = (Q15i5 + Qorir;) aid; Since a; and b; are arbitrary unit vectors, it is at once clear that wR; = Q1dij + Qorir; (6-15) This may be identified with Eq. 5-3 if Q:1 and Q, are related to f(r, ¢) and g(r, t) as follows: Q=wg9, Q= ype = Z (6-16) C,7. Dynamics of Isotropic Turbulence. The dynamics of isotropic turbulence is governed by the Navier-Stokes equations of motion OU; Otieee 1 Op Bi We mia + vAUu; (7-1 ) ( 208 ) C,7 - DYNAMICS OF ISOTROPIC TURBULENCE where p is the density of the fluid, p is the pressure, v is the kinematic viscosity coefficient, and A is the Laplacian operator. It might be ex- pected that one could, from Eq. 7-1, derive the equations governing the behavior of all the statistical properties of turbulent motion, such as the level of turbulence, the correlation functions, etc. However, as one pro- ceeds to construct the equations for such purposes, it becomes at once clear that we are always faced with the difficulty of having fewer equa- tions than unknowns, caused primarily by the nonlinear terms in the differential equations. Unless additional assumptions are introduced, de- ductions from such an approach are quite limited. In this article, we only discuss the results following the formal construction of the equations for the change of the correlation functions. The necessary additional assump- tions will be taken up later (Art. 16 and 17). To obtain the equation for the change of the double correlation func- tion, one may multiply Eq. 6-1 by uw, and add to it a similar equation obtained by the interchange of the role of the points P and P’. The lead- ing term of the combined equation is then / ee ee Upon averaging, this yields an equation for the time rate of change of the double correlation tensor u?R;,. However, there are clearly terms of other types appearing in the equation. The nonlinear term on the left of Eq. 7-1 gives rise to triple correlations and the pressure term gives rise to a pressure-velocity corre- lation. It can be easily verified by using Eq. 6-13 that the pressure term vanishes identically, and the equation finally reduces to fe) ~ Qe i oa ry (2 in) ue or; (Vige + Piz) = QvwARs (7B) 4) The appearance of the triple correlation in Eq. 7-2 would suggest the attempt to establish a relation governing its time rate of change. This can be done by combining three equations with leading terms , IH) pos Oly. , OU; Uz,Uj, “Ot 2) i ; obtained by multiplying the equations of motion at P, P’, and P”’ respec- tively by suitable factors. Upon averaging, an equation for (0/dt) (u37’:x,1) is obtained, but this equation also involves correlations of the fourth order and pressure-velocity correlations. The latter can be eliminated in terms of velocity correlations by the following process. From Eq. 7-1, one may obtain, by taking its divergence, a Poisson equation __ dus an Ox; Ox; Ap (7-3) ( 209 ) C- STATISTICAL THEORIES OF TURBULENCE relating the pressure to the instantaneous velocity. One may then ob- tain, for Senne, Apu;u; in terms of velocity correlations and attempt to calculate pujul’ by integration. Thus, one may expect that, in general, a system of differential equa- tions can be obtained for the correlation coefficients of different orders, each involving velocity correlations of one order higher. To obtain a de- ductive theory, it would be necessary to interrupt this process by some judicious assumption (suggested by experimental information or other theoretical considerations) connecting higher order correlations with ones of lower orders. (See Art. 16 and 17.) Many of the existing investigations involve only the dynamical equa- tion (Eq. 7-2) for double correlations. The equations for higher corre- lations have been exploited only recently. Now, Eq. 7-2 represents a sys- tem of six equations. However, since Ri, and 7; are each determined by a single scalar function (Eq. 6-3 and 6-10), it should be possible to re- duce Eq. 7-2 to a single equation. In fact, von Kdrman and Howarth [17] found it to be = (u*f) + 2u (2 +r =) = 2yu? 2 2S we) r or A direct Dei verification of this equation has been made by Stewart [//]. If one expands both f and h as power series of 7, one obtains a series of relations among the derivatives of those functions. The first of these is commonly written in the form 2 = = 10)— (7-5) where \ is Taylor’s vorticity scale defined by Le OF een -(#) | (7-6) The relation (Eq. 7-5) essentially gives the rate of decrease of kinetic energy. It was first established by Taylor [15], both theoretically and experimentally. The equations corresponding to the higher powers of r will be discussed in connection with the small scale structure of turbu- lence (Art. 18). C,8. The Spectral Theory of Isotropic Turbulence. The early adoption of statistical correlations for the description of isotropic turbu- lence is at least partly due to the fact that they are relatively easy to measure. Another powerful method for describing a fluctuating field is to analyze it into Fourier components, i.e. to adopt the spectral approach. It is well known that the spectral theory and the correlation theory are ( 210 ) C,8 - SPECTRAL THEORY OF ISOTROPIC TURBULENCE intimately connected with each other by simple mathematical transfor- mations. Physically speaking, however, the two methods of description put different emphasis on the different aspects of the same phenomena. The spectral theory is often found to give a clearer description of the basic mechanism of turbulence. Spectral analysis has long been used for the study of electromagnetic waves, such as the radiation of heat and light. It was first introduced into the study of turbulence by Taylor [1/9]. Taylor made spectrum measure- ments, behind a grid in a wind tunnel, of the velocity fluctuation as regis- tered by a hot wire fixed in the wind tunnel. This is a fluctuation in time. But Taylor assumed’ that “‘the sequence of changes in wu at the fixed point are simply due to the passage of an unchanging pattern of turbu- lent motion over the point.’’ The variation is then essentially the same as that in space, and the spectrum he observed corresponds to a one- dimensional Fourier analysis of the field of turbulence in the direction of the wind. . The field of turbulence in the wind tunnel is obviously not homogene- ous in the direction of the wind. However, in developing the theory, we shall consider a homogeneous field and its Fourier analysis. In isotropic turbulence, the analysis would be the same in all directions, provided we are always dealing with the component of velocity in the direction chosen for the analysis. The transverse component in general has a different spectrum whether the turbulence is isotropic or not. In the case of turbulent motion, we may formulate the Fourier trans- form relations between the power spectrum and the correlation function as follows. If 4Fi(x)dx is the amount of kinetic energy per unit mass, associated with the longitudinal component of the velocity, and lying in the range of wave numbers (x, « + dx), then F1(x) is related to the longi- tudinal correlation function f(r) by the pair of Fourier transform relations: AP) = is F(x) cos xrdx ae (8-1) Fi(k) = — il f(r) cos xrdr t Jo It is clear from the first formula in Eq. 8-1 that yw = ih ° Fi(x)de (8-2) recapitulating the original physical interpretation of F(x). To clarify our concepts, a derivation of Eq. 8-1 will be given in the next section. 7 A theoretical analysis justifying Taylor’s assumption was given by Lin [20]. A thorough experimental investigation, including measurements of velocity correlations involving both time and space separations, was made by Favre, Gaviglio, and Dumas (see [21] and the references quoted). ( 211 ) CQ - STATISTICAL THEORIES OF TURBULENCE Taylor made use of Eq. 8-1 to connect the observed time spectrum with the spatial correlation function by way of his assumption. To do this, the spatial distance r is replaced by Ut and the time frequency n is related to x by KU = 2rn Then, u*f(r) | F(n) cos — dn ; (8-3) FG) = a f(r) cos OO dr These relations were actually well verified, justifying his assumption ex- perimentally. (See Fig. C,8; after Stewart and Townsend [22].) From Eq. 8-3 we can calculate the rate of energy dissipation in terms of the spectrum. It is easy to show that wf"(0) = =: [ Tieaben (8-4) which is proportional to the rate of energy dissipation (cf. Eq. 7-5). This formula shows that the high frequency components are more important for the dissipation of energy. In fact, Taylor found from an analysis of his measurements of spectrum that the dissipation of energy is practically all associated with such high frequency components which contain a negligible amount of energy. This has a very important bearing on later develop- ments (see Art. 13 on Kolmogoroff’s theory). The above spectral considerations do not give a proper representation of the energy distribution among various scales. For theoretical purposes, one should then consider spectral functions obtained by a three-dimen- sional harmonic analysis. As it will be shown in Art. 10, the three-dimen- sional spectrum F(x) is connected with the one-dimensional spectrum F1(x) by the relation [cf. Eq. 10-3], F(x) = gle°Fy (x) — «Fy («)] (8-5) The kinetic energy per unit volume, for each component of the motion, lying in the range of spacial frequencies (x, x + dx) is now given by’ 4pF (x)dx, the functions F(x) and F,(x) being both normalized to give “2 = iL “dle = iL ° Fi(ede (8-6) It is now easy to obtain the equation for the change of spectrum. We take the cosine transform of the K4rmdn-Howarth equation and then apply the operation Lie Catan) 3 ie ae Ok (12) C,8 - SPECTRAL THEORY OF ISOTROPIC TURBULENCE Spectral energy density, arbitrary units Frequency, hertz Fig. C,8. Experimental verification of the Fourier transform relation between space correlation and time spectrum for turbulent fluctuations behind a grid in a wind tunnel (after Stewart and Townsend [22]). This leads to an equation of the form + W = — 200 (8-7) In the above equation, W(x, ¢) is connected with the triple correlation function h(r, t) by the following relations: K We, t) = AH) — HO] (8-8) Gao) C : STATISTICAL THEORIES OF TURBULENCE where Hae il Dri Sn era rae (8-9) h(r) = if * eHy(x) sin xrde It is clear that the quantity W(x, t) in Eq. 8-7 represents the transfer of energy among various frequencies. The above formula for W(x, ¢) also shows that ih Wide = 0 (8-10) which means that no energy is generated or lost while it is redistributed among various scales. The rate of dissipation is obtained from Eq. 8-7 by integrating it with respect to x from x = O tox = o: erie emer an Obie avert | bs eS ae a | a= 2» | KF dk (8-11) Exactly as in the case of the correlation theory, one cannot proceed much further with the basic equation (Eq. 8-7) without a more specific knowledge of W. However, with the physical interpretation that W(x, t) represents the transfer of energy among various frequencies, it has been found possible to obtain certain plausible formulas connecting W(x, 2) with F(x, ¢) and to make reasonable deductions. (Cf. Art. 17.) C,9. Spectral Analysis in One Dimension. We shall now develop briefly the one-dimensional spectral analysis of a field of turbulence and derive the Fourier transform relations (Eq. 8-1). In a homogeneous (not necessarily isotropic) field of turbulence, let u(x) be the velocity at the point z in the direction of the x axis. It re- mains finite as x > +. This makes its Fourier analysis more difficult than that of a function which vanishes rapidly at infinity. For such a function, ¢(x), we have the pair of Fourier transform relations o(x) = oe a(x)e-*dx ne) = 5 fi be d(a)e*da where a(—x) is equal to the complex conjugate a*(x) for real ¢(a), and |a(x)|? is a measure of the energy content associated with the wave num- ber or spatial frequency x. However, since the velocity fluctuation u(x) in a homogeneous field of turbulence does not approach zero asx— + ©, we cannot put u(x) in place of ¢(x) in the above relation. Instead we ( 214 ) C,9 - SPECTRAL ANALYSIS IN ONE DIMENSION first consider es a(x, X) = — / u(ax)e"*dx (9-1) Dae | ase and then try to adopt a suitable limiting process as X — o. In fact, we want to consider first the amplitude not at x but associated with a finite range of values of x. We integrate Eq. 9-1 between x and x + Ak, obtaining BANG. 30) = iL Ee NOV 1 T Oe Here, we may take the limit as X — o, and obtain pres) exeda[ei(4d2 — 1] AA(k) = - fe oe e*dz[etA92 — I] (9-2) since the integral is now convergent. We now form the expression for the measure of energy AA (x) - AA*(x) and calculate its statistical average. Then AA(k) - AA*(k) = = iL — [ei(Awe ee 1Jdz cL = wR(ax’ — x)[e*49” — 1]dx’ where RF is the statistical correlation between u(x) and u(x’). The inner integral can be transformed by replacing x’ — x by &. Then it becomes enix ie Criss w?R(£)[etnz-tt — I ]dE —-o 7 + eae A and we obtain AA(k)AA*(x) ae Ge = = a(Ak)e We shall now divide both sides by Ax and replace (Ax)z by a new varia- ble ¢. Then we obtain a measure of the ‘‘density of energy”: —i(Akx)z—i(Ak)& 1] AA (kK)AA*(x) ue u? si d¢ ; este (An) — ee Ol ee Oe It is easy to see that the right-hand side has a limit as Ax — 0. We there- fore have the interesting situation that AA(x) - AA*(x) is of the order of Ax and not of the order of (Ax)?. Let the limit be denoted by F1(x)/2. Then F(x) aie if * —ik 9 ra el R(é)e dé ( 215 ) CGC: STATISTICAL THEORIES OF TURBULENCE and wWR(E) = 4 [™, Falettde It is clear that F1(«) must be even when R(é) is real, and the above equations become the same as Eq. 8-1. C,10. Spectral Analysis in Three Dimensions. The one-dimen- sional spectrum, however, does not give an exact representation of the distribution of energy among the scales. Consider a simple harmonic vari- ation with wave number « in a direction making an angle @ with the z axis Fig. C,10. Diagram illustrating the relationship between one-dimensional and three-dimensional Fourier analysis of a field of turbulence. (Fig. C,10). Its period in the x direction would be longer and the wave number in a harmonic analysis in the z direction is Kz = kK COS 0 (10-1) Thus a modified picture is obtained of the energy distribution among the various scales. In the case of isotropic turbulence, as we shall demonstrate below, it is easy to establish the relation between the one-dimensional spectrum F';(x) and the spectrum function F(x) corresponding to a three- dimensional Fourier analysis. The relation is Fi) = 3 i : _ (2? — Fe’) (10-2) or, upon differentiation, F(x) = g3lePFy' (x) — «Py (x)] (10-3) Note that wu? = fPF(«)d« = JRF il(x)dk. ( 216 ) C,10 - SPECTRAL ANALYSIS IN THREE DIMENSIONS The analysis of the y component of the motion in the z direction leads to a spectrum mui) = 2 i : _ (2 + x!) F(x’) (10-4) by combining Eq. 10-2 and 10-4, we obtain, after a little calculation, Fe) = — 505 Fate) ” 5 F(0) | (10-5) This relation is more convenient for obtaining F(x) from experimental data. It is numerically more accurate than Eq. 10-3 since only one differ- entiation is involved. To establish the relations (Eq. 10-2 and 10-4) let us write*® the three- dimensional Fourier analysis of the velocity in the following form: us = Y Aalneteore (10-6) Then, for a wave in the direction of the vector «;, the equation of con- tinuity gives Kj Aj == (() (10-7) This means that all the motion associated with the vector wave number x; must be perpendicular to this vector. Consider now the contribution to the spectrum of a Fourier analysis in the x direction of a component of turbulent motion with vector wave number «;. In the first place, the motion appears to have a space frequency kz defined by Eq. 10-1. Secondly, the motion has in general all three com- ponents. The z component is (cf. Eq. 10-2 and Fig. C,10).' ui(x;) = —A(k) cos ¢@sin 6, A? = A;A; where @ is the angle between x; and the z axis, and ¢ is the angle which the velocity vector A; makes with the plane containing «x; and the z axis. Thus, averaging over the angle ¢, we have (cf. Eq. 10-1) ey, = Sn al Ky [wi(xy)|? = |AC)| 9 = |A(x)| A - 4) Consider now a distribution of energy in the « space. Let the total kinetic energy per unit mass and per unit volume of the « space be $®(x;) ; i.e. 4@(x;)dkidk2dx3 is the energy contained in the range «, x + dx. To obtain the energy per unit mass 3F'1(«:) lying between «1, «1 + dx; and associated with one component of the motion, one must multiply this ex- pression with the factor (1 — «2/«?)/2 and then integrate for all values of 8 The reasoning here is essentially that used by Heisenberg [23]. (217 C - STATISTICAL THEORIES OF TURBULENCE kg ad x3 while keeping «x: constant. Thus, the one-dimensional spectrum is F4(k1) = // : (1 , 3) ®(k;)dkedks3 Now, the three-dimensional spectrum is isotropic, so that 2 F(e) = ae considering all «;’s with the same magnitude x. We have finally F (x1) = F4(«1) -- F4(—k1) = I ( = ) ae dkodks This is easily transformed into Eq. 10-2 by carrying out the integration in a polar coordinate system in the plane of ka, xs. C,11. General Theory of Homogeneous Anisotropic Turbulence. The above development of the theory of homogeneous isotropic turbu- lence can be generalized to remove the restriction of isotropy. Such a generalization is necessary because anisotropy of turbulence, particularly in the largest eddies, does occur in practice. We shall outline here only the main features of the developments and conclusions, pointing out especially the difference between the isotropic and anisotropic cases. The concept of correlation functions requires very little modification, although it is now obviously impossible to represent the double corre- lation functions, for example, in terms of a single scalar function. The spectral function must be replaced by a spectral tensor, which may be defined as the three-dimensional Fourier transform of the double corre- lation tensor. Thus, ®:z(kj;) = 57 i | / Rix (1m) et" dr (11m) (11-1) and Rou il if i Bis( tem) Oi») dr (Ke) (11-2) It can be shown that ©,; represents the energy density in the wave num- ber space. In the case of isotropic turbulence, F = Arx’°,;; (11-3) Because of the condition of vanishing divergence of the correlation tensor, we obtain D3 5k; = 0 (11-4) ( 218 ) C,12 - LARGE SCALE STRUCTURE OF TURBULENCE and ®,; can be expressed in the form Big = W(Km)(K? — Kitty) + xi(Km) x7 (Km) (11-5) where y(x,,) is a scalar function of the vector kn, xi(km) is a vector perpen- dicular to km, and x} is its complex conjugate. When the turbulence is isotropic, x; = 0, and y(«,,) is a function of the magnitude « only. The form (Hq. 11-5) is due to Kampé de Fériet [24]. The dynamical equations for anisotropic turbulence are more compli- cated than those for isotropic turbulence, among other things, by the presence of the pressure terms in the equations of the change of double correlations. In the correlation form, the equations are of = Pe, de fie de Dos (11-6) where NY rere CL eer Ra 5 ( ar, pur, ai 0 ui) (11-7) 0 aay | APR Tn = —— (Uv; — We u,) (11-8) Or p In the spectral form, we have OD, aL = ID jz + Ox = 20K F iy (11-9) where II, and 0, are respectively the Fourier transforms of Px, and 7’. Obviously P;; = 0, so that II,;; = 0. Thus the pressure fluctuations have no effect on the total energy density F;,; their influence produces a re- distribution of energy among the various directions. It is not immediately evident whether the net effect is to make the turbulent field more or less isotropic, but general evidence seems to indicate that the former is the case. The above developments are mostly due to Batchelor [25]. Other detailed studies of anisotropic turbulence have been by Batchelor [26], Chandrasekhar [27], and others. The reader is referred to the original papers. CHAPTER 3. PHYSICAL ASPECTS OF THE THEORY OF HOMOGENEOUS TURBULENCE C,12. Large Secale Structure of Turbulence. In the following articles, we shall make use of the methods developed above—the corre- lation and spectral theories—to study the nature of turbulent motion. As pointed out above, the theory by itself allows us to reach only partial ( 219 ) C - STATISTICAL THEORIES OF TURBULENCE results. Some theoretical speculation and assumptions will therefore be introduced in the following discussions for the purpose of reaching definite conclusions. We shall begin by considering the large scale structure of turbulence, which is associated with small values of « in the spectral representation and large values of r in the correlation representation. Let us now consider the second equation in Kq. 8-1, Pe ee Fi(x) = = f(r) cos (xr)dr (12-1) 0 and expand cos (xr) into a power series. We obtain Fwy = #(5,-2% aot -) (12-2) where In = ii f(r)rndr (12-3) Such a step is justified only when the function f(r) vanishes sufficiently rapidly at infinity (e.g. as a negative exponential function) so that the integrals J, are convergent. In that case, one may derive from Eq. 12-2 a power series expansion for the three-dimensional spectrum F(x) by using Eq. 10-3. This gives 2, Oe ae e fae. ) (12-4) Similarly, assuming that h(r) also vanishes sufficiently rapidly at infinity, one can show that the transfer function W(x, t) behaves as x® for small values of «. The spectral equation (Eq. 8-7) then shows that d 2 — d p, " 4 = di (wJ4) = i E i f(r)r ar| = (12-5) It then follows that u? ik f(r)r4dr = J, a constant (12-6) Thus, the large scale motions are permanent in the sense that the princi- pal part of F(x) for small values of « remains unchanged. The above derivation (including explicit statements of the necessary convergence assumptions) was given by Lin [28] for the spectral interpre- tation of the parameter J, which was first obtained by Loitsiansky [29] from the Kérmén-Howarth equation. Indeed, if one multiplies that equa- tion by r* and then integrates it with respect to r from zero to infinity, one obtains : E ile feryrae | = 2u® lim (74h) (12-7) Tr 0 ( 220 ) C,13 - SMALL SCALE STRUCTURE OF TURBULENCE provided the integral involved is convergent. If, in addition, h(r, ¢) van- ishes sufficiently rapidly at infinity so that jimh —_ 0 (12-8) the relation (Eq. 12-5) is obtained. : It must be noted that there is no a priori reason® for the convergence of the integrals (Eq. 12-3) and the validity of Eq. 12-8. As a matter of fact, recent investigations of Batchelor and Proudman [3/] show that even if f(r) is exponentially small at infinity at an initial instant, because of the influence of the long range pressure forces, one can only be sure that it will be no larger than O(r—*) when r is large, although the possi- bility of an exponentially small behavior is by no means excluded. We are therefore only assured of the leading term in Eq. 12-4 and the existence of the Loitsiansky parameter, jf SUF i f(r)r4dr (12-9) However, the constancy of J depends on the relation (Eq. 12-8), which is shown to be not generally true by the analysis of Batchelor and Proud- man. On the other hand, for low Reynolds numbers based on the turbu- lence level u, the term on the right-hand side of Eq. 12-7 becomes negli- gible, and the Loitsiansky parameter is indeed approximately constant. (Cf. Art. 14 and 15 for the part dealing with the final period of decay.) From a physical point of view, any prediction of the behavior of the largest eddies must be regarded with some reserve, since it is expected to be dependent on the experimental apparatus. If the general scale of turbu- lence is much smaller than the dimensions of the experimental apparatus, it would appear that this complication may be avoided by a proper interpretation of the above results. The integrals (Eq. 12-3) may, for ex- ample, be considered as extending over a distance much larger than the scale of turbulence but still much smaller than the scale of the apparatus. Generalization of the above discussions to the anisotropic case has been made by Batchelor [25]. The earlier conclusions are again modified by the work of Batchelor and Proudman [31]. In fact, in the anisotropic case, the correlation tensor R;; is shown to be in general of the order of r—*, so that even the existence of a Loitsiansky parameter is in doubt. C,13. Small Scale Structure of Turbulence. Kolmogoroff’s Theory. We now turn to consider the small scale structure of turbu- lence. Here the formal relations analogous to Eq. 12-1, 12-2, 12-3, and 12-4 are obtained by expanding cos (xr) into a power series in the first equation in Eq. 8-1: wh(r) =f,” File) cos («r)de (13-1) 9 Cf. Birkhoff [30]. (221) C+ STATISTICAL THEORIES OF TURBULENCE We then obtain a power series for f(r) in the form fy = 14 EO ay... (13-2) with (—1)"u2f2(0) = Io, = i ” nF i(x)dx, m=0,1,... (13-3) In terms of the three-dimensional spectrum, these integrals become if KF’ (x) dx (13-4) 33 (2n + 1)(2n + 3) I Here it is useful to recall that Io is proportional to the energy, and that I, is proportional to the rate of energy dissipation. Consider now the dynamical relations in the correlation theory. We expand both f(r, t) and A(r, t) in power series of 7, Go ee Or h’"(0 (13-5) h(r, t) = = ae Se and substitute them into the K4rm4n-Howarth equation (Eq. 7-4). As observed before (Art. 7) the terms independent of r give the energy relation. The terms in r? give the vorticity equation in the form 2 — — 70hi"'y? = = les (13-6) or 2 ann aie 2 = = 2eoioe 5 = 1055 (13-7) where w; is the vorticity vector, w? is the mean square value of one com- ponent of the vorticity, and \2 is defined by 1 2 Boe ees of = er Phe © Nf, Ola biar SL. (18 8) The second term on the left side of Eq. 13-7 represents the change of vorticity due to stretching or contraction of the vortex tube without the action of viscosity. It is well known that, in a perfect fluid, the circulation around a vortex tube is permanent and hence the vorticity increases at a rate in proportion to its rate of stretching. The right-hand side repre- sents the dissipation of viscosity by viscous forces. Taylor [32] suggested that this relation represents one of the basic mechanisms in the process of turbulent motion. The rotation of the fluid is being slowed down by the effect of viscosity. This loss is partly com- ( 292 ) C,13 - SMALL SCALE STRUCTURE OF TURBULENCE pensated, or even over-compensated, by the stretching of the vortex tubes, due to the diffusive nature of turbulent motion. (Hence one may expect more stretching of the vortex tubes than compression.) Taylor calculated the relative magnitudes of the various quantities by deter- mining fj’ and fi’, and he found that all the three terms in Eq. 13-6 are of the same order of magnitude for his experiments. Such measurements were more accurately made later by Batchelor and Townsend [33] and by Stewart [//]. As noted before (Art. 8), in many experiments the dissipation of energy is practically all associated with the high frequency components which contain a negligible amount of energy. Combining this fact with the mechanism of vortex-stretching just discussed, one can form a reason- able picture of the process of turbulent motion. There are the large energy- containing eddies which contribute very little to the viscous dissipation directly. By their own diffusive motion, small eddies are formed, i.e. the kinetic energy of turbulent motion goes down to smaller scales. It is at these small scales that viscous forces become most effective and the pre- dominant part of the energy dissipation occurs. Thus one forms the pic- ture of an energy reservoir in the large eddies, and a dissipation process in the small eddies which may be presumed to depend very little on the structure of the large eddies except to the extent of the amount of energy supplied to them. This forms the physical basis of Kolmogorofi’s theory of locally isotropic turbulence [34]. Before we go on with the discussion of his theory, it should be empha- sized that the picture is correct only when the diffusive mechanism is strong; i.e. when the inertial forces are large compared with the viscous forces. In other words, the Reynolds number of the turbulent motion must be relatively large. This is well illustrated by the detailed calcu- lations made by Taylor and Green [35] on a model of isotropic turbu- lence.!° Indeed, they found that for very low Reynolds numbers of turbu- lence, defined by Ry = ~|5 the stretching mechanism is not strong enough, so that the magnitude of the vorticity decreases steadily. On the other hand, if the motion starts out at a fairly high R,, the mean square vorticity (and hence also the rate of energy dissipation) first increases to several times its original value due to the stretching mechanism. The kinetic energy of the motion, how- ever, decreases steadily. Eventually, it becomes very low, and the stretch- ing process is so weakened that the vorticity of the motion also decreases steadily. Kolmogoroff’s theory. In line with the above ideas, Kolmogoroff pos- tulates that, at large Reynolds numbers of turbulent motion, the local 10 See also Goldstein [36]. ( 293 ) C - STATISTICAL THEORIES OF TURBULENCE property of turbulent motion should have a universal character described by the following concepts. First, it is locally isotropic whether the large scale motions are isotropic or not.!1 Second, the motion at the very small scales is chiefly governed by the viscous forces and the amount of energy which is handed down to them from the larger eddies. The large eddies tend to break down into smaller eddies due to inertial forces. These in turn break down into still smaller eddies, and so on. At the same time, viscous forces dissipate these eddies at very small scales into heat. In the long series of processes of reaching the smallest eddies, the turbulent mo- tion adjusts itself to some definite state. The further down the scale, the less is the motion dependent on the large eddies. Furthermore, in line with Taylor’s experimental findings, Kolmogoroff essentially postulates that practically all the dissipation of energy occurs at the smallest scales when the Reynolds number of turbulent motion is sufficiently high. To formulate these concepts mathematically, he introduced the corre- lation functions of the type OE tie LC) which is the mean square value of the relative velocity of turbulent mo- tion. The introduction of the relative velocity stresses the local nature. The moments (uw — u’)” would then be emphasized instead of the usual correlations at two points. (In fact, the third moment (uw — w’)* is pro- portional to k(r).) The second step in the formulation of the theory is to introduce the assumption that, for small values of r, these correlation functions depend only on the kinematic viscosity v and the total rate of energy dissipation e. This is in accordance with the previously discussed physical concepts. One can then make some dimensional analysis and construct universal characteristic velocity and length for motion at very small scales. Indeed, from ¢ and », one can only construct the length scale Ae @) (13-9) and the velocity scale = (ve)? (13-10) We may then write (w’ — u)? = (ve)? Baa (*) (13-11) (u’ — u)® = (ve)*Baaa (") (13-12) where Baa and Baa are universal functions for small values of r. 11 See Sec. B on shear flows for the experimental confirmation of this fact. ( 224 ) C,14 - CONSIDERATIONS OF SIMILARITY For very high Reynolds numbers, Kolmogoroff visualizes that, at the larger end of the universal range, there is a range of r for which the vis- cosity coefficient does not play an explicit role. This range may be con- veniently referred to as an inertial subrange. The above relation then implies that (u’ — u)? ~ (er)? (13-13) A definite form of the correlation function is thereby obtained. The concept of Kolmogoroff can also be introduced into the spectral formulation. Thus, at high Reynolds numbers the spectrum F(x) at very high frequencies can be expressed as F(x) = v?nf(«n) (13-14) where the function f(z) has a universal form for large values of z. For the inertial subrange, the spectral function can again be deter- mined completely from dimensional arguments. This gives F(x) ~ &x3 (13-15) This form was first given by Obukhoff [37]. It has received some experi- mental support at high Reynolds numbers.!? With a spectrum of this form, it can be explicitly demonstrated that the dissipation of energy lies essentially in the universal range of Kolmogoroff (cf. [39]). The actual form of the spectrum in the universal range is obviously of basic theoretical interest. By following the general ideas discussed in this section, Townsend [38] developed a more concrete model giving a definite form for the spectrum of the small eddies. The results are in general agreement with experimental observations. The scales 7 and v defined above also occur in the study of the small scale structure even when the Reynolds number is not high. This cannot be interpreted on the basis of Kolmogoroff’s theory, but follows from considerations of self-preservation during the process of decay (see next article). C,14. Considerations of Similarity. As noted above, the general theory of turbulent motion, as developed in Chap. 2, cannot lead to specific predictions without auxiliary considerations. For this reason, von Kérmén and Howarth [17] introduced the idea of self-preservation of correlation functions.!* In terms of the spectral language, this states that the spectrum remains similar in the course of time. Since the energy distribution among the various frequencies is changing through the trans- fer mechanism, this may be reasonably expected provided that there is enough time for the necessary adjustments. In this article, we shall con- 12 Cf. [99] and [101] for detailed discussions. A different form of the spectrum has been recently obtained by Kraichnan [100]. 13 This article follows closely the treatment of von Kérm4n and Lin [39, p. 1]. (295) C : STATISTICAL THEORIES OF TURBULENCE sider the theoretical aspects. Comparison with experiments will be made in the next article. Let us consider the equation (Eq. 8-7) for the change of spectrum ot W = -2ntF and try to find a similarity solution. If V is a characteristic velocity, and J is a characteristic length, then, from dimensional arguments, i Ven), \W = Vee). = Kl (14-1) Thus, the above equation becomes dl. 21 dV yg ie k oq lV® +¥Ol+ BS vO +O = - FeO 42) If the similarity solution is to be valid, one must have ral V dt = (Ohi) (14-3) 21 dV Tea ee aie Vv V1 = 3 (14-5) where ai, @2, and a; are all constants. Eq. 14-2 becomes arey’(é) + (ai + aa)P(E) + 2aséY(E) + w(é) = 0 (14-6) Besides Eq. 14-3, 14-4, and 14-5, it is evident that the mean square value u? and the rate of energy dissipation have to satisfy the relations (ef. Eq. 8-6 and 8-11) w= V? [> vede (14-7) Cue WV Et — 48 ay YL” eupae (14-8) Finally, if the convergence criteria for Loitsiansky’s relation (Eq. 12-6) are assumed to be valid, we have Ve im ne - J (14-9) This system of equations presumes that the transfer term in Hq. 14-2 is considered generally of equal importance with the term expressing the viscous dissipation. It has been shown by Dryden [40] in the equivalent problem of self-preserving correlation functions that such a solution is connected with the statement that the square of the characteristic length is proportional to the time ¢ and the law of decay is expressed by u? ~ (71. ( 226 ) C,14 - CONSIDERATIONS OF SIMILARITY Heisenberg [4/] indicated an equivalent solution for the spectral problem. It is easily seen that these solutions are at variance with Eq. 14-9. In other words, full similarity is only possible when we reject Loitsiansky’s theorem. In addition, experimental evidence clearly indicates that the law of decay and the behavior of the characteristic length during decay exclude the possibility of adopting full similarity as a generally valid assumption for all decay processes. Let us now consider two opposite approaches. In the first approach, we assume that Loitsiansky’s invariant exists and that it plays a role in the similarity of the spectrum. In the second approach, we assume that similarity of the spectrum is occurring only in the eddies contributing appreciably to the dissipation process, and that the largest eddies play no role in determining the similarity of the spectrum. Clearly, the first approach will not yield valid results unless Loitsiansky’s invariant does exist. This is definitely known only in the decay of zsotropic turbulence at very low Reynolds numbers (case (a) below). The second approach is naturally independent of Loitsiansky’s invariant. Let us consider now two opposite specific cases in the first approach: (a) the transfer term is negligible for all frequencies, and (b) the influ- ence of viscous dissipation is restricted to high frequencies whereas for low frequencies the transfer term is the prevailing factor. Case (a), w() = 0, leads to a solution of Eq. 8-7 which has full simi- larity for all frequencies and also satisfies Loitsiansky’s relation. One ob- tains with ¢ = «land 1 = Vt F = const V2lé4e—2” (14-10) or Fi—ACOMSty Viele Cmax (14-11) By using the definition of J in Eq. 14-9, we write Toi mare (14-12) The corresponding correlation function can be easily shown to be NG, Oi Ce (14-13) by using Eq. 8-1 and 10-3. This correlation function was noted by von K4rmé4n and Howarth [1/7], and discussed by Millionshchikov [42], Loitsiansky [29], and Batchelor and Townsend [43]. Karman and Howarth also obtained a more general self-preserving solution in terms of the Whittaker function, with a spectral form F = Cx"e-?”"*, It can be easily shown that the solution must specialize into Eq. 14-13 if the Loitsiansky invariant is to be finite. The law of decay in this case is the five-fourths power law: u? ~ (t — to), dh? = Av(t — to) (14-14) This law of decay and the corresponding correlation function have been ( 297 ) C - STATISTICAL THEORIES OF TURBULENCE verified experimentally by Batchelor and Townsend for the final stage of decay (see Art. 15 for further details). Case (b) has also been treated in the theory of self-preserving corre- lations by von Kaérmdén and Howarth [17] and later by Kolmogoroff [44]. The former authors came to the conclusion that any power law for the decay-time relation may prevail in the decay process. Kolmogoroff pointed out that if one assumes the validity of Loitsiansky’s theorem the relations wi constiimn and) Avi (14-15) must apply.14 Von Karmén [45,46] dealt with the corresponding spectral problem in two communications assuming the specific decay law (Eq. 14-15). It should be reiterated, however, that this first approach, especi- ally in case (b), can only be regarded as tentative because of the un- certainty in the constancy of the Loitsiansky integral. Consider now the second approach. Clearly, the idea of complete similarity, with the rejection of Loitsiansky’s relation, belongs to this case. However, there are physical and mathematical reasons for believing that the large eddies do not play a significant role in the determination of the similarity characteristics in the smaller eddies. We therefore con- sider cases where the similarity requirement is relaxed for an increasing range of frequencies at the end of largest eddies. Case (c). We first consider the assumption that similarity extends over the whole frequency range, with the exception of the lowest. More specifically, we assume that the deviation from similarity shall occur for such small values of « that, whereas the contribution of the deviation is negligible for computation of « (Eq. 8-11), it enters in the calculation of energy (Eq. 8-6). It is easy to see the corresponding assumption in the correlation for- mulation by using Eq. 13-2 and 13-4 in the following form: ino) 2 fs (ere 1 a oe vt —s@l = — Vg ne ay | Feed 1416) n=1 The above assumptions imply that all the higher moments of F(x) are not appreciably influenced by the deviation from similarity. Hence, they are all proportional to V2/-2. Similarity is therefore assumed for u*[1 — f(r)]. This form of the similarity hypothesis was introduced by Lin [48]. Assuming the self-preservation of we FC (u — u’)® = 12u%h(r) 14 See Frenkiel [47] for some discussion of the comparison of Eq. 14-15 with some experiments. and ( 228 ) C,14 - CONSIDERATIONS OF SIMILARITY he derived the law of decay u? = a(t — to)! + 6 (14-17) where a and 0 are constants, with a > 0. This law can be easily obtained from the general relations (Eq. 14-3, 14-4, 14-5, and 14-8), which are valid for any similarity hypothesis. One obtains the positive and nega- tive half-power laws for the change of the characteristic length and the characteristic velocity V, and the inverse square law for the rate of dissi- pation e. To be more specific, one finds that 1 and V may be identified with Kolmogoroff’s characteristic quantities (cf. Eq. 13-9 and 13-10) n= ey and v = (ve)t (14-18) It can easily be seen by introducing these relations into Kq. 14-3, 14-4, and 14-5 that the law of decay is of the form of Eq. 14-17. It is convenient to rewrite the results as follows, with definite physical interpretations attached to the constants. The law of decay is given by {| Do\ 5 pee [LOW Dees _ 10uy = (Pr) Poa = (Pr): up, or rAX* = 10rt {1 Dy (14-19) where w%, is the additive constant giving the departure of the energy con- tent from that in the case of similarity, and Dp is the initial diffusion coefficient 2) 2 olin t20 «26? (14-20) defined according to a formula of the kind suggested earlier by von Karman [7/3]. The changes with time of the characteristic velocity and scale, and of the Reynolds number of turbulence are given by 2 vy? = (10)-#Ryovt—!, 9? = (10)#Retvt, Ry = Ryo (1 4 “eee ) (14-21) 0 where Ryo is the initial Reynolds number of turbulence yy = Tena 2 (14-22) to0 PV It is evident from Eq. 14-19 and 14-21 that the solutions obtained can only be applied to an early stage of the decay process, in which 10u3,t/Do remains small. Case (d). The above assumption is based on the idea that the low frequency components do not have the time to adjust themselves to an equilibrium state. (An investigation of such a concept was made by Lin, and will be briefly presented in Art. 17.) It is specifically assumed that « may be calculated by a similarity spectrum. Goldstein [49] further ( 229 ) CGC: STATISTICAL THEORIES OF TURBULENCE relaxed the requirement and assumed that the similarity spectrum might be adequate only for the calculation of higher moments of F(x). If the similarity spectrum is accurate only for the calculation of ih x4F (x)dx and higher moments, Goldstein shows that the law of decay becomes u(t ae to) =a-+ b(t = to) ae c(t iat to)? (14-23) This includes one more constant than Eq. 14-17. Further generalization involving higher powers of t — ¢ is immediate. Comparison of the laws of decay with experiments will be made in the next article. C,15. The Process of Decay. We shall now examine the whole process of decay and compare the above theoretical laws with experi- ments, whenever such evidence is available. 0.16 0.12 ye Giie 0.08 0.04 x/M Fig. C,15a. Change of vorticity scale during a decay process at low Reynolds numbers. For small Reynolds numbers of turbulence, the nonviscous range can- not be expected to occur. The process of decay may be described by adopting the law (Eq. 14-17) for the first part of the decay process and the law (Eq. 14-14) for the last part. This is shown by the experimental results!® of Batchelor and Townsend (Fig. C,15a). The slope of the curve (\2, vt) begins with a value 10 and ends with a value 4. 15 The experimental agreement in this case should be accepted with some reserva- tion, since so little data are available. See [30,31] for detailed discussions. ( 230 ) C,15 - THE PROCESS OF DECAY For large initial Reynolds numbers of turbulence, von Kdrmdn and Lin [39] made a tentative proposal to divide the process of decay into three stages: (1) the early stage in which the law (Eq. 14-17) holds, (2) the intermediate stage, in which the law (Eq. 14-15) holds, and (3) the final stage in which the Reynolds number is very low and the law (Eq. 14-14) holds. For estimates of the length of the three periods, we refer to the original article. Here it suffices to say that there is as yet no ex- perimental result available to check the theory for the intermediate stage, and that the recent doubt cast on Loitsiansky’s invariant tends to change the basis for such an assumption. Detailed discussions will therefore be given only for the early and the final stages. Final period of decay. When the Reynolds number of the turbulent motion is very low, as it must eventually happen in the final period of decay of a homogeneous field of turbulence, without external supply of energy, the inertial forces are negligible and only the viscous forces are effective. Case (a) discussed in Art. 14 then applies. On the other hand, the problem now admits of an explicit solution. In fact, if the quadratic terms are neglected from the equations of Navier-Stokes, we have Ou; pes I @) = ape tPA (15-1) By the equation of continuity, this leads to Ap = 0 Now the only solution of a Laplace equation which is finite throughout the whole space is a constant. Thus the pressure must be independent of po- sition, and the equation for u; becomes the equation for heat conduction!® OU; Ot => vAu, (15-2) The solution of the initial value problem of this equation is well known to be 1 (2) co [- 2} u(x, Y, @, t) aa ene u(X, a Z, 0) exp | - ed dXdYdZ (15-3) From this, the properties of the motion can be explicitly calculated. In 16 Reissner [50] was the first to attack the problem of turbulence by using the explicit solution of Eq. 15-2. He obtained results analogous to the observed laws of decay. They are, however, more adequate for the discussion of temperature fluctua- tions, and will be taken up again in that connection. The following development is due to Batchelor [51]. ( 231 ) C - STATISTICAL THEORIES OF TURBULENCE particular, it is found that the correlation function is R(E, n, £, b) = gp |[[ 2@t.0.0 exp | - ee | dadbde (15-4) This last formula can be used to evaluate the asymptotic behavior of the correlation function for large values of t. A simpler approach to the problem is to use the spectral tensor. In fact, Eq. 15-2 shows that the pressure terms P;, must be dropped when the nonlinear effect represented by 7, is negligible in the spectral equa- tion (Eq. 11-6), which becomes simply oF Dap: = —2°F (15-5) The general solution of this equation is Fij(km, t) = Fj(km, to) e— 27K? (tbo) (15-6) From this, we may calculate the correlation tensor by a Fourier trans- formation. For large values of ¢ — to, only small values of « are important. Thus one may try to expand Fy, in powers of x, and retain only the lowest terms. Following this method, Batchelor and Proudman [3/] found that the longitudinal correlation coefficient f(r, t) is of the form (Eq. 14-13) for isotropic turbulence and certain very special cases of anisotropic turbu- lence. Previous to this investigation, Batchelor and Townsend [43] com- pared the experimental curve for f(r, ¢) with the Gaussian curve (Kq. 14-13) and found good agreement. At that time, this agreement was ex- plained by assuming F;(km, £) to be essentially expandible as a Taylor series IN km. Since this assumption is now found to be not true in general, other tentative explanations are suggested by Batchelor and Proudman [31]. A critical examination of this problem is clearly warranted. Early period of decay. Much experimental information is available during the early part of the decay process. Recently, Stewart and Town- send [22] summarized their results and compared them with some of the above self-preserving hypotheses. They cautioned against the assump- tion of complete self-preservation, but did not include case (c) in their discussion, which seems to fit all their experimental findings. In Fig. C,15b, the law of decay observed by Stewart and Townsend [22] is presented. Although the variation of \? and u-? both follow the linear law, as they would in the case of complete similarity, the origin of time (or x axis) must be taken differently for the two straight lines. It ( 232 ) Ci5 ° HHE PROGESS OF DECAY can easily be seen that the Reynolds number of turbulence A) steadily decreases in the case shown in the figure, contrary to the law of decay for complete similarity. The earlier experiments of Batchelor and Town- send [33] also show a definite trend for the decrease of Ay. It should be noted that the curves for \? and w-? versus time both have fairly large slopes, and it is therefore more difficult to detect any slight deviation 0.4 Via © Slats grid iS x Parallel cylinder grid U=620cm/sec 0) 50 100 150 200 x, Clin Fig. C,15b. Decay of turbulence behind grids of differing shapes. (After Stewart and Townsend [22].) from a straight line. On the other hand, R, should remain constant according to the assumption of complete similarity and is more sensitive for detecting the departure from such laws. The comparison of the inter- cepts of the straight lines in Fig. C,15b is also a very sensitive method for detecting the same effect. On the basis of the discussions of case (c) of Art. 14, it may be expected that R, should decrease linearly in 4, if u2, > 0, ie. if the energy in the large eddies is smaller than that corre- sponding to full similarity. The observed decrease of R, indicates that this is indeed the case. (6233) C - STATISTICAL THEORIES OF TURBULENCE A more definite verification of case (c) is provided by their measure- ment of the spectrum, which is reproduced in Fig. C,15c and C,15d. Fig. C,15c¢ gives the one-dimensional spectrum F(x) which shows a large departure from similarity at low values of x. Fig. C,15d shows that, for x/M Rau 60 80 7 2625 100 40 60 ¢5250 80 30 40 +10500 60 30°, 21000 e a x 10) ® ® i} B & rN Pete x S : : Q.2 K/ Ks Fig. C,15c. Spectrum of wu fluctuations [ks = (8¢/2v3)4]. Of2 ° Te) a + W mises (0) w— @ oS @ = ic] ~~ 0.1 = os és ¥) x Fig. C,15d. Spectrum of du/dt fluctuations. each experiment, all the points for «?/'1(x) fall on a single curve. Similar results are obtained for «4F'1(x) and «°F'1(x) by Stewart and Townsend [22]. Thus, all the moments ik ” 2rFi(x)de, nZl ( 234 ) C,15 - THE PROCESS OF DECAY es) — ie) Aue S< eo S) oO (X — 0.06)", me Fig. C,l5e. Top, decay of turbulence behind two grids My and M,. with mesh widths 5 em and 1 cm, respectively (M2 downstream from M)). x is the distance behind | the second grid M,., x, is the distance between the grids, U is the mean velocity, and wu? is the mean square turbulence. (After Tsuji and Hama [62].) Data show departure from the simple law of decay u? ~ #71. : Bottom, same data as presented in Fig. C,15e, upper, replotted to show conformity with Eq. 14-7 (after Tsuji and Hama [52)). ( 235 ) C- STATISTICAL THEORIES OF TURBULENCE can be calculated from a hypothesis of self-preservation. This provides an experimental basis for Lin’s earlier hypothesis [48] which was obtained from general considerations influenced by the theory of Kolmogoroff. It may be noted here that, at least in these experiments, there is as yet no need for generalizing the hypothesis further in the line indicated by Goldstein [49], although the need for such generalization is not ex- cluded. (See also Art. 17.) Goldstein also proposed the measurement of the law of decay of turbulence behind a grid when another grid of larger mesh is placed upstream. In this case, the large eddies from the first grid tend to cause the turbulent motion behind the second grid to depart greatly from similarity. Such experiments were made by Tsuji and Hama [52], showing strong departure from the law of decay u? ~ t~1 (Fig. C,15e, upper). On the other hand, the more general law of decay (Kq. 14-17) is verified with the additive term b # 0 (Fig. C,15e, lower). More recently, Tsuji [53] examined the spectral distribution of the turbulent motion behind the second grid and obtained results in agreement with the above concepts. When the second grid is 70 mesh widths behind the first grid, the similarity of the vorticity spectrum is not found to be accurate, as one may also expect from the fact that a well-developed turbulent mo- tion is not yet formed behind the first grid. C,16. The Quasi-Gaussian Approximation. As pointed out in Art. 7, it is possible to obtain an infinite system of differential equations for determining the correlation functions of all orders. In order to obtain a ‘deductive theory,” a closed system of a finite number of partial differ- ential equations is needed. For this purpose, some approximation has to be made. Now it is known that the probability distribution of the ve- locity components at a given point is approximately the normal distribu- tion. If this were true for the joint probability distribution at several points, the triple correlation function would vanish, while the correlation functions of the fourth order would be related to those of the second order by the relation uuu” = wah uu” + uu wl” + uh” uly (16-1) It is known that triple correlations do not vanish in homogeneous turbu- lence, but it is still natural to speculate whether Eq. 16-1 may still be true or remains a good approximation without the vanishing of the triple corre- lations.!” There is some support of such a step from experimental obser- vations (ef. Fig. C,8a and C,38b). To give an idea of the application of the hypothesis (Eq. 16-1), let us indicate how the pressure correlation at two points may be derived. One 17 A hypothesis of this type was first introduced into the theory of turbulence by Millionshchikov [42]. ( 236 ) C,16 - THE QUASI-GAUSSIAN APPROXIMATION first notes that 2 Ap = eae (uss) (16-2) so that ApA’p’ = NEN UUjUjzUy (16-3) Or,0rjor.0r, | ° * | We now break up the quadruple correlation by Eq. 16-1. In this manner, one finds that pp’ = 2u! i ‘ (« — =| Lf’ (6 Pdé (16-4) Eq. 16-4 was given by Batchelor [1/2] while an equivalent relation in the spectral formulation was obtained earlier by Heisenberg [23]. An interesting observation may be made here. If the relation (Eq. 16-4) is accepted, it is possible to show that, at high Reynolds numbers of turbulence, the magnitude of the pressure term in the Navier-Stokes equations is much smaller in magnitude than either the local acceleration or the convective acceleration taken individually [20]. With the help of Eq. 16-1, one can derive a system of partial differ- ential equations with the same number of equations as unknowns. The equation for the change of double correlation functions involves the triple correlation functions. The equation for the change of the triple correlation function involves the fourth-order correlations, which may be reduced to the double correlations by Eq. 16-1. In this manner, a closed system of equations is obtained. Such a theory was independently developed by Proudman and Reid [54] and by Tatsumi [55] for decaying isotropic turbulence in the spectral formulation. Without giving a detailed account, a few of the outstanding features are discussed in the following paragraphs. 1. The above reasoning for the establishment of a closed system was obviously only true for multipoint correlations and not for two-point correlations alone. While correlations of the fourth order are repre- sented in terms of two-point correlations by Eq. 16-1, the triple corre- lations must be kept in the general form of a three-point correlation. In an isotropic case, there are then three independent space variables, e.g. the three sides of the triangle with vertices at the points in ques- tion. In the spectral formulation, the final equations contain three independent wave numbers. 2. According to the experimental results of Stewart (Fig. C,3b) the hypothesis (Eq. 16-1) may become poor for small distances. Thus it would be desirable to examine the behavior of the small eddies accord- ing to this theory, and to compare it with the Kolmogoroff-Obukhoff spectrum «x? in the case of infinite Reynolds number. Such a com- parison has not yet been carried out. ( 237 ) C - STATISTICAL THEORIES OF TURBULENCE 3. On the other hand, the theory gives the definite prediction that there is no permanence of the largest eddies. Thus, there is a definite contra- diction between Eq. 16-1 and the hypothesis leading to Loitsiansky’s invariant. This led Batchelor and Proudman [3/] to the extensive investigation mentioned in Art. 12. Chandrasekhar [56] used a different approach. He limited his study to two-point correlations, but considered a difference in time. Thus Eq. 16-1 is replaced by relations of the kind usta’, t)uj(a’, VU )un(e”, tule”, t”) = Qi Qn + QQ + Qz(0, 0)Qi2(0, 0) (16-5) where Qu = uz’, U)u;(@", v”) (16-6) In this approach, the number of space variables is reduced, but another time variable is introduced. Chandrasekhar then introduced the concept of ‘stationary homogeneous and isotropic turbulence,” and assumed that the double correlation (Eq. 16-6) (and similar third order correlations) depend only on the space vector 2’ — z’ and the time interval |t’”” — ¢’|. In this way, the theory leads to a single partial differential equation in two independent variables, and a number of deductions were made. In particu- lar, a discussion is given to show its compatibility with Kolmogoroff’s theory. C,17.. Hypotheses on Energy Transfer. Another method for arriving at a theory capable of yielding definite deductions is to assume, on the basis of physical arguments, a relation between the spectrum function F(x, t) and the transfer function W(x, ¢). Various hypotheses of this type were proposed by Obukhoff [37], Heisenberg [23], von Karman [45], and Kovasznay [57]. It should be recognized that there is no a priori reason that such relations should exist. However, the success of these hypotheses seems to indicate that this type of theory does give a reasonably accurate description of the physical process. Heisenberg argued that the transfer mechanism is essentially similar to viscous dissipation with the smaller eddies corresponding to molecular motions. This is reasonable provided the smaller eddies are very much smaller than the eddies from which they take energy. If this point of view is accepted, the rate at which energy is lost to the smaller eddies is proportional to ==) F(«)dx = E ail zie) a" | i OF (x!)k2dx’ (17-1) t Jo F K 0 ( 238 ) C,17 - HYPOTHESES ON ENERGY TRANSFER where C is a constant. The term =C ie jae i) dx” (17-2) represents an apparent kinematic viscosity coefficient associated with the motions of wave number above x. Von Kaérm4n proposed the form Ip We SG { A le («")Jex’#de | { if d P(e!) ex Fde’ | (17-3) for the transfer function. It reduces to Heisenberg’s form for a = 3, 6 = —8. It also reduces to a modified Obukhoff form for a = 1, B = 0. In the present discussion, we shall restrict ourselves to the Heisenberg formula and use Eq. 17-1 as the basis for determining the spectrum F(x, t) at any future time from its present knowledge. For large values of x, the rate of loss of energy to still larger wave numbers is expected to be very small. Consequently, the left side of Eq. 17-1 is negligible. It then follows (see Chandrasekhar [58]) from Eq. 17-1 that i 1 F(x, t) = const (* ) eee (17-4) where xo is a constant and «x, is inversely proportional to Kolmogoroff’s scale y. For large values of x, P(x) ~ «-?. However, theoretical and ex- perimental considerations indicate that F(x) probably decreases faster than «7 at high wave numbers. If we now introduce the hypothesis of similarity, we can determine the spectrum function completely. In fact, if the hypothesis of complete similarity is used, we have F(x, ¢) in the form P(x, 1) = alels 3 Sol (17-5) where ko and fo are certain constants. According to Eq. 17-1, f(x) satisfies [ “H@)de = 5 af(@) = lat He He) a'| iP 2f(a")e!%da" (17-6) where R = 1/vx2to. This equation has been solved numerically by Chan- drasekhar [58] for a series of values of Ry (Fig. C,17). Proudman [59] computed the correlation function from these spectral functions and found quite good agreement with experiments. The spectrum defined by Eq. 17-6 has a relatively long range, at low values of x, for which F(x) ~ «x. However, an examination of the stability of the spectrum [60] indicates that this part of the spectrum is not stable. At the Reynolds numbers obtained in usual experiments, this unstable { 239 ) CC: STATISTICAL THEORIES OF TURBULENCE range of wave numbers corresponds to about one third of the total energy. However, the contribution to the rate of dissipation from this range is negligible. This agrees with the concept of similarity of the vorticity spectrum x?/'(x, ¢), and lack of similarity for the energy F(x, ¢) for low wave numbers (case (c) of the similarity hypothesis spectrum). Fig. C,17.. The decay spectra for various values of R. The curves marked 1, 2, 3, 4, 5, 6, and 7 are for R-! = 1.65, 1.34, 0.98, 0.55, 0.22, 10-3, and 0 respectively. The curve for R = o is the decay spectrum for infinite Reynolds number and becomes asymptotic to the Kolmogoroff spectrum f(z) = 3x~3 for z— ©. It should be pointed out that this case has a special bearing with Heisenberg’s formula. If we differentiate Eq. 17-1, we obtain a —2fto pf fGD ar] army +20 [PO [erm enae at eu NOE ee (17-7) Thus, the behavior of the spectrum at frequency « depends on the lower frequencies only to the extent of Sox 2F'(x’)dx’. If this integral can be accu- rately calculated by the similarity spectrum for high values of x, Eq. 17-7 is consistent with the hypothesis of similarity. Otherwise, there is some doubt that similarity is possible at all, even at high frequencies, until the last term in Eq. 17-7 becomes negligible. CHAPTER 4. RURBOLENT” DIFRO SION AND TRANSFER C,18. Diffusion by Continuous Movements. One of the most striking properties of turbulent motion is its diffusive property. Obser- vation of smoke from a chimney stack gives a general idea of such a ( 240 ) C,18 - DIFFUSION BY CONTINUOUS MOVEMENTS process. Temperature measurements in the wake of a heated wire, such as those made by Schubauer [6/], and more recently by Uberoi and Corrsin [62], give a more quantitative description of the phenomenon. A complete theory of the phenomenon of turbulent diffusion is, however, not available, even in the simplest case of homogeneous isotropic turbu- lence, because of some inherent difficulties. In the first place, the usual concept of a diffusion coefficient can in general be at best a crude first approximation, because the variation of the statistical properties of inter- est in turbulent diffusion (or transport) occurs over scales comparable to that of the scale of the turbulent motion itself. The analogy in the molecu- lar case would be variations at scales comparable to the mean free path. Secondly, the mathematical difficulty encountered in trying to develop a detailed theory is extremely heavy. Indeed, a theory of diffusion dealing with the transport of material particles from one point to another sug- gests the use of the Lagrangian description. This in itself makes the theory difficult. On the other hand, the eventual diffusion of a certain physical property, such as temperature, must be accomplished by the molecular process, which is more conveniently described by the Eulerian method. In the face of these difficulties, most of the existing theories are far from being complete. In the present treatment, we shall therefore limit ourselves to a brief account of some of the elementary concepts de- veloped and some of the issues examined. For a more detailed treatment, the reader is referred to the recent article by Batchelor and Townsend [63, pp. 352-399]. A fundamental approach to turbulent diffusion was advanced by Taylor [64] in 1921. While the theory deals with an idealized situation, it does reveal some of the essential features of the process, and forms the starting point of many later developments. In the simplest form of this theory of diffusion by continuous movements, we restrict ourselves to the idealized case of a homogeneous isotropic field of turbulence which is not decaying. We consider diffusion from a plane x, z where all the diffusing particles are concentrated at time ¢ = 0. If Y is the coordinate of a particle at time T, then Y = {jvdt, and _——— EWR Ae Didi sain, one Hh =o f dot (18-1) 0 where the average is taken in the statistical sense or over planes parallel to the x, 2 plane. We may now introduce the correlation coefficient R(r) by TODO) = WIN@), n= f= (18-2) Then Eq. 18-1 becomes ig) ee pile a, i Rar (18-3) C - STATISTICAL THEORIES OF TURBULENCE and 1y? = 2 i * dt! iE " R(x)dr = 2? ip ‘(T — #)R(t)dt (18-4) In general, we may expect R(r) to decrease with increasing 7. Suppose that, for all times 7 greater than Ti, P(r) is practically zero. Then iL ' R(n)dr = if PRG@\dr «(forse Ty (18-5) and Eq. 18-4 gives Ve a ti R(r)dr + const (18-6) The integral (Eq. 18-5) is a measure of the time scale of diffusion, and D=yY ih ° R(t)dr (18-7) may be regarded as a diffusion coefficient, since for molecular diffusion, Y? = 2DT. Thus the concept of a diffusion coefficient is justified for large values of time of diffusion. On the other hand, for small values of time t (¢ much less than the time scale defined by Eq. 18-5), R(r) ~ 1, and Eq. 18-4 gives Y2=yT? or VY? =0T (18-8) It appears that when T is small, Y? is proportional to T? instead of T, as in an ordinary diffusion process. This is clearly so, because over the time interval in which R(r) is nearly equal to unity, the velocities of the particles are nearly constant so that for each particle Y =0f (18-9) In this case, therefore, not only is Eq. 18-8 valid, but the frequency dis- tribution of Y is the same as the frequency distribution of v. We shall now apply these ideas to the problem of the spread of heat behind a heated wire. If heat is spread from a concentrated plane source, after an interval of time t, the distribution of temperature according to the usual process of molecular diffusion is proportional to ¢~! exp [—y?/4ké], where y is the normal distance from the plane source. The temperature distribution behind a heated wire corresponds to such a problem, if there is only molecular diffusion. The distribution is given by the above ex- pression with ¢ replaced by «/U, where z is the distance downstream from the wire and U is the speed of the wind. In a turbulent stream, diffusion due to turbulent motion must be superposed on molecular dif- fusion. In fact, Schubauer has observed the error law for the distribution of temperature, and it would appear that the phenomenon of turbulent diffusion can be described by an adequate diffusion coefficient associated with the turbulent motion. However, the above analysis shows that the ( 242 ) C,19 - ANALYSIS INVOLVING MORE THAN ONE PARTICLE use of a diffusion coefficient is valid only for large distances downstream (provided the idealization used is valid). Close to the source, there is no basis for using a diffusion coefficient. Indeed, Taylor [75] pointed out that the Gaussian distribution of temperature near the source is not associated with a usual diffusion coefficient but should be accounted for by Gaussian distribution of the velocity of fluctuation (cf. Eq. 18-9). If the frequency distribution of velocities had obeyed some other law, the distribution of temperature near the source would also have deviated from an error curve. On the other hand, the temperature distribution very far from the source must necessarily fit an error curve, whatever the frequency distribution of velocities may be. In reality, however, the analysis at very large distances downstream is complicated by the fact that the turbulence dies away downstream so that the above analysis is not accurate. For other approaches to the problem of turbulent diffusion, see Frenkiel [65], where some semiempirical calculations are given. C,19. Analysis Involving More Than One Particle. In the above analysis, we do not consider the joint configuration of a number of fluid particles. This is of course necessary if we wish to get a more complete description of turbulent diffusion. Indeed, one may consider a large num- ber of particles, and their joint statistical behavior during the course of time would give an almost complete statistical description of the turbu- lent motion in the Lagrangian scheme. In practice, one is often limited to the consideration of the separation of two particles. If s; is the separation between two particles, the rate of variation of the statistical average s? is clearly a measure of the rate of diffusion. We shall now consider a special case where a concrete formula can be obtained for the variation of s? as a function of the time of sepa- ration tr = ¢ — to between the initial instant to and the present time 1. If r is sufficiently large, then the influence of the initial separation must be negligible. If, furthermore, the separation between the particles lies in the range of scales of turbulence for which the spectral law (Eq. 13-15) holds, there is only one parameter—namely the rate of energy dissipation e—characterizing the properties of the turbulent motion. Thus, dimen- sional reasoning shows that s? (which depends only on 7 and e) must be of the form s? ~ er? (19-1) or = ~ er? ~ é(s?)3 (19-2) This means that the dispersive effect becomes larger and larger as the particles separate further and further from each other, the diffusion coef- ( 243 ) CQ: STATISTICAL THEORIES OF TURBULENCE ficient increasing as the 4 power of the separation.1® The law (Kq. 19-2) was obtained experimentally by Richardson [66] and deduced theoreti- cally by Batchelor [67] in a somewhat different manner. However, the agreement is partly fortuitous since the length scale involved in Richard- son’s data does not fulfill the requirements imposed in the theory. More detailed analyses of the magnitude of separation can be carried out when the distance of separation is small. Such analyses can be used to examine the deformation of a fluid element—material lines, surfaces, and volumes. An interesting question is whether a material volume will eventually be stretched into a needle-shaped line or a disk-shaped surface. For the details of such investigations, the reader is referred to the original articles of Batchelor [68] and Reid [69] or to the article of Batchelor and Townsend [63]. C,20. Temperature Fluctuations in Homogeneous Turbulence. The above concepts can be extended to a continuous distribution of sources. The results are particularly instructive when the distribution has a uniform gradient. Following Corrsin [70], let us consider a dis- tributed heat source in the plane x = O with a linear distribution in the y direction: T =T1+ ay (20-1) Consider the flow of fluid with nondecaying isotropic turbulent motion past these sources with a uniform mean speed U which is much higher than the velocities of the turbulent motion. If molecular conduction is omitted, the instantaneous temperature at any point downstream is de- termined by that of the fluid particle present at that instant. This in turn depends on the location where the particle crossed the plane x = 0. Since a point may be reached by a fluid particle from above and from below with equal probability, it is clear that the mean temperature dis- tribution (Eq. 20-1) persists downstream. Consider now a fixed point x > 0 in the plane y = 0 (which is in fact a typical plane), and the particle occupying that point at any instant. If the level of turbulence is low, this particle passed by the plane x = 0 at the time 7 = x/U earlier. The position of the crossing point is given by — Yo, where Yo= [i v@’at! (20-2) t being the time under consideration. This introduces a temperature deviation 6= —aVYo (20-3) 18 This result can also be obtained by formally using Heisenberg’s formula (Eq. 17-2). { 244 ) €)21- STATISTICAL THEORY OF SHEAR FLOW The statistical average rate of temperature transfer is therefore —v6 = wYo (20-4) The value vY» can be evaluated in a manner very similar to that used in Art. 18. We obtain in this manner vYo = fi. o@o@ ae’ = 7 f” R(')dr' (20-5) It should be noted that although this formula is very similar to Eq. 18-3, the physical interpretation is different. The rate of temperature transfer may now be written in the form —v0 = D! oe DD! = 7? if R(r’)dr’ (20-6) dy 0 where D’ is a “diffusion coefficient”’ in that it gives the rate of increase of the mean square deviation (Art. 18). It is proportional to xz at first, and approaches a constant value D after a sufficient distance downstream. Similarly it can be shown the standard deviation is given by G2 ety? ik (@ = pine (20-7) and that it becomes infinite as the first power of ¢ or the distance down- stream. In reality, this will be limited by molecular diffusion. In contrast to the above problem of heat transfer, an analysis of tem- perature fluctuations in a statistically homogeneous field can be carried out in much the same way as for homogeneous velocity fields. This was done by Corrsin [77], who found that in the final stage of the decay process the mean square of the temperature fluctuation decreases as the inverse 2 power. This is different from the case of velocity fluctuations, and the reason for this difference is the absence of the equation of continuity in the present case (cf. Art. 15). Such a law was first obtained by Reissner [50] in his asymptotic solution of the heat equation. C,21. Statistical Theory of Shear Flow. Although studies of turbulent flow with shear date back further than studies of isotropic tur- bulence, a complete statistical theory has not yet been developed. The classical ideas of Reynolds still stand out as the best description of the basic mechanism of turbulent shear. The mixture length theories,'* while useful for practical purposes, are obviously not adequate statistical theo- ries. Deviations from the classical concepts are especially evident near the edge of the turbulent boundary layer and in turbulent jets and wakes. Intermittency of the turbulent motion appears to be quite predominant in such phenomena. It appears from these intermittent phenomena that 19 For details, see Sec. B. ( 245 ) C - STATISTICAL THEORIES OF TURBULENCE a statistical theory of shear flow can be developed only after adequate descriptions are obtained, both for motions on a small scale and for motions on a scale comparable with that of the mean flow. The fact that turbulent transfer is most effectively carried out by such large scale mo- tions is in direct contrast to the phenomenon of molecular transfer. In that case, the mean free path is much smaller than the scale of the mean motion, and a definite coefficient of transfer is established. The difference in scales of the random motion responsible for the transfer mechanism in the two cases makes the analogy imperfect, and is at the root of the difficulties in developing a theory of turbulent transfer. For steady flow through pipes and channels, the phenomenon is sim- pler in the sense that intermittency is not apparent. Attempts to develop a statistical theory, based on the use of correlations, have been made by Keller and Friedmann [72], von K4rm4n [13], Chou [73], and Rotta [74]. While the theory predicts the mean velocity and turbulence level in reasonable agreement with experiments, the presence of arbitrary con- stants shows their weakness. In the following, only an indication of the approach is given; the reader is referred to the original papers for the details. The equations of motion for the turbulent fluctuations can be ob- tained by subtracting the Reynolds equations (Eq. 2-3) from the com- plete equations of motion (Eq. 2-2), Ou; Ou; OU; vat 1 Op 1 OT; Bi cape he es OE MNOE OU; ot + U; + vAu; (21-1) where p is the fluctuation of pressure, and 7;; are the Reynolds stresses. The velocity fluctuations wu; also satisfy the equation of continuity, <~==0 (21-2) Instead of dealing with the velocity fluctuations themselves, one may attempt to deal with the statistical correlation of the velocity fluctuations in analogy with the study of homogeneous turbulence. To simplify mat- ters, one may also eliminate the pressure fluctuation by using the Poisson equation obtained by taking the divergence of Eq. 21-1: Lap = 9. 2Un dln | 0 p P OLA NOLre 1) OG nOln (Un — UmUin) (21-3) and solving it under appropriate boundary conditions. However, the fun- damental difficulty encountered in the homogeneous case—that higher correlations are invariably brought into the picture when an equation is constructed for dealing with correlations of a given order—also occurs here. Approximations are therefore introduced by the various authors at this stage, and the theory is not entirely free from arbitrariness. ( 246 ) C,22 - TURBULENT MOTION IN A COMPRESSIBLE FLUID Because of the difficulties encountered in the development of the statistical theory of shear flow, several authors studied the more special- ized problem of homogeneous turbulence in a field of uniform velocity gradient. Application of the concepts of Art. 17 was made by Reis [75] and later by Burgers and Mitchner [76] with almost identical assump- tions and results, although the work appears to have been done inde- pendently. Application of the concepts of Art. 16 to this case has recently been carried out by Craya [77]. An entirely different approach to the problem of turbulent shear flow has been proposed by Malkus [78]. The reader is referred to his original paper for the details. CHAPTER 5. OTHER ASPECTS OF THE PROBLEM OF TURBULENCE C,22. Turbulent Motion in a Compressible Fluid. When a com- pressible gas is in turbulent motion, there are density and temperature fluctuations as well as velocity fluctuations. At any instant, the velocity fluctuation may be decomposed into two parts, ue = UP + u® (22-1) such that div u® = 0, curl u® = 0 (22-2) The rotation of the fluid is given by the first part wi? and the com- pression is given by the second part u”. In general, there is a continuous conversion between the rotation component and the compression component of the velocity fluctuations. This additional degree of freedom in the compressible case naturally makes the theory of turbulence much more difficult. In the case of small disturbances from a homogeneous state these modes are separable from each other.*° The study of small disturbances superimposed on a shear flow is treated in connection with the instability of the boundary layer at high speeds. Attempts have been made to extend directly to the compressible case the various approaches to the theory of turbulence in the incompressible case: e.g. the study of isotropic turbulence by Chandrasekhar [80], and the consideration of von Kaérm4n’s similarity theory by Lin and Shen [87] for shear flow. The method discussed in Art. 21 can also be extended to a compressible gas. Obviously, such approaches cannot go beyond the limitations in the incompressible case. It is therefore natural that the more fruitful theoretical results on turbulent motion in a compressible fluid are obtained in connection with the study of the influence of com- 20 See [79] for a detailed discussion of this case. ( 247 ) C - STATISTICAL THEORIES OF TURBULENCE pressibility on turbulent motion, principally for small Mach numbers of turbulence. It is then plausible that the chief influence of compressibility is that acoustic energy is constantly being radiated, causing the turbulent motion to dissipate faster than in the incompressible case. Lighthill [82] has shown that, in the absence of solid boundaries, turbulent motion acts as quadrupole sources of sound. He also showed that the amount of energy radiated per unit volume of turbulence is proportional to pV*/a*l, where p is the density, V is a typical velocity of the turbulent motion, a is the acoustic speed, and / is a typical linear scale. Since the rate of energy conversion in turbulent motion is propor- tional to pV 3/1, the acoustic efficiency is proportional to the fifth power of the root mean square Mach number. At low Mach numbers, this would be a very small amount indeed, if it were not for a numerical factor of proportionality of the order of 40, as shown by Proudman [83]. In the cases where the theory is applicable, the experimental results bear out the general theoretical conclusions. If solid boundaries are present, such as in the problem of the noise from the boundary layer of a flat plate, Phillips [84] found that dipole sources are present if the plate is semi-infinite. Acoustic sources are again of the quadrupole type if the plate is infinite and the motion is statisti- cally the same along the plate. The scattering of energy due to the interaction of turbulence with sound or shock waves has been considered by Lighthill [85] and others. All of the above results are for low Mach numbers of turbulent mo- tion. At the present time, only speculation can be made for the cases of higher Mach numbers where shock waves may appear. C,23. Magneto-Hydrodynamic Turbulence. In astrophysics, one important problem is the turbulent motion of an electrically conducting gas in the presence of magnetic fields. One is then dealing with the con- version of energy from the mechanical form to the electro-magnetic form. There is an extensive and rapidly growing literature on this subject, and it is perhaps inappropriate to try to survey it at the present time in a volume on high speed aerodynamics. One of the central problems at issue is the partition of energy between the two modes. Batchelor [86] noted that the equation for the magnetic field is exactly the same as that for vorticity, and suggested that the energy spectrum of magnetic energy is proportional to «F'(x). This would mean that there is little magnetic energy in the large scales. Other authors, however, contended that there should be equipartition of energy of the two modes. Recently, Chandrasekhar [87] undertook a systematic de- velopment of the theory of turbulent motion for magneto-hydrodynamics along the lines of Art. 16 and 17, and found solutions which are in agree- ment with the latter opinion. However, since his assumption limited him ( 248 ) C,24 - SOME AERODYNAMIC PROBLEMS to moderate and small wavelengths, the former opinion is not yet ruled out. For a more detailed discussion of the arguments for and against the two standpoints, see [88, pp. 93-98]. Work decisively distinguishing be- tween them is clearly needed. In this connection, it would perhaps be worthwhile to obtain some special solutions in magneto-hydrodynamics analogous to those obtained by Taylor and Green [35] in the ordinary case to get an idea of the validity of the existing arguments. C,24. Some Aerodynamic Problems. There are a number of aero- dynamical problems associated with turbulent motion in which its dif- fusive nature takes on a secondary role. The random nature of turbulent motion still makes it necessary to use statistical treatments. In this cate- gory of problems, we briefly discuss (1) the dynamical effects of turbulent motion, (2) the effect of contraction on wind tunnel turbulence, and (3) the effect of damping screens. Dynamical effects of turbulent motion. Dynamical effects caused by turbulent motion are often treated by statistical methods. For example, in the case of a pendulum suspended in a turbulent wind, the spectrum of the motion of the pendulum can be calculated in terms of that of the turbulent motion and the dynamical characteristics of the pendulum [89]. Recently, Liepmann [90] tried to apply these methods to the buffeting of airplanes moving through a turbulent stream. Effect of wind tunnel contraction. The effect of wind tunnel contrac- tion on the intensity of turbulence has been studied by Prandtl and Taylor [91, p. 201]. Recently, Ribner and Tucker [92] applied Taylor’s ideas to the study of the influence of the contraction on the spectrum. The combined effect of damping screens and stream convergence have also been studied by Tucker [93]. The reader is referred to the original papers. An experimental investigation of the detailed behavior of the tur- bulent fluctuations during contraction has been made by Uberoi [94]. Effect of damping screens on homogeneous turbulence. Damping screens have long been used for the reduction of turbulence level in the wind tunnel. While these screens no doubt act also as a grid in producing turbu- lence, the scale of such turbulent motion is usually so small that it damps out at a comparatively small distance behind the screen. The resistance of the screen to the flow, on the other hand, tends to reduce the large scale turbulent motion already existing in front of it. The characteristics of a damping screen are usually described in terms of two force coefficients Ks and Fy. If the screen is placed with its normal at an angle @ relative to a stream of speed U, there is a drop of pressure across it, given by p2 — pi = Ko: spU? (24-1) where pi and p» are the static pressure upstream and downstream of the ( 249 ) CC: STATISTICAL THEORIES OF TURBULENCE screen. At the same time, there is a side force in the plane of the screen per unit area, given by S Sty Sauer (24-2) Experiments by Schubauer, Spangenberg, and Klebanoff [95] at the National Bureau of Standards (NBS) show that the coefficients F's and Ko are related for usual wire gauze screens. Dryden and Schubauer [96] proposed the relation Fo 4Keo 6 8+Ks CrP which agrees with experiments for Kg < 1.4. Taylor and Batchelor [83] fitted the NBS data with the empirical formula V1 + Ke which appears to be a reasonable approximation for 0.7 < Ks < 4. Schubauer, Spangenberg, and Klebanoff also found that K,/cos? 6 can be uniquely related to R cos 0, where FR is the Reynolds number. This means that the pressure drop depends essentially on the normal compo- nent of the velocity. Theoretically, it is useful to introduce a ‘‘refractive index” a. If the departing stream makes an angle ¢ with the normal to the screen, then = $/6. For small angles, MR Ror: Fe Ol ae S (24-5) This leads to oa ae TOL a he —aules: “4 (24-6) a — a for 0.7 < K < 4 ten Dryden and Schubauer [96] found that the kinetic energy of turbu- lence is reduced by the factor (1 + A)~—! after passing through the screen. This result does not distinguish between the longitudinal and lateral com- ponents of the velocity. It has been verified by the more careful measure- ments of Schubauer, Spangenberg, and Klebanoff for flow Reynolds num- bers above a certain critical value. For lower Reynolds numbers, the reduction factor is found to be lower. A theory developed by Taylor and Batchelor [97], however, predicts different reduction factors for the longitudinal and the lateral compo- nents. They also predicted a reduction of the turbulence level immedi- ately in front of the screen. ( 250 ) C,25 - CITED REFERENCES Townsend [98] performed experiments to check the theoretical predic- tions and reached the following conclusions: 1. The reduction of turbulent intensity on the upstream side of the gauze is described by the Taylor-Batchelor theory with adequate accuracy. 2. The reduction of the total turbulent intensity in passing through a gauze is consistent with the theory except for small values of x, when the observed reduction is greater than predicted. 3. As predicted by the theory, the turbulence emerging behind the gauze is anisotropic, with the intensities of the lateral components exceeding that of the longitudinal component. The degree of anisotropy is less than the theory predicts. The smaller degree of anisotropy is ascribed by Townsend to a possible rapid initial adjustment toward isotropy in the zone of influ- ence of the gauze. Downstream of the gauze, Townsend found that the recovery of isotropy is very slow, but local isotropy is valid within the range of observation in the sense that au\" _1(av\" dx) 2\dzx There is some obvious disagreement between the results of Townsend and the NBS data. Such differences are, however, not unexpected in view of the experimental difficulties attending such measurements. While the effect of damping screens has probably been determined with sufficient accuracy for practical application, further experiments are desirable for checking the theory. The NBS data show that the scale of turbulence is not changed by the screen. No extensive experimental results are yet available on the change of spectrum. Some of the results obtained by Townsend do not agree with the theoretical predictions. In making such a comparison, it should be borne in mind that the theory is linear and the change of spectrum downstream of the screen is not considered in the theory. C,25. Cited References.?! 1. Batchelor, G. K. The Theory of Homogeneous Turbulence. Cambridge Univ. Press, 1953. Lin, C.C. J. Aeronaut. Sci. 23, 453-461 (1956). . Burgers, J. M. Adv. Appl. Mech. 1, 171 (1948). . Burgers, J. M. Proc. Natl. Acad. Amsterdam 58, 247, 393, 718, 732 (1950). Hopf, E. J. Rat. Mech. and Anal. 1, 87 (1952). Hopf, E. Commun. on Pure and Appl. Math. 1, 303 (1948). DP Ov ye 9 bo 21 References [100] and [101] appeared after the present article was completed. Kraichman’s paper [1/00] deals with an approach to the theory of turbulence (both nonmagnetic and magnetic) which differs substantially from those discussed earlier in the text, and is indeed contradictory to the Kolmogoroff theory. Corrsin’s paper [101] considers the question of local isotropy also discussed in [99]. ( 251 ) C - STATISTICAL THEORIES OF TURBULENCE . Blanco Lapierre, A., and Fortet, R. Théorie des fonctions aléatoires: application a divers phenoménes de fluctuation, Chap. 14. Masson, Paris, 1953. . Reynolds, O. Phil. Trans. A186, 123 (1895). . Simmons, L. F. G., and Salter, C. Proc. Roy. Soc. London A145, 212 (1934). . Townsend, A. A. Proc. Cambridge Phil. Soc. 43, 560 (1947). . Stewart, R. W. Proc. Cambridge Phil. Soc. 47, 146 (1951). . Batchelor, G. K. Proc. Cambridge Phil. Soc. 47, 359 (1951). . von Karman, Th. J. Aeronaut. Sci. 4, 131 (1937). . Robertson, H. P. Proc. Cambridge Phil. Soc. 36, 209 (1940). . Taylor, G.I. Proc. Roy. Soc. London A151, 421 (1935). . 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Frenkiel, F. N. J. Appl. Mech. 15, 311 (1948). . Lin, C. C. Proc. Natl. Acad. Sci. 34, 540 (1948). . Goldstein, S. Proc. Cambridge Phil. Soc. 47, 554 (1951). . Reissner, E. Proc. 5th Intern. Congress Appl. Mech., 359 (1938). . Batchelor, G. K. Proc. Roy. Soc. London A195, 513 (1949). . Tsuji, H., and Hama, F. R. J. Aeronaut. Sci. 20, 848 (1953). . Tsuji, H. J. Phys. Soc. Japan 2, 1096-1104 (1956). . Proudman, I., and Reid, W. H. Phil. Trans. A247, 163 (1954). . Tatsumi, T. Proc. Roy. Soc. London A239, 16-45 (1947). . Chandrasekhar, 8. Phys. Rev. 102, 941 (1956). . Kovasaznay, L. 8. G. J. Aeronaut. Sci. 15, 745 (1948). . Chandrasekhar, §. Proc. Roy. Soc. London A200, 20 (1949). . Proudman, I. Proc. Cambridge Phil. Soc. 47, 158 (1951). . Lin, C. C. Proc. Fourth Symposium Appl. Math., Am. Math. Soc., 19 (1953). . Schubauer, G. B. NACA Rept. 524, 1935. ( 252 ) C,25 - CITED REFERENCES . Uberoi, M.S., and Corrsin, S. NACA Rept. 1142, 1953. . Batchelor, G. K., and Townsend, A. A. G.I. Taylor Anniversary Volume. Cam- bridge Univ. Press, 1956. . Taylor, G.I. Proc. London Math. Soc. 20, 196 (1921). . Frenkiel, F. N. Adv. Appl. Mech. 3, 61 (1953). . Richardson, L. F. Proc. Roy. Soc. London A110, 709 (1926). . Batchelor, G. K. Quart. J. Roy. Met. Soc. 73, 133 (1950). . Batchelor, G. K. Proc. Roy. Soc. London A213, 349 (1952). . Reid, W. H. Proc. Cambridge Phil. Soc. 51, 350 (1955). . Corrsin, 8S. J. Appl. Phys. 28, 113 (1952). . Corrsin, 8S. J. Aeronaut. Sci. 17, 417 (1950). . Keller, L., and Friedmann, A. Proc. First Intern. Congress Appl. Mech., 395 (1924). . Chou, P. Y. Quart. Appl. Math. 3, 198 (1945). . Rotta, J. Z. Physik 129, 547 (1951); 131, 51 (1951). . Reis, F. B. Ph.D. Dissertation, Mass. Inst. Technol., 1951. . Burgers, J. M., and Mitchner, M. Proc. Koninkl. Ned. Akad. Wetenschap. 56, 228, 343 (1953). . Craya, A. Compt. rend. 244, 560, 847, 1448, 1609 (1957). . Malkus, W. V. R. J. Fluid Mech. 1, 521-539 (1956). . Kovdsznay, L. 8S. G. J. Aeronaut. Sci. 20, 657-674 (1953). . Chandrasekhar, 8. Proc. Roy. Soc. London A210, 18 (1951). . Lin, C. C., and Shen, 8. F. NACA Tech. Notes 2541 and 2542, 1951. . Lighthill, M. J. Proc. Roy. Soc. London A211, 564 (1952). . Proudman, I. Proc. Roy. Soc. London A214, 119 (1952). . Phillips, O. M. Proc. Roy. Soc. London A234, 327-335 (1956). . Lighthill, M. J. Proc. Cambridge Phil. Soc. 49, 531 (1953). . Batchelor, G. K. Proc. Roy. Soc. London A201, 405 (1950). . Chandrasekhar, S. Proc. Roy. Soc. London A233, 322, 330 (1955). . Cowling, J. G. Magnetohydrodynamics. Interscience, 1957. . Lin, C. C. Quart. Appl. Math. 1, 43 (1943). . Liepmann, H. W. J. Aeronaut. Sci. 19, 793 (1952). . Goldstein, S. Modern Developments in Fluid Dynamics, Vol. 1. Clarendon Press, Oxford, 1938. . Ribner, H. S., and Tucker, M. NACA Tech. Note 2606, 1952. . Tucker, M. NACA Tech. Note 2878, 1953. . Uberoi, M.S. J. Aeronaut. Sci. 23, 754-764 (1956). . Schubauer, G. B., Spangenberg, W. G., and Klebanoff, P.S. NACA Tech. Note 2001, 1950. . Dryden, H. L., and Schubauer, G. B. J. Aeronaut. Sci. 14, 221 (1947). Also appendix to [97]. . Taylor, G. I., and Batchelor, G.K. Quart. J. Mech. and Appl. Math. 2, 1 (1949). . Townsend, A. A. Quart. J. Mech. and Appl. Math. 4, 308 (1951). . Tchen, C. M. Phys. Rev. 93, 4 (1954). . Kraichnan, R. H. Phys. Rev. 107, 1485 (1957); 109, 1407 (1958). . Corrsin, S. NACA Research Mem. RM58B 11, 1958. ( 253 ) SECTION D CONDUCTION OF HEAT M. YACHTER KH. MAYER CHAPTER 1. INTRODUCTION D,1. General Remarks. The mathematical description of tempera- ture distributions, which arise in jet engine walls and other jet engine elements subjected to severe heat transfer rates, is of importance in the development of rational procedures for designing such elements. Owing to uncertainties in numerical data on thermal properties and heat trans- fer rates, it is usually sufficient in practice to calculate the temperatures for somewhat idealized geometric representations of the jet engine ele- ments and for idealized heat transfer conditions, which nevertheless repre- sent in good approximation the actual physical conditions. Thus the repre- sentative geometric models of particular interest are plane parallel slabs, cylinders, and cylindrical shells, in which heat flow is principally one- dimensional. Heat transfer rates across the boundaries are assumed to be of the Newtonian form; i.e. proportional to the difference between the temperature at the boundary and the temperature of the surrounding medium. Insofar as heat transfer through the boundaries occurs primarily by convection, rather than by conduction or radiation, the use of the Newtonian form appears justified. Radiation rates, governed by the Stefan-Boltzmann law, are of some importance in rocket chambers where they may contribute up to one third of the total heat transfer rate, and therefore they are only crudely represented by inclusion in the Newtonian form. The problems discussed in this section have been selected on the basis of their immediate applicability to heat flow in combustion chambers and nozzles, in skins of high speed aircraft, turbine blades, etc. Emphasis is placed on simple, yet representative, model problems leading to analytic results, which allow an insight into the role played by geometry, heat transfer coefficients, and thermal properties, and which therefore lend themselves to generalizations not readily inferred from the complicated results of more difficult problems. Attention is focused on transient tem- perature problems which are of importance in uncooled combustion ( 254 ) D2 -MATHEMATICAL FORMULATION chamber walls. In regard to steady state problems which are important in cooled units, it is merely noted that, on the one hand, their solutions correspond in general to limiting cases of transient heat flow for long times; on the other hand their solutions may be obtained by special, elementary methods abundantly illustrated in the literature [/,2,3,4,5]. In the paragraphs below we summarize the general mathematical formulation of the heat conduction problem with emphasis on those features of the solution which have a special bearing on the particular problems presented in this section. D,2. Mathematical Formulation. For our purposes it is sufficient to consider the differential equation of heat conduction in an isotropic two-dimensional domain G without internal heat sources or sinks [6]: ) oT ) oT oT Bie (i eco ROL (ec J rad i] ax (: ) 7 Ba (: x) fatale Cy) In this equation T is the temperature, p, c, and k are the density, specific heat, and thermal conductivity of the medium in domain G, while z, y, and ¢ are, respectively, the two rectangular coordinates and the time. As stated in Art. 1, we shall consider heat transfer only by conduction and approximate radiation described by the Newtonian form with a heat transfer coefficient h independent of the temperature or time. Thus k = + h(a, y)[T — T,] = 0 onthe boundary IL of thedomainG (2-2) where, under the conditions stated, either 7, = flame or gas temperature when h ~ 0 or otherwise h = 0. A derivative with respect to the (outer) normal to the boundary is represented by 07'/dn. The initial condition is LG, Y, to) TE F(a, y) where, without loss of generality, we may write to — 0 Actually we are interested primarily in the cases where F(z, y) = const, corresponding to ambient temperature. The conductivity k, the density p, and the specific heat c may be functions of position as well as of the tem- perature 7. The boundary condition may be even more general, as in the case of surface melting. In that case a term involving the heat energy absorbed in the change of state will also appear in the boundary con- dition, while the boundary IL itself will be changing with time (cf. [7]). It is seen that the general differential equation and boundary con- ditions are nonlinear and, except for some special cases, cannot be dis- cussed in any general manner. We shall discuss below at some length ( 255 ) D - CONDUCTION OF HEAT only two nonlinear problems: (1) The problem of surface melting as a factor in erosion and (2) the problem of heat conduction in a material with thermal properties (k and c) which vary with the temperature. Finally, among the linear problems we shall discuss primarily those which involve only one space coordinate. In order to reduce the boundary conditions to homogeneous form we make the transformation Th cae 1 Then the linear problems to be discussed are included in the following formulation: 5) eet Wael) nar BBs ihe: we =) + ay ¢ =) Nad 2-la gO EMS) § 29 Sn Ce 6, = Ai, y))| vatit =20 where s is the are length on IT measured from a reference position and k, p, and c are functions only of position. Detailed discussion of the mathematical system represented by Eq. 2-la is given in texts dealing generally with the theory of partial differential equations. Here we merely summarize what may be considered as the most important characteristics of the system, particularly so in relation to analysis of heat conduction in solid mediums. 1. It is always possible to eliminate the time either by separation of variables or by the Laplace transform [6]. Thus by the method of sepa- ration of variables, if we write 8 = o(x)yP(t) BLE oe) ey es ey oa ales ¢ x) + ay (: ay Nae aa » must be independent of xz, y, and ¢, i.e. a constant, since the left- hand side is independent of ¢t while the right-hand side is independent of x and y. Moreover, it can be shown that \? is positive [8, Vol. 1, p. 252]. We obtain firstly then and secondly the eigenvalue problem 9 (99) | 9 (1,99 a ey ae a (: ae) 4 2 (: 22) + pee 0 mG (2-2) k Oe he =0 onYl on P ( 256 ) D,2 - MATHEMATICAL FORMULATION In general, the eigenvalue problem leads to an infinite set of eigenfunc- tions ¢, with a corresponding infinite set of eigenvalues \, which are ob- tained by satisfying the boundary conditions. 2. The eigenfunctions ¢,, when multiplied by +/pc, form an orthogo- nal set. Since this is perhaps the most important property of the eigen- functions we shall indicate briefly the proof of this important result: If k, p, and ¢ are continuous with continuous derivatives in the domain G, we have by a generalized Green’s formula [8, Vol. 1, p. 239]: f (ey = ike | vk (+ Som _ oy i) ds (2-3) G where ds is an element of arc on I, and (eX) fe) (eX) Os (i se) a dy ¢ =) Substituting the differential equation and boundary condition (Eq. 2-2) into Eq. 2-3, we get ==) J i/ PCOmendzdy = iL R(omen — Ondm)ds = 0 (2-4) Hence | ecemendady = 0 for mAn, Nn F~XAn G so that the functions Bn = VC on (n=1,2,...) form an orthogonal set. It often happens that the thermal properties k, p, and c are continu- ous in different subdomains Gi, G2, etc., but are discontinuous across the boundaries T'1,Ts, . . . separating the domains, as is the case in composite rocket walls. In this case the general Green’s formula must be applied to the subdomains Gi, G2, etc., sepa- rately and the results added. Suppose for the sake of simplicity that the total domain G is thus divided into only two subdomains, Gi and G2, separated by the boundary I; (see Fig. D,2)- Let the mth and nth eigenfunction in G be r Pim) Pin in Gi g= a Pam; Pam I 2 Fig. D,2. Division of the : : domain G into the subdo- Then, applying Green’s formula in both do- aing Gl phat @s tye dhe mains separately and adding, we have boundary 1. ( 257 ) D - CONDUCTION OF HEAT (ONS Na.) ( \ / P1C1P1meindady + il i prcremendizdy ) Gi Ge 4 m fe) nr 0 ™m Ce) Nr -[. E oe — Yim me — ke & = — Yom = )| ds re OPim ny OPom = Je. { (Hoge) ~ om (re) OP1n Ofon = EB (i — P2m @ an ) || ds the integrands on the exterior boundary vanishing identically as in Eq. 2-3, n being a normal to the boundary. Hence the function set Bn = VC On (Co 1, go) will be orthogonal if Pin = P2n ky at = ke ue on T, (2-5) Now, the above conditions represent the physical requirements of con- tinuity of temperature and heat flux through any internal boundary, and therefore the above relations represent the boundary conditions which must be used in internal boundaries separating domains of different mediums. It is seen, therefore, that by the imposition of the physical requirements, the orthogonality property of the function set ~/pc ¢» is always automatically assured. A particular solution is then 0,(z, Yy; t) = on(2, y)e™t and, since the system is linear, the complete solution is a linear sum of particular solutions: (7,1) =) Ane. = Aeneas (2-6) n=1 n=1 O(a, 4,0) = ) Anvalz, y) = F(, ») n=1 The amplitudes A, still remain to be determined. 3. Now, a further theorem [8, Vol. 1, p. 319] states that if F(z, y) isa continuous function in the domain G with continuous first and second derivatives and in addition satisfies the boundary conditions of the problem, then F(z, y) can be expanded in an absolutely and uniformly convergent { 258 ) D,2 - MATHEMATICAL FORMULATION series in terms of the eigenfunctions F(a,y) = ) Anga(a, y) n=1 where, in view of the orthogonality property, J | pcF e,dxdy An = (2-7) [[vceteas i pcy,dady Actually, the function F(z, y), representing the initial temperature dis- tribution in the domain, seldom satisfies the boundary conditions, the usual form being F(z, y) = const. However, this is not a serious diffi- culty, for there always exists a function F(z, y) which satisfies all con- ditions of the theorem and is, nevertheless, arbitrarily close to F(z, y). This can be seen immediately if z = F(z, y) is regarded as the equation of a surface. The only consequence is that, in general, there will not be uniform convergence but rather convergence “‘in the mean,’”’ that is: tin ff[ Anes) — Peon aety = m=1 which is sufficient for all practical purposes (cf. [8, Vol. 1, p. 43]). The practical use of the Fourier series solution (Eq. 2-6) for transient temperature calculations is contingent on rapid convergence of the series, which is assured for sufficiently large times ¢ by virtue of the decreasing exponential factors e-*". At small times, however, the convergence may become slow and calculations with Eq. 2-6 may become cumbersome. In this case, special approximate procedures, based on Laplace transform, source and image methods, etc. [6], prove to be more convenient. We shall have occasion to employ such procedures in the limit of small times when use of the Fourier series is not practical (cf. Art. 5). It is seen then that the general problem reduces to the solution of the eigenvalue problem, or what is usually known as the third boundary- value problem in mathematical physics. It may be worthwhile to mention that the boundary-value problem can always be formulated as an equiva- lent isoperimetric problem in the calculus of variations, designated by Riemann as Dirichlet’s principle (cf. [8, Vol. 2, Chap. 7]). In problems of vibration of beams, membranes, and plates, in aeroelastic problems, etc., this principle, also known as the Rayleigh-Ritz method, is a very powerful tool of analysis. However, in the case of heat conduction this method does not prove to be so useful. ( 259 ) D - CONDUCTION, OF HEAT D,3. Thermal Property Data and Range of Heat Transfer Coeffi- cients. For purposes of orientation, Table D,3a below lists some thermal properties of typical materials employed in the construction of experimental and practical rocket engine walls. The listed data represent averaged properties between temperatures at ambient and 2000°F, as ob- tained from various sources. Table D,3a Metals Refractories Mapenal Low | Stain- Molyb carbon | less |Copper d or°YP- | Graphite; Alumina] Zirconia steel steel Sa p, lb/ft? 485 485 550 563 140 250 360 BTU/hr —— 2 1 21 2 1.2 ha Pit 0 5 0 80 60 5 c, BTU/Ib °F 0.15 | 0.15 | 0.10 0.08 0.17 0.30 0.16 Melting tempera- ture Tn, °F 2700 | 2700 | 1980 4750 es 3110* | 3850* * Failure temperature under 40 lb/in.? Table D,3b below shows the range of gas heat transfer coefficients h(BTU/hr/ft? °F) occurring in various phases of jet propulsion engi- neering. Table D,3b Ramjet and Supersonic and hypersonic Chamber | Nozzle throat turbopump elements missile skin friction 100-400 700-4000 100-1000 100-2000 These ranges of h are based on both experimental data and theoretical calculations. CHAPTER 2. ONE-DIMENSIONAL HEAT CONDUCTION IN A HOMOGENEOUS MEDIUM D,4. Slab of Thickness d. The differential equation governing heat flow in the solid is k= = pe (4-1) D,4 - SLAB OF FINITE THICKNESS In accordance with the remarks in Art. 2 the boundary conditions (see Fig. D,4) are: I —) aT k—=h(T,—T)| at « = 0 atc — a. The initial condition is T = 0 at ¢ = 0. Introducing a transformation to nondimensional variables: T x Ke wy, i — _ — a iaca2 a2 (4-2) ee ae where « is the thermal diffusivity of the material, «x = k/pc, we obtain the differential equation and boundary conditions in nondimensional form: HE) GE) af OF ee) c eel 00 hd =e | eh = 0 | Oe 1s : (4-4) 4 = 0 at &=1 with the initial condition Cala a — 0 (4-5) When the system is solved by separation of variables as discussed in Art. 2, we obtain the following results: The eigenfunctions are ig. D,4. gn(t) =cosu(l1— 8) (=1,2,...) ee where the eigenvalues yu, are the roots! of the eigenvalue equation p tan p = 2 (4-6) The complete solution is iL = A(é, T) TE T. = 1 at O(é, T) = 1 oF ye dahon cos ACL ir E)e7 tn (4-7) i n=1 where the amplitudes, determined from the initial condition 0(£,0) = —1, are 1 - f, COS un(l — E)dé 4 sin pn An = 1 = = 5 5) (4-8) if cos? un(l — é)dé Zin + SID 2un It is seen from the nondimensional formulation (Hq. 4-3, 4-4, and 4-5) 1 The first six roots of this transcendental equation, for 0 < hd/k S ~, are given in [6 (Appendix IV, Table 1)]. ( 261 ) D - CONDUCTION OF HEAT that the problem involves only one essential parameter, namely, the Biot number _ hd 7 (4-9) In the solution (Hq. 4-7) the Biot number enters via the eigenvalues which depend on N, according to Eq. 4-6. Except for small values of 7, corresponding to @ <1, the series in Eq. 4-7 converges rather rapidly and one or at the most two terms (in case of large N) are sufficient for practical purposes. When the conver- gence of the series is slow, good approximations of the temperature dis- tribution in the slab can be obtained by application of results for the semi-infinite solid discussed in the next article. D,5. The Semi-Infinite Solid. The heat flow problem for this case is formulated by the differential equation (Eq. 4-1) with its initial and boundary conditions for d— o. Its solution [6, Chap. 2] can be put in the form, 6(N.,7*) = erfc = — e(Nzt*") erfe - + vi) (5-1) - where erfc z is the complementary error function, 2 z erfec z = 1 — — = edz A/a i while the dimensionless variables are ei hee Ealeuthy 1 OAEL uhh iar Ree (5-2) Curves of 6(N;z, 7*) vs. 1/7* with N, as parameter are shown in Fig. D,5. In application to slabs of thickness d, the significant feature of these curves is that 0 at x = d in the semi-infinite solid is close to zero, until some time 77 when the temperature at the position represented by Nz=a begins to rise rapidly. For shorter times r* < 77, the temperature dis- tribution in the slab is approximately represented by that in the semi- infinite solid with increasing accuracy as x = éd approaches the flame side (— 0). However, as — 1 the slab temperature given by Eq. 4-7 rises above that for the semi-infinite solid at Nzo:a, given by Eq. 5-1, owing to the influence of the boundary at xz = d where heat flow is blocked in accord- ance with the boundary condition 07 /dx = 0 for the slab. The effect of this thermal barrier can be represented by supposing that, in addition to the heat source of strength h(T, — T) at x = 0, the region defined by 0 S x Sd in the semi-infinite solid receives heat from an zmage source ( 262 ) D,6 - APPLICATIONS of this strength located at x = 2d, i.e. a distance x = (2 — £)d from the plane at x = éd. It can be shown that, with two such sources located sym- metrically about x = d, the conduction of heat in the region 0 S$ x S dis governed by the same differential equation and initial and boundary con- ditions as in the slab during a time 7* < 73,, when the temperature rise Fig. D,5. Temperature transients in semi-infinite solid (cf. Eq. 5-1). at x = 0 is not yet influenced by the image source at x = 2d. The tem- perature in the slab given by Eq. 4-7 is therefore accurately represented at short times 7* < 7%, by the superposed effect of the two sources: O(E, rT) = O(Namta, T*) + O(N2=(2-5 a, T*) (5-3) where the conversion from 7* to 7 for the slab is tT = 1r*/N2_j. In practice, Eq. 5-3 is accurate within 1 per cent of @ up to times when the Fourier representation (Eq. 4-7) requires only one term for 1 per cent accuracy. Computations of 6(N,, 7*) in Eq. 5-1 and 5-3 are facilitated by tables of e” erfc z and appropriate asymptotic expansions given in [6]. D,6. Applications. The temperature distribution in the slab given by Eq. 4-7 may be regarded as a superposition of damped spatial modes, cos un(1 — &), with decreasing amplitudes A, and increasing exponential { 263 ) D - CONDUCTION OF HEAT damping due to the factor e-“:" as the mode number increases. Insofar as the fundamental mode n = 1 is the dominant term in the Fourier series, the physical role of the material and heating parameters (h, T,) in the development of temperatures in the slab can be deduced from the ex- pression for this mode. If only the fundamental mode is retained in the Fourier series of Eq. 4-7, the space average of the temperature ratio 6(&, rT) is 1 6 A(r) = if XS Nelts 1k Bae one ea: (6-1) 1 For small Biot numbers N <1 the eigenvalue equation yields N= wa tam yt at = and therefore the exponential time term becomes while the amplitude in the mean temperature formula (Eq. 6-1) is m2 Ai = = a eo} (6-2) Hence for N <1, the mean temperature is given by 2 _ ft 6(7) ~1—e ved (6-3) showing that, for low Biot numbers, the slab has a time constant i, = ue N <1 (6-4) which, in contrast with time constants of vibratory mechanical or electri- cal systems, depends not only upon the system structure (thermal proper- ties and thickness) but also upon an external factor, namely the heat transfer coefficient. For larger N the amplitude given by Eq. 6-2 is still approximately —1, and a more general expression, analogous to Eq. 6-4, is obtained for the time constant: a 6-5 ie SS = Ky (Gp) where the heat transfer coefficient enters ¢, via the Biot number implicit in 1. The dependence on N becomes weak for N > 2 as ui approaches its asymptotic value of 7/2 for large N. In application to rocket wall design, Eq. 4-7 is useful in determining ( 264 ) D,6 - APPLICATIONS the thicknesses of relatively thin cylindrical elements in which heat flow is principally in the radial direction (cf. Art. 7). The design problem usu- ally consists of the determination of d for given material constants and heat transfer parameters h and T',, subject to the condition that, at the end of a specified duration time t,, a given critical temperature T’., shall be attained at a position z,,. Thus 7, might be the melting temperature and the position might be at the flame boundary z,. = 0. In thin-walled rocket chambers, 7, usually corresponds to a space average temperature at which the strength of the material begins to de- crease appreciably. From Eq. 6-1 and 6-2 we obtain for this case the following expression for ¢, in terms of t, and @,. = T../T;,: 1 ta =, In =r (6-6) For small Biot numbers the above criterion yields from Hq. 6-4 d= oe a (6-7) pc In =Ae and for greater Biot numbers we have from Eq. 6-5 Kta ace Ih == Gr. d=~wm (6-8) In The latter is applicable in the approximation that the temperature drop across the slab at the time ¢; as computed from Eq. 4-7 is small com- pared with T,,: T (0, ta) al T (d, ta) << die (6-9) Under many conditions the inequality (Eq. 6-9) cannot be well satis- fied by use of the criteria, represented by Eq. 6-7 and 6-8, based on the mean temperature formula (Eq. 6-1). Thus when h and T, are sufficiently large, there may be large temperature drops across the wall at the time ¢,; then T’,, is usually specified at (or near) the flame side. Here either one or two terms in the Fourier series of Eq. 4-7 or the use of Eq. 5-3 may suffice to determine d. It may happen, however, that the desired duration time t, is so large with given large heat transfer parameters (h, T,) that 6,. will be exceeded regardless of the thickness d, i.e. Eq. 4-7 or Eq. 5-3 cannot be solved for d. This can be ascertained by noting that the flame side temperature transient for the semi-infinite solid @(0, r*) in Eq. 5-1 rises slower than that of any finite slab with the same h, 7’, and material constants. In other words, when the following condition, 9(0, r*) = 1 — evr erfc W/7* > On ( 265 ) D - CONDUCTION OF HEAT with holds, then the desired duration time t, cannot be attained for any slab thickness whatever. CHAPTER 3. TRANSIENT RADIAL HEAT CONDUCTION IN A HOMOGENEOUS HOLLOW CYLINDER D,7. Classical Results for Newtonian Heat Transfer. Let the inner and outer radii of the hollow cylinder be a and }, respectively, as Vis Fig. D,7. shown on Fig. D,7. The temperature distribution 7'(r, £) satisfies the dif- ferential equation in polar coordinates, (SF pt) me (7-1) en on We consider a Newtonian heat input at r = a from a flame at temper- ature T’, oT ko | =KT(,1) - T,) r=a and no heat transfer through the boundary at r = b he or aor a 0 ( 266 ) D,7 - CLASSICAL RESULTS The initial condition is, as usual, T(r, 0) = 0. The dimensionless varia- bles appropriate to the cylinder problem are ’ Ua = we (7-2) by which Eq. 7-1 and its initial and boundary conditions become ao , 100 a0 ans nan ar. 2) 00 ha 90 = at = 6 dw te a = —1 at t,=0 (7-5) From this nondimensional form it is seen that the solution depends essentially on only two dimensionless parameters, namely a type of Biot number, NV, = ha/k and the ratio b/a, which we shall designate as Q, the thickness number. When the system is solved by separation of varia- bles as discussed in Art. 2, the results are expressed in terms of the following combinations of Bessel functions J and Neumann functions Y: Jo(uw) — Yo(uw)’ Ji(uQ) = Vi(uQ) Ji(uw) — Yi(uw) Ji(uQ) =“ Yi(uQ) where the general eigenvalue yu satisfies the characteristic equation [9], oe Ri(u) eas y. Ro(u) jl Ro(uw) = Ri(uw) = (7-6) the eigenfunction corresponding to the nth root yu, of Eq. 7-6 is: gn(w) = Ro(unw) with the orthogonality condition in this case (i.e. polar coordinates) ik Taboaloads =O) (Gee a (7-7) The complete solution in nondimensional variables is: 2) Plaine 7 Th Oey le valores. | (Ges) n=1 ( 267 ) D - CONDUCTION OF HEAT where the amplitudes [9] determined from the initial condition O(w, 0) = —1 are ia Ro(unw) wd 2N aRo(un) : = - —___+________ 7 JP Riluna)edo (un 9) °R3(4n) — (uh +N) RF(Hn) Aes Analogous expressions for temperature transients due to Newtonian heat input at the outer radius of the shell, r = b, are given in [10]. All of the above results can be readily deduced from the somewhat more general result, i.e. the heat transfer through both boundaries, obtained in [6, p. 278]. It is seen that the formal expressions for the hollow cylinder are con- siderably more cumbersome than those for the slab. It is therefore of practical importance to ascertain the range of wall thickness to radius ratio, so that the problem can still be treated in a good approximation by expressions for an equivalent slab. This point will be discussed in more detail in the next article. D,8. Applications. Thermal Stresses. For a hollow cylinder of rather large thickness number 2 = 2 and aslab of thicknessd = b — a = (Q — l)a = a, the flame side temperature transients are shown in Fig. D,8 for hd/k = h(b — a)/k = 1. Initially, the temperatures rise at the same rate in both structures; subsequently, as the larger heat capacity of the cylinder is utilized by the heat flow, the temperature falls below that in the slab. Thus, in relatively thick (refractory) cylinders, use of the slab formula may lead to considerable error in the calculation of temperatures. However, for relatively thin cylinders (metallic walls), the heat capacities of the cylinder and slab of same thickness are approximately equal, in the ratio ; ean 2 pe TO ©). 50(b — a) = 5 (241) (8-1) It is to be noted further that the effect of the larger heat capacity on the temperature is to some extent counteracted by the Newtonian type of heat input, which is greater at lower wall temperature. By detailed studies of temperature distributions based on Eq. 7-8, it is found [9,11] that the equivalent slab solution is applicable within the approximation with which h and the material data can be specified in practice, provided the thickness number © does not exceed 1.2. This result justifies the use of slab formulas in application to thin-walled cylindrical structures, such as rocket chambers, metallic nozzle walls, etc. Larger thickness numbers © occur for refractory shells used as inserts in rocket chambers and nozzles. With such refractory liners the problem of thermal stresses becomes important in design considerations. In the approximation that the material behaves elastically in the temperature ( 268 ) D,8 - APPLICATIONS. THERMAL STRESSES range of interest, the transient thermal stresses in the cylinder can be calculated by use of Eq. 7-6 in the thermoelastic equations [12], relating the stresses to instantaneous temperature distributions. This has been done in [11] with the following principal results for the long hollow cylin- der heated by Newtonian heat transfer through the inner boundary. At any instant ¢ > 0 the transient hoop stresses a(r, t) have extreme values at the boundaries, there being maximum compressive stress ou(t) = o(a, t) at the inner boundary and maximum tensile stress o,(¢) = o(8, f) at the outer boundary. Similarly, axial stresses also have extreme values s fam pe a if fi fl 2 og ke fap) ig = ta a ee ee 7 a go A Se a a Ge — ktiioed? Fig. D,8. Comparison of temperature transients in plane parallel slab and hollow cylinder. at any instant at the boundaries, r = a, b, where these extremes are equal numerically to the local hoop stresses o.(t), oo(t) respectively. Radial stresses are zero at the boundaries and rise to a maximum within the shell, but this maximum is small compared with boundary stresses at any instant. The thermoelastic equations [11] lead to rather simple ex- pressions, as given below, for the boundary stresses o.(¢) and o,(¢). These boundary stresses in turn possess absolute maxima attained at some time during the heating process. We define the dimensionless stress 7 by be o(1 — v) aT. (8-1) ( 269 ) D - CONDUCTION OF HEAT where v, EH, and 6 represent Poisson’s ratio, Young’s modulus, and the coefficient of thermal expansion, respectively. By the use of Eq. 7-8, the mean dimensionless temperature in the cylinder is i i Eee) 2 C 0(ra) = 7(b? — a?) i T. 2rrdr = || O(a, Ta) wdes Accordingly, solving the thermoelastic equations for the heated cylinder [11] we obtain: ne = ag = Ble) — 01,70) (8-2) my = A = We) — 0(9, 7) (8-3) Thus the dimensionless stress on the boundary is numerically equal to the difference between the mean dimensionless temperature and the boundary temperature, with y S$ 0 indicating compression and tension, respectively. In the course of heating from initially uniform temperature, the bound- ary stresses rise from zero, pass through respective maxima, and subse- quently decrease toward zero as the shell approaches thermal equilibrium at @ = 1. These maxima depend only on the Biot and thickness numbers. Analogous expressions for thermal stresses in a cylinder with Newtonian heat flux at the outer radius r = 0 are given in [13]. D,9. Remarks on Thermal Shock. A serious difficulty in the use of refractories under rapidly varying temperature conditions is their tend- ency to crack, chip, or spall due to the presence of thermal stresses set up by large spatial variations of the temperature within the material. The phrase ‘thermal shock”’ is commonly employed to describe the failure of the material under transient stress conditions associated with large tem- perature gradients. In the literature on the subject, various criteria have been proposed to express the resistance to thermal shock in terms of material properties and size factors. Recently [14] an attempt has been made to include the effect of heating parameters in the criterion for resist- ance to thermal shock. On this basis it can be shown that previously pro- posed criteria are special cases corresponding to various regimes of the heat transfer coefficient h. The physical role of material and heating parameters in the occurrence of thermal shock can be described, assuming that the material behaves elastically, by comparison of transient thermal stresses with maximum stresses such as, say, the yield stress of the material. This has been done in [14] on the model of a slab of thickness 2d heated (or cooled) symmetrically by Newtonian heat transfer through its ( 270 ) D,9 - THERMAL SHOCK boundaries. The results given below from [1/4] can also be deduced from Eq. 8-2 and 8-3 for the hollow cylinder,? in the limit 2 — 1. At any instant ¢t > 0 the thermal stresses have maximum values at the boundaries (compression on heating) and the midplane (tension on heating). Let the dimensionless stresses at the boundaries and midplane be denoted by 70(7) and 7»(7) respectively. As functions of 7 these have absolute maxima, |7o|max 2Nd |7m|max, the magnitude of which depend only on the Biot number N = hd/k as shown in Fig. D,9. If S, and S, denote, Tension on midplane : ae (Seer Te “ hd/k Fig. D,9. Maximum dimensionless stresses |n|max in symmetrically heated plane parallel slab vs. Biot number. respectively, the yield stress in compression and tension, the correspond- ing allowed dimensionless stresses are on S.(1 — v) i EgT, AS S,(1 —— v) EpT, Thus, in heating the slab, the resistance to thermal shock is measured by 2 The midplane divides the slab into two regions, in each of which the thermal stress distributions correspond to limiting cases of the thin hollow cylinder with d=6-a, Q—1. om \ D :- CONDUCTION OF HEAT the ratios Tree cata = st I Rpeceitogs = as (9-1) \no max Poe which involve, in addition to material properties, the heat transfer param- eters h and T,. On symmetric cooling of the slab, analogous relations are [14] ye eee —= als Pie taste rsien — aes (9-2) |10| max 7a Comparisons of experimentally rated values of resistance to thermal shock with analytically deduced values show a one-to-one correlation of Reension in Eq. 9-2 with experimental data [1/4]. It is conjectured therefore that, under the usual test conditions of alternate heating and cooling of refractory bricks, failure should be expected primarily in tension. Some evidence supporting this conjecture is provided by experimental data in Vio MO The appearance of the Biot number in the above thermal shock re- sistance formulas shows the varying importance of thermal conductivity and size factor in different regimes of the heat transfer coefficient h. De- tailed discussion of the resistance formulas (Eq. 9-1 and 9-2) is given in [14]. For further treatments of thermal shock based on the use of transient temperature formulas substituted in available thermal stress equations see [16,17]. CHAPTER 4. TRANSIENT HEAT CONDUCTION IN A UNIDIMENSIONAL COMPOSITE SLAB D,10. General Results for Newtonian Heat Transfer. The problem of heat flow in a composite rocket wall may be treated approxi- mately on the basis of heat conduction in a composite slab (Fig. D,10) con- sisting of an inner layer between e = —d,and x = 0, possessing uniform thermal properties (distinguished by subscript 1) and an outer layer be- tween « = 0 and x = dy with uniform thermal properties (distinguished by subscript 2). The inner material usually serves as a thermal shield of low inherent strength (refractories) preventing excessive temperature rises in the outer material (metals, Fiberglas at low temperatures) which must withstand the combustion pressures developed during the operation of the rocket engine. In accordance with the results in Art. 2, we formulate the problem for the dimensionless temperature defect 0 = (T/T,) — las 8 = gay yH)=e” 6 ( 272 ) Sb ein 0) D,10 - GENERAL RESULTS ~~ > 8 3 | o + oO = E is = aS jag i | | % = = 6h <0 = 2 ios DMC: where the eigenvalue differential equation for ¢(z) is d*p opty Se k FES ar pcd\79 = 0 (10-1) ra of ge BORA AS ) [== [be eS yon f = he = == eg = o1(a) —d,i<2<0 p1C1 ke t—sks (to = yyy, fi == hp = e = ¢2(z) Os2zed, p2Ce2 with exterior boundary conditions ky a = hei v= —d, (10-2) des ith Le i and interior boundary conditions f1 = $2 x=0 (10-4) d¢1 is dye ma a The eigenfunction obtained from Eq. 10-1 is g1(x) = 1 COS Ae + bi sin AM —-diszr<0 Ki V/k1 g(t) = < : (10-6) zh + be sin Z O Ta 1 =| (10-9) where, without loss of generality we have set a1, = 1, since this constant can always be included in the amplitudes A, which appear in the general solution (cf. Eq. 2-6), Oe, 1) =) Ane, = ) Anen(a)e a ke) ued 2188) ( 274 ) D,10 - GENERAL RESULTS By the results of Art. 2 we determine the amplitudes A, upon expansion of O(a, 0) = —1 in terms of the eigenfunctions ¢,(x). Thus we obtain from Eq. 2-7 0 dz pegndr prCigindx + p2C2P2nd& Lal eh / Ray Joe Ir i| pcg,dx Hs picigi dx + ip P2203 ,0X At this point it is convenient to introduce the following nondimen- sional quantities: 4 0 4 0 Silman age Aime: Te mg? Me de Ki MinT1 i M5 nT2 im Nea (2 a di «:) Kin 1 Ke Then the characteristic equation for yw (or pe) is N, tan wi + m1 _ ka K2 hdy Nee ata evan tan pe = Ths kL Ni = ki (10 7a) and the expression for the amplitude in Eq. 10-6 becomes Gin + p2C2Gon YApspo a ng ECL 10-10 picili, a= p2C2H on ( ) where d ; Gin = = [sin Hin + bin (cos le aa 1)] GCG. = a [SiN pon — ben (COS Hon — 1)] ee so (Gl hme, eR) Sica re = Oe ane reel d : : Hon = a [(1 + 03,)Men + (1 — 63,) Sin Hon COS pon + 2bon Sin? pon] Thus we obtain for the temperature distribution: GD) ee 6(&, 7) 1 + y An (cos Mink + Din sin winks) e7Bin™! —1 = 1 s 0 (10-11) 1+ )) An (COS Hans + ban sin pane”? OS S1 where the pin’S (u2n’s) are computed from the eigenvalue equation (Eq. 10-7a), the amplitudes A, are determined by Eq. 10-10, and the b,’s are computed from Eq. 10-8 for ai, = 1. The first two eigenvalues satisfy- ing Kq. 10-7a are obtainable from curves in [18]. ( 275 ) D - CONDUCTION OF HEAT Computation of temperatures from Eq. 10-11 are generally rather cumbersome. However, for two limiting cases of importance in rocket engineering, i.e. thin and thick shielding layers, computations based on Eq. 10-11 can be considerably simplified, as shown in the next two articles. D,11. The ‘*Thin’”’? Shield. In practice, some thermal shielding of rocket walls is effected by very thin refractories or protective paints on the inner boundary under conditions when, while N,; > 1, the thickness d, is sufficiently small to give rise to the following relations in Eq. 10-7a: hd Min < Pon, pu Stanwu, w,) Hence, when the higher Fourier terms in Eq. 10-11 are negligible, the temperature of the shielded material given by this equation reduces to the simple slab formula (cf. Eq. 4-7 and 4-8): 4 sin M21 2ue1 + sin 2p21 0(£s, tT.) =1 cos pail = £5) @—Ha™ (11-3) where, in view of the eigenvalue equation (Eq. 11-2), the effective “‘ Biot” number is _ ids i No ke i.e. the shielding reduces the heat transfer coefficient to the shielded material to an effective value Nett = AION 1+N, It is thus seen that the problem of the composite slab reduces, in this limit, to the case of the simple slab with a reduced effective heat transfer coefhicient. Furthermore, Eq. 11-4 shows that even a thin thermal shield can cause a large reduction in the heat input rates if h is large, which is hess as (11-4) ( 276 ) D,12 - “THICK” THERMAL SHIELDS indeed the case for rocket nozzles. For example let the allowable temper- ature be 6. = 7.,/T, = 0.5 at the flame side boundary of a low carbon steel? wall of thickness d, = 0.25 in. = 0.0208 ft with heat transfer coef- ficient at the nozzle throat h = 1500 (BTU/hr) /ft?°F. The Biot number is N = N2 = hd2/kz = 1.56, and the duration time tf, = d374/x2, computed with two terms in the Fourier series for the simple slab equation (Eq. 4-7) in this case a(0, Ta) a Bor =. il a 0.625e—1-9%a = 0.095e—12-57a = 0.5 is fg = 1.25 sec. If a zirconia layer of thickness d; = 0.02 in. = 0.0017 ft shields the steel, the effective Biot number is N2/[1 + (hdi/ki)] = 0.50. Eq. 4-7, in this case, 0(0, 74) = Or = 1 — 0.850e~% #274 — 0.086e71°-8a = 0.5 gives for the duration time tj = 7.1 sec. The expression for h,,; in Eq. 11-4 can be deduced from steady state considerations given in [6, p. 15] in a discussion of thin layers of oxide, grease, or scale. The application to transient states is justified in the approximations of Eq. 11-1. D,12. ‘‘Thick’? Thermal Shields. Another important limiting case arises when the temperature in the shielded material has no appreciable gradients either because of large conductivity in the latter, ky > ki, or because of large insulation thickness d; >> dz. By a limiting procedure for ko— © (i.e. 2 0, Din > pin(p2C2d2/picid1), etc.) Eq. 10-7a can be re- duced to an eigenvalue equation for pin: Na = 1H py tan wi = eee (12-1) where y is the ratio of heat capacities at p2Co2dl2 picid The temperature distribution (Eq. 10-10a) becomes Tae, ) TL g = 6(&, 7) 1 =e ae A,(cos Hin€1 =F bin sin ceo ee = Il ss £1 Ss 0 jet » Aare = te » AneHin™! 0S 81 3 Material data and units are given in Table D,3a. ( 277 ) D - CONDUCTION OF HEAT where A,, in this limiting case, is ri 2(sin Hin -+ YHin COS Hin) [1 + (ymin)? Juin + [L — (ypan)?] sin pin COS pin + 2YH1n COS? Hin (12-3) An = Eq. 12-2 shows that the shielded material is at a uniform temperature equal to the interface temperature 0(0, 7) at any instant. It is to be noted that the result (Eq. 12-2) represents, for —d; S & S 0, the solution of the simple slab problem with Newtonian heat input through one boundary oT Lire aE itl) at x = —di and heat transfer through the other boundary governed by oT oT ki = p2tole —, at oO In view of the latter boundary condition the shielded material behaves essentially as a thermal capacity of magnitude p2ced2 per unit area of the slab. D,13. Design Criterion for Minimum Weight. The considerations of the preceding article can be applied to the design of uncooled com- posite rocket chamber walls in which the outer material (envelope) pro- vides structural strength, while the inner material acts as a thermal shield to prevent the attainment of excessive temperatures in the envelope dur- ing the firing period ¢,, which may be of the order of a minute in duration. In the approximation that the composite thickness d: + d:2 is small com- pared with the chamber radius, and ked; > kid2, the envelope temper- ature 7’. is given by the interface temperature in Eq. 12-2 ae = r = 6(0, 7) (13-1) Because of strength requirements, the hoop stresses in the envelope must not exceed a critical stress Se, say the yield stress of the material, which is specified as a function of To, i.e. Se = S2(T2). This requirement is ex- pressed by pk S S82(T2)de (13-2) where p is the combustion pressure in the rocket chamber and RF is the chamber radius. A second requirement is that the composite weight per unit area of the wall, w = pidi + pod» (13-3) shall be a minimum. Now, S2(7'2) generally diminishes as 7’; increases and therefore it diminishes also as ¢ increases during the firing period. { 278 ) D,13 - DESIGN CRITERION FOR MINIMUM WEIGHT For any desired duration time f,, however, it is possible to select an infi- nite set of thicknesses (di, d2) large enough to assure that the inequality in Eq. 13-2 holds up to ¢,. In the set (di, dz) there is, in general, a particu- lar pair of dimensions which makes the weight w a minimum. The prob- ae Sea peels oA kiss Ri ae eee alos Composite slab weight per unit area w, Ib/in? Metal ticles de, in. Fig. D,13. Weight per unit area of composite slabs (zirconia-metal) vs. metal thickness [19]. (Cf. Art. 13.) lem of optimum design is to determine, for specified materials and heating parameters, the pair of dimensions (di, d2) which minimize w and lead to equality in Eq. 13-2 at the time fu. Fig. D,13 shows curves of w vs. dz for combinations of zirconia shield- ing aluminum and molybdenum envelopes with the following data: 279») D - CONDUCTION OF HEAT h = 300 (BTU/hr) /ft?°F TS — ely) Oks) Woy WW t; = 60 sec The critical stress vs. temperature curves employed for aluminum 24ST and molybdenum are given in [20] and [21] respectively. It is seen from Fig. D,13 that for certain combinations of metal and refractories there exist rather sharp minima, so that a considerable weight saving can be achieved by the optimum choice of thicknesses. For other combinations, however, as in the case of zirconia and aluminum, the choice of thicknesses from the point of view of minimum weight is not so critical. D,14. Remarks on the Composite Hollow Cylinder. On the basis of the discussion in Art. 8, the transient temperature distribution in a relatively thin shell (Q S$ 1.2) can be computed from an equivalent slab formula. This rule appears justified for composite media also, and there- fore the results of Art. 10, 11, 12, and 13 can be applied in many com- posite cylinder problems which arise in rocket engineering. For cylinders of greater thickness number (Q2 > 1.2) the use of these results leads to conservative estimates from the design point of view, since these results, based upon the composite slab, predict higher temperatures than would actually occur in cylinders. Accordingly the solution for the latter case is not discussed here because of its complexity. For related problems the interested reader is referred to [22,23]. It may be noted further that, for the combination of a thin refractory with a thick metal shell, a reduced effective heat transfer coefficient can be computed from Eq. 11-4 and this h,; applied in the formula for the temperature transients in a simple cylinder as given by Eq. 7-8. CHAPTER 5. SOME SPECIAL PROBLEMS D,15. Variable Thermal Properties. In metals, c increases while k may decrease or increase with temperature. In the latter case the effect of the variation of the thermal properties is at least partially neutralized, as can be seen from the expression for the time scale (i.e. the conversion factor from the dimensionless to the physical time ¢ = {,7), D,15 - VARIABLE THERMAL PROPERT. In the former case, however, the effect on temperature t considerable. Accordingly we shall discuss only the for To a first approximation, c and k may be considered with the temperature, ey Kegs: sc Brewery a (0 < ki < ko), (0 T, I The differential equation and initial and boundary conditions for a homo- geneous slab (unidimensional) are o pat) = 2 oF aN Ge) Oe Gp Ch 1p} gf Ato Ox i —"() om £S0 Let i ME hae) Co Nats Cals ee Se St waley CS e ths = Prema ths 7 TP 1 (15-3) Substituting Eq. 15-1 and 15-3 into Eq. 15-2 we obtain after some alge- braic manipulation a°fi(8) _ dfe(@) 0 OT 098 hd OS 0 bp 5, 138) eins =) (15-4) ele) ae 0 at &=1 Oat, — 0, (= 0 = 0) where ‘S) fi(9) = e(1 = px®), f2(®) ae e(1 ain pO), F3(®) fava ee 27,0” 1 ky if C1 1 < —i— ¢ = — <= — -_ OS PD 2 ko — ky’ OS Be 75 > Cy, =?) Can In the purely mathematical range, —-© < 0S ~, the functions fi, fo, and f; are, of course, nonlinear, so that in principle the differential equa- tion and boundary conditions are nonlinear. However, the actual entire physical range of 9 is —1 S 0 S O, and in this range it develops that in the practical ranges of p, and p, the three functions are rather close to straight lines. ( 281 ) D - CONDUCTION OF HEAT In Fig. D,15a the graphs of f1, f2, and f3 vs. 8 are shown in the entire range —1 <0 <0. These graphs are drawn for p, = 3 and p, = ¢. These values correspond to 662-per cent increase in c and 333-per cent decrease in k from their initial values up to the melting point, on the basis that the melting temperature 7’, is $ of the flame temperature T,. These values represent a rather wide range of variation in thermal properties. f,(O) f,(O) f,(0) he ae Maye iia sedy oD) rey gic A ee a FET ea | el ae p [|p i] falar txt Belhce See ea ae A a | ee eee AV Aae: ale oul ieee OS Fig. D,15a. It is neither necessary nor useful to calculate temperature transients beyond the time at which the fire side wall reaches the melting temper- ature. Beyond this time the phenomenon of heat conduction changes in character, and the results which are based upon the boundary condition in Eq. 15-2 are no longer valid. This point will be discussed in more detail in the next article. Hence one needs to consider only the range -1S08 0, where hs ol oy Thus if, say 7’ = $7, then the range of interest is -1ses-} ( 282 ) D,15 - VARIABLE THERMAL PROPERTIES It will now be observed from Fig. D,15a that in this range the curves can be approximated rather closely with straight lines. If this is done, then the differential equation and boundary conditions become linear and the solution can be written immediately on the basis of the simple slab solution with constant thermal properties. The essential feature of this method of linearization is the fact that now there exists at least a quali- tative criterion whereby one may judge the accuracy of the solution; in general, the closer the fit between the straight lines and the curves in the range —1 < 0 S 0, the more accurate the solution. However, it is not necessary to carry out the solution by this linearization method because of the following result. 0 0.2 0.4 0.6 0.8 1.0 Dimensionless time Fig. D,15b. Comparison between exact results and results based on average c for small N. Suppose that two straight lines are drawn so as to minimize the mean square error between these lines and the curves representing f:1(0) and fo(®) in a range —1 < © S © S O. Suppose further that a third straight line, line A in the figure, is drawn, passing through the origin (0 = 0) and intersecting the curve f;(0) at 9 = —(1 — 6) (ie. at © correspond- ing to the midpoint of the range). Then it is found that the results based upon this linearization are exactly the same as the results which would be obtained if average thermal properties were assumed at a temperature corresponding precisely to the mid-interval. This then is the geometric meaning of the use of average thermal properties. Observing then that the generally used method of average thermal properties corresponds to the fitting of certain straight lines to the graphs { 283 ) D - CONDUCTION OF HEAT representing f1, fe, and f;(0) and observing further from Fig. D,15a that these fits appear to be quite reasonable, it follows that the method of average thermal properties cannot lead to excessive errors. Indeed this conclusion can be verified exactly in the case of small Biot numbers, for in the latter case it is easy to obtain an exact solution. In Fig. D,15b a comparison is shown between the exact solution, in a range 0 Gy 9 \ Ghee cr (le ae Pt ae 2-5 Tw é Pipa | dy* ee Ghis k 1 Pp Cp Ew afl" Ow l= Pr, =F Pw Oe a | dy* (2 6) where ul u RV on O = % | — * — Nip @ (uw/Pw) * = Cs = Des and a= & ) GeiN/ ny pe éu The subscript , refers to values at y = 0, i.e. at the wall. i 1 Terms involving pv’ are sometimes included in Eq. 2-1 and 2-2. These terms can, however, be combined with pd in the equations of momentum, energy, and continuity to give pv. The variable pv, rather than d, then appears in the conservation equations. Consideration of pv’ would be necessary only if it were desired to calcu- late 3. ( 289 ) E - CONVECTIVE HEAT TRANSFER AND FRICTION Probably the first attempt to predict heat transfer from Eq. 2-5 and 2-6, or Eq. 2-3 and 2-4, was made by Reynolds [4, pp. 81-85]. He assumed constant properties, that 7/tw = q/qw, that the molecular shear stress and heat transfer terms in the equations are negligible compared with the turbulent terms, and a = 1. With these assumptions, Eq. 2-5 can be divided by Eq. 2-6 and integrated to give 7* = u™. If the temperatures and velocities are weighted in the same way to calculate bulk temper- atures and velocities, Tf = uf, or guy Sea ei aati (2-7) Up Eq. 2-7, which relates the heat transfer to the shear stress, is usually called Reynolds analogy. It applies reasonably well to gases, which have Prandtl numbers close to one, but fails for liquids. In fact, Eq. 2-7 follows from Eq. 2-5 and 2-6 if Pr = 1, even if the molecular terms are not neglected. A number of refinements of Reynolds’ original analysis to make it more general are given by various authors. Prandtl and Taylor [5, pp. 110-113] introduced a laminar layer near the wall and obtained better agreement with the data for fluids with Prandtl numbers close to 1. Von Karman [6] added a buffer layer between the laminar layer and the turbulent core and thus extended the analysis to somewhat higher Prandtl numbers, i.e. to liquids. Further improvements in the theory are given in [3,7,8,9,10,11,12,13,14,15]. It is desirable to obtain relations for the heat transfer which do not contain the shear stress or friction factor, as does Eq. 2-7. This is especi- ally true in the case of variable fluid properties, where the friction factors may be no better known that the heat transfer coefficients. In order to obtain such relations it is necessary to make an assumption for the eddy diffusivity e,, in Eq. 2-5 and 2-6. E,3. Expressions for Eddy Diffusivity. Several assumptions to relate the eddy diffusivity in Eq. 2-5 and 2-6 to the mean flow have been made by various investigators [7,/1,13,15,16,17,18]. Reasonable assump- tions for the variation of e, follow. For purposes of analysis, the flow is divided into two portions termed the ‘‘region away from the wall” and the ‘‘region close to the wall.” Region away from wall. In the region away from the passage wall, it is assumed that the turbulence at a point is a function mainly of local conditions, that is, of the relative velocities in the vicinity of the point [19, p. 351]. This is probably not a good assumption near the passage center where considerable diffusion of the turbulence occurs [20]. How- ever, in that region the velocity or temperature gradients are so small ( 290 ) E,3 - EXPRESSIONS FOR EDDY DIFFUSIVITY that the error in calculated velocities or temperatures should not be large. A Taylor’s series expansion for wu as a function of transverse distance (changes in the axial direction neglected) then indicates that e, is a func- tion of du/dy, d?u/dy?, d’u/dy’, etc. If, as a first approximation, we con- sider e, to be a function only of the first and second derivatives, and apply dimensional analysis, there results aD du d’u\ __, (du/dy)® wm 0 Fae) = 8 ear ay where «x is an experimental constant. This expression was obtained by von K4érm4n in a somewhat different manner and is generally known as the Kdérmdn similarity hypothesis [/8]. A critical examination of the K4rm4n hypothesis from the point of view of statistical turbulence theo- ries is given by Lin and Shen [21]. Eq. 3-1 can be written in dimensionless form, as a, Ged Un/Py (d2u*/dy*?)? (3-1’) Region close to wall. In the region close to the wall it is assumed that e, is a function only of quantities measured relative to the wall u and y, and of the kinematic viscosity. This assumption includes, to a first approximation, an effect of the derivatives of wu with respect to y. Since the flow becomes very nearly laminar as the wall is approached, the first derivative approaches the value u/y, and hence may be omitted since u and y already appear in the functional relation. The second deriva- tive approaches the constant value zero as the wall is approached. The kinematic viscosity is included, inasmuch as the ratio of viscous to inertia effects is high near the wall where the turbulence level is low. The eddies in that region are small, so that the shear stresses between the eddies and the viscous dissipation of the energy in the eddies are large. With these assumptions, and dimensional analysis, L 2 €u = Ex @ Y, ) = n?uyF ee The form of the function F cannot be determined by dimensional analysis. On the basis of simplicity, and the fact that F should approach zero at the wall (effect of u/p large) and should approach one asymptotically as uy/(u/p) becomes large (effect of u/p small), it is assumed in reference [15] that neuy F=1-—e #0 ( 291 ) E - CONVECTIVE HEAT TRANSFER AND FRICTION or nruy é, = nuy(l—e #“/?) (3-2) where n is an experimental constant.? Eq. 3-2 becomes, in dimensionless form, Hy, ss nu*y*(1 — @ | Pw # ) (8-2’) Eq. 3-1 and 3-2 for e, can be considered as reasonable first approxi- mations. Whether or not these approximations are adequate can, at pres- ent, be determined only by experiment. E,4. Analysis for Constant Fluid Properties. Velocity distribution data for flow without heat transfer have been used to evaluate the con- stants x and n in Eq. 3-1 and 3-2. Velocity distributions. Eq. 2-3, with Eq. 3-1 or 3-2, was integrated numerically or analytically for constant properties for the regions close to and at a distance from the wall in [/5] and [22]. The integration was carried out for both a constant and a linearly varying shear stress (r = 0 at passage center) with similar results for Reynolds numbers > 10,000, so that the effect of variation of shear stress is neglected in most of the following calculations.* In the region at a distance from the wall the molecular shear stress is neglected because it is small compared with the turbulent shear stress [/4, Fig. 12]. The familiar K4rmdén-Prandtl logarithmic equation is obtained in the region away from the wall: 1 * u* = = K In 7 + ut where y* is the lowest value of y* for which the equation applies and ut is the value of u* at y¥. In obtaining this equation, one integration constant was set equal to 0 by using the usual condition that du*/dy* = co for y* = 0 [18]. This assumption can be avoided by including the molecular shear stress and heat transfer in the region away from the wall and evaluating the constant by assuming a continuous velocity derivative at yj [14, Fig. 12]. This assumption gives essentially the same numerical results as that made above. 2 The quantity in parenthesis in Eq. 3-2, which represents the effect of kinematic viscosity on e,, becomes important only for heat or mass transfer at Prandtl or Schmidt numbers appreciably greater than one. For Prandtl or Schmidt numbers on the order of one or less, or for velocity profiles, e. = n?wy (the value of n differs from that in Eq. 3-2) is a good approximation for the region close to the wall [73]. 3 The variation of shear stress is neglected only for the purpose of simplifying the calculations, and as shown in [1/4, Fig. 11], this neglect has little or no effect on the results, even for variable properties. ( 292 ) E,4 - ANALYSIS FOR CONSTANT FLUID PROPERTIES Fig. E,4a shows, on semilogarithmic coordinates, velocity profile data from [20,22], together with the analytical curves obtained. The data are for fully developed adiabatic flow in tubes. The constants « and n in Eq. 3-1 and 3-2 were found, from the data, to have values of 0.36 and 0.124 respectively. The relation for e, from Eq. 3-2 applies for y* < 26, and that from Eq. 3-1, for y* > 26. In matching the two solutions it was assumed that the velocity is continuous at the junction of the two regions. The velocity distribution is often divided into three regions rather than two: the laminar layer, where turbulence is supposed to be absent, the buffer layer, and the turbulent core [6]. The use of Eq. 3-2’ for e, close Reference o [22] oO [20] 10,000 Fig. E,4a. Generalized velocity distribution for adiabatic turbulent flow (vertical line is dividing line between equations at yj = 26). to the wall eliminates the need for a laminar layer and reduces the num- ber of regions to two. For values of y* < 5, Eq. 3-2 indicates that e,,/ (u/p) <1, so that the flow is nearly laminar in that region. This can be seen from the plot of u* against y* in Fig. E,4a, where, for small values of OG 20 Several other analyses were made recently and might be mentioned at this point. Van Driest [23] used a modified form of Prandtl’s mixing- length theory which assumed that the presence of the wall reduces the universal constant « or the mixing length. By introducing a damping factor into the expression for shear stress, he was able to obtain a velocity profile which agreed with the data for the regions both close to and away from the wall. ( 293 ) E - CONVECTIVE HEAT TRANSFER AND FRICTION An analysis by Einstein and Li [24] assumed that the laminar sub- layer is basically unsteady; that is, the sublayer grows until it becomes unstable and then collapses, the cycle repeating itself. This model gave a velocity profile agreement with data for the region close to the wall. Temperature distributions. For extending the analysis to heat trans- fer an assumption must be made for a, the ratio of eddy diffusivities for heat transfer to momentum transfer, in Eq. 2-6. The relation for a has not been clearly established. It is a fact that analyses based on an a@ of one agree closely with experiment [6,14]. However, some attempts to measure a directly indicate values which, in general, are somewhat greater than one [25; 26, pp. 122-126], except in the case of low Peclet or Prandtl numbers where values of a less than one may occur [14; 27, pp. 405-409; 28). The direct measurement of a is difficult, especially in the important region close to the wall, because it involves the measurement of velocity and temperature gradients. Reichardt [3] proposed the hypothesis that a is one at the wall and increases as the distance from the wall increases. For turbulent flow the important changes of velocity and temperature take place close to the wall for Prandtl numbers on the order of one or greater, so that the assumption of a = 1, in general, gives good results. It is of interest that the Prandtl mixing-length theory [16], which assumes that an eddy travels a given distance and then suddenly mixes with the fluid and transfers its heat and momentum, gives a value of a equal to 1. Although the actual turbulence may be more complicated than indicated by that theory, it does indicate that a value of a on the order of one is not unreasonable. I As was the case with the shear stress, the variation of heat transfer per unit area with distance from the wall has but a slight effect on the temperature distribution, except for liquid metals [/4, Fig. 11]. With g/Gy = « = 1, and constant fluid properties, Eq. 2-6 can be integrated numerically for the region close to the wall (e from Eq. 3-2) to obtain a relation between 7* and y* [15]. For the region away from the wall (y* > 26), where the molecular shear stress and heat transfer are neg- lected, it is easily shown from Eq. 2-5 and 2-6 that T* —T =u* — ut (3-3) where 7'* and u* are the values of T* and u* at yf = 26 (Fig. E,4a). Calculated temperature distributions are shown in Fig. E,4b on log- log coordinates. The temperature parameter 7* is plotted against y* for various Prandtl numbers. The curves indicate that the temperature dis- tributions become flatter over most of the passage cross section as the Prandtl number increases. From Eq. 2-6, d7'*/dy* = Pr at or very near the wall, so that the slopes of the curves at the wall increase with Prandtl number. The slopes of the curves in Fig. E,4b near the wall appear ( 294 ) E,4 - ANALYSIS FOR CONSTANT FLUID PROPERTIES equal, because the curves are plotted on log-log coordinates (d(log T*)/ d(log y*) = 1 at the wall). Included for comparison is the temperature distribution for a Prandtl number of 300, calculated by assuming «. = n?uy (n = 0.109) close to the wall rather than Eq. 3-2. This expression is a good approximation for velocity profile but, as indicated in the figure, is not accurate at high Prandtl numbers. Somewhat better results were obtained in [11], where it was assumed that «, = const u?/(du/dy), but the analysis is again in- accurate at very high Prandtl numbers. The sensitivity of the temper- ature profile at high Prandtl numbers to various assumptions for the 10,000 Reference —— Present analysis ——- 13. = n?uy) 10 100 1000 ‘10,000 ~LVRe y vie Fig. E,4b. Generalized temperature distributions for various Prandtl numbers. %- turbulent transfer in the region close to the wall, compared with that of the velocity distribution, indicates that the region very close to the wall could be studied advantageously by measuring temperatures ‘at high Prandtl numbers, rather than by measuring velocities in that region. Some work along these lines has been reported in [1/2]. In that case, con- centration profiles, rather than temperature profiles, were measured for mass transfer at high Schmidt numbers. No evidence of a purely laminar layer (linear concentration profile) was found for values of y* as low as one. This result is in agreement with Eq. 3-2, which indicates that e, = 0 only at the wall. Relations among Nusselt, Reynolds, and Prandtl numbers. It can readily be shown from the definitions of the various quantities involved ( 295 ) E - CONVECTIVE HEAT TRANSFER AND FRICTION that the Nusselt and Reynolds numbers for a tube are given by ree. Iie = Aer (3-5) where r* = fy VTw/Pw/(Hw/Pw), 'w being the tube radius, * Teuk(rs — y*)dy* IP = se Pee SPEAR OU (3-6) NE 2 3 ae * Ht u*(r, — y*)dy and Zee ee a (cen ATEN Taye 2 Gay: i u*(ry — y*)dy (3-7) The Nusselt number in Eq. 3-4 is based on the difference between the wall temperature and the mixed mean or bulk temperature 7*. The relation 10,000 100 10 1000 10,000 100,000 1,000,000 Re = pu,D/ ee Fig. E,4c. Predicted fully developed Nusselt numbers plotted against Reynolds number for various Prandtl numbers. among Nusselt, Reynolds, and Prandtl numbers can be obtained from these equations and the generalized distributions given in Fig. E,4a and E,4b. The parameter r* appears in all the equations and is assigned arbi- trary values for plotting the curves. Predicted Nusselt numbers for fully developed heat transfer are plotted against Reynolds number for various values of Prandtl number in Fig. E,4c. Examination of the curves indicates that the slopes of the ( 296 ) E,4 - ANALYSIS FOR CONSTANT FLUID PROPERTIES various curves are approximately equal on a log-log plot. This result justifies the usual practice in heat transfer investigations of writing Nu = f(Re, Pr) as f(Re) X f(Pr) (usually as Re*Pr°). The same result does not hold for very low Prandtl numbers where the slopes change considerably. A comparison between predicted and experimental results is given in Fig. E,4d. Fully developed mass transfer as well as heat transfer data are included, inasmuch as an analogy exists between heat and mass transfer when the concentration of the diffusing substance is small. The Stanton number is plotted against the Prandtl or Schmidt number for a Reynolds number of 10,000. Similar results were obtained for Reynolds num- ber, of 25,000 and 50,000 [75]. The predicted Stanton numbers were Transfer Ref. Ethylene glycol Heat [29] and water —— Present analysis Seas Eq. 3-9 for very high ’ Prandtl numbers Transfer Ref. a 2 ) E - CONVECTIVE HEAT TRANSFER AND FRICTION 2000 1000 500 Nuy © B = 0.0077 — 0.0096 g 0.0054 — 0.0064 Q 0.0032 — 0.0047 x 0.0025 — 0.0029 100 20,000 50,000 100,000 500,000 Rey Fig. E,5h. —T7y = 130°F. Experimental and theoretical results for heat transfer to carbon dioxide in the critical region. Pressure = 1200 lb/in.? abs. From [62]. .0112 — 0.0143 .0070 — 0.0078 .0052 — 0.0067 100 20,000 50,000 100,000 500,000 Rew Fig. E,5i. —7w = 150°F. Experimental and theoretical results for heat transfer to carbon dioxide in the critical region. Pressure = 1200 lb/in.? abs. From [62]. ( 312 ) Ei wee O DIC CAMO E,6. Concluding Remarks. The calculation of turbulent flow and heat transfer in passages from the conservation equations of momentum, energy, and continuity alone has not yet been found practicable. By intro- ducing physical assumptions to relate the eddy diffusivity to the mean flow, however, and determining several dimensionless constants by meas- uring a velocity profile, heat transfer and friction in a number of circum- stances can be calculated. These cases include heat transfer and friction for various Prandtl numbers for fully developed flow, for the entrance region, for constant and variable properties, and for noncircular passages. In most cases where an experimental check is available the agreement between theory and experiment is good. A particularly good check for the case where the fluid properties are variable was obtained for heat transfer to carbon dioxide in the critical region. Although considerable progress has been made in turbulent forced convection heat transfer, much work, both analytical and experimental, remains to be done. The relation between eddy diffusivities of heat and momentum should be more clearly established. Experimental work on heat transfer for liquids with variable properties is desirable. Knowledge of local heat transfer in noncircular passages is still limited and more definitive research on liquid metals is needed. CHAR iia 2 SOkV ETS OF PROBLEMS IN BOILING HEAT TRANSFER R. H. SABERSKY E,7. Introduction. Boiling heat transfer is defined as the heat transfer from a surface to a liquid under such conditions that the temper- ature at and near the surfaces is sufficient to create the vapor phase. The temperature of the bulk of the fluid is equal to or below the saturation temperature, and the difference between the saturation temperature and the actual bulk temperature is usually called the subcooling. The exist- ence of this type of heat transfer has of course been recognized for a long time. The first technical discussion of the problem is probably that con- tained in a paper by Mosciki and Broder [63] in 1926. The interest in boiling heat transfer has increased, particularly in the last ten years, during which the problem has become of great technical importance in connection with the cooling of rocket engines, the construction of rapid response boilers, and the operation of nuclear power producers. Boiling heat transfer may be subdivided into problems occurring under conditions of free convection or forced convection. A typical experimental apparatus for studying free convection boiling may consist of a vessel filled with the test fluid into which an electrically heated metal strip is Wei) ) E - CONVECTIVE HEAT TRANSFER AND FRICTION submerged [64]. Instead of a strip a wire may also be used [65], although the curvature of the heating surface must then be counted as one of the variables. Boiling heat transfer with forced convection may be studied by pumping the test fluid through an electrically heated section of metal tubing [66]. If visual observations are to be made, the fluid can, for example, be made to flow in an annulus formed by an inner heating tube and an outer Lucite tube [64], or an axial heating strip may be en- closed in a Lucite duct [67]. Instead of electrical heating, other heating methods can be used, and some experiments have been performed using a condensing vapor as a heat source. Most workers in the field have pre- ferred electrical heating because of its ease of control and its flexibility. Many investigators have also found it necessary to study the vapor for- mation near the heating surface in detail, making use of high speed photographs for this purpose. Frame speeds up to 20,000 frames per second were needed in some of these investigations [67]. E,8. General Results. Typical results obtained from boiling heat transfer experiments are shown in a graph of heat transfer rate per unit Heat transfer Con- | Nucleate |Partial film} Complete vection boiling boiling | film boiling Wall temperature, °F Fig. E,8a. Typical curve for boiling heat transfer. area vs. wall temperature (Fig. E,8a). The same type of curve is ob- tained for forced or free convection conditions, with or without subcool- ing. The first part of the curve (A to B) corresponds to the usual con- vection conditions without boiling. When the wall temperature reaches a certain value (somewhat above the boiling point of the liquid) the heat { 314 ) E,8 - GENERAL RESULTS transfer increases sharply with the wall temperature until a maximum is reached at point C. A decrease in heat transfer occurs with further in- crease in wall temperature and only at very high wall temperatures does this trend reverse. If-electrical heating is used, the part from C’ to D is unstable and special care has to be taken to obtain measurements in this region [64]. If the heat transfer rate is increased beyond the value at C, the wall temperature has to jump to a value indicated by point F’. Since, in many cases, this temperature is higher than the melting point of the heating surface, failure of the heater may occur if the heat transfer at C is exceeded. Point C is therefore generally called the “burnout” point. This name is somewhat misleading since, MELEE Nucleate boiling depending on the fluid and the (1) material of the heater, physical destruction does not necessarily Partial film boiling occur. The existence of a maximum in the heat transfer curve, however, a is a significant engineering charac- teristic of boiling heat transfer, since UAE an abrupt temperature increase oc- (2) curs if an attempt is made to trans- fer an amount of heat greater than that indicated by this point. The Fulig Poulis determination of the burnout point is therefore important for the design of heat transfer equipment which is WM VLEET WM to operate in the boiling range. An eu exact knowledge of the shape of the Fig. E,8b._ Schematic representation of curve from B to C is often not re- typical bubble formations. quired because of the relatively small change in wall temperature which corresponds to this range. Visual examination [65,67,64] of the processes at the heating surface shows that, at point B, small bubbles appear at the surface (see Fig. E,8b (1)). These bubbles may grow and collapse without ever leaving the surface or they may leave the surface, depending on the conditions of the bulk fluid. As the temperature of the surface is increased, more and more bubbles appear. When it is increased beyond C, the bubbles become so numerous that several of them will merge into a larger vapor mass which may adhere te the surface for some time. Eventually it will detach and float into the bulk fluid (see Fig. E,8b (2)). Progressively more of the larger vapor masses are formed as the temperature is increased, until at point D a rather stable continuous vapor film is formed, which covers the entire heating surface (see Fig. E,8b (3)). The heat transfer mechanism corre- ( 315 ) E - CONVECTIVE HEAT TRANSFER AND FRICTION sponding to section BC of the curve in Fig. E,8a is called ‘nucleate boiling,” that corresponding to section DF (and beyond) is called ‘film boiling,’ and that corresponding to section C'D is called ‘partial film boiling,” in reference to the observed surface phenomena. In Fig. E,8c, E,8d, and E,8e some actual test results of the type de- scribed in the foregoing are shown. The test fluid in these cases is dis- tilled degassed* water and the pertinent test conditions are noted in the figures. In Fig. E,8c data for two different temperatures of the bulk fluid are given and it is seen that the burnout point has increased with de- creasing fluid temperature. The behavior in the pure convection region follows the usual heat transfer laws. It is interesting to note that nucleate xX 100°F subcooling O-—— 50°F subcooling Velocity = 1.1 ft/sec Pressure = 16 |b/in? abs Heat transfer, BTU/in? sec 0) 5)0) J 11510) 950 Temperature difference between wall and boiling point, °F Fig. E,8c. Effect of subcooling on boiling heat transfer. Distilled degassed water [64]. boiling begins only at a wall temperature of approximately 30°F above the ordinary boiling temperature. Fig. E,8d shows the effect of velocity, which again is to increase the burnout point. As before, the behavior of the curves at the lower temper- atures is explained by the usual theory of forced convection. Fig. E,8e illustrates the effect of fluid pressure. For better comparison, these meas- urements have been made at constant liquid subcooling (normal boiling point — liquid temperature) rather than at constant fluid temperature. The resulting data can be plotted on practically the same curve if the temperature difference between the wall and the liquid is taken as the abscissa. As a consequence it has frequently been assumed that the degree of subcooling rather than the absolute pressure itself was the important 6 The word ‘‘degassed liquid”’ is used to designate liquids in which the gas content has been reduced to a value of less than approximately 15 per cent of saturation. ( 316 ) E,8 : GENERAL RESULTS nat TMS is Qe Eo ed 5 ft/sec o——— 1.1 ft/sec Pressure = 16 |b/in2 abs Subcooling = 50°F Heat transfer, BTU/in? sec 10 100 1000 Temperature difference between wall and liquid, °F Fig. E,8d. Effect of velocity on boiling heat transfer. Distilled degassed water [64]. x 60 |b/in? abs O——— 16 |b/in2 abs Velocity = 1.1 ft/sec Heat transfer, BTU/in? sec Subcooling = 50°F 10 100 1000 Temperature difference between wall and liquid, °F Fig. E,8e. Effect of pressure on boiling heat transfer. Distilled degassed water [64]. characteristic in boiling heat transfer. On the basis of this assumption it has often been attempted to simplify the presentation of experimental data (see, for example, Fig. E,9f). This simplification, however, should be used with some caution, since it applies only when the pressure changes are relatively small. Csi E - CONVECTIVE HEAT TRANSFER AND FRICTION When the pressure reaches the critical value, the entire boiling phe- nomenon disappears, of course, and heat transfer is by pure convection. As the critical pressure is approached, the differences between the liquid and the vapor phase diminish. The vapor then becomes almost as good a heat transfer agent as the liquid and the decrease in the heat transfer rate with the appearance of film boiling should vanish. The curve of ‘“‘heat transfer rate”’ vs. “‘wall temperature” will then show a continuous in- crease, and there will be no more temperature jump when the vapor film forms. The effect of large pressure changes has been illustrated by a set of experiments with Freon (Fig. E,8f). In these experiments the burnout —t — (o) nN oO foe) o ‘ Maximum heat transfer, BTU/in? sec ro) fo) NO Oo 0 OM NOD TN OA) NOG: Oley TKO pe une Reduced pressure p/ Per Fig. E,8f. Maximum heat transfer to Freon as a function of reduced pressure. The fluid is Freon No. 114. Mass velocity 5.2 lb/in.?-sec. Annular cooling passage [68]. point was measured as a function of pressure at constant subcooling and constant fluid velocity, and the pressure was increased above the critical value. The resulting curve shows a continuous decrease until finally the value corresponding to simple forced convection is reached, a result which should be expected from the foregoing discussion. The disappearance of the temperature jump in the graphs of the heat transfer rate vs. temper- ature is well illustrated by a set of curves in Fig. E,8g. The data in Fig. H,,8g were obtained for a ‘‘jet fuel’ consisting of a mixture of hydrocarbons. The critical pressure in this case was approximately 600 lb/in.? abs. An extensive set of experiments showing the effect of pressure on the maxi- mum heat transfer rates of hydrocarbons is reported in [70]. In the following pages the information presently available on nucleate ( 318 ) Heat transfer, BTU/in?2 sec E,9 - NUCLEATE BOILING 8.0 i SPORE en ee ee eS sc i | ee oe Pa 4.0 N o) Heat transfer, BTU/in? sec 5 0.2 30 60 100 200 400 1000 1500 Wall temperature — fluid temperature, °F Len A 400 600 800 1000 1500 Wall temperature — fluid temperature, °F Fig. E,8g. Heat transfer to jet fuel [69]. boiling will be discussed in some detail. Only a short review of the problem of film boiling will be given, because experimental results for this type of heat transfer are still scarce. E,9. Nucleate Boiling. DESCRIPTION OF THE PROBLEM. In the previous section an over-all description of the problem of boiling heat transfer was given, and the effect of some of the variables was indicated. The information which may ( 319 ) E - CONVECTIVE HEAT TRANSFER AND FRICTION eventually be most useful to the engineer is a correlation of the maximum allowable heat transfer (burnout points) as a function of the fluid proper- ties. For some special cases a correlation of the heat transfer rate in the complete region of nucleate boiling may also be required, although this information is believed to be of lesser importance (see Art. 8). To the knowledge of the writer, satisfactory correlations of this kind are not available at present. For future development of such relations it is be- lieved that an understanding of the detailed processes taking place in nucleate boiling will be necessary, and for this reason the present con- «cepts of the mechanism of nucleate boiling will be discussed below. The process of nucleate boiling can, for convenience, be subdivided into three phases: the nucleation proper (i.e. the generation of the bub- bles), the growth cycle of the bubbles, and the effect of the bubble motion on heat transfer. Nuc LEATION Process. As the name “nucleate boiling” indicates, it is believed that the vapor bubbles in question originate from nuclei. These nuclei are imagined as consisting of small gas or vapor pockets stabilized on submicroscopic solid particles of low wettability. Upon heat- ing, part of the vapor (or gas) is forced away from the stabilizing particle. If the resulting gas or vapor mass is sufficiently large, the inside pressure overcomes the surface tension forces as well as the outside pressure, and the nucleus grows into a bubble. If the detached gas mass is too small, it collapses by the surface tension forces and no bubble forms. If the initial cavity is spherical, the relation between the surface tension forces and the pressure becomes simply Dr p= = (9-1) where p; is the pressure inside the initial cavity, p, is the pressure of the surroundings, r is the radius of the cavity, and o is the surface tension. If gas is present in the cavity in addition to the vapor, p, would be the sum of the vapor pressure and the partial pressure of the gas. In order to create a bubble, the temperature surrounding the nucleus has to be suf- ficiently high to create a pressure in the initial cavity larger than that indicated in Eq. 9-1. By measuring the temperature at which a bubble is observed and assuming the pressure p; to be approximately equal to the vapor pressure corresponding to this temperature, an estimate of the size of the initial cavity in terms of ‘‘equivalent spherical size’’ can be made. Nuclei of the kind described are believed to be present throughout the test fluids as well as on the heating surface. The existing nuclei cover a certain range of sizes, and a certain distribution curve of number vs. size can be imagined in each case. Nucleate boiling first becomes noticeable when the temperature near the heating surface becomes high enough to cause the growth of a significant number of the largest nuclei. As the ( 320 ) E,9 > NUCLEATE BOILING temperature is increased, smaller nuclei become capable of forming bub- bles, and the number of bubbles per unit area per unit time increases. The temperature may be raised until the bubble population becomes so high that the burnout point is reached. The number of bubbles per unit area per unit time at the burnout point depends on several conditions, such as the fluid temperature and velocity. As an example, for water at a velocity of 10 ft/sec, at a pressure of 25 lb/in.? abs, and at 155°F sub- cooling, a bubble frequency of 16 X 10° bubbles/in.? sec was measured near the burnout point [67]. According to the concept of nucleation, the wall temperature at the burnout point should depend on these same con- ditions. This dependence has also been observed experimentally [66]. The observed bubbles always occur at or in the immediate vicinity of the heating surface. One should be somewhat hesitant, however, to con- clude from this that the heating surface is solely responsible for the sup- ply of nuclei. Nuclei existing in the fluid would also begin to grow near the surface, because the temperature in this region is the highest. It is of interest here to cite an experiment in which distilled degassed water was heated by radiation, in such a way that the walls of the vessel would be below the bulk fluid temperature [7/]. In this case bubbles were created inside the water at a temperature quite similar to that of the heating surface in nucleate boiling. It is believed, therefore, that the nuclei responsible for boiling may come from either source. Whether the sources are of equal importance, or whether more nuclei come from the fluid itself than from the surface probably depends on the particular fluid and the particular surface. It may also be possible that the relative importance depends on the rate of heat transfer. The surface, for example, may be able to supply the nuclei for the rather low bubble frequencies required at low rates of heat transfer but, for the high nucleation rates occurring at high heat flow rates, the fluid might act as the main nucleation source. The fact that the surface can influence the results has been demonstrated by a set of experiments by Farber and Scorah [65]. In these experiments the heat transfer rates to boiling water were measured for a set of wires of different materials with results shown in Fig. E,9a. As mentioned previously, by measuring the wall temperature and using Eq. 9-1 an estimate of the size of the original nucleus in terms of “equivalent spherical diameter’’ can be made. From experiments with distilled degassed water at 1 atm [64], this size was found to be of the order of 10-* inch. Measurements for distilled degassed carbon tetra- chloride at 1 atm [64] indicate approximately the same size. The heating surface in both cases was a strip of stainless steel, type 347. Estimates of equivalent nucleus size made from cavitation experiments [72] lead again to the same order of magnitude. There is not, however, sufficient infor- mation available to draw any general conclusions. As seen again from Eq. 9-1, for a given nucleus size, lower surface (2m E - CONVECTIVE HEAT TRANSFER AND FRICTION tension should lead to boiling at a smaller temperature excess above the normal saturation temperature. Experiments with distilled degassed water and a degassed water-aerosol solution [64] tended to verify this result. In the first case a wall temperature of approximately 30°F above the normal boiling point was required to initiate nucleate boiling, whereas only about 15°F was required for the aerosol water solution, the surface tension of which was considerably below that of pure water. The presence of gas in the initial nucleus according to Eq. 9-1, should have the same effect as decreasing the surface tension. This fact has also been shown experi- mentally [64]. The concept that nuclei are responsible for the boiling as discussed in this section is the most widely accepted theory at present. The precise Chromel C Chromel A Tungsten Nickel XR Baan Al = cee ai 0 10 ae 1000 Wall temperature — fluid temperature, °F Heat transfer coefficient h, Ba U/nrmtt2 °F Fig. E,9a. Heat transfer coefficient for four different surface materials. Bulk boiling, 1 atm pressure; wire diameter 0.040 inches; wire length 6 inches. Distilled water, free convection, horizontal wire [64]. role of surface and liquid in supplying nuclei, as well as the size dis- tribution of nuclei in each case, is still not known. There is one other mechanism of bubble generation which has been considered. In any liquid, cavities are continuously formed due to the random fluctuations of the molecules. These cavities, if large enough, could grow into bubbles. The probability that cavities of sufficient size would form [73] to cause boiling at the observed temperatures, however, is practically zero. It has further been suggested that these cavities might form at the surface of ‘‘nonwettable”’ solids [74], in which case the re- quired size would be much smaller. The probability that the necessary fluctuations would occur at such specific points is, however, again ex- pected to be very small. Experimental evidence, which would clearly indicate which of the two ( 322 ) E,9 - NUCLEATE BOILING mechanisms is the essential one, is difficult to obtain, but the results of the following set of experiments may be pertinent. In these experiments [74] a vessel containing water and a submerged heating wire was subject to pressures of the order of 15,000 lb/in.? for approximately 10 minutes. The pressure was then lowered to the atmospheric value and the wire was heated electrically. It was found under these circumstances that the first bubbles would form only when the wire temperature was raised consider- ably above the usual nucleation temperature. Very similar results had been obtained previously by several investigators [76] who observed that a body of water, after being subjected to a pressure treatment as described above, could be heated to temperatures much higher than the normal boil- ing point before any bubbles would form. This observation was explained in terms of the theory of pre-existing nuclei. In accordance with this theory, the pressure treatment causes a decrease in the number and size of the initial nuclei and the smaller nuclei require a higher temperature before becoming capable of growing. The observed increase in nucleation temperature on the other hand would be difficult to explain in terms of thermal fluctuations. Since the same type of phenomenon was observed in the experiments with the heated wire, it is reasonable to assume that the presence of the wire did not change the nucleation mechanism and that pre-existing nuclei were again responsible for the bubble formation. In general, therefore, thermal fluctuations are not believed to play a major role in boiling heat transfer. The thermal fluctuations, on the other hand, are probably of essential importance in determining the maximum tensile strength of a perfectly pure liquid [73]. GROWTH AND CoLLAPsE Process. Bubble motion has been discussed in detail in [66,67,64] and the concepts given in these references will be used in this section, since they appear to be the most plausible ones at this time. In Fig. E,9b the typical stages of the growth and collapse of a bubble are shown schematically. In Fig. E,9b (1), a nucleus is shown sur- rounded by superheated liquid. The dotted line indicates the isotherm which is at the temperature of the normal boiling point. The pressure in the nucleus is essentially equal to the vapor pressure of the surrounding liquid plus the pressure exerted by any gas present. If the nucleus is of sufficient size, it begins to grow (Fig. E,9b (2), (3)). As the size increases, the surface tension forces decrease rapidly and further motion depends principally on the pressure inside the bubble. This pressure is a function of the rate at which vapor can be supplied to the growing bubble, assum- ing the fluid to be sufficiently degassed so that gas diffusion can be neg- lected. The rate of this vapor flow is determined by two processes: the heat transfer from the liquid to the surface of the cavity, and the evapo- ration from the surface. The temperatures influencing these processes are the temperature of the superheated liquid and the temperature of the vapor inside the bubble. The temperature of the bulk liquid should have { 323 ) E - CONVECTIVE HEAT TRANSFER AND FRICTION little influence at this stage of the growth. In Fig. E,9b (2) the bub- ble is shown after some growth has taken place. The isotherm of the normal boiling point has approached the bubble surface partly because of the stretching of the hot film of liquid over the bubble and partly because of the heat transfer from the film. The part of the film which is stretched over the top of the bubble is further cooled by heat transfer to the sur- rounding cold liquid. The superheat of the liquid surrounding the upper part of the bubble is finally removed completely (Fig. E,9b (3)) and vapor is condensed over this portion of the bubble. The rate of vapor removal depends on the rate of the condensation mechanism itself, as well as on the rate of heat transfer to the cold bulk liquid. The significant temperatures in this case are the temperature of the bulk liquid and the oe Ave é a RC CM (9) Fig. E,9b. Schematic representation of bubble growth. temperature of the vapor in the bubble. As the bubble grows, the evapo- ration rate over the lower portion of the bubble also changes, because of changes in the surrounding temperature and configuration. Eventually the heat transfer from the upper part of the bubble may become suf- ficiently large so that the condensation overcomes the evaporation. The pressure inside the bubble then decreases rapidly and falls below the pressure of the fluid, the resulting force reducing the momentum of the surrounding fluid. If large enough, it reverses the fluid motion and brings about the collapse of the bubble at the surface (Fig. E,9b (4 through 6)). If the heat transfer from the bubble is low, the resulting pressure de- crease in the bubble may not be sufficient to cause a reversal of the mo- mentum in the fluid. In this case the bubble will be carried with the fluid, away from the surface. It will then collapse in the bulk of the fluid, pro- vided the fluid is subcooled. Fig. E,9b (7 through 9) illustrates this latter ( 324 ) E,9 ; NUCLEATE BOILING process. For a particular combination of properties and temperature, bubbles may also become stationary on the heating surface. The tendency of the bubble to leave the surface, as well as its shape during the growth cycle, probably also depends on the contact angle between the fluid, the vapor, and the surface [77]. In the following, effects of some of the more important variables on bubble motion will be discussed in the light of the foregoing description. Comparisons with experimental data will be made whenever possible. EFFECT OF VARIABLES ON BUBBLE MOTION. Nucleus size and surface tension. As mentioned previously, the nu- cleus size determines the temperature at which boiling begins. Since this temperature also influences the initial growth rate (see Fig. E,9b), nucleus 0.020 ©o Oo on Bubble radius, in. 0 200 400 600 800 1000 1200 Time, microsec Fig. E,9c. Bubble radius vs. time. Distilled degassed water at 1 atm pressure and 78°F. Heat flux 50 per cent of burnout value. Free convection [64]. size and initial growth should be interdependent. Smaller nuclei should correspond to higher initial growth rates. In a very similar way, a reduc- tion in surface tension should lead to a decrease in the initial growth rate. This latter effect is believed to be principally responsible for the differ- ence in initial bubble growth rates in water with and without dissolved aerosol (see Fig. E,9c and E,9d). Shape of vapor pressure curve. The size of the nucleus and the surface tension actually determine the pressure which is required to initiate boil- ing; they only indirectly control the required temperature of the surround- ing liquid. The temperature, however, is an important factor in the growth of the bubble, and the relation between vapor pressure and temperature of the liquid therefore becomes a property affecting this growth. If the amount of superheat required to produce the necessary pressure in the ( 325 ) E - CONVECTIVE HEAT TRANSFER AND FRICTION nucleus is large, the bubble is surrounded by a wide region of greatly superheated fluid. The temperature differential for the initial heat trans- fer is then large and the bubble is expected to grow rapidly. A steep vapor pressure-temperature curve, on the other hand, should lead to slow bub- ble growth. The rate of change of vapor pressure with temperature for each liquid is a function of the absolute pressure. Growth rate measurements over a sufficiently wide pressure range, to indicate the effect of the slope of the vapor pressure-temperature curve directly, have not been made. The fact, however, that the superheat re- quired for boiling decreases with increasing pressure has been checked experimentally [78] and the results are shown in Fig. E,9e. The curve is Bubble radius, in. 0 200 400 600 800 1000 1200 1400 Time, microsec Fig. E,9d. Bubble radius vs. time. Distilled degassed water-aerosol solution at 1 atm pressure and 90°F. Heat flux 80 per cent of burnout value. Free convection [64]. partly influenced, of course, by changes in surface tension and possible changes in nucleus size, but these changes cannot fully explain the large variation in superheat. The change in slope of the vapor pressure curve is believed to be the principal factor [64]. Explosive boiling at very low pressure, a phenomenon well known to the chemical worker, is another instance which may be explained by the slope of the vapor pressure curve. For water at 0.1 atm pressure, e.g., the superheat necessary to produce a 15-lb/in.? over-pressure would be 100°F. A nucleus, growing in such a highly superheated liquid, would of course grow very rapidly, which could explain the observed results. In vacuum work it is often necessary to make a special effort to introduce large nuclei in order to avoid explosive processes. Thermal diffusivity. In the brief description of bubble motion given ( 326 ) E,9 - NUCLEATE BOILING above it was seen that the heat transfer from the liquid to the bubble played a role in determining the bubble motion. The thermal diffusivity of the liquid therefore should be an important property affecting this motion. A low diffusivity should favor slow bubble growth and collapse. It is difficult to isolate this effect because changes in other properties are usually involved when using fluids of different thermal diffusivity, but a comparison of experimental data obtained for carbon tetrachloride with those obtained with a water-aerosol solution is probably pertinent. The surface tensions of the two liquids are quite similar and the thermal diffusivity for carbon tetrachloride is only about > that of the water solution. Measurements under comparable conditions at a pressure of Te 100 3 Critical pressure " Wall temperature — saturation temperature, ° BE Aes 10 100 100 500 1000 Liquid pressure, Ib/in? Fig. E,9e. Excess of wall temperature over saturation temperature as a function of pressure for fully established boiling. Distilled degassed water. Heat transfer surface is SS347 except for the point at the lowest pressure, in which case the surface is SS304 [78]. 1 atm show considerably slower growth and collapse rates for the carbon tetrachloride than for the water solution [64]. Temperature of bulk liquid. The temperature of the bulk liquid influ- ences the heat transfer from the bubble to the bulk fluid, which in turn limits the growth of the bubble and is responsible for its collapse. If the temperature of the bulk fluid is low, the growth of the bubble stops early and the collapse is rapid. For free convection boiling of distilled water at 1 atm, e.g., the maximum bubble radius decreases from 0.022 to 0.014 inches as the fluid temperature is changed from 170 to 60°F [64]. If the fluid temperature is high, the bubble may grow to a rather large size. The under-pressure eventually created in the bubble may not be suf- ficient to reverse the momentum stored in the fluid during the growth and the bubble may detach itself from the surface. If, in the limit, the temperature of the bulk fluid is itself at the boiling point, there is no heat { 327 ) E - CONVECTIVE HEAT TRANSFER AND FRICTION transfer from the bubble to the fluid at all. The bubbles will then always detach themselves from the surface. Velocity. The effect of the velocity of the bulk fluid on bubble mo- tion is similar to that of the temperatures of the fluid. Higher velocities, like lower temperatures, improve the heat transfer from the bubble to the fluid and lead to smaller bubble sizes and faster collapse. It should also be mentioned that a moving bulk fluid exerts a drag on the bubbles, so that they have been observed to slide along the heating surface at velocities approximately equal to 80 per cent of that of the fluid itself [67]. Heat TRANSFER IN NUCLEATE BOILING. Effect of bubble motion on heat transfer. The last phase of nucleate boiling to be discussed is the effect of bubble motion on the heat transfer. In early analyses it was suggested that the increased heat transfer could possibly be explained by the fact that the vapor created near the heating surface absorbed a large amount of heat and that this vapor was then carried away with the rising bubbles. For the case in which the bubbles did not detach from the surface, the increased heat transfer was explained by the vapor flow from the lower to the upper regions of the bubble. Numerical estimates of the amount of heat that could possibly be re- moved in this way [77, Chap. 29; 79; 80], however, showed that this mechanism could not account for the observed heat transfer. There are at present two mechanisms which have been suggested as explanations for the heat transfer improvement. According to the first concept, which is held by many investigators [64,66,67,80], the improved heat transfer is caused by the agitation of the bubbles. The transfer proc- ess is essentially one of forced convection and accordingly it should de- pend on a characteristic Reynolds number and Prandtl number. The typical velocity in this case should be the average fluid velocity induced by the bubbles, and the maximum bubble diameter might be chosen as the typical dimension. Attempts at verifying this concept have been made with some success [64], although the available data do not cover a sufficient range of variables to allow any definite conclusions. The second concept [81] is based on the following idea: After the collapse of each bubble, relatively cold fluid is suddenly brought into direct contact with the hot heating surface at the point of collapse. The resulting large temperature gradients, momentarily and locally, cause ex- treme rates of heat transfer. Initial estimates [8/] indicate that these rates could increase the average heat transfer sufficiently to yield the values observed experimentally. So far it has not been possible to determine which of the two mecha- nisms is predominant. From experiments, the heat transfer—and in par- ticular the burnout point—seems to improve with increased bubble ac- tivity. Both concepts could serve as an explanation for this observation; therefore this fact alone is not sufficient to determine the actual process. ( 328 ) E,9 - NUCLEATE BOILING Because of the observed effect of activity on heat transfer, however, it is possible to make qualitative predictions for the effect of some of the variables on boiling heat transfer. The factors discussed below, which lead to increased bubble activity, should also lead to improved heat trans- fer. This view may be kept in mind when examining the experimental data that follow. EXPERIMENTAL RESULTS. Heat transfer. Having considered a number of the factors which may influence nucleate boiling, some experimental results as well as references for additional data will be given. A considerable amount of information Heat transfer, BTU/in? sec UG Tae “Sao. eo ai a Aon ae Saturation temperature — liquid temperature, oO Fig. E9f. Heat transfer vs. subcooling. Distilled degassed water. Pressure range from 15 lb/in.? abs to 164 lb/in.? abs [67]. on water can be found in [66,67,78]. Some results from [67] on the burnout points for distilled degassed water at various velocities are reproduced in Fig. E,9f. The extremely high heat transfer rates observed in some of these tests are certainly noteworthy. Data on burnout points with free convection for distilled water, distilled carbon tetrachloride, a water-aerosol solution, as well as aerated water, are shown in Fig. E,9g and E,9h [64]. An attempt may be made to give some explanation for the shape and relative position of these curves. The increase in heat transfer with decreasing temperature is probably caused principally by the increase in the temperature differential itself. The reason that the water-aerosol data is below that for distilled water may be traced to the difference in surface tension. Lower surface tension ( 329 ) E - CONVECTIVE HEAT TRANSFER AND FRICTION a) Oo co 0.6 (S ar 0 40 60 80 100 ~~ 120° "140 mies Liquid temperature, °F i) iN Maximum heat transfer fo) ie) (burnout point), BTU/in? sec Fig. E,9g. Maximum heat transfer values for distilled, degassed, and aerated carbon tetrachloride. Free convection, pressure 1 atm [64]. Maximum heat transfer (burnout point), BTU/in? sec OM MnISON 50 Water temperature, °F Fig. E,9h. Maximum heat transfer values for distilled degassed water, aerated water, and a water-aerosol solution. Free convection, pressure 1 atm [64]. ( 330 ) E,9 - NUCLEATE BOILING allows boiling at less superheat, the bubble velocity and agitation is less, and lower heat transfer results. The values for carbon tetrachloride are still lower because of its low thermal diffusivity which influences the heat transfer directly and also leads to lower bubble velocity (see above). As already stated, the effect of increasing gas content is similar to that of decreasing the surface tension. In addition, gas diffusion influences bubble motion to some extent and gas bubbles can rather easily become stable and adhere to the heating surface [64]. For both of these reasons the Burnout point 500 Ib/in? abs a Experimental ft/ n | Burnout point N — — Calculated from Ro) = Sieder-Tate equation oe feyline Glos ~ =) kK jaa) D 50 Ib/in2 abs = 15 ft/sec S ,e) _ ro ~ ©) oO AS E =) & x = 0 0 100 200 400 Weer rs ey le Fig. E,9i. Heat transfer to red fuming nitric acid. Forced convection-nucleate boil- ing. Burnout points are indicated. Liquid temperature 80°F. Heat transfer surface SS347. NO2 content of acid 64 per cent, water content 3 to 2 per cent [82]. burnout points of the two liquids, when containing gas, are below the values which are obtained when they are degassed. Fig. E,9h also con- tains some burnout points for aerated water which were not caused by the adherence of bubbles, a random process which generally cannot be controlled. For design purposes the lower curve would have to be taken. | Further experimental results on boiling heat transfer are given in Fig. E,9i, which contains information on nitric acid [82]. Burnout points for Freon have already been given in Fig. E,8f and for ‘“‘jet fuel” in Fig. E,8g. For information on the heat transfer to a number of hydrocarbons under bulk boiling conditions, the reader is referred to [70]. (e33ilo) E - CONVECTIVE HEAT TRANSFER AND FRICTION Friction. There is one other phenomenon applying to boiling heat transfer with forced convection which should be pointed out. As the heat transfer improves, the frictional pressure drop increases. Typical results are shown in Fig. E,9j. The increase in friction is certainly an important consideration in the design of heat transfer equipment. With the purpose of studying the inter-relation between friction and heat transfer, some experiments have been performed [83] in which water 20 _ On N Ft Heating tube pressure drop, Ib/in? 0 OD OMT O16 “Wiolse a opmenme Heat transfer, (BTU/ft? hr) x 1076 Fig. E,9j. Frictional pressure drop with boiling heat transfer. Distilled degassed water. SS347 tube, 0.226 I.D., 25 inches long. Data taken at constant subcooling of 100°F and constant mass flow of 3.81 X 108 lb/hr-ft? [78]. was forced through an electrically heated tube of about 3-in. diameter. Sufficient measurements were taken so that the local Stanton number SZ, as well as the local friction coefficient c;, could be computed. The two coefficients are defined by the equations soa PVC irs a eli) and dp ite Fe ; 2pV? E,10 - FILM BOILING ee local pressure gradient p = fluid density V = average fluid velocity (adjusted for bubble obstruction) Cy = specific heat of fluid d = effective diameter of unobstructed flow passage T, = wall temperature T, = bulk temperature of fluid. It may be pointed out that the average velocity V required in the above equations is not equal to the ratio of the volume flow rate to the unobstructed cross-sectional area of the pipe. The vapor bubbles on the wall may represent a significant reduction of the available flow area and the velocity V may differ substantially from the above-mentioned ratio, particularly where the diameter of the tube is small. The results of the experiments [83] indicated that the relation between the two coefficients was approximately given by the simple equation Cige os St This, however, is the relationship obtained from Reynolds’ analogy, either for the case where a laminar boundary layer at the heating surface is nonexisting or for the case where the Prandtl number of the fluid is equal to unity. For the temperatures of the water in the above experi- ments, the Prandtl number of the water was relatively close to unity, and any conclusion will have to be limited to that case. For this limited case, however, the experimental results indicate that the postulate which forms the basis of Reynolds’ analogy for forced convection still applies in the nucleate boiling region. This would mean that the mechanisms of heat transfer and momentum transfer are ‘‘similar,”’ within the meaning of Reynolds’ analogy. These results, if verified over a wider range of variables, would incidentally tend to confirm the point of view that the improved heat transfer in nucleate boiling is obtained by increased agi- tation rather than by the creation of periodic steep temperature gradients. E,10. Film Boiling. In addition to the nucleate boiling region, the film boiling region is of engineering importance, although considerably less experimental work has been published concerning results in this region than in the nucleate region. In film boiling, the heating surface is sepa- rated from the fluid by a continuous, stable vapor film. The film is in motion due to free or forced convection, and the flow of the film may be either laminar or turbulent. The heat from the surface is largely trans- mitted through the film to the liquid. Some of the heat serves to evaporate ( 333. ) E - CONVECTIVE HEAT TRANSFER AND FRICTION liquid and to provide vapor for the film, and some may be transmitted to the bulk fluid. No detailed investigations on the distribution of the heat or on the stability of the film have been carried out, but some calcu- lations, based on the assumptions that the film is laminar and all of the heat is used to evaporate fluid, have been performed. For this case the problem becomes very similar to the familiar one of the condensation on a surface [77, Chap. 30]. For the heat transfer coefficient h from a horizontal wire, with some additional simplification, the expression k®p(o. — p)gL |* can be obtained [84]. In Eq. 10-1, p is the density of the vapor, p; the density of the liquid, At the temperature difference between the wall and the liquid, D the diameter of the wire; k, u, and L are the thermal con- ductivity, the viscosity, and the latent heat of vaporization of the liquid respectively, and g the acceleration due to gravity. Heat transfer rates predicted from Eq. 10-1 and adjusted for radiation have been compared with experimental data on film boiling to both nitrogen and water. The agreement between the measured and predicted values was rather satisfactory. E,11. Closing Remarks. In the foregoing an attempt has been made to acquaint the reader with some of the problems concerning boiling heat transfer. The attention of investigators has so far been directed mostly towards nucleate boiling rather than toward film boiling. The principal aim of studies in nucleate boiling is to arrive at a method of predicting heat transfer rates—in particular at the ‘‘burnout point’’—as a function of fluid properties. This aim has not as yet been reached. In order to eventually arrive at a satisfactory method of prediction, it will first be necessary to determine which of the two proposed heat transfer mechanisms is essential. Then, it would seem possible, on the basis of the discussion on bubble motion, to select the significant fluid properties and to form the dimensionless groups on which boiling heat transfer should depend. The success of this approach, however, is some- what in question because the nucleation process plays a key role in boil- ing and for any predictions, therefore, some information on the nuclei distribution will be essential. Information of this kind is not available at present. It is known that the number and size of nuclei in a fluid depend on the previous history of this fluid and to some extent on the type and treatment of the heating surface. If the nuclei distribution should be very sensitive to outside influences, and if in addition various liquids should react in a markedly different manner to a given treatment, it may be practically impossible to predict this distribution. In that case it seems ( 334 ) E,12 - CITED REFERENCES AND BIBLIOGRAPHY doubtful that general rules for the prediction of boiling heat transfer can be formulated. If, on the other hand, the nuclei distribution is found to be predictable, a correlation for boiling heat transfer can probably be developed. Some encouragement for this latter view may be taken from the fact that results obtained with distilled water by different investi- gators and at different times have yielded essentially identical results. Further hope may be derived from the result that the effective nucleus size found in distilled carbon tetrachloride was about the same as that found in distilled water under similar conditions. In any case, further investigations on the problem of nucleation are believed to be necessary for a satisfactory solution of the problem.’ K,12. Cited References and Bibliography. Cited References 1. Howarth, L. Modern Developments in Fluid Dynamics. High Speed Flow, Vol. 1. Oxford Univ. Press, 1953. 2. Boussinesq, J. Essai sur la théorie des eaux courantes. Mémoires présentés par divers savants a l'académie des sciences 23, Paris, 1877. 3. Reichardt, H. The principles of turbulent heat transfer. Archiv. Ges. Wdarmetech. 617, 129-142 (1951). 4. Reynolds, O. On the Extent and Action of the Heating Surface of Steam Boilers. Scientific Papers, Vol. 1. Cambridge Uniy. Press, 1901. 5. Eckert, E.R. G. Introduction to the Transfer of Heat and Mass, 1st ed. McGraw- Hill, 1950. . von Karman, Th. The analogy between fluid friction and heat transfer. Trans. Am. Soc. Mech. Engrs. 61, 705-710 (1939). . Murphree, E. V. Relation between heat transfer and fluid friction. Ind. Eng. Chem. 24, 726-736 (1932). . Boelter, L. M. K., Martinelli, R. C., and Jonassen, F. Trans. Am. Soc. Mech. Engrs. 63, 447-455 (1941). 9. Martinelli, R. C. Heat transfer to molten metals. Trans. Am. Soc. Mech. Engrs. 69, 947-959 (1947). 10. Seban, R. A., and Shemazaki, T. T. Heat transfer to a fluid flowing turbulently in a smooth pipe with walls at constant temperature. Am. Soc. Mech. Engrs. Paper 50-A-128, 1950. 11. Rannie, W. D. Heat Transfer in Turbulent Stream Flow. Ph.D. Thesis, Calif. Inst. Technol., 1951. See also: Summerfield, M. Recent developments in convective heat transfer. Heat Transfer Symposium, Univ. Mich. Eng. Research Inst. 164-169 (1953). 12. Lin, C.S., Moulton, R. W., and Putnam, G.L. Mass transfer between solid walls and fluid streams. Ind. Eng. Chem. 46, 636-640 (1953). 13. Deissler, R. G. Investigation of turbulent flow and heat transfer in smooth tubes, including the effects of variable fluid properties. Trans. Am. Soc. Mech. Engrs. 73, 101-107 (1951). 14. Deissler, R. G. Heat transfer and fluid friction for fully developed turbulent flow of air and supercritical water with variable fluid properties. Trans. Am. Soc. Mech. Engrs. 76, 73-86 (1954). 15. Deissler, R. G. Analysis of turbulent heat transfer, mass transfer and friction ™The preceding chapter was revised in June, 1955. Several important advances have been made since that time. A very few references are mentioned in the bib- liography, which will serve to introduce the reader to more recent work. The list of references is not meant to be complete. For additional publications attention is directed to the bibliographies at the end of the cited references. (2) SG ( 335 ) 16. live 18. tO? 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. ol. 32. 33. 34, 35. 36. 37. 38. 39. E : CONVECTIVE HEAT TRANSFER AND FRICTION in smooth tubes at high Prandtl and Schmidt numbers. NACA Tech. Rept. 1210, 1955. (Supersedes NACA Tech. Note 3145, 1954.) Prandtl, L. Bericht tiber Untersuchungen zur ausgebildeten Turbudenz. Z. angew. Math. u. Mech. 5, 136 (1925). Taylor, G. I. The transport of vorticity and heat through fluids in turbulent motion. Proc. Roy. Soc. London A188, 1932. von Kaérmdn, Th. Turbulence and skin friction. J. Aeronaut. Sci. 1, 1-20 (1934). Goldstein, S. Modern Developments in Fluid Dynamics, Vol. II. Clarendon Press, Oxford, 1938. Laufer, J. The structure of turbulence in fully developed pipe flow. NACA Tech. Note 2954, 1958. Lin, C. C., and Shen, S. F. Studies of von Kaérmdan’s similarity theory and its extension to compressible flows. A critical examination of similarity theory for incompressible flows. NACA Tech. Note 2541, 1951. Deissler, R. G. Analytical and experimental investigation of adiabatic turbulent flow in smooth tubes. NACA Tech. Note 2138, 1950. van Driest, E. R. On turbulent flow near a wall. Preprints of Papers for 1955 Heat Transfer and Fluid Mech. Inst., Stanford Press, 1955. Einstein, H. A., and Li, H. Shear transmission from a turbulent flow to its viscous boundary sub-layer. Reprints of Papers for 1955 Heat Transfer and Fluid Mech. Inst., Stanford Press, 1955. Cavers, S. D., Hsu, N. Y., Schlinger, W. G., and Sage, B. H. Temperature gradients in turbulent gas streams. Behavior near boundary in two-dimensional flow. Ind. Eng. Chem. 465, 2139-2145 (1953). Seban, R. A., and Shimazki, T. T. Temperature distributions for air flowing turbulently in a smooth heated pipe. Proc. General Discussion on Heat Transfer, Inst. Mech. Engrs., London, Sept. 1951. Isakoff, S. E., and Drew, T. B. Heat and momentum transfer in turbulent flow of mercury. Proc. General Discussion on Heat Transfer, Inst. Mech. Engrs., London, 1951. Deissler, R. G. Analysis of fully developed turbulent heat transfer at low Peclet numbers in smooth tubes with application to liquid metals. NACA Research Mem. E52F06, 1952. Bernardo, E., and Eian, C. S. Heat transfer tests of aqueous ethylene glycol solutions in an electrically heated tube. NACA Wartime Rept. E136, 1945. Kaufman, S. J., and Isely, F. D. Preliminary investigation of heat transfer to water flowing in an electrically heated inconel tube. NACA Research Mem. E50G381, 1950. Eagle, A. E., and Ferguson, R. M. On the coefficient of heat transfer from the internal surface of tube walls. Proc. Roy. Soc. London A127, 540-566 (1930). Kreith, F., and Summerfield, M. Pressure drop and convective heat transfer with surface boiling at high heat flux; Data for aniline and n-butyl alcohol. Trans. Am. Soc. Mech. Engrs. 72, 869-879 (1950). Grele, M. D., and Gedeon, L. Forced convection heat transfer characteristics of molten sodium hydroxide. NACA Research Mem. E52L09, 1953. Hoffman, H. W. Turbulent forced convection heat transfer in circular tubes containing molten sodium hydroxide. Oak Ridge Natl. Lab. Rept. 1870, 1952. Barnet, W. I., and Kobe, K. A. Heat and vapor transfer in a wetted-wall tower. Ind. Eng. Chem. 38, 436-442 (1941). Chilton, T. H., and Colburn, A. P. Mass transfer (absorption) coefficients. Ind. Eng. Chem. 26, 1183-1187 (1934). Jackson, M. L., and Ceaglske, N. H. Distillation, vaporization, and gas absorp- tion in a wetted-wall column. Ind. Eng. Chem. 42, 1188-1198 (1950). Bonilla, C. F. Mass transfer in liquid metal and fused salt systems. U.S. Atomic Energy Comm. Tech. Information Service, First Quarterly Progress Rept. N YO-3086, Oak Ridge, Sept. 1951. Linton, W. H., Jr., and Sherwood, T. K. Mass transfer from solid shapes to water in streamline and turbulent flow. Chem. Eng. Progr. 46, 258-264 (1950). ( 336 ) 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. E,12 - CITED REFERENCES AND BIBLIOGRAPHY Lin, C. 8., Denton, E. B., Gaskill, H. S., and Putnam, G. L. Diffusion controlled electrode reactions. Ind. Eng. Chem. 43, 2136-2143 (1951). Hama, F.R. On the velocity distribution in the laminar sub-layer and transition region in turbulent shear flows. J. Aeronaut. Sci. 20, 648 (1953). Elrod, H. G., Jr. Note on the turbulent shear stress near a wall. J. Aeronaut. Sct. 24, 468 (1957). Latzko, H. Heat transfer in a turbulent liquid or gas stream. NACA Tech. Mem. 1068, 1944. Elser, K. Der Warmeubergang in Rohreinlauf. Allgemeine Wdrmetechnik 3, 1952. Poppendick, H. F., and Palmer, L. D. Forced convection heat transfer in thermal entrance regions, Part II. Oak Ridge Natl. Lab. Rept. 914, May 1952. Deissler, R. G. Analysis of turbulent heat transfer and flow in the entrance regions of smooth passages. NACA Tech. Note 3016, 1953. Deissler, R. G. Turbulent heat transfer and friction in the entrance regions of smooth passages. Trans. Am. Soc. Mech. Engrs. 77, 1221-1233 (1955). Langhaar, H.L. Steady flow in the transition length of a straight tube. J. Appl. Mech. 9, A55-A58 (1942). Sparrow, E. M. Analysis of laminar forced-convection heat transfer in entrance region of flat rectangular ducts. NACA Tech. Note 3331, 1955. Hartnett, J. P. Experimental determination of the thermal entrance length for the flow of water and of oil in circular pipes. Trans. Am. Soc. Mech. Engrs. 77, 1211-1220 (1955). Lyon, R. N. Forced convection heat transfer theory and experiment with liquid metals. Oak Ridge Natl. Lab. Rept. 361, 1949. Jenkins, R. Variation of the eddy conductivity with Prandtl modulus and its use in prediction of turbulent heat transfer coefficients. Preprints of Papers for 1951 Heat Transfer and Fluid Mech. Inst., Stanford, 147-158 (1951). Lubarsky, B., and Kaufman, S. J. Review of experimental investigations of liquid-metal heat transfer. NACA Tech. Note 3336, 1955. Deissler, R. G., and Taylor, M. F. Analysis of fully developed turbulent heat transfer and flow in an annulus with various eccentricities. NACA Tech. Note 8461, 1955. Eckert, E. R. G., and Low, G. M. Temperature distribution in walls of heat exchangers composed of noncircular flow passages. NACA Rept. 1022, 1951. Elrod, H. G., Jr. Turbulent heat transfer in polygonal flow sections. Nuclear Develop. Associates NDA-10-7, New York, 1952. Lowdermilk, W. H., Weiland, W. F., Jr., and Livingood, J. N. B. Measurement of heat-transfer and friction coefficients for flow of air in noncircular ducts at high surface temperatures. NACA Research Mem. L63J07, 1954. Knudson, J. G., and Katz, D. L. Fluid Dynamics and Heat Transfer, 1st ed. Univ. Mich. Eng. Research Inst., 1953. McAdams, W. H. Heat Transmission, 2nd ed. McGraw-Hill, 1942. Humble, L. V., Lowdermilk, W. H., and Desmon, L.G. Measurements of average heat transfer and friction coefficient for subsonic flow of air in smooth tubes at high surface and fluid temperatures. NACA Rept. 1020, 1951. Goldman, K. Heat transfer to supercritical water and other fluids with tem- perature-dependent properties. Nuclear engineering, Part I. Amer. Inst. Chem. Engrs., Chem. Eng. Progr. Symposium Series 50, 1954. Bringer, R. P., and Smith, J. M. Heat transfer in the critical region. Am. Inst. Chem. Engrs. J. 3, 49-55 (1957). Mosciki, I., and Broder, J. Heat transfer from a platinum wire. Roczniki Chem. 6, 319-354 (1926). Complete English translation on file at Eng. Research Lab. Expil. Sta., E. I. duPont de Nemours and Co., Wilmington, Delaware. Ellion, M. E. A study of the mechanism of boiling heat transfer. Calif. Inst. Technol. Jet Propul. Lab. Mem. 20-88, Mar. 1954. Farber, E. A., and Scorah, R. L. Heat transfer to water boiling under pressure. Trans. Am. Soc. Mech. Engrs. 70, 369-384 (1948). McAdams, W. H., Addoms, J. N., and Kennel, W. E. Heat transfer at high 33m) 67. 68. 69. 70. (AN 72. 73. 74, 75. 76. ile 78. ao: 80. 81. 82. 83. 84. 85. 86. 87. E : CONVECTIVE HEAT TRANSFER AND FRICTION rates to water with surface boiling. Mass. Inst. Technol. Rept. ANL-4268, Dec. 1948. Gunther, F. C. Photographic study of surface-boiling heat transfer to water with forced convection. Trans. Am. Soc. Mech. Engrs. 73, 115-123 (195i). Gunther, F. C. Private communication. Hatcher, J. B. Calif. Inst. Technol. Jet Propul. Lab. Progress Rept. 20-157, 1952. Cichelli, M. T., and Bonilla, C. F. Heat transfer to liquids boiling under pressure. Trans. Am. Inst. Chem. Engrs. 42, 411 (1946). Dergarabedian, P. The rate of growth of vapor bubbles in superheated water. J. Appl. Mech. 75, 537-545 (1953). Parkin, B. R. Scale effects in cavitating flow. Calif. Inst. Technol. Hydrodynam. Lab. Rept. 21-8, July 1952. Volmer, M. Kinetik der Phasen Bildung. Steinkopff, Dresden, 1939. Larson, R. F. Factors that influence heat transfer in boiling. Fluid Mech. Inst., Los Angeles, 1953. Sabersky, R. H., and Gates, C. W. On the start of nucleation in boiling heat transfer. Jet Propulsion 2, 67-70 (1955). Harvey, E.N. On the cavity formation in water. J. Appl. Phys. 18, 162 (1947). Jakob, M. Heat Transfer. Wiley, 1949. Buchberg, H., Romie, F., Lipkis, R., and Greenfield, M. Heat transfer, pressure drop, and burnout studies with and without surface boiling for de-aerated and gassed water at elevated pressures in a forced flow system. Heat Transfer and Fluid Mech. Inst., Stanford, 1951. Plesset, M. S. Note on the flow of vapors between liquid surfaces. Calif. Inst. Technol. Hydrodynam. Lab. Rept. 26-5 to Office of Nav. Research, 1951. Rohsenow, W. M., and Clark, J. A. A study of the mechanism of boiling heat transfer. Trans. Am. Soc. Mech. Engrs. 73, 609-620 (1951). Rannie, W. D. Private communication. Hatcher, J. B., and Bartz, D. R. High flux heat transfer to JP-3 and RFNA. Coke deposition of JP-3. Calif. Inst. Technol. Jet Propul. Lab. Publ. EP119, 1951. Sabersky, R. H., and Mulligan, H. E. On the relationship between fluid friction and heat transfer in nucleate boiling. Jet Propulsion 1, 9-12 (1955). Bromley, L. R. Heat transfer in stable film boiling. Chem. Eng. Progr. 46, 221- 227 (1950). Sparrow, E. M., Hallman, T. M., and Siegel, R. Turbulent heat transfer in the thermal entrance region of a pipe with uniform heat flux. Appl. Sct. Research 7, Sec. A, 37-52 (1957). Deissler, R. G., and Taylor, M. F. Analysis of axial turbulent flow and heat transfer through banks of rods or tubes. TI D-7529, Reactor Heat Transfer Conf. of 1956, Pt. 1, Book 1, 416-461 (1957). Deissler, R. G., and Taylor, M. F. Analysis of turbulent heat transfer in non- circular passages. NACA Tech. Note 4384, 1958. Bibliography Camack, W. G., and Forster, H. K. Test of heat transfer correlation for boiling metals. Jet Propul. 10, 1104-1106 (1957). Forster, H. K., and Greif, R. Heat transfer to a boiling liquid; Mechanism and correlations. Dept. of Eng., Univ. Calif., Papers Rept. 7, 1958. Griffith, P. The correlation of nucleate boiling burnout data. Am. Soc. Mech. Eng. Paper 67 HT-21, 1957. Zuber, N., and Tribus, M. Further remarks on the stability of boiling heat transfer. Dept. of Eng., Univ. Calif., Rept. 58-5, 1958. (£358!) SECTION F CONVECTIVE HEAT TRANSFER IN GASES KE. R. VAN DRIEST F,1. Introduction. Any discussion of convective heat transfer in gases is essentially a discussion of the characteristics of the boundary layer in a compressible real fluid subjected to arbitrary wall temperature. Since the density, viscosity, thermal capacity, and thermal conductivity of gases vary considerably with the temperature in the boundary layer, they thereby affect the rate at which heat may be transferred to or from the adjacent surface. The state of the boundary layer is important; it may be laminar, turbulent, or mixed, depending upon the Reynolds num- ber, the Mach number, and the wall-to-free stream temperature ratio. At high temperatures the gas may dissociate, or even ionize, the result of which would be a change in the physical properties of the gas. Owing to the variety of problems brought about by high temperature, a careful analysis of heat transfer is paramount in the successful design of high speed aircraft. Performance-wise, a knowledge of the temperature of the skin of a high speed vehicle is necessary for accurate calculation of skin friction. F,2. The Mechanism of Convective Heat Transfer. Convective heat transfer is heat transfer to or from a flowing fluid. The region of the flowing fluid which absorbs or gives up the heat is the boundary layer. Strictly speaking, the heat transfer to or from the boundary layer takes place by molecular conduction at the wall, whether the flow is laminar or turbulent; thus the transfer of heat from a wall to a flowing fluid, or vice versa, is the product of the thermal conductivity and temperature gradient in the fluid at the surface of contact of fluid and wall. Since, with fixed wall temperature, the heat transfer at the wall is proportional to the temperature gradient in the fluid at the wall, any means of increasing that gradient will increase the rate of heat transfer; and, of course, de- creasing the temperature gradient will decrease the heat transfer. { 339 ) F - CONVECTIVE HEAT TRANSFER IN GASES Now the temperature gradient at the wall may be steepened by either increasing the mass flow external to a given boundary layer or inducing transition from laminar to turbulent flow. At low speeds, the effect of increasing the external flow is to thin the boundary layer by inertial force, thereby steepening the temperature gradient. However, as the external speed continues to increase, direct compression as at a stagnation point, or dissipation of energy by internal friction, rapidly increases the temper- ature of the fluid within the boundary layer, thereby also steepening the wall temperature gradient; thus the effect of high speed is literally to cover the wall with a layer of hot fluid, between which and the wall the heat transfer then takes place. As a practical consequence, for example, Free stream — T g \Z y a irre Wall Fig. F,2. Schematic of boundary layer in a compressible viscous fluid. increasing the external flow speed will first cool, and later heat, a surface whose temperature is initially higher than the free stream temperature. Transition from laminar to turbulent flow effectively thins out the inner regions of the boundary layer by scouring action, thus greatly increasing the temperature gradient and consequent heat transfer for low as well as high speeds. Since, in high speed flow, the temperature of the boundary layer rises because of compression or energy dissipation, it follows that for a given speed there will be a certain wall temperature, above the free stream temperature, at which no heat transfer will take place. It is proper to consider the zero heat transfer temperature as the reference temperature. Thus, at low speeds, the free stream temperature becomes the reference temperature. When the wall is hotter than the reference temperature, heat flows from the wall into the boundary layer, whereas, when the wall is cooler than the reference temperature, the reverse is true (Fig. F,2). ( 340 ) F,3 - FLAT PLATE SOLUTION CHAPTER 1. SURVEY OF THEORETICAL RESULTS APPLICABLE TO AERODYNAMIC HEAT TRANSFER. STATUS OF EXPERIMENTAL KNOWLEDGE LAMINAR FLOW F,3. Flat Plate Solution. The investigation of the thin laminar boundary layer (cf. IV,B) in steady state on a smooth flat plate is of fundamental importance in aerodynamic-heating problems, because of its practical possibilities and relative simplicity of solution. Typical velocity and temperature curves across a thin compressible laminar boundary layer with heat transfer are shown in Fig. F,2. The heat transfer to or from such a layer in the steady state is obtained upon solution of the continuity, momentum, and energy equations, viz. Fe (ou) + 5 (wv) = 0 (3-1) ut + yp ta 2 (pM) _ @ (3-2a) Hs =0 (3-2b) pu dh + gy Mh my (A) 4 2 (n97) 4 (3-3) respectively. In these equations, u and v are the x and y components of the velocity at any point, the x axis being taken along the plate in the direction of the free stream and the y axis perpendicular to the plate. The symbols p, w, k, Cp, T, h, and p represent the density, absolute viscosity, thermal conductivity, specific heat at constant pressure, absolute temperature, enthalpy per unit mass, and pressure, respectively. Since c, = oh/dT, Eq. 3-3 can be written in terms of the Prandtl number Pr = c,u/k as follows: ah Ob (OON @. |) te Oe dp put + po Sh = (22) oe (54) + ue oe) in which the Prandtl] number is variable and a function of temperature. The equations include the variation of free stream velocity and surface temperature in the direction of flow. While the above equations have been studied by many investigators after Prandtl first announced the concept of the boundary layer in 1904, a complete solution of the equations is not readily available. However, under certain restricted conditions, such as constant free stream velocity C341) F - CONVECTIVE HEAT TRANSFER IN GASES and constant wall temperature, exact solutions can be obtained, and these have practical application for supersonic flow over cones and slender ogives, and over wedges and thin airfoils, especially when covered with thin skins. The analytical and numerical results of Crocco [1] for the case of con- stant free stream velocity, wall temperature, and Prandtl number are singular because of their extensiveness and applicability. Crocco not only developed an accurate method of numerical solution of the momentum equation (Eq. 3-2a), but also gave a practical solution of the energy equation (Eq. 3-4) for Prandtl number near unity. (For a review of the Crocco analysis and detailed calculations, the reader is referred to [2].) van Driest [3] in turn has extended the Crocco analysis to include variable Prandtl number in the solution of the energy equation. Owing to the importance of the numerical results derivable therefrom, the extension of the Crocco analysis to include variable Prandtl number will be outlined here with pertinent formulas. Following the procedure of Crocco, the independent variables z and y are first transformed to z and u by u = u(a, y) and « = x. Eq. 3-1, 3-2a, and 3-4 then become, upon elimination of v, 1 (CHE) ae ae NY f ee dx du \r a Ce 0/1 oh 1 oh\ dr Dilek ek ae Bi salt Pel jl B (3 a ny | ae ait @ =) Ou oh oh dp _ = py SE + (8 4 u) 2 =0 (3-6) where shear stress7 = p(du/dy). These equations are still in general form. However, when 0h/dx = 0 and dp/dx = 0, they simplify considerably. The enthalpy h is accordingly a function of u only, and for a perfect gas the density varies inversely with the temperature. Since p = wi(T) = u2(u), Crocco next showed, upon satisfaction of the boundary condition T— © as x— 0, that Eq. 3-5 becomes d? g om + puu = 0 (3-7) where g(u) = 7 (a, y) ~/2a. In dimensionless form, Eq. 3-5 and 3-6 are then, for dh/dx = 0 and dp/dz = 0, Ix Gx aie QU Px Mx = 0 (3-8) ee emery LONI ()sa-me(M)--2 ee Fj3°- FLAT PEATE SOLUTION with boundary conditions OG. =O ea (Oe vat wh, Gg, = 9, h,=1 ah Oh, = and in which u, = u/Ue, py = p/Pe, hy = h/he, and g, = 2 VW x/peet3 - 7. Subscript . indicates conditions at the outer edge of the boundary layer, and the primes denote differentiation with respect to u,. Inspection of Eq. 3-8 and 3-9 shows that the momentum equation (Eq. 3-8) is nonlinear, and besides, according to Eq. 3-10a and 3-10b, the boundary conditions are on opposite sides of the boundary layer. Hence the solution of Eq. 3-8 is expected to be somewhat troublesome. On the other hand, the energy equation (Eq. 3-9) is linear and first order in hi/Pr as a function of u,, so that the solution of Eq. 3-9 is readily Pe _ Ce aso ale p= exp| do v See Pr (0) 2 Ux 9x Enis i exp || (1 — Pr) os an,| (3-11) Re wo gx(0) Ix assuming that the shear distribution is known from the momentum equation. Further integration gives I co) (3-10a) (3-10b) | — hi, 0 Cas 9x il hy (Ux) = hy (0) + Pao) ‘ Pr: exp | - / A (1 — Pr) | dix D) Ux 9x =| Pr-exp| — [ a = pp ate he Jo gx (0) Ix Ux 9x dg exp (1 — Pr) —* | du,,{ du, (8-12) 0 gx (0) Ix or 1.0 2 he (te) = Ihg 0) + Fy Site) = FE RC) (3-13) where nye il "Pr exp | — i i (ies pr) {8 | dug (3-14) and R re Ux 9x dg (Ca) Pr:-exp| — (1 — Pr) = 0 gx (0) Ix Ux 9x dg. exp (1 — Pr) —* | du,,{ du, (8-15) 0 gx (0) 9 x Now, h,(1) = 1. Hence from Eq. 3-13, Pr(0) h;(0) = Sd) F =O) = i ro | (3-16) ( 343 ) F - CONVECTIVE HEAT TRANSFER IN GASES which, when substituted back into Kq. 3-13, gives hang) = he(0) — (hy 0) — 1) SO) 4 8 | $04) ey — Rea) | (3-17) S and R will be found to depend upon the free stream Mach number, the free stream temperature, and the plate temperature (i.e. heat transfer). When the Prandtl number is constant, S = PrI and R = PrJ, where ee sin) = fey) pte) ace at Hence there results Crocco’s original formula: and ate) = (0) — Uhe(0) — 1) MO) 4 pr ME | He) gay — yeu) | (3-20) Crocco tabulated J and J for various fixed Pr and the Blasius shear dis- tribution. Extensive calculations by Crocco had shown that J and J were approximately independent of Mach number and heat transfer for moder- ate supersonic speeds, regardless of viscosity-temperature law (i.e. py ws. variation) when the Prandtl number was not too far from unity. (Indeed, this is exactly true for Pr = 1.) Hence it was concluded that the Blasius (incompressible flow) shear profile, which resulted from the assumption that! p,u, equaled unity, was appropriate for the calculation of J and J and consequently the approximate enthalpy distribution from Eq. 3-19, given an average constant Prandtl number. The Blasius shear distribu- tion is tabulated in Table F,3a. The J’s and J’s, calculated by Crocco, are tabulated in Table F,3b. For moderate Mach numbers, the specific Table F,3a. Shear function gx, when pxux = 1. 0.27994 . 23881 . 19293 . 14097 . 11797 .09334 .06659 .03681 0 0 0. 0 0. 0 0. 0 0. 0 0. 0 0. 0 0. 0 0. 0 0. 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It is seen above that the energy equation can be written as an inte- gral, dependent, however, upon the momentum equation. On the other hand, the momentum equation can be written as an integral equation, dependent upon the energy equation. Thus if U: 5D) Qu (Ux) = fl digs i EE tty (3-21) Ux which can be integrated by the method of successive approximations, starting with the Blasius solution (Table F,3a), if none better can be assumed as a first approximation. However, as Crocco pointed out, the iterative process does not converge upon a single solution in general but rather yields values of g,, which oscillate about the exact value. For, if an initial value of the shear function g,.1, equal to Ag,.. in which A is a constant and g,.. 1s the exact value, is substituted into the right-hand side of Eq. 3-21, the new g,.2, obtained upon integration, will be equal to Juex/A. Resubstitution of g,.2 yields g,3 = Agyex = Jx1. Therefore it is seen that the next substitution should be Wgx19%2 = Jyex- In the iter- ative process of solving Eq. 3-8 (Eq. 3-21), it follows that successive values of g,, for resubstitution into Eq. 3-21 should be the geometric mean of the two previous substitutions. It is next observed that a singularity exists at wu, = 1, owing to the boundary condition g, = 0 at the outer edge of the boundary layer. Therefore, in the case of Eq. 3-21, numerical integration cannot be carried all the way across the layer but must stop at some point u,, = 1 — 7 just short of u,, = 1, where 7 is arbitrarily small. Hence Eq. 3-21 becomes 1=2% Ux2 Qu (Uy) = gx(1 — i) + / dlr if Sein Pile yg cstes | (BEB) ux ny) 9x (Ux) or, in order to avoid double integration, es 1-i pain) ah OEE a) i Lied dug = 4 i) Kite) au - Ee (1 — rug) LM) dy (3-23) ux G5 (ne where f(u,y) = 2Uy Py My. Since the first term, g,(1 —-2), on the right-hand side of Eq. 3-23 is unchanged by successive iteration, a method to adjust g,(1 — 2) in each iteration is necessary so that the boundary condition g,.(1) = 0 is more nearly approached. Following Crocco, one notes that for u, — 1, Eq. 3-8 { 346 ) F,3 : FLAT PLATE SOLUTION is approximated by Fig. F,3. 9, as a function of g,. 9x9 +2=0 : which can be integrated in closed form. The double bar is used to desig- nate the solution of this approximate equation. In terms of g4, the solu- tion of Eq. 3-24 which satisfies the condition g,.(1) = 0 is (3-24) where ¢ is the error function, viz. es AV He E 2 =e) o() Lae (3-25) “e-" df J 0 Now g% can be obtained from the requirement that the slopes of g, and @,, Kq. 3-8 that OU 0) = eG! 10) -{ fe flux) 0 Gx (Ux) we must match at u, = 1 — 7. Hence it follows from the first integral of ( 347 ) (3-26) F - CONVECTIVE HEAT TRANSFER IN GASES The ratio 2/g, is plotted versus g, in Fig. F,3. For algebraic calcu- lations, Crocco found that the combination (1/|9%|)[(G/z) — 1] can be represented by the linear expression 0.7828 + 0.0178 |g| over the practi- cal range of g} between —2.2 and —4 (see Fig. F,3). Thus, to within a slight error, viz. i = OC = 2) SL 9) (3-27) the laminar boundary layer on a flat plate can be solved by successive approximation. Crocco gives a method of computing the error 6g, in Eq. 3-27 (see [1]); however, according to the experience of the writer [2], 6g, was found to be negligible for all practical purposes, being of the order of 10~* for 7 = 0.02. Now that the momentum and energy equations can be integrated separately, it is necessary to iterate between the two integrals in order to obtain accurate enthalpy and shear distributions. Crocco found it suf- ficiently accurate for his purposes to calculate the enthalpy distribution only once using the Blasius shear distribution, whereupon that enthalpy distribution was introduced in the momentum equation to compute a new and final distribution. F,4. Heat Transfer. The rate of heat transfer to the boundary layer per unit area is oT qv = ti (7) (4-1) hy en kw dh ou Wi itew aaa Non) lig he in — Cpaitw te hi, (0)t~ Nica (se => —_— pe A. hy (0)T. (4-2) where subscript ~ refers to the wall. Substitution of Eq. 3-16 into Eq. 4-2 gives ee = ty er |1 + BR) — 40) | (4-3) are uw = — S(1) us B + 2R(1) = es he | SEO) Us 2 wae ee E + 2R(1) © he | (4-4) since = x Pelle © g,(0) = 2 —_ T and Re o nie? ( 348 ) F,4 - HEAT TRANSFER Defining 1 gx (0) t= —— 4-5 S(1) 2 ~~ Re (5) and U2 h, = h. + 2R(1) 3 (4-6) Eq. 4-4 becomes dw = —Stp.ue(h, — he) (4-7) The symbol Re is the Reynolds number. The dimensionless heat transfer coefficient Sé is called the Stanton number. Since heat transfer is proportional to skin friction by Eq. 4-2, it is sometimes desirable to write the heat transfer coefficient St in terms of the local skin friction coefficient defined by cy; = 27./p.u?, thus eee my Se The factor S(1) is called the Reynolds analogy factor and is denoted by the symbol s. Hence [3] 1 9x Q = Sh) = if Pr: exp E i (1 — Pr) a dus, (4-9) 0 gx (0) Ix Now the quantity [h. + 2R(1)u?/2] in Eq. 4-4 is equal to the total enthalpy of the free stream, except for the factor 2R(1). Furthermore, when the plate is insulated, i.e. when q, = 0, it follows from Kq. 4-7 that hy = h, + 2R(1)u2/2. For these reasons, the quantity [h. + 2R(1)u2/2] will be called the boundary layer enthalpy (or simply) recovery enthalpy, h,, and the factor 2R(1) the enthalpy recovery factor r. Therefore [3] 1 9x dg r= oR) =2 | Pr-exp| - | a — Pr) | 0 gx (0) g Ux 9% dg. exp (1 — Pr) —* | du, ; du, (4-10) 0 gx (0) 9x Eq. 4-9 and 4-10 reduce to Crocco’s results, viz. 1 Pr-1 ot UE ss j= Pr f || dus, (4-11) 1 Pr-1 Ux va Ix UE = Bis 2b rf Fro I Fro diy diy or when Pr is constant. It will be found that in general both s and r are functions of speed and heat transfer. However, for moderate speeds and heat transfer rates, St (4-8) and { 349 ) F - CONVECTIVE HEAT TRANSFER IN GASES Crocco has shown that s = Pr?, approximately, and r = Pr’, closely, where Pr is constant. Since the variation of the Prandtl number with temperature is ordinarily not great for common gases such as air [4], the approximate formulas s = Pr? and r = Pr’ can be used for practical pur- poses for moderate flight conditions when an average Pr is assumed. At very high speeds where skin temperatures become great, and for accurate experimentation, the more exact solutions are necessary. F,5. Numerical Results for Zero Pressure and Temperature Gradients along the Flow. Results of calculations of friction and heat transfer coefficients, as well as of recovery and Reynolds analogy factors 0.76 Ne 2 3 ——> Free flight ae 0.70 = a er 0.68 —=: NUMERICAL RESULTS assuming the Blasius shear distributions and constant Prandtl number are compared with results of Crocco, who used the Blasius shear distribu- tion with constant Prandtl number in his calculations. The numerical method of solution utilized in [3] was as follows: a mean constant Prandtl number was first estimated, whereupon an enthalpy dis- tribution was computed using Eq. 3-19 with the Blasius shear distribu- tion. A new Prandtl number distribution was then obtained from Fig. F,5a ' i. Crocco O Present report Fig. F,5e. Reynolds-analogy factor for constant Prandtl number and Blasius shear distribution. after converting the enthalpy to temperature by means of NBS-NACA Table 2.10. A new shear distribution was also computed with enthalpy distribution using the Crocco method for solving numerically the momen- tum equation. The new Prandtl number and shear distributions were then substituted in Eq. 3-14 and 3-15 to obtain the final recovery and Reynolds analogy factors. It was not necessary to iterate any further for both shear or enthalpy distribution, because of the rapid convergence of the iteration process. It is seen, however, that it was necessary to make one more ( 353 ) F - CONVECTIVE HEAT TRANSFER IN GASES iteration beyond Crocco for enthalpy, even for constant Prandtl number, in order to attain the accuracy desired; this is illustrated in Fig. F,5f and F,5g where the exact recovery and Reynolds analogy factor are plotted as functions of Mach number for both a true shear distribution, corre- 0 D 4 6 8 10 12 14 16 Mach number M, Fig. F,5f. Effect of shear distribution on recovery factor for Pr = 0.715. 0) 2 4 6 8 10 12 14 16 Mach number M, Fig. F,5g. Effect of shear distribution on Reynolds analogy factor for Pr = 0.715. sponding to a constant Prandtl number of 0.715 and T, = 400°R, and the Blasius shear distribution. Fig. F,5h shows typical shear and Prandtl number distributions for a complete calculation. Fig. F,5i shows the local heat transfer coefficient (multiplied by V/ Re) for a laminar boundary layer on a flat plate at zero angle of attack in free ( 354 ) F,5 > NUMERICAL RESULTS flight with a free stream temperature of 400°R and various wall-to-free stream enthalpy ratios. The recovery factor when the plate is insulated is plotted in Fig. F,5j. It is seen that the recovery factor follows Pr, up to about Mach number 3 after which r overshoots and remains con- sistently below Pr’. Fig. F,5k gives the recovery factor for different wall-to-free stream enthalpy ratios. Fig. F,51 is a cross plot of Fig. F,5k. 1.0 Shear distribution Prandtl number distribution Me —— 4 Te = 400°R 0 0.5 1.0 U, Fig. F,5h. Typical shear and Prandtl number distributions across a laminar boundary layer. Fig. F,5m gives the Reynolds analogy factor for the limiting case of the insulated plate. s has the same characteristics as r, overshooting Prt in the neighborhood of M. = 4. Fig. F,5n and F,5o0 gives the Reynolds analogy factor for various wall-to-free stream enthalpy ratios. Local heat transfer coefficients, recovery factors, and Reynolds anal- ogy factors for insulated flat plates in heated wind tunnels are presented in Fig. F,5p, F,5q, F,5r, F,5s, F,5t, F,5u, and F,5v. In particular, Fig. F,5q and F,5r show the relation of r,,, to Pr? and Pr’, and of s,,, to Pr? and ( 355 ) F - CONVECTIVE HEAT TRANSFER IN GASES = i sd. aie | Msuloked prs te 10 eck nee Me. Fig. F,5i. Local heat transfer coefficient for a laminar boundary layer on a flat plate in free flight for various wall-to-free stream enthalpy ratios. T, = 400°R. 0.81 0 z 4 6 8 10 12 14 16 Mach number M,. Fig. F, 5j. Recovery factor for a laminar boundary layer on an insulated flat plate in free flight. T, = 400°R. ( 356 ) F,5 - NUMERICAL RESULTS 0 2 4 6 8 10 12 14 16 Mach number M, Fig. F,5k. Recovery factor for a laminar boundary layer on a flat plate in free flight for various wall-to-free stream enthalpy ratios. T. = 400°R. 0.85 0.84 Fins 0.83 0.82 0.81 0 Enthalpy ratio h, (0) Fig. F,51. Recovery factor for a laminar boundary layer on a flat, plate in free flight for various Mach numbers. T, = 400°R. ( 357 ) F - CONVECTIVE HEAT TRANSFER IN GASES 0.82 0.81 0 2 4 6 8 10 12 14 16 Mach number M, Fig. F,5m. Reynolds analogy factor for a laminar boundary layer on an insulated flat plate in free flight. JT, = 400°R. 0 2 Ann ie 8 10 12 14. mG Mach number M., Fig. F,5n. Reynolds analogy factor for a laminar boundary layer on a flat plate in free flight for various wall-to-free stream enthalpy ratios. T, = 400°R. ( 358 ) F,5 - NUMERICAL RESULTS 0.82 0.81 0.80 Sins 0.79 0.78 Enthalpy ratio h, (0) Fig. F,50. Reynolds analogy factor for a laminar boundary layer on a flat plate in free flight for various Mach numbers. T, = 400°R. Mach number MM, Fig. F,5p. Local heat transfer coefficients for a laminar boundary layer on an insulated flat plate in a wind tunnel at various supply temperatures. ( 359 ) F - CONVECTIVE HEAT TRANSFER IN GASES 0.88 0.87 Mach number M, Fig. F,5q. Recovery factor for a laminar boundary layer on an insulated flat plate in a wind tunnel at a supply temperature of 100°F. Mach number M., Fig. F,5r. Reynolds analogy factor for a laminar boundary layer’on an insulated flat plate in a wind tunnel at a supply temperature of 100°F. ( 360 ) F,5 - NUMERICAL RESULTS Recovery factor Fins Mach number M, Fig. F,5s. Recovery factor for a laminar boundary layer on an insulated flat plate in a wind tunnel at various supply temperatures. Recovery factor Fins Supply temperature, °F Fig. F,5t. Recovery factor for a laminar boundary layer on an insulated flat plate in a heated wind tunnel for various Mach numbers. (361 ) F - CONVECTIVE HEAT TRANSFER IN GASES o fo) ow oS = 0 5100 mao. o CO = Reynolds analogy factor Sin; S) N (oe) 12 Mach number M.,z Fig. F,5u. Reynolds analogy factor for a laminar boundary layer on an insulated flat plate in a wind tunnel at various supply temperatures. Reynolds analogy factor Si, Supply temperature, °F Fig. F,5v. Reynolds analogy factor for a laminar boundary layer on an insulated flat plate in a heated wind tunnel for various Mach numbers. Pr’, for a supply temperature of 100°F. Fig. F,5t and F,5v are cross plots in Fig. F,5s and F,5u. All curves include variable specific heat from supply chamber to test section. F,6. Cone Solution. Hantzsche and Wendt [6] have shown that the equations for a thin laminar boundary layer on a circular cone at zero angle of attack in a supersonic stream with attached shock wave can be { 362 ) F,6 - CONE SOLUTION reduced by means of a simple transformation to equations of the same form as those for a flat plate. As a consequence, the local coefficient of heat transfer for the cone is ~/3 times the corresponding coefficient for the flat plate. In other words, since the local heat transfer coefficient 0 2 4 6 8 10 12 14 16 Free stream Machnumber M.,, Fig. F,6a. Local heat transfer coefficient for laminar boundary layers on insulated cones in free flight. T., = 400°R. 0.86 Free stream Machnumber M, Fig. F,6b. Recovery factor for laminar boundary layers on insulated cones in free flight. T,, = 400°R. varies inversely with the square root of the Reynolds number, the coef- ficient for the cone can be obtained by dividing the cone Reynolds num- ber by 3 and using flat plate results. Furthermore, there follows, as a result of the geometry, that the mean coefficient of heat transfer for the { 363 ) F - CONVECTIVE HEAT TRANSFER IN GASES cone is 2 1/3 times the mean coefficient for the plate based on the same slant area. It must be mentioned also that the coefficients for the cone are based upon the flow condition just outside the boundary layer of the cone and not in front of the attached shock wave. 0) 2 4 6 8 10 12 14 16 Free stream Mach number M, Fig. F,6c. Reynolds analogy factor for laminar boundary layers on insulated cones in free flight. 7’, = 400°R. Local Mach number M, Fig. F,6d. Local heat transfer coefficient for laminar boundary layers on insulated cones in a wind tunnel at various supply temperatures. Because of engineering interest in cones, the local heat transfer coef- ficient, recovery factor, and Reynolds analogy factor are plotted in Fig. F,6a, F,6b, and F,6c, respectively, for insulated cones at zero angle of attack in free flight. With experimental work in wind tunnels, Fig. F,5s, F,5t, F,5u, and F,5v can be used directly for recovery factor and Reynolds analogy factor ( 364 ) F,7 - STAGNATION POINT SOLUTION on cones, provided that the Mach number is the local value just outside the boundary layer. In the case of the local heat transfer coefficient, Fig. F,5p can be used also for cones except that the ordinate must be multi- plied by +/3 in accordance with the above discussion. Fig. F,6d repre- sents a corrected (by +/3) plot for moderate heat transfer to or from cones in heated wind tunnels. F,7. Stagnation Point Solution. Because of its importance in general missile design, heat transfer at stagnation points of cylindrical and spherical surfaces should be given a few words at this time. It may be desirable to round the leading edges of airfoils and the noses of bodies of revolution at high speeds in order to diminish the local heat transfer rates at those locations and to allow easier internal cooling, if necessary. The heat transfer problem for incompressible flow lends itself readily to analysis. For the cylindrical surface, Squire [7, p. 631] used Homann’s solution [8] of the momentum equation near the stagnation point and found, upon simultaneous solution with the energy equation, that the local heat transfer coefficient St, defined by dk = SSG jo (ig = ALS) (fal) may be expressed by the relation —0.6 0.5 —0.5 mi CoM BD pUD ss = oro (s)™* (00)"* (002 2 where D is the diameter of curvature of the cylindrical surface and B = (du./dx)z—-0 where x is measured along the body from the stagnation point. Subscript .. refers to the undisturbed flow. From incompressible perfect fluid theory, it is found that BD/u.. = 4. This relationship has been verified experimentally [7, p. 631]. For the spherical surface, Sibulkin [9] also used Homann’s results, and following the method of Squire [7, p. 631], obtained the formula —0.6 0.5 —0.5 ne Cpe BD pUD Si = 0.763 2) i (7-3) in which D is the diameter of curvature of the spherical surface. Also from incompressible perfect fluid theory, 8D/U = 3. Eq. 7-1, 7-2, and 7-3 can be used for approximate heat transfer calculation with supersonic flow about a body when it is remembered that the problem is strictly a local one and therefore the fluid properties c,, k, uw, and p in all three equations must now be taken at the stagnation temperature T° at the outer edge of the boundary layer. Thus, Eq. 7-1 becomes for gases du = St°p°U(h° — hy) (7-1a) ( 365 ) F - CONVECTIVE HEAT TRANSFER IN GASES with 0? o\—0-6 BD 0.5 °UD —0.5 St = 0.570 (=) & “ 5 (7-2a) for the cylindrical face, and cy —0.6 BD 0.5 p° UD —0.5 St = 0.763 ( jO ) e ic (7-3a) for the spherical nose, where the superscript ° indicates stagnation con- ditions. (Note that U in the equations is arbitrary, because the effect of speed is already accounted for in Cy, k, u, p, and 6.) However, in terms of undisturbed conditions, Eq. 7-la, 7-2a, and 7-3a become dw = = Stopol (h? Tat hz) (7-1b) 0.5 e057 JONG 0:67 Ng NOI ig NOE a Se S| _dp pus + pv al (se — Ve we ee ae = A? = eT and 3B T/T. 1 (11-25) F,11 - HEAT TRANSFER Finally, taking 1 = Ky and putting Eq. 11-24 into Eq. 11-20, witht = tw, yields the following velocity distribution result: Uu oe LS aa Qs 1 B 7 (Ey we ~' (BP + 44D) es i Tw Y 2 = |e +2 (ee) (11-26) where F is a constant and v,, the kinematic viscosity at the wall, is intro- duced because of its influence in the laminar sublayer. Eq. 11-24 and 11-26 can now in turn be substituted into the von K4érm4n momentum integral relation for a flat plate with zero pressure gradient, viz. sin pu(ue — u)dy (11-27) 0 to yield a complicated integral which, however, can be expanded into a series by means of integration by parts. Upon neglect of terms of higher order, the resulting series can be approximated by a simple expression which leads to an engineering formula for the local skin friction coef- ficient in terms of Reynolds number, Mach number, and wall-to-free stream temperature ratio when the final constant involving F is adjusted to reduce the formula to the von K4rm4n friction law for incompressible flow. In this way van Driest [2/] obtained the formula 0.242 f é Ad(T./To! (sin“! a + sin“! B) = 0.41 + log (Re - ¢,) 1 hee (41) me(2) cvs where i 2A B d me B oy (ame cS (Bisel and n is the exponent in the viscosity law » = const - 7”. For air, the exponent n ranges from 0.76 at ordinary room temperature to 0.5 at higher temperatures. If, now, one assumes the similarity law for mixing length, viz. l= —K(di/dy)/(d?a/dy”), instead of 1 = Ky, then, following the same procedure given above, one obtains ii A aay + sin-! 8) = 0.41 + log (Re - c,) ] is Aa(T,/T.)* S a-+ sin~? 6) = 0.41 + log (Ke: cy Oe (11-29) ( 383 ) F - CONVECTIVE HEAT TRANSFER IN GASES Fig. F,llh. Effect of heat transfer and Mach number on local skin friction coefficient according to Eq. 11-28 for a Reynolds number of 10’. which is different from Hq. 11-28 by the term 4 log (T../T.). When the plate is insulated, Eq. 11-29 reduces to the formula derived by Wilson [33]. Other formulas are readily derivable. For example, Cope [34] held the density constant and equal to the wall value in Eq. 11-20, so that, with l = Ky, he obtained for the velocity profile TD le ny To Y Ue = Ue Je | + K in( me 4) (11-30) When the density is allowed to vary according to Eq. 11-24, the von K4rm4n integral (Eq. 11-27) then leads to the following expression: 0.242 fh =——. = 0.41 + log (Re: cs) — ] AT./To! 0.41 + log (Re: cs) — (1 + n) log T. Cope originally derived this formula for the case of the insulated plate only, where (11-31) te Teer. yc 7S Ge 1+ 5 M? ie. B = 0 in Eq. 11-24. The most simple approach (and the first attempted) was that used by von Karman [35], who allowed the density to remain constant and equal ( 384 ) F,11 - HEAT TRANSFER to the wall value in both Kq. 11-20 (with 1 = Ky) and 11-27, i.e. through- out the entire analysis. One obtains, then, for the heat transfer case, 0.242 IES ci(T./T.)3 = 0.41 + log (Re : Cs) =F n log T. (11-32) which differs from Eq. 11-31 by the factor log (7.,/T.). Von Karman also originally derived his equation for the insulated-plate case, because he (like Cope later) desired to find the effect of speed, only, upon drag. Fig. F,11i. Effect of heat transfer and Mach number on local skin friction coefficient according to Eq. 11-29 for a Reynolds number of 107. The general question that must now be answered is: Which of the above formulas is the most valid for engineering purposes? Although the question can best be answered by experimental data, a preliminary check on the form of the equations can be made upon observation of the effect of heat transfer (7.,/T.) on the local skin friction coefficient. Eq. 11-28, 11-29, 11-31, and 11-32 are plotted in Fig. F,11h, F,11i, F,11j, and F,11k where the ratio of the compressible to the incompressible flow coefficient for one Reynolds number and n = 0.76 is shown as a function of Mach number for various wall-to-free stream temperature ratios. It is immedi- ately seen from Fig. F,11h and F,11i that, regardless of mixing-length theory assumed, Eq. 11-28 and 11-29 yield friction coefficients which are definite functions of Mach number for a constant wall temperature. On ( 385 ) F - CONVECTIVE HEAT TRANSFER IN GASES Fig. F,11j. Effect of heat transfer and Mach number on local skin friction coefficient according to Eq. 11-31 for a Reynolds number of 107. ye Twat \ \ 0.8 Fig. F,11k. Effect of heat transfer and Mach number on local skin friction coefficient according to Eq. 11-32 for a Reynolds number of 107. { 386 ) By ae 2) y Sas bine 2 ABR He Fe lbs lin Oi WOM Teeth? WEIN ars cOL X 4S F - CONVECTIVE HEAT TRANSFER IN GASES the other hand, Fig. F,11j and F,11k show that Eq. 11-31 and 11-32 yield friction coefficients independent of Mach number for a constant wall tem- perature. Since it can hardly be expected that the wall temperature has complete control over the variation of fluid properties, i.e. that dissipation can be neglected, it appears reasonable to rule out Eq. 11-31 and 11-32. For completeness, the results for both laminar and turbulent flow for near-insulated flat plates are brought together in Fig. F,111 using Eq. 11-2 and 11-29 and s (turbulent) = 0.825. F,12. Cone Solution. For geometrical reasons, boundary layers are thinner on cones than on flat plates and therefore it is expected that turbulent boundary layers will have greater heat transfer coefficients for cones than for plates. The von Kérm4n momentum integral relation for a boundary layer on a cone in a supersonic stream with zero angle of attack and attached shock wave is 6 6 oo i a —wldy\ 4S ih aula ady mee 0 x JO in which the coordinate distance x is measured from the cone apex along the cone and y is measured normal to the surface. Then, using Eq. 12-1 instead of Eq. 11-27, and following the same procedure as carried out fin the derivation of Eq. 11-28, van Driest has shown [36] that a simple rule exists for the transformation of turbulent heat transfer results from a flat plate to a cone in supersonic flight. The rule states that the local heat transfer coefficient on a cone is equal to the flat plate solu- tion for one half the Reynolds number on the cone, the Mach number and wall-to-free stream temperature ratio remaining the same; thus the turbulent flow rule is similar to that for laminar compressible flow where the cone solution is equal to the flat plate solution for one-third the Reynolds number on the cone. For turbulent flow, the correction amounts to only about 10 to 15 per cent, whereas for laminar flow it amounts to 73 per cent. F,13. Stagnation Point Solution. Although it is expected that the flow will be laminar in the immediate neighborhood of the stagnation region of spheres and cylinders, it is possible for the flow to become unstable and eventually turbulent with increasing distance from that region, owing to the low Reynolds number of the local flow there. A theoretical analysis can be made for a fully turbulent boundary layer near the stagnation point when it is assumed, as in flat plate flow, that the velocity profile remains similar with distance. Assuming a 1-power law for velocity distribution, the coefficient of heat transfer ( 388 ) F,13 - STAGNATION POINT SOLUTION St,. in the formula dw = — St..p.U(h, Td, he) becomes [37]: ae BD\' ae aay & =) (3) SO ee (5) ( fi ke (dey) bea) \ID i) for spheres. For cylinders, the constant is 0.040. For an approximate calculation over the face of a sphere, the con- stant 0.042 may be apportioned linearly with 6 to 0.030 for flat plates, Stagnation solution C10 —— Sites 0.005 0 0.5 1.0 x/D Fig. F,13a. Heat transfer on the face of a sphere in air. M,, = 3; Rep, = 10°. and the ratios p./p., Ue/H.o, aS well as 8 computed from Newtonian pres- sure calculations and isentropic expansion from the stagnation region. The heat transfer rate then becomes a maximum at about 40 degrees. Fig. F,13a, F,13b, and F,13c show the heat transfer on the face of a sphere in air with M,, = 3 and Rep, = p.»UD/u. = 10°, 10%, and 10’, respectively. Also shown in the figures are the variations of heat transfer for completely laminar flow using Eq. 7-3b. It is seen that the maximum turbulent heat transfer rate increases relative to the maximum laminar rate as Reynolds number increases. ( 389 ) F - CONVECTIVE HEAT TRANSFER IN GASES 0.010 St,, 0.005 Fig. F,13c. Heat transfer on the face of a sphere in air. M, = 3; Rep, = 107. ( 390 ) F,16 - STATUS OF EXPERIMENTAL KNOWLEDGE F,14. Effects of Variable Free Stream Pressure, Wall Tempera- ture, Ete. The effects of variable free stream pressure and variable wall temperature are generally qualitatively the same but relatively less for turbulent than for laminar boundary layers. Two references are Rubesin [38] for surface temperature variation and Clauser [39] for pressure gradients. Fluid injection in the stream through the wall is effective in reducing heat transfer to the surface from the boundary layer. For example, Rubesin [40] has developed a theory for gas injection into a high speed turbulent boundary layer; comparison of the theory at Mach number zero with data of Mickley, et al. [41] shows good agreement. (For detailed discussion see Sec. G.) F,15. Rough Walls. All of the aforementioned analyses had to do with smooth walls. However, the following formula, derived in the same manner as Eq. 11-28, will be indicative of local skin friction (and there- fore heat transfer) on rough plates [37]: 0.242 oy at By ape Z AAT ./T.)! (sin-1a@ + sin7! 8) = 1.40 + log (: 2) (15-1) where ¢ is the plate roughness and z is the distance from the plate leading edge. It is assumed, of course, that the roughness projections are great enough to disrupt the viscous influence of the wall and that the projec- tions do not reach the sonic line. As with skin friction, heat transfer rates for rough plates should be significantly greater than for smooth plates. F,16. Status of Experimental Knowledge. Skin friction. Since heat transfer is proportional to skin friction, and since friction is apparently easier to measure than heat transfer, it is proper to glean first the experimental data on skin friction so that more data may be made available to verify the theory. The data of Coles [31] and Korkegi [42] for local friction on insulated plates is plotted in Fig. F,16a. The data were obtained by direct force measurements. Also plotted are Eq. 11-28 and 11-29 for n = 0.76. Ap- parently both Eq. 11-28 and 11-29 are adequate for engineering purposes. However, for more precision when more definitive data are available, and assuming that Eq. 11-28 and 11-29 have the proper form, it may be sug- gested that an equation be written as follows: 0.242 IE A Ship RANT Inne in! = . dates, —— Ad(T./T.)3 (sin-! a + sin“! 8) = 0.41 + log (Re - cs) (p + n) log = (16-1) where p is an arbitrary constant to be adjusted to the data. ( 391 ) F - CONVECTIVE HEAT TRANSFER IN GASES The mean skin friction data of Sommer and Short [43] and Chapman and Kester [44] are plotted in Fig. F,16b. The former data were obtained from deceleration measurements of hollow cylinders in free flight, and therefore the wall-to-free stream temperature ratio remained low (ranging from 1.03 at Mach number 2.81 to 1.75 at Mach number 7). The latter data were the result of steady state, direct total force measurements of Re x 10-6 oO 4 AES X 6 ~~ Korkegi Coles CF Ge Fig. F,16a. Comparison of theory and experiment on local skin friction coefficient for turbulent boundary layers on insulated flat plates. the cylinder of a cone-cylinder combination under zero heat transfer con- ditions. The mean skin friction equation, viz. 0.242 Ts ACK T/T.) (sin-! a + sin-! 8) = log (Re: Cys) — n log Tr. corresponding to Eq. 11-29, is also plotted in Fig. F,16b for n = 0.76. It is readily apparent that Eq. 16-2 is verified and that the ruling out of Kq. 11-31 and 11-32 is justified. It will be noted (see Fig. F,16a) that the data of Coles show an effect of Reynolds number as predicted by theory, whereas the data of Chap- man and Kester yielded practically no effect of Reynolds number. Although the above data of Sommer and Short were gathered for supersonic speeds, it should be pointed out that heat transfer effects on skin friction can be studied at low speeds without a supersonic wind (16-2) ( 392 ) F,16 - STATUS OF EXPERIMENTAL KNOWLEDGE tunnel. For example, when the Mach number is zero, Eq. 16-1 becomes 0.242 2 oe T, (T./T.» B (V1 + B—1) = 041 + log (Re- cy) — (p +n) log Ge Is (16-3) in which p = 4forl = Ky and p = Oforl = —K(da/dy)/(@?a/dy?), but may be adjusted by the data. Experimental data under these conditions are apparently not available as yet. Heat transfer. The necessary ingredients, viz. r, s, and c;, have now been presented for the calculation of heat transfer q, from Eq. 11-1. 1.0 Re < 10-6 O 3-9 Sommer-Short O 6-16 Chapman-Kester Fig. F,16b. Effect of heat transfer and Mach number on mean skin friction coefficient according to Eq. 16-2 for a Reynolds number of 107. They have also been checked against experiment. Therefore, it is ex- pected that the resulting heat transfer calculations will be adequate for engineering purposes. A final check on the theory may be made by measuring the heat trans- fer rate into or out of the boundary layer, thus obtaining the value of St directly. In Fig. F,16c, F,16d, and F,16e are plotted some heat transfer coefficients obtained by Shoulberg and others [45] at the Massachusetts Institute of Technology for M. = 2.0 at 7,/T. = 2.1, M. = 2.5 at T./T. = 2.7, and M. = 3.0 at T,/T. = 3.3. Theoretical curves, derived from Eq. 11-2, 11-19 (corrected for Mach number and heat transfer effect, say s = 0.825) and 11-29, are also drawn in the figures. Good agreement ( 393 ) F - CONVECTIVE HEAT TRANSFER IN GASES Fig. F,16c. Local heat transfer coefficient for a turbulent boundary layer on a heated flat plate. M. = 2.0 and Ty/T. = 2.1. Fig. F,16d. Local heat transfer coefficient for a turbulent boundary layer on a heated flat plate. M. = 2.5 and Ty/Te = 2.7. is noted, which seems to justify the entire heat transfer analysis. How- ever, there is some variance in the experimental data of various labora- tories [45,46,47,48], as seen in Fig. F,16f, and therefore additional de- finitive data is needed. Fig. F,16f represents the condition of moderate (small) heat transfer. Further check with greater heat transfer rates would be useful. ( 394 ) F,16 - STATUS OF EXPERIMENTAL KNOWLEDGE Fig. F,16e. Local heat transfer coefficient for a turbulent boundary layer on a heated flat plate. M. = 3.0 and T,/T. = 3.3. Re X 10-6 O 2-3 Slack [46] (ee) Fallis [47] 4 1-10 Pappas [48] xX 1-18 Shoulberg et al. [45] St St 0 2 4 6 8 10 M. Fig. F,16f. Comparison of theory and experiment on local heat transfer coeffi- cient for turbulent boundary layers on near-insulated flat plates. { 395 ) F - CONVECTIVE HEAT TRANSFER IN GASES TRANSITION F,17. Stability of the Laminar Boundary Layer and Relation to Transition. That the heat transfer coefficient for turbulent flow is an order of magnitude (say 10 times) greater than the heat transfer coef- ficient for laminar flow is evident from Fig. F,11k. This difference is due to the fact that the velocity gradient at the wall in turbulent flow is con- siderably greater than the velocity gradient at the wall in laminar flow. Although the region of development of the boundary layer between the minimum critical Reynolds number (neutral stability for infinitesimal disturbances) and the Reynolds number of fully turbulent flow is truly the transition region, yet in this discussion the expression ‘‘transition”’ will refer to the beginning of fully turbulent flow. It will be found that the transition Reynolds number so defined will be many times greater (again perhaps 10 times or more) than the minimum critical Reynolds number. Since the heat transfer coefficients for turbulent flow are much greater than those for laminar flow, it is desirable to employ ways and means of delaying transition as much as possible. One method of delaying tran- sition is to draw heat out of the laminar boundary layer at the wall. By this means the minimum critical Reynolds number is increased. For two- dimensional infinitesimal disturbances, it was demonstrated by Lees [49] that (1) with subsonic free stream flow, cooling the boundary layer was stabilizing, although the layer would always become unstable for suf- ficiently high Reynolds number, whereas (2) with supersonic free stream flow, cooling was again stabilizing, yet it was possible through sufficient practical cooling to maintain stability for any Reynolds number however large. When the wall is insulated, an increase in free stream Mach num- ber is destabilizing for subsonic or supersonic flow. It was next shown by van Driest [50], through numerical calculation, that the region of complete stability (infinite Reynolds number) extended from Mach number 1 to 9 for air when the Prandtl number was taken as 0.75 and the Sutherland viscosity law was used with a free stream temperature of —67.6°F. The results are given in Fig. F,17a. The minimum critical Reynolds numbers other than infinity were computed using an estimation formula given by Lees in [49]. The cooling required for complete stabilization of the laminar bound- ary layer for air under various conditions is plotted in Fig. F,17b. The solid curves (the viscous solution) are the more accurate in that they include the viscous forces in the stability analysis, whereas the dotted curves (the inviscid solution) are stability criteria because they include only the pressure forces and not the viscous forces in the analysis. The condition (Pr = 0.75, p,u, = 1) should be used for ordinary wind tunnel work because at low temperatures the Prandtl number is approximately ( 396 ) F,17 - STABILITY OF THE LAMINAR BOUNDARY LAYER 0.75 and the viscosity is proportional to the temperature. The condition (variable Pr (see Fig. F,5a), Sutherland law, T, = 400°R) would best be used for slender bodies and thin surfaces in free flight. The condition (Pr = 0.715, piu, = 1) is applicable to cones or blunt bodies where the aS Wall-to-free stream temperature ratio T,,/T- (oe) — (Re — oo) OMe) oo a OS cunt eon. TO. ad, 12 Free stream Mach number M, Fig. F,17a. Minimum critical Reynolds number as a function of free stream Mach number and wall-to-free stream temperature ratio. Prandtl number 0.75 and Suther- land viscosity law. ul — ——-— Inviscid solution Viscous solution aS Wall-to-free stream temperature ratio Ty/Te NO w Free stream Mach number M, Fig. F,17b. Cooling required for complete stabilization of the laminar boundary layer for air. ambient temperature (just outside the boundary layer) is great so that Pr = 0.715 and p, = T%. The calculations resulting in the above curves were based on analyses of the boundary layer as given in [/,2,3]. Since transition from a laminar to a turbulent boundary layer is a consequence of instability of the laminar flow, heating the boundary layer (4897 ) F - CONVECTIVE HEAT TRANSFER IN GASES should promote, and cooling retard, transition. That this is the case has been shown experimentally by Scherrer [51], Higgins and Pappas [52], and Czarnecki and Sinclair [63] of the National Advisory Committee for Aeronautics, Eber [64] of the Naval Ordnance Laboratory, and van Driest and Boison [66,66] of North American Aviation, Inc. Plate F,17 is a set of photographs by van Driest and Boison showing the boundary layer when 12 —) oO Complete stability (2-dimensional disturbances) Transition Reynolds number Re... < 10° 0 1.0 1.8 22 2.6 3.0 3.4 Average wall-to-local stream temperature ratio e/a Fig. F,17c. Effect of surface cooling on transition Reynolds number for several local Mach numbers on a smooth 10° cone. it is distorted (magnified) 20 times normal to the flow by means of a cylin- drical lens built into the Schlieren system [57]. Transition and its delay by cooling is readily discernible from the photographs. The length of each photograph represents 16 inches of a smooth 10° (apex angle) cone, cooled internally with gaseous nitrogen; the left-hand edge of each photograph is located 4.5 inches from the apex of the cone. The Reynolds number per inch is 500,000. The effect of surface cooling [56] on the transition Reynolds number for several local Mach numbers on the cone in a low turbulence tunnel is indicated in Fig. F,17c. Lines of infinite minimum critical Reynolds number are also plotted in the figure. The dashed line ( 398 ) F,18 - EFFECT OF TUNNEL TURBULENCE of Fig. F,17c shows the effect of Mach number on transition for the zero heat transfer case. It is thus generally seen from the figure that transition seems to follow the same trends predicted for the stability of the laminar boundary layer, not only with cooling, but also with increase in Mach number for an insulated surface. F,18. Effect of Supply Tunnel Turbulence. The data in Fig. F,17c are for a smooth model (10-micro-inches) in a wind tunnel with Per cent turbulence O 0.4 ia) | 20 7A GeO) > cS ia) So S n o _ J a. E fe) O Ree/in. = 0.54 « 106 Vass SIZIR Cooling Zero cooling Tw/Te Fig. F,18a. Effect of supply turbulence on transition with cooling. Me = 1.90; 10° smooth cone. { 399 ) F - CONVECTIVE HEAT TRANSFER IN GASES Per cent turbulence O 0.4 Oo 2.0 A 9.0 Re./in. ——OO/, x*K 106 Te = 228°R Cooling | D = ze) fo} UO ° = o N 1.1 on 1.9 28 Fig. F,18b. Effect of supply turbulence on transition with cooling. M. = 2.70; 10° smooth cone. ( 400 ) Zero heat transfer, T,/T, = 3.31 Cooling) i/ i — 206 Cooling ie/ Te — 179 0 4 8 12 16 20 Distance from tip of cone, inches Plate F,17. Schlieren photographs showing delay of transition on a smooth 10° cone by surface cooling. M, = 3.65, Re./in. = 0.50 X 10°. B13) EPRPECT OF TUNNEL TURBULENCE Per cent turbulence oO 0.4 Complete stability Ree/in. = 0.50 « 106 Tes ID2IR Zero cooling ae alee Fig. F,18c. Effect of supply turbulence on transition with cooling. M. = 3.65; 10° smooth cone. (u’/u)o we 100 Fig. F,18d. Effect of supply-stream turbulence on transition as a function of Mach number. Zero heat transfer. { 401 ) F - CONVECTIVE HEAT TRANSFER IN GASES 0.4 per cent turbulence in the supply chamber. Fig. F,18a, F,18b, and F,18c show the results [56] of increasing the supply turbulence to 9 per cent for M, = 1.90, 2.70, and 3.65, respectively. It is immediately con- cluded that: (1) the effect of cooling in delaying transition decreases with Trip size No trip 0.0005 in. 0.001 in. 0.002 in. — > g= O 2) ~ n 8) _ ao) Qa S [e) O Ree at trip = 1.62 & 108 Ree/in. = 0.54 « 106 ig = IIR 0.5 oor att’ 1.3 1.7 Fig. F,19a. Effect of roughness on transition with cooling. M, = 1.90. Wire trips at Ree = 1.62 X 108; 10° cone. increasing turbulence, (2) the effect of supply-tunnel turbulence in pro- moting transition decreases as Mach number increases. The second con- clusion is again drawn from Fig. F,18d, which is a cross plot of Fig. F,18a, F,18b, and F,18c for zero cooling. The ordinate of Fig. F,18d is the ratio ( 402 ) F,19 - EFFECT OF SURFACE ROUGHNESS of the transition Reynolds number with variable turbulence Re... to that with 0.4 per cent turbulence Re,,, and the abscissa is the percentage ratio of the root-mean-square velocity fluctuation u’ to the mean velocity wu in the supply chamber. F,19. Effect of Surface Roughness. Transition promoted by surface roughness can still be controlled by cooling, depending, however, upon the roughness size. Data [56] obtained for wire rings 3 inches from the tip of a 10-degree smooth cone at local Mach number 1.90, 2.70, and 3.65 Trip size No trip 0.0005 in. 0.001 in. 0.002 in. 0.004 Ree X 106 Trip position 0.3 0.7 1.1 1.5 Io 23 Urals Fig. F,19b. Effect of roughness on transition with cooling. M,. = 2.70. Wire trips at Ree = 2.01 X 10°; 10° cone. are shown in Fig. F,19a, F,19b, and F,19c, respectively. These data show that: (1) for sufficiently small two-dimensional roughness, cooling can still delay transition as though the body were smooth, (2) sufficiently large roughnesses disrupt the flow to such an extent that cooling is no longer effective, (3) for intermediate roughnesses, a reversal in transition is apparently possible, during which transition is first delayed and then promoted by cooling, and (4) the effect of roughness in promoting tran- sition decreases as Mach number increases. The reversal may be explained by the argument that cooling first tends to stabilize the flow until the ( 403 ) F - CONVECTIVE HEAT TRANSFER IN GASES Trip Size, in. Ree, Shape No trip 0) 0.001 500 Circular 0.002 1000 cylinder 0.004 2000 0.008 4000 0.010 5000 0.020 10,000 0.037. 18,500 0.0105 5250 Sharp 0.012 6000 3-dimensional 1/4 30 aan — DT Ree atitrip = 1.50< 108 Ree/in. = 0.50 x 106 Ve = |522R 1.9 2.3 27, Fig. F,19c. Effect of roughness on transition with cooling. M, = 3.65. Wire trips at Re. = 1.50 X 105; 10° cone. Fig. F,19d. Effect of roughness on transition as a function of Mach number. Zero heat transfer. boundary layer becomes sufficiently thin that the roughness shows its effect. The fourth conclusion is again seen in Fig. F,19d, which is a cross- plot of Fig. F,19a, F,19b, and F,19c for zero heat transfer. The ordinate of Fig. F,19d is the ratio of the transition Reynolds number with a trip Re., to that for the smooth cone Re,,, whereas the abscissa is the ratio of the roughness height k to the boundary layer displacement thickness 6; at the trip, in accordance with the procedure of Dryden [58]. ( 404 ) F,20 - AERODYNAMIC HEATING OF HIGH SPEED VEHICLES CHARMER 2. APPLICATION OF THEORY) TiO ENGINEERING PROBLEMS AT HIGH SPEEDS F,20. Aerodynamic Heating of High Speed Vehicles. The rise in temperature of the air in immediate contact with the surface of a vehicle as a result of high speed causes transfer of heat into the vehicle, thus the expression ‘‘aerodynamic heating.’’ The temperature rise of the contact air may be caused by direct compression, such as at the nose of a blunt body, or friction in a boundary layer, or both. The glowing of meteorites is a manifestation of the high temperatures associated with aerodynamic heating. Reference material: Aluminum 2024-T4 at room temperature Weight ratio for tension 0 400 800 1200 1600 Temperature, °F Fig. F,20a. Effect of temperature on the weight ratio for the same tensile load for various metals. At low speeds, aerodynamic heating is usually objectionable only to the pilot. However, at high speeds, such heating actually dictates the design of the vehicle, not only for structural reasons, but also because of the problem of insulating vital compartments, such as for fuel and guidance equipment. Indeed, the design of hypersonic missiles, such as glide or ballistic rockets, awaits further research on boundary layer heat transfer rates and high temperature insulating and structural materials. Fig. F,20a and F,20b show the effect of temperature on the strength of aluminum alloy, stainless steel, and titanium using aluminum alloy at room temperature as the base [59]. While the loss in tensile yield-strength with temperature is indicated in Fig. F,20a, a more significant presen- tation is given in Fig. F,20b, which gives the relative weights of metal ( 405 ) F - CONVECTIVE HEAT TRANSFER IN GASES required to carry the same load for buckling. According to Fig. F,20a it is apparent, from a tension-load standpoint, that titanium is most suitable all the way up to about 800°F, above which stainless steel would be preferable. On the other hand, from a plate-buckling standpoint, Fig. F,20b shows that aluminum alloy is preferable up to a temperature of about 600°F, titanium is most suitable up to about 900°F, after which stainless steel would be desirable. At any rate, regardless of loading con- dition, it may be concluded that titanium should be the most useful of the three metals between about 600°F and 900°F. (Mach number ranges, corresponding to full boundary layer temperature rise from an ambient N oO 2.0 Reference material: 1.0 - Aluminum 2024-T4 at room temperature Weight ratio for flat plate buckling un 0 400 800 1200 1600 Temperature, °F Fig. F,20b. Effect of temperature on the weight ratio for the same buckling load for various metals. temperature of —60°F, are also indicated in Fig. F,20b.) At higher tem- peratures, say 1500°F, other heat-resistant alloys must be considered. For example, Hastelloy C is a recently developed high temperature, nickel-base alloy which has good inherent section properties and there- fore may prove suitable for the design of the main structure of high Mach number vehicles. Because the temperature of a body is greatest at the nose as well as at other protruding parts, it may be necessary to use an insulating material of low structural value, such as a ceramic, in those regions, especially when such regions may be backed up with suf- ficient supporting structure. The leading edges of wings may be treated likewise. Calculation of skin temperature. In the engineering calculation of the skin temperature of a high speed vehicle, the usual assumptions are: (1) the skin is so thin that the temperature gradient in the skin normal ( 406 ) F,20 - AERODYNAMIC HEATING OF HIGH SPEED VEHICLES to the surface is negligible, (2) heat conduction along the skin is negli- gible, (3) no heat transfer takes place to or from other parts of the missile, (4) the specific heat of the air in the boundary layer is constant, and (5) radiation emissivity and absorptivity of the skin are equal. Accordingly, the differential equation for skin temperature is pe 2 Somane, = To) SAGs —.C) (20-1) where, on the left-hand side of the equation, Ty, cx, pw, and 6, are the Incident radiation G, BTU/ft? hr Lower surface Upper surface Altitude, ft & 10° Fig. F,20c. Solar and nocturnal radiation to a black body as a function of altitude. temperature, specific heat, density, and thickness of the skin, respec- tively, and ¢ is time. On the right-hand side, the symbols T,, ¢, and ¢ represent the insulated-skin (recovery) temperature, skin emissivity, and the Stefan-Boltzmann radiation constant (0.173 X 10-§ BTU/ft? hr (°R)4), respectively. G signifies the incident radiation from solar, terres- trial, and interstellar sources. The specific heat at constant pressure of the air is c,, while the density and velocity at the outer edge of the boundary layer are p, and ue, respectively. St is the Stanton number for either laminar or turbulent flow or mixed. Fig. F,20c shows the variation of incident radiation with altitude [60]. Owing to orientation of the air- craft in flight, only a fraction of such radiation is received. If heat trans- fer to other parts of the vehicle is considered, then the rate of that heat transfer must be subtracted from the right-hand side of Eq. 20-1; if heat { 407 ) F - CONVECTIVE HEAT TRANSFER IN GASES Table F,20. Emissivities of a few materials. Emissivity Material Temperature range, °F 0.11 to 0.19 | Aluminum oxidized at 1110°F 390 to 1110 0.55 to 0.60 | Smooth sheet iron 1650 to 1900 0.66 Oxidized rolled sheet steel 70 0.62 to 0.73 | Stainless steel (8 per cent Ni, 18 per cent 420 to 980 Cr) after 42 hr heating at 980°F is transferred to the skin from other parts, then such transfer must be added to the right-hand side of the equation. The emissivities of a few important materials are given in Table F,20 [6/]. The solution of Eq. 20-1 is usually carried out by successive approxi- mation because St is an implicit function of T.,, Re., and M.. 600 500 450 Plate temperature, °F 400 350 0 4 8 12 16 20 Length of plate, ft Fig. F,20d. Temperature distribution along a flat plate moving at Mach number 3. ( 408 ) F,20 - AERODYNAMIC HEATING OF HIGH SPEED VEHICLES 200 160 80 Altitude, ft x 10° “) es (ft/sec) o 103 Fig. F,20e. Flat plate flight temperature at point 5 ft from leading edge. Laminar boundary layer. Rt Cee Nl Ae MH Aaa ariti/| | | | | Velocity, (ft/sec) « 1073 Fig. F,20f. Flat plate flight temperature at point 5 ft from leading edge. Turbulent boundary layer. Altitude, ft « 1073 40 Eq. 20-1 presupposes relatively slow change in motion of the vehicle, depending of course on how thick the skin really is. Apparently the method is satisfactory for the boosting of missiles (such as the V-2) when the skin thickness is of the order =, inch and the acceleration is about 5 g’s. For highly transient conditions, such as with ballistic dive-ins, it would be necessary to allow for the variation of temperature across the ( 409 ) F - CONVECTIVE HEAT TRANSFER IN GASES skin. Such a procedure becomes quite laborious because it usually involves the numerical method of finite differences [62]. Fortunately, it is still suf- ficiently accurate to consider only the heat transfer normal to the plate. In the steady state (cruise), the left-hand side of Eq. 16-1 is zero. The results of typical temperature calculations in the steady state are shown in Fig. F,20d for a flat plate at zero angle of attack for both laminar and turbulent boundary layers. The plate was assumed to be moving at Mach number 3 at elevation 50,000 ft. in the NACA Standard Atmos- phere. The emissivity was taken at 0.5 and the incident radiation was assumed to be 200 BTU/ft? hr. The heat transfer coefficients were ob- tained from Fig. F’,5i for laminar flow and from Eq. 11-2, 11-19, and 11-29 2500 2000 ° ville [e) o 5 1500 = iw O = 1000 z 5 es - y, avian es S Ome Fig. F,20g. Flat plate flight temperature at point 5 ft from leading edge. Altitude constant at 40,000 ft. Steady state. for turbulent flow. To facilitate calculations with turbulent flow, Eq. 11-29 can be put in nomographic form [63]. It is seen that the temper- atures decrease with distance from the leading edge and that they are considerably higher for a fully turbulent boundary layer than for a laminar boundary layer. Fig. F,20e and F,20f may be useful in hypersonic cruising-missile design, because they show at what altitude a missile must cruise in order to maintain a given temperature at a distance 5 ft aft of the leading edge, assuming the missile can be represented by a flat plate. The plate is at zero angle of attack, the radiation emissivity was again taken at 0.5, and the NACA Standard Atmosphere was used. Fig. F,20e is for laminar flow and Fig. F,20f for turbulent flow. Lines of constant Reynolds number per foot of length are also indicated in the figures. The temperature con- ( 410 ) F,20 - AERODYNAMIC HEATING OF HIGH SPEED VEHICLES tour 580°R represents the case where the radiation just balances the absorption of heat from the sun. This temperature (580°R) is then the temperature of a body at rest in the atmosphere or in motion out in space. It may also be the temperature of a body in motion in the atmos- phere when the speed is such that the boundary layer temperature is 580°R. The fact that there are lower temperature contours within the Complete stabilization due to cooling Skin temperature, °R 0 102 ON SO 40uCaNSO CO 70 Time, seconds after take-off Fig. F,20h. Correlation of theory with skin-temperature data from a V-2 rocket. Boundary layer not tripped. Data of Fischer and Norris. 580°R contour means that as a body starts from rest the temperature falls at first, because the ambient air is at a lower temperature than the body temperature, and then rises as the friction within the boundary layer increases. At a certain speed, the boundary layer temperature be- comes equal to the wall temperature, whereupon the wall radiates heat at exactly that rate at which it receives it from space. At higher speeds, the friction increases the boundary layer temperature and heat is trans- ( 411 ) F - CONVECTIVE HEAT TRANSFER IN GASES ferred into the wall from the layer, thereby increasing the wall temper- ature. The fall and rise in temperature of a flat plate increasing its speed from subsonic to supersonic speed while remaining at 40,000 ft altitude is shown in Fig. F,20g. It appears that the solar heat is balanced at about Mach number 1.5 for 50 per cent of direct solar radiation. The skin tem- Skin temperature, °R XJ 8 600 | Attached shock wave 0 IO 20 BO POR 30" GO” FY 70 Time, seconds after take-off Fig. F,20i. Correlation of theory with skin-temperature data from a V-2 rocket. Boundary layer tripped. Data of Fischer and Norris. perature then is about 120°F which would begin to produce discomfort for the pilot if one were present. Correlation of theory and experrment. Considerable skin-temperature data have been recorded during the flight of supersonic missiles. Of par- ticular interest are the data reported by Fischer and Norris [64,65], meas- ured at several points on the nose of a V-2 (German A-4) rocket during ascent at White Sands, New Mexico, on October 9, 1947. Fig. F,20h and F’,20i show the experimental temperature variation with time for stations ( 412 ) F,20 - AERODYNAMIC HEATING OF HIGH SPEED VEHICLES G and H located on opposite sides of the nose cone and 12 in. from the tip. The temperatures are plotted up to 61 sec, when the rocket motor burned out. Stations G and H were chosen for study because station H was located aft of a strip of two-dimensional boundary layer trip, whereas station G was not; therefore, the temperature data at those stations should show the difference between the normal boundary layer and a fully turbulent layer. For the purpose of comparing the data with the theory, theoretical temperature curves for laminar and turbulent layers are plotted in the figures. The heat transfer coefficients for the cone were obtained from flat plate theory by use of the rules stated above. During 10,000 Wall-to-free stream temperature ratio Tie / ales Free stream Mach number M, Fig. F,20j. Course of V-2 flight data. the period of the measurements, radiation was practically negligible. Also shown in Fig. F’,20h is the theoretical time at which the laminar boundary layer becomes completely stable (regardless of Reynolds number) for infinitesimal two-dimensional disturbances due to cooling caused by skin- temperature lag. This limit was obtained by following the course of the wall-to-free-stream temperature ratio in Fig. F,20j and using Fig. F,20k, which is included to show the variation of pertinent properties neces- sary in the heat transfer study for station G. The instant of absolute stabilization seems to correspond to the intersection of the temperature ratio and the stabilization curve in the figure. However, it is suspected that stabilization began before that time, as apparently happened in Fig. F,20h. ( 413 ) F - CONVECTIVE HEAT TRANSFER IN GASES That the theory for turbulent flow follows the data quite satisfactorily is shown in Fig. F,20i. Perhaps the slight deviation after 45 sec is caused by the strong stabilization effect of cooling in spite of the boundary layer trip. Fig. F',20h is generally interesting because it shows clearly the course of events which is typical of boundary layer development as a function of due to cooling S no) ~ oO bs i 2) ~_ wn vo: ~ a0) Qa € ie) O Lal 1.10 1.00 Time, seconds after take-off Fig. F,20k. Variation of flow parameters on nose cone of V-2 rocket in Fig. F,20h and F,20i. Reynolds number and wall-to-free stream temperature ratio (heat trans- fer). At first, the Reynolds number soon becomes large enough to make the boundary layer turbulent; therefore the data follow the turbulent trend for a while. However, as the speed increases, the boundary layer temperature increases, thus bringing about heat flow into the missile skin owing to heat capacity of the skin. The boundary layer cooling then tends to stabilize the layer; in fact, a rate of cooling is finally reached after which a laminar boundary layer is stable for any Reynolds number. The transition from turbulent to laminar flow is clearly seen in Fig. F,20h. ( 414 ) F,21 - HEAT TRANSFER IN ROCKET MOTORS It is to be emphasized that the strong tendency for stabilization exhibited in Fig. F,20h is caused by the transient heat lag of the skin and not by radiation, since radiation was almost negligible compared to the heat absorption of the skin. Thus, transient firings of the above type are of great importance in the study of boundary layer characteristics at super- sonic speeds. However, steady state temperature data, when available, will check radiation rates of cooling and their effect on boundary layer stability. Reliable data at hypersonic speeds and higher altitudes will be useful for studying the effect of slip flow on heat transfer. It is of interest to note in Fig. F,201 that after 70 sec of flight the flow for the one-foot station of the above-discussed V-2 had already entered the slip-flow regime defined by Tsien [66]. 5 a So = oO 2 / € fof Spc ilsee ae > Je 5 = ~ = UO ©} 1 eal] 0 Fig. F,201. Flow domain during flight of V-2 rocket no. 27 for 1 ft station. F,21. Heat Transfer in Rocket Motors. Another important heat transfer problem is that of the rocket motor in which the metallic walls must be protected against the high temperatures of the propellant gases. The problem is essentially the same as in the aerodynamic heating of high speed vehicles, except that in rocket motors the flow properties change much more rapidly. The ambient temperatures in the nozzle are always high, so that the nozzle wall, particularly at the throat, must be continually protected by either regenerative or film cooling. Only at hypersonic speeds does the boundary layer temperature on the outside of a vehicle become comparable to that in the nozzle. Regener- ative cooling is brought about by circulating some of the fuel in the motor jacket before injecting it into the combustion chamber. Film cooling is a ( 415 ) F - CONVECTIVE HEAT TRANSFER IN GASES technique, first used by the Germans in the V-2 rocket motor, by which small quantities of fluid, say the regenerative coolant, are permitted to enter the nozzle at many points on the interior surface and spread over the wall in a thin film. When the coolant is introduced in the nozzle through the porous wall, the process is called sweat cooling. Rocket motor Heat transfer, BTU/in? sec f ped a Fe 0 4 8 12 20 Motor length, in. Fig. F,21a. Heat transfer distribution in acid-aniline rocket motor. Experimental investigations. Because of the large variation of flow properties within the rocket nozzle, theoretical calculation of heat trans- fer in rocket motors has given way to experiment. A rather complete experimental investigation has been carried out by Boden [67] to determine the effect of many factors, including film cooling, on the heat transfer in a 1000-lb thrust rocket motor using 64 per cent red fuming nitric acid for the oxidizer and a mixed fuel of 80 per cent aniline and 20 per cent furfuryl alcohol. The combustion-chamber tem- perature was about 5000°F. The rate of heat transfer was obtained at various sections of the motor, including the combustion chamber, by ( 416 ) F,21 - HEAT TRANSFER IN ROCKET MOTORS measuring the change in bulk temperature of the jacket coolant. The film coolant used in these experiments was water. Fig. F,21a shows a typical distribution of heat transfer through the walls of the motor using fresh oxidizing acid in the propellant and no film cooling. It is at once observed that the rate of heat transfer is highest just upstream of the nozzle throat, probably because in that vicinity the boundary layer is the thinnest owing to the favorable pressure gradient. It is also seen that the heat transfer in the combustion chamber is rela- tively low. In this region where the gases are very hot, radiation from the gas contributes a good share (up to 30 per cent) of the heat transmitted to the walls, the remainder being caused by convection. In the nozzle where ambient temperatures are lower, radiation is considered unimportant. Fis _!lm <0olant injected at A Fis Mm ; wo lant injected at B 0 0.1 0.2 Ce ion 0.5 Film coolant, Ib/sec Heat transfer, BTU/in? sec Fig. F,21b. Effect of rate of film-coolant flow upon heat transfer at nozzle throat. It was found that fresh acid always produced low heat transfer. Also, the presence of iron, chromium, and nickel dissolved from the shipping containers could increase the heat transfer as much as 50 per cent. A factor which decreased the heat transfer rate was, of course, a decrease in temperature as indicated by a decrease in oxidizer-fuel mixture ratio. No significant loss in motor performance was observed with change in mix- ture ratio. The design of the propellant injector as well as the nozzle shape were other more or less important factors. How film cooling affects the rate of heat transfer at the throat section of the nozzle is shown in Fig. F,21b. Fresh acid was used in the experi- ments represented in this figure. The upper curve shows the decrease in heat transfer through the wall when film coolant is injected tangentially to the circumference of the combustion chamber at the entrance to the Cay F - CONVECTIVE HEAT TRANSFER IN GASES nozzle. The lower curve indicates an even greater reduction when the coolant was injected midway between the entrance and throat of the nozzle where the heat transfer was increasing rapidly. Observations of the nozzle after the tests showed that film cooling persisted throughout the nozzle throat. For the particular motor used, maximum cooling was attained when the coolant flow rate was 5 per cent of the propellant con- sumption rate. From the results of his experiments, Boden concluded that film cooling reduces the heat transfer in rocket motors up to 70 per cent, and that proper control of the operating mixture ratio and the propellant Enthalpy, BTU/Ib 0) PP 4 6 8 JO" 2 14. 16 18.) 20) . Doyeeoe Temperature, °R & 10° Fig. F,22a. Enthalpy of air as a function of temperature and pressure. composition would gain an additional 10 to 15 per cent reduction. Al- though the above results were obtained using water as the film coolant, other data indicated that fuel was equally effective. A similar set of experiments was conducted by Greenfield [68] to deter- mine the coefficients of heat transfer in a rocket motor designed to pro- duce a 1000-lb thrust with a liquid oxygen-ethyl alcohol propellant. The experimental procedure for measuring the heat transfer rate utilized the transient temperature rise of the five uncooled segments which comprised the walls of the motor nozzle. Two series of tests were carried out: one series using the liquid oxygen-ethyl alcohol propellant which developed an estimated 3000°F, the other using high pressure air at 1400°F. No film ( 418 ) F 22 - DISSOCIATION EFFECTS cooling was attempted. The following empirical formula fitted fairly well the data from the five nozzle-wall segments: 0.8 h = 0.029 oa Cpa”? (21-1) where h is the heat transfer coefficient in BTU/ft? hr (°F) and is equal to the product Sécp,pewe. Also in Eq. 21-1, G is the mass velocity (at mid- point of a segment) in lb mass/ft? hr, D the inside nozzle diameter at a segment midpoint in feet, c, the specific heat of the gas at the insulated- wall temperature in BTU/lb mass F, yu the viscosity of the gas at the insulated-wall temperature in lb mass/ft hr. It is interesting to note that in spite of the rapid changes in the flow properties throughout the rocket nozzle, Eq. 21-1 has the same form as empirical laws for turbulent flow in straight pipes [59]. F,22. Dissociation Effects. When the speed of an aircraft becomes so great that the temperature of the surrounding air (owing to com- Absolute viscosity, (Ib force sec/ft2) & 105 O. Uae Ge 1. ats Temperature, °R & 1073 Fig. F,22b. Absolute viscosity of air as a function of temperature and pressure... ( 419 ) F - CONVECTIVE HEAT TRANSFER IN GASES a 0 Temperature, °R & 10° Fig. F,22c. Prandtl number of air as a function of temperature at 0.1-atm pressure. Prandtl number, an) He Ni a Pas 14 12 je) Density ratio p°/ px Mach number M.. Fig. F,22d. Ratio of stagnation to ambient density across a normal shock for air. ( 420 ) F,22 - DISSOCIATION EFFECTS pression behind shock waves or friction in the boundary layer) becomes sufficiently high, the air components partially dissociate, and the compo- sition of the new air will be entirely different from that at low temper- ature conditions. For example, at a speed of about Mach 20 at 100,000 feet altitude, the composition of the air behind a normal shock would be Temperature T°, °R & 1073 Mach number M.,, Fig. F,22e. Stagnation temperature behind a normal shock for air. approximately 50 per cent atomic nitrogen, 24 per cent molecular nitro- gen, and 26 per cent atomic oxygen, compared to 78 per cent molecular nitrogen and 21 per cent molecular oxygen at low temperature. The degree of dissociation increases with decrease in pressure; hence the result- ing composition of the air is a function of pressure as well as temperature. An important effect of dissociation is an increase in specific heat ( 421 ) F - CONVECTIVE HEAT TRANSFER IN GASES resulting from the absorption of energy in the breaking apart of the air components. PROPERTIES OF DissociaTED AiR. Before any calculation of heat transfer under conditions conducive to dissociation can be undertaken, the thermodynamic, as well as transport, properties must be determined. For example, Fig. F,22a and F,22b give the variation of enthalpy [69] and viscosity [70], respectively, of dissociated and undissociated air for Viscosity ratio p°/ pL. Mach number M. Fig. F,22f. Ratio of stagnation to ambient pressure across a normal shock for air. various pressures. The Prandtl number, however, is not altered appreci- ably by high temperature, as indicated in Fig. F,22c [7/]. CALCULATION oF Heat TRANSFER NEAR THE STAGNATION POINT INCLUDING DISSOCIATION. Laminar flow. As mentioned in Art. 7, the results expressed in Eq. 7-1b, 7-2b, and 7-3b for laminar flow heat transfer at the stagnation point of a body at supersonic speed can be applied with some approximation to hypersonic flow. In Eq. 7-2b and 7-3b, 6 is again obtained from Fig. F,7a. The Prandtl number (c,u/k)° is taken from Fig. F,22c where it is seen to remain at a value of about 0.7 for air. The density ratio p°/p,. value of about 0.7 for ( 422 ) F,22 - DISSOCIATION EFFECTS air. The density ratio p°/p,. across a normal shock in free flight is plotted in Fig. F,22d; this ratio was computed through simultaneous solution of the continuity, momentum, and energy equations, along with the equation of state in [69]. Out of this calculation also comes the tempera- ture ratio (Fig. F,22e). From Fig. F,22e and Fig. F,22b is determined the viscosity ratio u°/u,, plotted in Fig. F,22f. The temperature ratio in Fig. F,22e is of extra interest because it indicates immediately what 1000 je) je) i. O Sea level 40,000 ft 100,000 ft Pressure ratio p°/ pa Mach number Mea Fig. F,22g. Ratio of stagnation to ambient viscosity across a normal shock for air. temperature an insulated body would acquire, or rather, what tempera- ture a body is subjected to, at very high speeds. The stagnation enthalpy h° is readily computed from 2 ho =h, + = (22-1) and is therefore independent of dissociation. It is noteworthy that the stagnation pressure behind a shock wave is apparently also independent of dissociation (see Fig. F,22¢). { 423 ) F - CONVECTIVE HEAT TRANSFER IN GASES The above simple procedure using Eq. 7-1b, 7-2b, and 7-3b checks shock tube experimental data of Rose and Riddell [72] very well for the stagnation point of a sphere as seen in Fig. F,22h. A more elaborate theory of Fay and Riddell [73], taking into account the effects of diffusion and atomic recombination, also fits the data well, so that it would seem that these latter effects do not significantly influence the heat transfer rate, at least according to the above experiments. (S) Oo Eq. 7-1b,7-2b,7-3b Stagnation point heat transfer qy, kw/cm? 16 20 24 28 Flight velocity, (ft/sec) & 1073 Fig. F,22h. Comparison of theory and experiment on heat transfer at the stagnation point of a sphere. While the above procedure produces good results for the calculation of heat transfer from a dissociated gas, the actual over-all effect of disso- ciation on the heat transfer rate for a perfect gas is shown in Fig. F,221. The calculation was made for a sphere at 100,000-feet altitude in the ICAO (International Civil Aviation Organization) atmosphere, and the wall temperature was assumed to be 2500°R. It is concluded that the effect of dissociation (real gas) on heat transfer, compared to the perfect gas solution, is not great. Turbulent flow. As with laminar flow, it can be shown that for turbu- lent flow the effect of dissociation, compared to the perfect gas solution, will also not be very great [74]. ( 424 ) F,23 - CITED REFERENCES —) Ov 3 Heat transfer rate qy/D & 102, BTU/ft? sec (e) (ee) 6 8 10 12 14 16 18 20 Mach number M. Fig. F,22i. Effect of dissociation on heat transfer at the stagnation point of a sphere at 100,000-ft altitude in the ICAO atmosphere. Wall temperature at 2500°R. F,23. Cited References. Ile © © 00 bo er) Crocco, L. Lo strato limite laminare nei gas. Ministero della Difesa-Aeronautica, Roma., Monografie Scientifiche di Aeronautica 3, 1946. Transl. in Aerophys. Lab., North Amer. Aviation Rept. AL-684, 1948. . van Driest, E.R. Investigation of laminar boundary layer in compressible fluids using the Crocco method. NACA Tech. Note 2697, 1952. . van Driest, E. R. The laminar boundary layer with variable fluid properties. Heat Transfer and Fluid Mech. Inst., Berkeley, 1954. . Keyes, F. G. The heat conductivity, viscosity, specific heat, and Prandtl num- bers for thirteen gases. Mass Inst. Technol. Project SQUID Tech. Rept. 37, 1952. . Tables of the Thermal Properties of Gases. Natl. Bur. Standards Circular 564, U.S. Government Printing Office, 1955. . Hantzsche, W., and Wendt, H. Jahrbuch deut. Luftfahrtforschung 76, 1941. . Goldstein, 8. Modern Developments in Fluid Dynamics, 1st ed., Vol. 2. Oxford Univ. Press, 1938. . Homann, F. Z. angew Math. u. Mech. 16, 159 (1936). . Sibulkin, M. J. Aeronaut. Sci. 19, 570 (1952). . Grimminger, G., Williams, E. P., and Young, G.B. W. J. Aeronaut. Sci. 17, 675 (1950). . Korobkin, I. Laminar heat-transfer characteristics of a hemisphere for the Mach number range 1.9 to 4.9. Nav. Ord. Lab. Rept. 8841, 1954. . Stine, H. A., and Wanlass, K. Theoretical and experimental investigation of aerodynamic-heating and isothermal heat-transfer parameters on a hemispherical nose with laminar boundary layer at supersonic Mach numbers. NACA Tech. Note 3344, 1954. ( 425 ) 46. 47. F - CONVECTIVE HEAT TRANSFER IN GASES . Goland, L. J. Aeronaut. Sci. 17, 436 (1950). . Levy, S. J Aeronaut. Sci. 21, 459 (1954). . Morris, D. N., and Smith, J. W. J. Aeronaut. Sci. 20, 805 (1953). . Chapman, D. R., and Rubesin, M. W. J. Aeronaut. Sci. 16, 547 (1949). . Lighthill, M. J. Proc. Roy. Soc. London A202, 359 (1950). . Smith, J. W. J. Aeronaut. Sci. 21, 154 (1954). . Eber, G. R. J. Aeronaut. Sci. 19, 1 (1952). . Shoulberg, R. H., Hill, J. A. F., and Rivas, M. A., Jr. An experimental determi- nation of flat plate recovery factors for Mach numbers between 1.90 and 3.14. Mass. Inst. Technol. Nav. Supersonic Lab. Wind Tunnel Rept. 36, May 1952. . van Driest, E.R. J. Aeronaut. Sci. 18, 145 (1951). . van Driest, E.R. 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Aeronaut. Sci. 19, 645 (1952). . Coles, D. J. Aeronaut. Sci. 21, 433 (1954). . Colburn, A. P. Trans. Am. Chem. Engrs. 29, 174 (1933). . Wilson, R. E. J. Aeronaut. Sci. 17, 585 (1950). . Cope, W. F. The turbulent boundary layer in compressible flow. Brit. Aeronaut. Research Council Repts. and Mem. 7634, 1943. . von Kaérmaén, Th. The problem of resistance in compressible fluids. V. Convengo della Foundazione Alessandro Volta, Rome, 1935. . van Driest, E.R. J. Aeronaut. Sci. 19, 55 (1952). . van Driest, E.R. Aeronaut. Eng. Rev. 16, 26 (1956). . Rubesin, M. W. The effect of an arbitrary surface-temperature variation along a flat plate on the convective heat transfer in an incompressible turbulent bound- ary layer. NACA Tech. Note 2345, 1951. . Clauser, F. H. J. Aeronaut. Sci. 21,91 (1954). - Rubesin, M. W. An analytical estimation of the effect of transpiration cooling on the heat-transfer and skin-friction characteristics of a compressible, turbulent boundary layer. NACA Tech. Note 3341, 1954. . Mickley, H. 8., et al. Heat, mass and momentum transfer for flow over a flat plate with blowing or suction. NACA Tech. Note 3208, 1954. . Korkegi, R. H. Transition Studies and Skin Friction Measurements on an Insu- lated Flat Plate at a Hypersonic Mach Number. Ph.D. Thesis, Calif. Inst. Technol., 1954. . Sommer, S. C., and Short, B. J. J. Aeronaut. Sci. 23, 536 (1956). . Chapman, D. R., and Kester, R. H. J. Aeronaut. Sci. 20, 441 (1953). . Shoulberg, R. H., et al. An experimental investigation of flat plate heat-transfer coefficients at Mach numbers of 2, 2.5 and 3 for a surface-temperature-to-stream total temperature ratio of 1.18. Mass. Inst. Technol. Nav. Supersonic Lab. Wind Tunnel Rept. 39, 1953. Slack, E. G. Experimental investigation of heat transfer through laminar and turbulent boundary layers on a cooled flat plate at a Mach number of 2.4 NACA Tech. Note 2686, 1952. Fallis, W. B. Heat transfer in the transitional and turbulent boundary layers of a ( 426 ) 48. 49. 50. 51. 52. 53. F,23 - CITED REFERENCES flat plate at supersonic speeds. Inst. Aerophys., Univ. Toronto, UTIA Rept. 19, 1952. Pappas, C.C. Measurement of heat transfer in the turbulent boundary layer on a flat plate in supersonic flow and comparison with skin-friction results. NACA Tech. Note 3222, 1954. Lees, L. The stability of the laminar boundary layer in a compressible fluid. NACA Rept. 876, 1947. van Driest, E.R. J. Aeronaut. Sci. 19, 801 (1952). Scherrer, R. Boundary layer transition on a cooled 20° cone at Mach numbers of 1.5 and 2.0. NACA Tech. Note 2131, 1950. Higgins, R. W., and Pappas, C. C. An experimental investigation of the effect of surface heating on boundary-layer transition on a flat plate in supersonic flow. NACA Tech. Note 2351, 1951. Czarnecki, K. R., and Sinclair, A. R. An extension of the effects of heat transfer on boundary-layer transition on a parabolic body of revolution (NACA RM-10) at a Mach number of 1.61. NACA Tech. Note 3166, 1954. . Eber, G. R. J. Aeronaut. Sci. 19, 55 (1952). . van Driest, E. R., and Boison, J.C. J. Aeronaut. Sci. 22, 70 (1955). . van Driest, E. R., and Boison, J.C. J. Aeronaut. Sci. 24, 885 (1957). . Buchele, D. R., and Goossens, H. R. Rev. Sci. Instr. 25, 262 (1954). . Dryden, H. L. J. Aeronaut. Sci. 20, 477 (1953). . Strength of metal aircraft elements. Munitions Board Aircraft Comm. U.S. Dept. of Defense, Rept. ANC-5a. Revised June 1951. . A design manual for determining the thermal characteristics of high-speed aircraft. Air Force Tech. Rept. 6632, Wright Field, Ohio, 1947. . McAdams, W. H. Heat Transmission. McGraw-Hill, 1942. . Kaye, J. J. Aeronaut. Sct. 17, 787 (1950). . Martin, J. J. J. Aeronaut. Sci. 20, 147 (1953). . Fischer, W. W., and Norris, R.H. Trans. Am. Soc. Mech. Engrs. 71, 457 (1949). . Fischer, W. W., and Norris, R. H. Supersonic convective heat transfer correla- tions from skin temperature. General Electric Project Hermes Rept. 55258, 1949. . Tsien, H.S. J. Aeronaut. Sci. 13, 653 (1946). . Boden, R. H. Trans. Am. Soc. Mech. Engrs. 73, 385 (1951). . Greenfield, S. J. Aeronaut. Sci. 18, 512 (1951). . Hilsenrath, J., and Beckett, C. W. Tables of thermodynamic properties of argon-free air to 15,000°K. Nail. Bur. Standards MIPR-AEDC-1, 1956. . Moore, L. L. J. Aeronaut. Sci. 19, 505 (1952). . Hansen, C.F. J. Aeronaut. Sci. 20, 789 (1953). . Rose, P. H., and Riddell, F. R. An investigation of stagnation point heat trans- fer in dissociated air. AVCO Research Lab. Research Note 32, Apr. 1957. . Fay, J. A., and Riddell, F. R. Stagnation point heat transfer in dissociated air. AVCO Research Lab. Research Note 18, June 1956. . van Driest, E. R. Transition and turbulent heating, including possible real gas effects. OSR-Convair Astronautics Symposium, Feb. 1957. ( 427 ) SECTION G COOLING BY PROTECTIVE FLUID FILMS Ss. W. YUAN G,1. Introduction. One of the most important current problems in aeronautical engineering is concerned with the flow of high energy gases. Such flow has been experienced in engines of the turbine or ramjet type, rocket motors, and nuclear reactors which use gases with high temper- atures but relatively low velocities to develop power and/or thrust. The combustion chambers, turbine blades, and afterburners are examples of components exposed to high temperature gases. Recently, much atten- tion is being given to the problem of aerodynamic heating in high speed flight, in which exterior surfaces of aircraft and missiles are exposed to gases with low temperatures but high relative velocities. In steady flight at Mach numbers of four or higher, such surfaces become heated to tem- peratures at which the strength properties of the strongest-known alloys deteriorate markedly. Moreover, the pilot and such critical cargo as instruments and explosives must be protectively cooled. In rocket motors, where combustion temperatures of 4000 to 5000°F are easily reached, cooling has been used for some time. The conventional method of cooling rocket motors is to use one of the propellants as a regenerative coolant which circulates in ducts around the motor and is then injected into the combustion chamber. This method limits the choice of many high energy propellant combinations such as the hydrogen- oxygen and hydrogen-fluorine systems, because they do not possess the desired physical properties for a satisfactory regenerative coolant. Fur- thermore, the inherent disadvantage of this method is that it is difficult to increase the heat transfer coefficient of motor wall-to-coolant to a value much higher than those values which exist in motors of current design. This is so because, in order to increase the liquid film coefficient, the velocity of the coolant must be increased at the sacrifice of increasing the pressure drop through the motor. Since the allowable value of pres- sure drop in the cooling jacket of a jet motor is limited, the coolant velocity and the liquid film coefficient are also limited. The heat transmitted from the combustion gas to the chamber wall can be considerably reduced by placing some thermally insulating mate- rial on the hot gas surface of the wall. Such an insulation has been tried in the form of ceramic coating, but the limited lifetime of refractory ( 428 ) G,1 - INTRODUCTION materials so far developed for chamber wall coatings has made it im- practical for rocket motors. Furthermore, there appears to be a limit to the improvement in materials. It is generally accepted that, ultimately, methods of cooling exposed surfaces must be used. A method of coating the chamber wall with a layer of fluid would insulate the wall better than a ceramic coating because of the much lower thermal conductivity of fluid. A promising means for controlling the heat flow to the wall of a rocket motor (first used by Germans in the V-2) is the technique of introducing small quantities of liquid at many points, distributed uniformly over the interior surface of the combustion chamber. The liquid so introduced is spread over the chamber wall in a thin film and eventually evaporates. The essential advantage of this film-cooling method is that the screening film of coolant fluid is permitted to vaporize, thus increasing its heat- absorbing capacity many times over that of a system in which the fluid remains in the liquid phase. It has a further advantage in that the fluid will form a heat-resistant layer which separates the hot gases from the chamber wall surface and in this way diminishes the heat transfer rate from the hot gases to the wall. A logical extension of the film-cooling process is to increase the num- ber of cooling orifices infinitely, i.e. to use a porous wall. The combustion chamber walls to be cooled can be made porous by powder metallurgy or the Poroloy process. A coolant in the form of a gas or liquid can be forced through the pores. Such a technique is often referred to as sweat or tran- sptration cooling. As the fluid passes through the porous wall in a direction opposite to the heat flow, heat will be transmitted from the wall to the fluid, the fluid forming a protective layer on the surface exposed to the hot gases similar to the case of film cooling. In the method of film cooling, the fluid film is gradually destroyed by turbulent mixing with the hot gases so that the effectiveness of the film decreases in the downstream direction from the point of injection. This disadvantage is eliminated in the transpiration-cooling method where the coolant is continuously in- jected along the entire chamber wall. In addition to this advantage, the method of transpiration cooling provides much greater surface area for heat transfer. It can be seen that the coolant absorbs heat as soon as it enters the abundant region of the porous wall. Because of the great sur- face area available for heat transfer, the method of transpiration cooling is particularly desirable when nuclear energy is used as the power source for rocket and jet motors. The purpose of this section is to present a critical review of the funda- mental aspects of cooling by protective fluid films. it must be realized that neither the theoretical nor the experimental aspects of this subject have been sufficiently developed to permit a logical presentation, starting from a basic assumption and progressing to the solution for engineering applications. Instead, it is found necessary to review the progress of this ( 429 ) G :- COOLING BY PROTECTIVE FLUID FILMS demanding subject made along several independent lines of attack which may serve as basic references for further research and exploration. Since the basic theories on heat transfer and fluid dynamics problems are treated at length in other sections of this volume, only the application of such theories to heat transfer in transpiration cooling is discussed in detail in the present section. G,2. Flow through Porous Metal. Porous metal. The porosity of a specimen may be defined by Porosity (per cent) _ specific gravity of the alloy — specific gravity of the specimen ry specific gravity of the alloy The specific gravity of the specimen can be determined by weighing and measuring the specimen after sintering. An ideal porous medium is a medium which is composed of innumerable voids of varying sizes and shapes termed pore spaces. Pores are interconnected to one another by constricted channels through which the contained fluid may flow under the influence of a driving pressure. A clear way to comprehend the porous medium is to visualize a body of ordinary unconsolidated sand. Porous metals can be produced by the powder metallurgy process. A method adopted by German scientists is to sinter the metal powder in a refractory container without any previous compacting pressure. The advantage of this method is that porous parts having complicated shapes which would be difficult to press in dies can be produced. However, the porosity of the finished product is very difficult to control due to the fact that only one variable, namely the particle size of the powder, seems to have a great influence on the porosity after sintering. An alternative method of preparing porous metals was developed by Duwez [1] at the Jet Propulsion Laboratory of the California Institute of Technology early in 1945. This method consists of mixing the metal powder with a certain amount of porosity-producing agent which is com- pacted at high pressure and then sintered at a high temperature. The formation of pores in the compact, interconnected by constricted chan- nels, is due to the fact that the porosity-producing agent decomposes into a gaseous state and must escape through the grain of the metal powder. Because of the shrinkage of the compact during sintering, some of these channels may close. However, a sufficient number of channels remain open to make the metal permeable. The technical details of the methods of preparation of porous metal have been described in [1]. As an illustration, the technique used for pre- paring porous stainless steel by the use of ammonium bicarbonate may be briefly reviewed. The porous specimens were prepared by mixing the metal ( 430 ) G,2 - FLOW THROUGH POROUS METAL powder with a certain amount of ammonium bicarbonate, compacting the mixture at 80,000 lb/in.?, and sintering in an atmosphere of pure hydrogen for 4 hours at 2300°F. The variation in porosity is from about 18 to approximately 52 per cent, for a variation of ammonium bicarbonate from 0 to 15 per cent. According to the results of experiments [1], the main variable factor is the amount of ammonium bicarbonate which controls porosity. The tensile strength of the porous specimens varies from 45,000 to 8000 lb/in.?, with the variation in porosity from about 17 to 54 per cent. From the viewpoint of strength it is advisable to use the maximum practical compacting pressure and to adjust the amount of ammonium bicarbonate in order to produce the required porosity. Recently a sintered-wire porous metal known as Poroloy was devel- oped by Wheeler and Duwez [2] at the California Institute of Technology. The wound-wire porous metal is made by wrapping a very thin and nar- row ribbon of flattened wire (composed of any sinterable metal) around a mandrel of any arbitrary cross section. After the wire has been wound on the mandrel to the desired depth, the mandrel and wire, as a unit, are placed in a controlled-atmosphere furnace and then sintered. Following sintering the mandrel is removed and the metallic shape is processed into a finished form. The porosity is formed by the space between the indi- vidual strands of wire, and with proper control the pores, of predeter- mined size, are interconnected and uniformly distributed, and form a predetermined passage for fluid flow through the metal. The advantages of Poroloy over ordinary sintered powder-porous metals are a higher strength for a given permeability because of its wire construction and a higher ductility because of the continuous strands of fine wire which bear a large portion of applied loads. In sintered porous metals, on the other hand, the entire load must be carried by the indi- vidual sinter bonds between the particles of powder. Furthermore, Poroloy can be made like plywood, a nonisotropic material having greatest strength in the direction of the bisection of the acute crossing angle, and the lowest strength at right angles to this direction. When the wire strands cross at right angles the material exhibits a uniform strength in all directions. Permeability of porous metal. An important problem in the design of transpiration-cooled parts is the study of the flow of fluids through porous metals. In other words, the permeability of the metal of which the parts are to be made must be known. The permeability of the metal expresses the capacity of a porous material to pass fluids when pressure differences exist. As a result of the complexity of the structure of porous metals, a complete analytical study of the problem of predicting the permeability is precluded. The following discussion is based on the experimental study of the flow of gas through porous metals [3,4]. ( ign G - COOLING BY PROTECTIVE FLUID FILMS For low values of velocity, Darcy’s law =P = const (2-1) gives the relationship between the pressure difference Ap acting on the two surfaces of a plain porous wall of thickness L, the viscosity yu, the velocity v, of the coolant flowing through the porous wall, and the length d characterizing the pore openings. This law is valid only if the pressure drop is the result of viscous shear in laminar flow. It gives a linear rela- tionship between pressure drop and velocity analogous to Poiseuille flow in a pipe. For high Reynolds numbers the pressure drop is proportional to the density of the coolant and the square of the velocity which can be expressed as follows: 2 “P = const a (2-2) In the flow through a porous medium, unlike the flow in pipes, there is no definite small range of Reynolds number to distinguish the laws given in Eq. 2-1 and 2-2. The gradual transition from the Darcy regime is due to the inertia of the fluid contracting and expanding through the pores. The inertia factor becomes progressively more important with increasing velocity. Hence, in the pressure drop equation, the loss due to both viscous shear and inertia effects must be included. The two foregoing equations can be combined in the following manner if the weight rate of flow G is introduced to take account of the compressibility effect. It becomes Ap ( 2pon 20 in which oo is the specific weight of the fluid at a reference pressure Do. The two coefficients, a and @, defined by Eq. 2-3, are independent of the nature of the fluid and have only the dimension of some unknown length characterizing the structure of the porous medium itself. Fig. G,2a gives typical curves of pressure-squared difference vs. the weight rate of flow from experimental results made with fine iron and fine ammonium bicarbonate powders. Fig. G,2b gives the relation between the strength, the flow rate, the relative density, and the pressure for Poroloy stainless steel with a 35° crossing angle, where p; and pz are in absolute pressure (Ib/in.”) and ZL is the thickness in inches. The viscous resistance coefficient a and the inertia resistance coefficient 6 of Eq. 2-3 can be determined from these experimental curves. The viscous resistance coefficient a is found to be inversely proportional to approximately the seventh power of the porosity. The variation of the inertial resistance coefficient 8 with porosity is rather complex and the only conclusion to be drawn is that it decreases with increasing porosity. The relation between ( 432 ) G,2 - FLOW THROUGH POROUS METAL the coefficients a and 8 and the percentage porosity of porous metals can therefore be established only for a given metal. A more general correlation of the measured permeability values obtained from different metals can- not yet be obtained. The complexity of the permeability problem of porous metals so far obtained lies in the fact that not all of the pores or channels are neces- (CREE oe rs I BRD ESean | | a | i i eee a Pressure-square difference, (Ib/ in)? = AOAC IAA CGM le be NTs Ome 10-5 Ome Ome 1Om Rate of flow, Ib/in* sec Fig. G,2a. Pressure-square difference vs. rate of flow of nitrogen for porous iron specimens prepared with fine iron and fine ammonium bicarbonate powders. Air at To = 540°R, uo = 0.017 centipoise. (From [4].) sarily continuous throughout the metal specimen. Furthermore the vague knowledge of the distribution of the pores and the complex passages inter- connecting the pores in porous metals makes it very difficult to obtain any quantitative correlation between the permeability coefficient and the porosity of porous metals. The pattern of the flow of gases leaving a porous metal surface was investigated experimentally at the Jet Propulsion Laboratory [5]. The ( 483 ) G - COOLING BY PROTECTIVE FLUID FILMS results indicate that the fluid leaves a porous surface in the form of a number of small jets which coalesce almost immediately to form a uni- form outward-moving layer. On the other hand, there is another type of flow pattern in which the fluid leaves the porous surface in the form of isolated jets which maintain their identity and create a very turbulent Thickness Pressure-square difference , OZ Os! ae Pe Ore 1Ons! Air flow (G), lb/in? sec Fig. G,2b. Relation between strength, flow rate, relative density and pressure for Poroloy with 35° crossing angle. (From [2].) boundary layer. However, the uniformity of the flow pattern can be con- trolled to a satisfactory degree by mechanical means during the fabri- cating process of a sintered-wire porous metal. G,3. Physical Nature of Transpiration-Cooling Process. In discussing the problem of heat transfer inside transpiration-cooled porous walls, it is desirable to consider the simplest possible case. Since the pattern of the porous passages is a very complicated three-dimensional network, the assumption of a network consisting of identical cylindrical channels running from one end of the specimen to the other is made. Fig. G,3 shows the variation of the temperature throughout the porous ( 434 ) G,3 - TRANSPIRATION-COOLING PROCESS wall and through the boundary layers. A cold medium flowing along the bottom surface of the porous wall is pressed through the pores in the wall represented by cylindrical channels in Fig. G,3 and leaves the wall on the upper side. Hot gas with a free stream temperature T’., flows along the upper surface of the porous wall and builds up a boundary layer which is usually turbulent. Within this turbulent boundary layer, a laminar sublayer forms in the immediate vicinity of the surface where the tem- perature drops rapidly to the value T., (temperature of the upper wall surface). The amount of heat flow qg per unit of time and surface area entering the wall through the upper surface is determined by the tem- perature gradient on the wall. Since the layers adjacent to the wall are Fig. G,3. Temperature variation between coolant and hot fluid. at rest, heat is transferred to the surface of the wall, essentially by con- duction. The heat transfer in the boundary layers is discussed in detail later. The investigation of heat transfer inside the transpiration-cooled po- rous wall was made by Weinbaum and Wheeler [6]. In this study it is taken that no change of state of cooling fluid occurs and that its direction of flow is opposite to that of the heat flow through the cylindrical bars of porous metal. It is further assumed that a steady state of heat flow is attained. The time rate of heat flow is, in the case of solid metal, given by the familiar Fourier equation dt qgq=—kA aE (3-1) where A is the cross-sectional area of the porous wall, k is the thermal conductivity of the metal, and ¢ is the temperature of the metal at any ( 435 ) G - COOLING BY PROTECTIVE FLUID FILMS given point inside the wall. This expression must be corrected to take into account the fact that the wall consists only partially of solid metal. If s denotes the porosity of the metal, then Eq. 3-1 can be rewritten as follows: g= -k- AS (3-2) The rate of heat transfer from metal to fluid is proportional to the area of contact and to the difference between the temperature ¢ of the metal and the temperature 7 of the fluid. Since both ¢ and T vary along the cylindrical bars, this heat transfer changes continuously along the width of the porous metal. For an infinitesimally small length dz, the following expression holds: dq = —hArNd(t — T)dz (3-3) where h is the heat transfer coefficient, N is the number of passages per unit cross-sectional area, d is the diameter of the cylindrical pore, and aNd is the total circumference at any cross section. The heat conduction from metal to fluid is used in raising the tem- perature of the fluid, hence dg — Qcrat (3-4) where Q is the mass flow of the cooling fluid through the cylindrical pores and ¢, is the specific heat of the cooling fluid at constant pressure. The solutions of the above three simultaneous differential equations give the temperature of the metal and the fluid at any point within the porous wall. The prescribed values of the temperature of the cooling fluid before its entrance into the cylindrical bars and the temperatures of the metal at both the hot and cold ends are used to determine the constants of integration. The resulting expressions show that the temperatures of the metal and the fluid become almost indistinguishable except within a very narrow range near the cold end of the wall. The temperature dis- tribution along the width of the porous wall is not linear, as in the case of a solid metal, but is an exponential function. The indistinguishable difference in temperature between the cooling fluid on its flow through the pores and the wall material can be realized from the fact that the metal surface area in contact with the cooling fluid is very great in the porous wall. The cooling fluid therefore leaves the porous wall with the wall temperature T., and with small velocity normal to the surface. In passing away from the surface, the cooling fluid picks up momentum from the gas flow until it finally reaches the outside gas velocity. At the same time its temperature increases either by conduction or by turbulent mixing until at some distance the gas temperature is reached. A counterflow is thus created between the heat flowing from the hot gas toward the wall and the cooling fluid flowing away from the wall. ( 436 ) G,4 - TRANSPIRATION-COOLED BOUNDARY LAYER The cooling fluid continuously absorbs heat from the hot gas and in this process the over-all heat transfer from hot gas to the wall is diminished. In the foregoing discussion the cooling medium is assumed to be a gas. If a liquid coolant is used and the mass flow of the coolant is great enough, then the liquid evaporates from the upper surface of the wall. The heat transfer from the hot gas to the wall is essentially the same as in the case when a gas coolant is used. It is evident that cooling with a liquid is more effective than cooling with a gas since considerable heat is absorbed by the vaporization process. There is a boundary layer on the coolant entrance side of the wall within which the coolant temperature increases from the initial value 7) to the temperature with which the coolant enters the pores. The thickness of this boundary layer and the temperature increase within it, however, are much smaller than on the hot side of the wall. G,4. Heat Transfer in Transpiration-Cooled Boundary Layer. GENERAL PRosieEms. It is well known that fluid flowing along a solid wall builds up a boundary layer along the surface of the wall. When a temperature difference exists between the fluid and the wall, a thermal boundary layer is built up along the wall, which, for gases, has a thick- ness of the same order of magnitude as a hydrodynamic boundary layer. The transfer of heat between a fluid stream and wall mainly takes place within this boundary layer. The boundary layers may be laminar or turbulent. Since the amount of coolant injection necessary to keep the same wall temperature is, for the turbulent boundary layer, about twice that for the laminar one, it is important to study the conditions of flow for each particular case. In the combustion chamber of jet motors, due to the rough combustion process, the flow is certainly of the turbulent type. Considering the high negative pressure gradient in the flow through the nozzle, the flat plate solution is assumed to yield some indication of heat transfer in transpi- ration-cooled turbulent boundary layer in combustion chambers and nozzles. Furthermore, due to the extremely high accelerations at the throat of the nozzle, the flow in the nozzle might be laminar in some cases. In addition to reducing turbulence, a negative pressure gradient tends to increase the stability of the laminar boundary layer in the nozzle. On the other hand, the flow along a gas turbine blade is expected to be laminar in the region around the nose of the blade. Due to the exist- ence of pressure gradient along the blade surface, the boundary layer solution for the flat plate can no longer be applied here. Although a positive pressure gradient in the flow direction would decrease the heat transfer from the hot fluid to wall, the stability of the laminar layer is decreased. The exact location of the transition to turbulent flow cannot be exactly predicted yet by calculation, although a reasonable indication can be expected from the stability analysis which is discussed later. The ( 437 ) G - COOLING BY PROTECTIVE FLUID FILMS influence of a pressure gradient in transpiration-cooled turbulent bound- ary layer theory is still uncertain; however, it is believed that the influ- ence is less on a turbulent layer than on a laminar one. Another important application of transpiration cooling is in reducing the aerodynamic heating problem in high speed flight. Since both heat transfer and drag coefficients are known to be lower for laminar than for turbulent flows, it is more advantageous to have a laminar boundary layer than the turbulent type. The solution of a transpiration-cooled boundary layer on a flat plate can be employed here with reasonably good approximation. The treatments in the subsequent articles are divided into approxi- mate methods for the solution of the laminar boundary layer, exact solu- tions of the laminar boundary layer, and approximate solutions of the turbulent boundary layer. The stream fluid and the injected fluid are assumed to be homogeneous. APPROXIMATE METHODS FOR THE SOLUTION OF HEAT TRANSFER IN THE LAMINAR BounpDARY LayeErR. The heat transfer in the laminar boundary layer of a transpiration-cooled wall in a flow can be solved by the von Kérmdn momentum and energy equations for the boundary layer. The basic derivation of these equations is treated at length in Vol. IV. For two-dimensional compressible flow with a pressure gradient and a uniform injection (or suction) at the wall, the momentum and energy equations for the boundary layer are given, respectively, as ones Boe ee du el pu(ue — u)dy + > i (pele — pu)dy = pwleYw + (. a) a oh d(CpT'e) yi ie @ 2 = || puc,(T. — AE faa eer i U(pe — p)dy + . Me ay dy ar Lrg tgp) ae (3 a (4-2) where the subscript . represents quantities at the outer edge of the laminar layer and , quantities at the wall. The other symbols are stand- ard and a sketch of velocity and temperature fields within the boundary layer along a transpiration-cooled wall is shown in Fig. G,4a. Incompressible boundary layer on a porous flat plate. It was men- tioned in the previous article that in many of the applications of heat transfer the boundary layer is turbulent. Nevertheless, it is interesting and important to understand the mechanism of heat transfer qualita- tively, as well as to clarify some physical quantities involved which can- not be easily interpreted in complicated cases. The following analysis is made by the investigation of the flow of a hot gas over a porous flat plate under the condition of uniform gas injection from the bottom of the plate. The assumptions made in the present investigation are: (1) the mass ( 438 ) G,4 - TRANSPIRATION-COOLED BOUNDARY LAYER density and viscosity of the fluid are assumed to be constant, (2) the flow is assumed to be laminar, and the fluid along the wall and the coolant flowing through the pores are assumed homogeneous, and (8) the wall temperature in the direction of flow is constant. In accordance with the above assumptions, Eq. 4-1 and 4-2 can be simplified considerably by dropping both the second terms on the left- hand side and taking out p, wu, and c, from the integrals. Furthermore, both u. and T. are equal to the constant quantities U and 7’,, in the free stream. The term giving the heat produced through internal friction in Eq. 4-2 can be neglected because it is comparatively small at the low speed considered here. U Too — —>— Fig. G,4a. Boundary layer along a porous transpiration-cooled wall. In order to solve these two simplified momentum and energy equa- tions for the boundary layer, polynomials of the fourth degree as approxi- mations of the velocity and temperature profiles are assumed. The coef- ficients of the fourth degree polynomials are calculated from the boundary conditions used by Pohlhausen, except that at the wall the velocity per- pendicular to the main flow is equal to the injection velocity v, instead of zero value. The results are, for velocity profile, o-[()-2(2) +) and for temperature profile, F-% -[@)--(+0)] G - COOLING BY PROTECTIVE FLUID FILMS where A = Vw6u/v and A, = Vwbn/v; 6, and 6, are thicknesses of the hydro- dynamic and the thermal boundary layers, respectively. With the aid of Eq. 4-3 and 4-4, the analytical solutions of the momen- tum and energy equations for boundary layers are obtained. The results can be expressed as follows: 2 peste) 1OnO Gye aypP es Tet te a oNeeoe nese we tan-! 2 +/3 ( -- 5) | (4-5) for the momentum equation and 5 Nan ii ia) (4-6) for energy equation where é = (Ux/v)(v./U)?, ¢ = 6,/6, and Pr is the Prandtl number. The complete expression for Eq. 4-6 is much too com- plicated to be presented here and [7] should be consulted. For a Prandtl number equal to unity Eq. 4-5 and 4-6 are identical. It is noted that the changes of d, for different Prandtl numbers are not appreciable within the range of £ which is of interest in the investigation of transpiration cooling. The results calculated by Eq. 4-5 and 4-6 indicate that the relation between é and , is linear except in the region where £ is less than unity. In other words, a linear relationship is approached between the bound- ary layer thickness (6, or 6,) and the length in the direction of flow x when 6, reaches a certain value depending on the magnitude of v,. The Blasius solution reveals, for an impermeable flat plate, that the boundary layer thickness is directly proportional to the square root of the length in the direction of flow as well as the viscosity of the fluid. In the case of flow over a flat plate with injection when 6, reaches a certain thickness, the effect of viscosity on the boundary layer becomes negligibly small, and the formation of the boundary layer is mainly due to the additional mass fluid injected into the main fluid. Hence the ratio of the boundary layer thickness to the length in the direction of flow is linearly propor- tional to the ratio of injected velocity to the main stream velocity. The instability of the laminar boundary layer may be interpreted from the inflection points occurring in the velocity and temperature profiles. It is found that the larger the value of v./U the farther the inflection points move outward from the plate. A discussion on stability considerations of the laminar boundary layer is given later on in this article. The temperature field in the laminar boundary layer described in the previous paragraphs may be used to determine the amount of coolant required to cool the wall to a predesignated temperature. From the bal- ance between the heat flow to the wall from the hot gas and the heat S18 eek ( 440 ) G,4 - TRANSPIRATION-COOLED BOUNDARY LAYER absorbed by the coolant, one gets k eal ee (re ey (4-7) Since the fluid layers adjacent to the wall are at rest, the heat flow from the hot gas to the wall must be transferred by conduction through these layers. This is represented by the term on the left-hand side of Eq. 4-7. The term on the right-hand side of Eq. 4-7 is the heat absorbed by the coolant. In the case where variation of wall temperature in the direction of flow is considered, an additional term representing the heat flow in the metal must be added in Eq. 4-7. Hence the thermal conductivity of the metal enters into the energy balance equation. At the suggestion of the author, Ness [8] made a theoretical investi- gation of the temperature distribution along a semi-infinite porous flat plate under the condition of uniform coolant injection. A heat-balance differential equation of the second order, including a term containing the physical parameters of the plate, is used in conjunction with the solution of the equations of continuity, momentum, and energy. The temperature distribution along the plate is obtained for the respective cases of thermal conductivity not equal to, and equal to, zero. Results show that the inclusion of the thermal conductivity term in the heat-balance equation eliminates the infinite temperature gradient at the leading edge. The total heat flow to the plate can be obtained by integrating Eq. 4-7 over the entire plate of length /. The relation between the wall temper- ature and the amount of coolant needed is then determined as follows: Pn = Thy 05 (on =) | ee Or Ce i, = Wy a DG IN a A GREE) oe ee Ar] —1 +35 VB tn 2-V3(x+3)1 | (4-8) where ; can be determined from the curve AX vs. é for a corresponding value of &, i.e. (Ul/v)(v./U)?. For a predesignated wall temperature and given Prandtl number and Reynolds number, the amount of coolant re- quired per unit time can be determined from Eq. 4-8, provided that the temperature of the hot fluid and of the coolant are known. The expression in Eq. 4-8 is derived for Pr = 1 and [7] should be consulted for Pr + 1. In Fig. G,4b the ratio of the temperature difference, (7... — Ts)/ (T.~ — To) is plotted against the coolant velocity ratio v,/U for Pr = 1. The influence of the Reynolds number on the coolant discharge and the wall temperature is rather appreciable. As the Reynolds number increases the coolant discharge decreases for a given wall temperature. The oppo- site is found to be the case for the Prandtl number. The above phenomena can be explained by the fact that heat transfer from a hot gas to the wall 0 ( 441 ) G :- COOLING BY PROTECTIVE FLUID FILMS is inversely proportional to the thickness of the thermal boundary layer adjacent to the wall. Since the thickness of the thermal boundary layer is directly proportional to the Reynolds number of length in the direction of flow, the first phenomenon is clear. As mentioned in the previous article the increase of Prandtl number does not increase appreciably the thick- ness of the thermal boundary layer. On the other hand, the increase in fluid viscosity due to the increases in the Prandtl number may give a sufficiently low Reynolds number of boundary layer thickness to increase the final heat transfer to the wall. 2.0 1.6 ie iillee” S) x OS 0.8 Réez=3 11102 Compressible 0.4 — — — Incompressible Ths Ty To Fig. G,4b. Temperature ratio vs. mass flow ratio. (From [7,9].) The film heat transfer coefficient h; between the hot gas and the wall can be determined by Eq. 4-8. The ratio of film heat transfer coefficient h, with transpiration cooling to the film heat transfer coefficient h with impermeable plate under the same conditions of flow over the plate as a function of v,/U ratio is given in Fig. G,4c. It is interesting to see that for an injected coolant velocity equal to 1 per cent of the hot gas velocity, the heat transfer to the wall can be reduced to about one-fifth of the value without transpiration cooling. Compressible boundary layer on a porous flat plate. In order to under- stand the phenomena of heat transfer in transpiration cooling in which large temperature differences occur across the boundary layer [9], the physical properties of the fluid must be taken into consideration. The ( 442 ) G,4 - TRANSPIRATION-COOLED BOUNDARY LAYER assumptions made in the present article are: (1) the inverse proportion between the mass density and the temperature inside the boundary layer is used, and the viscosity is assumed to be proportional to both the square root and three-fourths power of the temperature; (2) the flow is assumed to be laminar, and the fluid flowing along the wall and the coolant flow- ing through the pores are assumed homogeneous; (3) the Prandtl number is assumed to be equal to unity; and (4) the wall temperature in the direction of flow is constant. 1.0 0.8 0.6 J|> 0.4 0.002 0. tO 0.006 0.008 0.010 Q W Fig. G,4c. Ratio of heat transfer coefficient with and without transpiration-cooled plate vs. mass flow ratio. 0.2 Crocco [10] has shown that, for a Prandtl number equal to unity, the equation of motion in the boundary layer of a flat plate in steady com- pressible flow and the corresponding energy equation can be satisfied by equating the temperature 7 to a certain parabolic function of the ve- locity u only. This relation between T and w is Les a ee Ua iancaae sen) ale gl ron (G- )otterst-a) 4 With the aid of Eq. 4-9 the variation of mass density and viscosity inside the boundary layer can then be expressed as a function of the velocity u only. As in the case of incompressible flow a polynomial of the fourth degree as an approximation to the velocity profile is assumed and its ( 443 ) G - COOLING BY PROTECTIVE FLUID FILMS coefficients are calculated from the boundary conditions as described pre- viously. Eq. 4-1 can then be solved upon substitution of the velocity profile and the expressions for mass density and viscosity. The results can be expressed as follows: _ const pw A ys b= aaye pe (. Tr. w) (4-10) 2 Gile) Sgcemmelcne) depending on p ~ W/T orp ~ T?. The difference between Eq. 4-5 and 4-10 is that the latter contains the Mach number and the ratio of the wall temperature to the hot gas tem- perature which do not appear in the former equation. This is due to the fact that the variation of the mass density and viscosity as functions of temperature are taken into account in the solution of the momentum equation in the present case. The growth in boundary layer thickness with the increase of Mach number and the ratio of hot gas temperature to the wall temperature can be easily interpreted from Eq. 4-10. Since the effect of compressibility is to increase the heat transfer through the wall, and since the amount of heat produced in the boundary layer in- creases with speed, the effects of both the increase of Mach number and the ratio of hot gas temperature to the wall temperature to the boundary layer thickness are the same. The results as calculated from Eq. 4-10 also reveal that the temperature gradient at the wall increases as the Mach number increases, and decreases as vw~/U increases. This behavior indi- cates that the heat transfer through the wall increases as the compressi- bility of the flow becomes more pronounced and decreases as the injec- tion of coolant increases. From the balance between the total heat flow to the wall from the hot gas and the total heat absorbed by the coolant, one obtains l il ib (=) ip Ci ay (4-11) 0 OY Jw The temperature gradient at the wall can be obtained from Eq. 4-9 and 4-10. The relation between the wall temperature and the rate of coolant injection is then determined by the following expression: plate ls Spy & a 1) Hn Q i( Te, wr) (4-12) where feed lr 4h Lew VS ee where Q _ pats W p.U The influence of variation in the physical properties of the gas across the boundary layer to the transpiration cooling can be seen in Fig. G,4b. ( 444 ) G,4 - TRANSPIRATION-COOLED BOUNDARY LAYER It indicates that for zero Mach number, unless the temperature difference across the boundary layer is large, say 7',,/T'~ > 3, the increase of the rate of the coolant injection in order to maintain a predesignated wall temperature is about 10 per cent over the case in which constant physical properties of the gas are assumed. On the other hand, an appreciable in- crease of the rate of the coolant injection for maintaining a predesignated wall temperature is found between the case of I = 2 and M = 0. This leads to the conclusion that for a flow of subsonic speed and in which the temperature difference of the hot gas relative to the wall is not large, the physical properties of the gas may be regarded as constant in the appli- cation of transpiration cooling. As already mentioned, the relation between the rate of the coolant injection and the wall temperature is based on the average value in a flow of gas over a plate with a given Reynolds number. It must be borne in mind that the boundary layer thickness increases almost linearly with the length in the direction of flow, and the heat transfer to the wall decreases proportionally from the leading edge of the plate to downstream. This results in a longitudinal temperature gradient along the transpiration- cooled wall and, naturally, heat flow through the thermally conductive plate occurs. For this reason, as far as the laminar flow is concerned, the efficient method in transpiration cooling is to vary the rate of the coolant injection along the plate in accordance with the local heat transfer at the wall [17]. Compressible boundary layer on a porous wall with a pressure gradient. Flows with pressure gradients (favorable and/or adverse) are of con- siderable practical importance in connection with the transpiration cool- ing of turbine blades or airfoil surfaces in high speed flow (aerodynamic heating problem) [12]. The flow along a gas turbine blade is expected to be laminar at least in the region around the nose of the blade, while at supersonic speeds it may be possible to maintain a laminar boundary layer along aircraft and missile surfaces. Since the presence of an adverse pressure gradient has an effect similar to that of a normal injection mass flow, i.e. they both tend to increase the boundary layer thickness, it is the purpose of this article to determine the net effect of these parameters on the flow over a transpiration-cooled surface. The present investigation is based on the assumption that the coefficient of viscosity is linearly pro- portional to the absolute temperature and the Prandtl number is unity. In order to solve the momentum equation (Eq. 4-1) and the energy equation (Eq. 4-2) for the hydrodynamic and thermal boundary layer thickness 6, and 6;, respectively, it is convenient to replace the normal distance y by the variable ¢t, defined as follows: y = i ‘él dt (4-13) G - COOLING BY PROTECTIVE FLUID FILMS Eq. 4-2 can be further simplified if the stagnation enthalpy h® = (u/2)? + Cpl’ is used instead of the absolute temperature 7. If, with the above assumptions, both the velocity and stagnation enthalpy profiles are assumed as fourth degree polynomials in ¢, satisfying appropriate con- ditions at the outer edge of the boundary layer and at the wall, and these profiles are substituted into the modified equations (Eq. 4-1 and 4-2), then two ordinary differential equations in the nondimensional hydro- dynamic and thermal thicknesses are obtained. On the basis of a uniform wall temperature, general approximate solutions of these differential equa- tions for the boundary layer thicknesses are derived. These solutions are valid for a prescribed external flow as given by u./U and M,,, and for a given wall temperature and mass flow injection distribution. By this means the boundary layer characteristics can then be calculated with comparative ease. The following general conclusions are drawn from the above analysis: (1) in the region of an adverse pressure gradient, the cool- ing of the wall tends to delay the separation of the flow, (2) for a fixed wall temperature, normal mass flow injection tends to promote separation, although in the absence of an adverse pressure gradient, injection alone cannot cause separation; and (3) the effect of the wall temperature on the boundary layer characteristics depends on whether the axial pressure gradient is adverse or favorable. The skin friction tends to be diminished by a decrease in the wall temperature (for fixed injection) in a favorable pressure gradient, but tends to be increased in an adverse pressure gradi- ent. Similar conclusions hold for the Nusselt number but it is less sensitive to change in the wall temperature than the skin friction. In the preceding analysis, the wall temperature and the injection mass flow have been treated as independent quantities. Actually, however, a consideration of the heat balance at the wall indicates that the wall tem- perature and the amount of injection mass flow are related to each other through the temperature of the coolant. Thus, by considering such a heat balance, a new parameter involving the coolant temperature is intro- duced. The details of this analysis can be found in [13,14]. Exact SoLuTion or Heat TRANSFER IN THE LAMINAR BOUNDARY Layer. In the preceding articles, approximate methods for the solution of heat transfer in laminar boundary layers on a transpiration-cooled wall have been discussed. The solutions obtained by approximate methods have explained most of the physical phenomena in the transpiration- cooling problems, even though they satisfy the differential equations of boundary layer flow only on the average. The present article considers some exact solutions of the equations of boundary layer flow. The essential restrictions of the exact solutions are that they are based on the case in which the velocity outside of the bound- ary layer is proportional to a power of the distance along the main flow (wedge flow) and that the velocity of fluid injection is proportional to ( 446 ) G,4 - TRANSPIRATION-COOLED BOUNDARY LAYER the reciprocal of the square root of the distance from the leading edge of a flat plate. However, the velocity and temperature profiles obtained by this method are quite accurate. They can be used in a laminar bound- ary layer stability analysis. The equations of the laminar boundary layer for steady state flow of a viscous compressible fluid with heat transfer may be obtained from Vol. IV as Momentum equation: ou OUTS, du\ op Oo ALE a a (. ) ax ee) Continuity equation: O(pu) , A(pv) _ Bot ay =e (4-15) Energy equation: aT OR sO OW du\” dp co(ouae to Sr) = ay (BS) +S) tae 0 The boundary conditions are: when y = 0, w= OS 0] @)s 2 Sie ) and when y = ~, a aT eu U iad Ola — mee: ai U = Ue} 0 Incompressible boundary layer with constant fluid properties. In the case of incompressible laminar boundary layer flow with constant fluid properties (9 = constant and uw = constant) [15], the last two terms in Eq. 4-16 can be neglected. When the velocity outside the boundary layer is assumed to be proportional to a power of distance along the wall from the stagnation point (ue = cz”), the transformation methods of Schlichting [1/6] and of Falkner and Skan [1/7] can be applied. With the following changes in variables: 0 eZ (4-18) sas V vil a ne T where u = dy/dy, v = —dwW/dx and m is the Euler number, the momen- tum equation (Eq. 4-14) and the energy equation (Eq. 4-16) are trans- ( 447 ) G - COOLING BY PROTECTIVE FLUID FILMS formed into the following two ordinary differential equations with f and 6 as functions of 7 only: df aj en apy een F dn? lop 1+m (i) oa ne and d?@ dé Tip see ein Ea (4-20) With the boundary conditions given in Eq. 4-17 the above transformation is based on the assumption that the temperature of the wall is constant and the normal injection velocity at the wall is given by Sle Wl Oe eee: -,J $ fe = Narr const (4-21) and vy ~ 1/+/z for a constant Ue. The differential equation (Eq. 4-19) can be solved only numerically and Eq. 4-20 can readily be integrated if the function f is known from the solution of Eq. 4-19. Numerical results for a laminar boundary layer flow on a flat plate (m = 0) and for flow near a stagnation point (m = 1) were given in [/6]. With the aid of these solutions the heat transfer phenomena in the above two cases for Prandtl number equal to unity were calculated in [/5]. The results can be briefly summarized as follows: 1. In the case of flow on a flat plate, the heat transfer coefficient h de- creases rapidly for an increase in (v./ue) ~/ Re;. It is reduced to one tenth of the value without transpiration cooling for an average ratio of Va/Ue = | per cent (Re, — 1104). 2. For flow near the stagnation point (m = 1) it is found that the point of inflection does not appear in the velocity profiles. This is under- standable because the flow in the neighborhood of the stagnation point is under the influence of a favorable pressure gradient and therefore becomes more stable than the case on the flat plate. The heat transfer coefficient h diminishes practically to zero when the coolant injection parameter v~/+/ve reaches 3.2. 3. It appears that the required coolant injection to maintain a given wall temperature is much less in the present case than the result obtained in the approximate solution. This result is expected because in the approximate solution a uniform coolant injection was assumed, whereas in the present exact solution the coolant injection is proportional to the reciprocal of the square root of the distance from the leading edge of a flat plate. Compressible boundary layer with variable fluid properties. The simul- taneous effects of pressure gradient in the main stream flow over a porous ( 448 ) G,4 - TRANSPIRATION-COOLED BOUNDARY LAYER wall and property changes in the fluid due to large temperature differ- ences between the wall and the free stream [1/8] will be treated in the present article. In order to simplify the analysis, the following assump- tions are made: (1) The Mach number WM is small, (2) the Euler num- ber m is constant, (3) the wall temperature 7’, is constant, and (4) the fluid property variation is expressible as some power of the absolute temperature. DIP I UC CIN ie I (4-22) On the basis of assumption 1, the last two terms in Eq. 4-16 can be neg- lected and the quantities p, and JT. can be treated as constants. It was mentioned in the previous article that, for a wedge-type flow, the transformation methods of Pohlhausen [19] and of Falkner and Skan [20] can be applied. With the following changes in variables: ane Pwlle ) wl yo ae Pw V bw Pw lek the momentum equation (Eq. 4-14) and the energy equation (Eq. 4-16) can be transformed into two ordinary differential equations with f and 0 as functions of 7 only. With the boundary conditions given in Eq. 4-17 the above two equations can be solved numerically for any prescribed Kuler number m, Prandtl number Pr, temperature ratio (stream temper- ature divided by wall temperature), and coolant flow parameter f,. From the boundary condition at the wall, the following expression gives for a dimensionless measure of the coolant flow in terms of the coolant velocity: i 1+ mu jie = Rex (4-24) where f, is considered to be a constant which yields a constant wall tem- perature if the conduction along the wall and the radiation are neglected. The expressions for the local skin friction coefficient at the wall and the local heat transfer coefficient (Nusselt number) to the wall are obtained, respectively, as follows: Bye ey (4-25) ve = en (4-26) ( 449 ) and G : COOLING BY PROTECTIVE FLUID FILMS A balance between the heat flow to the wall from the hot fluid and the heat absorbed by the coolant at the wall yields TH Re 1+m a fr i Se kD Pre (6 /an)« (4-27) For a predesignated wall temperature and given Prandtl number, Euler number, and Reynolds number, the amount of coolant required can be determined from Eq. 4-27 provided that the temperatures of the hot fluid and of the coolant are known. In Fig. G,4d the ratio of the temperature difference, (JT. — Tw)/(Tw~ — To), is plotted against the coolant mass flow ratio (Q/W) ~/ Re. for Pr = 0.7. It is seen that the influence of the Fig. G,4d. Temperature ratio vs. mass flow ratio for favorable, zero, and adverse pressure gradients. pressure gradient on the coolant discharge and the wall temperature is appreciable. For a favorable pressure gradient (m = 1) the heat trans- fer to the wall is increased almost twice the value of the flat plate case (m = 0). On the other hand, the heat transfer to the wall is diminished by 25 per cent of the flat plate value with a small adverse pressure gradi- ent. The wall skin friction behaves in a similar manner but it will be more sensitive to change with the pressure gradient than the heat transfer to the wall. Furthermore, the displacement and momentum thicknesses are reduced to approximately one third of their flat plate values in the favor- able pressure gradient, but the thermal boundary layer thickness is changed only slightly in this case. Calculations of 58 velocity and temperature distributions were made for air which include the simultaneous effects of pressure gradients in the ( 450 ) G,4 - TRANSPIRATION-COOLED BOUNDARY LAYER main stream, the flow through a porous wall, and the large temperature variations through the boundary layer (Pr = 0.7, w = 0.7, e = 0.85, and a = 0.19 were used in Eq. 4-22). A complete tabulation of these calcu- lations can be found in [2/]. An extension of the above wedge-type solu- tion to the heat transfer in flow around cylinders of arbitrary cross section for transpiration-cooled surfaces was made in [22]. Wall temperature ratio Ty/T. Fig. G,4e. Effect of Mach number on surface temperature. (From [23].) An exact solution of the heat transfer of the compressible laminar boundary on a transpiration-cooled flat plate is made in [23]. The effect of Mach number on surface temperature for different rates of coolant injection are shown in Fig. G,4e. It is interesting to note that, for large rates of coolant injection, the wall temperature is less dependent on Mach number than for small rates of injection. STABILITY CONSIDERATIONS OF THE LAMINAR BoUNDARY LAYER WITH Coouant IngEcTION. The stability theory for the laminar boundary ( 451 ) G+: COOLING BY PROTECTIVE FLUID FILMS layer on an impermeable surface was developed by Tollmien, Schlichting, and Lin for an incompressible fluid. Lees and Lin [24] extended the theory to include the effect of compressibility. By application of the stability theory it is possible to determine, from the velocity distribution in the poundary layer, the local Reynolds number at which a flow with such a velocity distribution becomes unstable. Transition to a turbulent bound- ary layer may be expected to occur somewhere downstream of the point of stability. The works of the above investigators indicate that an adverse pressure gradient in the flow direction destabilizes the boundary layer and a favorable pressure gradient increases the stability. In the analyses of the stability of compressible laminar boundary layers [24,25], the results indicate that stability is greatly influenced by the heat transfer from the wall to the gas. In accordance with Eq. 4-14 at the wall condition of the flat plate it follows that the curvature of the velocity profile at the wall is proportional to a negative product of the temperature gradient and the velocity gradient at the wall. Then if the wall is hotter than the free stream fluid the temperature gradient at the wall will be negative, and in turn, the curvature of the velocity profile at the wall will be positive. It follows that in the boundary layer on a heated wall the velocity profile has a point of inflection which is a neces- sary and sufficient condition for the existence of amplified disturbance, hence, its instability. On the other hand, in the boundary layer on a cooled wall, the curvature of the velocity profile at the wall is negative and consequently the limit of complete stability increases. It is known that the effect of fluid injection is to destabilize the boundary layer in a way similar to the effect of an adverse pressure gradient. It can be seen that fluid injection (1) increases the boundary layer thickness (a growing boundary layer is more prone to become tur- bulent) and (2) fluid injection creates a velocity profile which is less stable than one without injection. Since cooling of the wall and fluid injection at the wall have opposite effects on the stability of the laminar boundary layer with coolant injection, it is desirable to determine the simultaneous effects on transition. Based on the improved viscous solutions of the stability equations [26], calculations of the stability of the compressible laminar boundary layer with coolant injection are made in [23] for the reduction of aero- dynamic heating in high speed flight. The results apply at moderate supersonic speeds and indicate the complete stability limits for two- dimensional disturbances. In Fig. G,4f, the complete stability curves for several rates of coolant injection are shown. For a given rate of coolant injection, each of the curves depicts the region of complete stability. For any given Mach number, the wall temperature must be below the curve in order to attain a completely stable laminar boundary layer. SUBLAYER THEORY IN TURBULENT FLow. The heat transfer in the ( 452 ) G,4 - TRANSPIRATION-COOLED BOUNDARY LAYER turbulent boundary layer is of much greater importance than in the lami- nar because it is more often encountered in engineering problems, yet little progress has been made in the development of methods for the calcu- lation of turbulent boundary layer even without transpiration cooling. The difficulty in this problem is that no precise knowledge of the surface shear and the shearing-stress distribution across the turbulent boundary Wall temperature ratio T,/T.. Free stream Mach number M., Fig. G,4f. Limiting wall temperature required for complete stabilization of boundary layer. (From [23].) layer (essential quantities in connection with the solution of boundary layer equations) is available. It has long been recognized that the momentum exchange in turbu- lent flow is impossible when the fluid stream approaches the vicinity of a solid wall where a thin laminar sublayer exists and that the transfer of shearing stress must depend on viscous action. Outside of this thin sub- layer only the turbulent exchange mechanism is effective while the trans- fer by molecular action may be neglected. In a like manner, the rate of heat transfer between a fluid stream in turbulent flow and a smooth wall ( 453 ) G - COOLING BY PROTECTIVE FLUID FILMS is largely controlled by the relatively high resistance of the laminar sub- layer next to the wall. The above concept was used by Prandtl in the investigation of turbu- lent flow in a pipe. Rannie’s extension of the concept [27] to heat transfer in transpiration cooling is discussed here. The assumptions made in this investigation are: (1) steady flow is assumed and all derivatives with respect to length in the direction of flow are zero, (2) the physical properties of the fluid remain constant across the sublayer, (3) the gas flowing along the wall and the coolant flowing through the pores are assumed homogeneous, and (4) the wall temper- ature in the direction of flow is constant. The velocity distribution in the laminar sublayer can be determined easily by integrating the Prandtl boundary layer equation with the aid of a continuity equation which may be expressed in the following form: Pwow U ae | a (4-28) lam Vv Ubtan PwUw E —1 where the boundary conditions at the wall (subscript w) and at the boundary of the laminar sublayer and turbulent layer (subscript lam) are applied. Eq. 4-28 indicates that the velocity distribution becomes linear when v. = 0 and the wall shear decreases as the rate of injection increases. In a like manner the temperature profile in the laminar sublayer can be derived from the energy equation which can be written in the form Prpwiw y T—T, nS —1 (4-29) Ti a ie Etbwow Slam é B Kq. 4-29 reduces to Eq. 4-28 for Pr = 1. The motion of molecules in laminar flow and the motion of eddies in turbulent flow by its transport of momentum are the causes of skin friction; the same motions also transport heat. Therefore a relationship should exist between skin friction and heat transport. Reynolds used this approach to obtain the following relation between momentum transfer and heat transfer across a turbulent stream: Qiam Tiam ACL: eat ae) a (u, - Utam) G a) where giam and 7,, are the heat transfer and shearing stress at the bound- ary between the laminar sublayer and turbulent layer and 7’, and wu, are the temperature and velocity at the center of the pipe. These can be evaluated from Eq. 4-29 and 4-28, respectively. Upon the substitution for quantities Gam) Tiamy aNd 7), in Eq. 4-30, the relation between the wall ( 454 ) G,4 - TRANSPIRATION-COOLED BOUNDARY LAYER temperature and the rate of coolant injection is determined as v ee aa Lb ee eG é TI) Coy ( Uc = ewiw Slam a) 1 + — —-1l})Q-e A Ulam Since experimental information on transpiration-cooled turbulent boundary layers has not been achieved, the variation of laminar-sublayer thickness with the velocity of injection remains unknown. It is assumed that the flow in the turbulent case is not affected by the velocity of injection, and hence that the shearing stress and the velocity uw, at the edge of the core are the same as for flow in a smooth pipe. The thick- ness of the laminar sublayer has been measured in smooth pipe and found to satisfy the relation a = y* (4-32) where u2 = 7~/p and y* = 5.6 is taken by Prandtl after examination of the velocity profile measured close to a wall. On the basis of Eq. 4-32 the following expressions are obtained: = apf 2 Omen vy Ne Tw p wherert. = c;pu?/2. For the Reynolds number range 5000 < Re < 200,000 the friction coefficient c; for smooth pipes satisfies the empirical relation Cf = 0. 046(Re)-5. The relation between the wall temperature ane the rate of coolant flow is calculated from Eq. 4-31 and shown in Fig. G,4b. The result reveals that for a designated wall temperature the rate of coolant re- quired for transpiration-cooled turbulent flow is almost twice as much as in the case of laminar flow. For an injected velocity equal to 1 per cent of the hot fluid velocity, the heat transfer to the wall is reduced to 70 per cent of the value without transpiration cooling. The comparison of this result with the result obtained under the same conditions in lami- nar flow is shown in Fig. G,4c. APPROXIMATE SOLUTION OF Hat TRANSFER IN TURBULENT BounD- ARY LAYER ON A Fuat Puate. The treatment above may be extended to the case of the effect of coolant injection on the behavior of a com- pressible turbulent boundary layer [28,29]. In order to simplify the analy- sis, the following assumptions are made: (1) the coolant fluid is the same as the boundary layer fluid, (2) the wall temperature in the direction of flow is constant, and (3) the Prandtl number is equal to unity. (4-33) = * Ujam aD y ( 455 ) G - COOLING BY PROTECTIVE FLUID FILMS The basic equations, which represent the principles of conservation of momentum, conservation of mass, and conservation of energy, for the compressible turbulent boundary over a flat plate, can be expressed as follows [Sec. B]: The momentum equation in the x direction is —oi ,— da 0 f do ——>\_ oF } re ee pra) = 2 (4-34) The continuity equation is A(pu) , A(pv) _ oe ae at 0 (4-35) The energy equation is —d(¢pT) ,—0@7) 9@f,9F — aa\" pu ty pe ay -2(. cpT’) + (u + 6) (3) (4-36) The quantities with bars represent time-average quantities, while primed quantities represent instantaneous values of fluctuating quantities. The specific heat is assumed to be a constant and the fluid properties of density, viscosity, and thermal conductivity are considered to vary with temperature. The boundary conditions are Atay — 0; Pi es a t—40) = pw; PU = Pwlw 4-37 a3 oS OF ( ) SI | At y = 6; P= pay) pl) = pave For a Prandtl number equal to unity, the relation between 7 and a can be expressed as in Eq. 4-9 in the following manner: (Napoli Tog i @ a TF. + (1 ~ _) U7 oY dp, £(1- =) (4-38) In order to make the solution of Eq. 4-34 possible, at least for practi- cal purposes, certain nonrigorous assumptions are made which are analo- gous to those used in the case of low speed, incompressible turbulent flow along a smooth wall without injection. The first simplification is to assume that the dependency of variable quantities with respect to x is negligible compared to their variations with respect to y in the neighborhood of a permeable wall. Secondly, Prandtl’s mixing-length hypothesis is assumed to apply for the present problem. Then in the turbulent region, Eq. 4-34 can be simplified as follows: du_dr_ad wera 4-39 rey = ay = an ORY Cae) | i G,4 - TRANSPIRATION-COOLED BOUNDARY LAYER The bars representing time-average quantities have been dropped since all the terms in Eq. 4-39 are mean values. The two boundary conditions are: (1) the shearing stress 7 = 7, when y — 0, and (2) the velocity dis- tribution from the solution of Eq. 4-89 must reduce to von Kdérmdan’s logarithmic velocity distribution law for T,/T.,, = 1, M., = 0, andv~ = 0. The equation of state for zero pressure gradient in y direction leads to (ie5 th ae (4-40) Upon the substitution of Eq. 4-40 and Eq. 4-88 into Eq. 4-39, the solu- tion which represents the velocity distribution of a compressible turbu- lent boundary layer with fluid injection at the wall is obtained: Oupwl ie Nee e(Ii—DK) (4-41) fe, TED Or where u/U oe ; Edz ‘ (4-42) a Pwlw ee yy ; ee) poe 49) f ee Cr 2 = / Oh us QT w Cf mie (4-44) and i Le (ah 2. = Mea eh T Vaal 2 - 5 een Wie ee a The empirical constants K and D can only be determined from the proper experimental velocity distribution. The skin friction coefficient can be determined from the momentum integral given in Eq. 4-1 as follows: (aed, 1 pu U “| pave = 22] a, | a (i-#)ae 205 (4-46) Eq. 4-46 can be simplified by substituting the expression for d(y/6,) from Eq. 4-41, after its differentiating with respect to u/U, along with the relation between p, 7’, and wu given in Eq. 4-38 and 4-40. The final ex- pression for local and average skin friction coefficient is In Ie =n =| aa iz) R,| + In iS = alin ‘el In DK = ith Ip (4-47) ( 457 ) G - COOLING BY PROTECTIVE FLUID FILMS Saat ; Sai (448) » [1 to eh [ ata «(3) | Mew = ben ie (4-49) and on : i Ree (4-50) From the assumption that the Prandtl number is equal to unity it can be shown that the simple relation between skin friction and heat transfer is ee. h ae Cr Sr a: (4-51) and C=& (4-52) Once local or average skin friction is determined from Eq. 4-47, the appropriate heat transfer coefficients can be determined from Kq. 4-51 and 4-52. In the case where the Prandtl number is not equal to unity the effect of transpiration cooling on the relation between the coefficients of skin friction and heat transfer may be found in [29]. In [30] the heat transfer and skin friction coefficients have been ob- tained from measured data for turbulent boundary layers at very low Mach numbers. In these tests both suction and injection were applied at the plane boundary of the stream. Skin friction coefficients were deter- mined from plots of the momentum thickness against longitudinal station, where the momentum thickness was determined by obtaining the velocity profiles at various stations. The heat transfer was determined by direct measurement. The Reynolds number range of this data was about 9 X 104 to 3.3 X 10° and the temperature difference between the wall and the free stream was about 30°F. In order to make a comparison between the theoretically derived local skin friction and heat transfer coefficients and the appropriate data in [30], the empirical constants K and D in Eq. 4-47 are determined by letting Cy = cys, Tv/T. = 1, M.. = 0, and v», = 0 and comparing the resulting expression with the following von Karman incompressible local skin fric- tion law 4.15 log (c;Rez) + 1.7 = on (4-53) from which K = 0.393 and D = 6.53. ( 458 ) G,4 - TRANSPIRATION-COOLED BOUNDARY LAYER COLA MO Sin kD) MAING. MDIONN 2.467218 ava SvGlnnae 2pww! P,,UCt, 1, where c and k are the constants of integration to be determined. Eq. 5-14 is an ordinary nonlinear differential equation of the third order which resulted from the Navier-Stokes equations and the continuity equation by the similarity transformation. With the aid of the four given boundary conditions an exact solution can be obtained and the constant of integration c determined. It can be seen that the limiting form of Eq. 5-14, by letting v, ap- proach zero, is the equation describing a flow through a circular pipe with impermeable walls. The solution of this equation which satisfies all the four boundary conditions given in Eq. 5-5 and 5-7 is the well-known Poiseuille law for pipe flow. If small values of \ are treated as a pertur- bation parameter, a solution of Eq. 5-14 can be obtained, which is dis- cussed later. On the other hand, if large values of \ are treated as a perturbation parameter the third order differential equation (Eq. 5-15) is reduced to a second order one. The solution of Eq. 5-15 can also be obtained in the same manner since all four boundary conditions given in Eq. 5-5 and 5-7 can be satisfied. ( 462 ) G,5 - TRANSPIRATION-COOLED PIPE FLOW The solution of Eq. 5-14 can be expressed for small values of \ (A S 1) by a power series developed near \ = 0 as follows: i= tins Wj ae ap 8 oe ae Nai (5-16) and € = Co + Aer + A2%co + --- +A"en (5-17) where f,’s and c,’s are taken to be independent of \. By substituting Eq. 5-16 and 5-17 into Eq. 5-14 and equating coefficients of like power of X, one obtains the following set of equations: Zfo’ + fo’ = Co (5-18) een ae, JO} o) = Ca (5-19) ofa! + fe’ — ffi + fof + fof’ = e2 (5-20) The boundary conditions to be satisfied by f,’s are iO) Sti Ws — V2 fi (z) = 0 for all n (5-21) fol) = 2 The second order perturbation solution of Eq. 5-14 obtained by solv- ing Kq. 5-18, 5-19, and 5-20 is given as follows: X ee ae ge ie) = 2-52 + — ag tag Fath as 83 LO ew II ag 1 ‘) : mse (fhe = yt ee Ta ae Teco) C= 82 he (5-23) It is seen from the foregoing equations that the second order pertur- bation solution is sufficiently accurate even for \ = 1. The velocity com- ponents in the axial and radial directions are obtained by substituting Eq. 5-22 and 5-10 into Eq. 5-9 as follows: a]; ee ee 18 5400 + Zupp (166 — 7602 + 8252 — 3002" + 7524 — 624 | (5-24) moo Bevel set x (- Az + 922 — 623 + 24) oe aes (1662 — 38022 + 2752* — 7524 + 1525 — 2 (5-25) ( 463 ) G - COOLING BY PROTECTIVE FLUID FILMS The pressure drop in the flow direction can be obtained upon the substitution of Eq. 5-22 into i 5-11 and 5-12, i.e. 1 oe ees — aes iS aE Fela Soe 70) ReR|R ee The coefficient of skin friction at the wall can also be obtained from Kq. 5-24, and can be written Oem 1 ha WIE oF ea Re ASE ee Pas all Cae l 18 ' 5400 In a like manner, the solution of Eq. 5-15 can be expressed for large values of \ (A > 1) by a power series developed near 1/A = 0. The com- plete treatment of this part of the work is given in [3/]. LOR Fig. G,5b. Velocity profiles vs. length in radial direction for various \(Re = 103, x/R = 10). (From [31].) The velocity distributions in the main flow direction at an arbitrary cross section of the pipe for \ = +1 and A = 10 are shown in Fig. G,5b. It was noted that when \ = 0, the profile becomes Poiseuille’s paraboloid, and for \ > 0 (fluid being injected through the wall) the axial velocity increases and the velocity gradient at the wall increases. For \ < 0 (fluid being withdrawn through the wall) both the axial velocity and the ve- locity gradient at the wall decrease as compared with Poiseuille’s case. The above phenomenon follows the law of conservation of matter. In the present case the radial velocity, which vanishes in Poiseuille’s case, has a finite magnitude except at the center of the pipe where it vanishes. ( 464 ) G,5 - TRANSPIRATION-COOLED PIPE FLOW One of the essential parameters in the present investigation is the skin friction at the wall. In Poiseuille’s flow the skin friction coefficient at the wall c; has a constant value of 4/ Re. The wall friction coefficient as calcu- lated from this analysis indicates that the effect of injection in a pipe flow is to increase the wall friction coefficient. In a boundary layer flow on a “O 0.002 0.004 0.006 0.008 Vw/ Uy Fig. G,5c. Variation of local wall frictional coefficient with fluid injection. (From [3/].) porous flat plate the effect of fluid injection at the wall is to increase the thickness of the boundary layer and to decrease the velocity gradient at the wall. Hence the wall friction decreases in this case. On the other hand, in a pipe flow the effect of fluid injection at the wall is to accelerate the main stream velocity. Hence the velocity gradient at the wall, which determines the wall friction, increases. For a fluid injection ratio v,/u1 = ( 465 ) G - COOLING BY PROTECTIVE FLUID FILMS 0.01 the wall friction coefficient increases by 85 per cent over the Poiseuille flow case. A comparison of the variation of local wall friction coefficient with fluid injection between the case of flow in a porous-wall pipe and on a flat plate was shown in Fig. G,5c. Temperature distribution and heat transfer. For an incompressible fluid, it can be shown that the terms in Eq. 5-4 due to the pressure gradient and the dissipation ¢ can be neglected, and furthermore it is assumed that the molecular heat conduction in the axial direction may be neglected in comparison with that in the radial direction. Hence Eq. 5-4 can be simplified in the following nondimensional form: v 06 u 00 1 1 oO 06 29 ue 1 ([22 (0%) re) where £ = 2/R,n = Vz =71/R, and 0 = (T — T,)/(T1 — Tx). Upon substitution of the velocity components from Eq. 5-24 and 5-25 into Eq. 5-28, one obtains 1 r 00 ri, 1 u\2 d ce) n ON ERE oe” Seay |) 2 18 1 1 a) 00 ——- a | -a—la 5-29 " Pre f'(n) : on ¢ =] Nae The energy equation written in the form of Eq. 5-29 yields a solution in the form of an infinite series Ne Naar = yy A; (1 ay eee :) * Min, ¢3) (5-80) j=1 where the M,(n, c;)’s are the particular solutions of the equation. aM 1 n®\ | dM dn * [5+ 2pa( Salen ae ja a eats x (Cael? a5 299 | M=0 (5-31) and c,’s are the eigenvalues of Eq. 5-31 which correspond to the boundary condition 0,-1 = 0. The series solution of Eq. 5-31 corresponding to the eigenvalue c;, which is free from singularities at 7 = 0 is M 1 ae 1 Cj 2 a 4 fee | a eh oq hi Uhl Ce ( 466 ) G,5 - TRANSPIRATION-COOLED PIPE FLOW Multiplying both sides of Eq. 5-31 by the appropriate factor, Eq. 5-31 can be written as the following Sturm-Liouville equation: d eB [m(n) Mj] + cp(n)M; = 0 (5-33) where m(n) — naa) pn) = m(a) | = 99) + 2 Cnt — Ont + 2n9 | Hence M;(n, c;) and M;,(n, c;) for ¥ 7 are orthogonal functions with re- spect to the weight function p(n); 1.e. i * p(n) MMjdn = 0 (5-34) The coefficients of the series expansion, A,;’s (Eq. 5-30) are determined from the boundary condition (Eq. 5-6) applied to Eq. 5-30, which is ), AMfiln, oi) = 1 (5-35) j=l1 Then multiplying both sides of Eq. 5-35 by p(n)M,(n, c;) and integrating from 0 to 1 1 f, PO) Mila, enddn ig = Sa (5-36) fy p@)M3Q, edn From the differential equation (Eq. 5-33), it can be seen that 1 Tey ee ‘ i p(n) M;(n, ¢;)dn = 3 € ie a“ (5-37) and ; 1 (& ) = 2 ‘ pas 3Prr [| 72-9 ise ec / p(n) Mj (n, cj) dn 2c; € dn we 0c pila (5 38) Thus Ae z (5-39) re i Oc c=c;,7=1 The heat transfer coefficient for the flow in a pipe is usually calcu- lated with the difference between the mean temperature of the fluid and the wall temperature. The mean temperature over the cross section con- sidered, weighted with respect to the axial velocity, may be defined by ( 467 ) G - COOLING BY PROTECTIVE FLUID FILMS the equation R i i ieee pe il, ik durdr Ou = RTS ie an (5-40) The heat transfer coefficient is then defined by q= (Tx — T,) (5-41) where heat flow per unit area at the wall is ¢= =k oe =-4(,-7,) ee (5-42) Nu is hD 2 (00 Ns ik} eo z(¥)__ » ee Ve = 52 {4:[1+ Pa (pte 5) * M4(1) + Aa[1 + 8P— (esl 4Prx Mi(1) +--+} (5-43) The temperature distribution given in Eq. 5-30 may be used to deter- mine the amount of coolant required to cool the wall to a predesignated temperature. From the condition of heat balance at the wall, one obtains aad ih k & a dz = —pv,c,(T, — Tol (5-44) Upon substitution of Eq. 5-30 into the above equation, one obtains the temperature difference ratio as follows: fy me Vee) (2 vy 1 ‘apa rom ~ (2) | ee Le) 4Prxr ce? =H EC) I tte (1 + ae 2) aha} | (5-45) 2 ~ 4Pry In Fig. G,5d the heat transfer coefficient or Nusselt number is plotted against the coolant Reynolds number. It indicates that the Nusselt num- where n=l ( 468 ) G,5 - TRANSPIRATION-COOLED PIPE FLOW ber decreases almost linearly with an increase of the coolant Reynolds number. This is due to the fact that the range of coolant Reynolds num- bers considered in the present investigation is rather small (A S$ 1). As the coolant Reynolds number increases further, the Nusselt number then decreases more gradually. The above phenomenon was also obtained in the case of nonisothermal flow over a plate with coolant injection. MARINE BIOLOGICAL ESET TIE I PI EDIE LE ED I DT IR 0 0.25 0.50 0.75 1.00 Fig. G,5d. Local heat transfer coefficient for various rates of coolant injection ((1/PrRe)(1/D) = 0.075). (From [32].) The ratio of temperature difference (T1 — T.)/(T~ — To) is plotted against the coolant flow ratio v,l/umD in Fig. G,5e. For a predesignated wall temperature the amount of coolant required per unit time can readily be determined provided that the entrance temperature 7; and the coolant temperature 7’) are known. The results obtained from the studies of nonisothermal laminar flow over a plate with coolant injection show that the friction coefficient at the wall decreases with the increase of coolant injection. They also indi- ( 469 ) G - COOLING BY PROTECTIVE FLUID FILMS cate that the heat transfer coefficient decreases with the increase of cool- ant injection. Thus there is a definite direct relationship between the friction coefficient and the heat transfer coefficient. On the other hand, the results obtained here indicate that in a pipe flow the effect of fluid injection at the wall is to accelerate the main stream velocity. Hence the velocity gradient at the wall, which determines the wall friction, increases. This phenomenon indicates that the analogue between the heat transfer and the momentum transfer does not exist in porous-wall cooling of pipe flow. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Ts came Thy Ry ae To Fig. G,5e. Temperature difference ratio for various coolant parameters ((1/PrRe)(1/D) = 0.075). (From [32].) APPROXIMATE SOLUTION OF TURBULENT PipE FLow witH CooLANT InsEecTION AT Wau. In the present treatment an approximate analysis for determining the effect of transpiration cooling on a fully developed turbulent flow in a circular pipe is given [34]. In order to simplify the analysis it is assumed that: (1) the fluid is incompressible, i.e. the fluid properties remain constant, (2) the fluid flowing in the axial direction and the fluid flowing through the porous wall are assumed homogeneous, (3) the fluid flowing through the porous wall is uniform throughout, and (4) the wall temperature is constant. If a curvilinear coordinate system is introduced with the origin at the center of the cross section of a circular pipe where z is taken in the direc- tion of the flow and r in the radial direction and ¢ is the azimuthal angle, then with axial symmetry of flow the Reynolds equations for the time- ( 470 ) G,5 - TRANSPIRATION-COOLED PIPE FLOW average velocities become ov OD. kop. loz Dap ae ee nee O20 where r = —pdu/dr — pu’v’. The continuity equation is GY)» OG). roar aR 0 (5-48) The eddy heat transfer equation excluding the dissipation term is aT oie ade eve Eas pc pu Oe +- pCpV Ep = ip or E ap — pc,rv fl | (5-49) The boundary conditions are p= Oe Uh = Ue 7=0 (5-50) IP = AN 8 & = (5-51) Velocity distribution and skin friction. The first simplification neces- sary in achieving an approximate solution for Eq. 5-46, 5-47, and 5-48 is to assume that v = —v,r/R and that in the region close to the wall is valid. With the above assumptions and with the aid of the continuity equation, Eq. 5-46 and 5-47 are reduced to OG?) BOO ye LOGe) River © pdx ~ p or (22) and dp ap 0 (5-53) According to momentum transfer theory this gives — du\? Wl eS PIE a v l ay (5-54) where 1 = K(R — r). After neglecting the viscous shearing stress and combining Eq. 5-54 and 5-52, one obtains, after integration with the aid of Eq. 5-50 K?(R — r)? a = = (U2 <= O20) (5-55) G - COOLING BY PROTECTIVE FLUID FILMS where u2 = —(R/2p)(dp/dx) has the dimension of the square of a ve- locity. The closed form solution for velocity distribution in a fully de- veloped turbulent pipe flow with fluid injection at the wall is obtained from Eq. 5-55 with boundary conditions given in Eq. 5-50. This is ty ey VA Maat) 2 ae | lace a rm E a 2¢ | AK E year: 2 | (5-56) where ¢ = (r/R)}, u, is the velocity at the center of the pipe, and K is an empirical constant to be discussed later. For a zero injection velocity Eq. 5-56 reduces to the form expressing the velocity distribution in flow through a circular pipe [36, pp. 340-344]. Vw/Uc 0.00837 0.00485 9 0 AI = Fig. G,5f. Effect of fluid injection on the velocity distribution in turbulent pipe flow. (From [36].) From Eq. 5-56 the mean flow velocity wm can be determined by inte- gration over the cross section. It is R wn =o f urdr Shee Wagan R? Jo Hence the empirical constant K can be readily calculated from Eq. 5-57 if the measured velocity distribution and pressure gradient are known. The average value of K is about 0.24 for the case of zero injection. How- ever, the values of K tend to increase with the increase of fluid injection. Additional data from further tests (now being conducted at the Poly- technic Institute of Brooklyn) are needed in order to form a definite relationship between the above two parameters. In Fig. G,5f the effect (5-57) ( 472 ) G,5 - TRANSPIRATION-COOLED PIPE FLOW of fluid injection on the velocity distribution of a fully developed turbu- lent flow in a circular pipe was shown. The theoretical curves were calcu- lated from Eq. 5-56. The experimental data were taken from a porous stainless steel pipe with a 5-inch diameter and a length of 20 inches. The range of Reynolds numbers was from 10° to 3 X 10°. The comparison between the theoretical curve and the experimental data shows close agreement. The momentum integral equation for the turbulent flow through a pipe with fluid injection at the wall is obtained by integrating Eq. 5-46 over the cross section from r = 0 tor = R. This yields ; i Da, a ae a aie E i (<) (i) “(a ine The integral in Eq. 5-58 can be evaluated by substituting the ex- pression for the velocity profile from Eq. 5-56. After a very tedious inte- gration, the resulting expression for the skin friction coefficient becomes 2 A 2 A ee ae! 9 it RYa) Uh in COO // Oe ci (2) ue, d(a/R) E [ Te a Rp 7a MEO) (OB) MO 3.235 Up Uw ee (4 = 5) a gKe wo te} | a) Temperature distribution and heat transfer. In order to achieve an approximate solution for the temperature distribution in a fully devel- oped turbulent flow in a pipe with coolant injection at the wall it is necessary to make some simplifications of Eq. 5-49. It is assumed that v = —vyr/R and that in the region close to the wall holds. The temperature distribution may be written es Al @) +6 iS (5-60) According to the mixture length theory, the eddy heat transfer term du| dT dr| dr v'T’ = —Ki(Rk — r)? (5-61) Following the above hypothesis and assumptions Eq. 5-49 can be simpli- fied, after integration, as follows: k dT r AN nha |= 2s K2(R — 1)? == a | a + vy (5) gE! = || rudr (5-62) The condition at r = 0 gives the zero constant of integration. ( 473 ) G - COOLING BY PROTECTIVE FLUID FILMS Upon substitution of the term of the velocity gradient from Eq. 5-56 into Eq. 5-62 it may be integrated to give T = [e~ jonas [| eJootary h(¢)dt + T,e — fe oar (5-63) 2 MS) = gm | %, PrRe sy see = ON i-e Up 1+¢ rn met Fuk (mj Lag 2°) ne = (44 f # sar] 4 (5-64) Ue 2 The term (um/w.)(2/PrRe) is due to heat transfer by molecular conduc- tivity which is negligibly small in comparison with the eddy heat trans- fer term except very close to the wall. The empirical constant K,; may be determined from the measured temperature profiles. In Fig. G,5g the effect of coolant injection on the temperature dis- tribution of a fully developed turbulent flow in a circular pipe was shown. The comparison between the theoretical curve calculated from Eq. 5-63 alig== "9 OUR O Experiment 8 O Experiment 10 Q/W = 0.00837. Q/W =0.00485 Theoretical ThA Fig. G,5g. Effect of coolant injection on the temperature distribution in turbulent pipe flow. (From [36].) ( 474 ) G,6 - COMPARISON WITH EXPERIMENTAL RESULTS and the experimental data show close agreement for small coolant injec- tion. Since the variation of the fluid properties was not taken into con- sideration in the theory it might be the main reason for the discrepancy between the theory and experiment, especially in the case of large coolant injection. The rate of heat transfer per unit area radially outwards at the wall can be calculated by integrating the energy equation (Eq. 5-49) across the radius of the circular pipe. R w=sa de i, pCpl'ur dr— DOr Dll (5-65) From the condition of heat balance at the wall (the heat transfer by the hot gas to the wall is absorbed by the coolant) one obtains Ge = pwalp,(Tn — Tr) (5-66) Combining Eq. 5-65 and 5-66, after integrating between two cross sec- tions, /; and J2, one obtains the temperature difference ratio as follows: eas @ ‘ a Nie te ie ue we LE YanWi|WefUm\ — [Um (5-67) T= To Le ZL CROW | Wa CUR): U./i where W = p.U., Q = pwz, and the quantities ( ):; and (_ ). represent the values at stations, /1 and ls, respectively. G,6. Comparison with Experimental Results on Transpiration Cooling. EXPERIMENTAL RESULTS vs. THEORY. The systematic experimental study of transpiration cooling was initiated at the Jet Propulsion Labo- ratory by Duwez and Wheeler in 1946 [37,38,39,40,41,42]. The experi- mental investigation was limited to the case of a cylindrical duct made of porous material through the walls of which the coolant was injected. The test section containing the porous-wall duct was one inch in diameter and eight inches long and a gasoline-air flame served as the source of hot gas. The gas temperature ranged from 1100 to 1900°F with Mach numbers approaching 1.0 and Reynolds numbers up to 140,000. For the detailed description of experimental equipment, [38] should be consulted. The main object of the experiments is to establish a relation between the surface temperature of the porous material and the weight rate of coolant flow for different conditions of temperature and velocity in the main stream of hot gas. Four significant variables were measured in the experiments, namely the temperature 7’, of the main stream of gas, the weight rate of flow of the hot gas W, the weight rate of flow of coolant Q, and the surface temperature 7. of the porous wall at several points along ( 475 ) G - COOLING BY PROTECTIVE FLUID FILMS the tube. For a given porous material and a given coolant, all the measure- ments can be reduced to the ratio (7, — T)/(T'~ — To) as a function of Q/W. A comparison of the experimental data with the theoretical results of [27] is given in Fig. G,6a in which (1, — Tw)/(T, — To) is plotted against Q/W. For a given porous material the measured values of (T, — Tw)/ (T, — To) and Q/W obtained within the range of temperatures and velocities covered in the experiments fall more or less on a single curve having the shape of the theoretical curves. But the theoretical curves indicate there is small dependency on the Reynolds number. This can be 1.0 0.8 Fig. G,6a. Comparison of theoretical and experimental results on temperature difference ratio vs. mass flow ratio. (From [27].) seen in Fig. G,6a. Since the theoretical analysis does not include the con- ductivity of the porous material, the result shows a fair agreement with the measured values obtained with a porous ceramic specimen and the poorest agreement with a porous copper specimen. Perhaps it should be mentioned that a very good agreement has been obtained between the experimental results of a stainless steel specimen and the theoretical curve computed from the laminar boundary layer theory by Yuan [7] for a Reynolds number of 3 X 104. This is given in Fig. G,6b. Although the agreement is for a particular case, it is interesting to note that qualitatively there is very good agreement between theory and experiment in transpiration cooling. An experimental investigation of the isothermal laminar boundary layer on a porous flat plate (injection begins at a distance parameter ( 476 ) G,6 - COMPARISON WITH EXPERIMENTAL RESULTS from the leading edge) was made at the Polytechnic Institute of Brooklyn [43]. Fig. G,6c shows that good agreement has been obtained between the measured velocity profiles and those calculated from the theoretical results given in [7]. The heat transfer theory in the laminar boundary layer can therefore be considered to have been verified, at least for a low rate of heat transfer. Due to the roughness and jet effect of the injected velocity of the porous plate, only qualitative agreement is obtained be- tween the measured transition Reynolds number and those predicted from the laminar boundary layer stability theory [25]. Experimental Data (CIT) Fig. G,6b. Comparison of measured data in a stainless-steel specimen with theoretical result [7]. (From [37].) An extensive experimental study of transpiration cooling of turbulent pipe flow with injection is being conducted at the Polytechnic Institute of Brooklyn under the auspices of Project Squid. The investigation has been guided by the theory discussed in Art. 5. The experimental apparatus for this investigation was designed and so arranged that fully developed turbulent flow would be established upstream of the test section. Hence any change in the flow conditions in the porous test section would be due to mass injection. The porous stainless steel segment, made by the Poroloy method, is a five-inch-diameter circular pipe, 24 inches in length. The velocity and pressure surveys were made at six stations along the axial direction of the porous-wall pipe. For zero injection at the wall the meas- ured velocity profiles (10° S Re S 5 X 105) are in good agreement with Nikuradse’s measurement for the smooth pipe. A comparison of the measured velocity profiles with various injection velocities to the theo- retically calculated velocity profiles is given in Fig. G,5f. ( 477 ) G + COOLING BY PROTECTIVE FLUID FILMS Radial temperature profiles were also measured at six stations along the axis of the porous-wall pipe. The inner temperature of the porous wall was measured at six axial stations by thermocouples of the integrating type. The results of the measured temperature profiles were also used in a comparison with those of the theoretically calculated temperature pro- files. This comparison was given in Fig. G,5g. The results of the present temperature-profile surveys for a fully developed turbulent pipe flow with coolant injection at the wall will be used to develop a semiempirical rela- tion of the heat transfer coefficient for transpiration cooling. The study has not reached the stage at which final results can be presented at this time. O Experimental points € = 0.497 [43] Theoretical curve Sr "0497 ii — —-— Theoretical curve p01 17/4) Fig. G,6c. Comparison of theoretical and experimental velocity profiles with injection for — = (v~¥/U)?Re, = 1.0. (From [43].) QUESTION OF SKIN FRICTION AND HEAT TRANSFER COEFFICIENT. As pointed out in Art. 4 the rate of change of momentum in the boundary layer due to mass fluid injection at the wall has the same effect as the rise of pressure gradient in the flow direction. Hence it is clear that the increase of fluid injection at the wall increases the thickness of boundary layer and decreases the slope of the velocity profile at the wall in the boundary layer. In the case of laminar boundary layer it is evident that the wall shearing stress and heat transfer from the hot gas to the wall decreases as the injected coolant increases. Since the heat flow from hot gas to the wall can be expressed by the following forms: q= ee = h(T, — Te) (6-1) G,6 - COMPARISON WITH EXPERIMENTAL RESULTS and the wall temperature decreases with the increase of coolant injection, it can be seen that the film heat transfer coefficient h, decreases as the ‘coolant injection increases. This is shown in Fig. G,4c. The above phenomena were also observed in the case of heat transfer in the turbulent boundary layer on a flat plate with coolant injection at the wall. This can be seen in Fig. G,4g and G,4h. The result obtained from the study of laminar pipe flow with fluid injection at the wall has shown that the effect of fluid injection at the wall is to accelerate the main stream velocity. Hence the velocity gradient at the wall, which determines the wall friction, increases. On the other hand, the result of the study of heat transfer of a laminar pipe flow with coolant injection shows that the heat transfer coefficient at the wall de- creases with an increase in the rate of coolant injection. This phenomenon thus indicates that the analogy between the heat transfer and momentum transfer does not exist in transpiration-cooled pipe flow. For turbulent pipe flow the connection between the pressure drop and the flow volume which, in turn, determines the wall shearing stress must be obtained from tests. Wheeler [44] has made some preliminary pressure- drop studies in a transpiration-cooled pipe. From the test data an empiri- cal expression is formulated ON hal: ae B. @) 7 (6-2) 2 er a where W is the weight rate of flow in the main stream, F is the gas con- stant, 7’, is the main stream temperature, g is the acceleration of gravity, and D is the diameter of the pipe. The first term on the right-hand side of Eq. 6-2 represents the head loss, which was observed with no coolant flow where f is the friction factor corresponding to Blasius formula for a smooth pipe. The second term expresses the additional loss which occurred when coolant was added. For isothermal flow the exponent n becomes unity. The parameter B seems to depend somewhat on the initial value of f and on the nature of the cooling gas, being greater for the less dense gas. The exact factors which control the parameter B and n cannot be determined from the present preliminary data. The importance of Eq. 6-2 is realized when one considers that the shearing stress at the wall +, may be determined experimentally by the measurement of the pressure drop. Once Eq. 6-2 is exactly established the shearing stress at the wall 7, and the heat flow to the wall from the hot gas gw can be determined in the transpiration-cooled turbulent bound- ary layer. GENERAL DISCUSSION ON TRANSPIRATION COOLING. On comparing the theoretical curves computed from [27] with the experimental results, it is seen that the shape of the curves is correct but that the theoretical wall temperature for a given coolant flow is lower than the measured fede g’?D { 479 ) G - COOLING BY PROTECTIVE FLUID FILMS values. The deviation appears to be more pronounced when the coolant flows become larger. It is believed that the discrepancies between theories and experimental data can be interpreted by the following discussions. Assessment of experimental errors. The errors inherent in the measure- ments of the main stream gas temperature were probably the most serious. It can be realized that the application of proper thermometric techniques is difficult without disturbing the velocity profile of the gas stream. The deterioration of thermocouples at high gas stream temperatures creates another serious problem. Furthermore the radiation heat loss from the thermocouple to the straightening-tube wall, the uncertainty as to the recovery factor of the probe, the error in the location of the thermocouple with respect to the temperature and velocity profiles, and the changes in the calibration of the thermocouple due to chemical reaction between the couple and the gas stream may contribute to error in the measurement of the true gas temperature. Next, the error in the measurement of the wall temperature due to the presence of the thermocouple tends to block off the flow of coolant between itself and the hot wall of the specimen. Thus the thermocouple indicates temperatures which are higher than those that would actually exist on the surface of an undisturbed wall. Disposition of carbonaceous materials on the specimen surface may cause the difficulty in obtaining reproducible experimental results. Since carbon deposition decreases the permeability of the specimen it would affect the coolant discharge pattern, the thermal conductivity of the specimen surface, and therefore the measured wall temperature. There is some radial heat loss from the back of the specimen to the holder which was not measured. The effect of this loss in general is to decrease the value of wall temperature for a given value of Q/W. Error arising from assumptions in the theory. As previously men- tioned, the theory assumed that there is no wall temperature gradient in the flow direction and hence no heat conducted along the wall. Actually, there is an ‘‘inlet length” for the porous material where the temperature distribution changes from that typical of flow in an uncooled pipe to the final distribution for porous-wall cooling. This inlet length depends on the conductivity of the porous material and is large for a material of high conductivity. Hence the theoretical results show a better agreement with measured values obtained with porous ceramic specimens than a porous copper specimen. Furthermore the measured data indicates that the dis- crepancy among various porous materials is reduced considerably when a longer specimen is used. In the theory the coolant flow normal to the wall was assumed uni- formly distributed over the surface of the specimen. It has been found in experiments that the coolant leaves the surface in the form of a number of isolated jets. Thus the transpiration-cooling process will be less efficient ( 480 ) G,7 - FILM COOLING in practice than in theory and the values of measured wall temperatures are correspondingly higher for a given coolant flow. It has been found in the experiments that the velocity of a stream of gas passing through a transpiration-cooled tube increases rapidly along the length of the tube. The rate of increase depends on the mass flow ratio Q/W and the density of the gas. The main stream velocity used in the theory is independent of the length in the direction of flow. Fundamentally, the influence of the rate of coolant flow to the shear- ing stress at the wall, to the main stream velocity profiles, and to the lami- nar sublayer thickness must be thoroughly investigated experimentally before an accurate theory in transpiration cooling can be realized. DIFFERENT PHYSICAL PROPERTIES BETWEEN COOLANT AND Hot Gas. When the physical properties of the coolant gas differ from those of the main stream gas, Eq. 4-31 can also be used to approximate the relation between the wall temperature and the coolant flow. It can be seen from Eq. 4-31 that the specific heat is the most important property, with a secondary effect due to the Prandtl number. The theoretical results com- puted for nitrogen and hydrogen coolants give a fair agreement with experimental data. The results of the experimental investigation of transpiration cooling by injecting water as a coolant indicate that there is a critical value in the amount of coolant, above which the surface temperature of the porous material remains near or below the boiling point of water, and below which the surface temperature increases very rapidly with decreased flow. The instability of water coolant flow may be explained by the evapo- ration of water inside the porous metal just below the hot surface. A film of vapor rather than a film of water was formed on the surface exposed to heat. It is undoubtedly true that this phenomenon would occur in the use of other kinds of liquid coolant. Unless some means of controlling the evaporation of the liquid coolant inside the porous metal is developed, a liquid coolant cannot be successfully used in transpiration cooling. G,7. Film Cooling and Its Comparison with Transpiration Cooling. General description. It was mentioned in the introduction that film cooling is a method of protecting a surface from a high temperature gas stream by separating the surface and the hot gas stream with a thin con- tinuous film of a liquid or gaseous coolant. The coolant is discharged by slots or orifices to the surface where the hot gas is flowing and is carried downstream by the flowing hot gas. In this way an insulating film is formed along the surface; however, the film is gradually destroyed on its way downstream by turbulent mixing. The coolant film has to be renewed at a certain distance downstream by injecting a new coolant through { 481 ) G - COOLING BY PROTECTIVE FLUID FILMS additional slots, because otherwise the wall temperature would eventually approach the temperature of the hot gas. The insulating effect of liquid film layer is better than the gas film layer due to the fact that the liquid coolant is evaporated by the heat from the flowing hot gas. The heat re- quired for evaporation keeps the temperatures of the liquid film and the wall to the evaporation temperature, until a point downstream from the slot is reached where the liquid is completely evaporated. In early work in Germany certain scientists employed the method of film cooling in rocket motors by injecting a coolant through small holes into the chamber and nozzle walls of a regeneratively cooled motor to cool the predetermined “‘hot spots.’”’ A series of holes generally uniformly spaced around the circumference of the wall has been employed if a com- plete wall is needed to be film-cooled. A different injection method is to introduce the coolant through inclined holes along the motor walls. This method has the advantage over the injection through radial holes in that it retains most of the coolant along the wall that is to be cooled, while injection through radial holes cannot avoid discharging a portion of the coolant directly into the main gas stream. Tangential holes which inject the coolant around the circumference of the motor, thereby taking ad- vantage of centrifugal force to hold the coolant on the wall, have also been employed. An investigation of film cooling based on the injection of the coolant through a slot around the circumference of the combustion chamber or nozzle is presented in [45]. The advantages of the radial injection of the coolant through a slot are: (1) the coolant flow rate may be accurately predicted in any section of the motor, and (2) a uniform film can be established at all sections of the motor. Film-cooling problems. The heat transfer process in film cooling is essentially the same as the process in transpiration cooling. One of the most important problems in film cooling is to determine the length of the fluid film along the wall. This depends on the quantity of fluid injected and the stability of the fluid film. Up to the present time there are neither theoretical analyses nor experimental investigations which yield a definite and general answer in regard to this important phenomenon. Hence it is limited only to particular results which will be presented here. In Fig. G,7a the wall temperature of a particular liquid-film-cooled tube is shown. The liquid-film-cooled length is indicated between the points A and C. At point C the wall temperature reaches the boiling temperature of the liquid and beyond the point C the liquid is completely evaporated, while the wall temperature increases rapidly to a value approaching the gas tem- perature. If a gas coolant is used instead of a liquid the length of the film would only exist for a short distance between B and C. This makes the use of a gas coolant in film cooling inadequate since numerous slots would be required. ( 482 ) G,7 - FILM COOLING Tsien [46] has indicated that, when the friction drag of a gas passing over a liquid film is of appreciable magnitude, the flow of a film coolant on a solid surface can be compared to the flow of the liquid near the wall of a pipe completely filled with a turbulent flowing liquid. Furthermore, when the film is of sufficient thickness to include a turbulent layer of coolant on the gaseous side of the film, instability occurs and some of the coolant breaks away in the form of droplets. The criterion of the insta- bility of the coolant film can be completely described by y*, a parameter indicating flow conditions near the wall. The above phenomenon was also interpreted by Rannie [47]. Knuth [48] has calculated some NACA experi- mental results on film cooling by Sloop and Kinney [49] and found that a definite value of y* exists for the criterion of the instability of coolant film. Hot gas temperature ee ee Duct wall temperature Boiling temperature of water Distance downstream of coolant injection Fig. G,7a. Liquid-film-cooled length in smooth duct (injection begins at A). (From [49].) An experimental study of the stability of the liquid film under various flow conditions is given in [44]. In this experiment, visual observation was made by injecting the liquid coolant through a slot into the test section where the air was blown through. At very low flow rates of the coolant, it entered the test section smoothly and flowed in a uniformly thin layer along the surface of the test section downstream from the injection slot. As the flow of liquid was increased, a point was reached where small air bubbles were formed immediately downstream of the slot. The mean ve- locity of the liquid in the slot corresponding to the condition where the bubbles first started to form was taken as the critical velocity of injection. If the liquid flow was increased beyond this critical velocity, the major portion of it separated from the wall of the test section. ( 483 ) G - COOLING BY PROTECTIVE FLUID FILMS The results of the experiments showed that the critical velocity of injection is increased when (1) the main stream air velocity is increased, (2) the coolant viscosity is increased, (3) the slot width is decreased, and (4) the density of the coolant is decreased. It further indicates that in- clined injection of the coolant increases the critical velocity of injection considerably more than the case of injection perpendicular to the main stream flow. Comparison of the effectiveness of film and transpiration cooling. Eckert [50] has made a comparison of the relative effectiveness of film and tran- spiration cooling which is based on a turbulent flow along a flat plate with constant gas velocity and temperature. The Reynolds number for the turbulent gas flow is assumed equal to 107 and the Prandtl number is equal to 0.7. Air is considered as the coolant, as well as the outside flow gas. The parameters used in the comparison are the ratio of the temper- ature difference, the difference of wall temperature and coolant temper- ature to the difference of hot gas temperature and coolant temperature; and the ratio of coolant mass flow to the gas mass flow. In the transpiration-cooling calculation the relation between the ratio of the temperature difference and the ratio of the mass flow is derived from the heat balance equation at the wall. The heat transfer coefficient used herein is obtained from sublayer theory in the turbulent flow [5/]. In the film-cooling calculation the Wieghardt method [52] is used to deter- mine the temperature difference ratio in relation to the mass flow ratio for a single slot. Although the temperature conditions within the bound- ary layer in [52] are opposite to the conditions found in film cooling, the results obtained in [52] can be used for the film-cooling process as long as the temperature differences are small enough to permit the gas property to be considered constant. A comparison of the relative effectiveness of the two cooling methods considered is shown in Fig. G,7b. Transpiration cooling is much more effective than film cooling with a single slot and gives much lower wall temperature for a specified coolant flow. In other words, transpiration cooling requires a much smaller amount of coolant to cool the wall to a predesignated temperature. It must be borne in mind that in the film- cooling calculation the wall temperature cannot be made constant as in the case of transpiration cooling, and it represents the highest temper- ature occurring within the wall. At smaller downstream distances, the temperature decreases toward the value 7’) obtained immediately behind the slot. The effectiveness of the film cooling therefore is increased by increasing the number of slots along the plate, and it is expected that the film cooling eventually transforms into transpiration when the num- ber of slots becomes very large. There are, of course, other considerations, aside from that of a minimum of coolant, which influence the choice of the cooling method for a particu- ( 484 ) G,8 - CITED REFERENCES AND BIBLIOGRAPHY lar application. A distinct advantage of film cooling for practical appli- cations is that it can be very easily adapted in most designs. In addition, film cooling, as explained above, appears to be a good method for thor- oughly cooling a specific location. In this connection, further intensive research on the problems of methods of injection, stability of coolant film, == 1lin == IG Teo o 0 = ——-— Transpiration [51] —— Film [52] 0 0.002 0.004 0.006 0.008 0.010 0.012 Coolant-flow ratio pwWw/p..U Temperature difference ratio Fig. G,7b. Comparison of transpiration and film-cooling methods (turbulent flow Re = 10’, Pr = 0.7). (From [50].) and the mass and heat transfer between a hot gas and a coolant film in film cooling is needed. G,8. Cited References and Bibliography. Oo bo IO o f Cited References . Duwez, P., and Martin, H. F. Preparation and physical properties of porous metals for sweat cooling. Calif. Inst. Technol. Jet Propul. Lab. Progress Rept. 3-14, July 1946. . Wheeler, H. L., Jr. Private communication, Apr. 1956. . Duwez, P., and Wheeler, H. L., Jr. An experimental study of the flow of gas through porous metal. Calif. Inst. Technol. Jet Propul. Lab. Progress Rept. 1-66, Aug. 1947. . Green, L., Jr., and Duwez, P. The permeability of porous iron. Calif. Inst. Technol. Jet Propul. Lab. Progress Rept. 4-85, Feb. 1949. . Wheeler, H. L., Jr.. and Myer, F.O. The pattern of flow of gas leaving porous metal surface. Calif. Inst. Technol. Jet Propul. Lab. Progress Rept. 4-83, Nov. 1948. . Weinbaum, S., and Wheeler, H. L., Jr. Heat transfer in sweat-cooled porous metal. J. Appl. Phys. 20, 113-122 (1949). . Yuan, S. W. A theoretical investigation of the temperature field in the laminar boundary layer on a porous flat plate with fluid injection. Project Squid Tech. Rept, 4, Sept. 1947. ( 485 ) 20. 21. 29. 30. G - COOLING BY PROTECTIVE FLUID FILMS . Ness, N. On the temperature distribution along a semi-infinite sweat-cooled plate. J. Aeronaut. Sci. 19, 760-768 (1952). . Yuan, S. W. Heat transfer in laminar compressible boundary layer on a porous flat plate with fluid injection. J. Aeronaut. Sct. 16, 741-748 (1949). . Crocco, L. Sulla Transmissione del Calone de una Lamina Piana un fluido scorrente ad alta velocita. Aerotecnica 12, 181-197 (1932). . Yuan, S. W., and Ness, N. Heat transfer in a laminar compressible boundary layer on a porous flat plate with variable fluid injection. Project Squid Tech. Mem. P.I.B.-16, Sept. 1950. . Morduchow, M., and Galowin, L. The compressible laminar boundary in a pressure gradient over a surface cooled by fluid injection. Proc. First Iowa Thermo- dynamics Symposium, State University of Iowa, 143-169 (1953). . Morduchow, M. On heat transfer over a sweat-cooled surface in laminar com- pressible flow with a pressure gradient. J. Aeronaut. Sci. 19, 705-712 (1952). . Morduchow, M. Laminar separation over a transpiration-cooled surface in compressible flow. NACA Tech. Note 3559, 1955. . Eckert, E. R. G. Heat transfer and temperature profiles in laminar boundary layer on a sweat-cooled wall. Army Air Force Tech. Rept. 5646, 1947. . Schlicting, H., and Bussman, K. Exakte Losungen fur die laminare Grenzschicht mit Absangung und Ausblasen. Schriften deut. Akad. Luftfahrtforschung 7B, 2, 19438. . Falkner, V. N., and Skan, S. W. Phil. Mag. 12, 865 (1931). Also Brit. Aeronaut. Research Council Repts. and Mem. 1314, 1930. . Brown, W. B. Exact solution of the laminar boundary layer equations for a porous plate with variable fluid properties and a pressure gradient in the main stream. Proc. First U.S. Natl. Congress Appl. Mech., Chicago, June 1951. . Pohlhausen, E. Der Warmeanstauch zwischen festen Korpen and Flussigkeiten mit kleiner Reibung and kleiner Warmeleitung. Z. angew. Math. u. Mech. 1, 115-121 (1921). Hartree, D. R. On an equation occurring in Falkner and Skan’s approximate treatment of equations of the boundary layer. Proc. Cambridge Phil. Soc. 33, 223 (1937). Brown, W. B., and Donoughe, P. L. Table of exact laminar-boundary-layer solutions when the wall is porous and fluid properties are variable. NACA Tech. Note 2479, 1951. . Eckert, E. R. G., and Livingood, J. N. B. Method for calculation of heat transfer in laminar region of air flow around cylinders of arbitrary cross section (including large temperature difference and transpiration cooling). NACA Rept. 1118, 1953. . Low, G. M. The compressible laminar boundary layer with fluid injection. NACA Tech. Note 3404, 1955. . Lees, L., and Lin, C. C. Investigation of the stability of the laminar boundary layer in a compressible fluid. NACA Tech. Note 1115, 1946. . Lees, L. The stability of the laminar boundary layer in a compressible fluid. NACA Rept. 876, 1947. . Dunn, D. W., and Lin, C. C. On the stability of the laminar boundary layer in a compressible fluid, Part 1. Mass. Inst. Technol. Math. Dept., Office Nav. Reserve Contract Ndori-07872 and Néori-60, Dec. 1953. . Rannie, W. D. A simplified theory of porous wall cooling. Calif. Inst. Technol. Jet Propul. Lab. Progress Rept. 4-50, Nov. 1947. . Dorrance, W. H., and Dore, F. J. The effect of mass transfer on the com- pressible turbulent boundary-layer skin friction and heat transfer. J. Aeronaut. Sct. 21, 404-410 (1954). Rubesin, M. W. An analytical estimation of the effect of transpiration cooling on the heat transfer and skin friction characteristics of a compressible turbulent boundary layer. NACA Tech. Note 3341, 1954. Mickley, H. S., Ross, R. C., Squyers, A. L., and Stewart, W. E. Heat, mass and momentum transfer for flow over a flat plate with blowing or suction. NACA Tech. Note 3208, 1954. ( 486 ) G,8 - CITED REFERENCES AND BIBLIOGRAPHY 31. Yuan, 8. W., and Finkelstein, A.B. Laminar pipe flow with injection and suction through a porous wall. Trans. Am. Soc. Mech. Engrs. 78, 719-724 (1956). 32. Yuan, 8. W., and Finkelstein, A. B. Heat transfer of a laminar pipe flow with coolant injection. Presented at the 1956 Heat Transfer and Fluid Mech. Inst., Stanford Univ., 1956. 33. Graetz, L. Ann. Phys. 18, 79-94 (1883); 25, 337-367 (1885). 34, Nusselt, W. Z. Ver. deut. Ing. 54, 1154-1158 (1910). 35. Yuan, 8. W., and Galowin, L. 8. Transpiration cooling in the turbulent flow through a porous-wall pipe. Jet Propul. 28, 178-181 (1958). 36. Goldstein, S. Modern Development in Fluid Dynamics, Vol. II. Oxford Univ. Press, 1938. 37. Duwez, P., and Wheeler, H. L., Jr. Experimental study of cooling by injection of a fluid through a porous material. J. Aeronaut. Sci. 15, 509-521 (1948). 38. Wheeler, H. L., Jr., and Duwez, P. A gasoline-air combustion chamber for the study of sweat cooling. Calif. Inst. Technol. Jet Propul. Lab. Progress Rept. 1-59, Nov. 1947. 39. Duwez, P., and Wheeler, H. L., Jr. Heat transfer measurements in a nitrogen sweat-cooled porous tube. Calif. Inst. Technol. Jet Propul. Lab. Progress Rept. 4-48, Nov. 1947. 40. Wheeler, H. L., Jr. Flow of gases through sweat-cooled tubes. Calif. Inst. Technol. Jet Propul. Lab. Progress Report 4-87, Dec. 1948. 41. Wheeler, H. L., Jr. Heat transfer in nitrogen and hydrogen sweat-cooled tubes. Calif. Inst. Technol. Jet Propul. Lab. Progress Rept. 20-160, Jan. 1952. 42. Wheeler, H. L., Jr., and Duwez, P. Heat transfer through sweat-cooled porous tubes. Jet Propul. 25, 519-524 (1955). 43. Libby, P. A., Kaufman, L., and Harrington, R. P. An experimental investigation of the isothermal laminar boundary layer on a porous flat plate. J. Aeronaut. Sci. 19, 127-134 (1952). 44. Wheeler, H. L., Jr. The influence of wall material on the sweat-cooled process. Calif. Inst. Technol. Jet Propul. Lab. Progress Rept. 4-90, May 1949. 45. Zacrow, M. J., Beighley, C. M., and Knuth, E. Progress report on the stability of liquid film for cooling rocket motors. Project Squid Tech. Rept. 23, Nov. 1950. 46. Tsien, H. 8. Research in rocket and jet propulsion. Aero. Digest 60, 120-125 (1950). 47. Rannie, H.W. Heat Transfer in Turbulent Shear Flow. Ph.D. Thesis, Calif. Inst. Technol., 1951. 48. Knuth, E. L. Mechanics of film cooling. Jet Propulsion 24, 359-365 (1954). 49. Kinney, G. R., Jr., and Sloop, J. L. Internal film cooling experiments in a 4-inch duct with gas temperature to 2000°F. NACA Research Mem. F50F19, 1950. 50. Eckert, E. R. G., and Livingood, J. N. B. Comparison of effect of convection, transpiration and film cooling with air as coolant. NACA Rept. 1182, 1954. 51. Friedman, J. A theoretical and experimental investigation of rocket-motor sweat-cooling. J. Am. Rocket Soc. 79, 147-154 (1949). 52. Wieghardt, K. MHot-air discharge for de-icing. Army Air Force Trans. F-TS- 919-RE, Dec. 1946. Bibliography Berman, A. Laminar flow in channels with porous wall. J. Appl. Phys. 24, 1232-1235 (1953). Clarke, J. H., Menkes, H. R., and Libby, P. A. A provisional analysis of turbulent boundary layer with injection. J. Aeronaut. Sci. 22, 255-260 (1955). Crocco, L. An approximate theory of porous, sweat, or film cooling with reactive fluids. J. Am. Rocket Soc. 22, 331-338 (1952). Donoughe, P. L., and Livingood, J. N. B. Exact solution of laminar-boundary-layer equations with constant property values for porous wall and variable temperature. NACA Tech. Note 3151, 1954. Dorrance, W.H. The effect of mass transfer on the compressible turbulent boundary- ( 487 ) G + COOLING BY PROTECTIVE FLUID FILMS layer skin friction and heat transfer—An addendum. J. Aeronaut. Sci. 23, 283-284 (1956). Duwez, P., and Wheeler, H. L., Jr. Preliminary experiments on the sweat-cooling method. Calif. Inst. Technol. Jet Propul. Lab. Progress Rept. 3-13, July 1946. Eckert, E. R. G. One dimensional calculation of flow in a rotating passage with ejection through a porous wall. NACA Tech. Note 3408, Mar. 1955. Eckert, E. R. G. Transpiration and film cooling. Heat Transfer Symposium, Univ. Mich. Press, 1953. Eckert, E. R. G., Diaguila, A. J., and Donoughe, P. L. Experiments on turbulent flow through channels having porous rough surface with or without air injection. NACA Tech. Note 3839, 1955. Ellerbrock, H. H., Jr. Some NACA investigation of heat transfer characteristics of cooled gas-turbine blades. General discussion on heat transfer. Inst. Mech. Engrs. London, Sept. 1952. Emmons, H. W., and Leigh, D. Tabulation of the Blasius function with blowing and suction. Harvard Univ. Combustion Aeronaut. Lab., Div. Appl. Sci., Interim Tech. Rept. 9, Nov. 1953. Green, L., Jr. Gas cooling of a porous heat source. J. Appl. Mech. 19, 173-178 (1952). Grootenhuis, P., and Moore, N. P. W. Some observations on the mechanics of sweat cooling. Seventh Intern. Congress Appl. Mech., London, 1948. Grootenhuis, P., et al. Heat transfer to air passing through heated porous metals. General discussion on heat transfer. Inst. Mech. Engrs. London, 405-409 (1952). Howarth, L. Modern Developments in Fluid Dynamics, High Speed Flow, Vol. II, Chap. XIV. Oxford Univ. Press, 1953. Ivey, R. H., and Klunker, E. B. An analysis of supersonic aerodynamic heating with continuous fluid injection. NACA Rept. 990, 1950. Jakob, M., and Fieldhouse, I. B. Cooling by forcing a fluid through a porous plate in contact with a hot gas stream. Heat Transfer and Fluid Mech. Inst., June 1949. Lees, L. Stability of the laminar boundary layer with injection of cool gas at the wall. Project Squid Tech. Rept. 11, May 1948. Lew, H. G., and Fanucci, J. B. On the laminar compressible boundary layer over a flat plate with suction or injection. J. Aeronaut. Sci. 22, 589-597 (1955). Livingood, J. N. B., and Eckert, E. R. G. Calculation of transpiration-cooled gas- turbine blade. Trans. Am. Soc. Mech. Engrs. 75, 1271-1278 (1953). Mayer, E., and Baras, J. Q. Transpiration cooling in porous metal walls. Jet Pro- pulsion 24, 366-368, 378, 386 (1954). Muskat, M. The Flow of Homogeneous Fluids Through the Porous Media. McGraw- Hill, 1937. Staniforth, R. Contribution to the theory of effusion cooling of gas turbine blades. General discussion on heat transfer. Inst. Mech. Engrs. London, Sept. 1951. Yuan, 8. W. Further investigation of laminar flow in channels with porous walls. J. Appl. Phys. 27, 267-269 (1956). Yuan, S. W. Preliminary investigation of heat transfer in turbulent boundary layer on a porous wall. Project Squid Tech. Rept. 19, 29-36 (1950). Yuan, S. W., and Barazotti, A. Experimental investigation of turbulent pipe flow with coolant injection. Heat Transfer and Fluid Mech. Inst., June 1958. Yuan, 8. W., and Chin, C. Heat transfer in a laminar boundary layer on a partially sweat-cooled plate. Project Squid Tech. Rept. 18, Aug. 1949. Yuan, 8S. W., and Whitford, C. Further investigation of heat transfer in a laminar compressible boundary layer on a porous plate with fluid injection. Project Squid Tech. Rept. 14, Sept. 1949. ( 488 ) SECTION H PHYSICAL BASIS OF THERMAL RADIATION S. $8. PENNER H,1. Introduction. The conventional and most successful approach to engineering calculations of radiant heat exchange is described in the following section. Included in the discussion are empirical rules and ex- trapolation procedures. In order to appreciate the limitations involved in the use of these empirical rules, it is essential to gain some understanding of the physical principles which determine emitted and absorbed radiant energies. Since a fundamental description of the phenomena involved is particularly simple for the equilibrium radiation of gases, we shall con- fine our attention to a brief survey of fundamental laws and to a quali- tative outline of the methods used for calculations on the thermal radi- ation characteristics of gases. H,2. Black Body Radiation Laws. A black body is defined as a body which neither transmits nor reflects any radiation which it receives; a black body absorbs all of the incident radiation. It can be shown that the equilibrium energy of radiation emitted from the unit area of a black body in unit time at a fixed temperature represents an upper limit for the thermally emitted energy from unit area for any substance which is at the same temperature as the black body. This definition of a black body and the quantum mechanics principle of equipartition of energy [/, Chap. 2; 2, pp. 546-550; 3, pp. 363-372] are sufficient to establish the Planck black body distribution law, which expresses the equilibrium rate at which radiant energy is emitted from a black body as a function of wavelength and temperature 7. The Planck black body distribution law has been abundantly confirmed by experiments. The spectral (or monochromatic) radiancy R&dd is defined as the energy emitted, per unit time, from unit area of a black body in the wavelength range between \ and \ + d) at the absolute temperature T’ (in °K), into a solid angle of 27 steradians. The Planck black body dis- tribution law is Ci DN nG OS | (2-1) R2dv\ = where c,/m and c2 are known as the first and second radiation constants, ( 489 ) H : PHYSICAL BASIS OF THERMAL RADIATION respectively. The quantities c,; and c2 may be expressed in terms of the fundamental physical constants c (velocity of light), h (Planck’s const), and k (Boltzmann const). Thus c; = 2mc*h & 3.742 & 10-° erg-cm?-sec—! and c. = hc/k & 1.439 em-°K. For \T Rodn’ have been tabulated [4] for ihe wavelengths and/or temperatures which are likely to be encountered in practice. The frequency v is related to the wavelength \ through the expression (2-5) H,3 - NONBLACK RADIATORS where c is the velocity of light (c = 2.998 101° cm-sec~!); the wave number w is the reciprocal of the wavelength, i.e. 1 => (2-6) G) From Eq. 2-1 and 2-5 it follows that the spectral radiancy in the fre- quency range between v and v + dy at the temperature T is given by the expression 2rhv? dy ROD = rea (2-7) Similarly R°dw is determined according to the equation Redo = Iherw ae (2-8) H,3. Nonblack Radiators. The (hemispherical) spectral emissivity € of a substance is defined as the ratio of the spectral radiancy for the given substance to the spectral radiancy of a black body. Thus the energy emitted from a nonblack substance, per unit area, per unit time, into a solid angle of 27 steradians in the wavelength range between \ and A + dv at the temperature T is Ryd = 6 Ridr if the spectral emissivity of the substance is e. The nonblack substance at temperature T' is said to have (total hemispherical) emissivity «¢ if the total emitted energy, per unit area, per unit time, into an angle of 27 steradians is Wii—selVe— leas) The definitions of the spectral and total emissivities apply to dis- tributed sources of radiation as well as to surfaces. A discussion of the thermal radiation characteristics of gases (Art. 5) involves essentially the development of basic procedures for the calculation of « and e for equilibrium systems of pure gases and gaseous mixtures. A simple example for the use of theoretical spectral emissivity rela- tions is provided by Drude’s law [5, Chap. 1] for pure metals, which holds within a few per cent for wavelengths longer than about 2u.! Drude’s law relates the spectral emissivity to the electrical resistivity r (in cm) and the wavelength \ (in cm), viz. € & 0.365 ae (3-1) 1 The following wavelength units are often used in discussions of radiant heat transfer: Angstrom unit A (1 A = 107-8 cm) and micron p (1p = 10-4 cm). The wave number (reciprocal of the wavelength) is customarily expressed in cm~!. The frequency is given in sec7!. ( 491 ) H - PHYSICAL BASIS OF THERMAL RADIATION Since the electrical resistivity is roughly proportional to the first power of the temperature for many metals, it follows that the total emissive power of many metals varies as T within the range of validity of Eq. 3-1. H,4. Basie Laws for Distributed Radiators. For thermodynamic equilibrium one can deduce Kirchhoff’s law, which states that the spectral radiancy of any substance equals the product of the spectral absorptivity Pj, and the spectral radiancy of a black body.? In other words, the spectral emissivity ¢, and the spectral absorptivity P’, are identically equal. It is convenient in practice to introduce the product of two-dimensional parameters for the dimensionless spectral absorptivity P’,. Following customary procedure we write for distributed radiators P! = P,dX Gell) where P,, is termed the spectral absorption coefficient and is expressed in em—!-atm7! or in ft~+atm~!, with the pressure referring to the actual >| dX = pdX \ ee ee Fig. H,4. Schematic diagram for the determination of the basic spectral emission law for distributed isothermal radiators. (X represents optical density, p equals the partial pressure of the radiator, and / and dz are geometric lengths.) pressure of radiators responsible for absorption at the wave number w; correspondingly, the optical density dX, which represents the product of a geometric length and the partial pressure of the radiators, must have the dimensions of cm-atm or ft-atm, respectively. Consider now a system of isothermal radiators at pressure p dis- tributed uniformly through a region of geometric length J. The optical density of a region of infinitesimal geometric length dz is dX = pdx; the optical density of the region of geometric length / is X = pl. A schematic diagram is shown in Fig. H,4 in which the abscissa has the dimensions of optical density. It is desired to obtain an expression for the total spectral radiancy from the isothermal distributed radiators located in a column of geometric length J. Let the spectral radiancy incident on the face A be fk... The change in spectral radiancy corresponding to the region of optical 2 The choice of the wave number w for identification of the spectral region is, of course, arbitrary. Either the wavelength \ or the frequency »v may be used if desired. The statement that P/, is independent of the intensity of the incident radiation may be regarded as an experimentally established fact. This result follows also from molecular considerations concerning the relation between transition probabilities and absorption coefficients. ( 492 ) H,4 - BASIC LAWS FOR DISTRIBUTED RADIATORS depth dX is then Oi Bey 3 (CE ODOM EID. O) a (4-2) where the first term on the right-hand side of Eq. 4-2 represents the emitted spectral radiancy in dX, and the second term measures the attenuation by the absorbers of radiation in dX. Since Rk, = 0 for X = 0 it follows from Eq. 4-2 that 15 = LE GL Se ee) (4-3) Eq. 4-3 is the basic phenomenological law for the emission of radiation from distributed sources. It is apparent that if an external light source is used such that R, = Ri, for X = 0, then Eq. 4-3 should be replaced by the expression Jon = IBA Se) Se inaeers (4-4) In those cases where the first term in Kq. 4-4 is negligibly small (i.e. negligible emission of radiation from the region under study), Eq. 4-4 re- duces to the Bouguer-Lambert law of absorption Iie = Remto~ (4-5) In absorption studies it is customary to choose 1/2zw as the unit of length and to introduce an extinction coefficient x through the relation i ef ~ Arw (4-6) For absorbing liquids (and sometimes also for gases), a specific absorption coefficient 8 may be introduced through the relation Pe p=" (4-7) Cc where c’ is the concentration of the absorber. If c’ is expressed in mole cm—* and Pp in cm™, then @ has the dimensions mole—! cm? and the absorption law may be referred to as Beer’s law of absorption. For ideal gases, sets of units involving mass absorption coefficients k,, (in cm?-g—!) are sometimes used; in this case k, = P,,.R’T where R’ is the gas const per gram. The quantity P..X is now replaced by k.,pl with p representing the gas density. Reference to Eq. 4-3 shows that the spectral emissivity ¢, of the uni- formly distributed radiators is given by the relation & = 1 — e-Pox (4-8) Similarly, the total emissivity « used in engineering calculations on radiant heat transfer is ls ‘ 0 EE IE eXG = = 7) Be € oO 7) 10) O = S NSS INS SS NS YY YP IK Y YIN IN XN NS \ N SS ES as LD DTEINGIRSEAIRSSINEARS SE TSS (07 cn ENING ONIN SS So 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Temperature, °R Fig. I,3c. Emissivity of water vapor. e, for H.O (each evaluated as if the other gas were absent) to obtain the e, due to the two together is read from Fig. I,3e. The same type of cor- rection applies in calculating a,. Recapitulating by the use of subscripts on e, indicating in sequence the gas (c or w), the temperature on the plot, and the value of pL at which ¢ is read, one has & = €o,T Pe Ce + Ew,T gs PyLU w =" Aer, (3-3) m 0.65 T 0.45 4 & a, = €0,T,PeLT \/T g (7) C2 = €w,T1,PwLT\/T g (Fz) Cz = Aar, 1 1 ( 516 ) 1,3 - RADIATION FROM FLAMES AND GASES The formulation of radiant interchange between a gas and a black surface completely enclosing it, when the gas contains CO», and H,0, is then = GH ev) (3-4) If the surface is gray, multiplication of the right side of the above by €:(= a1) would make proper allowance for reduction in the primary beams from gas to surface and surface to gas, respectively; but some of the gas radiation initially reflected from the surface has further opportunity for absorption at the surface because the gas is but incompletely opaque to the reflected beam. Consequently, the factor to allow for surface emis- sivity lies between e, and 1, nearer the latter the more transparent the gas — eed foe) oF -N/ Correction factor C, Ss © On Fig. 1,3d. Correction factor Cy for converting emissivity of H2O to values of pw and P; other than 0 and 1 atm, respectively. (low p,,) and the more convoluted the surface. If the surface emissivity is above 0.8, use of the factor (e: + 1)/2 cannot be greatly in error. If €, is smaller, the more nearly rigorous method of Art. 5 may be used. Although a,: approaches e, as 7; approaches T,, q/S of Eq. 3-4 does not in consequence reduce to oe,(T4 — T'}) for values of 7; and T, close together. If over a restricted range of variables e, is assumed proportional to (p,L)*(T,z)®, then ag: is proportional to (p.L)*(T1)7t* “Te where c is the previously encountered exponent used in evaluating absorptivity— 0.65 for CO. and 0.45 for H.O. It may be shown that, when 7, and T, are not too far apart, de=1 wie sear * + a z b— | (T! shit T’) (3-5) Cae I - ENGINEERING CALCULATIONS OF RADIANT HEAT EXCHANGE o > I) Sonipee! jo) —— ee — SS eer Bes ction to gas emissivity due to spectral overlap of water vapor and carbon dioxide. Fig. I,3e. Corre ( 518 ) 1,3 - RADIATION FROM FLAMES AND GASES in which &,,, is the gas emissivity evaluated at the arithmetic mean of T, and T;. Values of a and b, which represent 0 In ¢,/d In p,L and @ In ¢,/ 0 In T,, respectively, are given in Fig. I,3f for CO. and H.O. When a mixture of the two gases is present, it is recommended that mean values of a, b, and c be used in Eq. 3-5, the factors for the different gases being weighted in proportion to their emissivities. (A roughly approximate value of (a + 6 — c) suffices, however, since an error of 0.1 in it produces an error of only 2.5 per cent in g/S.) The use of Eq. 3-5 when T, and 7, do not differ by a factor greater than 2 leads to results of accuracy com- parable to Eq. 3-4, and will usually save time as well by eliminating the necessity for the fairly tedious evaluation of a,1. Mean beam length. The preceding expressions were formulated for the case of interchange between a gas hemisphere and a spot on its base, Gas emissivity € 1000 2000 3000 4000 5000 1000 2000 3000 4000 5000 Temperature, °R Temperature, °R Fig. 1,3f. Rate of change of emissivity with 7 and p,L, for CO2 and H.O. i.e. for the case in which the path length L of the radiant beam is the same in all directions. For gas shapes of practical interest, it is found that any shape is approximately representable by an equivalent hemisphere of proper radius, or that there is a mean beam length L which can be used in calculating gas emissivities and absorptivities from Fig. I,3a and I,3c [30,31]. As pL approaches zero, the mean beam length approaches as a limit the value 4z (ratio of gas volume to bounding area). For the range of pL encountered in practice, L is always less; 85 per cent of the limiting value is generally a satisfactory approximation [32]. Table I,3 summarizes the results of tedious graphical or analytical treatment of various shapes. Temperature variation along flow path. If gas radiation occurs in a system in which there is a continuous change in temperature of the gas or surface, or both, along the gas flow path from one end to the other of the interchanger, and if the dimensions transverse to flow are small enough to make radiant exchange in the flow direction relatively un- ( 519 ) I - ENGINEERING CALCULATIONS OF RADIANT HEAT EXCHANGE Table I,3. Beam lengths for gas radiation. Factor by which z is multiplied to obtain Characterizing} mean beam length L Shape dimension ii When | For average pL = 0 | values of pL Sphere Diameter 0.67 0.60 Infinite cylinder Diameter 1 0.90 Semi-infinite cylinder, radiating to center of base Diameter 0.90 Right circular cylinder, height = diameter, radiating to base at center Diameter 0.77 Same, radiating to whole surface Diameter 0.67 0.60 Infinite cylinder of half-circular cross section. Radiating to spot on middle of flat side Radius 1.26 Rectangular parallelopipeds 1 cube Edge 0.67 1:1:4, radiating to 1 X 4 face Shortest edge | 0.90 “ce a3 1 s< 1 ce 0.86 oh “ all faces 0.89 iShe taoinen 1:2:6, radiating to 2 X 6 face Shortest edge} 1.18> ; ve “at pease “ce “16 “ 1.24 in table ag a3 1 x< 2 ce 1 : 18 4 “ all faces 1/20 1: ©: 0, Infinite parallel planes Distance be- 2 tween planes * For parallelopipeds, multiply beam length suitable for pL = 0 by the following ratios [32]: When plz or pw = 0.01 0.1 1 Ratio for CO: 0.85 0.80 0.77 Ratio for H,O 0.97 0.93 0.85 ( 520 ) 1,3 - RADIATION FROM FLAMES AND GASES important, exact allowance for stream temperature variation can be made by conventional graphical integration. Let the heat transfer surface per unit length be designated by P, the local gas and surface temperatures at length x by T, and T,, the mass flow rate and heat capacity of the gas by m and cy, and the local heat transfer rate—by whatever mechanisms, SS EE 0.20 LL TU YY 0.10 <= S : SS = SN = 005 = = = kK Leal p= 0:02 0.01 0.005 2 >< 1012 10!8 10!4 Bo Fig. I,3g. Plot for calculation of mean gas temperature Re in countercurrent gas radiant heat exchangers, in terms of arithmetic mean gas and surface temperatures 7’, and 7; and gas-temperature change 7; — T.2. (Cohen [33].) and expressible as a function of T, and T1;—by (¢/S)hoca. Then one may write mepAT, = (9/S)tocalax (3-6) If the gas temperature at length B is 7’,,z, the above yields on integration ie etude =a —{*— 3-7 MCp ie (G/S) toca Cw Then a plot of the reciprocal of the local transfer rate vs. gas temper- ature yields a curve, the area under which is proportional to the length of the exchanger. This graphical procedure can be avoided, however, by evaluation of suitable mean gas and surface temperatures for use in Eq. 3-4 or 3-5 to obtain (q/S).,. Fig. 1,3g, from Cohen [33], gives the mean gas temperature for radiation T’ as a function of the four terminal tem- peratures. The mean surface temperature 7’ is that corresponding to the point in the exchanger where the gas temperature is Tj, and may be obtained by an enthalpy balance. Rigorous allowance for temperature variations in a gas within the (soe) I - ENGINEERING CALCULATIONS OF RADIANT HEAT EXCHANGE field of view of an element of radiation-receiving surface enormously complicates the problem; see end of Art. 4. Application of the principles discussed is illustrated in the following example: Nonluminous combustion products of a hydrocarbon fuel, con- taining 10 per cent CO. and 15 per cent H.O by volume, flow through a black wall cylindrical chamber 15 inches in diameter at a total pressure of 5 atm. In the region where the gas temperature drops from 2700 to 2000°F and the wall temperature is substantially constant at 1200°F, estimate the average heat transfer rate due to radiation from CO, and H.O. The arithmetic mean gas and surface temperatures are (2700 + 2000) / 2 + 460 = 2810°R, and 1660°R; 28104 — 16604 = 0.548 X 10!4. From Fig. I,3g, (7, — T))/T, = 0.016. The mean radiating gas temperature is then 2810(1 — 0.616) = 2765°R. Mean gas and surface temperatures differ sufficiently little to justify use of Eq. 3-5. T. = (2765 + 1660)/2 = 2212°R (3-8) pl =0.10X5 (33) xX 0.8 = 0.50 (3-9)? Pel = 0.15 X 5 (33) xX 0.8 = 0.75 (3-10)? From Fig. I,3a and I,3b, e, (at T = 2212°R, p.L = 0.5) = 0.119 K 1.15 = 0.187 (@=11) From Fig. I,3¢ and I,3d, é, (at T = 2212°R, pL = 0.75) = 0.167 X 1.6(?) = 0.267 (8-12) 0.404 From Fig. I,3e, the superimposed radiation correction’ = 0.057 Total gas emissivity, «, = 0.347 From Fig. 1,3f, a and 6 for COz2 = +0.35 and —0.40; for H.O, +0.52 and —0.85. For the mixture, a = 0.46, b = —0.7, c = 0.52. Then, from Eq. 3-5, (2) = 0.171 0.347 ar Lule at cs ad (27.654 — 16.6) = 24,500 BTU/ft? hr 2 A smaller factor is used than for an infinite cylinder; the length of the system has not been specified. 3 The pressure correction for H2O is far beyond the end of the plot; the extrapolation is based on the assumption that the pressure effect levels off above a few atmospheres pressure. 4In the absence of a recommendation, in the literature, as to the effect of total pressure on the correction Ae, it may be assumed to be increased by the average of C. and Cy. The effect is negligible in the present example. ( 522 ) I,4 - RADIANT EXCHANGE A convection coefficient of about 22 BTU/ft? hr °F would produce the -same transfer rate. Radiation from other gases. Gas radiation plots similar to those for CO, and H.O have been prepared for SOx, CO, and NH; [5, Chap. 4]. Infrared spectroscopic data available on many gases can serve as a basis for estimating radiant heat transfer although the data, particularly on the effects of temperature and total pressure, are often found to be in- adequate. For a discussion of the methods of calculation the reader is referred to Sec. H. 1,4. Radiant Exchange in an Enclosure of Source-Sink and No-Flux Surfaces Surrounding a Gray Gas. One of the most complex problems of heat transmission is the evaluation of heat transfer in a com- bustion chamber, where all of the mechanisms of radiation so far dis- cussed are operating simultaneously. Allowance is to be made for the following: (1) the combined actions of direct radiation of all kinds from the flame to the heat sink, (2) radiation from flame to refractory sur- faces, thence back by reradiation or reflection through the flame (with partial absorption therein) to the sink, (3) multiple reflection of all non- black surfaces, (4) convection, (5) external losses, and (6) for the fact that the refractory surfaces take up equilibrium temperatures which vary continuously over their faces. Allowance for space variation in temper- ature and emissivity of the gas introduces major complications, and the problem is here limited to consideration of a gas mass of uniform concen- tration and temperature equal to some set of mean values (see, however, the last part of Art. 4). The problem is otherwise capable of a solution free from seriously limiting assumptions. It will be convenient to group surface zones into two classes. Source-sink zones, such as a fuel bed, a carborundum muffle, a row of electric resistors, a liquid- or air-cooled surface and stock on a furnace hearth are designated by subscripts 1, 2, 3, . . . ; surface 1 has area Si, temperature 7'1, emissivity e1. Completing the enclosure are the insulating refractory connecting walls, which are heat sinks only to the extent that they lose heat by conduction through the walls. If the difference between gas convection to the inside of such a wall and conduction through the wall to the outside is small compared to the radiation incident on the wall inside, then the assumption that the net radiant heat transfer at the wall surface be zero is an excellent one. It enormously simplifies the problem of source-to-sink heat transfer and the effect thereon of the refractory surfaces. All such zones will be re- ferred to hereafter as no-fluz surfaces, with the understanding that refer- ence thereby is to radiant heat transfer alone, and the letter subscripts R, S, T, . . . will be appended to their properties. Several restrictions are imposed at this point, some of them to be re- moved later. (1) All of the sources and sinks, including the gas as well { 523 ) I - ENGINEERING CALCULATIONS OF RADIANT HEAT EXCHANGE as the surfaces, are assumed gray. A gray gas is one which exhibits, for radiation from whatever source, an absorptivity a equal to its emissivity e. Its emissivity and absorptivity vary, however, with path length. If the gas transmissivity 7, equal to 1 — a, is established for radiation from one zone to another, the transmissivity for twice the mean beam length be- tween zones is r?, a conclusion which can be true only for a gray gas which does not produce a change in quality of the radiation transmitted by it. (2) Such reflection as occurs, whether at a source-sink or at a no-flux surface, must be diffuse reflection, a term describing reflection which, like black radiation, obeys the cosine principle that appeared in Eq. 2-1; and emission from any surface must likewise obey the cosine principle. Non- metallic and oxidized metallic surfaces do not depart greatly from this characteristic. (3) A zone of the no-flux surfaces, or those of the source- sink surfaces which are not black, must be chosen small enough so that the intensity of radiation leaving the zone in consequence of irradiation by some other zone is uniform over the zone. This completes the assump- tions. The additional nomenclature needed is the representation of trans- mittance from one zone through the gas to another zone by 7 with ap- propriate subscripts to indicate the two zones involved. Flux between gray source-sink surfaces; the factor S12. The net flux between S; and S2 occurs by a complex process involving multiple reflec- tion from all source-sink surfaces as well as both reflection and reradiation from the no-flux surfaces; and one might at first consider the contribution ~ of Sr, Ss, . . . to the net flux between S; and S»2 impossible to disen- tangle, since the equilibrium temperature of Sr, for example, depends on contributions from S3, Su, . . . as well as from S; and S2. The new con- cept necessary here is that the refractory zone Sz can be thought of as having a partial emissive power due to the presence of each of the source- sink zones and the gas, and a total emissive power equal to their sum. Thus the term qi=2 represents net flux between Si and S2 consequent solely on their respective emission rates and includes, in addition to direct interchange S1F j2.€1e.0(T} — T)r12, the contributions due to multiple re- flection at all surfaces, as well as such contributions by reradiation from the no-flux surfaces as are consequent on their partial emissive powers due to the existence of S; and S» alone as net radiators in the system. This is the necessary meaning of qi—2 if it is to become zero when 7; = T2. It is apparent that qi22. must take a form equal to o(7? — T$) multiplied by some factor which depends on the geometry of the whole enclosure, the emissivity of its source-sink surfaces and the transmittance of the gas, and that it can be expressed in the form jaz = SiFne(T} — T2) = S2Fn0(Ti — T?) (4-1) The problem is to evaluate the new factor S, called the over-all exchange ( 524 ) I,4 - RADIANT EXCHANGE factor. Plainly, it cannot depend on any system temperatures. Conse- quently, if the gas and all source-sink surfaces except S; are kept at abso- lute zero, and qi=2 (which now becomes simply q1_,2) is evaluated and used to determine § in Eq. 4-1, that value of § will be generally applicable regardless of the particular combination of temperatures of the source- sink surfaces. Space does not permit presentation of the detailed deri- vation here [5, Chap. 4; 34]. Briefly, there are as many unknown emissive powers (due to reflection alone at all source-sink zones except one, and to reflection and/or emission at the no-flux zones) as there are zones; an energy balance may be written for each zone, thereby permitting a solu- tion. The evaluation of & necessitates the use of a determinant D, sym- metrical about its major diagonal, and of order equal to the total number of zones into which the enclosing surface has been divided. To simplify the expression of D and the solution for 5, a shorthand nomenclature is desirable. Let Sif i271r (which also equals SF zi7k1) be represented by 1R (or its equivalent R1, although the convention is adopted of mentioning small numbers first, and numbers before letters); because reflectivity (1 — e) appears so often, replace it (except in any final simplification) by p. Then ii S1 12 13 IR 1S eI: P1 phate S Las i a Raves 12 TS 23 2R 28 p2 13 23 2 ee 3R 38 ps D= IR 2R 3k RE aise es 1s 38 RS SS — Ss (4-2) With D defined, Sr5mn (= SnFnm) may be evaluated. Sen ae ea enn enn Dinn (4-3a) Pm Pn D n nn Din Siren aa enn («. Se enn 3) (4-3b) Pn Pn where D/,,, is the cofactor of row m and column n of D, defined as (—1)™*™ times the minor of D formed by crossing out the mth row and nth column. ( 525 ) I - ENGINEERING CALCULATIONS OF RADIANT HEAT EXCHANGE Then Sidhe SiSe Pi p2 12 3 4 OR aS 13/33. > *. 34 3R 38 2 Tron tag eee aR is p4 (4-4) We | Oe aR RSS, LUReS 1s 38 4S RS SSimuSs D The number of unique view factors F necessary for evaluation of § by Eq. 4-4 may be determined. By noting that in a p zone system there are p? F’s in the determinant D but that (a) D is symmetrical and (b) any row or column of F’s adds to 1, it is seen that the number of unique F’s necessary is p(p — 1)/2. If, in addition, each n of the zones cannot see itself, the number is further reduced by n. Eq. 4-4 can be used to make allowance for any degree of complexity of an enclosure, and to approach the true solution to any degree of approximation dependent on the number of zones into which a surface is divided. The guiding principle in deciding upon the number of zones necessary is that any reradiation or reflection must come from a zone small enough so that different parts of its surface do not have a signifi- cantly different view of the various other surfaces. Black source-sink sur- faces need be zoned only according to temperature, but light gray ones may require further subdivision. As one of the simpler examples of application of the determinant method, consider a system containing no emitting or absorbing gas. One case may be presented which covers a wide range of practical situations, i.e. the case of an enclosure divided into any number of no-flux zones but only two source-sink zones S; and S» (an especially justifiable assumption if the emissivities of S; and Sz are so high as to make reflections from their surfaces relatively unimportant in the over-all heat transfer). From Eq. 4-4 it may readily be shown that for this case 1 eff Lal 1 Te oa (he «al ety sulk lee : ly free Si eG ) a So (= ) 1 lee free, & 5) lack surfaces ( 526 ) I,4 - RADIANT EXCHANGE Black source-sink surfaces; the factor Fy.. Further to indicate the tech- nique of application of the determinant method, let Eq. 4-4 be used to determine the interchange factor when all source-sink surfaces are black. The interchange factor for this case has been designated by F, to indicate that it covers a more complex situation than F but a less general one than %. Let the problem be to evaluate F'12. Setting into Eq. 4-4 the con- dittonmiihatier|— ier —e7)-.4 2 — 2 onthat p| = ips — ps) 7) — Os onecan eliminate all rows and columns containing reference to any source-sink surface except 1 and 2 as follows. Cancel S;/p; out of the numerator and multiply the denominator first column by p:/S, which, being zero, makes the first column —1,0,0,0, . . . and reduces the order of D by one. Similarly, multiply the second column of the numerator and the second column of the new denominator by p;/Ss, making them become 0, —1,0,0, . . . each. Similarly, eliminate all terms containing numbers other than 1 and 2. One thus obtains 1D 2k 28 IR RR-—Sp RS 1S RS SS — Ss Shine = SS (4-6) If there is but one refractory zone, all rows and columns mentioning others may be crossed out, and Eq. 4-6 yields (LR) (2R) SiPis = 12 + 7 ee (4-7) This simple result, an approximation because all no-flux surfaces are assumed to be in equilibrium at a common temperature, is often adequate for estimating the transfer between S, and S». If no emitting or absorbing gas is present in the system, Eq. 4-7 yields Pal Rut pee (4-8) gas free The factor F1. for systems containing no interfering gas has been deter- mined exactly for a few geometrically simple cases [35]. If S: and S2 are equal parallel disks, squares, or rectangles connected by nonconducting but reradiating walls, F's. is given by Fig. 1,2b, lines 5 to 8. a2) I - ENGINEERING CALCULATIONS OF RADIANT HEAT EXCHANGE Flux between gas and surfaces; the factor 1,. The treatment of the general problem of radiation in a gas-containing enclosure was not com- pleted by presentation of Eq. 4-3 giving SnFmn. In addition to the various values of g,-, for different pairs of source-sink zones, one is interested in the net transfer between any one of such zones and the gas, which may well be the primary heat source of the system. The net radiant flux from the gas is given by deg, net = Qe=1 ata Ge=2 aE rms (4-9) and q,=1 is given by Qeai = SiFi1.0(T?2 — T?) (4-10) The problem is to evaluate 5,. Let S$; be the only original emitter in the system; all other source-sink zones and the gas are kept at absolute zero. S; radiates at the rate Sie; per unit value of black emissive power at tem- perature 7;. Radiation streaming away from Se, S;, ..., Sr, Ss, .. - is due solely to reflection at S2, S3, ... , and to reflection and/or re- radiation at Sr, Ss, . . . . Of the total emission Sje,, the amount S151, re- turns to and is absorbed be S1, SiS12 goes to and is absorbed by So, . . . Si51n is absorbed by S,. The residue must have been absorbed by the Bes since all other surfaces are nonretaining. Then SiSies On (Cie One SO toe eee i) (4-11) Since it will have been necessary to evaluate all of these $’s except 511 in fixing the radiant interchange in the system, 51; is the only new factor requiring evaluation to determine Si5,,. Another approach to the problem [5, Chap. 4; 36] yields the following direct formulation: Sen EDF Sang = HE (4-12) in which ,D, is obtained by inserting, into the nth column of D (liq. 422), the terms) —iSiysn@l tl2) =r) oes Qi 4.22:7023 > ee ), |, Sep CRI - + RR-+ -- -). Any one of these expressions may be written —S,[1 cas (Faitn1 + Fate + ‘one + Fartar To a i the parenthetical term of which equals the weighted-mean transmissivity or transmissivity for the total radiation arriving at or leaving S, (and therefore identifiable with a single subscript, 7,). The complement of 7, is the gas absorptivity and, because it is gray, the gas emissivity én. Note that «1 and e» differ only because the mean path lengths through the gas to S; and S» differ. Eq. 4-12 may of course be shown to be the equiva- lent of Eq. 4-11. Temperature of no-flux zones. Before the use of Eq. 4-11 is discussed, ( 528 ) 1,4 - RADIANT EXCHANGE the problem of interchange in a gray-gas-containing system requires one more item for completion. Near the beginning of Art. 4, the concept of partial emissive powers of a no-flux surface due to the separate effects of the various sources and sinks was presented. This leads to an evalu- ation of the equilibrium temperature of a no-flux surface. It may be shown [5, Chap. 4; 36] that Dal} + 2DeT$+ +> +> +.DeTi +--+ + DoT! Th = 5 (4-13) Since the sum of the De’s in the numerator can be shown to equal D, Kq. 4-13 states that T} is a weighted mean of the fourth powers of the various original emitters present, including all source-sink surfaces and the gas—as it must of course be. Application of this relation to determine the equilibrium refractory temperature in a simple system consisting of a gray gas enclosed by a single heat-sink zone S; and a single no-flux zone Spr, with all values of e«, taken to be the same, yields 1 Te = Ti- (T!- TY (4-14) bruit Mien I 3 1— ¢€ 1 €g Sr + af él a 3 +1) When 8; is black, the last term in the denominator drops out. Application of the factor 5,,. Returning now to the use of 5, con- sider the same simple system just used in illustrating the calculation of Tr. From Eq. 4-11 SiSic = Sie — Sifu = pe Se (al ee Hime (4-15) PAS a TR 1R R— Sr or 1 ey /ull if SiS. Si G ) i S,;- 11+ 1R’/(RR — Spr) ( ) When «; becomes 1, $1, by definition becomes F;, and, from Eq. 4-16, ae ee Siltie = (Si = We ae —- 4-17 Fi = Si mes (4-17) and 1 IL il 1 = —1 4-18 SiFic Si (" ) a; Sif i, ( ) I - ENGINEERING CALCULATIONS OF RADIANT HEAT EXCHANGE For this system all direct view factors can be expressed in terms of the single one Fr; as follows: IR = Sif irtir Fay SrF ririr iii = Syuru = Sil = Fyr)t1 — (Sy = SrFravru RR = SrFeetere = Srl — Faitee Substitution of these values in Eq. 4-17 and replacement of + by 1 — «, gives Sr Be Si Pie en en/err — €11 iy Aly Shen 1—eqn Fri ae (Sr/Si)F in is Nal Fri + err(1 — Fri) (eu. + err — 2er1 + €%: — €11€rr) (4-19) in which double-subscript e’s refer to an evaluation of gas emissivity based on a path length specific to the two surface elements mentioned. The problem, simple as it appeared to be, has a solution rather formidable for engineering use. If the various gas emissivities are assumed to be alike or if each one is replaced by their average value, called ¢,, it will be noted that the second term on the right of Eq. 4-19 will vanish, giving Sr Fi. =e 1+ : i (4-20) or Weary Some practical consequences of Eq. 4-18 and 4-20 are these: increasing the flame emissivity increases the heat transmission, but not proportion- ately; decreasing surface emissivity ¢: (and absorptivity) from one, when the flame is very transparent, produces but little effect on the heat trans- mission; but decreasing ¢: from one, when the flame is substantially opaque (e, = 1), produces a proportional decrease in heat transmission. The limitations on the validity of Eq. 4-18 and 4-20 must be borne in mind. They are restricted to a one-zone sink, a one-zone refractory or no-flux surface, and a gray gas. The first two assumptions are rigorously justifiable only when each element of Si (or Sz) shares its “‘view”’ of its own zone and of Sz (or 8) in the same ratio as every other element; and this in turn is true only when the two kinds of surfaces are intimately mixed in the same ratio on all parts of the enclosure, forming what one might call a ‘‘speckled”’ enclosure. Under those circumstances, the as- sumption made in going from Eq. 4-19 to 4-20, that the various «,’s are representable by an average value, becomes valid; and /'p1 becomes S;/ { 530 ) 1,5 - ENCLOSURE OF GRAY SOURCE-SINK SURFACES (S, + Sr). If Eq. 4-20 is used as an approximation for a system which does not have a speckled enclosure, however, use of the true value of Fri is preferable. Kq. 4-18 and 4-20 are well-known solutions, available and in use for many years [5, Chap. 3; 37] before the determinant method of derivation was available; and their derivation from first principles was perhaps as simple as the one here presented, but only because of restriction to a two-zone system. With the new method available, summarized in Kq. 4-3, the decision as to the number of zones of heat-sink or refractory area into which the enclosure should be divided can be made to depend, as it should, on the importance of the particular problem and the time available for handling it, rather than on whether the engineer can see his way through a multizone solution. Allowance for space variation in gas temperature. Many problems of heat exchange between combustion gases and their enclosing walls may be satisfactorily approximated by using mean values of gas temperature and composition. Where an accurate solution to the problem is of suf- ficient importance, however, allowance can be made for gradients in tem- perature and gas composition provided that knowledge is available of the flow pattern and progress of combustion; but the method is time-con- suming [38]. For orientation as to the need for allowing for radiation due to gas temperature gradients, a simple solution is available [39,40,41,42, 43\ for the following special case: When a unidirectional temperature gradient exists in the interior of a strongly absorbing gas far from its bounding walls, the radiant flux density ¢/S is given by q Aiea th) Outn dT Leg Sia reg eas Bde) 2 5 Ge LE a S 3h” dz 3 rae dx eo) where k is the absorption coefficient of the gas (see Art. 5), and p is the refractive index of the medium (1 for gases). To minimize the effect of a wall at distance L from the plane of flux, the value of kL must be greater than about 3; and to satisfy the condition of strong absorption Radiation exchange between a plane wall and an overlying gas, the isothermal surfaces in which are parallel to the wall, has also been treated [5, Chap. 4]. 1,5. Enclosure of Gray Source-Sink Surfaces Containing a Real (Nongray) Gas. In the derivation of interchange factors for gray gas systems the single value of transmittance t12 applied to all radiation leaving S; for So, whether originally emitted by Si or reflected from it after any number of passages through the gas. Only for gray gas is this ( 531 ) I - ENGINEERING CALCULATIONS OF RADIANT HEAT EXCHANGE true. For a real gas, with its characteristic absorption in certain spectral regions, the absorbable wavelengths are filtered out after a number of passages through the gas, and the transmittance of the gas for the re- mainder of the radiation approaches 1. The gray gas assumption thus leads to prediction of too large an interchange between gas and sinks, and too small an interchange between the source-sink surfaces. In ob- taining a value of Si51. applicable to a real gas, it is desirable to retain the mechanics of gray gas formulation. Fortunately, this is possible. For a gray gas the transmittance for the absorption path length repre- sented by p,L is e~*?s“, where k is the absorption coefficient of the gas, a constant independent of wavelength and therefore applicable to the integrated spectrum; and the absorptivity and emissivity equal 1 — e~*?24, The relation of transmittance to p,L for a real gas can be represented to any desired degree of accuracy by T= we berlh 4 ye hPL ae ge-kepL 1 1. - (5-1) and the emissivity relation by ee a(1 a e—kepL) AU y(1 mal e—kypL) ate 2(1 oo e—k=pL) BIE belo. (5-2) with k, representing the absorption coefficient applicable to fraction x of the total energy spectrum of the gas, and with the condition CO hie Ce eee ce Oe (5-3) Representing e—*:?4 by 7,, Eq. 5-1 yields for the transmittance Tn, of n layers of gas each of absorption path length p,L: Trek Tea ier cin Clune dae: (5-4) The components of which the total real-gas transmittance is composed are thus a series of gray body transmittances, each used with a weighting FACTORL 2: Yemen a Consider now an enclosure of surfaces which aid in the transmission of radiation from the gas to S,; only by the process of reflection at the other surfaces, i.e. a system of gray source-sink surfaces and completely reflecting, or white, no-flux surfaces. A little consideration will show that the real-gas solution for S,5;, is obtainable as the weighted sum of a num- ber of gray gas solutions, using successively 7,, 7,, and 7, for the gas trans- missivity and weighting each solution by the factors 2, y, 2, etc.; or Fig , e5i||) on a5 a on a my ae (5-5) use of rz use of ry and similarly Fi2 u oie |, on aw y5ie|, on (5-6) use of rz use of 7, ( 582 ) I,5 - ENCLOSURE OF GRAY SOURCE-SINK SURFACES The reason for the restriction on the validity of Eq. 5-5 or 5-6, that any no-flux surfaces, if present, must be white rather than gray, needs consideration. A white refractory surface reflects all incident radiation without changing its quality, i.e. without changing the fractions of it for which the gas will exhibit absorptivity 1 — 7,, 1 —7,, etc. But a gray refractory surface, to the extent that it absorbs and re-emits, changes the quality of the radiation. If, for example, a beam of radiation incident on Sr from the gas has an emissivity of } in one half of the energy spectrum, or a total emissivity and absorptivity of 4, the resulting radiation leaving Sez would be half absorbed by the gas on next passage through it if it left Sez by reflection without change in character, and only + absorbed if it left Sr by emission as black radiation. Since the derivation of S151, when Sr, Ss, . . . are present is based on attenuation by the gas in an amount independent of the history of a beam of radiation, the nongray gas solution represented by Eq. 5-5 applies rigorously only to systems in which any no-flux surfaces present are perfect diffuse reflectors. If allow- ance must be made for the grayness of any no-flux surface, it must be reclassified as a source-sink surface, say S;, of unknown temperature, the value of which is obtained by introducing the condition that the sum of the interchanges of S; with the other surfaces and with the gas must be Zero. The point has been emphasized that for gray systems the two terms each representing one-way flux in the expression Qiaz = SiFwoT} — S2Fo0T 3 differ only in temperature, that the SS product may be factored out. The imposing of the nongray gas condition makes this no longer true. Gas absorptivity for radiation from a source at 7’; is no longer necessarily equal to gas absorptivity for radiation from a source at 7's, except in the limit as T; approaches 7’,. Rather than use sequence of subscripts to indi- cate direction of the radiation, it is preferable to use an arrow and retain the equality of SiS12 and SoF21 as a matter of definition; but S151. does not now equal S151. except in the limit. F;2 is evaluated by use of the absorptivity of gas at T, for radiation from a black or gray source at 7; Fi-2 uses gas absorptivity based on emission from 7. Similarly the net interchange between gas and surface S; must now be written = = GASnatere Lie ras Si eee) (5-7) with S:5i_, based on the gas emissivity and S51, based on gas absorp- tivity for black or gray radiation from T). Plainly, however, if T, > 71, $1, is the term to evaluate rigorously, and it may be used with small error to represent both-way radiation. If T, and T are not too far apart, $1, evaluated by the use of an effective emissivity given by the bracketed term in Eq. 3-5 will probably suffice for both $i_, and 51... ( 533 ) I - ENGINEERING CALCULATIONS OF RADIANT HEAT EXCHANGE Although Eq. 5-5 or 5-6 can in principle be used to handle an en- closure of any degree of complexity as to zoning, and filled with gas of radiating characteristics producing any shape of curve of ¢, vs. p,L, a little consideration shows what an enormous amount of effort is involved if these expressions for § contain many terms. A simplification is mandatory and, fortunately, feasible. If wall reflectivities are not very large, a beam of radiation from the gas is rapidly attenuated in its succession of reflec- tions and transmissions, and the fitting of the ¢, — p,L curve is important only for a few units of p,l. The transmittance 7 given by Eq. 5-4 can be made to equal true transmittance at 0 and at two integral multiples of p,L by assuming the gas gray throughout the energy fraction x and clear throughout the fraction y + 2+ -:-:- =1-—4g, 1.e. by assuming that all the 7’s but 7, are zero and that an asymptotic transmissivity of 1 — x is approached as p,.l. = ©. An examination of Eq. 5-5 now indicates that, since all values of 5;, on the right-hand side except the first are for non- absorbing gas and therefore are zero, Fie a 75 | ea on (5-8) use of rz For source-sink surface interchange, Eq. 5-6 yields Sis oF | og on a (1 bi 2) 510 |, sea on (5-9) use of rz clear gas, T= There remains only the evaluation of x and 7, from a gas radiation plot such as Fig. I,3a. Let the objective be to fit the ¢,, p,.L curve at 1 and 2 units of p,L, and call the corresponding e’s read from the plot, ¢, and €2... From Eq. 5-4 Wes Vee — ae tl) =) and 1 — ye SS Oy = are + al = x) Solution of these gives 2 = = (5-10) and Ze, ran €2.g (5-11) t =~1l——=1- €g Recapitulating, 5;, equals x times a value of 5), from Eq. 4-12 (or Eq. 4-18 and 4-20) using a transmissivity of 1 — (e,/x) or an emissivity of e,/x, with a defined by Eq. 5-10. The determination of « and 7, from values of e at lp, and 2p,L is recommended when SS, has a low reflectivity and/or when p,L is large; but a small enclosure with heat-sink zones of high reflectivity may make ( 534 ) I,6 - APPLICATION OF PRINCIPLES the many-times-reflected radiation relatively more important. In that case some other pair of e, values may be used, such as e, and e€3.¢, OF €2.2 and €4... seldom is it necessary to add an extra term to Kq. 5-5. 1,6. Application of Principles. The procedures discussed above permit allowance for the effects of factors often casually handled in the past. The relations presented are not as easy to use as the relations of convective heat transmission; but this is because the mathematics of radiation in an enclosure, where every part of the system affects every other part, is intrinsically more complicated than the mathematics of heat transfer processes capable of expression in the form of a differential equation. With a little practice in manipulation of determinants® the reader should be able to evaluate & factors for systems of a considerable degree of complexity in a reasonable time. If the higher order determi- nants encountered are evaluated numerically for the specific example of interest rather than algebraically to obtain results like Eq. 4-19, the time required for a solution is not prohibitive. Some of the special cases en- countered are used so frequently, however, that algebraic formulation of their general solution is desirable. A few such cases are presented here. Real gas, gray sink Si, and reflecting no-flux surface Sr. The gray gas solution for this case, simplified by the use of a single path length and therefore a single 7, for all zone pairs, was given in Eq. 4-18 and 4-20. Modification to allow for nongray gas gives x SiS1ce¢ = il 1 1 (6-1) Si G We 1) u € Sr s Sia €,/X 1 1 a 1-— e/t Fri with x equal to e2/(2e, — e2,), and with «, evaluated for a path length given by Table I,3. If Sid. is wanted, ag: replaces ¢,. Real gas, enclosed by 2 gray sinks S; and Sz, and no Sz. This vari- ation on the previous case has interest for at least two reasons. Consider a gas, a primary heat sink S; and a refractory surface, the external loss from which is so large that it cannot be treated as a no-flux surface (the term no-flux still refers to radiant heat transmission only), because the internal gain by convection is so much less than the loss through the wall. Then the refractory surface becomes a secondary heat sink and is S» rather than Sz. Or consider a gas, a heat sink S; and a no-flux surface which is not justifiably classed as completely reflecting. As the discussion 6 Evaluation of higher order determinants by mechanical computers is of course feasible. The labor of evaluation with pencil and slide rule has been so greatly reduced by the method of Crout [44], however, that fifth or sixth order determinants need no longer be considered formidable by the engineer not equipped with the newer devices. ( 535 ) I - ENGINEERING CALCULATIONS OF RADIANT HEAT EXCHANGE in Art. 5 indicated, the no-flux surface must be treated as a source-sink type surface S,. Three S’s are necessary for a complete solution. Based on the simplifying assumption that the 7’s between zone pairs are all alike, Eq. 4-12 and 5-8 yield layin 1 1 A NO AG? lh Sh SENN ‘ ') Slee es iy a 1 1 1 pio Winey oie ahic/AN cotta So i Si 3 SiF yrs Fo, is obtained from the above by the interchange of subscripts 1 and 2. S151. is obtained from Eq. 4-4 and 5-6, which yield pak SiS12 Pree \ ry 1 1 1 1 (Ge co) team) denen) Ga) So + Si A SyP yrs laa = Ta 1 fe : Gl Way oD ae pee 1 2 1 pP2 S, wt agsieTRa as Sie These three §’s are for use in the heat transfer equations G1 = S153,0(T s = iT) 4 hiSi(T, ae T) (6-4a) (eS = Sitios = Is) ap liasCle = Ih) (6-4b) Qo=1 = SiFw0(T? — T}) (6-4c) (The difference between $,., and S,_, is here ignored for simplicity of treatment.) If surface S2 is losing heat to the outside at a rate S,U(T, — To), a heat balance on S2 yields Gem = G21 + S2U(T2 — To) or SsFoo( 14 =o T3) = (SiS (2) Sate SoU la = Lo) — heSa( ae) (6-5) Assuming the source temperature 7’, and the primary sink temperature T, to be known, Eq. 6-5 permits a solution for the unknown refractory temperature 7’. (but trial-and-error because mixed in first and fourth powers). The other application mentioned for this system of equations was to ( 536 ) I,6 - APPLICATION OF PRINCIPLES make allowance for grayness of a refractory surface. In this case S2 is truly a no-flux surface, with S,U(T2 — To) = hoS2(T, — T2). Then Eq. 6-5 can be readily solved for T3. If this is put into Eq. 6-4a and 6-4b and the radiation terms of those two equations are added, one obtains 1 Fz, net loss = q1, netgain — SiS 1. = by radiation by radiation 1 —|oPi-7) 6-6) Sifts T SoFo¢ The bracket, allowing as it does for the radiation from gas to S; with the aid of So, is like the term S151, for the system, gas—S:-Ser (with S2 repre- senting Sr) except that it now allows for the grayness of S». If, in Eq. 6-6, S» is assumed to be a white surface (op: = 1), it may be shown that the bracketed term reduces to the S15;, of Eq. 6-1. Estimation of heat transfer in a combustion chamber. Although rela- tions have been presented for evaluating radiant heat transmission in chambers filled with the combustion products of fuels, those relations have been restricted to idealized cases in which the gas temperature was uniform or was changing in one dimension along a flow path long com- pared to the transverse dimensions. Plainly, the average combustion chamber, in which combustion and mixing are occurring simultaneously and in a complicated flow path which involves recirculation as well, is far from typical of the idealized systems discussed. Those systems can never- theless provide an indication of the performance to be expected and can in many cases be used for quantitative prediction. The simplest case to discuss is the limiting one in which all dimensions of the chamber are of the same order of magnitude, and in which the mixing energy provided in the incoming fuel and air produces a turbulent gas mixture uniform in temperature throughout and equal to the temperature of the gas leaving the chamber. Assume the problem to be the determination of heat trans- fer in the chamber, given the mean radiating temperature 7; of the stock or heat sink, the chamber dimensions, and the fuel and air rates. Let the unknown mean gas temperature (and exit temperature) be 7’,. Then the net heat transfer rate from the gas is given by Coney = (Sree UE — I)) ae lonsh(Gieg 16) ae UrSr(T, — To) (6-7) The sink area S/ at which convection heat transfer occurs is indicated as possibly different from the area S; at which gas radiation occurs, because heat sink surfaces such as a row of tubes covering the gas outlet from the chamber, and therefore receiving by convection no heat which affects the mean gas temperature in the chamber, should be included in S; but not in S{. Convection from gas to Sz has been assumed equal to the loss through Sp, which replaces it in the equation. With the outside air tem- perature 7’) known, Eq. 6-7 expresses a relation between two unknowns, ( 537 ) I - ENGINEERING CALCULATIONS OF RADIANT HEAT EXCHANGE Jz, net and T’,. The other relation is an energy balance, such as Ye, net = b= mel, Ti To) (6-8) where 7 represents the hourly enthalpy of the entering fuel, air, and re- circulated flue gas, if any, above a base temperature 7’) (water as vapor); and c, represents the heat capacity (mean value between 7’, and 7'y) of the gas leaving the chamber, at hourly mass rate m. Eq. 6-7 and 6-8 may be solved, usually by trial and error, to give gz, n-, and 7. The limitation on dz, that it does not include gas convection at area S — S’, must be borne in mind. The pair of equations just discussed applies strictly to one of two limiting combustion chamber types—that one in which the assignment of a mean flame temperature equal to the temperature of the gases leaving is justifiable. The method consequently predicts the minimum heat trans- fer of which the system is capable. Better agreement between predicted and experimental results is obtained on some furnaces when the assump- tion is made that flame temperature and exit gas temperature are not the same but differ by a constant amount. In a number of furnace tests used to determine what value of this difference produces agreement be- tween experiment and the equations recommended, the difference was found to be about 300°F. The other extreme in chamber types is that one in which combustion occurs substantially instantaneously at the burners (through complete premixing of fuel and air); the temperature attained is that generally known as theoretical flame temperature or adiabatic combustion tem- perature; and the temperature falls continuously as the gases flow from burner to outlet. When such a chamber is long compared to its cross sec- tion normal to the direction of gas flow, Eq. 6-7 may be considered as applying to a differential length, and the remarks in connection with Eq. 3-6 are applicable. One must, however, be prepared to examine the validity of the assumption that radiant flux in the gas flow direction is of secondary significance. Allowance for the improbability of attainment of adiabatic flame temperature at the hot end of the chamber may be made, though somewhat arbitrarily, by use of what Heiligenstidt [45] calls a pyrometric efficiency, the factor by which to reduce the adiabatic flame temperature to obtain the true value. If the gases are assumed constant at this temperature from burner inlet until, by the calculation method just outlined, they have lost enough heat to equal the difference between their entering enthalpy and that at their assumed temperature, and they are allowed thereafter to cool in step with their heat transfer rate, better agreement with experimental data can of course be obtained—provided there is knowledge of what to use for the pyrometric efficiency. It varies primarily with burner and chamber design and fuel type. A value of about 0.75 has been used in application to steel reheating furnaces. The chief ( 538 ) I,7 - CITED REFERENCES function of the pyrometric efficiency concept is its use in fitting heat transfer theory to heat transfer data on combustion chambers. In other systems not too different in design, the same pyrometric efficiency can profitably be used. The derivation of many of the relations presented in this section has been prevented by the space limitation. These derivations, together with an application of some of the principles to the solution of numerical prob- lems, will be found in [5, Chap. 4]. I,7. Cited References. . Foote, P. D. J. Wash. Acad. Sci. 5, 1 (1915). . Schmidt, H., and Furthmann, L. Mitt. Kaiser-Wilhelm-Inst. Eisenforsch. Diisseldorf, Abhandl. 109, 225 (1928). . Michaud, M. Sc.D. Thesis, Univ. Paris, 1951. . de Corso, 8S. M., and Coit, R. L. Mech. Eng. 76, 682 (1954). McAdams, W. H. Heat Transmission, 3rd ed. McGraw-Hill, 1953. . Binkley, E. R. Heat transfer. Am. Soc. Mech. Engrs. 40-46, 1933-1934. . Sieber, W. Z. tech. Phys. 22, 130-135 (1941). . Person, R. A., and Leuenberger, H. Union Carbide Co. private communication, 1955. . Hottel, H. C. Trans. Second World Power Conf. 18, Sec. 32-243, 1930. . Hottel, H.C. Mech. Eng. 52, 699-704 (1930). . Hottel, H.C. Trans. Am. Soc. Mech. Engrs. FSP 58, 265-273 (1931). . Seibert, O. Wedrme 54, 737-739 (1931). . Hooper, F. C., and Juhasz, I. S. Fall Meeting Am. Soc. Mech. Engrs. Paper 52F 19, Sept. 1952. . Hamilton, D. C., and Morgan, W. R. Radiant-interchange configuration factors. NACA Tech. Note 2836, 1952. . Intern. Comm. on Flame Radiation. J. Inst. Fuel London 24, 8 (1951); 26, 8 (1952). . Intern. Comm. on Flame Radiation. Journée d’ Etudes sur les Flammes, Mar./June, 1953. . Sherman, R. A. Trans. Am. Soc. Mech. Engrs. 79, 1727-41 (1957). . Hottel, H. C., and Broughton, F. P. Ind. Eng. Chem., Anal. ed. 4, 166-175 (1932). . Senftleben, H., and Benedict, E. Ann. physique 60, 297 (1919). . Yagi, S. J. Soc. Chem. Ind. Japan 40, 50B (1937); 40, 144 (1937). . Wolfhard, H. G., and Parker, W. G. Proc. Phys. Soc. London B62, 523 (1949). . Hottel, H. C., and Mangelsdorf, H. G. Trans. Am. Inst. Chem. Engrs. 31, 517-549 (1935). . Hottel, H. C., and Smith, V. C. Trans. Am. Soc. Mech. Engrs. 57, 463-470 (1935). . Howard, J. N.,et al. Near-infrared transmission through synthetic atmospheres. Air Force Cambridge Research Center Geophys. Research Paper 40, 1955. . HoLeong, E. Mass. Inst. Technol. Chem. Eng. Dept. Internal Rept., Feb. 1957. . Wu, W. Mass. Inst. Technol. Chem. Eng. Dept. Internal Rept., June 1957. . Schmidt, E. Forsch. Gebiete Ingenieurw. 8, 57 (1932). . Schmidt, E., and Eckert, E. Forsch. Gebiete Ingenieurw. 8, 87 (1937). . Hottel, H. C., and Egbert, R. B. Trans. Am. Inst. Chem. Engrs. 38, 531-565 (1942). . Hottel, H.C. Trans. Am. Inst. Chem. Engrs. 19, 173 (1927). . Hottel, H.C. Ind. Eng. Chem. 19, 888 (1927). . Port, F. J. Sc.D. Thesis in Chem. Eng., Mass. Inst. Technol., 1940. . Cohen, E. 8. M.S. Thesis in Chem. Eng., Mass. Inst. Technol., 1951. . Hottel, H. C. Notes on radiant heat transmission. Mass. Inst. Technol. Chem. Eng. Dept., 1951. ( 539 ) - ENGINEERING CALCULATIONS OF RADIANT HEAT EXCHANGE . Hottel, H. C., and Keller, J.D. Trans. Am. Soc. Mech. Engrs., Iron and Steel 65-6, 39-49 (1933). . Hottel, H. C. Notes on radiant heat transmission. Mass. Inst. Technol. Chem. Eng. Dept., 1953. . Hottel, H. C. Notes on radiant heat transmission. Mass. Inst. Technol. Chem. Eng. Dept., 1938. . Hottel, H. C., and Cohen, E. 8. A. I. Chem. Eng. J. 4, 3-14 (1958). . Shorin, S. N. Jzvest. Akad. Nauk S.S.S.R., Otdel. Tekh. Nauk 3, 1951. . Kellett, B.S. J. Opt. Soc. Amer. 42, 339 (1952). . Genzel, L. Z. Physik 135, 177-195 (1953). . Konakov, P. K. Jzvest. Akad. Nauk S.S.S.R., Otdel. Tekh. Nauk 3, 1951. . Filippov, L. P. Izvest. Akad. Nauk S.S.S.R., Otdel. Tekh. Nauk 1, 155-156 (1955). . Crout, P.D. Trans. Am. Inst. Elec. Engrs. 60, 1235 (1941). . Heiligenstadt, W. Arch. Hisenhiittenw. 1, 25, 103 (1933). { 540 ) INDEX absorption coefficient, average, 499 specific, 493 absorptivity, spectral, 492, 494 surface, 504 aerodynamic heating, 54 calculation of skin temperature, 406 correlation of theory and experiment, 412 high speed vehicles, 405 slip flow, 415 Allen, C. Q., 49 amplification of small disturbances, 68 Ashkenas, H., 157 Atsumi, 8., 25 Bailey, G. W., 6 Barnes, H. T., 39 Batchelor, G. K., 219, 221, 227, 230-232, 237, 244, 250 Beer’s law, 493 Bertram, M. H., 57, 67 Betz, A., 69 Biot number, 262, 264, 271 Birkhoff, G., 221 black body, energy distribution in radi- ation from, 502 black body radiation, 489 Blasius, H., 4, 344, 479 Blasius flow, 4 Blasius shear distribution, 344 Blasius solution, 440 Boden, R. H., 416 boiling heat transfer, dimensional groups in, 334 mechanism, 328 nucleation in, 320 typical results, 314 with forced convection, 313 with free convection, 313 Boison, J. C., 67, 398, 402, 403 Boltz, F. W., 48 Boltzmann equation, 98 Bouguer-Lambert law, 493 boundary layer enthalpy, 349 boundary layer on flat plate, equations for turbulent compressible flow, 89 boundary layer parameters, equilibrium profiles, 137 boundary layer thickness, displacement, 131, 450 boundary layer thickness, hydrodynamic, 440 thermal, 440 breakdown of laminar flow, 27 Bringer, R. P., 310 Brinich, P. F., 57, 61, 62 bubble growth, 323, 327 bubble population, in nucleate boiling, 321 bulk convection and gradient diffusion, 169 Burgers, J. M., 196, 247 burnout point, 315, 321 effect, of pressure on, 316 of temperature on, 316 of velocity on, 316 in aerated water, 331 in carbon tetrachloride, 329 in Freon, 318, 331 in jet fuel, 318, 331 in nitric acid, 331 in water, 329 in water-aerosol solution, 329 Bursnall, W. J., 24, 25 carbon tetrachloride, bubble growth in, 327 burnout point in, 329 centrifugal field, stabilizing action of, 51 Chandrasekhar, S., 219, 238, 239 Chapman, D. R., 107, 368, 392 Chou, P. Y., 246 Clauser, F. H., 130, 132, 134, 135, 144, 149, 391 Coker, E. G., 39 Colburn, A. F., 381 Coles, D., 57, 131, 182, 142, 391 Coles’ wake function, 140 combustion chamber, estimation of heat transfer in, 537 composite hollow cylinder, heat flow in, 280 composite wall, heat conduction in, 272 minimum weight criterion, 278 convective heat transfer, mechanism of, 339 convergence in the mean, 259 coolant, regenerative, 428 coolant Reynolds number, 468 Cope, W. F., 384 { 541 ) INDEX corners, flow and heat transfer near, 303 correlation, statistical, 199 correlation function, 199, 202, 203, 205, 206, 208, 209 relation to spectral function, 211 correlation tensor, double, 203, 208 triple, 205 correlations, in boiling heat transfer, 320, 335 involving pressure, 206, 209, 236 Corrsin, 8., 52, 164, 166, 168, 171, 176, 206, 241, 245 Craya, A., 247 critical pressure, in boiling heat transfer, 318 Crocco, L., 342, 443 Crocco transformation, 342 Czarnecki, K. R., 59, 62, 66, 398 Darcy’s law, 482 decay of homogeneous turbulence, early period of, 232 final period of, 231 laws of, 230 Deissler, R. G., 291 density gradients, stabilizing and de- stabilizing effects, 53 Diaconis, N. 8., 67 diffusion of energy, viscous, 84 dimensionless groups, in boiling heat transfer, 334 displacement thickness, 131, 450 dissociated air, properties of, 422 dissociation effects, 419 of heat transfer near the stagnation point, laminar flow, 422 turbulent flow, 424 double velocity correlations, longitudinal, 202 transverse, 202 Drougge, G., 32 Drude’s law, 491 Dryden, H. L., 10, 12, 29, 226, 250, 404 Dumas, R., 211 Dunn, D. W., 63, 65, 67 Duwez, P., 430 Eber, G. R., 66, 369, 398 Eckert, E. R. G., 310, 484 eddy diffusivity, 289, 291, 292, 305 ratio, 294 eddy Reynolds number, 146 eddy viscosity, in boundary layers, 143 eigenfunctions, 257 eigenvalue problem, 257 eigenvalues, 257, 466 Einstein, H. A., 294 Ekman, V. W., 39 Elrod, H. G., 298 emissive power, 490, 492 of metals, 492 emissivity, 502 spectral, 492, 493 total, 491, 493, 499, 505 Emmons, H. W., 6, 69 energy dissipation in isotropic turbu- lence, 212 energy spectrum, stability of, 239 energy transfer hypotheses in homo- geneous turbulence, 238 enthalpy recovery factor, 349 entrance heat transfer, effect of various factors on, 301 entrance length, comparison of laminar and turbulent, 300 effect of Prandtl number on, 300 entrance region, heat transfer in, 298 equation of energy and enthalpy, 85 equilibrium boundary layer according to Clauser, 135 equilibrium profiles, 136 error function, complementary, 262 Euler number, 447 exchange factor in radiant heat transfer, application to simple enclosure, 529 from gas in enclosure, 528 modification for real gas, 532, 534 over-all, 524, 525 extinction coefficient, 493 Fage, A., 9, 11, 23, 47, 48, 49 Favre, A. J., 211 Feindt, E. G., 8, 11, 12, 18 Fila, G. H., 53 film boiling, 316, 333 film cooling, 429, 481 critical injection velocity, 483 film heat transfer coefficient, 442 Fischer, W. W., 412 Flachsbart, O., 23 flatness factor, 200 flow separation, 130, 446 fluctuations, role of, in generating bubbles, 322 form of turbulent motions, general, 78 Forstall, W., 171, 181 Fourier equation, 435 Fourier series, 259 Fourier transform relation, 211, 218 derived, 215 free stream boundaries of turbulent flow, 127, 163 free stream boundary, spreading, 166 mechanism of { 542 ) INDEX free turbulent flow, behavior in terms of an eddy viscosity, 163 definition of, 158 general characteristics of, 158 laws of spreading and decay, 159 similarity conditions, 161 simplified equations, of heat transfer, 162 of motion and continuity, 160 transport processes, 168 Frenkiel, F. N., 228 Freon, burnout point in, 318, 331 frequency distribution of turbulent fluc- tuations, 198 frequency of radiation, 490 friction, in boiling heat transfer, 332 friction velocity, definition of, 120 function set, orthogonal, 258 gas radiation, 513 carbon dioxide, 513 effect of temperature variation along flow path, 519 gases other than carbon dioxide and water, 523 interference of CO, and H.20, 516 water-vapor, 514 gas transmittance, gray, 532 real, 532 gas turbine blade, cooling of, 445 Gault, D. E., 25, 26 Gaussian distribution, 199 Gaviglio, J. J., 211 Gieseler, L. P., 56 Goland, L., 368 Goldstein, 8., 52, 223, 229, 236 Gortler, H., 8, 46, 174, 180 Gortler instability, 8, 46, 69 Gottingen equivalent sand-grain rough- ness, 18 gray body, definition of, 504 Green, A. E., 223 Greenfield, S., 418 Gregory, N., 52 Grimminger, G., 366 eal PAGVAS Sik Hama, F. R., 10, 236 Haimmerlin, G., 46 Hantzsche, W., 362 Haslam, J. A. G., 38 heat conduction, differential equation of, 255 radial, 266 variable specific heat, 281 variable thermal conductivity, 281 heat conduction, in composite slab, 272 in nozzle walls, 285 heat flux, Newtonian, 254 heat transfer, across a turbulent stream, 454, 466 film cooling, 416, 418, 482 liquid metal, 301 mechanism of, in boiling heat transfer, 328 heat transfer coefficient, 260, 270, 448, 467, 468, 478 effective, 276 heat transfer and skin friction, Reynolds analogy, 104 heat transfer in a laminar boundary layer, exact solution, 446 Heisenberg, W., 217, 227, 237, 238 Hermann, R., 53 Higgins, R. W., 65, 398 Hill, J. A. F., 369 Hinze, J. O., 170, 174 Hislop, G. §., 31 hollow cylinder, heat conduction in, 266 Holstein, H., 9, 16 Homann, F., 365 homogeneous anisotropic turbulence, 201, 218 dynamical equations for, 219 homogeneous isotropic turbulence, dynamical equations for, 210 kinematics of, 202 homogeneous medium, heat conduction in, 260 homogeneous turbulence, 200 Hopf, E., 196 horseshoe vortex, 30 Howarth, L., 210, 227 image source, 262, 263 independence principle, turbulent flow, 157 instability, turbulent boundary layer, 139 integral methods, 299 application of, to turbulent boundary layers, 153 intermittency factor, 164 invariant theory of turbulence, 206 isotropic turbulence, 200 dynamics, 208 energy dissipation in, 212 kinematics, 202 spectral theory of, 210 Tuchi, M., 11, 12 Jack, J. R., 67 Jacobs, E. N., 25 ( 543 ) INDEX Jedlicka, J. R., 66 jet, angle of spreading, 176 effect, of density, 176, 179 of free stream, 179 on heated jet, 182 experiments on free stream effect, 181 theory of free stream effect, 180 jet fuel, burnout point in, 318, 331 Johansen, F. C., 23 Jones, B. M., 24 Kampé de Fériet, J., 196, 219 Kenyon, G. C., 49 Kester, R. H., 107, 392 Kistler, A. L., 164, 166, 206 Klebanoff, P. S., 6, 15, 30, 45, 250 Knuth, E. L., 483 Kolmogoroff, A. N., 212, 223, 225, 228 Kolmogoroff’s scales, 224, 229 Kolmogoroff’s theory, 221, 223 Korkegi, R. H., 391 Korobkin, I., 367 Kovasznay, L. 8. G., 238 Kuethe, A. M., 159, 180 laminar flow, cone solution, 362 effect of variable free stream pressure and variable wall temperature, 368 experimental knowledge, 368 flat plate solution, 341 heat transfer in, 341, 348 heat transfer coefficient, 349 for heated wind tunnels, 355 free flight, 354, 355 numerical calculations, 350 stagnation point solution, 365 laminar layer, 293 laminar mixing region, 52 laminar sublayer, 435, 452, 454 Lange, A. H., 56 large scale structure of turbulence, 219 Laufer, J., 56, 57 law of the wake, according to Coles, 139 illustration of, 142 physical interpretation of, 142 law of the wall, 122 fully rough, 148 Lee, R. E., 56 Lees, L., 56, 63, 396, 452 Lessen, M., 23 Levy, S., 368 Lewis, J. W., 50 Li, H., 294 Liepmann, H. W., 7, 8, 53 Lighthill, M. J., 248, 368 Lin, C. C., 56, 63, 65, 67, 211, 220, 225, 228, 229, 231, 236, 247, 291, 452 Linke, W., 21, 25 liquid metal heat transfer, 301 Loftin, L. K., Jr., 24, 25, 45 logarithmic velocity law, derivation of, 125 range of validity of, 125, 132 rough wall, 151 Loitsiansky, L. G., 32, 220, 227 Loitsiansky parameter, 221, 226 luminosity, large particle, 512 oil flame, 512 soot, 511, 512 Luther, M., 62 MacPhail, D. C., 50, 205 Maekawa, T., 25 magneto-hydrodynamic turbulence, 248 Malkus, W. V. R., 247 Mangler, W., 48 Mangler transformation, 48 Marte, J. E., 56, 57 Martinelli, R. C., 302 mass transfer, 297 mean beam length, 519 mean values in turbulent motion, 81, 197 Mickley, H. S., 391 Millikan, C. B., 124 Millionshchikov, M., 227, 236 Mitchner, M., 6, 247 Mituisi, 8., 10 mixing in jets and wakes, 168 mixing length, 101, 143, 146 mixing length theory, 168 mixing zone, effect of density, 178 mixing zone of supersonic jet, 178 Moeckel, W. E., 57 momentum thickness, 110, 121, 131, 450 momentum transfer, across a turbulent stream, 454, 471 Morris, D. N., 368 Munk, M., 68, 69 Navier-Stokes equations, 82, 197, 461 Newtonian flow, 366 Nikuradse, J., 147, 375, 477 nitric acid, burnout point in, 331 nonblack radiator, 491 noncircular passages, 303 Norris, R. H., 412 nozzle walls, heat conduction in, 285 nucleate boiling, 316 bubble population in, 321 nucleation, in boiling heat transfer, 320, 323, 335 ( 544 ) INDEX nuclei, source of, in nucleate boiling, 321 nucleus, equivalent spherical diameter of, 320, 321 nucleus size, effect on bubble growth, 325 Nusselt number, 296, 306, 446, 449, 468 O’Brien, V., 206 Obukhoff, A. M., 238 Obukhoff spectrum, 225 optical depth, 492 origin of turbulence, 67 oscillations, Tollmien-Schlichting, 5 Pappas, C. C., 65, 107, 398 partial film boiling, 316 permeability of porous metal, 431 Phillips, O. M., 248 Planck’s radiation law, 489 Pohlhausen, H., 439, 449 Poiseuille flow, 461, 462, 465 Poroloy, 431, 477 Poroloy process, 429 porosity, 430 porous medium, 430 Potter, J. L., Jr., 60 power formula, limitations, 122 power law, in turbulent flow, basis of, 119 boundary layer thickness, 121 skin friction, 120 velocity distribution, 120 Prandtl, L., 49, 53, 122, 168, 290, 294, 341, 455 Prandtl mixing length, 382, 456, 473 formula, 108 Prandtl number, 341, 440, 441 turbulent, 102 pressure changes across turbulent bound- ary layer, 90 pressure correlation, 206, 209, 236 pressure gradient effect, on boundary layer, 130 on skin friction, 133 pressure gradient parameter for equilib- rium profiles, 146 Preston, J. H., 48, 49 probability distribution, 198 joint, 199 propellant, 428 Proudman, I., 221, 232, 237, 239, 248 quasi-Gaussian approximation, 200, 236 Quick, A. W., 11, 29 radiant beam length, mean, 519 radiant exchange, among wall elements of black enclosure, 510 black source sinks, no gas, 527 radiant exchange, real gas, gray sink and no-flux surface, 535 real gas, 2 gray sinks, no-flux surface, 535 system containing gray gas, 523, 528 two source sinks, no gas, 526 radiant flux, 490 radiant interchange, nongray body in enclosure, 504 Rannie, W. D., 309, 454, 483 Rayleigh-Jeans radiation law, 490 real gas, 531 reattachment of separated boundary layer, 41 recovery factor, 54, 92 effect of, Mach number on, 94 Reynolds number on, 92 recovery temperature, 54 reference temperature, 306, 308 reflector, diffuse, 533 Regier, A., 52 Reichardt, H., 53, 294 Reid, W. H., 237, 244 Reis, F. B., 247 Reissner, E., 231 Reynolds, O., 288, 290, 349, 378, 454 Reynolds analogy, 290 inboiling heat transfer, 333 Reynolds analogy factor, 349 Reynolds equations, 197 Reynolds number, 296, 349, 441, 455 local boundary layer, 20 Reynolds stresses, 197 Riabouchinsky, D., 52 Richardson, L. F., 244 Riddell, F. R., 157 Rivas, M. A., Jr., 369 Robertson, H. P., 203, 206 rocket motors, effect of film cooling on heat transfer in, 416, 418 heat transfer in, 415, 419 due to radiation, 417 Rosenhead, L., 23 Rotta, J., 246 roughness, 147 aerodynamic effect, 147 effect on shape parameter, 152 effect on transition, 8, 38, 49, 62 equivalent sand-grain, 147 limit, for aerodynamically smooth condi- tion, 149 for fully rough condition, 149 sand-grain, 18, 45 spherical, 45 ( 545 ) INDEX roughness, three-dimensional, 45 roughness Reynolds number, 147 Rubesin, M. W., 368, 391 scales of turbulence, 78 Scherbarth, K., 11 Scherrer, R., 65, 398 Schiller, L., 10, 21, 25, 39 Schlichting, H., 5, 32, 53, 447, 452 Schlichting’s computation of transition, 37 Schréder, K., 29 Schubauer, G. B., 5, 6, 15, 25, 30, 41, 45, 241, 250 secondary motion, 40 Seiff, A., 66 semi-infinite solid, heat conduction in, 262 separation, 21, 29 separation ‘‘bubble,’’ 24, 29 Shapiro, A. H., 171, 181 shear flow, definition of, 76 relation to turbulence, 76 statistical theory, 245 shear layer, reattachment of, 24, 41 shear layers, transition of, 21, 29 shearing stress, 454 Shen, 8. F., 247, 291 Short, B. J., 392 Shoulberg, R. H., 369, 393 Sibulkin, M., 362 similarity in isotropic turbulence, 224, 225, 229 Sinclair, A. R., 59, 62, 66, 398 skewness factor, 200 skin friction, compressible flow, basis of theories, 107 empirical laws, 113 experiment and theory compared, 116 for transpiration-cooled walls, 446 fully rough wall, 148 skin friction coefficient, 449, 457, 464, 478 definition of, 113, 120 determination from velocity profile, 134 skin friction laws in compressible flow, von K4rm4n and Prandtl hypotheses compared, 112 skin-friction logarithmic law, Prandtl-Schlichting formula, 128 Squire-Young formula, 128 von Karmdn formula, 127 von Kaérmdn-Schoenherr formula, 128 skin friction measurements, incompressi- ble flow, 129 Skramstad, H. K., 5, 6 Slack, E. G., 58 small scale structure of turbulence, 221 Smith, A. M. O., 39 Smith, J. M., 310 Smith, J. W., 368 Snodgrass, R. B., 67 Sommer, S. C., 392 Spangenberg, W. G., 250 spectral analysis of turbulence, one-dimensional, 214 three-dimensional, 216 spectral function, relation to correlation function, 211 spectral radiancy, 489 spectrum, equation for change of, 213 of disturbances, 68 Squire, H. B., 180, 365 stability in accelerated and retarded flow, 32 stabilization, cooling required for, 396 effect of centrifugal field, 51 effect of density gradient, 53 stagnation enthalpy, 446 stagnation point, 447 instability, 46 Stalder, J. R., 58 Stanton, T. E., 349 Stanton number, 104, 349 statistical averages, 198 statistical theory of turbulence, 196, 201 Stefan-Boltzmann constant, 490 Stefan-Boltzmann law, 502 Stephens, A. V., 38 Sternberg, J., 55 Stewart, R. W., 200, 232 Stine, H. A., 368 structure of turbulence, large scale, 219 small scale, 221 Stuart, J. T., 52 Stiiper, J., 11 subcooling, 313, 316 supercritical carbon dioxide, heat transfer in, 310 supercritical fluids, 309 surface erosion, 284 surface melting, 284 surface of discontinuity, 23 surface tension, boiling heat transfer, 321, 322 effect on bubble growth, 325 sweat cooling, 429 Szablewski, W., 180, 182 Tani, I., 10-12 Tatsumi, T., 237 ( 546 ) INDEX Taylor, G. I., 30, 39, 50, 205, 210, 212, 222, 223, 241, 250, 290 Taylor turbulence parameter, 31, 57 Taylor’s cellular ring vortices, 50 temperature, defect, 272 difference ratio, 468, 469 distribution, 294, 295, 308 effect, on bubble growth of, 327 on radiation of space variation in, 531 mean value in gas radiation, 521 of no-flux walls in enclosure, 529 temperature-velocity relationship in tur- bulent boundary layer, 91, 95, 109 Tetervin, N., 130, 154 Theodorsen, T., 30, 52 thermal boundary layer, 487, 442 thermal capacity, 278 thermal diffusivity, growth, 327 thermal properties, variable, 280 of various materials, 260 thermal shield, thick, 277 thin, 276 thermal shock, 270 thermal stresses, 268 thermoelastic equations, 269 thickness number, hollow cylindrical shell, 268 three-dimensional effects, significance in boundary layers, 156 yawed flow, 157 Tidstrom, K. D., 15, 45 Tollmien, W., 5, 452 Tomotika, S., 47 Townsend, A. A., 144, 159, 164, 166, 168, 171, 172, 206, 225, 227, 230, 232, 251 transition, at boundary of a jet, 52 at hypersonic speed, 67 at supersonic speed, 54 determination by local parameters, 37 dimensional analysis, 19 effect of, cooling, 397 curvature, 8 cylindrical wires, 10, 49 distributed roughness, 16 flat ridges, 11 heat transfer, 53, 54, 63 Mach number, 55 noise, 46 nose shape, 54, 57 pressure gradient, 6 roughness, 8, 38, 49, 62 effect on bubble transition, effect of, sand-grain rough- ness, 18, 45 scale of turbulence, 31, 47, 57 shock waves, 55 single roughness elements, 8, 61 spherical roughness elements, 15, 45 supply turbulence, 399 surface roughness, 403 surface temperature, 53 three-dimensional roughness ments, 15, 45 turbulence, 30, 41, 47, 55 two-dimensional roughness elements, 10 waviness, 8, 38, 45 in flow between rotating cylinders, 49 in pipe, of annular cross section, 40 of circular cross section, 39 of rectangular cross section, 40 of square cross section, 40 with curved axis, 40 with sharp-edged entrance, 40 in propeller wake, 46 in rough pipes, 40 near rotating disk, 52 on airfoils, 32, 38, 41, 58 effect of, angle of attack, 43 roughness, 43 flight data, 45 on airplane configurations, 45 on airplanes in flight, 45 on bodies of revolution, 46, 47, 49 on concave surfaces, 8 on cone, 55 on cone cylinder, 59 on convex surfaces, 8 on elliptic cylinder, 41 on flat plate, 31, 57 on hollow cylinder, 8, 11, 55, 61 on ogive cylinder, 59 on paraboloid cylinder, 65 on prolate spheroid, 48 on Rm-10 model, 59, 65 on sphere, 46 physical mechanism of, 28 relation to stability of laminar bound- ary layer, 396 transmittance, evaluation from gas radiation chart, 534 gas, 532 transpiration-cooled boundary layer, heat transfer in, 437 laminar, approximate solution, 488 compressible, 442 pressure gradient, 445 ele- { 547 ) INDEX transpiration-cooled pipe flow, heat transfer in, 460 perturbation parameter, 462 transpiration cooling, 429, 438 transport coefficients, turbulent, 102 transport processes, fundamental con- siderations, 97 transport theory, comparison with experiment, 170 critical examination of, 169 triple velocity correlations, 205 Trouncer, J., 180 Tsien, H. S., 415, 483 Tsuji, H., 236 turbulence, origin in shear flow, 76 sustaining mechanism, 77 turbulence, statistical theory of, correlation function, 199, 202, 203, 205, 206, 208, 209 relation to spectral function, 211 correlation tensor, double, 203, 208 triple, 205 correlations involving pressure, 206, 209, 236 decay of homogeneous turbulence, early period of, 232 final period of, 231 laws of, 230 diffusion (see turbulent diffusion) double velocity correlations, longitudinal, 202 transverse, 202 energy dissipation bulence, 212 energy spectrum, stability of, 239 energy transfer hypotheses, 238 flatness factor, 200 Fourier transform relation, 211, 218 derived, 215 frequency distribution of turbulent fluctuations, 198 homogeneous anisotropic turbulence, 201, 218 dynamical equations for, 219 homogeneous isotropic turbulence, dynamical equations for, 210 kinematics of, 202 homogeneous turbulence, 200 invariant theory of, 206 isotropic turbulence, 200 dynamics, 208 energy dissipation in, 212 kinematics, 202 spectral theory of, 210 von Kérman-Howarth equation, 210 Kolmogoroftf’s scales, 224, 229 in isotropic tur- turbulence, Kolmogoroff’s theory, 221, 223 Loitsiansky parameter, 221, 226 mean values, 81, 197 Obkhoff spectrum, 225 pressure correlation, 206, 209, 236 probability distribution, 198 joint, 199 quasi-Gaussian approximation, 200, 236 Reynolds equations, 197 shear flow, 245 similarity, 224, 225, 229 skewness factor, 454 spectrum of turbulence, equation for change of, 68 relation to correlation function, 211 statistical averages, 198 structure of turbulence, large scale, 219 small scale, 221 vortex-stretching, 223 vorticity scale, 210 turbulent ‘“‘bursts,’’ 6 turbulent diffusion, 240 coefficient of, 242 Gaussian distribution associated with, 243 general behavior and effect, 79 involving more than one particle, 243 time scale of, 242 turbulent energy transfer among various frequencies, 214 turbulent exchange coefficient, 102, 170 boundary layer, 145 compared to kinematic viscosity, wake and jet, 175 turbulent flow, 27, 370 cone solution, 388 cones, von Kdérmén momentum inte- gral, 388 effect of variable free stream pressure and variable wall temperature, etc., 391 equation of continuity, mean motion, 82 total motion, 82 flat plate solution, 370 friction coefficient, 381 heat transfer, 288, 372 kinetic energy equation, mean motion, 83 total motion, 84 momentum equation, mean motion, 82 nature of, 76, 196 Navier-Stokes equation, total motion, 82 properties by analogy to laminar flow, ( 548 ) INDEX turbulent flow, recovery factor, 372 Reynolds analogy factor, 378 rough walls, 391 stagnation point solution, 388 status of experimental knowledge, heat transfer, 393 skin friction, 391 turbulent heat transfer, 288 turbulent motion, dynamical effects, 249 effect of damping screens on, 249 in compressible fluid, 247 in wind tunnels, 249 mean value defined, 81, 197 turbulent Prandtl number, 371, 373 turbulent shear flow, statistical theory of, 245 turbulent spots, 6, 69 turbulent structure of bibliography for, diffusion and heat transfer, 188 free flows, 188 instrumentation, 189 statistical theories, 186 vorticity and structure of turbulence, 185 wall-bounded flows, 186 shear flows, Uberoi, M.S., 171, 176, 241 universal skin friction law, 151 van der Hegg Zijnen, B. G., 170, 174 van Driest, E. R., 67, 293, 342, 349, 350, 370, 373, 374, 379, 382, 383, 388, 389, 391, 396, 398, 402, 403 vapor flow, in boiling heat transfer, 328 vapor pressure, effect on boiling heat transfer of, 325 variable fluid properties, analysis of heat transfer for, 303 effect of, on heat transfer in air, 304 variable properties, effect in liquid heat transfer of, 307 velocity, effect on bubble growth, 328 velocity correlations of higher orders, 206 velocity-defect law, 123 velocity distribution, turbulent, 292, 305 velocity-distribution formula, plane jet, 174 plane wake, 172 round jet, 174 velocity profile, effects of free stream, 125 H-parameter family, 130 view factor, 507 viscous sublayer, 374 von Doenhoff, A. E., 25, 45, 130, 154 von Doenhofi-Tetervin integral method, 154 von Kérmén, Th., 122, 203, 205, 210, 225, 227, 229, 231, 238, 246, 290, 291, 374, 458 von Kaérmdén logarithmic skin friction law derivation, 127 von Kérmd4n logarithmic velocity, 457 von Kérmd4n momentum equation, 134, 154, 438 von Kdérmén momentum integral, 383 von Kaérmdén similarity hypothesis, 291 von Kaérmdén similarity law, 382 von Kaérmdén vortices, 27 von Kérmdn-Howarth equation, 210, 220, 222 von Kérm4n-Prandtl logarithmic equa- tion, 292 vortex layer, 21 vortex pattern, regular, 27 vortex-stretching in turbulent motion, 223 vorticity scale of turbulence, 210 Walker, W. S., 52 Wanlass, K., 368 water, aerated, burnout point in, 331 boiling heat transfer to, 329 water-aerosol solution, bubble growth in, 327 burnout point in, 329 wave length of radiation, 489 wave number of radiation, 491 wedge-type flow, 449 Weinbaum, §., 435 Wendt, F., 50 Wendt, H., 362 Wheeler, H. L., 431, 485, 479 Wieghardt, K., 484 Wien’s displacement law, 490 Wien’s radiation law, 490 Wilkins, M. E., 66 Williams, E. P., 366 Wilson, R. E., 384 Wright, E. A., 6 Yamamoto, K., 11, 12 Young, G. B. W., 366 Yuan, S. W., 476 ( 549 ) Te Hid Fj markt ae th weap a) RY, iv 4 i isa ae (iti aaa ‘oO Midht Brig glieaki : aay ie 4 My Nove reine Birt JE a hh orl ntiy HEREAD th HTT Ne j id Me Sas BES bar ‘s Nae S pberaaahs hit “yaar ay, iy aes winatoned Wats oeesiw) BEng » eral a PE is WA an, a Rai iet baht) . - es a Ree VMS! Ve ane uy 4 4 er Le iba PY Ry em Re Hiiiieh: Pr ee rar ir cer By ae ET vibe te x Vem oan oad ea th fa rs : AL ae 4 f th na p \ } Ne “it n a Boe, ; er “iy Tearaes oy SAE ays No ‘ak; heat, Sa py. JOTRGN ho! heer salah beh eerie ah aie sh ; NUS, eM , i 5 4 no PA Tiree tn math Pee Md Te Phey sy BP Fea Ty ws aR ey i i { ay ARE thts hp Ass: alu a ba te view HBR Fad tit ae ; | ve he they » ey aie | a my ae 2 Phe ome Be . 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