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MeGRA W-HItr, CIVIL ENGLNEERING SERIES HARMER E. DAVIS, Consulting Editor

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BABBI'IT '. Engineering.in Public Health BENJAMIN' Statically Indeterminate St~uctures Cnow . Open,-cha,nnel Hyqraulics DAVIS, TROXELL,'AND WrsKoCIL . Tl1e Testing and Inspection of ' Engineering Materials DUNl'iAM . Foundations of Structures DUNHAM' The Theory and Practice of Reinforced Concrete DUNHAM AND YOUNG.' Contracts, Specifications, and Law for Engineers GAYLORD AND GAYLORD' Structural Design HALLERT 'Photogrammetry HENNES AND EKSE . Fundamentals of Transportation Engineering KRYNINE AND JUDD' Principles of Engineering Geology and Geot.echnics LINSLEY AND FRANZINI . Elements of Hydraulic Engineering

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VEN TE CHOW, Ph.D.

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Proiessot of Hydtaulic En(finl!erin(f University of Illinois

INTERNATIONAL STUDENT EDITION

LmsLIDY, KOHLER, AND I'A ULHUB ' Applied Hydrology LINSLEY, KOHLER, AND PAULHUS' Hydrology f9r Engineers LU:8DER . Aerial Photographic Interpretation MA'l'SON, SMITH, AND HURD' Traffic Engineering MEAD, MEAD, AND AKERMAN' Contracts, Specifications, and Engineering Relations NORRIS, HANSEN, HOLLEY, BIGGS, NAMYET, AND 1fINAMI . :Structural Design for Dyiramic Loads PEURIFOY' Construction Planning, Equipment, and Methods' PEURIFOY' gstimating Constructi()u Costs • TROXELL AND DAVIS' Composition and Properties of Concrete TSCHEBOTARIOFF . Soil Mechanics, Foundations, and Earth Structures

HYDR~AULICS

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KOGAKUSHA·COMPANY, LTD.

URQUHART, O'ROURl<::t, AND WINTER' Design Concrete Structures WANG AND ECKEr. . Elementary Theory of Structures

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Exclusive rights by Kogakusha Co., Ltd., for manufacture and export from _. Japan. This book cannot be re-exported from the cou\ltry to which it is consigiled by Kogakllsha Co., Ltd., or by McGraw-Hill Boak Company, Inc., or any .of its subsidiaries.

Humanity and Human Welfare

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Copyright © 1-959 by the McGraw-Hill Book Company, Inc. All rights reserved. This bo"k, or parts thereof, may not be reprodlJced in any for~ without pClmissioll of the publishers. Libra,,' of COI1g1·es.r Catalog Card Number 58-13860

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In recent years water-resources projects and hydraulic engineering works' have been developing rapidly throughout the world. The knowledge of .open-chp,nnel hy~raulics, which is essential to the de~ign of many hydraulic structures, has thus advanced by leaps and bounds: To the students and engineers in the field of hydraulic engineering, such valuable new knowledge should be .made available in suitable book form. It is therefore not that some 'new books have already appeared. Howi:lver, most of them a.re present-ed in limited scope and 1111 2.re written in foreign In the· English langu!l.ge, the t\>,ro well-known books, respectively' by Bakhmeteff and by 1VoodwI1l'd and Posey, were published nearly two decades ago. 2 This book gives broad coverage.of recent developments; it should meet the present Ilead. It is designed as a. textbook fo\, both undergrachui.te and graduate stuuents .and also as a compendium for practicing <:;,ll'=>lU"'" Emphasis is given to the qualities of "teachability" and" practicability," and t1tt,empts were made in preaenting the material to bridge the gap which is generally to exist between the theory and the practice. In order to achieve these objectives, the use of .advanced mathematics is deliberately avoided as much as possible, and the exp~a.nation of hydraulic 1 Such. as: Etienne Orausse, "Hydrauliquc des can~ux decouvel·ts en regime permanent" (" Hydraulios of Open Channeld with Steady Flow"), Editions Eyrolles, Paris, 1951; R. Silber, "Etude at trace des ecouleroents permanents en canaUl( et rivieres" ("Study and Sket
viii

PREFACE

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theories is greatly simplified as far as practicable, ~Illustrativ.e example~ . are to show the application of the theories, and practical problems are provided Jor exercises. Furthermore, short historical accounts are given in footnotes in order to stimulate the reader's interest, and ample references are supplied for his independent .studies. Some references, hO'\'8ver, may not be readily available the reader, but they are liste:d for academicaud historical intercst. In essence, the book is the outgrowth of the author's 20 year,,' experience as a student,teacher, engineer, rese.iucher, and consultant in' the field of hydra\llic engineeTing, The manuscript of the book was drafted for the first time in the academic year of 1951-1952 for use in teaching the students of civil, agricultural, and mechanical engineering and of .. theoretical' ahd applied mechanics at the University of Illinois. Since then several revisions have been made. In .the beginning, the material was prepared solely for graduate students. Owing to the general demand for a book on the design of hydraulic structures for undergraduate studies, the manuscript was expanded to include more fundaments;! principles and design procedures. At the sa.me time, most of the advanced mathematics and theories were either omitted or replaced by more practic.'l..l approaches using mathematical operations of a level not higher than calculus.. Fro!!. 1951 to 1955, the author made several special visits to many major engineering agenCies and firms in the United Stat,es tQ discuss problems with their engineers. As a result, a Yl'.st fund of information on hydraulic design pnictices was collected and incorporated into the manuscript. Thereafter, the author also visited many hydraulic institutions and laboratories in other countries and exchanged knowledge with their staff members..In 1956 he visited England, France, Belgium, the Netherlands,.Germany, Italy,
to

l'REFACE

ix

In Part I on basic principles, the type of flow in open channels is cla~i-· fied according to the vliriation in the parrunetersof flow with respf.ct to space and time. For simplicity, the depth of flow is used as the flow parameter in the classification. The state of flow is classified according to the range of the invariants of flow with r~spect to viscosity and gravity. The flow invariants used are the Reynolds number and the Froude number. Since the effect of surface tension of water is insignificant l.n most engineering problems, the Weber nU!l1b~r as it flow invariant is not introduced. In fact, the state of fiow c~n be further classified for its stability in u.ccordu.nce with the Vedernikov number or other suitahlf?l:rlt!:!rla. However, ::mch a criterion h.as not been weli established in engil1eering praatice, and therefore it is taken up only briefly later in Chap,8. FOUl' coefficients for velocity and pressure distributions are introduced, particular, the f.mel'gy coefficient is presented throughout the book. This coefficient is usuHlly ignored ill most books on hydraulics. In practical applications, the effect of the energy coefficient on computations and hence on is quite significant f:Ll1d therefore should not be overlooked, e,reHl though the value of the coefficient miJ,y not always be determined accurately. The energy and momE,ntum principles constitute the basis of interpretation for most hydraulic phenomena. A thorough treatment of the two prfnclples is in Chap. 3. SilIce the book is intended for the use of practicing engineel's, the treatment of a problem i~ in m013t cases based on a ollEl- or two-dimensional flow: . In Part II 011 \ll1ifol'm flow, eeveral uniform-flow formulas are. introduced. ,Despite many new proposals for a formula having a theore~ical background, the Manning formula still holds its indisputable top position in the field of practical a;pplicatioI)s. This formula. is therefore used extensively in the book. In certain specific problems, however, the Che2iY formula is used occasionally. ' . The design for uniform flo\v covers nonerodible, erodible, and grassed channels. The erodible channels in general may be classified under three types: chapl1els which scour but do ilOt silt, channels which silt but do not ~cour, and channels which ;icour and silt simultlllleously. In channels of tlie second and third types, it is nec€6sary for the walter to carry sediments, As will be stated later, the sediment transportation il'; consid~red as a subject in the domain of river hydraulics, Therefore, only the channels: of the first type, which :carry relatively clear wa~er in stable condition, are treated in this bodk ,. Iii Part III ongl'adually va-ried flow, se,,-eral methods for the computation of flow profi1.~s are discussed. .A new method of direct integration is introduced which requires the use of a, varied-flow function table first

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developed by Professor Boris A~ Bakhmeteff in 1912. 1. The table given in Appendix D of th IX. book ]s an ~xten8ion of the table to n~arly: three times its original size. This extended table \1nd a table for negative slopes were prepared during 1952 to 1954 by the author for teaching purposes at the : j Universjty of Illinois,2 For the computation of fiow profiles in circular , e) conduits, a varied-ft.ow function table is also· provided in Appendix E. : The method of singular point is a powerful tool for the analysis of flow , • profiles. Since this method requires the use of advancedmatheml1tics} it is described only briefly in Chap. 9 for the purpose of st.imulating £ur. the I' interest in the theoretical study of flow problems. . alit In Part IV on rapidly varied fiow, the treatmellt of the problems is 1 I largely supported by experimental data, because this t.ype of flow is so ~:, complicated that a mere theoretical analysis. in most cases will not yield sufficient information fO!' the purpose of practical design. The use of the ~!i fiow-n!;i metllod and the method of characteristics is mentioned but no . details are given, becallse t.he former is so popular that it, can be found .. in most hydraulics books; while the latter requires t.he knowledge .0£ i I advanced Tiiathematics beyond the scope of this wbrk. In Pnrt V on llnsteady:fiow, the keatment is general but practical. It \ Iii I·1 should be recognized that this type of f10wia a highly specialized subj ect. 3 The knowledge of advanced mathematics would be required if a compre. hensive treatment were given. i It should be noted that the subject matter of this book dwells mainly t/) on the flow of water in channels where water contains little foreign material.· Consequently, problems related to sediment trAnsportation and I)) air entnl.inment are not fully discussed. In recent years, sedirrient tr!lnsi portatim in channels has become abroad subject that is generally COvered -'~ a1 in the study of river hydraulics, which is often treated iI).depc:mdently.1 •

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1 Boris A. Bakh~ete.ff,"O Neravnomernom Dvizhenii Zhidkoati v Otkrytorn Rusle" {"Varip.d Flow in Open Channels"),·St. Petersburg, Russia, 1912. , Ven Te Chow, Integratmg the eC\uation of gra~ually varied /lOw, paper no. 838, Proc!;ecHngs, A.men·caT' Society of Civil ETl{/in~er3, vol. 81, pp. 1-32, November} 1055. Closing disc~ssioil by the author in J01trnal of HydrauHcs Division,vol. 83, no. HY1, paper no. 1177, pp. 9·-22, February, 195;'. : ' • Spedal references are: J: J. Stoke". "W';'ter Waves," vol. IV of "Pilre and Applied Mathenlatics," Interscience Publishers, New Yorle, 1957; V. A. ArkhangelskiI, "Ra~chety Neustanovivsh~gosia pvizheniill. v Otkrytykh Vo40tokakh" ("Calcula.tion of Unsteady Flow in Op£,n Channels"), Academy of ScieRces, U.S.S.R., 1947; and ·8. A. Khristianovich, "N eustanovivsheiesia d vizhenie v kana.lakh i rekakh" ("Upstea.dy Motion ill Channels and.Rivers"), in "Nekotoryie!Vopl'osy Mekhaniki Sp19~hnol Sredy" (" Several Que~tions on the Mechanics of Oontinuous Media "), Aca~emy of Sciences, U.S.S.R., 1938, pp. l3-154. .. • ~pecial references on the subj'eet of river hytlrauiics are:· Serge Lelia.vsky, "An Introduction to Flilvial Hydrauli~s," Constable and Co., Ltd.; London, 1955; and T.Blench, "Regime Dehaviour of yanllis and Rivers," Butterwor;th & Co. (Publishers) Ltd" London, 1957. .

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Similarly, the transient flow in chanI).els subject to the iilfiuence of the tides is a special topic in the rapidly developed fields of tiditl hydraulics and cOD,stalel1gineering and is therefore b9yond the scope of this book. In a science which has reached so advanced a state of development, a large portion of the work is necessarily one of coordination of' existillg cOlltributions. Throughout the text} the auth6r hr,s attempted to make specinc acknowledgment regarding the source of m:lter.ial employed} and any failure to do so is o.n unintention:al oversight. . In the preparation of this \:lOok, engineers and a,dministrators in many engineering agencies hl1ve ~!1thusi<:1st.ically furnished information and extend~d cooperation. The author is especially indebted to those in the U.S. BUl'cau of Reclamation, U.S. Geological Survey, U.S. Soil Conser,ration Ser~ice. U.S. Agricultural Research Service, U.S. Army Engineer. . WiLterways E~periment Stat.ion, Offices .of the Chief Engineer and District EnD'ineers of the U.S. Army Corps of Engineers, U.S. Weather Bureau} U,S~Buref~u of Public Roads; and the Tennessee Vilrey Authority . Also, mn.ny friends and colleagues have kindly. supplied information and generously offered suggestions. In particular, the author wishes· . to thank Dr. Hunter Rouse, Professor of Fluid Mechanics and Director of Iowa Institute of Hydraulic Reseal'ch, State Unive·rsity of Iowa; Dr. Arthur '1'. Ippen, Professor of Hydraulics and Director of Hydrodynamics Laboratory, Massachusett.s Institute of Technology; Dr. Giulio De Marchi, Professor of I:1ydraulics and Director of Hydraulic Laboratory, Institute of Hydraulics and Hydraulic Construction, Polytechnic Institute of Milan,Italy; Dr. Roman R. Chugayv, Professor and Head of Hydraulic Con:;truction, Scientific Research Institute of Hydraulic Engineering, Polytechnic Jl1sti;tute of Leningrad, U,S.S.R.; Monsieur Pierre Danel, President of SOGREAH (Societe Grcnobloise d'Etudes et d'Applications Hydrauliques), France, and President of the Interun,tional AssociatiO!l of Hydraulic Research; Dr. Charles Jaeger, Special L.ectul'er at the Imperial College of Science and Technology, Univemityof Londoll, and COllsulting Enginee!' of. The English Electric Company, Ltd., England; Professor L. J. Tison, Director of Hydraulic Institute, University of Ghent, Belgium; Dr. Tojil'O' Ishihara, Professor of Hydraulics and Dean of Facult.y of Engineering, 'Kyoto University~ Japan; and Dr. Otto Xirschmer, Yrofessor 'of Hydraulics and Hydralllic Structures, Technical Jnstitute of Darmstadt, G e l ' m a n y . . , Special acknowledgments are due. Dr. Nathan M. Newmarlt, Professor and Head; ofthe Department of Civil Engineering} University; of Illinois} for his encouragement and unfailingisupport of this project; D~. James M. RobertsoiJ., Professor of Theoretica~ and Applied Mechanics, ;University of Illinoi.':\, for his review of and c~mmellts on Chapter 8 onitheoretical concepts;: nnd Dr. Steponas Kolu~aila, Professor of Civil Engineering,

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PREFACE'

University of Notre Dame, for h.is readi~g of the entire manuscript and his valuable suggestions, Dr. Kolupaila' also helped in interpreting and collecting information from the hydraulic literature written ill Russian, Polish, Lithuanian, and several. other languages which are unfamiliar to theauthol', . The author also wishes to express his warm gr[l.titucie t,o those whQ have constantly shown a keen intei'est ill .his work, as this iuterest lent a strong impetus toward the completion. of this volume.

CONTENTS

Ye.nTe Chow Prefacs

. PART I,

Chapter 1.

BASIC PRINCIPLES

Open-channel Flow !llld Its Classiffcations

1-1. D~scriptiori . 1-2. Types 'Of Flow . 1-3. State Qf FlQW 1·4. Regimes Qf Flow . Chapter 2.

2-6. 2·7. 2·8, 2-9. 2·10.

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Open Channels and Their·Properties

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Energy and Momentum Principles.

3-1. Energy in Open-channel Flow. 3·2.. Specific Energy. 3-3. Criterion for a CriticnJ State of Flow 3-4. Inte~pretation of Local'Phenomena 3-5. Energy in Nonpri~atic Chl!.nnels 3-6. Momentum in Open-charinel Flow 3-7, Specific Force. . ,. 3-8. Momentum Priu¢iple Applied to Nonprisffiatic, Channels. Chaptet 4.

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Channel Geometr)~ . Geometric Elements of Cha.nnel Section, Velocity Distribution in a. Channel Section. Wide Open Channel Measurement of Velocity , Velodty-dbtribution Coeflicienu, . DeterminaHon of Velocity-distribution Coefficients PrllS
Chapter 3.

3 3

2-1. Kinds of Open bl~annel

2-2, 2-3, 2-4, 2-5.

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Critical Flow: Its CQl~nputation and Appllcations .

4-1. Critical Flow 4-2. The8ection Factor for Crit.ical-fiow Computa.tion . for Criticru-fiow Computation _ 4-3. The HydrILulic' 4-4. Computation of Flow. 4-5. Control of Flow. 4-6; Fl9W Measurement.

20 22

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26 27 27 28

30

32 39 39 41 42

43 46

49 5~

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63

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5f! 59 70 r ) 74

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CO!-T'l'ENTS

PART 11. Chapter 5.

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Development ()f Uniform Flow and Its Founulas

Computation of Uniform Flow .

The Conveyance of a Channel Section The Section Factor [or Uniform-fio;; Computation . . The Hydraulic EXjlQnent for Uniform,.flow COmputation. . Flow Chua.cteristics in Jl. Closed Conduit with Of/en-channel Flow Flow in a Channel Section' with Composite Roughness Determination of the Normal Depth and Velocity. Determination of the Normal a.nd Critice.l Slopes 6-8. Problems of Uuitorm-fiow Computa.tion. 6-9. Computation of Flood Discharge. 6-10. Uniform Surface Flow . Chapter 7.

81l

89 89 91

93 94 98 10l

101 108 114

D<"sign of Channels for Uniform Flow

128 ,12.8

131 134

136 140

142 144

146 148 157

A. NONERODIBLE CHANNELS

157 157

7-1. The Nonel"Ddlble Channel.

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. 7-2. Nonerodible r-,·laterilll and Lining.

158 158

7-3. The Minirmlm Permissible Velocity

7-4. Channel Slopes . . 7-5. FreebofJ.l"d 7-6. The Best Hydraulic Section 7-7. Determination of Section Dimensions

B,

159 160

162

ERODIBLE CHANNELS WHICH SeOUR BU'l'

7-9. The Maximum Pel'missible Velocitv . 7-10. Method of Permissible Velocity . 7-11. The Tra.ctive· Force. 7-12. Tractive-force Riltio 7-13. Permissible '1;'raotive Force 7-14. Method of Tractive Force. 7-15. The Stable Hydra.ulic Section.

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164 165

7-8. MethOds of Approach .

C. GRA8SEDCHANNELS 7-16. The Gra.ssed. Channel . 7-17. The Retardance Coefficient 7-18. The Permissible Velocity

Chaptllr 8. Theoretical Concepts of Boundary Layer, Surface Roughness,. .-192 Velocity Distribution, and Instability of Uniform Flow.

167 168 . 170

172 175

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179 .179

184

192 194

8-1. The Boundary Layer . 8-2. Concept of Surface Roughness 8-3. Computation of Boundary Layer. S.4. Velocity Distribution in Turbulent Flow 8-S. Theoretica.l Uniform-flow: Equations. . 8-6. TheOl·etica.l Interpretation of Manning's Roughneas Coefficient 8-7. ::Vlethods for Determining MalUling's Rougbneas Coefficient 8-8. Instability of U:uiform Flow . . PART III.

128

6-1. 6-2., 6-3. 6-4. 6-5. 6-6. 6-7.

184 184

7-19. Selection of Gr8.53. . . 7~20. Procedure of Design

UNIFOR.M FLOW

5-1. Qualifications for Uniform Flow : 5--2. Establishment of Uniform Flow . 5--3. Expressing the Velocity of a Uniform Flow. 5-4. The Chezy Formula 5-5. Determination of Ch,s'Oy's Resistanoe Factor 5-6. The Manning Formula. 5-7. Determination of l\-1anning's Pw:mghneas Coefficient 5-8. Factors Affecting Manning's Roughness Coefficient 5-9. The Table of :VIa.nnillg's Roughness Coefficient 5-1.0. Illustro.tions of Chimnels with Various Roughnesee3 , Chapter 6.

xv

CONTENTS

198 200

202 205

206 210

GRADUALLY VARIED FLOW.."

Chapter 9. . Theory and Analysis. 9-1. Basic Assumptio!l.9 . 9-2. Dynamic Equation of Graduallv Varied Flow 9-3. Che.ra.cteristiDs of Flow Profiles' 9-4. Cla.ssification of Flow Profiles. 9-5. Analysis of Flow Profile 9-6. Metnod of Singular Point. 9-7. The Transitio,nal Depth

Chapter 10.

Methods of Computa.tion

10·1. 10-2. 10-3. 10-4, 10-5.

The GrD.phical-int!lgration Method Method of Direct Integration. The Direct Step Method The Standard St.ep Method Computation of iii Family of Flow Profiles 10-6. Th~ Standard Stap Method for Natural Channels. 10,7. The SLage-fall-discharge Method ior Natural Channell! 10-8. The E.m. Method for Natural Channels.

Cb,apter 11.

Practica.l Problems .

11·1. Delivery of a Canal for Subcritical Flow 11-2. Delivery of a Cana.! for Supercrit.ical Flow 11-3. Prublems Related to Cana.! Design . . 11-4. Computation of Flow Profile in Nonprismlltic Channels 11-5. Design of Transitions . 11-6. Transitions between Canal and Flume or Tunnel 11-7. Transitions between Cana.l and Inverted Siphon 11-8. Backwater Effect of a Da.m 11-9. Flow P!l.SSing Isla.nds . . 11-10. River Confluence Chapter 12.

Spatia.!ly Vflried Flow

12-1. Basic Principles and' Assumptions 12-2. Dynamic Equa.tion for Spa.tia.lly Varied Flow

217

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217 218 222 227 232 237 242

249 249 252 262

265 268 274 280 284 297 297 302 303 306 307 310 .317 319 320

321 327 327 329

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12-3. 12-4. 12-5. 12-6.

Analysis of Flow Pro5Ie . . Method of Numerical Integration The Isoclinal flilethod . . Spati~.lly Varied Surface flow PART IV.

Chapter 13.

RAPIDLY VARIED FLOW

IntroducUon.

13-1. Characteristics of the Flow 13.2. Approa.ch to the Problem . Chapter 14.

Flow over Spillways

14-1. The Sharp-crested Weir 14-2. Aeration of the Nappe . 14-3. Crest Shape of Overflow Spillways 14-4. Discharge of the Overflow Spillway '14-5. Rating of Overflow Spillways. ' . ' . . 14-6. 'Upper Nappe Profile of Flow over Spillways 14-7. Effect of Pi~rs in Gated Spillways' .14-8. Pressure on Overflow Spillways . . 14-9. Drum Gates. . 14-10. Flow at the Toe of Ov.erllow Spillways 14-11. The Ski-jump Spillway 14-12. Submerged OverfloW' Spillways Chapter iii.

Hydra.ulic Jump and Its Use as EneIgy DissipatoI

15-1. The Hydra.ulic Jump . 15-2. Jump in Horizontal Rectangular Channels 15-3. Types of Jump . 15-4. BllSic Cha.racteristics of the Jump 15-5. Length' of Jump. 15-6. The Surface Profile .. 15-7. Location of Jump .' 15;8. Jump as Energy Dissipator 15-9. Control of Jump by Sills . 15-10. Control of Jump by Abrupt Drop 15-11. Stilling Basins of GeneraU"ed Design 15-12. The SAF stilling Basin 15-13. USBR Stilling Ba.3in II 15-14. USBR Stilling Basin IV 15-15. The St.l'aight Drop Spillway 15-16. Jump in Sloping Channels. 15-17. The Oblique Jump. Chapter 16. 16-L 16.2. 16-3; 16-4. 16-5; 16-6: 16.7.

333 341 346 347

Flow in Channels of Nonlinear Alignment

Nature of the Flow . Spiral Flow . Energy Loss. Super elevation Cross Waves. . . . . : . . . . . Design Considerations for Subcritical Flow. Design Considerations for S~percritical Flow

357 3.57 '357

360 360 362 363

365 368

Chapter 17.

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xvii

CONTENTS

CONTENTS

xvi

Flow through Nonprismatic Channel Sections

461 '--:,....

17-1. 17-2. 17-3. 17-4. 1t-5.

17-6. 17-7. 17-8. 17-9. 17-10. 17-11. 17..12. 17-13. 17-14.

Sudden Transitions,. Subcritica.l Flmv through Sudden Transitions Contractions in Supercritical Flo''''' Expansions in Supercritical Flow. Constrictions Sl!bcritical Flow through Constrictions Backwater Effect due to Constriction Flow through Culverts. Obotructions. ro'loW' between Bridge Piers Flow through Pile Trestles Flow through Trash Racks Underflow Gates Channel Junctions

461 464 468 470 475 476 490 493 499 501 506 506 507 512

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370 374 380 382 384 385 393 3113 395 395 396 398 399 399 404 408 412 414 415 417 422 423 425 429 439 439 439 441 444 448 455 456

I

PART V.

UNSTEADY FLOW l~.

Chapter 18. 18-1. 18-2. 18-3. 18-4. 18-5. 18-6. 18-7. 18-8.

Gradually Varied Unsteady Flow

525

Continuity of Unsteady Fiow Dynamic Equation 'for Unsteady Flow,. Monoclinal Rising Wave . Dynamic Equa.tion for Uniformly Progressive Flow Wave Profile of UnIformly Progressive Flow Wave Prupagation . Solution of the UnsLeady-flow Equations Spatially Varied Unsteady Surface Flolv

Chapter 19.

Rapidly Varied UnsteadY Flow

.( ( '--'

l_

554

19-1. Uniformly Progressive Flow 19-2. The Moving Hydraulic Jump, 19-3. Positive Surges . 19-4. Negative Surges. 19,5. Surge in Power Canals. 19-6 .. Surge in Naviga~ion Canals 19-7. Surge through Channel Transitio!IS 19-8. Surge at Channel.Junctions 19-9. Pulsating Flow. Chapter 20.

,~

525 626 528 631 533 537 640 543

..

Flood Routing

-.'

554 557 559 566 568 572 575 578 580

C '-...,'

l. I.. i, __

586 (

20-1. 20-2. 20-3. 20-4. 20-5. 20-6.

Ii

,.

Routing of Flood Method of Characteristics. Method of Diffusion Analogy. Principle of Hydrologic Routing Methods of Hydrologic Routing A Simple HydrologIC Method Routing

of

"~._~ • • ' _ _ _ _ V~·

y.'.----.-... ---

586 587 601 604 607 609

'-

\ ~,

i ,--

\

I

CHAPTER

,1

OPEN~C:HANNEL FLOW AND ITS CLASSIFICATIONS

1-1. Description.

The flow of water in a conduit may be either openThe two kinds of flow are similar in many ways but differ in ohe important respect. Open-channel flow must have a free surface, whereas pipe flow has none, since the water must fill the whole conduit. A free surface is subject to atmospheric pressure. _:PiR~ , flow, being connnedin a .closed "£Q~1duit, x~rts irect atl!lo,§ heri pressure but hydraulic pressure only. ., , "--,, ,,The t\VO kinds of flow are compared in Fig. 1-1. Shown on the left side is pipe flow. Two piezometer tubes are installed on the pipe at' sections 1 and 2. ,The water levels in the tubes are maintained by the pressure in the pipe at elevations represented by ~he so-cag~a.lYJdr,guliq g.rade line. The pressure exerted by the water in each section of the . pipe is indicated in the corresponding tube by the height y of the watet c;olumn above the center line of the pipe. The tIJt.al energy in the flow of the section with reference to a datum line is the sum of the elevation z of the pipe-center line, the piezoine'tric height y, and the velocity head V2/2g, where V is the mean velocity of flow. l The energy is represented in the figure by what is called the energy grade line or simply the energy line. The loss of energy that results when w:ater flows from seqtion 1 to sectiori 2 is .represented by hi. A similar diagram for open-channel flow is shown on the right side of Fig. 1-1. For simplicity, it is assumed . that the flow is parallel and has a uniform velocity distribu,tion and that the slope of the channel is, small. lP,..i,his c!1;,§e, the 'Y~~!t~ §~rj~ce is ,the .hldraulic m£l~dine, and ~~~.1t£.Qf..1~j~OIUjUg",,~ piezometric [email protected] '., . . ~ Despite the similarity between the two kinds of flow, it is much more cl.ifficult to solve problems of flow in open channels than.in pressure pipes. ~low conditions in open c~annels are complicated Py the ',fact that the channel flotu or 'Pip~ flow.

---'-->

~

!

It is here assumed that the velocity is uniformly distrib\.ited across the conduit othenvise a correc,tion ,,",ould have to be made, such Ilf is desaribed in Art. 2-7 fbr open c h a n r r e l s . ! , : 2 If the flow were curvilinear or if the slope of the channel w~re large, the piezometric height would be appreciably different from the depth of flow (l
-"1"'-

3

: 1

s~ction;

;~

I" .,

I

,,--.. :

.

'-....

" .~

xviii

CONTENTS

Geometric Elements for Circular Channel Sections

625

Appendix B. Geometric Elements for. Trapezoidal, Triangular, and Parabolic Cb-annel Sections

629

Appendix C.

640

Appendix A.

, Appendix D.

NQIDographic Solution of the Manning Formula Table of the Varied-flow FuncH,ms

.

F(u.,N)

Appendix E.

r " I-UN dlt

= Jo

and

F(u,N)_s,

=

r"

dtl )0 l+'uN

Table of the Varied-flow Functions for Circular Sections

,,

641

!

'~

·PART I

BASIC PRINCIPLES

657

Name Index.

663

. Subiect Index

669

(

I

)

r

,- .

)

4

""

BASIC PRINCIPLES

OPEN-CHANNEL FLOW AND ITS, cL'ASSlPICATI0NS

position of the free surfa.ce is likely to ch~nge with respect to time .and space and also by Lhe fact that the depth of flow, the discharge, and the slopes of the channel bottom and of the free surface are interdependent. Reliable experimental data. on flow in open channels are usually difficult to obtain. Furthermore, the physical condition of open channels varies -'much more widely than that of pipes. In pipes the cross section of flow is fixed, since it is completely defined by the geometry of the .conduit. ; _I The cross section of a pipe is generally round, but that of an open channel may be of any the circular to the irregular forms of natural streams.' In pipes, the interior Surface ordinarily ranges in roughness

Steady Flow and Unsteady Fww: Time as the Criterion, Flow in an open channel is said to be steady if the depth of flow does not change or if ' it can be assumed to be constant during the time interval under consideration. The flow is unsteady if the depth changes with time. In most open~channel problems it is necessary to studyftow behavior only under sready conditions., .If, however, the change in flow condition with respect . to time is of maior concern) the flow should be .treated as unsteady. In floods and surges, for instance, which are typical examples of unsteady now, the stage of flow changes instantaneously as the Wf1ves PMS by, and the time element becomes vitally important in the design of control structures. For any flow, the discharge Q at a channel section is expressed by

Q

'VA

(I-I)

where V is the mean velocity and A is the flow cross-sectional area normal to the direction of the flow, since the mean velocity is defined as the discharge divided by the cross-sedional area. In most problems of steady flow the discharge is constant throughout the reach of the channel under cOllsideration;in other words, the flow is continuous. rrhus, using (1-1), (1-2) Pipe flow

Open-chann,1 flow

FIG. 1-1. Comparison between pipe flow and open-channel flow.

from that of new smooth brass or wooden-stave pipes, on the one hand, t,o.that of old corroded iron or steel pipes, on the other. In open channels the surface varies from tha.t of the polished metal used in testing flumes to that of rongh inegular river bed'l. :.vloreover, the roughness in an open channel varies with the position of the free surface. Therefore, the selection of fdction coeffic~el1tS is attended by greater uncertainty for ~ open channels than for pipel$. In general, the treatment of open-channel flow is 30mew~at morl:l~Ip.Eirical than that of pipe flow. The empirical ) method is the best available at present and, if cautiously applied, can yield results of practical value. The flow in a. closed conduit is not necpssarily pipe flow. It must be " classified as open-channel flow if it has a free surface. The storm sewer, . .1 for example, which is a closed conduit, is generally designEld for openchannel flow because the flow in the sewer is ekpected to maintain a free surface most of .the .time. '. 1-2. Types at Flow. Open-channel fiow call be classified into many '\ types and descri:be'd in various ways, The following classification is made according to the change in flow depth with respect to time and space.

,

...

where' the subscripts designate different cllannel sections. .This is the '/::::,.. continuity equation for a Qontinuous steady flow. ' -------~----.-,.-~-"'- .. -.-,..-'-.-.--.~-- . --~~~----Equation (1-2) is obviously invalid, however, where the discharge of a steady flow is nonuniform along the channel, that is, where- water runs in or ont along th.&..Q~...QLfloW'. This type of flo,!!L. kn2wn as_~/Ltiql.l'!l ~!ied .!!!...!ii8c
.P. • ....

I,

6

I

i .~"

BASIC PRINCIPLES

is a practically impossible condition. The term «uniform flow" is , , therefore, used hereafter to refer only to steady uniform flow. Flow is varied if the depth of flow changes along the length of the channel. Varied flow may be either steady or unsteady. ~nce unsteady 2l~fI.~~e, the t~,rm II unsteady flow" is used hereafter to designate unsteady varied flow exclusively:-'-. ._ ... Varied. flow roay be furt~; da.ssified as either rapidly or gradually varied. The flow is rapidly; varied if the depth changes abruptly over a comparatively short distance; otherwise, it is gradually varied. A rapidly varied flow is also known as a local phenomenon,' examples are the hydraulic jump and the hydraulic drop.

OPEN-CHANNEL FLOW AND ITS CLASSIFICATIONS

I

For clarity, the classification of open-channel flow is summarized as follows:

f'

I

I I I

'{'

I r

I Chong'e of depth from time time

to

Uniform 'flow - Flow in a laboratory' channel

G.V.F.

RV.F.

Sluice

Confraction below the sluice

----:---~-

-

~ -'.

,

//

G.v. F. - Flood wove

R.V.F. - Bore

lJnsteqdy flow •

FlG. 1-2. Various types of open-channel Bow. RV.F. ~mpidly varied flow.

G.V.F. = gradually varied flow;

t ~'.

The flow is turbulent if the viscous forces are 'weak relative to the inertial fDrces. In turbulent flDW, the water particles move in irregular paths which are ,neither smooth nor fixed but which in the aggregate still represent the .forward motion of the entire stream. Between the laminar and turbulent ,states there is a mixed, or transitional, state. The effect .of viscosity relative to inertia can be represented by the Reynolds number, defined as

Varied flow

'

A. Steady flow L Uniform flow .2. Varied flow c. Gradually varied flow b. Rapidly varied flow B. Unsteady flow L Unsteady uniform flow (rare) 2. Unsteady now (i.e., unsteady varied flow) a.Gradually varied unsteady flow b. Rapidly varied unsteady flow Various types of flow are sketched in Fig, 1-2. For iIlu.strative purposes, these diagrams, as well as other similar sketches of open channels in this book, have been draw.l to a greatly exaggerated vertical scale, since ordinary channels have small bottom slopes. 1-3. State of Flow. The state 01' behavior of open-channel flow is governed basically by the effects of viscosity and gravity relative to the in.ertial forces of the flow. The surface tension of water may affect the behavior of-flow under certain circumstances, but it does not piay a significant role in most open-channel problems encountered in engineering. , Effect of Viscosity, Depeilding on the effect of viscosity relative to inertia, the flow may be laminar, turbulent, or transitional. The flow is laminar if t.he viscous forces are so strong relative to 'the inertial forces that viscosity plays a signiftcant part in determining flow behavior. i~ laminaLfl.mY...Jh~~arti.91esa,ppeal' to IAove in deE!lite smooth paths, or streamlines, and infinitesimally thin layers of. fi»id se~11l To slia:e ove~\:n~Q~!.lt l~ye.r~: . - . - . - - - . - - . - - - - . - . - - -

Unsteady unifarm flow - Race

RV.F.

7

I

1 i I I

II

i

j.

R == VL JJ

(1-3)

where V is the velocity of flow in fps; L is a characteristic length in ft, here consider'ed equal ,to the hydraulic radius R of a conduit; and 11 (nu) is the kinematic viscosity of water in fV/sec.

The kinematic viscosity

8

\..J I

,

"-

.

"BASIC PRINCIPLES

'

OPEN-CHANNEL FLOW AND ITS CLASSIFICATIONS

ill ft2/sec is equal ~o the dynamic viscosity, f.I (mu) in shlg/ft-sec divided by the mass densl(;Y p (rho) in slug/ft,·. For water at 68"F (2000) 5 f.I = 2.09 X 10- and p = 1.937; hence, 11 1.08 X 10-$. " An open~chan?,el flow is laminar if the ReYl10lds number R is small and tUl'bt~lent if. R IS ,large. Numerous experinlents have shown that the flow 10 n. pIpe . c~an~es, from laminar to tul'l)ul~l1t in the range of R betwee~ the cntlCal val.ue~&Q.Q. ~n~ a valuethfl,t inay be as high as 50,000. . I~ th~se exp~nment.s the dIameter of the pipe was taken as the chal'act~rIStlC. le~gth m defining the Reynolds number. When the hYdra~11C radlUS 15 taken as the characteristic length, the corresponding' range IS from ,500 to 11,50q, ,. since th.e~!.fl:,l!l~~~_ of '!..,Eipe is fo~r times its hydraulic mdlUs. ' --'--.•-.-.--~--'-''''------

I I

r I i

,

I

I

,,I i

;t

I

(1-4)

,where 14, ~ the fl'1::Lional los; i.n ft. for flow in the pipe, i is the friction fact~~. L L~ the l~ugth of the pIpe III ft, do is the diameter of the pipe in V IS the velocIty of flow in fps, and g is the acceleration due to gravity In ftfsec~. '

:t,

hJlL, the above equation

1. .."

(1-5)

I

. Th f-R[IiJrelati~l1S~pJ.~smo~t~J;_~~~ can ~~~e,!2I'.!lssed bv th, e Blas:ius equatton ' . '. - ----"-,---- , .. .---'- ., 0.223

?

=

RO.~'

(1-6)

t

.1 t8 a result ~f Darc(s. stud.: [21: ~n flo~ in pipes, his name is commonly associated wlthithat ~ WelS~o.ch [31 deslgna,tJng~hlS. equation which Weisbach firstformulated..

in

r'l

From Eq;S. (1-3)• and (1-5) it can be shown that i

In

Act~IlYI dAubulSson [4] presenteji, prIOr to Darcy, a formula. that can be reducEid to the form of Eq. (1-4). i . , t tn. this equation, ~he hydraulic :radius is used as the characteristic lengtll in defin.. -.. mg the Reynolds .number. If the; diameter of pipe were used as ~he characteristic length, the numerlllal constant of ~he numerator in this equationfwould be 0.316.

(1-7)

basis for comparing flow conditions in open channel.s. It may be noted that corresponding equations forflO'wi'n open channels have been derived byJf~~~l~L§>lld ~.epear to be very_similaLtQ_ihrnQe-fiow t:lJl'\l.!:l:jJ.QTI!3~ giYQ!L:;tj;lov:e. It must be remember.ed, however, that, owing to the free "mlrface and to the ~~oUhe }u~draulic ~us. discharge... and slope, the f-R relationship inopen-channel flow does not follow exactlyJ;htl. , sinlpleConcepts thathohlfor pipe flow. Some specific features of the '.f-R relationshiP open-chann.erfloware described below. ,t Experimental data available, for the .determinationof the f-R relationship in open-channel flow can be found in various publications on hydmu-' which plots the relationship for flow iIi lics. 1 Figure is based on data developed at the University of HUn an University of Minnesota [20]. In this plot the following features may be noted: l.The 'plot shows clearly how the state of flow changes from laminar to turbulent. as the Reynolds number, increases. The discontinuity of the plot and the spread of data characterize the transitional regIon, as they do in ,the Stanton for fio'w in pipes. The transitional range, however, so as "pipe flow. The iower critical Reynolds number depends to some extentoa channel shalle. The value :ia~0rQD1 Q.OQJ;9~-(i60;oemgg~i;~rally l~g~;-th-;:;-th-; value 'for pipe flow. For practical purpoSes,thetransitioUal range of:R Iol:-opei1='channel -~.~ flow mfty be a.~sumed to bc..§QQJ.Q..~..QQP. ' It should be noted, however, that the uppe.r value is arbitrary, since there is no definite upper limit ' for all flow conditions. 2. The: data in the laminar regiQu can be defined by a general equation 'I( f F -R • (1-8)

,

;. It snould be noted that there is actually no definite upper limit..

,!

+ 0.4 '

---

:'hls equation may a.lso be applied to uniform ahd nearly uniform flows channels. ' , - - -_________

~/-m...0pen

f

2 log (R ...;'1)

The resulting Prandil-:von

I),

and transitional stat.es ot open-.channel flow c~m be expressed by. a.~lagram thatshows a relation between the Reynolds number and the fnctlOn factor of the Darcy-Weisbach formula. Such 9~ dlUgra~, g~I1el'ally knoWn us the BtanloTL4:f.ggn;,m Ill, has beeg.
Since da = 48 and the ~ne~gy gradie.u,t S may be rewritten for the friction factor

Nikuradse 8 ~

-'--~Eciuations (1-6) and (1-7) will be used i'n the follo\ving discussion as

~Th~-i~mi~~:, turbul~ntJ

,

v..'ith the data obtained b d1'man equ~..!4sm Is 1

9

~

, I ,

I

I Y I

I

K = 8gR 2S ! .V'

(1-9)

Se~ [l(~1 to [23].' ! ' : . The da.;ta. for the rectangular channel; were furnished through the cdurtesy of Professor W. :/;1. lAnsford ann processedJoi the present purpose by the author. 1

2

11 10

BASIC PRINCrPLES

Since V and R have speeific ~'alues for any given chantiel shape, K is a purely numerical factor dependent only on channel shape. For laminar . flow in smooth cha.nllE)ls, the value of J( call be determined theoretically [20} .. The pJot in Fig. 1-3 indica.tes that Kia approximately 24. for tht> reetanguJar channels and 14 fol' the triangular channel under consideration.

e

R

o

F: '.

(C

$

',{I td

a.o AI

R ., 4,01;:11

a til

•

a ..

e

R • 0.0

~

to

..

R • 3,0*

(I

R

..

4,

~4 tK

tloll

A ... 1.:2 C!A

eta -i.lli:vtRS;'TY n,= fLl!NOlS DATA G

0••

fI£CTAlfGlA.AR c.HAflNa~,1.5Ff \\'10£. wlTK GL,us WALlS &. POI.JSH(O 9Fl4SS PLATE

0.2

!~

lhjrro~,

e . TRtANCiULi\J'i CHNfh£i..'(P VE~TEX MOLt. wr:'H SMOO'lM

.,..

•

RECrANGtJt.AI\ C:Io!IAIfN!:l..1.7 1M WfDt~ 'Wt!'H SJ,t0Q'l',.. SlJFl~

FACE; o

or

Sl'Flt;CTtiRAL

srUL

nU4NCAA...A.R CHA'HN£l.,l'd" TO

fsd''V~Rrt)t ,t~I.ES,WlrH

0.08

f

0.1

om

f

O.Oi

0.06

().04 Q

k

100.0rt9fM

.. k.

0.02.831"

O.qz 0.02

\

,

1

O.OIl-----4----+-....:..--·t---~·'r-~

0.01

0.000 0.008

O.OO41--+-+++4-+-~-+-'f+-+-'-JHh-+-'-I--+-l-+I--!·-++H

i

4

.J ~

•

$1'0'

I

FIG. 1-3. The f-R relationship for flow in smooth channels.

3. The data in the turbulent region correspond closely to the Bllisius:j=lrandtl-vo1n Karman curve. This ~ndicates that the law for turbulent fiowin smaoth pipes may be approximately representative of a.Usmooth channels. The plot also shows that the shape of the channel does not have an important influence 011 friction in turbulent flow, as'it does in ," laminar fl o:W. The dataJor laminar flow obtained at the University of Minnesota [20J and the da~ for turbulent flow cdllected individually by Kirschmer l

.' . h channelS. . Bo.zin's channeis; No.4, FIG. 1-4. The f-R. ~elatlonshlp for Row In rOU~ed wood; No. 14, 'unpolished wo~d gravel embedded In aement; No. Il,. unpo m lon 10 mm high, and·lO mm In roughened by tra.nsverse wooden stripS 27. sp.~in'" of mm; No. 24, cem, ent .. 7 Ne 14. except WIt .1 a te· : spacmg; No.1, same B.S. ' d K' her's cha.nnel: smooth concre . lining; and No. 26, u~p<Jhshed woo. IrSC In . "

;n

R ;

"--'

10

\

\ I,

50

.Q

"

. . . (23] are shown in the diagram for fio~ in (15,16}, Elsner [221, and Kozelnny f the' data channel roughnes.~ is (Fig. 1-4). 13Qme 0 ' • f I _ .r . • h' . e8.SUre of the roughness fll1rttcles or;m k whic IS a SIze m . f t ' s'. Y , . represen d' . illustrates the followmg ea ur.e ing the channel surface. The lsgram d fi d b Eq (1-8) In this • • . . th d til. can be e ne Y . . . L In the lammat reg~on e ~ly h' h than it is for smooth channels region, the value of K l.S genera 19 e r . .

\, I

'-.---

..

~

~-"

t

IJ. )-J (J.

r

L

r

.",

)

(

'ru

f'J I' )--

")

Ir}

~".

i

."-"'

12

~.-.-.---

-_._._-,_

t j) -1. [

and ranges bfltween 60 and 33, indicating the pronounced influence of the channell'oughnesson the friction factor. . 2. In the turbulent region the channel shape has a pronounced effect on the friction factor. It is believed that, when the degree of roughness i~ const.ant the fdction. factor decreases roughly in the order of rectangular, t:'iangular, trapezoidal, and circular channels. At the suggestion of Prandtl, Kirschmer [15,lDj explained that the effect of channel shape may be due to the development of secondary flow, which is ~pparently more pronounced in rect(1ngular channels than in,say, triangular channels: The secondary flow is the movement of water particles on /l, cross section normal to the longitudinal direction of the channel. A high secondary flowrinvolves high energy loss andthns r.,ccounts for high chaI!!lE11.J:{:!§.i..frul.nce. . . . . 3. In the turbl,1lent region most plots appear parallel to the Prandtlvon Karman curve, This curve serves as an approximate limiting position toward which a plot moves as the over-all resistance becomes less, Accoi'dingtQ a concept advanced by Morris [241 (Art. 8-2), the rise of the plots above the smooth-conduit curve may be explained as a resul~ of additional energy loss generated by the roughlless elements. When the Reynolds number is very high, some plots become essentially horizont,al, reaching a stllte of SO-Galled complete turbulrnce. At this state the value of fis independent of Reynolds number and dep~nds solely on roughness, hydraulic' radius, .and channel shape . . 4. The' plot of Varwick's data [16J for a give~ roughness,- hydraulic radius, and channel sha~ star~ off from ll. curve parallel to thLPran..Q.tj~ Voill'{arman curve, then rises as the Reypolds n~mber incre~§, and final1y becomes horizo'ntal as a state of complete turbulence is reached. The rise of the plot is a peculiar phenomenon which demands explanation, I and, since this finding has not been verified by other data, mor.e experimental studies .seem necessary to substantiate it. ~-'-~It should be noted that the ab'Jve descriptions are limited to lowvelocity, 01' subcritical, flow (which will be defined later in this.-ar.ti.cl.e)_ and to Row onwhich 8.ll:.rface tension dOEls not have a §1IDlificant influence. .. _._._----;---,. .. - .... -In most open channels laminar flow occurs very rarely. ,Th~!!l;~t tfu;.1 the surface of a streilr!!:J!:eE!~!g_~<JmlOoth and glassY_.JQ..Jtn ob.sru:.v.ru.2...by no mea~__a..lL.i.ll.dicati2.~~inar; }llOID;' prohllJily, it indicates that the sU11aqe vel.QQi!! is lower than that required for capillary wav~Qnn.· :J;.,.aminat open-cha~Jm9,!..n t'O exist, however, ~~::wJ1~~ th.in sheets...2f_ ~t?l' flow over the grq,,';1nd Dr where it is created deliberat~ ~testlng ~~!lnels,

-

---

According to thelloncept of Morris [241. thIS phenomenon probably represents a tJ'ansition of thefiow to a.nother type of flow having higher energy loss. As the Reynolds nu~ber increases, the fio'IV may be cha.nging from quasi-smooth flow to wakeinterference flow, and then tomolatecl·rou.ghness flow (~t. 8-2). 1

"

BASIC PRINCIPLES

-

OPEN-CHANNEL FLOW. AND ITS Ci..AftSIFICATIONS

13

As the flow in most channels is turbulent, a model empl.oyed to simulate a prototype channel should be designed so that the Reynolds number of flow of the model channel is in the turbulent range. E.ffect oj Gravity. The effect of gravity upon the .state of flow is represented bt aratio...91 ine~al forces to gravity forc~ This l'atio is,given by the FrQuile numbe1'z' defined» ',.

v

(1-10)

where, V is the mean velocity of fiow in fps, g is the acceleration of gra Yity in ft/sec z) and Lis a.characteristic length in ft. In open-channel fiow the characteristic leng~h is made equal to the lmdraulic deptAJ2., which is defined as the cross-sectional area of the water normal to the direction of. flow in the channel divided by the width of .the fr¢e surface. For rectangular channels this is equal to the depth of the flow' section, When F is equa.l to unity, Eq. (1-10) gives V

=

.../gD

(1-11)

and the flow is said to be in a critical sta.te. If F is less thaJ1 unity. or V < V(jJ5, the flow is s'libcriticaZ. In this state the role played. by gravity forces is mOfe pronounced; so the flow has a low velocity and is often described as tranquil a.nd streaming. If F is greater than unity, or V > the flow is 8upercriticaZ. In this state the inertial forces become domL'lant; so the flow has a high velocity and is usually described as rapid, shooting, and torrentiaL IE the mechanics of water wave~Jhe.critic& velocity v'gD is ideirtified as the celerity of the small gl'avitywaves that~ccu;fns-hallo;v wnterm channels as a result of any momentary change in the local depth of the . _water (Art. 18-6). Such a change may .Q~opecLbjullsJ;urha.ll.c~~u;l.!: obstacl~in...thJL!lb.!IJlp.el that cause a di§placement of waj;.!'lr.J.!.b.
v'{jD;

>IL

n-2

I O~ber dimensionless ratios used for the. sa.me purpoae.include (1) the lcinenc-flow factor}. VI/uL ... FI, first tlsed by Rehbock [251 and then by Ba.khmetefi' f26Ji (2) the Bouuinesq number B "'" V / v'2UR, first used by Engel [27J; 8.nd (3) the kinelicity or velocitY-head ratio 11; = V'/2gL, proposed by Stevens [28] alld Posey [29J respecti vaLr. . •

-I

itJ

'J 14

.

r

BASIC PRINCIPLES .

designed for this effect ;that is, the Froude llumhflr of the flow in the model' channel must be made "qual to tha,t of the flow in the prototype channeL 1-4. Regimes of Flow. A combined effect of viscosity and gravity may produce anyone of four 'regimes of flo7JJ in an open channel, namely, (1) 8ubcritical-larninar, when F is less than unity and R is in the 19,ininal' range; (2) 8upel'cl'itical-laminar, when F is greater than unity and R is in the laminar range; (3) ::mpercritical-turbulBnt, when F ia greater than unity

OPEN-CHANNEL FLOW AND ITS CLASSIFICATIONS

15

I I

1 \........,;;

~

1

1

I

1

,_I

I

'1

(

1

'-J

~\

I

j

,-,'

i

I

~J

f,

)

i I

C:--'\

I

\..-

\

!

~.~

Velocily, Ips

FIG.

1~5.

Depth-velocity relationships for four regimes of open':channel flow.

,_-

(Afler

Roberlson and Rouse [3D].)

and R is in the turbulent range'; and (4) subcritical-turbulent, when F is less than unity and R is in the tui:bulentrange. The depth-velocity relationships for the four flow regimes in a wide open channel can be shown' by a logarithmic plot (Fig. 1-5) [30]. The heavy line for F = 1 and the shaded band for the laminar-turbulent transi.tional range inters,ect on . th~ graph and divide the whole area into four portions, each pf which repres~nts a flow regime. The first two regimes, sub critical-laminar and ,supercritical-Iaminar, are not commonly encountered in applied openchannel hydraulics, since the flow is generally turbulent in the channels considered in engineering problems. However, these regimes occur : frequeI).tly where then~ is very thin depth-this is known as sheet flow. and they become significant in such problems as the testmg of hydraulic , models, the study of overland flow, and erOSlOn cOlltrol for such flQlY:• Photographs of the four regimes of flow are shown in Fig. 1-6. In each

'~

I

L Fw. 1-6. Photographs showing four flow regimes in a laboratory cll!1nnel. of H. Rouse.)

i

.

.

..

(Courtesy

photograph the direction of flow is from left to right. All flows are uniform except those on the right side of the middle and bottom views. The top view represents uniform subcritical-laminar flow, . The flow is su.b:ritical, since the Froude number was I',djusted to slightly below the cntical value; and the streak of undiffused dye indicates that it is laminar. Th~ middle ~i~w shows a uniform supercritical~laminar fl'ow changing to v~r~ed subcntical-turbulent. The bottom view shows a uniform superc:'1.tlCal-turbulent flow changing to varied subcritical-turbulent. In both cases, the diffusion of dye is the evidence. of turbulence.

{_.

,

16

I

(

\.

~l

,

BAstC ' PRINCIPLES

It is'believed th~t gravity action may have a definitive effect upon the flow resistance in cliurmels at the tut'bulent-flow range~ The experi,mental data studied by Jegorow [311 and Iwagaki [32J for smooth rec'tangular channels. and by Hom-:ma [33J for rough :channels have shown that, ~n the supol'critical-turhulent regil;l1e of flo~, the friflj;ion fact
( I

where A 1 and A. are the cross-sectional areas of the flow at sections' 1 and 2, respectively, a.nd lJ.y'is the drop in water surface between the sections. 1-2. Verify Eq. (1-10). 1~3. Verify by computation the depth-velocity' relationships shown in Fig. 1-5 £0: the four flow regimes in a wide rectangular open channel. The temperature, o,f the water is tnken as 68°F. . 1-4. A model channel is used to simulate a prototype channel 100 ft wi~e, carrying ;t disclmrge of 500 ds aL. a depth of 4 ft. The model is designed fOI' gravity effect, Itnd a: tUl'bu[ent-fiow condition is asaured. Determinet.he minimum size of the model and the scale ratio, BSSuming the upper limit of the \'r1lnsitionru-flow region to be R = 2,000. The sCII.le ratio is the ratio 'of the linear dimension of the model tot,ha.t of the prototype. . REFERENCES

'\

I

1. T. E. Stanton and J. R. Pannell: Similarity of, motion in relation to surfa.ce ftiction of fluids, PMlosophical1'r(l?!sactions, Roya! S()ciety of LaMon, vol. 214A, pp; 199224, 11114. : . ' : 2. H. Darcy: Sur des rechercbes experimentales relatives au mouvement des eaux dans les tuya~ (Experimental researches on the floW' of water in pipes), CiJmptes rendus des 8~an;ces del' Aoodemie des Sciences, vol. 38; pp. 1109-112~, June 2tlj 1854. 3. Julius Weishach: "Lefu-bueh der Ingenieur-! und Maschinemnechanik" (,ITextbook of Engineering Mechanics"), Brunswidk, Germa.ny, 1845. ' j. 4. J. F. d'Aubuisson de Voisins: "Tl"aite d'hydrl:\.ulique" ("Treatise on Hydraulics"), Levrant, Pari", '2d ed., 1840; trll.nslated int(l) English by Joseph Bennett, :Little, Brown &: COllj,pany, Boston, 1852, pp. 202-211. '

I

I ! !I ,I

OPEN-CHANNEL FLOW. AND JTS CLASSIFICATIONS

17

5. H. Blasius: Das Ahnlichkeitsgesetz bei Reibungsvorgiingenin Flilasigkeiten (The In.w oLsimilitude for: frictions in fl\1ids), Forschungsheft des VereitUl deutscher .' " Il1gimieure, No. 11:11, Berlin, Illi3. 6. Theodor von Ruman:. Mechauische Ahnlichkeit und Turbulenz (Mechanical similitude and turbulence), Proceedings afthe 3d In/ema/irma! ConfITBsl!liar Applied M ecka1!.ics, Stockhol1n, vol. I, pp. 85"-93, 1930. 7. L. Prandtl: The mechanics of viscous fluids, in W. F. Durand (editor-in-chiei): "Aerodynamic Theory./' Springer-Verlag,Berliu, 1935, vol. III, div. a,.p. 142. 8. J. Nikuradse: Gesetzmlissigkeiten der t,Ul'bulenten Stromung in glatten Rohren (Laws of turbulent flow· in smooth pipes), Forsck'ungsheft des .Vereins deutllcher Ingenieur~, No.'3M, Berlin, 1932. 9. Garbis H. Reulegan: Lews of turbulent flow in open channels, paper RP1151, Jou;'!1.al of Research, U.S. Na.t.iollaL Bureau of Sl.c.ndards, vol. 21, pp. 707-741, Dllcember, 1938. 10. J. Allen: Streamline and turbulent, flow in open channels, The London, EdinbuTgh and Dublin Philosophical Magazine and Journal of SciencfI, ser. 7; vol. 17, pp. 1081-1112, June, 1934. . , 11. H. Ba.zin: Recherches experimentrues, sur l'ecoulement de l'eau dans les canaux decouveris (Ex.perimental researches on the flO\v of wa.ter in open cha.nnels),' M emClires pT~senUs par d'!IIBrS savnll!~ d l' Academie des Sciences, Paris, voL 19, 1865. 12. Studies of river bed materials and their mQvement, with special referance to the lower Mississippi River, U.S. Waterways Experiment Sta.tion, Technical ' Paper 17, January, 193.5. 13. 8. P. Raju: Versuche uberden Stromungswidersta.nd gekrummter o:ffener Ka.n!!.le (Study of flow resistance in curved open ehannels), Miu"ilunge:TL des hydro.uli8chen In~tiluts der lel:h'rr.ischrn Hochschule Muncher£, no. 6, pp. 45-60, Munich, 1933. English translation by Clarence E. Bardsley: Resistance to flow in curved open channels, Proceedings, American Society of Civ'ii Engineers, vol. 63, pt. 2, p. 49 after p. 1834, Novembet:, 11)37. 14. Lorenz G. Strllub: Studies of the transition-region between laminar and turbulent flow in open channels, Transactions, American GeQphY~icat Union, vol. 20J pt. iv, pp. 649-663, 1939. 15. Otto Kirachtner: Reibungsverluste in Rohren und Kanalen (Frictional losses in pipEllj l1nd channels); Die Wass~rwirtschaft, Stuttgart, voL 39, no. 7, pp. 137-142, April; no. 8, pp. 168-174, May, 1949. 16. Ott.o Kirschmer: Pertes de charge dansles conduites forcees et lea eaI!o.ux decouVllJ'ts (En:ergy losses in pressure conduits and open channels), RllIIue q6n~rale de l'hydrauUllue, vol. 15, no. 51, pp.1l5-138, May-June, 1949. 17. Yuichi Iwaga.ki: em Lllmina.r to turbulent Row in a wide open channel, by W. hi. Owen, Tran8actions, .Ameriron Society of Civil .Engineers, vol. 119, pp. 1:165-1166, 1954.

•

.

18. Horace William King: :"Handbook of HydmIlIics," revised by Ernest F. Brater, McGrll.w-Hill Book Co,npany, Inc., New York, 4th ed., 1954, p. 7-35. 19. F. Bettes: Non.uniforln flow in cli.annels, Ciuil Ertgineering and P,.blicW01'ks Review, London,· vol. 52, no. 609, pp. 323-324, M~rch; no. 610, pp~ 434-4,36, April, 1957. , . 20. Lorenz G. Straub, Edvtard Silberman, and Herbert O. Nelson: Open-channel flow at. small Reynolds nUfnbers, Transacfions, American Society of Civil Engineers, vol. 123, pp. 685-706, ,1958. ! .21. Walll1ce M. Lansford ~nd James M. Robertson: DiB~lIssion·of Open-channel flow at small ReynCllds nUl)lbers, by.Lorenz G. Stra.ub, Edward Silberman, and H"r-

.j 18

BASIC PRINCIPLES :

b!lrt C. Nels01l, Trqnsaction8, American So.ciety o.J Civil Engineers, vol.. 123, pp. 707-712, 1 9 5 8 . . . 22. FtlLnz Eisner:- Offene Gerinne (Open channeL), sec. 4 of vol. IV, "Hydro- und Aerodynamik," in \V. Wien and F. Hanns (editors-in-chief): "Handbuch der Experimentalphysik," Akademische Veda.gsgesellschaIt mbH, Leipzig, 1932, p.298. 23. Josef l(ozeny: "Hydraulik" ("Hydraulics"), Springer-Verlag, Vienn9., 1953, p.574. . 24. Henry M. Mcirris, Jr.; Flow in rough condl,lits, Transactions, American Society oj Civil Engi1J,ee1's, vo.l.120, pp. 373-398, 1955. Discussio.ns on pp. 3.99-410. 25. Th.R.ehbock: Zur Frage des Briickenstaues (On the prDblem Df bridge constrictions), Zenlralblail der B(11!verwaZtun!l, BerUn, vol. 39, ~o. 37, pp. 197-200, 1919. 26. Boris A. Bakhmeteff: "Hydraulics of Open Channels,"McGraw-Hill Book Com. pany, Inc., New York, 1932, p. 64. . 27. F .. V. A. E. Engel: Non-uniform flow of water: Problems a.nd phenomena. in open channels with side cont.ractions, The Engineer, vol. 155, pp. 392-394, 429-430, 456-457, 1933. . ... . 28. 1. C. Stevens: Discussion Dn The hydraulic jump in sloping channels, by C. E. Kindsva.ter, TramactiO'nll, American Society of Civil Engineers, vol. 109, pp: 11251135, 1944. 29. C. J. Posey: DiscussiDn on The hydraulic jump in sloping channels, by C. E. Kindsvater, TransactioTLI/, American Society oj Civil Engineers, vol. 109, pp. 11351138, 1944. 30. J. M. Robertson and lIunter Rouse: On the four regimes· of open-channel flow, Civil E'lI.{Ji11eering, vol. 11, no.. 3, pp. 169-171, !vIa.rch, 1941. 31. S. A. JegorDw: Turbulente 'Oberwellenstromung(Schiesscnl in offenen Gerinnell mit gla.tten Wanden (Turbulent 5up'
I·I

\..:.·,1

I

\

i

i, CHAPTER

2

OPEN CHANNELS AND THEIR PROPERTIES

2-1. Kinds of Open Channel. An open channel is a conduit in which water flows with a iree surface. Classified according to its origin a channel may be either natural 01' artificial. ( Natural channels include all watercoUl:~ that exist naturally on the <......1 \ earth, varying in size from tiny hillside rivulets, through brooks, stl·eams,. smaH and large rivers, to tidal estuaries. Underground streams ca~ water with a fr~e surface are also considereclrulturul open chann~. The hydraulic properties of nll,tural channels are generally very irregular. In some cases empirical assumptions reasonably consistent with actual obs:ervl;ttiol1s and experience may be made such that the conditions of flow. in these channels become amenable ~o the analytical treatment of theoretical hydraulics. A comprehensive study of the behavior of flow in natural channels requires knowledge of other fields, such as hydrology, geomorphology, :sediment transportation, f1tc, . It constitutes, in' fact, a subject of its own, known as river lLJlP,raulics, Artificial channels are those constmcr,ed or developed by human effort: navigation channels, power canals, irrigation canals f),nd flumes, drainage ditches, trough spillways, floodways, log chutes, roadside gutters, etc., as well as model channels that are built in the laboratory for testing pm·poses. The hydraulic properties of such channels can be either controlled to the extent desired or designed to meet given requirements. The application ofhydra.ulic theories to artificial channels will, therefore, produce results . fairly close to actual conditions and, hence, are reasonably accurate fOl' ,......... practical design purposes. . Under various circumstances in engineering practic\'l the.artificial open channel is 'given different names, such as "canal," :'\ftume," "chute," "drop," "culvert," "open-flow tunnel," etc. These names, however, are used rather loosely and can be.defined only in a very genernl way. The canal is usually a long and mild-sloped channel built in, the ground, which may be unlined C!r lined with stone masonry, concrete, cement, wood, or bituminous materials. The flume is a channel of wood, metal, concrete, or masonry, usually supported on or above the surface of the ground to carry water across a depression. The chute is a channel having steep 19

20

(

BA.SIC PRINCIPLES

slopes. The drop is similar to a. chute, but the change in elevation is effected in a shott distance. The culvert flowing partly full is a covered channel of comparfl,tively short length installed to drain water through highway and railroad embankments. The open-flow t'unnel is a comparatively long covered channel used to ca.rry water through a hill or any obstruction on the ground. 2-2. Channel Geometry. A chann\'ll built witlulllY.a:t+ing cross sectiOll ti..'1dc~bottom slope is ~~lled a 'B.risma(ikma1Jnel. Otherwise, the channel is nonprismaticj an example is a trough spillway having variable width and curved alignment. Unl\lss specifically indicated, the channels dGScl:ibed in this book are prismatic. The term channel Bection used in this book refers to the cross section of a channel taken normal to the direction of the flow. A l'ertical channel s8ctio-n, howeve".is the vertical section passing through the lowest or bottom point of·the channel section. For hvl'izontal channels, therefore, the channel section is always a vertical channel section. Natural channel sections are in general very irregular, usually varying from an approximate parabola to an approximate tmpezoid. For streams subject to frequent fioods, the channel may consist of a main channel section carrying normal discharges and one or more side channel sections for accommodating overflows. Artificial channels are USUally designed with sections of regular geometric shapes. Tablc 2-1 lists seven geometric shapes that are in common use. The trapezoid is the commonest shape for channels with unlined earth banks, for it provides side slopes fol' stability. The rectangle and triangle are special cases of the trapezoid. Since the rectangle has vertical sides, it is commonly used for channels built of stable materials, such as lined masonry, rocks, metal, or timber. The triangular section is used only for small ditches, roadside gutters, and laboratory works. The cirde is Hle popular section for sewers and culverts of small and medium sizes. The parabola l is used as an approximation of sections of small and medium-size natural channels. The round-cQrnered rectangle is a modification of the rectEmgle. The round-bottom triangle L':! an . approximation of the parabola; it is a form usually created by excavation with .shovels. Closed geometric sections other than the circle are frequently used in sewerage, particularly for sewers enough for a man to enter. These sections are given various names according to their form; they may be

I

I

!

I I i i

I

r 1

i

I !

.i!:

+

I;'.>

+

~

1 The side slope z; 1 of a pa.rl!.boli~ section at the intersection of the sides with the free surface can be computed easily'by the simple formula z = Russian e!lgineers [11 also use semielliptica! and para.bolic of higher order: y azP with 11 .;, 3 Qr 4. The constant CL is computed fl'om the side slope assumed· at the' free surface ..

21

22

BASIC PRINCIPLES

egg-shaped, ovoid, semielliptical, U-shaped, catenary, horseshoe, baskethandle, 'etc. The complete rectangle and square are also common for large sewers. Dimensions and'properties of sewer sections may be found in textbooks on sewerage.! A special geometric section kilOwn as hydrostatic catenary or lintMrw [4,5J is the shape of the cross section of a trough, formed of flexible sheets assumed to be 'Iveightless, filled with water up to the top of the seCtion, . and firmly supported at the upper edges of the sides but with no effects of fixation~ The hydrostatic catenary has been used for the design of the sectiolls of some elevated irrigation flumes. These flumes are constmcted of metal plates so thih that their weight is negligible, and are :firmly attached to beams at the upper 2-3, Geometric Elements of Channel Section.' Geometl"l:c elements are pro'perties of a channel section tha.t can be defined entirely by the geometry of the section B.nd the depth of flow. These elemerits are very importa.nt and are used extensively in flow computations. For simple regular channel 'sections. the geometric element.s can be expressed mathematically ill terms of the depth of flow and other dimensions. of the sectioll. For complicated sections and sections of natural streams, however, ,110 simple formula can be 'written to express these elements, but curves representing the relation -between these elements and the depth of flow can be prepared for use in hydraulic computations. The definitioris of several geometric elements of basic importance are given below. Other geoIlwtric elements used in this book will be defined where they first appear. The depth of flow y is the vertical distance of the lowest point of a /' 'chanilel section from the free surface. This term is often used inter-' changeably with the depth offlow section d. Stridly sp~aking, the depth of flow section is the depth of flow normal to the direction of flow, or the height of the channel section containing the water. For a channel with a longitudinal slop~ angle e, it can be 8een that the depth of flow is equal to the depth of flow section divided by cos e. In the case of steep channels, therefore, the two tei'ms should be used discriminately. The stage is the elevation or vertical distance of the free smfa-ce above 'a' datum. If the lowest point of the channel section is chosen as the datum, the stage is identical with the depth of flow. The top width T is the width of channel section at the free surface. / The wafer al'ea A is the cro~s-sectional area of the fi~w norlD,al to the / direction of flow. The wetted perimeter P is the length of the line of intersection of the channel wetted surface' with a cross-sectional plane normal to the direction bf flow. ' I Many typica.! seWer sections arEi described in [2J nnd [3].

OPEN CHANNELS AND THEIR PROPERTIES

23,\

The hydr.aulic radiu,$ R is the ratio of the water area to its wetted perimet.er, or. A R (2-1)

'1

The hydr~ulic de1Jth D is the ratio of ~le wa.ter area to the top width~ or D

=

T

(2-2)

The slJction factor for critical-flow contputtllion Z is the product of the . water area and the square root of the hydraulic depth, or

Z=A

(2-3)

.j

. The section factor for uniform-flow compulaUon A]z% is the product of the water area and the two-thirds power of the hydraulic radius. .

\

"

I

.!

.

FIG. 2-1. Geometric elements of a. circular section.,

I

~lements

Tabie 2-1 furnishes 11 list of formulas for six basic geometric of' 1 seVf\11 . commonly' used channel sections, For a circular section, the curves in Fig. 2-1 represent the ratios of the geometric elements of the I s~ction to' the corresponding elements when t11e section is fl~wing full.] These curves are prepared from It table. given in Appendix,A. For cer- '. tain trapezoidal, triangular, and parabolic sections commonly found in practical uses, the diag.rams given in Appendix B provide a convenient Ii means of determining the geometric elements. . '

I

'{ [

·1

,\

\ ' \

i

--

24 )

), ,-

"

~',.#

, Example 2-1. Compute the'hydraulic radius, hydraulic dep~h, and section factor.?: Of the~ra.pezoidal chUlluel section in Fig. 2-2. The depth of flow is' 6 ft. ,

'

~~' , '

'

Q,?!/' ~ '

\

~~ .~' C~b=20':, .1' \ '~//l7~~0~

.

....&..

T:44"

P=46,8 -

25

OPEN CHANNELS ,AND 'TIIEIR PROPERTIES

BASIC PIIINCIPLES

,

1.5

t~.5'

,

.

'.

I

Tropezuidal chonnel

Triangular channel

,

FIG. 2-2. A channel crosS'section. $oluti{)ll. By formulas given in Table 2-1, the following are computed: P = 20 +' 2 X 6 ";5 = 46.8 ft; A = 0,.5(20 + 4'!L0 6 = 192.0 ft>; R = Hl2/46.8 = 4.10 ft; D = 19%4 = 4.37 ft; and Z = 192 -./4.37 = 401 ft,·,:

Sholtow ditch

.

NarroW' (@c!anglJlar 5~ction

a

),

\'I

2-4. Velocity Distribution in Channel Section. Owing to the pres~ ence of a free surface and to the friction along the Ghannel wall, the velocities in a channel are not uniformly distributed in the channel section. The measured maximum velocity in ordinary channels usually appears to occur below the free .sUlface at a distance of 0.05 to 0.25 of the depth;

Pipe Natural irregular channel

FIG. 2-4. Typical curvea of equal velocity in various channel sections.

FIG. 2-3. Velocity distrib,ution in'·a. rectangula; 'channeL

I r

r I, .

the closer to the banks, the deeper is the maximum. Figure 2-3 illustrates *e general pattern of veJocity distributio;l over \;al'ious vertical and horizontal sections of a rdctangular channel sectio~ and the curves of equal velocity in the cross section. The general patterns for velocity distribution in several channel sections of other shapes are illustrated in Ji]ig. 2-4. ; , I The velocity distributfoJ in a channel section depends also on other f('l.clors, such as the unusu~l shape of the section, tlle roughness of the

channel~ <md the pJ·esence of bends. In a broad, rapid, !111d, shallow stream,or in a very smooth channel, the maximum velocity may often be found l,Lt the h'ee surface. The ,roughness 6f the channel will cause the curvature of the vertical-velocity-distribution curve to increase (Fig. 2-5). On a bend the velocity increases greatly at the convex side, owing to the centrifug!11 action of the flow. CCfmtrary to the usual belief, a surface wind has very little effect i on velocity distribution. '/ As revealed by cv.refullaboratory investigations, ~ROugh' bed the flow in a straight prismatic channel is in fact three-dimensiqnal, manifesting a spiral motion, although the velocity component in the transvel:se 'FIG., ,2-5. Effect of chann~i section is usu!),lly small, and insignificant roughness on velocity compMecl with the longitudinal velocity com- distribiltion in an open , chann~I. ponents. Shukry [6] found that, in short labora. tory fl~mes, a small disturbance ~t the entrance, which is usually unavoidable, is sufficient to cau()e the zqne of highest water level ito shift to one side, thus giving rise to a single ~piral motion (Fig. 2-6). : In a long and uniforrri reach femotefl'om the entrance, a double spiral mqtion will oc~ur to per¢it equalization of shear stresses on both sides of th~ channel [7,8J.

26

OPEN CHAN)lELS AND THEllR PROPERTIES

BASIC PRINClPLES

The pattern will include. one spira}.on each side of the center line , where . th€ water level is the highest. In practical considerations, it is quite safe to ignore the :;piral motion in straight prismatic channels. Spiral flow in curved channels, however, is an important phenomenon to be considered in design add wHl be discussed later (Art. 16-2).

(cleonlcur Jines

'or

equal componenl lv'l

(o)Con!our lines 01 equal ve c for (v)

Co I Con lou, lines of equal componenHv,1

(d) ConI our lines 01 . equal component (vyl

(e) Dire clio n Jines and

Jr

ma9niludes of the IOlerol currenlslvor '

FIG. 2-6. Distribution of the velocity components, fa.cing downstream a.t the midsection of a. straight flume. Voloeitiestl.re In em/sec (= 0.0328 fps); y/b = 1.0; R "" 73,500; and Q = 701iters/s.ee (= 2.47 cfs). (Afler .A. ShlLkry [6J.)

2-5. Wide Open Channel. Observations in very wide open channels have shown that the velocity disttibution in the central region of the section is essentially the same as it ;would be in rectangular channel of infinite width. In other words, under this condition, the sides of the !iliJ!imel h~i practically DO influence on the velocit.y distribution in the C~gi~l, and. the flow i~_Y;~~.Eehtral region c~n therefore be regarded as two-dimensional in hydraulic ana:1yses. Careful experiments indicate, further, that this central region '~Xi~t8 in rectangular channels only when

a

~

I

27

the width. h,grea~.E 5 to 10 times. the depth <:>f flow ...4~IJe~on the .2_'?~ion of l:lu):fac~ roughness. Thus, a wide Open channel can safely be defined as a rectangular channel whose width is greater than 10 tiri1es the dept.h of flow. For either experimental or anaJytical purposes, the flow in the cen tral region of a wide open channel may be' considered to be the same as the flow in !1 rectangular channel of infinite width. 2-6. Measurement of Velocity. According to the stream-gaging procedure of the U~S. Geological Survey,'lthe channei crosssecticH1 is-ciivid~d -~rti6a1 strips by a number of successive vert~cals, and mean velocities in verticals are determined by measuring t.he velocity at 0.6 of the depth J!-l_~Gh vertical, or, where morc reliable results are required, by taking the average of the velocities at 0.2 and 0.8 of the depth. When the stream i.s covered with ice, the mean velocity is no longer close t.Q 0.6 of the water dept.h, l;mUI!e average at 0.2 and 0.8 of the water depth still gives reliable resul.1§. The average of the ~ean velocities in any two adjacent verticals multiplied by the are:1- between the verticals gives the discharge through this vertical strip of the cross section.: ThG sum of discharges through all strips is t.he total discharge. The mean velocit.y of the who.Ie section is, therefore, eq ua! to the total discharge divided by the whole· area. It should be noted that the above methods are simple and approximate. For precise measurements more elaborate methods must be used, which are beyond the scope of this book. 2-7. Velocity-distribution Coefficients. As a result of nonuniform 9.istributiol1 of velocities over a channel section/the velocity hea.d of an open-channel flow is generally greater than the value computed according to the expression V 2 /2g, where V is the ri;lea.n velocity. When the,energy principJe is used in computation, the true velocity head may be expressed as a V 2/2g, where a is known as the energy coefficient or Co-riolis coe.fficient, in honor of G .. Coriolis [12J who first proposed it. Experimental data indicate that. the value of a varies from about 1.03 to 1.36 for fairly straight pri!:imR.tic channels. The value is generally higher for small channels and lower for large streams of considerable depth. The nOllUniform di.stribution of '/elocities also affects the co.rnputation of momen~umin open-channel flow. From the principle of mechanics, the "momentum of the fluid passing through a channel section per unit time is expressed· by {JwQVj(J, where f3 is known as·the mom.entum coefficient or Boussinesq coe:ffi.cient, after J. Boussinesq [13] who fii'st proposed it; w.is the unit weight of water; Q is the discharge; and V i.'3 the mean velocity. It is ,generally found that the value of (3 for fairly straight prismatic cha.nnels varies approximately from 1.01 to 1.12. The two velocity-distribution coefficients are always slightly larger than the limiting value of unity, a.t which the velocity distribution is 1

For details see

!9J

to [11 J.

.

_,. t:../

_

)

'I

!

) .

· ,

\

28

,

BASIC PRINCIPLES

OPEN CHANNELS AND THEIR PROPERTIES

strictly uniform across the channel section. For channels of regular cross section and fairly straight alignment, the effect of nonuniform velocity distrihution on the computed velocity head and momentum is small, especially in comparison with other uncertainties involved in ~he computation. Therefore, the coefficients are often assumed to be unity. In channels of complex cross section, the coefficients for energy and momentum Can easily be great as 1.6 and 1.2, respectively, and can vary quite rapidly from section to section in case of il'I'egular alignment. Upstream' from weirs, in the vicinity of obstructions, or near pronounced irregularities in alignment, values of a greater than 2.0 have been observed. 1 , Precise studies or analyses of flow in such channels will require measurement of the'actual velocity and accurate determination of the coefficients. In regard to the effect of channel slope, the coefficients are usually higher , in steep channels than in flat channels. For practical purposes, Kolupaila [16J proposed the values shown below for the velocity-distribution coefficients. Actual values of the coefficients for a number of channels may be found in {l7) and [18J.

as

I

"I

I!

Vo.lue of", '

Value of {j

Channels Min

Av

'Max

lVIin

Av

Ma.x '-"0-

Regular channels, flumes, spillway:;: .... , N o.tural streams and torrents ..... . . Rivers.under ice cover., ..... ....... River vlllleys" ove,flooded ..... ...... ~

r

I,

I r I

~

1.10 1.15 1.20 1.50

1.15 1. 30 1.50 1.75

1.20 1.50 2,00 2,00

1.03 1.05 L05 1.10 1.07 '1.17 L 17 1.25

1.07 1.17 1.33 1.33

2-8. Determination of Velocity-distribution Coefficients. Let LlA be an elementary area in the whole water area A, and w the unit weight of water; then the weight of water pasSing LlA per unit ,time with a velocity v is 'wv LlA. The kinetic energy of water passing AA per unit time is wv 3 AA/2g. This is equivalent to the product of the weight wu AA and the velocity head v2/2g. The total kinetic energy for the whole ,WB.ter area is equal to 2:wv ll t1Aj2g. Now, taking the whole area as A, the mean velocity as V, and the 1 A va.lue of", = 2.08 was computed by Lindquist [141 uslng da.~a from Wl}it' measurements made by Ernest W. Schoder and Kenneth B. Turner. In the case;of clolled conduits, much larger values of ,. have been obS,erved [15]. A value of '" = a.87, observed at the outlet section of a. dro.ft tube in the Rublevo power pmnt, is probably the largest known value obtained from actual measurements; the re",l value there must have been still lar!!~r-,-10,2% more, if the effect of a. IS" curvattire of the streamlines is taken into account. The largest known value from laboratory meO:suremen1;$ is believed to be '", = 7.4, which was derived by V. S. Kv:ia.tkovskii in 1940 in the VIGM (All-Union Institute for Hydraulic Machinery, U.S.S.R.) for the spiral flow under !l. model turbine wheeL

29

2

corrected velocity head for the whole area as a V /2g, the total kinetic energy 'is ,aw A/2g. Equating this quantity with 2:wv 3 AA/2g and reducing, J1]3 dA L;'~! LlA (2-4) V3A ,'"'"

va

The momentum of water passing AA, per unit time is the product of the mass W
, Jy 2 dA {J

=

V2 A

2:v 2 LlA "". V2 A

(2-5)

O'Brien nnci Johnson [19] used a graphical solution of the above formulas sa follows: From the measured velocity-distribution curves, the area ivithin each curve of equal velocity is planimetered. Taking the velocity indicated by each equal-velocity l!'!!rve as v, a curve of 1]3 against the corresponding planimetered area is constructed. . It is ,evident that the area beneath this v3 curve is the integral 2:v 3 AA, which can be obtained by planimetering again. Similai:ly, integrals 1:v 2 Ail. and 2:v LlA can also be obtained. The integral 2:v AA divided by A gives V. With these quantities determined, the above equations can be solved for the coef- . ficients a and p. For approximate values, the energy and momentum coefficients can be 'compl.lteaDytne following formulas: 1 Ct

'p

= 1 + 3.% 1

+

f2

2t 3

(2-6)· (2-7)

where Ii = v.u/V 1, VM being the maximum velocity and V being the mean velocity; . Computation of the velocity-distribution coefficients for. irregular natural channels will be discussed later (Art. 6-5). In most pl'nctical problems dealing with regular cha~nels it is not necessary to consider the varia.tion of velocity throughout the cross section, since use of the average· veiocity will give the accuracy required. The expressions V2/2g and wQ V j g a~e used extensively in this book with the understand:ing either that these items have been cOlTected for the effect of the non~ uniform velocity dis~ributionj Of that a value of unity is asiumed. 2 I These formula.s a.re obLained by I1ssuming a. logtLrithmic distribution Qf velocity (Art. 8-5, Prob. '8-9). Assuming a lincar velocity distribution, Rehbock [201 obtained c< = 1 f" I1nd.{j = 1 "/3. . • For discussions on this subject, the reader may look into. 121J and [22]. However, he should use' judgment in reading these references because they contain erroneous

+

+

OPEN CHAN,lIjELS AND THEIR PROFER'I'IES

BASIC PRINCIPLES

30

2-9. Pressu're Distribution in a Channel Section. The pressure at any point on the crO,';fi section of the flow in a channel of small slop: can be , measured by the height of the water column in a piezometer tub~ Instayed at the point (Fig. 2-7). Ignormg mInor disturbances due to turbulence, etc., it is apparent that this water column should rise from the point of measurement up to the hydraulic grade line or the water surB face. Therefore, the pressure at any point on the section is directly propor(al tional to the depth of the point below the free ~ui'face 9,nd equal to the hydrostatic pressure corresponding to this depth. In other words, the distribution of pressute over the cross section of the channel is the 'same as the distribution of hydrostatic pressure j that is;the distribution is linear and can be represented by a straight line AB (Fig. 2-7a). This is known ,as the, hydrostatic latO of preSS1lrB distribution . ./ Strictly speaking, the application of the hydrostatic law to the pressure distribution in the ,crQ§s se~tiQn of a flQwing , 8 s' channel is valid only if th&.!o\v filaments (c 1 have no accele~~n coinponertts in the FIG. 2-7. Pressure distribution in straight and curved channels of ·plane·gf cross sectioE' This type of flow small or horizontal slope at the ~oretically known as parallel flow, l section under considera.tion. h = that is, such th'at the streamlines ha VI! piez~metric head; h. = hydrosta.tic head; and c "" pressure-bead cor, neither substantial curvature nor diverrection for curvature. (a) Par~ gence. ~~uently, there are n.o agallel flow; (b) convex flow; (c) preciablc acceleration component.§..JlQ!:= concave flow. mal to the direction of flo'\' that \VO~ disturb the hydrostatic-pressure d~J;.iQ.u in thfLcros~ctien of-flt--

)

31

In actual problems uniform flow i:s practically parallel flow. Gradually varied flow may also be l'egarded .as parallel flow, since the cha.nge in depth of flow is so mild that the streamlines have neither appreciable curvature hoI' divorgence; that is, the curvature and divergellce are so small that the effect of the acceleration components in the cross-sectional plane is negligible. For practical p1.rposes, therefore, the hydrosta.tic law . of pressure dislribution is applicable to gra,dually varied flow as well as to 1miform flow. If the curvature of stl'eamlines is substantial, the flow is theoretically ~-> known as curvilinear flow. The effect of the curvature is to produce appreciable ELGceleration components or cell trifugal forces normal to the direction of flow. Thus, the pressure distribution over the section deviates from the h}'drostatic if ourvilinear flow occurs rn the vertical plane:' Such curviiinear flow may be either convex or concave (If;lg~nd c). In both cases the .nonlinear pressure distribution is represented by AB' instead of tlte straight distribution AB that would occur if the flow wel'e parallel. It is assumed that ail streamlines are horizontal at the section under consideration. In concave flow the centrifugal forces are pointing dDW:lward to reinforce the gravity action; so the resulting pressure is greater than the otherwise hydrostatic pressure of a parallel flow. In convex flow the centrifugal forces are acting upward against the gravity action; consequently, the resulting pressure is less than the otherwise hydrostatic pressure of a parallel flow. Similarly, when divergence of streamlines is great enough to developappl'eciable acceleration components normal to the flow; thehydrostati'c pressure distribution will be disturbed accQrdingly. Let the deviation from an otherwise hydrostatic pressure h. in a curvilinear flow be designated by c (Fig. 2-7b and c). Then the true pressure or the piezometric height h = h; + c. ' If the channel has a curved longitudinal profile, the approximate centrifugal pressure may be computed, by Newton's law of acceleration, ~~,~he E~~~~~ oL~L@iWt d and a croSQ..§ection. Qf 1 sq ft, that is, wd/g, and the centrifugal ac~ v2/r; or ' _----7

~.-~---

, ~l"tlow.

",/'

statements. Some authors have proposed the use of the mOluentUnl coeffic~en.tto , r'epJace the energy coefficient even in computations based on the energy PrJ?ClpJ~, " t ec't V'Nhether the energy coefficient or the momentum . , coeffiCient . . o1 ,IS d TI 1IS IS no carr . to be used depends on whether the energy or the moment?m prw::lple -:,e . The two coefficients are derived independently from baslca.lly different principles (Art. 3-6). Neither of them is wrong and neither ca.n be replaced by t~e other; both should be used in th~ correct sense. ' " ' I Specific qualifica.tions for parallel flow were clea:rly stated for the first tlme by Belanger [23]. '

=

wd~~ g

r

)

(2-8)

where w is the unit weight of water, g is the gravitational acceleration, v is the velocity of flow, and'r is the radius of curvature. The pl'esstirehead correction is, therefore, d v2 (2-9) c=-gr

IS Ill:r

;

\

.

p

~

.I

)

,

For computing the value ofc at the channel bottom, r is the radius of cu::',rature of the bottom, d is the depth of flow, and for practical purposes

J,

(

.

. i

32

"

BASIC PRINCIPLES

OPEN CHANNELS AND 'l'HEIR PROPERTIES

v m~y be ass~med equal to the average velocity of the flow. Apparently, c is positive for concave. flow, negative for convex flow, and. zero for parallel flow. ,. In parallel flow the pressure is hydrostatic, and the pressme head may be represe11ted by the depth of flow y. For simplicity, the pressure head of a curvilinear fiow may be represented by City, where a'is a correction ~oefficiimt for the curvature effect. The l:orrecti.on~oeffici~nt is referred to as apressure-dislribtttion coejfiy'ient.. Since this coefficient is applied to a pl'esswe head, it may be specifically SLa~'/m1~:!t!~~!1fficienl. It can be shown that the pressure coefficient is expressed by a'

/., i

I\ I

r.

~y}o

A

1

hI! dA

+

1

r

A

cv dA

sure head at any vertical depth is equal to this iiepth multiplied by a correction fact·or cos 2 e. . Apparently, if the angle (J is small, this factor will not differ apprecill.bly .fl'Om unity. In fact, the correction tends' to decrease the pressure head by an amount less than 1 % until e is nearly , 6" i a slope of about 1 in 10. Since the slope of ordinary channels is far less than 1 in 10; the correction foi: slope effect can usually be safely _ _-:i"" ignored .. However, when the cha~l_slope is large and its eff5l9t becoroe~ appreciable, thecorrection should be made if Mcumte comput~ttion is -'-~-.----.

-'

Pressure di.stribution

FrG. 2-8. Pressure tlistribution in parallel How in cha.nnels of lurge slope.

desired. A channel qf this type, say, with a slope gre~ter than 1 in 10,1.'.3 hereafter called a channel of large slope. Unless specifically mentioned, all chMlnels descrlbeanereafte-r are ~onsidered to be channels of small slope, where the slope effect is negligibie. If a channel of large slope ,has a. 10ngitudin!11 vertical profil£,; of appreciable. curvatul'e~ the pressure head should be cOl;rected fo]' the effect of the curvature of streamlines (Fig. 2-9). In simple notation, the pressure head may be expressed as g'y cos 2 Jt., """~~~~S== In channels of large slope the usu . andhighel' thl1n the critical velobity. When this velocity reaches a certain riHl.gni, "tude, the flowing water ,vill entrain nil', produ'cing a swell in its volume:i .and Ml increase in depth.l For this rel1son the pressUl'C computed uy Eq. {2-11) or (2-12) 4,~n shown in several gases to b.e higher than th~

h = Y cos 2, e h = d 90S e

'

.

'on. ~ertical section A'C

where d = ~ cos e, the depth measured perpendicularly from the water surface. It should be noted from geometry (Fig. 9-1) that Eq. (2-11) does not apply strictly to varied flow) piwticularly when 0 is v~ry large, whereas Eg. (2-12) still applies. Eq~lition ·(2-11) states that ,the presor--,

-

(2-10)

}o '

where Q is the total discharge andy is the depth of flow. It can easily be seen that a' is greater than 1.0 for concave flow, less than 1.0 for convex. flow, and equal to LO for parallel flow. . For complicated curved profiles, the total pressure distribution can be determined approximately by the fjow~net method or DlOre exactly by model testing. , In ra..mr!l.t.Y!1ried flow theghange in depth of fl.oJYJs so rapid and abl'Upt .:that th
or

1

=~

33

JAil' becomes entraine~.in ",,,,ter generally...!!! v"e19\l~ of !>~t go Ips and higher.[ Besides velocity, however. other factors such'as entrance condition, channel rough., ness, distance .traveled, cha.nnel cross section. volume discharge, etc •• all have some; bearing on a.ir entrainment. ' . ;

,

M. Hasurrh has measured the distributibn of pressure along the slopihg faces of we~ [241. The data. obtained from these ekperimenta have verified Eqs'. :(2 .. 11) and (2-12) very sa~Jsf!l.Ctorily (25]. . 1

or

i

I

I

35

BASIC PRINCIPLES

OPEN CHANNELS AND 'rHEIR PROPERTIES

actual measured pressure obtained 'by model testing. If the average density of the air-water mix~ure is known, it shouM be' used to, replace the density of pnrc water in the computation when air entrainment is expected. The actual density of the mixture varies from the bottom to: the surfafaTor the floV\L For practical purpo;es, however, the density may be assumed constant; this assumption of I!!llform air distributi_o_I?:_i~_

struct curves showing the relationships between the depth y and the section elementS A; R, D, and Z; and (b) determine, from the curves the geometric elements for y = 4.

34

Distance f,om a reference point neClr left bank, ft Left bank: -5

Stage, ft

5.6 4.6 4.0

-4 -2

1.9 0.8 0.2 0.3 0.2

0 1

2

3 5

'1

Distance fro~ a referenc(J point near .left bank, ft 7 9 11 13 15 17

Stage, ft -0.1 -0.1 -0.4 -0.1 0.7

19 Right bank: 20

3.2 4.1

j

\\

2.6

j \

2-6. The hydrostaLil" cll.tenary may be plott.ed for [Lny given depth l/ and slDpe angle O. at its ends by the following two approximate equations: :1:)=

fk [(1 -

%k' - 1)1.g,k4).p

+ (%k' + %2k4) sin 21/>

Concave flow

FIG. 2-9. Pressure distribution in curvilinear flow in channels of large slape.

the cross section wiII simplify cQmputation, with, the errors on the safe side.

.[

PROBLEMS 2-1. Verify the formulas far geometrio elements of the seven channel sections given in Tahle 2-1. $-2. Verify the cUrY'es shown in Fig. 2-1. 2-3. Construct curves siinilar tQ those shown in Fig. 2-1, for a square channel section. 2-4. Construct cu'rY'es similar to those shown in Fig. 2-1 for an equilateral triangle with one side as the channel bottom. 9.-5. From the data g(ven below on the cross section 1 of a natural stream con·

. I (2-13)

I

l

(2-14)

YI = if cos 10

Convex, flow

~ %56k~ sin 4.j.]

where :1'1 and YI, respectively, are the ordinate and abscissa measured from t,h.e midpoint of the free surface; k = sin (110/2); '" = sin- 1 f [sin (1/>/2)l!kl; and 11 is the slope angle at the point (XI,l!I), varying from 0 at the bottom of the curve to 9, at the ends. The above equations will define' the. cross section when the flow is at its full dept.h. The slope angle at the ends of a hydrostatic catenary of best hydraulic efficiency is found mathema.tically to be II, = 35'37'7". (a) Plot this section with' a depth y = 10 ft, and (bl determine the values of A, R, D, and Z at the full depth . 2-7. Estimate the Ylllues of momentum coefficient (j for., the- given values of energy c(lefficient ex = 1.00, 1.50, and 2.op. , 2-8. Compute the energy and mo~entum coefficients of the cross st'ction shown in Fig. 2-3 (a) by Eqs. (2-4) and (2-5), and (b) by Eq~; (2-6) and (2-7). The cross section and the curves of equal velocity can be transferred to a piece of drawing paper and enlarged for deSired ll.ccuracy. 2-9. In designing side walls steep chutes and overflow spillways, prove that the overturning moment due to the pressure of the flowing water is equal to Yswy' cos' 9, wherew.is the unit weight of water, y is the vertical depth of the flowing water, and 9 is t,he slope angle of the channel. 2 ..10. Prove Eq. (2-10). 2-11. A high-head overflow spillway (Fig. 2-10) has a 60-ft-radius flip bucket u.t its downstream end. The bucket is not submerged, but acts to change the direction of the flow from the slope of the lipillway face to the horizontal and to discharge the flov1 into the air' between vertical training walls so ft apart. , At: a discharge of 55,100 ds, ;the water surface at the vertical section OB is at El. 8.52. 'Verify t.he curve that represents the computed hydraulic ,pressure acting on the training wall at section DB. The computatiQn is bailed an Eq,' (2-9) and on'the following assumptions: (1) the velqcity is uniformly di~tl'ibuted across the section; (2) the vo.lu,e used for r, fQr pressur~ values near the wall base, is 'equal to the radius of the bucket but, for other pre;isure values, is equal to the radius of the concentric flow lines; and (3) the flow is entto.ined with air, and the density ,of the air-water mixtureca~ be estimated by the

(,

"

, i

of

/a)

1 It is common practioe to show the cross section of a stream in a direction looking , downstream and to prepare the lQngitudinal profile qf a channel so that the wate~ flows from left to right, ;unless this arrangement would bit to show the feature to b~ illustrated by the cross'section and profile. This practice is generally fqllowed bt most 'engineering offices. However, for geographical reasons or in order to depict clearly the location and profile of a stream, the profile may be shown with water ftow~ ing from right to left and the cross section ma.y be shown looking upstream. This happens in ma.ny drawings pre'pared by the TennesseEj Valley Authority, because the Tennessee River and most of its tributaries flow from:east tQ west, and so are shown with the direction of flow from right to left on a, conventional map.

1

f

)

j )

.1

,.

! 36

BASIC PRINCIPLES

OPEN CHANNELS. AND THEIR PRO'PERT1ES

Douma. formula,' that is, 'U -

10

~0.2V: . gR

- 1

(2-15)

where u is the percentage of entrained .air by voiume, V is the velocity of flow, and

R Is the hydraulic radIus. . 2-12. Compute the wall pressure on the section OA (Fig. 2-10) of the spillway described in Prob. 2-11. that at section DB.

It is assumed that the depth of tlow section is the same!l.S

/

\«

$PilIW1!Y

Iraining wall,

eo II

cpO!1

;:;

C .2 ::l

.)

:OJ

t;j

i. 2

~ I

/, i

4

,a Un;l pressare, II 01 woler

\I

FIG, 2-10. Side-wall pressures on the flip bucket of a spillwa.y. 2-13. Compute the wall pressure on the section OA (Fig. 2-10) of the spillway descrIbed in Prob. 2-11 if the bucket is submerged with a tailwater level at EL 75.0. It is !l.SSulned that the pressure resultbg from the centritugal force or the submerged jet need not be considered beca.use the submergence will reault in a severe reduction in velocity.

REFERENCES 1. S. F. Averillnov: 0 gidravlicheskom raschete rusel krivolineinoI formy poperech,

2.

3. .

4.

nogo secheniia (Hydraulic design of channels with curvilinear form oithe crosS section), lzvestiia Akademii Nauk S.S.S.R., Otdelenie ~'ekhnic"eskfk;h Nauk" Moscow, no. 1, pp. 54-58, 1956. Leonard Metcalf and H. P. Eddy: "American Sewerage Pra.ctice," McGraw-Hm Book Company, 1M., New York, 3d ed., ,1935, vo!. 1. Harold E. Babbitt: "Sewerage and Sewage Treatment," John Wiley &: Sons, Inc., New York, 7th ed., 1952, pp. 60-:.66. H. M. Gibb: Curves for solving the hydrostatic oatenary, Engineering News, vol. 73, no .. 14, pp. 668-670, Apr, 8, 1915.

I This iormull!. [26J is based on da.ta obtained from actual· conorete and wooden chutes, involving errOnl of ±10%. '

37

5. George Higgins: "Water Channels," Crosby, Lockwood &: Son Ltd., London, 1927, pp.15-36. . . 6. Ahmed Shukry: Flow around bends in an open flume, Transactions, AmericilTl Society of Civil Engineers, vol. 115, pp. 751-779, 1950.· ' 7. A. II. Gibson: "Hydraulics and Its Applications,'" Constable &: Co., Ltd., London, 4th ed., 1934, p .. 332. , 8.J. R. Freeman: "Hydr·a.ulic Laboratory Practice," Amedcan Society of Mecha.nical , Engineers, New York, 1929, p. 70: ' 9. Don M. Corbett and ot.hers:8trealn-ga.ging procedure, U.S. Geologicnl SlI1vey, Water Supply Paper 888, 1943. 10. N. C. Grover and A. W. Harri'ngtoo.: "S.ream FlOW," John Wiley &; 80·ns, Inc.) New York, Hl43. 11. Standards for methods and records of· hydrologi~ measurements, United Natio7ls Economic Comm.isslcn for Asia: and the Fa:r Ei.I$~, Flood Control Series, No.6, Ba.ngkok, 1954, pp. 26-30. , 12. G. CorioUs: Sur.l'etablissemellt de Ill. formule qui donne la figure des remons, et .sIU· 12. ilorrection tiu'on doH y int,roduire POllr tenir compte des diffel'ences de vitesse dans les diVers points d'une marne section d'un COUl'ant (On the ba.ckwater-curve equation a.tid the corrections to be introduced to !lccount for the difference of the velocitie$ at different points on the same cross section), Ivnmoire No. 268, ..,l,n'nalca du punts et chaw;sees, vol. 11, ser. 1, pp. 314-335, 1836. 13. J. Boussinesq: Esg's'i sur la theorie des eaux courantes (On the theory of flowing waters), M~moire& ]fr/;sentes par diven savants ri l'Academie des Sciences, Paris, 1877. . . 14. Erik G. W. Lindquist: Discussion un Precise. weir measurements, by Ernesf W. Schader andT(ennethB. Turner, 1"7'(tllaac:l.ions, American Society of Civil Engineers, vol. 93, pp. 1163-1176, 1929. 15. N. M. Shcha.pov: H Gidrometriia Gidrotelchnicheskikh SoorllllheniI i Gicir,omashin" (" Hydrometry of Hydrv.lllic Structures and MacJ:Jnery ") I Gosenel'goizciat, . . Moscow, 1957, p. 88. 16. Stcponas Kolupaila: Methods of determin!l.tion of the kinetic energy facto!', The Port Engineer; Calcutta, India., vol. 5, no. I, pp. 12-18, Januo.ry, 1956. 17. M. P. O'Brien and G: H. Hickox: "Applied Fluid Mechallics," McGraw-Hill Book Company, Inc., New York, 1st ed., 1937, p'.272. ' 18, Horace William·King; i'Handbook of Hydraulics," 4th ed., l'evised by Ernest F. Brater, McGraw-Hill Book Company, Inc., New York, 1954, p. '7-12. 19. Morrough P. O'Brien and Joe W. Johnson: Velocity-head correction for hydrau1ia flow, Engineering News-Record, vol. 113, 0.0.7, pp. 214-216, Aug. 16, 1934. . 20. Th. P..ehbQck': Die Bestimmung der I,age der Energielinie bei ftiessenden Gewfulsern mit HilIe des GeschwindigkeitshOhen-Ausgleichwertes (The determina.tion of the position of the energy line in flowing water with the o.id of velocity-head a.djustment), Der Bau.ingenieuT, Berlin, vol.. 3, no. 15, pp. 453-455,·· Aug. 15, 11122. 21. Boris A. Bak&meteff: CorioIis and the energy principle in hydraulics, in "Theodore von !Urman Anniversary Volume," California. Institute of Teohnology, Pasadena, 1941, pp. 59-65. 22. W. S. Eisenlohr: Coefficient's for velocity distribution in open-Channel flow, Tra.nsac:I.ior/.$, American. Socie4/ of Civil Enginee7's, voL 11:0, pp. 633-644, 1945. Discussions, pp. 645-668. 23. J. B. Bela.nger: "Essai sur la solution numeriqne de ql,lelques problemes relatifs au mou.-ement permanent des eaux courantes" ("Essa.y on tIle Numerica.l Solution of Some Problems Relative to Steady Flow of Wa.ter"), Carilian-Goeury, Paris, 1828, pp. 10-24.

\

,

38

BASIC PRINCIPI..ES

24. It Ehrenberger: Versuche Iiber die Verteilung der Drucke an Wehrriicken infolge des I1bsturzcnden '.Vassers (Experiments on the distribution 'of pressuresa\ong the f~~e of w(d ..;; resulting from the impact of the fa.lling water), Die W IMJserwirtschaft, Vienna, vol. 22, no. 5, pp. 65-72, 1929. 25. 'H&rald Lauffer: Druck, Energie und Fliesszustand in Gerinnen mit grossem Gefiille (Pressure, energy, and flow type in channels with high gradients), Wasserkrafl, und Wasserwirtschaft, Munich, vol. 30, no. 7, pp. 78--82, 1935. 26. J. H. Douma: Discussion on Open channel flow at high velocities, by L. Standish Hall, in Entra.inment of atr in flowing water: a symposium,T1'ansactions, American Society of Civil Engineers, vol. 108, pp. 1462-1473, 1943.

LeI!.'foI.l-

.2.

.

Pit' 67/\1c: ,Il.1 ,f;.-N

h}jc/rct,;.d.i :/'t'--10 f? .

d

,

.ftu;o 4,4ff luitf&

Yl

tU. ,lillJ-' foe-t ~

.

3

1

ENERGY AND MOMENTUM PRINCIPLES

eN

c: c ,'" Y . 5 . W/ibn

CHAPTER

(,

3-1. Energy in Open-channel Flow. It is known in elementary hydraulics that the total energy in foot-pounds per ponild of water in any streamline passing through 9, channel section may be expressed as the total head in feet of water, which is equal to the sum of the elevation

<

7 tUi, rv:5 (id..

t

Y-'.c,tt/).Je1'

@ ,

..k.": In a CCt·n~l.pdjW 4t- /~1{."fp"U.1/f'A(iCL

_ P ,~,> J: .. ::>..;t;.1".!/f"''' V"'r1.tl.-7

I

HI]~II tt'l"1

.

c. "e J' .,t:;

'I

VW' _..:!.~,~" .it.-.:.~(j

~'i._ .:di.l:£nnd~~L4;'1~ -t&.4i-:5 p,ttt'a . e>J rret ,/·1/0 fall . ..vV ,he. ellA-of /:J~"m ;$ dtre,·r/6..,;tLul ·'n'YII!'. .s~t:rt, Vr';,t.. cd cbf #.. is /l. J'Vi4 v..u ¢ ,pver.( e @'we <5 (I!

~y.& o,C;...l."< ~

"'"'t

iJ d /,1 (f?)e.

'!, 6 rtf..~'( • .§

/u..-",,' fYr..f t! Ci.~"Vl/ r1 f~J .

/

/J

/

'.

IS

r1.!1 /!.~Ci//. bkrl~)~""'p1)

t·'1' .

'7....

('/.

d

Saelioil 0

~

I

I

I

"

.) • (

ftvtAP. f

t'k,~e l',., ctylv) '

Jt#rry.: (IU9A .~) . ~it..d-(c4blh ( .bq:...... e It:;rY:J

< .5q,-'Ui4'V"t

d.c;ot/..,

I

1.__ ~:lum _1_1. 1.~_ __ •

FrG. 3-1. Energy in grad'.lo1ly varied 'open-channel Row.

above a datum, the pressure head, and the velocity head. For example, with respect to the datum plane, the totltl head H at a section 0 conta.ining point A on a streamline of flow in a channel of large slope (Fig. 3-1) may be written H

!/YiM..ltU... (l!':71tdVt) ~-;P. T>/~~cf

I

CttYD.iJ,.u.:; r;~yo;o (}v'Iv: vI<

1

= ZA +

d A, cos

8+ V2;

2

(3-1)

where ZA is the elevation of pain t A above the datum plane, dA is the depth of point A below the, water surface measured along the channel section, 8 is the slope angle of the channel bottom, and V.{2j2g is the velocity hea~ of the flow in the streamline passing through A. In general, every streamline passing through a channel section will have . 39

_. r"',

• i

,

40

BASIC PRINCIPLES

ENERGY AND MOMENTUM PRINCIPLES

~ different velocity head, owing to· the nonuniform vel~ity distribution ~ctual fl~w, .Onl~ in un ideal pal'a:lel fl~w of uniform velocity distributl.on can .he velocIty head be truly IdentlCal for aU points on the cross sectlOn. In the ~ase of gradually varied flow, however,· it may be assumed, for pr~ctlCal purposes, that the velocity heads for all points on the ch~nnel sectIOn are equa.l , and the el~ergy coefficient may be used ·to correct for the over-all effect of the nonuniform velocity distribution. Thus, the total energy at the channel section is

-3-2. Specific Energy, Specific energyl in a channel section is defined as the energJi per pound of water at any section of· a channel measured with respect to the channel bottom. Thus, according to Eq. (3:-2) with Z = 0, the specific energy becomes .

U1

H

=

z + rl cos () .

For channels of small slope, 0 = O. nel section is

S. = d cos or, for 8: channel of small slope and a

2

.'

+ c ;P2g .-

(3-3)

Consid~r !lOW a prismatic channel of large slope (Fig. 3-1). The line r~pl'eSelltlltgJhL~_~~va~ion of the tota.Lhea..cLO-f.Jlow is the lill.&gy_line. ;rhe slope of the hne IS known as the energy g'radient, denoted by Sf.

The slope oLthe water surface is de;noted by Sw and the slope of the channel bottom l by So = sin () . In uniform £10'" - . () . ", S J-- S w-- S o-Sln. According to the principle of conservation· of energy, the total energy head at the upstream section 1 should be eql.lal to the· total energy head at t~e downstream seetion 2 plus the loss of energy hf between the two sectIons; or ...

)

I,

Zl

+d

l

V;2

+ al ~2·g .

cos /1

.

=

Z2

V"

+ d z COs /1 + az ---.l: 2g

.L

,

. h

'f

(3-4)

This equation applies to parallel or gradualJy varied flow. For a channel of small slope, it becomes .

.

ZI

V; + YI .+ al -2 = g

Z2

. + Y2 + a2 -V~ +h 2g .f

Either of these tW? equations is known as the energy equation. = .a2 = 1 and hr = 0, Eq. (3-5) becomes

(3-5)

Wh~n

al

. +

V I2

. . Yl+ -2 g

Zl

.·.

=

Z2

+y~

V 22

+ -2g

const

(3-())

This is the well-known BernoHl,li energy equation.2 1 The slo~e is generally defined as tan O. For the present purpose, however it is defined as sm O. ' 2 .. It ~s .believed th~t this equil-tioni is ascribed to the Swiss mathematician Daniel Berno\llh only lllf~l'ence, to gjve recognition to his pioneer achievement in hy.dro1yn~mICs, m ,partlcular the in~roductio'n of the concept of "~ead." Actuall}', tlu~ e~uatlOn was first formulated by; Leonhard Euler a.nd later popularized by Julius WelSbl\ch [1 J. .. . .

, \ 'JI

?y

+

V2

a 2g

,.

)

(3-7)

1,

=

E =y+-

(3-2)

Thus, the tots'! energy at the chan..

H = z +d

8

V!

V + a --.:. 2g

41

(3-8)

. 2g

which illdic!l.tes that the specific energy is equal to the sum of the depth of water and the velocity head. For simplicity, the following discussion' will be based on Eq. (3-8) for a channel of small slope. Since V = Q/ A, Eq. (3-8) may be written E = y + Q2/2gAz. It can.be seen that, for a given channel section and discharge Q, the specific energy in a channel section is a function of the depth of flow only. When the depth of flow is plotted against the specific energy for a given channel section and discharge, a specific-energy CUTve (Fig. 3-2) is obtained. This curve has ,.two limbs AGand BG. 'rh~ .lifl.lb AC.~J)proache;3the ,horiz.ontall\-xis asymptoticallx toward the righ.t. The limb Be approaches the line OD as. it extends l.lpward and to the right. Line OD is a line that passes through the origin and !ul:,s _ananglfi Qf lr~eJlP.ati.QJUiQl!j11 to '!5~!- For a channel of large slope, the angle of inclination of the line OD will be different from 45°. (Why?) At any point P on this curve, the ordinate represehts the depth, and the: ,abscissa represents the specific energy, which is equal to the snm:6f the pressure head y a.nd the'velocity head V 2 /2g. . The curve shows that, for a given specific energy, there are tW9 possible depths,. for instance, the low stage YI and the high stage y~. The low stage is· called the alternate depth· of the high stage, and vice vel'sa. At point G; the specific energy is a n~inimum. It will be proved Inter that this condition of minimllriJ. specific energy corresponds to the critical state of fio.v. Thus, at the critical state the two alternate depths apparently become one, which is known as the critical depth 1/e. When the depth of ;fiowis greater than the critical depth, the velocity of flow is less than the critical velocity for the given! discharge, and, hence, th:e fiow js sub criticaL When the depth of flow is less than the critical depth, the flow is supercritical. Hence, YI is the. depth of a supercritical flow, and 1/2 is the depth .of a subcritical flow. ' ' If the discrarge changes, the specific energy will be changed· aucordcurves A' B' and A" (Fig. 3-2) represent positions of ingly. The

1wo

1

B."

The concept of specific energy was first ir~troduced by Bakhrneteff [21

id 1912.

42

BASIC PRINCIPLES

,the specific-energy curve when the discharge is less 'and greater, respectively, than the discharge used for the construction of the curve AB. 3-3. Criterion for a Critical State of Flow. The critical state of flow ' has been defined (Art. 1-3) as the condition for which the Froude number is equal to uni~y. A more common definition is that it is the state of flow at which the specific energy is a minimum for a given discharge. 1 A

ro----

At the critical state of ftowthe specific energy is a dE/dy =0. The above equation, therefore, gives

VI D 2g - 2

min~mum,

N\':;" ,~

4

or

(3-11) j Supercritica I

For a channel of large slope angle 8 and energy coefficient for critical flow can easily be proved to be

)

ct

Vl 2g

=

D cos 2

e

ct,

the criterion

(3-12)

where D is the hydraulic depth of the wateI: area normal to the channel bottom. In this case, the Froude number may be defined as ,Fw. 3-2. Specific-energy curve.

(3-13)

theoretical criterion for critical flow may be developed from this definition as follows: Since V = Q/ A, Eq. (3-8), the equation for specific energy in a channel of small slope with a = 1J may be written Q2 217A -

E=y+--,

It should be noted that the coefficient a of a channel section actually ·varies with depth. In the above derivation, however, the coefficient is assumed to be constant; therefore, the resulting equation is il0t absolutely exact. 3-4. Interpretation of Local Phenomena. Change of the state of flow from subcritical to supercritical vice versa occurs frequently in open channels. Such change is manifested in a corresponding ch&nge in the depth of flow from a high stage to a low stage or vice versa. If the change takes place rapidly o'/er a relati velyshort distance, the flow is rapidly and varied and is known as a local phenomenon. ' The hydraulic drop , . hydraulic jump are. the two types M local phenomena, and, may be . described as follows: Hydraulic Drop. A rapid change in the depth of flow from a high stage to a low stage will result in a steep depression in the water surface. Such a phenomenon is generally caused by an abrupt change .in the channel slope or cross section and is known as a hydraulic drop (fig. 1-2). At the transitory region of the hydraulic drop a reverse curve usually

(3-9)

or

Diffe~entiatiilg with respect to y and noting that Q is a constant,

.

dE = 1 _ ~ dA = I _ VI dA dy gA3 dy gAdy . "

The differential water area dll near the free surface (Fig. 3-2) is equal to T dy. Now dA/dy = T, and the hydraulic depth D:= A/T; so the above equation becomes ' dE V2T 'V2 -=1--=1-dy 'gA , gD I Th~ concept of critica.l depth bas'ed on the theorem of minimurh energy introdl.).ced by BOss [3J. '

Wall

~

first

, ,<

,

(j

(3-10)

This is the criterion for critical flow, which states that at the critical state of flow, the velocity head is equal to half the hydraulic depth. The above eq~ation may also be written V!...;gJ5 == 1,which means F = 1; this is the definition of critical flow given previously (Art. 1-3). If the above criterion is to .beused in any problem, the following conditions must be satisfied: (1) flow parallel or gradually val:ied, (2) channel 01 small stope, alld (3) energy coefficient assumed to be unity, If the energy coefficient is not assumed to be unity, the criti~al-flow criterion is

y -----I

~~1

43·

ENERGY AND MOMENTUM PRINCIPLES

{

() I, 1

"

i.,'

I'

j

,

,I

I

44

\I, \

BASIC PRINCIPLES

ENERGY AND

appears, connecting the water surfaces before and after the 'drop.' The point of inflection on the reverse curve marks the approximate position of the cdtical depth nt which the specific energy is a minimum and the flow passes from a "SUbcritical state to a supercritical state. The free ove'rfall (Fig. 3-3) is a special case of the hydraulic drop. It occurs 'where the bottom of a fiat' channel is discontinued. As the free overfall entel'S the air in the form of a nappo, there will be no reverse curve in the water surface until it strikes some object at a lower eleva.tion. It. is the law of nature that, if no energy were added from the outside, the y

v'{ijl,/::~LJ)

'7

f"'~'''1i

MOME~TUM

PRINCIPLES -

45

.~~h~~ the brin~.

T~actual situation i~ that the brink section is thg true section of minimum energy, but it is not the critical section as comjnite(fEyt'te prineiple based on the parallel-flow assumpti~ Rouse [~l ... found that for small slopes the computed critical depth is about 1.4 times the brink depth, or Yo = 1.4yo, and that it is located about 3y< to 4yc "behind the brink in the channel. The actual water surface of the overfall is shown by the full line (Fig. 3-3). ,..-It should be noted that, if the change in the depth of flow from Do high stage to a low stage is gradual, the flow becomes a gradually varied flow

-,

o

-::!~

_ Ii

""c.
Y

Y

o "

" "" o.tl: /Theore.licOi water surfoce

---"""",-""-,,,,- ==:::t' ~s~~nLP~~e.!...! I~ _ _ _ -'Actual WQter surface - -- - -'\. _ _ _ _ _ _ _ _ _ ~

E

Ir.ilial deplh

Specific-ene'g)'

FIG. 3-4. Hydro.ulic

ju~np

CUrye

Hydraulic jump

Specific-·force

Culye

interpreted by.specific-energy and specific-force curves.

~

FIG. 3-3. Free overfa.ll interpreted by specific-energy c~rve.

\

'

1

water sl1T~!:\cewould seek its lowest possible position corresponding to the least possIble content of energy dissipation. If the specific energy at an upstream section is E, as shown On the specific-energy curve, it will COl1ti~u.e to be dissipated on the way downstream and 'Will finally reach a mllllmum energy content E",,,.. The specific-energy curve shows that the section of minimum energy or the critical section should occur at the brink. r~~~~tie less_tha!Lth.~,J~titical depth _because !Er~he:~ecl'eas.e," In deJ2th woul£.!eguire a.TI, increo.se in sgecific energy, ~.pos~ll>l~o.!ll.p~nsating external en~,ls sup~lie
having a prolonged reversed curve of water surface; t.his.phcnomenoll may ~Clllled ~ grad'unl hud7'a1t~ic drop and is no longer a local phenomenon~ Hydraulic Jump. When the rapid change in the depth of fiow is from a low stage to a. high stage, the result is usually an abl'llPt rise of water ~e. (Fig;. 3-4, in which the vertjc!l,l scale is exaggerated). This local' phenomenon is known a~J.he hydraulic j1tmp. It occurs frequently in a canal below a regulating sluice, at, the foot of a spillway, or at the place where a steep chann!)l slope suddenly turns jlat.' If thejump is low, that is, if the change in depth is small, the water will not rise obviOtisly and abruptly but will pass from the low to the high stage through a series of undulations gradually diminishing in size. Such a low jump is called an uooular jtimp. . When the jump is high, that is, when, the change in depth is great, the jump is called a direct jump. The direct jump involves a relatively large amount of energy loss thro,ugh dissipation in the turbulent body of water in the jump. Consequently, the energy content in the flow after the jump is appreciably less than that before the jump. . It may be noted that the depth before the jump is always less than the

46

BASIC PRINCIPLES

depth after the jump. The depth before the jump is called the initial .depth y 1 and that after the jump is called .the .sequent depOt Y2; The initial and sequent depths VI and Y2 are shown on the specific-energy curve (Fig. 3-4). They· should he distinguished from the altern!J.te. A~p~hs YI and Y2'! whi(J~~e the two nossible depth~ fo!:....the same specific ener~l' _.~~itial and s~lent depths are actual depths before and after a jump in which ~rgy lOss b.E is invoh-ed. In other words, the specific -energy E 1 at the initial depth Vl is greater tha.n the specific energy at the sequent depth .y~ by an amount equal to the energy loss AE. II there were 110 energy losses, the initial and sequent depths would become identical with the alternate depths in a prismatic channel. 3-5. Energy in NQnprismatic Channels. In preceding discussions the channel has been assumed prismatic so t.hat one specific-el~ergy curve could be applied to evil sections of the channel. For non prismatic channels, however, the channel section varies along the length of the channel and. hence, the specific-energy curve differs from section to section. This cbm'plication can be seen in a three-dimensional plot of the energy curves along the given reach of a nonpl'ismatic channel. For demonstrative purposes, a nOllprismatic channel with variable slope is taken as an example, in which agradually varied flow is carried from a, stibcritical state to a supercritical state: (Fig. 3-5) .. The vertical profile of the channel along its center line is plotted on the Hx plane with the x axis chosen as the datum. For a variable-.slope channel, it is more convenient to plot the total energy head H = z + y + V Z/2g, instead of the specific energy, against the depth of flow on the By plane. For simplicity, the pressure correction due to the slope a.ngle and curvature of flow is ignored in this discussion. An energy line is then plotted on the Hx plane below a line parallel to the x axis and passing through the initial total head at the H axis. The exact position of the energy line depends on the energy losses along the channel. Four channel sections are then selected and four energy curves for these sections are plotted in the Hy planes ~ shown. The initial section 0 is an upstream section in the sUbcritical-flow region. The two depths corresponding to a given total energy H 0 can be obtained from the energy curve. Shice this section is in the subCl'itical-flow region, the high stage yo should be the actual depth of flow, whereas the low stage is the alternate depth. Similarly, the alternate depths in other sections can be obta,ined. In the downstream sections rand 2, the low stages Yl and Y2 are the actual depths of flow singe they are in the supercritical-flow region. The critical depth at each se ytion can also be obtained from the energy curve at the point of minimum energy. At &ection C the critical flow occurs, and the depth y, is the critical depth. On the H x plane, varioUs lines can finally be' plotted; showing the channel bottom, water surface, critical-depth line, and

ENERGY AND MOMENTUM PRINCIPLES

47

alternate-depth line. At the critical section, it is noted that the three lines, namely, the .yater surface, the critical-depth line, and the alternate-depth line, intersect at a single point. It is seen' tho.t, in passing through the critical Reotion, t,he water surface entei'S the supercriticalflow region smoothly. The· three-dimensional plot of energy curves is complicated. The description given here is used only for helping the reader to visualize the problem. In actual applications, the energy C'lrves may be constructed

1 \

II

I I i I !

I

I

! II

X

FIG'. 3-5. Energy in a non prismatic channel of variable-slope, carrying gradually va.ried flow from sllbcritical to supen::ritical state.

separately on a number of tWo-dimensional Hy planes for the chosen sections. The data obtained from these curves are then used to plot the water surface, critical-depth line, and alternate-depth line on a twodimensional Ex plane. For simple channels, the energy curves are not necessary because thi;l critical depth and alternate depths C3.n easily be computed directly, Eumple3-L ·A rectangular cha.nnell0 ft wide is narrowed down to 8 ft by a contrMtion 50 ft long, built of straight wa,lls and a horizontal fiooc. If the discharge is . 100 cfB and the depth of Bow is 5 ft On the upstream side of the transition section, determine the flow-surface profile in the contra.ction (0.) allowing no gradual hydraulic drop in the contraction, Ilnd(b) a.llowing a gradual ,hydraulic drop ha.ving its point of inflection a.t the mid-sectioll of the contra.ction. Th!Jrip.tionalloss through the contraction is negligible. .

,

.

{ \

:l

48

BASIC PRINCIPLES

ENERGY AND MOMENTUM PRINCIPLES

. , . Sol1ttiDn. From the given data; the total'energy in the approaching flow meas~red above the channel bottom i~ u: = 5 {100/(5 ?< 1O)1'/2g = 5.062 ft. This energy is kept constant throughout l,lIC contractIon, since energy losses are negligible. A horizontal energy line showing the elevation'of the tobl head is, therefore, drawn on the channel· profile (Fig. 3-6). . "

49

., critical depth at t}Jis section is equal to the total head divided by 1.5 (Prob. 3-3), or 5.062/1.5 = 3.375 ft. By :r::q, (3-10), the critical velocity ts'eqllnl to V, = V3.375g = 10.4.5 fps. Hence, the width of this' critical section should be 100/(10.45 X 3.38) = 2.83 fi. ' With the size of the mid-section determined, the side walls of the contraction can bo drawn in ,with straight lines. The lm~ and high stages at each section are then computed by the equation previously'given, As the flow upstream from the critical section is subcritical, lts water surface should follow the high stage. Downstream from the critical secti6n, the flow is sllpercritical and its Burface profile 'follows the low-stage line. The criti.::al-depth line is shown to sepai'ate the high from the low stage or the subCl'itical from the stipercritical region of flow. On the basis of Eq. (3-10), the critica.l depth can be computed from the equation

+

(lOO/by,),

2g

y,

=

'2

. y = {/lO,OOO , "gb%

or

where' b is the width of the channel, which can be measured from the plan, It should be noted :that the vertical scale of the channel profile is greatly exaggerated. Furthermore, the outline of the gradual hydraulic drop is only theoretical, based on the theory of parallel flow. In reality, the flow near the drop is more or less curvilinear, and the .actual profile would deviate from th/'! theoretical one.. . This example also serves to demonstrate 11 method of designing a channel transition (Arts. ll-5 to 11-7), The designer may fit any type of contraction walls he desires to suit a given flow profile, or vice versa, .

3-6. Momentum in Open-channel Flow. As stated earlier (Art. 2-7), the momentum of. thefiow passing a channel section per 'unit time is expressed by pwQV /y, where p is the momentum coefficient, w is the l.mit and V is the mean weight of water in lb/ft i , Q is the discharge in velocity in fps, ,-----?'?>- According to Newton's second law of motion, the change of momentum per unit of time in the body of water in a flowing channel is equal to the . resultant of all the external forces that are acting on the body,Applying this principletO channel of large slope (Fig. 3-7), the fplfo~~ing expl:ession. for the momentum change per unit tirnein;the body of water enclosed betv.:een sections 1 and 2 may be written:

FIG. 3-6. Energy principle applied to a channel contraction (a) without gr:adual hydraulic drop; (b) with gradual hydraulic .(hop. '

cfs,

The alternate depths for the given tot ..l energy. e..n be computed by Eq. (3-9) as follows: . ' " 100' 5.062 = Y + 2g(by)' or

. , - 5 06'> " y . ~Y

+ 155.25 b'.

=

a

0

This is a cubic equtl.tion in which b is the width of the channel. At the entrance sec-' tion, where b = 10 ft, its s61ution gives two positive roots: a low stage 'YI = 0.589 ft, which is the altemate depth; and ahigh stage Y2 := 5.00 ft, which is the depth of flow. At the exit section ,where 11 = 8 ft, this. equation gives a low ati\ge YI =. 0.750 ftiand a high stage Y. = 4.964 ft. ! When no gr~dual hydraulic drop is allowed in'the contraction (Fig. 3-6a), the1depth of flow at the exit section.should be kept at the high stage, as shown, The high stages for other interm£ldiate sections are then compute~ by the above equation, whicli giveB the flow-surface ptofile. Similarly, the low stages are computed by the aboye procedure and indicated by the alternate-depth line~ ; When a grodua:~ hydrauiic drop is desired in the contraction (Fig. 3-6b), theldepth of flow at the exit ~ection should be at the low stage, Since the point of inAection of the drop or 11. critipal section is maintained at th~ mid-section of the c~ntracti
.

'~l 0(

(3-14)

J(.. 1 •.;.. "'>l·~)(qt

-

cit!

)>1'

_, Yn

I

,I

I !

!,

C(

, . , where Q, w, and 1:::' are :as.. previously defined, with subscl'ipts refe1'l'in'g to ' sectionr-l~nd P! and P 2 are the resultants of pressures acting on the two sections; W is the weight of water enclosed between the sections; and F! is the total external force QL friE,tion and reslstanc~~ing-,ilong the lLUliace of conta,Qt bet\V'een.the water and the cha;nl)el.The above equa-' tion. .is known as the m;omentum equation, l I · ,

2;

1

The application of the inomentum principle was first suggested by Belanger [5J.

,

50

r-

51

ENERGY AND )';IOMENTUJI{ PRINCIPLES

BASIC PRINCIPLES

rectangular channel of small slope and wi~lth b (Fig. 3-7),

For a parallei or gradually varied flow, the values of PI and P 2 in the momentum equation may be computed by assuming a hYdrosta.tic distribution of pressure. For a cur:-vilinea.r or rapidly varied flo"l,' however, the pressure distribution is no longer hydrostatic; hence the ~alues of PI and P z cannot be so computed but must be cOl'rected for the curvature effect of the streamlines of the flow. For simplicity, P 1 and P2 may be replaced, respectively, by {)!'P 1 and f1~'P2' where {:Jt' and {)z' are the correction coefficients at the two sections. The coefficients are referred

PI ;'~Wb1l1:l. P z = 7§.wb yz2 FI = wh/by

. and Assume

L

where h/ is the friction head and ii is the avera.ge depth, or (YI+ Y2)/2. The discharge ~hrcillgh the reach may be taken as the produot of the average velocity and the avera.ge area, or

I

Q

I

~ ~i(Vl

+ V~)bii

Also, it is evident (Fig. 3-7) that eha weight of the body of water is W = wbfjL

and

sin

{j

=

Substituting :111 the abo,'e expressions for the corresponding items in Eq. (3-14) and simplifying, .

.\

(3-16)1

I Co t1r I

1.

Fro. 3-7. Application of the momentum.principle.

to as pressure-dislribution coefficients, Sjnce and P2 are forces, the. coeffi(Jients may be specificall~r (JaIled force coefficients. It can be shown thu,t the force coefficient is expressed by {:J' =

1-:: fA AZ)o

hdA= 1

+~ fA AzJ~

cdA

(3-15)

where z is the depth of the centroid of the wa.ter area A below the free surface, h is the pressure head on the elementary area dA, andc is the pressure-head correction [Eq. (2-9)1, It can easily be seen th'1t pI is !.Treater than 1.0 for ooncave flow, less than 1.0 for and equai =~~ to 1.0 f <2.r...1lllJ:allel flow: It can be shown that the momentum e.quation is similar to the energy equation when ~pplied to certain flow problems. In this case, a gradually varied flow.is considered; accordingly, the pressure distribution in the sections may be assumed hydrostatic, and (1' = 1. Also, the slope of the . channel is ass~ed relatively small. 1 · Thus, in the short reach of a 1 If the slope a.!l~le 8 is large, then PI = }iwdl~ cos 0 and P. = ;fW~1 cos O,where ell and el, are the Beptbs of flow section and cos IJ is a. correction factor (Art. 2-10). , ,a. tI

/::}. A j'(,h .{- c)id JI. = " j.Ad4 . 1 <.l(cd .4 A >(:i:.;. ! c dA

() j:

•

IC!cr-'T(~ 0

'tl.

t'Yltkfr; ~t·-Z,a:k.~ 1.iJ" ~ e"" t Jt-l'!' ~.~~?}

i

i

This equation appes,rs to be practically the same as the energy equation (3-5). The.oretically sp'3aking, however, the two equations not only use different velocity-distribution coefficient,s, although these are nearly equal, but al~o invol;e different meanings of the frictiollltllosses. ,II!. the energy !ill.!ll!-t!.on, the Item hr mea::;yres t.he intl?rna.l energy dissipat.ed in the whole ~.~s of the water iI!tbe reseh, whereas the item hi in the momentum equatirJll measures the losses due to external forces exerted 011 the water by the walls of the channel. Ignoring the small difference between the coefficients a and fJ, it seems that, in gradually varied flow, the internalellergy losses are practically identical with the losses due to external forces. In uniform flow, the rate with which surface forces are doing work is equal to the rate of energy dissipation, In that case, therefore,. a dist,inction between hI and h/ does not exist except in definition. The simila.rit,y between the applications of the energy and momentum principles may be confusing: A clear ullderstandingof the basic differences in their constitution is important, de.'lpite the fact that mmany instances the two principles will produce practically. identical results. The inherent distinction between the two principles lies in the fact that energ'J is a i5calar quantity whereas momentum is a vectC)r quantitT also· the ~nergy equation contains a term for' internal losses, where:s th~ momentum equation contains a term for external resistance. Generally speaking, the energy principle offers a simpler and clea~er I;

r +~ (colA 1/ z

) '"

I

52

~.

BASIC PRINCIPLES

53

ENERGY AND MOMENTUM PRINCIPLES

explanation than does the momentum principle. But the momentum principle has certain advl.I.ntages in application to problems involving high internal-energy changes, such as the problem of the hydraulic jump. If the energy equation is applied to such problems, the unkriown internalenergy loss represented by h, is indeterminate, and· the omission of this term would result .in considerable errors. If instead the momentum equation is applied to these problems, since it deals only with extemal forces, the effects of the internal forces will be entirely out of considerfltion and need not be evaluated. The term for il'ictionallosses due to ex~ernal forces, on the other hand, is unimportant in such problems and can safely be omitted, because the phenomenon takes place in a short reach of the channel mid the efiect due to external forces is negligibl~ compfkreq with the internal losses. . Further discussions on the solution of the hydraulicjump problem by both principles will be given IELter (Example 3-3). An example showing the application ·of the momentum principle to the problem of It broad-crested weir is given below. Example 3-2. Derive the discharge per unlt width ofa broad-crested weir across a rectangular channel.

approach section 1 and the downstream section 2 at themininium depth of the weir, the following equation may be written:

qw g

(!1.Y2 _!J_) YI

= HWYI' -

MWy2 2

-

OIl

the top

Hwh(2y, - h) .

.

where q is the discharge per unit width of th~ weir'. /' Experiments by Doeringsfeld and Barker 181 ·have shown that, on the/average, VI - 11, = 2Y2. III that case the above equation can be simplified ani1 solved for q, q = 0.433 .

V2Y (-4-h ) l~ H%

I

·Y'T

(3-17)

//.

Conslderillg the limit.of h from ,zero to Infinity, this equation variesjfo~'!l'lR.~L tv q = 2.4.6RH. It is interesting to note that the practical range·1lf tfie coefficient to H~" obtained by actnal observations' is from 3.05 to 2.67. In applying the momentum principle tiJ"this problem, .it can be seen that knowledge of the internal-energy losses due to separation of flow at the entrance and to other causes ·is not needed in the ,analysis. .

3-7. Specific Force. In applying the momentum principle to a short horizontal reach of fl. prismatic channel, the extern:al force of friction and the weight effect of water can be ignored. Thus, with (J = 0 and F, = 0 and assuming also {31 = {3z = 1, Eq. (3-1.4) becomes

=

Qw (V2 -- 17 1) g

PI - P2

The hydrostatic forces PI and 'P 2 may be expressed as

PI = wi 1 A. 1

a.nd

P 2 = wi 2 A 2

where i l a~d i2 are the distances of the centroids of the respective wa.ter areas Al and Az below the surface of flow. Also, Y I = Q/ Al a.nd V 2 = Q/ A z• Then, the above momentum equation may be written

ir. + ilAl = il.... + ZZA2 gAl

FIG. 3-8. Momentum principle ~pplied to flow over a broad-cregted weir. Boilltiun. The assumptions to be made in this solution (Fig. 3"8) are (1) the frictional forces Fr' and F," are negligible; (2) the depth y. is the minimum depth on the weir; (3) at the channel sections under consideration there is parallel flow; and (4) ~~~r ,pressure £w'on ~e weir surface is equal to the total hydros~.tic p~.!!.!!re , meD.S\.Ired below the upstreamwatei: surface, or..

p ..

=

Hwh[y,

+- (YI

- 11,)] =

Hwh(2YI - h)

The accuracy (If the last assumption has been checked ~xperimentally [61. If the momentum equation (3-14) is ·applied to the body of water between the upstream J

..

r0 0 1">'1 e'Y°;t~,- [1""< a-"1,..""; 7 (~~4li/M ::

,

. '- 'Y)

c'Af:i

j .. /.::

.r-?

,rgf1lJ,o/,J<.

(){;.:aty/'b .d:./n Ct"7fd-t. 'c.~ t5 If) f3 L

/, .ve-.le, i:-~ ' r · »( , I
g.fi 2

(Qr //.,-c 1-1,) (3":18)

'The v:llue of the coefficient actually depends on many factors: mainly, the rounding of the upstream corner, the length and slope of the weir crest, and the height of the weir. Many experiments on bl'oad~crested weirs haye been performed. From several of the wen-known experiments King [7J has interpolated the data and prepared tables for the coefficient uncle·r various conditions. A comprehensive analysis including more recent data and a presentation of the results for practical applications were made by Tracy [8J. The well"known experiments all broad-crested weirs are (1) Bl].zin tests performed in Dijon, France, in 188.6 [9]; (2) U.B.D.lV.Bi Cornell tests performed at Cornell University in 1899 by the U.S. Deep Waterways Board under the direction of G. W. Rafter, and U.S.G.B. Corncllle.sts performed·by the U.S. Geological Survey under the direction of Robert E. Horton in 1903 [101; (3) Michigan tests performed at the University of Michigan during H)28-1929 [11]; and (4) MinneBola and· Washington tests performed,'respectively, at the university of Minnesota and Washington State University [6]. For some formulas and coefficients of discharge developed in the U.S.S.R., see [12]. For an analytical treatment of the problem, lIee· '[13].

55

BASIC PRINCIPLES

ENERGY AND MOMENTUM PRINCIPLES

The two sides of Eq. (3-18) are analogous and, hence, may be expressed for ahy channel section by a general function " Qz'

For a minimum yalue of the specific force, the first derivative of F with respect to iJ should bl') zero, or, from Eq. (3-19), '

54

.

F

- i t zA

dF

"(3-19)

u4'_

' This function cOMists of two terms. The first term is the momentum of the flow passing through the channel section per unit time per unit weight of water, and the second is !:.l!.e for.J<.ELrillr....unit weigb.:Lm w~t.!lr. Since , both terms al~e essentially force per unit weight of water, their sum may , , be ,c~lled the 8PlJcifjE.l(jr.c~.1 ,Accordingly, (3-18) may be expressfld

Q2 dA ,d(zA) dy T --elY

gP

'

,

For a change dy in the depth, the correspo,nding change 'd(iA) in the static'moment the water area. about the free surface is equal to [A (2 + dy) + T(dyP/2] - zA, (Fig. 3-9). Ignoring the differential of higher degree, that is, assuming (dy) 2 = 0, the change in static moment becomes d(iA) = A dy. Then the preceding equation may be written

of

dF

_ Q! dA

gA2 dy

B

Since dA/dy reduced to

c· o

~=-:-_. 45° for ()

0

= T,

Q/ A

V,and AfT

+A =

i"

!i

Ez '£lEiE:, channel of ,......, zero or small slope (0]

l.bl

3-9. Specific-fofce curve supplemented \~ith specific-energy curve. energy curve; (b) channel section; (e) specific-force curve.

(a) Specific-

'

as F 1 = F 2. . This means that the specific forces of section,s 1 and 2 are equal, p.r:ovided t~t the e~rnal forces and the weight effect of water in the..rea<;h between the two s~ctiQns can be igno];ed. By plottillg the depth against the spec~fic force for a given. channel section and discharge, a ~pecific-jorce CU'fve is obtained (Fig. 3-9). This curve has two limbs AC and BC. The limb AC approaches the horizontal axis asymptotically toward the right, The limb BC rises upward and extends indefinitely to the right. For a given value of the specific force, the curve has two poss(ble depths Yl and Yl' " A13 will be shown later, the two depths constitute the il1it~al and sequent depths of a hyqraulic jump. At point· C on the curve the two depths become one, and the specific force is a minimum. The following argument shows that the depth at the minimum value pf the specific force is eqv,al to the critical depth.~ , 1 This has been variously called the "force plus momentum,'~ the "momentum ftux," the "total force;" or, briefly, the "force" pf a stream (see pp. 81 and 82 of [14J). The {unction represented by Eq. (3:'19) was formula.ted by Bresse [15J for the study of the hYdra.ullc jump to be described in Example 3-3. : i The conoept of critic!!.1 depth based on tbe theorem of momentum :is believed to ha.ve:been developed by Boussinesq [16).

I,

D, the above equation may be (3-10)

i

I I (

,I

= 0

This is the criterion for the criMeal state of How, derived e~rlier (Art. 3-3). Therefore, it is proved that the dep~h at the minimum va-lueaf the specific force is the critical depth.I It may also be stated that at the critical state of flow the specific'jorce is a minim.'um for the given discharge.

NQw, comp£1,re the specific-force curve with the specific-energy curve, (Fig. 3-9). For a gi;iell specific energy Ell the specific-energy curve indicates two possible depths, namely, a low stage YI in the supel'critical flow region and a high stage yz' in the subcritical flow region. 2 , For a given v~lue of F1, the specific-force curve also indicates two poosible depths, namely,.ail initial depth lit in the supercritica.l region nncla sequent depth ljz in the sub critical flow region. It is assumed that the low stage and the initial depth are both equaJ to YI' Thus, the two curves indicate jointly that the sequent depth Y2 is a.lways less than the high stage 1/2'. Furthermore, the specific-energJ' curve shows that the energy content E2 for the depth Vi is less than the energy content El for the depth Y2'. Therefore, in order to maintain a constant value of F1, the depth of flow may be changed from Yl to Y2 at the price 'of losing a certain amouilt of energy, which is equal to El - E, = I:J.E. One example o( this is the 1 It should be noted that the above proof is based on the assumptidrul of para.liel flow and uniform velocity distribution. However, the concept of !lritico.l depth is: a genera.l concept, tl:..a.t is valid for aU flows, whether derived from energy or from momentum considerations. This validity has beeJ;!. proved by Jaeger [14,17,18), and the proof is known llS the J a.eger theorem {I9]. ' ~ In order to make 2. clear distinction between the sequent depth and the high of tile alternate depths, the sequent depth is designated by y. and the ):ligh sta.ge lit. In some other places in thLs book, however, both are designated by 1/1'

'.. ,I "'i

f

56

!'

BASIC PRINCIPLES

hydraulic jump on a horizontal floor, in which the specific forces before and after the jump are equal and the loss of energy is a consequence of the phenomenon. This will be explained further in the following example,' It may be noted at this point, however, that the depths YI and yz' shown by the specific-energy curve are the alternate depths; whereas the ?epths. Yl and Y2 shown by t.he specific-force curve are, respectively, the mitinl depth and the sequent depth of !1 hydraulic jump. ' Example 3-11. Dedve a relationship b~tween the initial depth and the sequent depth of 8: hydraulic jump on a horizontal floor in a rectangular channel. ' , Sob/ion. The el:ternal forces of friction and the wet'lH effect of Wil.Wr in the hydl'~ulic jump o~ a horizontal floor are negligible, because~ the-jump't.;;:kes place in a relatively short distance and the slope angle of the ,horizontal fioor is zero. The' specific forces of sections 1 and 2 (Fig. 3-4), respectively, before and after the jump, can,therefore be considered equal; that is,

i \ I

ENERGY AND MOMENTUM PRINCIPLES

57

section.' In applying the mornentur~l principle to nonprismatic channels, therefore,a three-dimensional plot similar to that shown for the application of the energy principle (Fig. 3-5) can be constructed. For practical purposes, however, this is rarely necessa.ry. Where there is no interyention 'of external forces or where these forces are either negligible or given, the momentum pdriciple can be applied to its best advantage to problems, such as t.he hydr.alllic jump, that deal ;"!I"':';...~;.tJ,,""'!j12!JiJ."'J="_<".bJ'-JlJ""'''''' th.at..!:~l!!!i~~. jUhe energy following example shows how the momentum design of a channel transition in which a hydraulic jump is involve(i. Example 8-4., A rectangular channel Sit wide,.earrying 100 cfs at a depth of 0,5 ft, is connected by a str:oight-wall transition to Il. channel 10 it wide, flowing nt.s. depth.

(3-18)

For ll. recLanguinr channel of width b, Q = V,A, = V.A.., Al = by" A, = bYt, = yi/2, and ii' - y~/2. Substituting these relations IlJld F, = V d v'Uih in the above equation and simplifying,

2,

- (2F,' + 1) (J!J)+ 2F,' 0 (:1!!)' y,. +. ~ - 2F,'] (~ - 1) - 0 [(1Lt)' YI YI YI '

Factoring, Then, let

~V + 1l! _ ( Y.:/ y,

(3-20)

=

~

2F • = 0 I

TIle solutjon of this quadra.tic equation is (3-21) .

For a. J:!.iv:? ,F'roude ?u~ber FI of the apprQaching flow, the xatio of the sequent. depth to the IlUt1l3l depth IS given by the above equa.tion. It should be understood that the momentum principle ill used in this solutiou because the hydraulic jump involves a high amount of internal-energy losses whioh cannot be evaluated in the energy equation., ' .The jOilit. UIlB of the specific-energy curve aud the specific-force curve helps to d~ter ~ne gra.plllcally the energy loss involved in the hydraulic jump for II. given appro8.chmgflow. For the given approaching depth 11., points P, ana. P,' are located on the spe~itic-force curve and the spec~c~energy curve, respectively (;Fig, 3-4). The point Pi'!gives the initial energy content E , . Dralv a vertical line, passing through the P?int PI and intercepting the upper limb of the specific-force curve at point P" which gives the sequent depth 1)2. 'Then, draw a horizontal line passing through the point P a and intercepting the specific-energy curve a.t point P.", which: gives the energy content E, after the j';lmp. The energy loss in the jump is then equal to El - E" represented by / l E . : '

a-8.Momentum

priIicipl~

Applied to

Nonprisrnati~

Channels.

The

sp~cifj.c force, like the specific' energy, varies with the snape ofthechannel

FIG. 3-10. (mnergy and'momentum principles applied to II. channel expansion (a.) with hydraulic jump; (b) without hydraulic jump. . .:

o~ 4 n (Fir;. 3~lO). Determine the flow; profile in the transition if th~ frictional loss through tl;\e tra.n&'ition is negligible. If, a hyqraulic jump occurs in the transition, how C!l.n i~ be eliminated? -:' , S"l'Uti"n~ From the given data, the tptal energy with respect to the channel bot~' tom in the approaching flow is E ~ 0.5 + [IOO/(O.S X 8))'/2g = 10.207 It, and in the dowiistrea.i:n, E - 4,0 + (100/(4 X 10)Ji/2y = 4.097 ft. ,It is appnhnt that this , ,

.

58

'BA.SIC PRINCIPLES

ENERGY A.ND MOMENTUM PRINClPLES

IlMrgy difference 01 6.110 It must be dissipated through the transition by ~ome means, since the frictlontll lru;s is negligible. Furthermore, the Fro.ude numbers 6.24 and 0.22 of the 8.pproaching a,nd downstrea.m flo'YS are, respectively, greater and less than unity, indicll.ting IJ. change of the flow from supercritical to Bubcritica.l. Therefore, a hydra.ulic jump can be expected to occur to dissipate the energy difference and to effect a change in the flow st,ate, Whether this jump will occur within the transition or in the upstream or. the downstream channel is, however, to be disclosed by fur~her analysis. .

for instance, by bolting orOM ,!limbers to the bottom. of the transition. It can be !LSSumed ill this a'(ample that the energy diff-crcnce of 6.1 10 ft is dissipated uniformly in tlte transition by artificial roughness. TllUB, the energy line in the tl'ansition is simply a straight line joining the total heads of the two end sections (Fig. 3-lOb). For. deSign p1l1-poses, it is convenient first to assUlne the lIow profile and then to proportion the dimensions of the tranllition so 'that the jump can be eliminated, In proportioning the transition, the jump is eliminated either by varying the wid~h or by raising the bottom of the transition. In this example. it is assumed that the bottom is to be raised, 01' "humped" (Fig.3-10b). The subsequent procedure of the compuGlttion is to (1) a,!lsume the flow profile; (2) compute the velocity head, which is equal to the difference. between the total heo.d B.nd the water-surface eleva.tion, at a llumbe!' of sel",cted sections; (3) compute the velocity and then the water area and depth of flow foreayh section; (4) determine the elevation of the bottom or the transition, which is equal to the devation of the water surfar.:e minus the depth of flow; (5) compute the o.lternate depth, ~ince the bottom of the transition is fixed; and (6) compute F'l and F, lines fo!' the low and high stages, an.d plot them on II convenient Sl'ale. It can be seen that the two F' lines inter3ect and' become tangent to each other a.t a critical section, where the :flow cha.nges from low to high stage, that is, from 8upercritical to subcritical stale. If the critic31-depthlinc is plotted, it will intersect the alternate-depth line and th~ water surface simultaneously at the critical section. Bl\Scd on the critica.l-depth line, a line of minimum speCific energy can also be construc,ted. This line should be tangent to the total-energy line at the critical flection.

TABLE 3-1.

CObfPUTATION FOR A CHAI'1N1:lL EXF.Ai'/SlON

EXAMPLE

Section width b, ft

3-4

Low stage 'II" ft

PI

0.500

78.6 78.7 78.8 78.8 78.8

for E = 10.207

8.00 8,50 9.00 9.50

0.419

10,00

0.398

o 470 0.4'.1:3

DESCRIBED

IN

., High stage 1/2, ft for E = 4.097

3.940 3.960 ,3.979 3.987 4.000

F, 71.9 75.9 79.9 83.6 87.8

Take 'five sections of the transition with their widths shown in Table 3-1. For the total approaching enel'gy of 1O.20i ft,.the low stage YI for each section call be computed by means of Eq. (3-8) or (3-9), or "':"""-1::--=::'-

where b is the width of the section. 4.097 can be computed from

+ VI,:""

59

PRODLEMS

3-1. With reference to a channel of small slope !l.ud a section shown in Fig. 2-2, (a) construct a family of specific-energy cllrves f(or Q = 0, 50, 100, 200, 300, and 400 ds, (b) draw the locml of the critical-depth point on these curves, (c) plot a curve of the critical depth against the'diilcharge, and (d) plot a family of curves of alternate depths, Yl vs. V~, for the given discharges_ . 8-2. Construct the specific-energy curve for a 36-in. pipe carrying 3:n open-channel flow of 20 cis (a) on It. flat slope, and (bj on a 30 slope. S-B. Show that at the critical state of How tha specific-energy head in a rectangular channeli;; equal to 1.5 times the depth of flow, assuming zero slope and a = 1. 3-4. Derive the equations for the lOCI15 of the critiMl-depth point on the specificenergy curve and for the curve of critical depth vs. discharge, as oota.ined in Prob. 3-L 3-6. Prove (3~12). 3-6. PrC/ve Eq. (3-13). 3-7. Prove that at the critica.l state o( flow the discharge is II ma::dmum for a given specific energy. 1 . 3-8. Show that 'the relation between the altc;rnate depths Y1,and 1/. in !1 rectangular channel ca.n be expressed by .

10.207

Similarly, the high st,age 1/2 for a total ene1'gy of

0

+ V2 =

4.097

Thelow- and high-stage linea are then construeted slong with t·lle energy lines (Fig. 3-lOa). After these stage and energy lines are determined, the specific forces P! and F. for low and high stages, respectively, at ea.ch section are computed and plotte~ to any convenient scale and datum. The hydraulic jump must, occur where the specific 'forces for the low and IUgh stages are equal, or at the intersection of the F lines. At. this
as

(3-22) where Y. is the critical depth. Using values of y.!Yc as ordinates !tud of Yz/Yt as abscissas, .construct a dimensiouleSiil graph for the above equation rand study its characteristics. ' The .concepG of critical depth based. on the theorem of maximum diacharge was first introduced by Belanger [20J. 1

r ~ '1

I \'

.1

\

".!

60

!!ABIC PRINCIPLES ,

ENERGY A.ND MOMENTUM PRINCIPLES

3-9. Solve the problem given in Example 3-1 (a) if there is a. total energy 108s of 0.60 ft unifol'Jnly diStributed throughout the lel!gth of the contraction, and (0) if a gradual hydraulic drop is desired with its point of inflection at !l. diStance 20 it upstream from tlie exit section. 3-10. Applying the momentumpl'inciple and the continuity equation to the analysis of a submerged hydrllulic jump which occurs at the sluice outlet in a rectangular channel (Fig. 3-11), prove that

where y. is thesubme!"ged depth; y, is the height of sluice-gate opening; y, is th~ tailwater depth; und F,' g~/gy!3, IJ being the discharge per unit width of the channel. Neglect the channel-bed friction i,.

FIG. 3-11. A submerged hydraulic jump at sluice outlet.

3-11. Prove tlmt the energy loss in a. horizont,.l hydraulic jump is

AE

()J2

-YI)' ./'

4j/lY2

_(3-24)*

3-12, If !I. hydrAulic jump is forllled on the ,horizontal floor at the toe of the spillway descl'ibed in Prob. 2-11, determine the sequent depUl and the energy loss involved in the jllmp. 3-ill. ,With reference to a Ilha.nnel of small slope and Ll. section shown in Fig. 2-2, (a) construct afamily of specific-force cUI'ves for Q = 0, 50, 100, 200, SOO, and 400 cfs, and (b) plot a family of curves of initial depth against sequent depth for the given discharges. 3~14. Construct the specific-force curve for a 36.-in. pipe c,nrying an open-channel flow of 20 cfs on Ii small sIope. 3-lD. Prove Eq. (3-15). 3-16. Using the momentum principle, show th!Lt the Froude number of a parallel or gradually Vllried flow in a cha.nnel of slope !Lngle IJ may be defined by

F

=

V VgD cos 8/13

61

'

(3-25)

* This formula wliU, shown by Brcsse early in 1850 115J. At the slime time Bresse illtrodu,ced the concept of critical depth, as a. dllpth at which the subcritical flow chl>nges to BUpel'criticai, or vice versa.

where V is the mean veloility, D is the hydraulic depth of the section, and t3 is the 'momentum coefficif;nt for nonuniform velocity distribution. ' 3-1'1. For eliminating the hydraulic jump in Example 3-4, the flow profile is assumed to be composed of two Feversed circular curves tangent to el1ch other at the middle aection of the transition and also to the water surfaces "in the connecting channels at the two ends of the transition. Verify the computation (shown in scale on Fig. 3-100). 3-18. A frictional loss of 1.0 ft is p,Ssumed to be uniformly distributed along the, ' length of the tra.nsition in EXMllple 3-4. Determine the flow pro.file in the tmnsition. REFERENCES

1. 'Huntel' Rouse and' Simon Ince: "History of H:rdraulics," Iowa Institute of Hydraulic Research, Iowa City, IOlVa, 1957. 2. Boris A. Bakhmeteff: "0 Neravnomernoln Dvizltenii Zhidkosti v Otkrytom Rusle" (U V9.r!ed, Flow in Open Channel "), St,. Petersburg, Russia, 1912. 3. PItHI Boss: "Berechnungder Wasserspiei\'ellage beim Wechsel des Fliesszustu,udes" ("C'Almputation of Water Surface with Change of the Flow Type"), SpringerVerlag, Berlin, 1919, pp. 20 and 52. 4. Hunter Rouse: Discharge characteristics of the fre.€; overfall, Civil E1I{}ineering, vol. 6, no. 7, pp. 257-250 .. April, 1936. 5. J. B. Belanger: "Resume de lec;.ons" ("Summary of Lectures"), Paris, 1838. 6. H. A, Doeringsfeld and C, L. Barker: Pressure-momentum theory a.pplied to the broad-created weir, Transactions, American .societv of Civil Engineers, vol. 106, pp. 93~-946, 1911.' .r , 7. Horace William King: "Handbook of Hydraulics," 4.th ed., revised by Emest F. Brater. McGraw-Hili Book Company, Inc" New York, 1954, pp, 6-1 to 6-Ht 8. H. J. Tracy: Discbar~e characteristics of broad-crested weirs, U.S. Geologica! Sm'vcy, Cir~ltlar 397, 1957. 9. H. Bazill: Experienceli nouvelles sur l'ecoulement en deversoir {Recent experiinents on tile flow of water over weirs}, 111"tmO'ires et Dor:u:mlmts, Anna/as des pants III cha1IssJ.S, 2e semestre, pp, 393-448, October, 188B. English trllnslatioll by Arthur Ma.richal and John C. Tl'antwine, Jr., Proc~ed:in.gs, Engingers' Club of Philadelphia, voL 7, no. 5) p.p. 259-310, 1890; Vi.)!. 9, no. 3, pp. 231-24~, and no. 4, pp. 287-319, 1892; and vol. 10, no. 2, pp. 121-164, 1893. 10, R. E. HOI·ton: Weir experiments, coefficients, and formul!J.s, U.S; Geologi.cal Survey, Waler S1I'pply and Irrigation Paper 150, 1906; revisedas Paper 200, 1907. 11. James G. Woodburn: Tests of broad-erested weirs, Trarlsuctian.1, American Society of Civil Engi1lee,s, vol. 96, pp. 387-416, 1932. 12. M. A. Mostk~w: "Handbuch der Hydraulik" ("Handbook of Hydraulics"), VEB Verl£1,g Technik, Berlin, 1956, pp. 188-195. 13. L. J. Tison: Le deversoir l\.seuil epais (The broad-creste'd weir), La' Houille blanche, Grenoble, 5th yr., no. 4, pp. 426-439, July-August, 1950. 14. Charles Jaeger: "Engineering l;'luid ~[echanics/' translated from the German by P. O. Wolf, Blackie & Son, Ltd., London and Glasgow,·pp. 98-112, 1956. 15. 'J. A. Ch. Bresse: "Cou~s de mecanique appliquee," 2e partie, Hydraulique (" Course in Applied Meeha:nics," pt. 2, Hydralllics), M"tlet-BacheUer, Paris, 1860. 16. J. V. Boussinesq: Essai sur Ill. theorie des eaux eonrantes (Essay on thli theory of water flow), Memoiresp1'sslmtes par diver~ savants a l'Academie des Sciences, Paris, vol. 23, ser. 2, no. 1, pp. 1-680, 1877.

62

BASIC PRINCIPLES

17. Cll!irles Ja.eger: Contribution a l'etude des courants liquides a 8urf8.c.oe libre (Contributil1nto the St11dy of Iree-surface liquid flo;,vs), RIWWi genlrale .de L'hydrauliq1l.{l PI'tTis, yol. 0, no. 33, pp. 111-120; no. 34, pp. 139-153, 1943. 18. Charles Jae~e.l·: De l'impulsion toto.le at de ses rapports a,yac I'energie totaled'un cou:u,nt liquide a surface libre (The tota.l impulse and its relations with the toto.l energy of a fl"ee-Ilurface liquid flow), RlWue gtntrale de l'hydravliq1l.e, Paris, voL 13, . no. 37, pp ..12-1\); no. 38, pp. 86-87; no. 39, pp. 143-151; no. 40, pp. 191-197; llO •. 41, pp. 257-2iH, 1947. HI. Etienne Crausse; "lIydr3ulique des canaux deco\lverts en regime permanent" ("Hyciro.ulio5 of Open Ch!lJlneL'! with Steady Flow"), Editions Eyrolles, Paris, 1951, pp. 111-112. . , 20. J. B. Bela.nger: Notes sur Ie cours d ' hydraulique (Notes on the course in hydrl:lulies), llHmoire, gmt" Na.tionale de$ Pont8 e~ ChaU$88I3$, Paris, 1849-1850, pp~ 32-33.

CHApTER

I~-

4

CRITICAL FLOW: ITS COMPUTATION AND AP:PLICATIONS

4-1. Critical Flow. As described in the previous chapter, the llritic!1! stf1.te of flow through a channel section is characterized by severa.l important conditions. 1 Recapitulating, they are (1) the specific energy is a minimum for (l. given discharge; (2) the discharge is a maximum for a given specific energy (Frob. 3-7); (3) the specific force is a minimum for a discharge; (4) the velocity head is equal to half the hydraulic depth in a channel of small slope; (5) the Froude number is equal to unity; and (6) the velocity of flow in a channel of small slope with uniform velocity distribution is equal to the celeri~y of small gravity waves in shallow wat,el' caused by local clisturbances. Discussions 011 critical state of fiow have referred mainly to a particular section of a channel, known as the critical sectiQu. If the critical state of flow exists throughout the entire lellgth of the channel or over 11 reach of. the cha.nnel, the flow in the channel is a criticaL flow. Since, as indicated by the critical-flow criterion Eq. (3-10), the depth of critical floW' depends on the geometric elements A' and D of the channel section when the discharge is constant, the critica.l depth in a prismatic channel of unilorm slope will be the same in all sectio1l3, and cl'itteal flow in a prismatic 'channel should, t.herefore, be uniform flow. At this condition, the slope of. the channel that sustains a given discharge at a uniform and critical depth is called the critical slope S~. A slope oithe c;tiannelless than the critical slope will cause a slower flow of subcritica,l state for the given discharge, as will be shown later, and, hence, is called a mild or subcritical . slope. A slope grel1ter th~\n the critical slope will result in a faster fiow of super critical state, and is called a steep or supercritical slope. A flow at or near the critical state is ullstable. This is becau~e a minor in specific ellergy at. or close to Cl'itical state will cause a major chl111ge in depth. This.fact can o,lso be recognb>ed in the specific-energy As the curve is almost verticalliear the critical depth, . curve (Fig. a slight in energy would change the depth to a much smaller or much greater alternate depth corresponding to the specific energy after l

For a historic~l account of the theory of critical £low, see [1J. 63

.,)

I.

(

'.

64

I , \ I

BASIC .PRINCIPLES

the change. It can be observed· also that, when the flow is near the critical state, the water sUl'face appears unstable and wavy. Such phenomena are generally caused by the minor chnnges in energy due to variations in channel roughness, cross section, slope, 01' deposits of sediment or debris. In the design of a channel, if the depth is found at or neart,he critical depth fOl" a great length of the chinnel, the shape or slope of the channel should be altered,if practicable, in order to secure greater stability. The criterion for a critical state of flow (Ar~. 3-:3) is the basis for the computation of critical flow/which will be explained in subsequent articles. Two major applications of critiClll-fiow theory nre flow control and flow measurement, which will also be discussed in this chapter. 4-2. The Section Factor for Critical-flow Computation. Substituting V Q/ A in Eq. (3-10) and Simplifying,

·11 . II

tJ .

(4-1)

...g.

.
~~,..

. Q

z=v'g

"'1 .., c

--:.~

0,-,

'"!"

-0

";l '-'

:3

... 't :5 ..,... . ..c::..," 'w .5 ... ..,..':! .~... Q

When the energy coefficient is nO& assumed tobe unity,

tj)

. Q

z=-v'g!cY.

('1-2)

"'0 ... "OJ

'0

In the above equations, Z A.v' D; which is the t;ection factor Jor critical-flow compu.tation [Eq. (2--3)]. Equation (4-2) states that the section facto!: Z for a channel section at the critical state of flow ~s equal to the discharge divided by the squaJ.~e root of U/Ol. Since the section factor Z is !l. function of the depth, the equation indicates that there is only one possible critical depth for maintaining the given discharge in channel and similarly that, when the depth is fixed, the~'e can be 'only one discharge that maintains a critical flow and makes the depth critical in the g'iven channel section. Equation (4-1) or (4-2) is ll. very useful tool for the computation and analysis of critical 'flow in 1111 open channel. When the discharge is given, the equation gives the critical section factor Zc and, hence, the critical depth Yo_ On the other hand" when the depth and, hepce, the section factw' are given, the critical! disharge can be compu~ed by Eq. (4-1) 'in the following form:

....o

Vg

fIl

0

-<

.j,

ci ~

(5

g

(4-3)

or by Eq. (4-2) in the following form: .

N;'

.s

OJ

I

Q=z:(2

1>

a '" ~... dt5;;:.. ::>

a.

Q =Z

"...

N

'0

(4-4) ci OpI ~ puo' q/,( lO SGnlOI\

65

66

. . CRITICAL :FLOW: ITS C.OMPUTATION AND. APPLICATIONS

BASIC PRINCIPLES

A subscript 0 is sometimes used to specify the oondition of oritical flow. Formulas fOI' t,he section factor Z of seven common channel sections are given in Table 2-1. The Z values for a circular section can be found either from the curve in Fig. 2-1 or from the table in Appendix A. In Ol·aer to simplify the computation of critical flow, dh:nensionless curves sho.wing the· relatiOll between the depth and section factor Z (Fig .. 4-1) have been prepared for rect<:.ngular, trapezoidal, 'and circular channels. .These self-explanatory; curves ·help to determine the depth y for a given section factor Z, and vIce versa.

This equation indicates that the value of M for the trapezoidal section is a function of z and y/b. For values of z = 0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, and 4.0, a family of curves for 111 versus y/b are constructed (Fig. 4~2). These curves in.dicate.that the value of JI.f varies in a range froni 3.0 to 5.0. S.C , . . - - -.......--.,-----,---7""',--,"'7'1rm-"----,

5.0 4.0 I-----.... ~-'-.-.----- --'-----+.........,f'--I'--7'-HI+-+------j

will

1,0

o.a 0

~ ,.. '0

c 0

......a,..

...

(4-6)

dOn Z)

3 l'

--ay- = 2

J.-

= JL .A

.2

~

=

3[1

+ 2z(y/b)P - :2z(y/b)[l + z(y/b)J . [1 + 2z(y/b>:J[1 + z(yjb)]

·r

I

o,oe O.os 0.05 0.04

vi A/T, and

0.03

(4-8) 0.02 . 25

\

3.0

3.5

4.0 Volues of M

4.5

5.0

5.5

,!

FIG. 4-2. Curve:; of Mvalues.

(4-9)

A curve for a circular section with M plotted against 1Ildo, where d u is t,he diameter, is alsoflhown (Fig. 4-2). This curve .w~ developed by a similar procedure but constructed from a much more complicated formula. The curve shows that ~he value of M varies within a rather narrow range for values of yjd D less than 0.7 or so, but increases rapidly as the value of y/do becomes greater than 0.7. The significance of this

This is a general equation for the hydrauUc exponent M, which is a func~ tion: of the channel' section a,nd the depth of flow. For a trapezoidal section, the exptessions for A and T olJtained from Table 2-1 are SUbstituted in Eq. (4-9); the resulting equation [2] is simplified arid Mcomes M

0.2

0.1

(4-7)

1 dT 2T

(31' -'- .:1-TdydT).

0.3

."

Equating the r~ght sides of Eqs. (4-7) an~ (4-8) and solving for M, M

0.6 0.5 0,4

0

.,.

where C is a coefficient and M is a parameter called the hyd"a-ulic ·expo- . nent Jor critical-flow comp-utation. Taking logarithms on both sides of Eq. (4-6) and then differentiating wi th respect to y, . d(ln Z) IvI Now, taking logarithms on both sides of Eq. (2-3), or Z = A then differentiating with respect to 1/, .

1

2.0

4.,.3 .. The Hydraulic Exponentfo.r Critical-flow Computation. Since the section factor Z is a function of the depth of flow y, it may be assumed that

2y

!

5.0

Example 4-1. Derive arl equatioI! showing critic:u dillcharge through a rectangular cha.nnel section in terms of the cha.nnel width and the total head. Sol~tion. For the rectangular section, Tal?le 2-1 gives the sQcti6n fantor Z = by!.J;. At the critica.l state'of flow, the depth y HI1.5 (see Frob. 3-3). Substituting these expressions in Eq. (4-3), using· g = 32.16, and simplifying, we find that the cri~icar discharge is Q. 3.087bHu . (4-5)

dif" "'"

67

. (4-10)*

I z(1/(b) may be constructed. It is obvious that this curve: be identical with the 1 in Fiilj. 4-2. Fo, convenience in application, however, a. family of eurve for z

• This equl).tion was a.lso developed indepe~dently by Chuga8v 13J. In this eq~e, tion, M ca.n be regarded as a function of z(ylb):; a.ccordingly, a single curve aiM versus

ourves of M versusy(b an shown, using "

I.

~ I

If.

as

&

pammeter.

·1

58

BASIC,P~INClPLES

CRITICA.L FLOW: ITS COMPUTATION .AND APPLICATIONS

characteristic is that, when the depth of flow in a circular section approaches the top of the circle, the section factor and with it the critical discharge, as shown hy Eq; (4-3) , become indefinitely large. In other words,it is practically impossible to maintain a critical flow in a circular . conduit at a depth approaching the top of the section. In fact, the wavy, surface of the c!'itical flew will touch the top of the conduit before it ' act\lally comes so near as to approach ~he top. A similar characteristic and phenomenon occur also in other types of closed conduit with graduI! ne pial bee om .. curved' "~.n Ihe depth approach.. I ally closing crown, when the water the Q1oduo II}' converging crown of a closed conduit P surface approaches the crown of the i:~g<~~~;o conduit. ot P For channel sections of other than trapezoidal or circular shape, exact 1.1=2. lon9 values of M can be computed di~ rectly by Eq. (4-9), pi'ovided that' N the derivative dT jdy can be evalu'" ~ ated. Approximate values of Mfor any channel section, however, may ,be obtained . hom the following equation

M

69

4 ...4. Computation of Critical mow. Computation of critical flow' involves the determination of critical depth and velocity when the discharge and the channel section are known. Three methods illustl'ated by simple examples will be given below. On the other hand, if the critical depth and channei section are known, the critical discharge can be determined by the method described in Art. 4-2. A. Algebra·ic Method. For It simple geometl'icchannel section, the critical flow can be determined by an algebraic computation using the basic equations. The method has already been used (Example 3-1), but the follOWing e.:1-:.ample is given for further illustration: Example 4-2. Compute the critical depth and velocity of the trapezoid"l channel (Fig. 2-2) carrying ll. disoha.rge of 400 ofs. ' Solution. The hydraulic depth and water area of the trapezoidal section arc eXIueflSed in terms oithe depth lras D

= 1[(10

10

+ 1/)

+ 2y

and

A = V(20

+ 2y)

The velocity is

Substituting the a.bove expressions for D and l'in Eq. (3-10) a,nd simplifying, 2,484(5

+ Y)

[!I(10

+ II)]!

Solving this equation for y by a trial-and-error procedure, Ye '"' 2.15 ft. This is the critical depth. The corresponding area is A. 52.2 iV, !l.lld the critical velocity is V< = 400/52.2 = 7.66 fps.

(4-11)

where Z 1 and are section factors for any two depths Yl and Yz of the given section. This equation can eesily be derived from Eq. (4-6). In applying Eq. (4-11), a graphical m~thod is recommended instead of direct computa.tion. This involves a logarit.hmic plotting of Z as ordinate against the depth as a.bscissa (Fig. 4~3). For most channels, except for closed conduits with depth approaching !J, gradually closing crown and some channels of peculiar shapes, the plot takes a more or less straightline fOl'm. The hydraulic exponent equal to twice the slope of the plot.ted stra.ight line. For a. depth approaching the gradually closing ci'own of a closed conduit,the plot becomes a curve, and the hydraulic exponent of a given depth is equal to twice the slope of the tangent to the curve at that depth, The hydraulic exponent M is described here only as a characteristic v.alue of a channel section under the condition of critical flow. The a:pplication of .this exponent will be further described in the computation of gradually vari~d flow (Art. 10-2), log y'

B. Graphical Method. For!l. complicated or natu~al channel section, a graphical procedure for critical~flow cOll)putation is genernlly employed. By this procedure a curve of yversus Z is constructed. The value of Q/ Vg is then com.puted. . Using Eq. (4-1), the critical depth may be obtained directly from the curve, where Z == Qjv'g.

FIG. 4-3. Graphical determination of the M value.

Example 4-3. A 36-in. concrete circular culvert carries a. dischD,rge of 20 cfs. Determine the critical depth. Solution. Construct a ~tlrve of Y VB. Z (Fi.g. 4-4.). Then compute Z == Q/-vii == 20/-vii 3.53. FrOIXl the curve the critical depth for this value of Z is found to be '!Ie 1.44 ,ft. The dimensionless curve (Fig, 2-1) or the table in Appendix A for the geometric , elements of a circular section might· also be used to solve this problem. Sinee d. 3.0 ft and· do" s == 15.6, Z/d. u = 3.53/15.6 = 0,226. From the dimensionless curve or from the table, Vida = 0.48, ,and so y. = 0.18 X 3 == 1,44. ft.

is

C. Method of Design Chart. The design chart for determinin.g the critical depth (Fig. 4-1) can be used with great expedienCY. In Example 4-2, Z = 400/ v'i "" 70.5 (Eq. (4-1)J. Thtl value of Z IbM is 0.0394this value, the chart gives '!I/b = 0.108 or '!I. ":' 2.Hi ft.

1:<'01' f,

I t

70

BASIC PRINCIPLES

. CIUTICAL FLOW: ITS COMPUTATION AND APPLICATIONS

,

!

71

If the channel has a critical slope (middle sketch in Fig. 4.5), then the flow is initially uniform and critical throughout the channel. In the presence of the dam, however, the flow through the pool Will be subcritical and the pool surface will approach the horizontaL At the dowllstream end a so-called drawdown ettrve will be developed, extending upstream Flo'..,. oon
"I

rr---_ ,

j

BOCkWO!il(

curve

~

Add~d depth due h:I backwater effect

I

1 Drowdawn

Zld~'"

= 0.226.

For this value the clHLrt. gives yld

= 0.48

~

---.

---~

FlO: 4-4. Curve Df y versus Z for a. cir(lular 5C)ction. In Example 4-3, y. = 1.44 ft.

r

Depth of subcriticol flow without dam

or

1

L

l.critIC<J~..s..2~..!.e_ctions;

4-5. Control of Flow. The control of fim:v in 3,11 open channel is defined loosely in many v'ays. As used here the tertn means the establishment of a definitive flow condit.ion in the channel or, more specifically, a definitive relationship between the stage and the discharge of the flow. When the contror of flow is achieved at a certain section of the channl?l, this section is a control section. It will be shown later that the control section controls tbeflow in such a way that it restricts the transmission of the effec~ of changes in flow qondicion either in an upstream direction or in a downstream direction depending on the state of flow in the Chlll111el. Since the control section holds a definitive st,age-discharge relationship, it is always a suitable site for a gaging station and for developing the dis.charg6 raturtg Gwrve, a curve representing the depth-discharge relationship at the gaging station. At the critical state of flow a definitive stage-discharge relationship can be established and represented by Eq. (4-1). This equation shows that the stage-discharge relationship is theoretically independent of. the channel roughiless and other tlllcontrolled circumstances. Therefore, a critical-flow section is a control section. , The location of the control section in a prismatic channel is generally governed by the state of flow, wmchin turn is determined by the slope of the channeL Take for an example a long straight prismatic channel in which a pool is created bya dam across the chalUlel and the water flows. over the dam through an overflow spillway (Fig. 4-.5). Three flow conditions in the channel are shown, representing the subcriliical, critical, and supercritical flows, respectively. The slopes of the channel in the three cases are, correspondingly, mild or ilubcritical, critical, and steep or 8upercritical.

I

,\

I

.

L

Row condition contrnl1ed Qt lhe upstream enc

Depth of supercrilicol flow wilhoul dam FIG,

4-5. Flow conditions in a long prisma.tic ohaullel.

!rom a section near the spillway crest and becoming asymptotic to the pool level. . If the channel has a sub critical slope (top sketch in Fig. 4-5), the flow is initially subcritical. In the presence of the dam, the pool surface will be fllrther raised for a long distance upstream from the pool in a so-called backwater curve. The additional depth of water is required to build up enough head to give the increased velocity necessary to pass water over the spillway. This effect of backing up the water behind tb.e dam' is

i

t

.~

I

J

·{l

72

i

BASIC PRINCIPLES

known as the backwater effect. At the downstream end the backwater curve is conllected. with a !':imooth drawdown curve which leads the water over the spillway. If the channel has a supercritical slope (bottom sketch in Fig. 4-5) 1 the flow is initially supercriticaL In the presence of the dam, the backwater

r

1J .~

i ~

Ii ,

><:

* ''.N

'

S;:-MMc<:)M:r.I~MP)(o)Mp')

,...r:OOOMp')~¢I'lO~tO
'<J' "\!1 "¢I

r:- . . .

ae

Q)

co

QO Q

000'0

----roI.<;q~:1N~

..

..100C::OOOOQOOOOO

~_-:-l_."C"')C'}(t)~.:";ImMI.'I?

N-N ~ 0

;; " " 0 0

0

t: : : ;) ""'" -

C'l N

~

<:;I 0

C'1

0: 0

~'~

0- <:)

("1 cq

~

~oo(Oooooooooo

SECTION L-L

I

i

a:-t;~~

FIG, 4-6, Plan, elevation, ana dimensions o.f ,he Parsholl flume. (U.s. SlIil Conservation 8811>ilie [26J.} Plan and elevation of a concrete Parshall measuring fIl\me showing lettered dimensions as follows.:· W - size of flume in in, 01' ft; A = length of side wall of converging seetion; ,%A = dista.nce back"from end of orest to gage point: B Ilxiallength of converging section; (} = wid.th ofdownstl'ea.m end of flume; D = width of upstrea.m end of flume; E = depth of flume; F length ofthl'oat; G "" length of diverging section; K = difference in ele.... ation between lower eml oj' flume and crest; ,yI = length of approach /lOOI'; N ~ depth af depression in throat \:lelow crest; P = width between ends of curved wing wa.lls; R radius of curved willg wall; X = hol'izontn.l distanoe to E. gage point from low point in tlu'Oll.t; Y "" verticil.l dilltance to. Eo ga.ge point from low point in throl1t. See the tlible on the next page for actual dimellsions for various sizes of flume."

(I'lC'lVlr')MC<')(?tnC")

. '"

..:; ..... ;:;-CO)oc)OOOQOOO ~Q..-t ..... NC'-lC')"Iq'IV)~,....OOCl! ,

~

q;)

~. ~

Q

S

~~~~~~;::~~ 'otI t--<,

2 '" ::.•~ e ~ S'

C"ol C'.J "'tt "ttl
~ CC .....

1:--

I~""~~~------------------I

~

'"

effect originlj.tillg from ~hepool will not extend far upstream. Instead, the flow in the upstream channel will eontinue in:the downstz-eam direction at a supercritical state until the flow-s]lrface profile is actually below the poolle'Tel;1 the~ .it will rise abruptly to the pool elevation in a hydrau-

.:~i:f

~

.=~M.~O

.• O.

l~ .... ·,......-tto':l(#)MM~"~"'')LI':l

ii~~~

~tO

:~

1 Itshoul:l: be noted that; the pool level in this case is ~ot. horizontal but curved. The curved water surface has an 81 profile, which will be :described later (Art. 9-4).

73

l

:::fOOlQIt)OIOOf;!)Q

..:. Col ,,' 'ttl

~

l.O·tn 0: 10 ..,;. l'-<

co

74

BASIC PRINCIPLES

CRITICAL FLOW: ITS COMPUTATION AND APPLICATIONS

lie jump. The backwater effect will not extend upstream through the hydraulic jump. The fiowupstream from the jump.is governed entirely by t.he Upst.l·cam conditions. The above example explains the important fact that on subcritical slopes the effect of change in water-suI:face elevation downstream is transmitted upstream by a ba:ckwater curve, whereas on supercritical slopes the effect cannot. be transmitted far upstl:eam. The flow condition in a sub critical channel is affected by downstream conditions; but, in a supercritical channel, the flow condition is dependent entirely upon the condition upstream or at the place where water enters the channel. Accordingly, tile control of flow is said to be at th,~ downst.ream end for channels ,,,ith sub critical slope and at the upstream end £01' channels with supercritical slope. When t.he challnel is on a subcritical slope a control section at the downstream end may be a critical5('1ction, sunh as that created on the top of an overflow spillway. On a supercritic~l slope, the r;ontrol section at· the upstream end may also be a critical section, 2.S shown in the figure. A sluice gate or an orifke or other control structure may also be used to create a control section. It should be noted that whether the channel slope is critical, subcritical, or supercritical will depend not only on the measure of the actual slope.but also on the discharge 0; the depth of flow. 4-6. Flow Measurement. It was mentioned in the preceding article that, at a critical control section, the relationship between the depth and the discharge is defillitive, independent of the channel roughness and other '.Incontrollable circumstances. Such a definitive stage-discharge relationshipoffers.r. theoretical basis for the measurement of discharge in open channels. Based on the principle of critical flow, various devices for flow measurement have been developed. In such devices the critical depth is usually created either by the construction of a low hump on the channel bottom, such as a weir, or by a contraction 'in the cross section, such as a criticalflow fiwne. The use of a weir is a simple method,. but it causes relati vely high head 109s. If water contains suspended particles, some will be deposited in the upstream pool formed by the weir, result.ing in a ,gradual change in the discharge coefficient. .These difficulties, however, can be overcome at least partially by the use of a critical-flow flume. ,The critir.al-flow flume, also known as the Vent~ri flume, has been designed in various forms.l : J+, is usually operated with an unsubmerged

or free-flow condition having the critical depth at a contracted section and a hydraulic jump in the exit section. Under certain conditions of flow however, the jump may be submerged. . . ' One of the most extensively used critical-flow flumes is the Parshall flume l (Fig. ,1-6) which was developed in 1920 by R. L. Parshall. The depth-discharge relationships of Parshall flumes of' various sizes as calibrated empirically, are represented by the following equations: '

,. I'

ThrQat width

Equ.a.lion

3"

Q = 0.9921I.u4T

(4-12)

6"

Q= Q= Q~ Q=

{4-13} (4-14)

9" 12" to 8' 10' to 50'

J

75

2.06H.ua 3.07 H.l.63 4WH.uuw'·'" (3.6875W -I- 2.5)H.1.G

(4-15) (4-16)

In the above equations Q is the free discharge in cfs Wis the width of ft, ~nd Ha is the gage reading in ft. Whe:l the ratio of gage readmg Hb (FIg. 4-6) to Ha exceeds the limits of 0.6 for 3-, 6-, and 9-in. flumes, 0.7 for 1- to 8-ft flumes, and 0.8 for 10- to 50-ft flumes, the flow becomes submerged. The effect of submergence is to reduce the discharge. In this case the discharge computed by the' above equatiollS ~ust be correc~~d by a negative quantity. The diagrams in Fig. 4-7 gIVe the correct1Ons for submergence for Pa.rshall flumes of various sizes. The correction for the I-ft flume is made applicable to the larger flume's by multiplying the correction for the 1-ft flume by the fador given . below for the particular size of the flume in use.

.thro~t in

Size of flu.me W, ft

1

1.5 2 3 4 6 8

Correction factor

1.0 1.4 1.8 2.4 3.1

4.3 5.4

Similarly, the correction for the 10-ft flume is made applicable to the I Experin).ents on this type of mea.suring. device, then called the Ventu.ri flu rite, were began by M. Cone at,the hydraulic laboratory of the Colorado Agricultural Experiment Station, Fort Collins, Colo.' The initial studies were reported ill [20[ and {211. ~he name ." Parshall measuring flume" was adopted for the device oy the Execu~Ive. Commlttee of the !rrigation Division; American Society of Civil Eq.gineers, durmg 1~.Dec~mber meetIng of 1929. Further developments on the Parshall flume arc d~scflbed by R. L. Parshall in [22] to [26J. :

y.

The critical-flow flumes mentioned in the text are those developed and studied in 'the United States. Outstanding designs of critical-flow flumes were aIso developed and tested by ,Jamcson [4,5J, Engel [ft,7], and Linford f8] in England; byCrump [9] and Inglis [10] in India; by De Marchi [11,12], Contessini[ll], Nebbia [13-15], and Citrini [16,17J in Italy; by Khafagi [18] in Switzerlandj and by Balloft;et [19J iIi. Argentina. I

.i

..;. ~.

"

·r

1 t

r

~l

]

76

BASIC PRINCIPLES

GRITICAL FLOW: ITS OOMI'iJTATIONAND APPLICA.TIONS

i

U

1.9 ZD

Discharge, cIs

I

77

Correction, cIs (tt)

(OJ

Discharge, cIs Correction. cis

( b)

(e)

Fro. 4-7. Diagrams ior computing Bubme.!"ged flow through ParshaU flumes of various sizes. (Colorado Agricullural Ezperime:n.t Sta.tion [25J a.nd U.S. SlYil Conservation Service [26).) (a) Diagram showing the rate of submerged flow. in cubic feet per ~econd,through a 3-in. Pa.rshall measuring flume. (bl Dia.gram s!lOwing the rate of submerged flow, in cubic feet per second, through a. 6-in. Pars!lall measuring flume. (el Diagram showing the rate of submerged flow, in cubic feet per second, through a 9-in. P!lrshall measuring flume. (d) Dia.gra.m for computing the rate of $ubmerge.d flow, in eubic feet per second, through a loft Parshall measl.lring flume,' (e) Diagram for determining the correction in cubic feet per second Plll': 10 ft of crest for subDlergedflow discharge.

b

'" aa

f

c.

4.0 Oischarge, cIs (~)

FIG. 4~7

4.5

5.0

5.5

6.0

CRITICAL FLOW: ITS COMPUTATION AND APPLICATIONS

BASIC PRINCIPLES 78 larger flumes by multiplying the correctioll for the lO~ft flume by the factor given below for the paJ,ticular flume ill use.

Size rJ/ flume W, f/,

CorrectiOlt factor

10 12

LO 1.2

15

1.5

20

2.0

25 30 40

2.5 3.0 4.0 5.0

50

.

;;

~

:

79

\

1 I

I I

[

j

1

II ,

It is desiiable to set the crest of the Parshall flume so that free flow will occur. If conditions do not permit free-floW operation, the percentage of submergence Hb/H" should be kept, whenever possible, below the practical limit of about 95 %, since the flume will not measure dependably if the submergence is greater. The size and elevatiori of the crest depend upon the discharge to be measured and upon the 5i;(;e of the flume and, consequently, upon the loss of head through the flume. The loss or head can be determined from ~he diagrams in Fig. 4-8. A practical example (Example 4-5) will be given' to show the deterrcinatiol1 of the size and elevat.ion of the flume crest. Because of the contraction at the throat, the velocity of water flowing through the flume is higher than that of the flow ill the channeL For this. reason any sand or silt in suspension or rolled along the bottom call. be carried through, leaving the flume free of deposit. When a heavy burden of erosion debris is pl't;lsent in the stream, ho\yever, the Parshall flume will become invalid like the weir, because deposition of ths debris will produce undependable results. For use under such circumstances, a modified Parsh.aLl flume known as the San DimGSfiume (27,28] has been developed, which has the advantage of a self-cleaning mechanism for heavily debris-laden flows in the stream. For measuring open-channel flow in closed conduits, such as sewers and covered irrigation canals, critical-flow flumes of special designs have been proposed. Palmer and Bowlus [29-31J have developed several of these flumes, including one'which is simplr a flat slab on the bottom and has no. side contractions, one with a rectangular croSs section, and several witp trapezoidal-shaped throats. Stevens [32], recommended a critical-flow flume in which he used a blister-shaped' hump control on the bed of the conduit to produce a critical flow over it. The frictional.1osses in this design are believed to be very small. Like. many measuring devices, the critical~flow flume has certain disadvantages. The flume CD.nnot be used directly with or, combined with a head gate. "It is more expensive to build and requires more accurate

I,

II l

,

I

.:~.

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~~:s' 4-8;cD()lio!l.Tgar;~sAf{)O~ del~ermalirunE' 'g th.s los,s h~!l.d t(hrough Parshall flumes of various Seroi~e ~61.) .ur Slat~{)'11 251 and U.S. Sml Ccm.ssnJl1tion of

lW

• • ell

,:z:pmmen

-\

"

80

BASIC PRINCl;l'LES

CRITICAL FLOW: ITS (TO¥PUTATION AND APPLrcATIONS

workmanship in its construction than other commonly used devices, such as weirs or submerged orifices. Technical information on other kinds of open-channel measuring devices and methods can easily be. found in ml1ny textboQks and handbooks on :hydraulics (such as [33J to [35]). . Many culverts along modern high ways can be used as or converted to critical-flow flumes for measuring runoff from the adjoining agricultural lands. This idea was first suggested by Mavis [361 and others and was later studied at the Oklahoma Agricultural Experiment Station [37,381 by carrying out extensive experimental tests on rectangular highway culverts. The results of this experimental study indicate that the culvert can be used as a flow-rate measuring device if it flows part full and has free outlet fall. A weir sill should be installed, however, to improve the accuracy ofthe measurement in the low-flow ,range. In this investigation, a standard Villemonte-type weir silP was developed and its location on the culvert floor determined. Head-discharge relationships were also determined for various flow ranges.

,

{

=

l

=

V.y, =%H.

V%i'if.

= 3.09H.1.S

,

Similarly, try 2" and 3-ft flumes. It is found that the respective crest elevo.tion~ are L53 and 1.23 ft and that the respective upstream water depths £1.1'(> 2.98 and 3.12 ft. In deciding them,!st practical aize of flume to use, it will btl necessary to examine the freeboard or the ch!LIInel and the effect of rise of the water surface upon the flow through the h~adgate. If these conditions arc satisfactory, the 2-ft flume will be the.

I

I j

,

4:-9.

FIG. 8.9ction of a Parshall Ilume illustrating ,th,e determination of the' proper crest elevatlOn [26]. most eoonomical because of its small dimensions. However when the width of the channel is considered, the final selection may be in bvor of th~ 3- or 4-ft flume because moderat~ or long wing wall.s may be required for a smeoll structure. Usually, the throat wldth of the flume Will be from one-third to one-h!'\lf of the channel width.

(4--17)

This is a theoreticru discharge equation i1) which H. is uncertain since the critical section is usually difficult to locate. For' practical purposes, however, the t\quation is generally written q = CH'Jl, where H is the elevation of the upstream wa~er surface iLbovethe weir c~est. This is the form described earlier (Example 3-2). If an ...!'sated free overfall exists at the downstrea~end of the weir, the a\'ove equation ca.n be expressed in termso! the brink depth Y" which can easily be measured. Since y, :. l.4y~ (Art. 3-4), the equ!l.tion required is

q ,

=

9.38y ol.l

PROBLEMS . 4-1. Prove the following critical-discharge equations for the triangular trapezoidal ' aud circular' sections: I ' , , Channel 8e£!lion Triangular

(4-18) ,

Experiments h9.V~ shown that, when the head on the broad-crested w!:!r is greater than about 1.5 times the length of the crest, the nappe of the free overfall becomes detached and the weir is in effect a sharp-crested weir. Example 4-6. Design ft. Parshall flume for handling 20 cfa of now in a.: chann'cl of moderate slope when the water depth in the channel is 2.5 ft." Solulimt. The discharge given can be m!"llsured by flumes of several si~es, but the best selection' is the flume of most practical' a.nd economical size. i ' Assume W .. 4 it ,and H,/H. 0.7, For Q = 20 cfs, Eq, (4-15) gives' H. ~ L15 ft. Hence, H, = Q,81 f t . ; , ' At 70% submergence, the 1vater surface ih the throat, at Hb gage. is essentially level with the surface of the tailwater. Under :this condition of flow, sllown lin Fig. 4-9, 1 The Vilhimonte weir sill consists or two; tri!LIIgular. tapered cotlvergin~ sills placed on the culvert Hoor with an opening left between them (39}. ' ' • This example is sdopted from [26J.

,

~+~=~~.

Example 44. Using the theory of criticaillow, derive an equation for the discha,ge over a broad·crested weir. Solution. COllsiaer the section on the weir crest where critical flow occurs. At this section, y, = 2(V,'/2g} HJ1.5 or V, "" VUH,/1.5, where H. is the specific-energy head at the section. The discharge per foot width of the weids, therefore, equal to

g

81

the tailwater depth D ~ 2.5 ft, and the elevation of the c~est above the channel b;t;.. torn is X 2.5 - 0.81 = 1.69 ft. . . ' From Fig. 4-8, the head loss corresponding to H~I H u = 0.7. Q = 20 cfs, and W = 4 rt is 0.43 ft. .Therefore, the depth of water upstrcuim from the flume will be

i

,

Equation Q,

=

2.295zH..·•

Trapezoidal

Q _ 5.671{(b + zy)yJu , - -(I) + 2zy)',$ -

Circular

Q.= 0.251(0' - sin OJ''' d

Parabolic

Q.

(sin

=

Yz 9) 0,6

2.005TH.L5

(4-19)

(4-20) U n

(4-21) (4-22)

In the above equation~, IX = 1 and H, is the specifiiJ..energy head; other notation' . follows that of Table 2-1; ; ' ' 4-2. C~mpute the hydr£lulic exponent M of the trapezoidnl cha.nnel section (lfig: 2-2) ~avlDg a flow dept!h of 6 ft, using (a) Eq. (4-10), (0) Fig. 4-2, and (e) the graphlCru method based on Eq. (4-11).. • 4.-3. Compute .the hydraulic. exponeD.t M of a 36"1n. circular conduit having a,; flow depth of 24 In. above the Invert, uSing (a) Fig. 4-2 and (b) the graphical method; based on Eq. (4-11). ' ' 4-4. Prove that the critical depth and velocity for a rectangular channel are'

82

BASIC FRINCIFl;IllS CRITICAL FLOW; ITS COMPUTATION AND ,APPLICATIONS

expressed by

83

(4-23)

and

(4-24)

where Q is the. discharge, b is the channel width, and "dll the energy coefficient. 4-5. A rectangular channel, 20 ft wide, carries g, discharge of 200 cfs. Compute the critical depth and velocity. 4-6. Solve Example 4-2 by various methods if the discharge is 300 cfs. 4-7. Solve Example 4-3 by various methods if tr.e discharge is 15 cis. 4-8. An' a.pproxhnate but practiclJ.I fonnula for the critical depth of a circular section of
is y.

= 0.325

(£t +

[).083d.

I

(4:-25)

I

is

which accurate only when 0.3 < y./d Q < 0.9. ' Solve Example 4-3 and Prob. 4-7 by this formula, 4-9. Referring to the naturalchaonel given in Frob. 2-5, construct a CUrve of critical depth aga.inst discharge, ranging from 0 to 400 crs. 4-10. Prove that the section of a cha.nnel in which the flow is critical at any st'lge takes the form expressed by

I

(

-{ !

J

(4-26)

whhle z is half the top width and II is the di5~ance of the water surface below the energy line. Draw II skeLch of the section and d!lscripe its properties. Is this channel possible? If not, how could it be made pOllSible? Is this channel practicable ind the flow stable? 4-11. Verify the computa.tions for the 2- and :HtPa.rshal! flumes tried in Example 4-5. 4-12. Determine the dischltrge through the 4i-ft Parsha.ll flume described in Example 4-5 if the percentage of submergence is .:l00/". 4-13. Determine the discharge measured by a 10-ft Parshall flume if the gage rea.ding H. is 3Al ft at a. free~flow condition. . 4-14. Design a Parshall f!IJme to mell.sure 10 ers of flow in J1 channel ha.ving a depth of flow equal to 1.5 ft. ' 4-16. A uniform flow of 300 cfs occurs'a.t a depth of 5 ft in a long rectan!!;lliar channel ,10 ft wide, Compute the mini,mum IH~ight of a. fiat-top hUmp that ca.n be built on the floor of the channel in order to produce' a critical depth. What wilLresult if the hump is lower or higher tha.n the computed minimum height? 4-16. If the critical depth in the above problem is produced by a contraction of the channel, wha,twill be the maximum contracted width? 4-17. A low da.m 5 ft, high having a broad horizontal crest is built in a. t"6ct~ng\llar channel 20 ft wide. Assuming that a depth df 2.5 ft measured on the crest is the critical depth, compute the discharge and the depth of flow upstrea.m from the dam. 4-1B. On the basis-of the theory of critical flow, Stevens [32] has derived the rating curves for the blister-lShaped critical-flow flume tlll~t he proposed for use in circular conduits (Fig. 4-10). In the deriva.tion, it is as~umed (1) that there is no energy loss from !ll to !It, (:~) that the !;I.pproachingvelocity in the pipe is eq ual to the discharge I

'

1 FIG. 4-10. Rating curves of ;l. critical-flow flume proposed for a clos"d conduit. J. C. Stevens [32J.) d. ""diameter of the cor.duit. '

I I

I 1 I f

I

I

(After;

dh'ided lJy the water area. corresponding to th'e energy head instead or'the actual area.: corresponding to Y" and (3) that the critical-flaw section i3 at the maximnm ~leight of. t~e cDntrol "hump." The second assu.mption elimina.te,s a. trial procedure In determmmg the velocity hegd of the a.pproa.ching flow !l.nd, furthermore, tends to , co,"?pensatc for the error involved in the first a,'55umption. Verify anyone of the , ra tmg curves.

REFERENCES 1. Charles Jaeger: "~ngineering Fluid Mllchanics,~) trrulsu;,ted from the German by P. O. Wolf, Bla.ckle & Son, Ltd., London and Glasgow, 1955, pp. 93-119. ' 2. Ven Te. Chow: Integrating the eq1!aticns of gra.du!l.lly varied flow, paper 833, Proceedtngs, Am.erican, Society of Civil Engi1l.llers voL 81 pp. 1-32 November 1955. ' , , ' ,

3. R., R. C~uga~v: Nekutorye vop.osy neravonomernogo dviphenna. vody v o tkrytykh pnz~tlchesJ{lk~ ruslakh (About some questions concerning n'Jnuniform flow of wat~r m op~n pnsma:i~channels), I zVcsliia. V sesoi11.Z1logo N a.uChno-l ssledClValel' alcaga Imt';.llila Gidrlliekhntk~' (Tramactians, All-l"lnian Scientific. Research Institu.te of Hydra.ulic Engineering), Leningra.d, vol. 1, pp. 157-289, 1931.

.1

84

CRITICAL FLOW: ITS COMPUTATION AND APPLICA~ONS

BASIC PRINCIPLES

4. A. H. Jameson: The Venturi flume and the effect of contractions in open channels, 1'ransactions, Institul-iolt of Water Engineers, vol. '30, pp. 19-24, June 30, 1925. 5. A. H. Jameson: The (il.rvelopment of the Venturi flume, Wawr and Water Engineering, London, voL 32, no. 375, pp. 105-107, Mar. 20, 1930. 6. F. V. A. E. Engel: Non-uniform flow of water: Problems and phenomena in open ehannels with side contrn.c.tions, The Engineer, vol. 155, pp. 392-394, Apr. 21; pp. 429-430, Apr. 28; pp. 45[':'457, May 5, 1933. 7. F. V. A. E. Eagel: The Venturi flume, The EngincliT, voL 158, PP: 104-107, Aug. 3; pp. 131-133, Aug. 10, 19&4.' 8. A. Linford: Venturi flume flow meter, Civil Engineering and Publ'ic W01'ks Rev-iew, London, vol. 36, no. 424, pp. 532-587, October, 19'11. An abstract is given in Journal, Americar;. Waler Works Association, vol. 34, pp. 1473-1475, September, 1942: 9. E. S. Crump: Moduling of irrigatioIl channels, Pt"dab Irrigation Brar4ch Publications, Paper Nos. 26 and 30A, Lahore, India, 1922 and' 1\)33. . lO. C. C. Inglis, NoteB on standing wave flumes and flume meter baffle falls, Pl:bUc Works Department, Government of Bombay, Techn·ical Papers, No. 15, India, 1928. 11. Giulio De Marchi (anthoT, pis. I and III) and FraMesco Contessi~i (author, pt. II): Dispositivi per la misura della pOl·tatll dei canali con minime perdit,e di quota: Nuove ricerche sperimentali slii misuratori a risalto !draulico (C'l.llnli Venturi); Part,e I, Esame del pro(;esso idmulico; Parte II, Descri,.ione delle esperienze; Pal·te III, Risultati delle esperienze [D.evices for measuring discharge in cannls with minimum loss of level: New expaTimentnl researches on standing wave flumes (Venturi flumes); pt. 1, AnnJysis of the hydralilic process; pt. II, Description of the experiments; and pt. lII, Results of the experiments], L' Enfn'gia . elettrica, Milano, vol. 13, no. 1, pp. 6-15, Jil.nnary, 1936; vol. 13, no. 5, pp. 236-244, May, 1936; vol. 11, no. 3, p'p. 189-214, March, 1937. Reprinted as lstituto di Idraulica e Co,~truz'io7'li JdJ'auliche, .Milano, Memorie e sludi Nos. 17, 25, and 26, 1936-1937. .12. Giulio Dc Marchi: Nouvelles recherches experimentales sur Ie jaugeur !I. ressaut hydra.uliqne, canal Venturi (New expel'imental researches' on standing-wave .flume, Venturi flume), Ministry of Agriculture, Pa.ris, France, 1937. This is an abstract of pt. I of \11]. .' la. Guido Neuhia: Ventul'imetri-per canali a sezioni di forma generica (Venturi meter for canals with cross sections of general forms), Acqua e gas, vol, 25, no. 11, pp. 2'rD-291, November, W36. . 14. Guido Nebbia: VeJ,lturimetri per canali a sezioni di tipo monomio (Venturi 'meter for canals with cross sections of monomial type), Acqua e gas, vol. 25, no. 12, pp. 326-333, December, 1936. 15. Guido Nebbia: Venturimetri per can ali a sezione di forma. generica: Primi rj;,'Ultat~ sperimentali (Venturi meter for cano.ls with cross sections of geneml forms: Preliminary experimental results), Aequo e (las, vol. 27, no. 5, pp. 155-181, May, 103S; vol. 27, no. 6, pp. 199-214, June, 1938. 16. Duilio Citrini: Miauratori a risalto (Standing-wave flumes), L' Energia elettrica, Milano, voL 16, nu. 10, pp. 758-763, October,1939; reprinted as Istitutodi Idra1,&lica e Costruz'ioni Idrauliche, Milano, .Memorie e studi No. 35, 1939. -17. Duilio Citrini: Modellatori a risalto: Guida al progetto (Standing-wave meters: Direction~ for design), Centro studi per Ie. aPl?lica.zioni dell'ingegneria all'agricoltura, Sindaeato ingegn.eri di :A{ilano, Se;oorate Paper No.5, Milan,. 1941 ; reprinted as I stituio Ji I draulica e Cos!ruzione' I drauliche, Milano, JIIlemorie e studi No. 44, 1941. lB. Anwar Khafa.gi: Der Venturikanal: Them'ie und Anwendung (The Venturi flume: theory and application), Eidgenossisehe Itchnische Hochscitu/e Zurich, Mitteilungen der Versuchsallslaltfv.r Wasserbau und Erdbau, No. I, Zurich, 1942.

85

19. Arma~do Ball.offet: C:i~ical fl~w meters (Venturi flumes), paper 743, Proceedings, Amencan Soc.ety of Cwtl Engmeers, vol: 81, pp. 1-31, JlIly, 1955. 20. V. M. Cone: The Ventnri flume, Journal of Agricultm·aIResea·rch, voL 9, no. 4, pp. 115-129, Apr. 23,1917. . . . , . 21. Ralph L. Parshnll and Carl Rohwer: The Venturi flume, Coloraclo Agricultural' Experiment Station, Bulletin No. 265 February 1921 .. 22. R. L. Parsho.ll: The improved Vent~ri flume, Trans~clions, AmeJ";cr!.n SlJciety lJf Ciuil.Enyineers, vol. B9, pp. B41~B51, 1926. 23.R. L. Parshall: The Parshall mell-suring flume, Colorado Agricultural Experimen l Siation Btdlet'in No. 423, March, 1936. ' 24 R. L. Parshall: Measuring water in irrigation channels, U.S. Department lJf Ag"i. ~ulh!rB, Parmer's Bulletin No. 1883, January, 1932; revised, October, 1941. . 25. R. ~. Par.sl;al~: Parshall flumes of large size, Colorado A(Jricul/ural Experiment Statton, BttLettn No. 3R~, May, 1932; revised as Bulletin No.· 426A, March, 1953. 26. H. L. Parshall: Measurmg i.vater in irrigation channels with Parshall flumes and s:nalt weirs, U.S. Soil Consenlation Senlicc, Circula?' 843, May, 1950. This Circular supersedes [24], 27. H. G. Wilm, John S. Cotton, and H. C. Storey: Measurement. of debris-laden stre~m flow with critical-depth flumes, Transactions, Ami-rican Society of Civil Eng-meers, vol. 103, pp. 1237-1253, 1938. . . 2B. K. J. Bermel: Hydraulic influence of modifications to the San Dimas criticaldepth measuring flume, Transactions, American Geophysical Union, vol. 31 no. 5 pp. 763-768, October, 1950. . ' , 2fl. Harold K. Palmer and Fred D. Bowlus: Adaptation of Venturi flumes to flow measurements in conduits, Transactions, American' Society of CilJU E7tginee;s vol. 101, pp. 1195-1216, 1936. . ' 30. John H. Lu~wig and Russel G. Ludwig: Design of Pltlmer-Bowlus flumes, Sewage ~ndltst"al Wastes, vol. 23, no.,g, pp. 1096-1107, September, 1951. 31. Edwl~ A. Wells, Jr., I'.nd Harold B. Gotaas: Design of Venturi flumes in circular condUits, Transactions, American Society of Civil ElIginee1's, vol. 123, pp. 749-771, 1958. 32. C. Ste;ens: Discussion on Adaptation of Venturi flumes to flow measurements III c~ndUlts; by Ha.rold re. Palmer and Pred D. Bowlus, Transactions, American Society of Ciuil Enginee1's, vol. 101, pp. 1229-1231, 193fl. N 33. Herbert Addison: "Hydmulic Measurements" JohnWile v & Sons In . ' J . ' c., ew York, 1941.

a;:d

!.

34. "Wate'l' 'tvIell.sunnnent Manual," U.S. Bureau of Il.eclamo.tioll May W53 43-58. ' , , pp.

, I

l

35. Horace William King: "Handbook of Hydraulics," 4th ed., revised by Ernest F , . Brater, Mct;lraw-Hill. Book Company, Inc., New York, 1954. ' '.. ;;6. F. T. MaVIS: Reducmg unknowns in small culvert' design, Engine~rin(l NewsRecord, vol. 137, no. 2, pp. 51-52, July 11, 1946. 37. W; O. Ree and F. R. C~ow: Measuring runoff rates with rectangular highway culverts, Oklahoma AIlT!CUU'Ural Experiment Stat'ion, Technical Bulletin T-51 November, 1954. . ' 38. W. O. Ree ~nd ~. R. Crow: Culverts as Wo.teT rUIloff measuring' devices, AgriC1!/tural Eng~neenng, \"01. 35, no. I, pp: 28-31 and 39, January, 1954. 39. James R. Villemonte: New type gaging station for small streams; Enginee1'ing News-Record, vol. 131, no. 21, pp. 748-750, Nov. 18, 1953. 40. C. D. C. Braine: Draw-down and other factors relating to the design of stormw2.ter outflows on sewers, Journal, Institution of CiviL Engineers, London vol. 28 no. 6, pp. 136-163, April, 1947. . ' "

.PART II

UNIFORM FLOW

j 1 i.

1 I

'1

;

. \

\

. j,

CHAFTER

5

DEVELOPMENT OF UNIFORM· FLOW AND ITS FORMULAS

.

I

J 1

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,I

II

I

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89

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l

I~~, '

'"

,.: ~..

.)

15-1. Qualifications for Uniform Flow. The uniform flow to be considert:d has the following mn.in features; (1) the depth, water area, velooity, und discharge at every section of the channel reach I.1r,e constant; and (2) the energy line, water surface, and channel bottom are allparallei; that their slopes are all equal, or Sj = S'" = Sn = S. For practica.l pm-poses, the requirement of constant velocity may be liberally interpreted as requiremel1t thtl.t the flow possess a constant mean velocity. Strictly speaking, however, this should mean that· thB flow possesses a constant, velocity at every point on the clHmnel section within the uniform-flow reach. In other words, the velocity distributIon across the channel section is uualtered in the I'each. Such fl. stable pattern of velocity distribution call be attained when the so-called" boundary layer" is fully developed (Art. 8-1). ' Uniform flow is considered to be steady ouly, since unsteady uniform flow is practically nonexistent. In natural streams, even steady uniform flow is rare, for rivers and streams in natural states seanleiy ever experience a strict uniform-flow condition .. Despite this deviation from the truth, the uniform-flow condition is frequently assumed in the computn:tiOD of flo\v in natural streams. The results obtained from this assumption are understood to be apPl·oximo.teallq. general, but they offer a relatively simple and satisfactory solUtion to many prnctico.l problems. As turbulent tmiform flow is most commonly encountered in engineering problems, it will be discussed e,.'(tensively il1 tbe following chapters. Laminar uniform flow has limited engineering applications, and will be described only in Art. 6-10. " It should be noted that unifor:m flow cannot occur at very high velocities, usually described as ttltTarapid. 'This is because, when uniform flow reache~ i certain high velocity, It becomes very unstable. At highel' vetoci~ies' the fl~ "'1ill eventually entrain Edr and becbme unsteady. The criterion fOl: instability of u,niform flow will be discussetl1i:1':A:rt:1f-8: 5-2. Establishment .of Uniform Flow. When flow OCCtll'S in an open channel, resistu.nce is encountered by the water as it flO\vs downstream.

,

'

DEVELOPMENT OF 1}NIFORl'rl FLOW AND ITS FORM,'()'LAS

This resistance is generally cO,unteracted by the compon~llts of gravity forces acting on the body of the water in the direction of motion (Fig. 5-2). A uniform flow will be developed if the resistance· is balanced by the gravity forccs, The magnitude of the resistance, when other physical factors of the channel are kept unchanged, depends on the velocity of flow.

may again be exceeded by gravit:r forces •.apd the flow may become varied again. For purposes of explanation. a long channel is showll with three different slopes: sub critical, critical, and supercritical (Fig. 5-1). At the subcritical slope (top sketch in Fig. 5-1) the water surface in the transitory zone appears tmdulatoi:y. The flow is uniform in the middle reach of the channel but varied at the two ends. 1 At the critical slope (middle sketch in Fig. 5-1) the water surface of the critical Row is unstable. Possible undula.tions may occur in the middle reach, but Ql1 the average the depth is constant and the flow may bf'. considered uniform. At the sl.lpc,rcritical slope (bottom sketch 'in Fig. 5-1) the transitory water surface passes from the subcritical stage to the supercritical stage. through a gradual hydl'aulic drop. Beyoml the transitory zone the flow Is approaching uniformity. The depth of a uniform flow is caned the nonnaJ depth. In all figures the long dashed line represents the llormal-depth line, abbreviated as N.D.L., and the short clashed or dotteclline represents the critioal-depth line, or C.D.L. The length of the transitory zone depends on the discharge and on the phj"Sical conditions of the channel, Buch as entrance' condition, shape, slope, and roughness. From a hydrodynamic standpoint (see Art. 8-1), the length of the transitory zone should not be less than the length required for the full development of the boundary layer under the given conditions. 5-3. Expressing the Velocity of a Uniform Flow. For hydraulic computations the mean velocity of a turbulent uniform flow in open channels . is usually expressed il,pproximately by a so-called unifol·tn-flow formula. Most practical uniform-fiow form1.lia,s ca:n be expressed in the following general form:

90

Ivorled~w

I

Tronsitor

zone

un[f~rm

flow on the ove''::Qe ,J

yt--I I .. ' . ' I

I

r~~;::~1 t;"ifOftl1

fl 0 "'_ _ _-I.1

I

i

I I

I I

I

I

t

I

!

"

1 l

1 )

.\ t

(5-1)

FIG. 5-1. :&!ts.blishment of uniform flow in

;1'

91

UNIFORM FLOW

Il

where V is t,he mean velocity in ips; R is the hydraulic rf;1,dh15 in ft'; S is the energy slope,2 x and yare exponents; and, C is a factor of flow resist~ ance, varying with the mean v:elocity, hydraulic channel roughness, viscosity, and many QtheJ: factors. For practical purpos~s, the fiow in a natural channel may be assumed l,miform under normal conditions, that is, if there are 110 flood Rows or markedly varied flows caused by. channel irregularities. 'In applying

long ohe.nne1.

If the water enters the channel slowly, the velocity and hence th(;l resista~ce are, small, and the resistance is outba,lanced by the gra;~ity forces, resulting:in an accelerating flo~v in the upstr~am reach. The velocity and . the resistance will gl·adually 'increase until a balance between resistance aDd gravity forces is reached. At this moment alld' a.fterward the flow becomes uniform. The upstream reach that is required for the establishment of uniform flow is known as the transilory zone;' In this zone the flow is accelerating and varIed. If the channel is shorter than the transitory length required by the given conditions,. unIform flow cannot be.' . attained. Toward the downstream end of the channel the resil5tance

1 Theoretically speaking, the varied depth a.t each end approaches the uniform dept.h in the middle llSymptotically and gradually. For praotica\ PUfilvS\lS, however, the depth ma.y be considez:ed constant if the variation in depth is within a certain margin, say, 1 % of the lwerage uniform-fiow depth. Z In. uniform flow f S :'" Sf =' S,. = So. When the uniform-flow formula is applied t.o the computation of energy slope in a gradually varied flow, the energy slope will be denoted specifically by Sf instead of S. .

t

. .l·.

I

. i

-\

92

93

UNlF'ORM FLOW

DEVELOPMF.NT OF UNIFORM FLOW AND 1'1'13 FORMULAS

the uniform-flow formula toa natural stream, it is understood that the result is very approximate' since: the flow condition is subject to more imcertnin betors th}l[l would be involved in it regular artificial channeL As pointed out by Schneckenberg [1]; a good uniform-flow formula for an alluvi.al channel with sediment transport and turbulent flow should take equal aecount of aU the following variables;

. 5-4. The ChezyFormula. As early as 1769 the French engineer Antoine Chez)" was developing probably the first uniform-flow formula, the famous CMzy furmula l which is usually expressed as follows:

A .V V ma P R y SUI

n

Q. Qb 'IL

the water area the mean velocity the maximum surface velocity' the wetted perimeter the hydraulic radius . the maximum depth of w-ater area the slope of the water surface a coefficient representing the channel roughness, known as the coefficient of 7'oughness 1 the sllspended sediment charge the bed load the dynamic viscosity of the water' the temperature of the water

T There have been dev,eloped and published a large number of practical uniform-flow formulas, t but none of these formulas meets the qualifications of a good form'lIla as defined above. The best known and most widely used formulas are the CMzy and Manning fOl'mulas, which will be described in the' following articles alid used extensively in this book. Theoretical uniform-flow formulas have also beell derived on the basis of a theOl'etir.AI velocity distribution across the channel section, wl1i~h will he discussed later (Art. 8-5). A different approach to the determina.tion of the velocity in a natural channel has been attempted by Toebes [5]. In this approach amultiplecorrelation analysis is applied to the following significant factors affecting the velocity in a gi.ven alluvial channel: water area, maximum surface velocity, wett~d perimeter, maximum depth,slope of water surface, coefficient of roughness, and temperature of water. By this method it is , possible to e"aluate the independent individual influence of each variable on the magnitude of the velocity. When such an evahULtion is made, the velocity under any givell condition of the variables is simply equal to the algebraic summation of the individual contl'ibut,ions as <\ffected by each variable. However, this method applies only to the streams in the geographical region for which the analysis is made; hence, its applica.tion . cannot be generalized. In British literature the term "l'llgosity coefficient IJ is used. , A number of weH-known uniform-flow formulas are given Md discussed in [2] to [5J. . 1

(5-2)'

where V is the mean velocity in fps, R is the hydraulic radius in ft, S is the slope of the -energy line, and C is a factor of flow resistance, called

ChezY'$ G.

..

,FIG,

. .\,.-,',

, '

,

5-'.l. Derivation of the CI~ezy forml;!a. for uniform flow; in open channel.

The CMzy formula C.:l.n be'derived mathematically from two assumptions. The first assnmption was made by CMzy. It states that the foi'ce resisting the flow per unit, area of the stream bed is proportional to the square. of the velocity; that ls, this force is equal to X"V2 where ]( is a constD.nt of proportionBlity. The surface of contact of the flow with the stream bed is equal to the product of 'the wetted peri~eter and the length of the channel l'efLClh, or PL (Fig. 5-2). The total foroe resisting the flow 2 is then equal to ](VSPL. 1 The source of this famous formula. is.not mentioned in most hydraullcs textbooks. In fact, this knowledge has long been sought for. In 1876, the German engineer Gotthilf Heinrich Ludwig Hagen mentioned in his 'work [7] that Geapard de Prony , had stated tha.t Chezy set up this formula in 1775, on the occa$ion of a report that Chezy made on the Canal de j'Yvette in conjunction with Jean-Rodolphe Perro net, . "But," says Hagen, "I have sought in vain for further information on the' Bubject." Then, in 1897, the American engineer Clemens Herschel through the assistance of a friend in Paris traced the original Canal de I'Yvette report to its hiding place, then translated the portion relating to the formula, IlInd published it in [8]. Chezy's report revealed that the formula' was developed and verified by experiments made on an earthen canal, the Courp!1let Canal, and on the Seine River in late 1769 .. • Thi~ channel resisting iorce may alao be ,explained by the principlea of fluid dynamics. The open channel can be coneeived as a. ftnt plate warped into a oylinder

i

94

i

UNIFORM FLOW

The second asst1l~ption is the basic principle of uniform flow, which is believed to have been claimed first by Brahms [9] in 1754. It states that,' in uniform flow, the effective component of t~e gravity force caus~ng the flow musl', be equal to the total force of reslStance. The effectIve gravity-force component (Fig. 5-2) is parallel ~o th~ chal~nel bott.o~ and equal to tuAL sin e = wAL8, where w is the Ulllt weIght Ol water, A lS the water a;'ea, fJ is the slbpe angle, and 8 is the channel slope. l Thence, wAL8 = KPPL. Let, AlP = R and let vw/K be replaced by a factor C; then the previous equation is reduced to the Chezy formula, Of

V

=

v(wIK){AIP)S

= C vRS.

.

DEVELOPMENT OF UNIFORM FLOW AND ITS FORMULAS

j !

i

i I

I

the formula appears cumbersome, it usua-lly produces satisfactory results. It has been so widely used that many tables and charts are available for its application; so the use of the, formula itself is seldom found necessary in engineering offices. Figure 5-3 gives a popular chart for the solution of the G..K formula. .~.~ B. The Bazin Formula.. In 1897, the French hydraulician H. Ba'lin l proposed a formula according to which Ch~zy's C is considered a function of R but not of 8. Expressed in Engli8h units, this formula is C

I

attempts have been made to de~ermine the value of Chezy's C. Three important fOI'mulas developed for this purpose will be given in the next article , 5-5. Determination of Chezy's Resistance Factor. Three importltnt formulas for the determination of CMzy's C are given as follows: . ~ ~ICk'l11Jj,la~_ In 1859, tw~ Swiss engineers, ?[mguillet and . Kutter [10], published a formula expressmg the value of C m terms of the slope 8, hydraulic radius R, aild the coefficient of roughness n. In English units, the formula is .0.00281 + 1.811

M~my

.41.65

C -

1

+

8

-n°

n + ( 41.65 + 0.00281) S vB,

(5-3)

The co~ffi~ient n in this formula is specifically known as [(utter's n. The value of n will be discussed in.Ads. 5-7 and 5-8. . The G. IL formula was derived elaborately from flow-measurement data in channels of ..various types, including Bazin's gagings and the gagings of many European rivers and of the Mississippi River. 2 Although but unclosed on one si.de which corr'osponds to the free surbce of the. ~~en-channel flow. A fluid flowing ill the uncloaed cylinder will create a drag or resls"mg ~orce on the inside surface.. This force is equal to tha drag created by the flow of .flUld along a flat plate whose tl\,O surfaces offer resistance to th.e flow. The la~ter' IS equal.t,) CdP V'PL/2, wherp. Cd is tbe coefficient of drag and P IS the ma~s de~lty _of the flUid .. Thus;the factor CdP/2 is equivalent to the constant. of proportlonabt;' .K.. . . 1 The' slope under consideration is defined as the sme of the angle 0, inclinatIOn, or S = sin 11. . , The Mississippi River gagings were made by Humphreys and Abbot on the I.ower Mississippi RiYer between 1850 and t860, and the data thus o.btained were published in a report submitted to the U.S. Army Corps of Topographica'J.Engin~ers in 1861 (11]. The term containing S was introduced into the G. K. formula Simply m order to make the formula ~gree with the Humphreys and Abbot d.ata. Thi~ se.ems somewhat ridiculous nolY, .becllu.se these data are known to have been qUite maccurate (see pp. 133-136 of (2]). Some authors have suggested that the slope term 0.00~81/S of

95

= __ 157.t'_·_ 1

+ m/vR

1

(5-4)

>;".here m is 9. coeiIicient of roughness whose values proposed by Bazin are given in Table ·5-1. TABLE

5-1.

PROPOS!lD VALUES OF BAZIN'S 171.

Description of chanael

Bazin's.1n

Very smooth cement of planed wood ............... . Uuplaned wood, concrete, or brick: ............... . Ashla.r, rubble masonry, or poor brickwork .......... . Earth channels in perfect condition ........... : .... . Earth channels in ordinary condition .............. . Earth channels in rough condition ................ .

0.11 0.21 0.83 1.54 2.36

The formula was developed primarily from data collected from small experimenta.l channels; hence, its general application is found to be less satisfactory than the G. K. formula. The Miami ConservancY'District [2] has made a study comparing the v!l.riations ill Chezy's C, Bazin's m, and I{utter's n for Bazin's experimental data and several naturaL streams. The results based on this study are showll in Table 5-2. The values of the average variation indicate that Bazin's formula is not as good as Kutter's even for his own measurements.

.~ C. The. Powell Formula. In 1950; Powell [14) suggested a logarithmic. formula for the roughness of artificial channels. implicit function of C, is C =

-42 (4i + It) log

This formula, an (5-5)

-----------------------------------

the G . .K. formula be omitted in order to simplify the aP?lmranCe of the formula. a.nd even to make -the general results more satisfactory. 1 From 1855 and 1862 an exten.~ive seiie:s of experiments on open-ch,mnel flow were first begun by H. Darcy and then completed by Bazin. The results were published by Bazin in 1865 (12). On the basis of the accumulated date., Baziu finally proposed the formula in 1897 (13).

-\ 1

. I

J j

OJ

.S. tQ

.> ,..

"'

'0

u..

0

'"s<
i'l

c

~

DEVELOPMENT-OF UNIFORM FLOW AND ITS FORMULAS

1l 'tl

where R is the hydraulic radiusin H; R is the Reynolds numbel';and E is a measure of the channel roughness, having the tentative values shown in Table 5-3. For rough channels, the flow is generally so turbulent that R beoomes very large compare~~th C; thus, Eq. (5-5) approximates the. form

.~

"'" .S 0

t1. t:i A

A

Q)

r:I

'"

A, j;:

il:

~-

p"

~

~

..,:il

'"

...'"

til

T.tUILE

'"

Q)

.c: .,

c := ;i!

0

OJ

..,

1:j

,S tQ

:S'"

-

Average vaiue3 C

Bazin's Series 6 .. , .......... " .. "'...

" "' .~

.... Q)

A

o

., c

oil

~g'O

~ l:t:1 ill' ",.",

'"""""

:El~

en

.q

'" '0 .!!l

~

'" :El

S.AZ~H:>

2~!!!~g.~E~i.~ii!~~

m

C

. . . . 0.185 O. 0121 ..

8 ........ " ........ " ...... , ..... 9 .................. " .... ' ..... 10 ............................... 11., ............................ 12 ............ , ................. Ill ........................... 14., ............... , ........ , .. 15 .............................. 16 .............. ·~ ............... 17 ....... :: ............... 1.... 32 .......................... j.... 33 , ., ..••.......... ,. . .... , .....

Q

~~]

varia.tions, %

7 .... , .................. , ... , .. , 0,156'0.0120 .. .

"I

,~

m,

n

t:i

~~

'"

BAZIN'g

Measurements

0

~

C,

AND KTJT'J'ER'S 11

g tJ" is ... "'-~ .,.,.., .gj

.J:!

;0;:

COMPARISON OF VARIATH)NS IN CHi:z.y's

c

III

<J

5-2,

oil

oil .. :a :a .... ... !5 .., .i1III ....'" .~'" ... .!!l 0-

97

.8

g :El'" ...,'"

'" ill

U en "'-

...

'A

'" .,m-

., 0; " 5.,'" ... ..::; '"~

'0

'"

'-'

Q)

J

I lI

.,

!J

0,1420.0116 .... . 0.1990.0130 ... .. 0.1440.0117 .... , 0.129 O.OU3 .... , 0.324,0.0151. 1.6 OJHI0.0148 .... :j2.7 0.3210.0150 .... 4.4 0.'7150.0209 . . . . 4,2 0,7110.0212 ..... ' 5.7 0.7210.0215 16.1 0.4240.0' 1.8 0.444 0.0 3 .1

1 1 1.0 2,5 1.2 1.4 3.8 1.0 1.2

1.8 1.2

1.5 2.2 0.4 L2 8,8 5.7

. 44, ..... .' ..................... 0.8580. , ... ;1118:6 45 .. , .......... :.; .............. 0.7040. . .... 11.1 Mia.mi River at Tadmor, Ohio, 1915-1916". 167.4* L9S O. 4.0810.9 4.9 Bogue Phalia River, Miss .• 1914 .... " ....... 63.3· 4.09 0.070424.20.35.1 22.2 Arkansas Drainll.ge Onnals, Ark., J915 ....... 165.9· 2.12 O. :3 18 4.8 Mississippi River, Carroltoll, La., J912 ..... , ...... 1.33 1.30 5.4 MiSSissippi River, 'Oarrolton, La., 1913 ... , ....... , 1.46 12.8 IrraWllddy River, Burm!l ......................... 1. 35 23.0 Volgu. River at Sa.mara., R.ussia. ... ,.......... .58 0.0311 1.8713.0 Volga River at Zhiguly, RUl~sia., ........ " . ,76 0.0363 18.8036,5 Average va.riation.. .. . " .:, ... , .. , .. ,

gg

fL~.tG,~I~,LIW.!~I~!~I'L1UT~.ll~ttLI~I~.llk'~~~~~~WW~~WU~~~~o S3l'iNlmlO .:Ici SIXV

• Values a..veraged by the author.

I

C = 42 log (R/e). For smooth channels, the iurface roughness may be: . so slight that e becomes negligible compared with R; then the formula approaches the fbrm,C = 42 log (4R/C). Sinoe Chezy's is expressed implioitly in the Powell formula, the solution of 'the formula for C requires: !l. trial-and-error procedure. . . The Powell formula was developed from limited laboratory experiments: on smJoth and rougn channels and from the theoretical velooity distri-;

, (

a

i'

96 .

:1\

."

~

,',

.;

98

UNIFORM, FLOW

DEVELOPMENT OF UNIFORM FLOW AND ITS' FORMULAS

butioll studied by Keulegan (Art. SA). The practical application of this formula is limited, since further investigation is needed for determination of the properva.[ues of E. TAIILll

5-3.

TllNTATlYE VALUES OF POWELL'S.

Powell's. Description of channel

Neat cement surface ............... . Unplaned-plank flmlles .... ; ............ . Concrete-lined channels. . . . . . . .. . ..... . Earth, straight and uniform ............ . Dredged ear~li channels ................. i

New

Old

0.0002 0.0010 0.004 0.04 0.10

0.0004 0.0017 0.006

Example 5-1. Compute the velocity and discharge in the trape~oidal channel described in Example 2-1, hONing a bottom width of. 20 ft, side slopes 2: 1, and a depth of wat,er 6 ft. Given: Kutter's n = 0.015, and S = 0.005. Solution. From Example 2-1, A = 192.0 it' and R = 4.10 It. Using the G. l{.. formula, the value or Ch6zy's C i"

0= Then, by the

Chc~y

41 65 + 0.00281 + 1.811 0.005 0.015 . . '41 6- + 0.00281) 0.015 +( • 0 0.005 V4J.O

=

1242 .

formula,

V

=

124.2 V4.10 X 0.005 = 17.8 fps

Therefore,

Q = 192.0 X 17.8

=

3,420 ds

5-6. The Manning Formula. In 1889 the Irish engineer' Robert Manning I presented a formula, which was later modified to its present 1 Manning first presented the formula in a paper read on December 4, 1889, at a meeting of the Institution of Civil Engine:ers of Ireland. The paper ,...-as Io.ter published in the Transa.ctions of the Institution [15J. The formula \V'as first given in a complicated form and then simplified to V '= CR?~S'\, where V is the mean velocity, C is a factor of flow resistance, R is the hydraulic radius, and S is the slope. This was further modified by others and expressed in me.tric units as V = (1ln)R~\SH. Later, it was converted back again to English uni~,resulting in V = (1.486/n)R;~S'~. In this conversion, as iIi the conversion of the: GanguilLet and Kutter formula, tlie numerical value,of n is kept unaffected. Cons:equently, the same value of n is widely used in both systems of units. In the vielV of modern flllid mechanics, which pays much attention to dimensions, the 'dimensions of n become a matter of consideration. Directly from the Manning formul!l, the din).ensions of·n are seen to be TL-H. Since it is unreasonable to suppose that the roughness coeHicient w01..lld contain the dimension T, some, authors assume that the, numerator contains -../g, thus yielding the dimensions cif i>~ for n. Also, for physical reasons, it will be seen that n = ((Rlk)Jk>~ IEq. (B-26)J, \~here k is

99

well-known form (5-6)

where V is the mean velocity in fps, R is the hydraulic radius in ft, S is the slope of energy line, and n is the coefficient of roughness, specifically known as 111anning' 8 n. This formula was developed from seven different formulas, based 011 Bazin's experimental data, and further verified by 170 observatipns} Owing to its simplicity of form and to the satisa linear meas,ure of roughness and (Rlk) is a iunction of Rlk. If
-1

j l.

\

I

J

,

100

i

I

UNIFQRMFLOW ,

DEVELOPj\HlNT OF UNIFORM FLOW A.ND ITS FORMULA.S

i

factory results it lends to practical applications, the Manning formula has become the most widdy used'of all uniform-flow formulas for 6pen-channel flow computations.; A nomographic solution of the formula is given in Appendix C. Within the normal ranges of slope and hydraulic radius; the values of Manning's n and Kutter's n are generally found to be numerically very close. For practical purpose~ the two values may be considered identical when the slope is equal to or greater than 0.0001 and the hydl'aulic radius is betwe~n 1.0 and 30 ft. Typical values good fer both Kutter's nand Manning's n are shown in Table 5-0 and illustrated in Fig. 5-S. Comparing the Chezy formula with the Manning' formula, it can be seen that

c

)

-I

(5-7)

This' equation provides an important relationship: between Chezy's C and :Manuing's n. The exponent of the hydraulic radius in the Manning formllla is !'.ctnally not a constant but varies ina range depending mainly on the channel shape !l.nd roughness (see 11 previous footnote). For this reason, some' hydraulicians prefer to use the. formula with a variable (lxponent. For example, the unifomi-flow formulil. widely used in the U.S.S.n. is of this type; this is the Pavlovskiiformttla [21], proposed in 192£$.* This formula in mei)'ic linits is (5-8) (5-9)

. where

and where C is'the resistance factor in the Chezy formula expressed in 'metric units. The exponent y depends on the Ioughnesscoefficient and hydraulio radius, Tile formula is valid Jor R between O.i'and 8.0 ill and ,

"

basis of other later studies, some authors Stiggested ll. value oi % [20], and others suggested a variable depending on Rand 11. [21J. . l The Manning formula was suggested for intern~tionaJ use by Lindquist [3j at the Scandinavie:Sectional Meeting oi the World Power Conierenee in 1933 in Stockholm. The final recommendation for such use was mlide by the Executive Committee at the 3d World Power Conference in 1938 in Washington, D.C. . 2 On account of this relationship, the Manning formula is sometimes oonsidered a variation'oi the CItezy fortimla with Chezy's C defined by Eq. (5~7). *' The Pa.vlovsIl.il formula. was published in severa! editions of Pll.v!ovskiI's "Ha.ndbook of Hydraulics" [211_ An article o.bout this formula. entitled Ft)1'l1m!a dlia kt)effitlrien/.ll Chazy (Formula fur a Chtzy coefficient) is given in pp. 140-149 of the 193T edition or the book. A footnote in this article reads: "'1'.he formula. was proposed in 1925."

101

for n between 0.011 and 0.040. For practical purposes, the following approximate forms of Eq. (5-9) are generally suggested for use: Y= 1.5 Y"" 1.3

vrn

vn

for R for R

< >

1.0 m 1.0 m

15-7. Determination of Manning's Roughness Coefficient, In applying Manning formula or the G. 1C. foi'Inula, the greatest difficulty lies iIi" the determination of the roughness coefficient n; for there is no exact of knowledge, to method of f'electing the n value. At the present select !1 value of n actually means to estimate the resistance to flow in a given channel, which is really a matter of intangibles. To veteran engineers', this means the exercise of sound engineering judgment and experience; for beginners, it. can be no mOl'e than agness, and different individuals will obtain different results. . .III Qrder to give guid::mce in the proper determination of the roughness coefficient, four general approaches' will be discussed; namely, (1) to understand thefa,ctors th!l.t affect the value of n and thus to acauire a basi~ knowledge of the problem and l~arrow the wide range of gues;work, (2) to consult a table of typical n values for channels of various types, (3) to examine and become acquainted with t.he appearance of some typical ehannels whose roughness coefficients are known, and (4) to determine' the value of n by an rmalytical procedure based on the theoretical velocity distribution in the channel cross section and on the data of, either velocity or roughness measurement. The first three approaches will be given in the next three articles, and the fourth approach will be· taken up in Art. 8-7, 5-8. Factors Affecting Manning's Roughness Coefficient, It is not uncommon for engineers to think of a channel as having a single value of n for aU occasions. Ii1 reality, the value of n is highly vELdable and depends on a number of factors. In selecting a propel' vaiue of n for various design conditions, fl,basic knowledge of these factors should be found very usefuL 'The factors that exert the greatest influence upon the coefficient of roughness in both artificial and natural channels are therefore described below. It should be noted that these factors are to a Certain extent int/ilrdependent; hence discussion abou,t one factor may be repeated in connection 'with anothel'. ' A. Surface. Rm€ghness, The surface roughness is represented by the size and shape of the grains of the material forming the wetted perimeter and producing a retftrding effect on the flow. This is often considered the only factor in selecting a rOllghness coefficient, but it is actually just one of several major factors. Generally speaking, fine grains result in a relatively low value of n and coarse grains, in a high value of n. In alluvial streams where the material is fine in grain, such as sand, t~e

DEVELOPMENT OF UNIFORM FLOW AND ITS FORMULA.S

UNIFORM FLOW 102 clay, loam, or silt, ,the retardiI).geffect is much less tha;n whe:e t?e material is coarse such as gravel;; or boulders. When the materIal IS fine, the value of ~ is low and relatively un.affected by change i.n flow stage, 'Yhen the material consists of gravels and boulders, the value of 71 is generally high, particularly at low or high stage. Larger boulders usually collect at the bottom of the st.ream, making the channel bottom rougher than the. banks and increasing the value of 71 at low stages; At high stages, a . portion of the energy of flow is used in rolling t.he boulders downstream, thus increasing the value of n. A theoretical discu,ssion of surface roughness will be 'given in Art. 8-2. B. T1 egetation. Veget.ation may be regarded as a kind of surface roughness, but it also markedly reduces the·capacii~ ~f the c~annel. an~ retards the flow. This effect depends mainly on heIght, denmty, dlstnbutiol1, and type of vegetation, I;1nd it is very important in designing . small d[ainl;1ge channels. At the University of Illinois an investigation has been made to determine the effect of vegetation on the coefficient of roughness [22]. On one of the drainage ditches in central Illinois under investigation, an avera~e n :value of 0.033 was measured in March, 1925, when the channel was ill good condition .. In April, 1925, there were bushy willow? a~d dry w~eds on the side slopes, and n was found to be 0.055. ThIS ll1crease m. 71 represwts the result of one year 's growt~ of vegetation. D~ring the summers of U)25 and 1925 there waS a thICk growth of cattaIls on the bottom of the channel. The n value at medium summer stages w~s about 0.115, l,tnd at a nearly bankfull stage it was 0.099. The cattaIls inihe channel were washed out by the high water in September, 1926; the average value of 71 found after this occurrence was 0.072 .. The conclusions. drawn from this investigati'on were, in part, as follows: 1. The minimum value of 11, that should be used for designing drain~ge ditches in· centml Illinois is .0.040. This value is obtainable at high· stages duri!1g the summer months in th~ most carefully n:aintained ch~n nels where the bott.om of the channel IS clear of vegetatiOn and the. Side slop~s are covered with grass or low weeds, bU,t no bushes. This low value of n shonld not be used unless the channel IS to be cleared a:nnuaUy of all weeds and b u s h e s . . . 2. A value of n = 0,050 should be used if the channel is to be cleared in alternate years only. Large weeds and bushy willows from 3 to 4ft high on the side slopes will produce this value of n. 3. In channels that are not cleai'ed for a number of .years, the growth may become so abundant that values of n· ~ 0.100 may. be found. ,. . 4. Trees from (3 to 8 in. in diameter growmg on the Side slopes do not impede the flo\v so much as do small bushy growths, provided overhang. ing branches are cutoff,

'.'

j

i ",1\.,'.,..

."'

'"

103

The U.S. Soil Conservation Service has made studies on flow of water in small shallow channels protected by vegetative linings (Chap. 7, Sec ..0), It was found that n values for these channels varied with· the shape and cross section of the channel, the slope of the channel bed, and . the depth of flow. Comparing two channels, all other factors being equal, the lesser average depth gives the higher n value, owing to a larger proportion of affected vegetation. Thus, a triangular channel has a higher n value than a trapezoidal channel, and a wide channel has a lower n value th11D. a narrow channel. A flow of sufficient depth tends to bend over and submerge the vegetation and to produce low n values. A .steep slope causes greater velocity, greater flattening of the vegetation, and low n valu,<::s. . The effect of vegetation on flood plains will be discussed later in item H. . C. Channel Ir:regularity. ChanneUrregularity comprises irregularities in wetted perimet.er and variations in cross section, size, and shs.pe along the channel length. In na.tural channels, such irregularities are uSl,lally introduced by the presence of sand bars, sand waves, ridges and depressions, and holes and humps on the channel bed. These irregul9.rities definitely introduce. roughness in addition to that caused by surface roughness and other factors. Generally speaking, a gradual and uniform change in cross. sectioll, size, and shape will not a.ppreciably affect the value of n, but abrupt changed or alternation of small and large sections necessitates the use of a large value of 71. In this case, the increase in n may be 0.005 or more. Changes that cause sinuous flow from side to side of the channel will produce the same effect. D. Channel Alignment . . Smooth curvature with large radius will give a relatively low value of 71, whereas sharp curvature with severe meandering will increase 71. On the basis of flume tests, Scobey [23] suggested that the value of 71 be increased 0.001 for each 20. degrees of curvature 111 100 ft of channel. . Altholagh. it is doubtful ,..hether curvatllre ever increases 11, mOfe than 0.002 or 0.003, its effect should not be ignored, for curvature may induce the accumulation of drift and thus indirectly increase the value of n. Generally speaking, the increase of roughness in unlined channels carrying water at low velocities is negligible. An increase of 0.002 in n value would constitute an adequate allowance for curve losses in most flumes containilig pronounced curvatures, whether built of concrete or other materials. The meandering of naturai streams, however, may increase the 71 value as' high as 30 %. E. Silting and Scouring. Generally speaking,.8ilting may change a very irregular channel into a comparatively uniform one and decrease n, 'whereas scouring may do the reverse and increase 71 • . However, the dominant effect of silting will depend on the nature of the material deposited. Uneven deposits such as sand. bars and sand waves 'are

\

1

-1

1

I 1

I

.

\

104

i

UNIFORM FLOW

l'

, channel irregularities and will increase the 'roughness. 1 The amount · uniformity of scouring will depend on the material forming the wetted · perimeter. Thus, a sandy or gravelly bed will be eroded more uniformly than a clay bed. . The deposition of silt eroded from the uplands will tend to even out the irregularities in:a channel dredged through clay, The energy used in eroding and carrying the material in suspension or rolling i~ along the bed will also increase the n value, The effect of scouring is not significant as long as .the erosion on channel bed caused by high velocities is.progressing eyenly and uniformly. F. ObstrucHon. ,The presence of log jams, bridge piers, and the like · tends to increase n. The amount cif increase depends on the nature of the obstructions, theil' size, shape, numbel', and distribution. G. Size and Shapl'J of Channel. There is no definite evidence about the size and shape of a channel as an important factor affecting the value of n. An increase in hydraulic radius may either increase or de.creasen, depending on t,he condition of the channel (Fig. 5-4). H. Stage and Di,scha1'{}e. The n value in most strearas decreases ,vith increase in stage and in' diseharge. When the water is shallow, the irregularities of the. channel bottom are exposed and their effects become pronounced. However, the n value may be large at high if the banks are rough and gra.ssy. and When the discharge is too high, the stream may overflow its a portion of· the flow will be along the flood plain. . The n v3,lue of the flood plains is generally larger than that of the yhannel proper, and its magnitude depends on the surface condition or vegetation. !f the bed and banks of a channel are equally smooth and regular and the bottom. slope is uniform, theyalue of n may remain almost the same at all so a constantn is usually assumed hi the flow computation. happens mostly in artificial channels. On flood plains the value of n usually varies with the stage of submergence of the vegetation at low stages. This cnnbe seen, for example, from Table 5-4, which shows the· n values for various flood stages according to the type of cover and depth TAriu:l 5-4.

n

VALUES OF

FOR VARIOUS' STAGES

m

f

!

!I i

I

I

I

;DEVELOPMENT OF UNIFORM FLOW AND ITS FORMULAS

105

of .inundfltion, as obseryed in the Nishnabotna River, low-a,fof the average gfowing season [24]. It should be noted, however, that vegetation has a marked effect only up to certain stage and that the roughness coefficient can be considered to remain constant for pl'actical purposes in determining overbank flood discharges.

a

50 o

Mississippi River oetweenMemphis and Fulton, Tennessee . o U,S, Geologicol Survey doto .. U.S, Corps of Engineers dato

i!:

~ ~O

.'2

g"

'"

20

;:;:

o

I

O.k02;;;5.----:O':::-:03~O"..----;:O,..,.Oi,.3.",5-.---,;,.,!-,=---=~::,! n value

Tennessee River at Cholloooogc, Tenn,

• 0.045·

n value

FIG. 5-4, Va.riations of the n value with tile.me!tn stage or depth.

THE NI5HNJ..BOTNA RIVEI!.,

IOWA, FOR THE AVlllRAca; GROWING SEl.SON.

Flood-pilli~

Depth of water, it

Channel section

Under 1 1 to 2 2 to 3 3 to 4 Over 4

0.03 0.03 0,03 0,03 0,03

Corn

Pasture

Meadow

i

0,10

0,06 0,06

0.07

I

0,07 0,05

O,O~

0.04 0.04 0,04

cover

0.07

0,06 0.05

Small gt'ains

Brush and waste

0,10 0.09

0,12 0.11 0,10 0.09

0,08 0,07 0,06

O,Oi

I I

!

1

Curves of n v!\lue versus stags (Fig. 5-4) in streams have been given by Lane (25\; showing how vahle of n varies with stage three river channels. For the roughness of large canals, it study in connection with the design of the Panama Canal ,yas made by IVley~r& and .Schultz [26).1 . The two most important conclusions reached from this study were (1) that the n value fol' a river !channel is least when the stage is at 01' somev;rhat above normal bankfull stage, and tends to in~rease for both 1 A tJble of ; vJlues for eleven lar~e cbanneJs at tbe most efficie[lt depths and tlle curVel, showing th.e variations 01 11 value with hydra.ulic radius in eight river channels are also given in this reference.

DEVELOPMJilNT OF UNIFORM FLOW AND ITS. FORMULAS

UNlFORM FLOW

:106

In selecting,. the. value of nl, the degree of irregularity is considered smooth for surfaces comparable to the best attainable for the materials involved; minor for good dredged chm1l1els, slightly ei'oded or scoured side slopes of canals or dra.inage cha.nnels; moderate for fair to peor dredged channels, moderately sloughed or eroded side slopes of canals cr drainage .channels; and severe for badly sloughed banks of natural strea.ms, badly eroded or sloughed sides. of canals or drainage channels, and unshaped, . jagged, and irregular surfaces of channels excavated in rock. III selecting the value ofn2' the character of variations in size and shape of cross section is considered gmdlLal when the change in size or shape occurs gradually, alternating occasionally when lal'ge and small se·::tions alterna.te occasionally or when shape changes r,ause occasional shifting of main flow from side to side, and alternating frequently whim iarge arid small sections alternate frequently or when shape changes cause frequent shifting of main flow from side to side, The selectioll of the value of n3 is based 011 the presence and characteristics of obstructions such as debris deposit", .stumps, exposed roots, boulders, and fallen and lodged logs. One should recall that conditions considered in other steps must not be reevaluated or' double-counted in this selection, In ,iudging the relative effect of obstructions; consider the following: t.he. extent to which the obstructions occupy or reduce the average water area, the character of obstructions (sharp-edged or angular objects induce greater turbulence than curved, smooth-surfaced objects), and the position and spacing of obstructions transversely and longitudi. nally in the .reach under consideration. . In selecting the value of n~, the degree of effect of vegetation is considered' . (1) Low for conditions comparable to the following: (a) dense gl'owths of fla"{ible ~urf grasses Qr weeda,of which Bermuda and blue grasses are examples, where the average depth of flow is 2 to 3 times the height of vegetation, and (b) supple seedling tree switches, such as willow, cottonwood, Or salt cedar where the average depth of flow is 3 to 4: times the height of the vegetation. (2) 1l1edi11.?n for conditions comparable to the following: (a) turf grasses where the average depth of flow is 1 to 2 times the height of vegetation, (b) stemmy grasses, weeds, or tree sCl'ldlings with moderate cover where the average depth of flow is 2 te 3 times the height of vegetation and {c) brushy growths, moderately dense, similar to willows 1to2 ~ean; old, dormant season, along side slopes of a channel with no significant vegetation along the channel bottom, where the hydraulic radius is greater than 2 ft. . '(3) High for conditions comparable to the f;llowing: ea) turf grasses where the average depth of flow is about equal to the height of vegetation,

;higher and lower stages; and (2) that the bankfulLn values do Dot vary . greatly for rivers and canals in different kinds of material and in widely. separated locfl.i;ions. For circular conduits, Camp [27,28] was able 'to show that the n value for 11 conduit flowing partially fuBis greater than that for a full conduit. Using measurements on clean sewer pipe and drain tile, both clay and concrete, from 4 to 12 in. in size, he found an increase of about 24% in the n value at the half-depth (Fig. 6-5).l The n value for the pipe flowing full was found to vary from 0.001:15 to 0.011. Taking an average value of 0.0103, the nvalue at half-depth should be about 0,013. This is identical with the usual design value,which is based largely on measured values in sewers Hov\e'ing partially fulL . I. SeasonoJ Change. Owing to the seasonal growtih of aqus,hicplants, grass, weeds, willow, and trees in the channel or on the banks, the value of n may increase in the growing seasOl]. and diminish in the dormant season. This seasonal change may cause changes in other factors. J, Suspended Material and Bed Load. The suspended material and the bed load, whet.her moving or not moving, would consume energy and cause head loss or increase the apparent channel roughness. All the above factors should be studied I1nd evaluated wit,h respect to conditions regarding type of channel, state of flow, degree of mainten~~ce, and other related considerations. They provide a basis for determmmg the proper value of n for a given problem. As a general guide to judgment it may be accepted that conditions tending to hldtlce turbulence and c~use retardance will increase 1. value and that those tending to reduce turbulence and retardance. will decrease n value. Recognizing several primary £actors affecting the roughness coeffic,el1~, . Cowan [32J developed a procedure for estimating the value of n. By thIS procedure, the value of n may be computed by <

n

=

(no

+

",1

+ n2 + nl + n()m5

(5-12)

where no is a basic n value for a straight, uniform, smooth channel in the natural ~aterial:; involved, nl is 8, value added to no to correct for the effect of surface in'egularities, 'lt2 is a value forvf.tl'iations in shape and size of the channel cross sectiol1, Tt3 is a value for obstructions, n4 is a value for vegetation and flow conditions, and mil is a correction factor fllr meanderiIlg of channeL ' Proper values of. no to n, and m, may be . selected from Table 5-5 according to the given conditions. . I The n/n~ curve was based O.D measurements by Wilcox [29] on 8-in, clay and con~ ~rete sewer pipes a.nd by Yarnell a.nd Woodwa.rd 130] on open-butt-joint concrete . ahd cla.y drain tiles 4. to 12 in. in eiZe. For depths less thaD about O.I5d., the eurve was verified by the data of Johnson [311 forlarge sewer!!, .

., ,j

I !

t

t

~""---'"~--'.".-~.-.-,'~

.

r-

1

,-I

J --\ \'

-\ ,1 ~!

:~'

,:i:; 'r.~.'

107

'.

~~.

I

'I

'\

,

!

4;1

I'

UNIFORM: FLOW

I,

(b) dormant season-willow or cottonwood trees 8 to 10 years old, inter-

I

grown with .some v,l:eds and brush, none of thp- vegetation in foliage, where the hydraulic radius is greater than 2 ft, and (c) growing seasonbushy willoW's about 1 year old intergrown with some weeds in full foliage along side slopes, no signifieant vegetation along channel bottom, where' hydraulic radius is greater than 2 ft. . (4) Very high for conditions comparable to the following: (a) 'turf . grasses where the average depth of fio~ is less than one-half the height of vegetation, (b) growing season-bushy willows about 1 yellr old, intergrown with weeds in full foliagealon.g side slopes, or dtmse growth of eat tails along Ghannel bottom, with any value ofhydratilic radius up to 1001' 15 ft, aIld (c) growiIlg season:-trees intergro,vn with weeds and brush, all in full foliage, with any value of hydraulic radius up to 10 or 15 ft. Ir:. selecting the value of ms, the degree of meandering depends all the ratio of th.e meander length to the straight length bf the channel reach. The meandering isconsidel'ed minor for ratios of 1.0 to 1.2, appreciable for mtios of 1.2 to L5, and severe for ratios of 1.5 and greater. In applying the above method for determining the n value, several things should be noted. The method dQes not consider the effect of suspended and bed loads. The values given in Table 5-5 were developed from a study of some 40 to 50 cases of small ftnd moderate channels. Therefore, the method is qU<jlstionable when applied to large channels whose hydraulic rn,dii exceed, say, 15 ft. The method applies only to unlined natural streams, Rood ways, and drainage channels and shows.il. rninimum value of 0.02 fol' the 11, value of such channels. The minimum value of n in general, however, may be as low as 0.012 in lined channels and as 0.008 in artificiH1 laboratory flumes. 6-9. The Table of Manning's Roughness Coefficient. Table 5-6 gives a list of n values for channels of various kinds. I For each kind of channel the minimum,. nOimal, and maximum values of 11, are ShO\VIl. The normal values for artificial channels given in the table are recommended only for channels with good maintenance. The boldface figures are values generally recommended in design .. For the case in which poor maintenance is expected in the future, values should be increased accordiQ.g to the situation expected. Table 5-6 will be found very usp-Iul as a guide to the quick selection of the 11. value to be used in !\, given problem. A poplllur table of this type was~prepn.red by Horton [34) from an examination of the best available experiments at his time. 2 Table 5-6 is compiled

fi

108

" )

L The minimum value lor Lucite was observed in the Hydra.ulic Engineel'ing La.bol'aLory nt the University or Illinois [33). Such a low n value Inay perhaps be obtained also for smoot\) brass o.nd glass, but no observntions have yet been reported. t A table showing n values and other elements hom 269 observations made on many existinga.rtifiCia.l cha.Ilnels is also given by King [35J.

DEVELOPMENT OF UNIFORM FLOW AND ITS FORMULAS 'fAllLl!l

5-5.

VALUES FOR THE CO]>tl'UTATlON OF THE ROTlGHNESS COE·FFICI.!!lNT

:BY

I,

EQ. (5-12)

Channel conditionB

I

Values'

Earth M.ateriul in vo I.... ed

.,

109

Degree of irregularity

0.020

Rock cut

0.025 nQ

Fine gravel

0.024

Coarse gravel

0.028

Smooth

0.000

Minor

r.,

0.005

Modernte

0.010

Severe

0.020

Gradual Va.ria.tions. of channel cross Alternating oceaSionally section Alternating irequently

0.000

j

n2

O.010~.015

Negligible Relative effect of obstruutions

0.000

Minor 1\...."

. 0.005

O. 01O-{}. 015

..

'11.. ~

0.030-{}.030

.~

0.040-0.080 Low

0.005-0.010

Medium

0.010-0.025

High

0.02S-{}.050

Very high

0.050-0.100

VegeLfLtion

Minor Degree of meandering

Appreciable Severe

1.000 m.

1.150

1.300

I

) 110

UNIFORM FLO>'i

Typ~ of channel and description

I Minimum I Normal

----~~~~~~~~~-I

II

Maximum

. I

CLOSED CONDUITS FLOWING. PaRTLY FULL

A-I. Metal a. Brass, smooth b. Steel 1. Lockbar and welded 2. Riveted and spiral c. Cast iron 1. Coated 2. Uncoated d. Wrought iron 1. Black 2. Galvanized e. Corrugated met.al 1. Sub drain . 2. Storm drain A-2. Nonmp.tal. a. Lucite

b. Glasll c. Cemen't 1. Neat, surface 2. Mortar d. Concrete 1. Culvert, straight and free of debris 2. Culvert with bends, connections, and some debris 3. Finished 4. Sewer wit.h manholes, inlet, etc., strl!-ight 5. Unfinished, steel form 6. Unfinished, smooth wood form 7. Unfinished, rough wood form e. Wood 1. Stave 2. Laminated, treated f. Clay 1. Commondrain8.ge tile 2. Vitrified sewer 3. Vitrified.sewer with manhol,~s, illlet, etc. 4. Vitrified subdrain with open joint g. Brickwork 1. Glazed 2 .. Lined with cement mortar h. S'anitary sewers coated with sewage slimes, with bends and connections i. Paved invert, sewer, smooth bottom j. Rubble masonry, cemented

111

DEVELOPMENT OF UNIFORM FLOW AND ITS FORMULAS

. TABLE 5-6. VALUES OF THE ROUGHNESS COEFFICIENT n (Boldface figures are values generally recommended in design)

A.

-(

I

0.009

0.010

0.013

O.OlD

0.013

0.012 0.016

0.014 . 0.017 .

I

0.010 0.011

0.013 0.014

0.014 0.016

I

0.012 0.013

0.014 0.016

0,015 0.017

0.017 0.021

0.019 0.024

0.021 0.030

0.003 0.009

0.009 0.010

0.Of3 .

0.011

0.011 0.013

0.013 0.015

0.010 0.011

0.011 0.013

0.013 0.014

O.OlD

O.OlD

I ! f

iI

0.011 0.013

0.012 0.015

0.. 014 1).017

0.012 0.012 0.015

0.013 0.014 0.017

0.014 0.016 0.020

O.OlD

0.015

0.012 0.017

0.014 0.020

0.011 0.011 0.013

0.013, 0.014 Q.015

0.017 0.017 0.017

0.014

0.016

0.018

0.011 0.012 0.012

0.013 0.015 0.013

0.015 0.017 0.016

0.016 0.018

0.019 0.025

0.020 0.030

J. f

!

. TABLE 5-6. VALUES OF THE ROUGHNESS COEFFICIENT n

Type of channel and description

-----D.

Minimum

(continued)

Normal Maximum

1

LINED OR .BUILT-UP CHANNE1.S

. B-l. Meta! a. Smooth steel surface 1. Unpainted 2. Painted b. Corrugated B-2. Nonmet.al a. Cement 1. N el1t, surf!lce 2. Morta-r. b. Wood L Planed, untreated 2. Planed, creosoted 3. Unplan(ld 4. Plank with battens 5. Lined with roofing paper c. Concrete 1. Trowel finish 2. Float finish 3. Finished, with gravel on bottom 4. Unfiliished . 5. Guuite, good section 6. Gunit8, wavy section 7. On good excavated rock 8. On irregular excavated rock d. Concr'lte bottom float finillhed with sides of 1. Dressed stone in mortar 2. Random stone in mortar 3. Cp,ment rubhle masonry, plastered 4. Cement rubble maSQnry 5. Dry l'Ubble or riprap e, Gravel bottom with sides of 1. Formed cOllCrete 2. Random stone in mortar 3. Dry rubble or riprap f. Brick 1. Glazed 2. In cement mortar 11. Masonry 1. Cemented I'Ub ble 2. Dry rubble h. Dresse.d ashlar i. Asphalt 1. Smooth 2: Rough . j. Vegetal lining

0.011 0.012 0.021

0.012 0,013 0.025

0.014 0.017 0.030

0.010 0.011

0.011 0.013

0.013 0.015

O.OlD

0.012 0.012 0.013 0.015 0.014

0.014 0.015 0.015 0.018 0.017

0.013 0.015 0.017 0.017 0.019 0.022

0.015 0.016 0.020 0.020 0.023 0.025

0.011 0.011 0 .. 012 0.010 0.011 0.013 0.01.5 0.014 0.016 0.018 0.017 0.022

0.027

0..015 0.017 0.015 0.020 0.020

0.017 0.020 0.020 0.025 0.030

0.020 0.024 0.024 0.030 0.035

0.017 0.020 0.023

0.020 0.023 0.033

0.025 0.026 0.036

0.011 0.012

0.013 0.016

0.015 0.018

0.017 0.023 0.013

0.025 0.032 0.015

0'.030 0.03.5 0.017

0.013 0.015 0.030

0.013 0.016

O.OZI)

J

'-,

0.500

·fI

, .·'h·

I:

I

.

t,

:.::."

,.,,,""::".:

~; I

112

UNIFORM FLOW

DEVELOPMENT OF UNIFORM FLOW' AND ITS FORMULAS

'l"\13LE. 5-6: VALUES OF TliE Ro'l1GllNESS COEFFlCllilNT Type 01 channel and description

i'

I

·Minimum

C. Exe.WATIilD OR DREDGED a. Ea.rth, straight and uniform .1. Clean, recently completed 0.016 2. Clean, e.fter weathering 0.018 3. Gravel, uniform section, clean 0022 4. With short gr!l.8S, f.ew weeds 0.022 b. Earth, winding and sluggish 1. No vegetation 0.023 2. Grass, some wGeds 0.025 3. Dense weeds or aquatic plants in 0.030 deep channels 4. Earth bottom lind rubble sides 0.028 5. Stony bottom and weedy banks 0.025 6. Cobble bottom nnd clea.n sides 0.030 c. Dritgline-eltcavated or dredged 1. No vegetation 0.025 2. Light brush on banks 0.035 d. Rock cuts 1. Smooth and uniform 0.025 2. Jagged and irregular 0.035 e. Channels not maintained, wc.eds and' brush uncut . l.Dense weeds, high as flow dept.~ 0.050 : 2. Clean bottom, brush on sides 0.040 3. Same, highest stage of flow 0.045 4. Dense brush, high stage 0.080 D. NATuML S'IREAMS' D-l. l\llnnr streams (top width at flood stage <100 it) a. Streams on plain Clean, straight, fun stage, no rifts or 0.025 deep pools I 2. Same as ..bove, but more stones and 0'.030 I weeds 3. Clean, winding, some pools and 0.033 shoals . 0.035 . 4. Same ail above, but llome weeds a.nd stones . , 1 5. Same as above, lower stages, more 0.040 ineffective slopes and sections 6. Same as 4, but more stones 0.045 7. Siuggish reaches, weedy, deep pools 0.050 13. Very weedy rea.ches, deep pools, or 0.075 ftoodways with heavy stand of tim- 1 ber a.nd underbrush

.

n (continued) Normal Maximum

I

I:

I I

I II

1·

I J

I

! I

I'

0.018 0.022 0.025 0.027

0.020 0.025 0.030 0.033

0,025 0.030 0.035

0.030 0.033 0.040 .

0.030 0.035 0.040

0.035 0.040

0.028 0.050

0.033 0.060

0.035 0.040

0.040 0.050

0.080 0.050 0.071iJ 0.100

0,120 0.080 0.110 0.1<10

0.030

0.033

0.035

0.040

0.040

0;04:>

0.045

0.050

0.0'48

0.055 .

0.050 0.010 0,100

0.060 0.080 0.150

0~050

TABLE5c6. V,u,UES

OF THE ROUCnl'NESS COE)"FICIENT 7i

ll3

(continued)

description ~inimum Normal Maximum' --~--------------~---I·----~I b. Mountain strea.ms, ·no vegetation in cha.nnel, ba.nks usually steep, trees and brush along banks submerged· at high sta.ges . 1. Bo~tom: gra.vels, cobbles, and fetv 0.030 0.040 0.050 boulders 2. Bottom: cobbles with large boulders 0.040 0.050 0.070 D-2. Flood plains 1.00 it). The 11. vallie is Jess than that for minor streams of similar ,description, becll.use banks offer less effective resistance. a. Regular section with no boulders or /1 0.0215 0.060 brush

I

____b_._I_f_re_g_u_la_r_a_:n_d_ro_U_g""h_s_e_ct_i_on_ _ _ _...i:-_O_,0_3_5_'11___L_0 .100,

114

fl'

UNIFORrk FLOW

from up-to-date information collected from various sources ([34,36,38]' and unpublished data) ; hence it is much broader in scope than the Horton table. 5-10. Illustrations of Channels with Various Roughnesses. Photo-, graphs of a number of typical channels, accompanied by brief descriptions of the channel conditions and the corresponding n values, are shown in Fig. 5-5. These photographs' are collected from different sources and arranged in order of increasing magnitude of the n vslues. They provide a [eneral idea of the appearance of the channels having different n values 8.nd so should facilitate selection of then value for a given channel condition. The n vaJue given for each che-nnel represents approximately the coefficient of roughness when the photograph was taken. The above type of visual aid is also employed by the U.S. Geological Survey. The Survey has made several determinations of channel roughness in streams, mostly in the northwes~ern United S~ates. These include measurements of cross-sectional area.) width, 9.epth, mean velocity, slope, and computation of the roughness coefficient. The reaches were photographed in stereoscopic color, and the photographs have been circulating among the district offices 'of the Survey as !!- guide in evaluating n.

l

I

-l

I

J

I

I

I

I

i

I

1 -
I 1 5-5. Typical channels ~howing different n values. (These photo h ' ft~m [3~J and [38J with the permi,nion of the U.S. Departme~~~f ~ ~~l~!t~:,: ~1~1~rtlj.m~1 rc~ure~ used Jo~ reprodu.ctil]ll purposes were supplied thTUugh th~ courtes; ~, •.rr" C' E' ";0 ey tofr ph,otogrhrkphs 1 10 14 rkncl phol()gl'aph 19, rknd lhrou(lh the courtesy FIG.

duced ~

,,,.

•

. ,.am.!er or l!e 01 !Jrs.)

,

1

-'

1

.) 't '. : .~

115

,-1

~l' !I I

I

t

1 i j I

(1)

!

(4)

)

1i I i

I! !

t

\'

(

I ;

,

I

.\

~ii)

!

I

I

I

(6)

FIG. 5.-5 (1-3) 1. n = 0.012. Canal lined with concrete slabs havir;lg smooth neat cement joints and very smooth surface hand-troweled a.nd with cement wash on concrete ba.!Ie. 2. n == 0.014. Concr~te canal poured behind s~reeding and. smoothing platf?rm. 3. n _ 0.016. Small concrete-lined ditch, straight ~nd u;Uf~rm, bottom slightly dished, the sides and bottom covered with 0. rough depoSIt, which Increases the n va.lue.

ll6

.

FtG. 5-5 (4-6) 4. n = O.OlB. Shot-concrete lining without smooth trea.tment. Surface covered with fine a.lgae and bottom with drifting sand dunes. S. n = 0.018. Earth channel excavated in a clay loam, with deposit of clelln sand in the middle and slick silty mud near the sides. 6. n = 0.020. 'Concrete lining made in a. tough lava-rock. cut, clean-scoured very Tougb, and deeply pitted. . '

.

117

\

J

(8)

(9)

.

IL :; ,. '\.

~ ~1

'I

I·

7. 1'1 = 8. 1'1 the earth 9. 1'1 =

.

1"10. 5-5 (1:"'9)

.

.0.020. Irrigatiollcanal, straight, in hard-packed smooth sand. of' .0.022.. ~ent-plaster lining applied directly tp the trimmed suria.ce channel. With weeds in broken places and looStl ssnd on bottom. 0.024. Canal excavated in silty clay loam. Slick and hard bed. US

FiG. 5-5 (10-12) 10. n "" O.024.l:!i~c~ lined on both sides a.nd bottom with dry-laid un~hinked rubble. Bottom qUite lrregular, with sca.ttered loose cobbles. 11. 1'1 = 0.026. Canal excavated on hillside, with upper bank mostly of willow roots and lower bank with well-made concrete wall. Bottom co:"ered with coarse gravel. 12. 1'1 = .o.O~8. Cobbl~bottom channel, where there is. insufficient silt in the wa.ter Or too hIgh a velOCIty, preventing formation of a graded smooth bed. . . lW

(13) (16)

(17) (14)

!

I

I

l i

(18)

I,

t (15)

t.

I,

!

!

13. 11 = ().029. Ea.rth c~nal excavated in alluvial silt 'soil, with deposits of sand on. bottom and growth of gi-ass. 14. 11 = 0.030. Canal with large-cobbleswne bed. 15. 11 = 0.035. Natural channel, somewhat irregular side slopes; fa.irly even, clean a.nd regular bottom; in light gray silty cla.y to light tan silt loam; very little varia.tion . in cross section.

12Q

FIG; 5-5 (16-18) 16. 11 ,= 0.040. Rock channel excav!\ted byexpl~sives. . 17. 7L;-= 0.040. Ditch in clay and sar.dy loam; irregular side slopes bottom !Lna cross section; grass on s l o p e s . : :' , 18. 11"" 0.045. Dredge channel, irregular side slopes .and bottom, in bla.ck, waxy clay at}op to yellow clay at bottom, sides covered with small'saplings and brush slight and gradual variations in cross section. ' 121

(22)

(19)

./

I

(20)

(23)

(21) (24)

I

I

I

I ~

.'{1

i

!

Flo. 5-5 (19-21) '19. 11. = 0.050. Dredge channel ~ith very irregul~r side slopes ap.d. bo~tom, in dark-colored waxy clay, with growth of weeds and gra.ss. Slight vanatlOn In shape of cross section for variation iin size. · 20. 11 = 0.060. Ditch in heavy silty clay; irregular side slopes and bottom; practi· cally entire seotion filled with large-size growth 'of troos, principally willows and cottonwoods. Quite uniform cross seotion. 21. 11. = 0.080. Dredge dianne! in black slippery clay and gray silty clay loam, : irregulrtr wide slopes and bottom, covered with dense gro\vth of bushy ~illOWIl, some · in bottom; remainder of both slopes c017ered with weeds and a soatteflng growth of willows and poplars, no' foliage; some silting on bottom... . . . . . 122

I

I i,

.

FIG.

.22. n = 0.110.

5-5 (22--241

.

8~me as (21), but with much foliage and covered for about 40 It

WIth growthresembllllg smart weed. . . 23. n = 0.1~5. Natural channel floodway hi median line sand toj fine clay none sId~ s!oPe:'; fairly e17ell and regular bottom with occasional Ba~ bpttom sl~ughs' l~ariation Hi depth; practically 17irg;in timber, very little undergrowth except occa~ slUnal dense patches or btlllh~d a.~d small trees, some logs and dead fallen trees. . 24. n ... 0.150. Natur:J.1 river In sandy clay soil. Very crooked course, irregular sl~e slopes and uneven bottom. Many roots, trees .and bushes, large logs and otb r drift on bottom; trees continua.Uy falling into ch",rmel due to bank caving. e .

123

f

.;;." Jt

124

DEVELOP!'.{ENT OF UNIFORM FLOW AND ITS FORMULAS

UNIFORM FLOW

PROBLEMS 5-1. Expiain why a uniform flow cannot occur (a) in Il frictionless channel, and (b) in a. horizontal channel. ' , ' 6-2. When Chezy's C determined by the G. Ie formula becomes independent of the slope S, show that the value of R 3.28. Find the corresponding relation between C all.:! Kutter's n. ., li-S. For tbe conditions given in Example 5-1, compute the values of Ba.zin'i! m and Powell's ~. 6-4. Compute the velocity and discharge of flow in a new etu'th canlll having the same shape, si;;e, slope, and depth of flow as the channel given in Example 5-1. Use (a) tile G. K. formula, ,assuming'Kutter's n "" 0.022; (b) the Bazin formula, selecting !l p'roper value of 11>; and (e) the Powell formula, selecting a proper value of •. 5-6. Taking Manning'S n as the given value of Kutter's n, solve E):ample 5-1 by the Manning formula. , ' 5-6. If the coefficient of l'oughness 11 is unknown for the channe.l ill Example 5-1, but a discharge of 2,000 cfs is observed under the given conditions, compute the v!Liues of l{utter's 1~ and Manning's 11. 5-7. From the MllJlning formula (using a consta.nt of 1.486 instead of 1.49 for theoretical a~curacy) and the Chazy formula, determine the reilltion betweeu CMzy's C and Manning's n for the condition de:;ocribed in: Pl·ob. 5-2. This \\'iii show that the G. K. formula and the Manning formula are theoretically identical at the condition when ,Cl,ezy's 'C is independent of the slope S. , 6'-8. Prov~ that the friction factor f in the Darcy-Weisbach formuir.,Eq. (1-4), is related to .Manning's n by f = 116n~/R~~. 1i-9. Run 12-4 of Bll.zin's teats [12J was made on a rectangular plank flume ,fi.±4 ft wide with wooden strips 1 em thick and 2.7 cm wide nailed crosswise on the bottom and ~ides at a spacinll; of 3.7 em center to center strips. This fiume gave, a mean velocity 0[3.33 fps at a flow depth of 1.02 ft and a slope of 0.0015. The t.emperature rending WIl.S 8.5·C. Determine Manning's n, and compute (a) CM.2y'S C, (b) Kutter's n, (c) ,Bazin's m; and, (d) Powell's •. " 6-10. Run 15-·1 of Bazin's tests was the san1e as run 12-4, described in the preceding problem, ex~ept that the spacing of stripa was increaaed to 7.7 ern. 'Using the same disc,harge as that of run '12-4., the depth of flow wn.s found to be 1.33 :it. Determine Mnnning's IL, and compute (a) Chez)"s C, (b) Kutter's 7'" {~} Bllzin's m, and (d) Powell's ..Oomptu'e the values of • obtained from runs 1274 and 15-4 with the height of the strips, and explain I,he effect of roughness in both cases. '6-11. U!iing the Manning formula., construct 0. discharge-rating curvel for the natural chlll\uel section given in Prob. 2-5. The slope ii! 0.0016, and, 11; ;= 0.035, Extend the sides of the channel by ~traight lines at high stages if necessary. '6-12. Tile actual rating curve of the channel section in Prob. 2-5 is described below. Construct a CUTve showing the' variation in Manning's r. with respect to the stllg'l above the datum. '

Discharge, cis

0.3 0.4 0.5

1.'0 2.3 4,.6 7.8 1l.0 15.0 20.0 25.0

'0.6

0.7 0.8 0.9 1.'0 1.25

Stage,

It

Discharge, cfs

1.50 1. 75 2.00 2.25 2.50 3.00

50.0 62.0 75.0 88.0 10'2.0

132.0 164.0 199.0

3.50 4.00

38.0

6-1S: By the Cowan method, estimate the n value for channel 21 of FIg. 5-5. '

!l.

slightly curved rellch ill

REFERENCES Schnack~nberg: Slope discharge formuiae for alluvial streams and rivers, Proceedings, New Zea.land·lnsWt!lion of Enginesrs, vol. 37, pp. 340-409, Wellington, 1951. Discussions, pp. 4.10-4.4.9. IVllJl E. Houk: Calculation of flow in {)pen channels, Miami Conservancy District, ' Technical Report, ·Pl. IV, Dayton. Ohio, 191B. Erik l,indquist: On velocity formulas for open channels and pipes, Transactions of the World Power C01~feTimce, Sedional Meeting, Scand'inavi(l, Stockholm; vol. I, pp. 177-234, 1933. ' Philipp FOfchheimer, "Hydraulik" (" Hydraulics JI), Teubner Verlagsgesellscha.ft, Leipzig e.nd Berlin, pp. 139-163, 1930. Zivko Vladislavlievitch: Apen;u critique sur le~ formules pour la predetermination de Ie. vitesse moyenne de I'ecoulement uniforme (Critical surveyor the formula() for predetermination of mea.n velocity of uniform flow), Transacli(]11)1 of th~ lst Congress, InternationaL Com'mislrion on Irrigalion and Drainage, New Delhi, voL 2, rept. 12, question 2, pp. 405-4.28, 1951. Oornelis. Toehes: Stres.mfiow: Poly-dimensional treatment of vari!).ble factors affecting the velocity in alluvia.l streams and rivers, Proceedings, Inslittttion of Civil Engineers, Lut,do'/1., vol. 4., no. 3, pt. Ill, pp. 900-93S, December, 1955. , G. H. L. He.gen: "Untersuchungen fiber die gleichformige Bewegung des Wassers" ("Research'!S on. Uniform Flow of Water"), Berlin, l~7f.L , ' Clemena Herschel: On the origin of the Chezy formula, J(l,.rnal, Association of Engineering Sooel:ie3, vol. 18, pp. 363-368. Discussion, pp. 368-369, JalluaryJune, '1897. A. Brahms: "Anfangsgriinde dcr Deioh- und Wasserbaukunst" ("Elements of Dam and Hyd!,aulic Engineering"), Aurich, Germa.ny, 1754 and 1757, vol. I,

L E. C. 2. 3. 4. 5.

or

1 It should be noted that the synthetio rating curve thus obtained is very approxima.te, parti~ularly for a natural channel, beoause the n vDJue is a.ctually not !l. constant but a function of th.e depth (see Art. 5-8). '

Stage, ft

125

6.

7. 8.

9.

p.l05.

,

'

10. E. Ge.nguillet llJld W. R. Kutter: Versuch zur Aufstellung einer neuen allegemeinell. Formel fUr die gleichfllrmige Bewegung des Wassers in CaniLlen und Fliissen (An investigation to establish a new general formula for uniform Bow of water in canals a.nd rivers), Zeitsclirift cIes Oeslerreichischel~ Ingenieu.r- u.nd Architekten Vern1les, vol. 21, no. 1, pp. 6-25; no. 2-3, pp. 46-59, Vienna, 1869. Published as e. book in. Bern, Switzerland, 1877; translated into English by Rudolph Hering a.nd John C.

126

UNIFORM FLOW

Trautwine; Jr., 1.\5 "A general Formula for the Uniform Flow of Water in Rivers and Other Glumnais," John 'Wiley &; Sons, Inc., Nevv York, 1st ed., 1888; 2d ed., 18!H and i\)Ol. 11. C:.\pt:.\in A. A. Humphrey.s and Lieut. H. L. Abbot, U.S. Army Corps of Topographical Engineers: "Report upon thB physics and hydraulics of the Mississippi River; U!Jon the protection of the alluvi!>l region a,gainst overflow; and upon the deepening. of the mOllths; based upon surveys and investigations . . . ," J. E·, Lippincott Company, Philadelphia, 1861; reprinted in Washington, D.C., in 18157, and as U.B. Army Corps (Jf Ellllineel's, PtofC8sionai Pap.,. No. 13, 1875. 12. H. DI.\!'cy a.nd H. Ballin: "Recherches hydra.uliques,'· lre partie, Recherches experimenta.les sur l'ecoulement de Feau dans ies canaux decouverts; 2e partie, Recherches experiment&les rel().tives aux ramous et il. In. propaga.tion des ,andes ("Hydralllic Resea.rches," pt. 1, 'Experimental research on flmv of water in open channels; p~. 2, Experimenta.l research on 'bMkwater and the propagation of 'wave:;), ACMiemie des Sciences, Paris, 1865. 13. H. Bnzin: li:tude d'one nouvelle fonnuls pour ca.lculer Ie debit des canaux deoollverts (A llllW formula.' for the calcula.tion of discharge in open channels), Mtmoire No. 41, Allnal6~ deB ponti I~t cha!,~~ees, vol. 14, seT. 7, 4me trimestrs, pp. 20-70, 1897. . 14. Ralph W. Powell: Resi~tance to flow in l'ough cha.nnels, 'l',u,naadion'5, American Geophysical Union, vol. 31, [lO, 4, pp. 575-582, August, 1950. 1.5. Hobert Manning: On the flow of water in open channels and pipes, Transactions, ·I1/.~titution of Civil Ellgine(Jl'8 'of Ireland, vol. 20, pp. 161-207, Dublin, lSIH; supplement, voL 24, pp. 179-207, 1895, 15. Ven Te Chow: A note on the Manning formula, Transadidns, American Geophysical U'~ion, vol. 36, no. 4, p. 688, August, 1955. 17. Allen J. C. OUlmingham: Recent bydrl'luIic experiments, Proceedings, Institution . of Civil Engineel's, London, voL 7l, pp. 1-36, 1883. . 18. Ph. Gauckler: Du 1l10UVeme[lt d~ l'eau da.::ls les ()onduites (The flow of water conduits), Annates de~ pOJ~ts .1 chriu.ssees, vol. 15, ser. 4, pro 229-281, 1868. 10. A. St.ridder: J3eitriige zur Frage der Geschwindigkeitsformr.i und der Rauhi,gkeitsia.hlen fur Strome, !{a,nii.le und ge.schbssene Leitungen (Some contributions to the problem of velocity fonnul" and roughness c(lI;lfficient for rivers, canals, and dosed conduits). Jfillcil!l.nge'l des eidgen6:isischen Amteofiir }Vasserwirtachajt, Bern, Swit~erland, no. 16,1923. 20. Thomas Blench: A new theory of turbulent fto,v in liquids of small viscosity, Journal, b'stittlt'ion of einil Engineer., Lolkion, vol. 11, no. 6, pp. 611-612, April, • I~~ . 21. N. N. Pavlov'skiI: "GidravlkheskiI Spravochnik" ("Handbook of Hydraulics"). This book ha.~ many editions: (1) "Giclrl\vlicheskiI Spravochnilc," Put, Letlingrad, 1924, 192 pp.; (2) "Uchebnyl GidravlicheskiI Spravoclwik'" (for schools),.!~ubuch, 'Leningrad, lng, 100 pp.; 2d ed, W3l, 168 pp.; (3) "Gidrnvlicheskii Spravochnik," Onti, Leningrad and Moacow, H}37, 890 pp; !lnd (4) "Kratki! GidravlicheskiI . Sprflvochnik," (concise version), Gosstrolizd'at, Leningrad and Moscow, 1940, . 314 pp. , 22. George W; Pickets: Ru·n-off investigat.ions in central IllinQis, University of Illinois, Engineering Experim.ent Sta.iion, Bulletin 232, ,vol. 29, no. 3, September, 1931. 23. Frederick C. Scobey: The flow of water in flumes, U.S. Departm.ent of ;Agriculture, Technical Bulletin No. 393, December, 1933. 24. Methodology for cmp and pasture inundation damage appraisal: "Training manual for hydrologists on watershed protection and flood preventiQn work pIau

DEVELOPMENT OF UNIFOIUI{ FLOW A~m ITS FORM (JLAS

127

parties," prelimi~ary ~raft, U.S. Soil Conservation Sel'vice, Milwaukee, Wis., 1954. 25. E. W. Lane: DISCUSSIon on Slope discharge formulae for alluvial streams a.nd by E. C. Schnackenberg, Pr(JcllJldin(J~, New Zealand 1~lItitu.l.ion of Enqillee!'s vol. 37, pp. 4.35-438, Wellington, 1951. '. ) 25. J•. S. ~ey~!,~ :'I,nd E. A. Schultz: P~na;Ha Canal: The sea-level pI'oject, in A symposIUm. Tlda.l currents, Tran,~act!OnB: Ametican Society of Civil Engineers voL 114, pp. 668-571, ~g49. ' 27. Thomas R. Camp: Design of sewers to facilitate flow, Sewage 'Work.:; Joumal, voL 18, pp. I-HI, .January-December, 194.6. 28. 1i"lomas It
1

..~

. r

COMP!UTATION OF UNIFORM FLOW

(6-6), this factof may be expressed as

CHAPTER·

6

6-1. The Conveyance of a Channel Section. The discharge of uniform flow in a channel may be expressed ns the product of the veloc:ity, represented by EC[. (5-1), and t.he water area, or Q = VA = CAR-S· = KS"

(6-1) (6-2)

where The term J{ is known as the comeY(1,nce of the channel section; it is a measure of the carrying capacity of the channel seDtion, since it is directiy proportional,to Q . . . . . I· When either the Ch~zy formula or the Mmmmg formult\ IS used D.s t"le. uniform-flolv formula, i.e., when y = 72, the discharge by Eq. (6-1) becomes

Q= K',IS

(6-8)

and, from Eq. (6-4),

COMPUTATION PF.UNIFORM FLOW

(6-3)

Primarily, Eq. (1)-8) applies to a channel s~ction when the flow is uniform. The right side of the equation contains the values of n, Q, and S; but the left side depends only on the geometry of the water area. Therefor'}, it shows that, fora giv~n condition of n,Q, and S, there is only one possible depth for maintaining a uniform flow, provided that the value of AR~i always increases with increase in depth, which is true in most. cases. .This Q~h is the normal depth. Wyen nand S are known at . a .()1~~!l!!!)1 section, it can 'be seeii1l1ilXl.f1Ji}]6-8) that there can beomyr!ne discharge for maintainin a unjform flow through th~ sectlOD.~provlaect ..-:Ui8:tARW'a ways increases withincrease of depth.l ThTs discharge 18 . 'tfi'e-normal d~8~'i.a.r.gr----·· .,._ ....,-.--.. :-_._-.... ,._-,-.' ...------.--:.-

''Equation'T6-8) is avery useful tool for the computation and analysis of uniform flow. When the discharge, slope, and roughness are known, this equation gives the section factor A"R"H and hence the norml\.l depth Yn' . On the other hand, when n, S, and the· depth, hence the section factor, are given, the normal discharge Q" can be computed from this equation in the following form: .

and the conveyance is l(

=

Q.yS

129

~

Q = 1.49 AR%

(6-4)

. n

v'S

(6-9)

.

'

where C' is Chezy's resistance facto::. Similarly, when the Manning formula is used, (6-6) K = 1.49 AR~ n

Thisis essentially thc product of the water area and the velocity defined by the Manning formula. The).'UP-.?cript ~_g,--.§ometill!!lL1L~.d t
The above two equations are used to compute the Conveyance when the geometry of the water area and t1')e resistanc~ factor or rou~hlless coefficient are :given. Since the Manning formula IS used extensively, mos~ of the folldwing discussions and computations will be based on:Eq. (6-6). 6-2. Th~ Section Factor for Uniform-flow Computati<:n. '11h: ex~re: sion AR~i is called the section factor for uniform-flow. eomputatwn; It IS an important element in the computation of uniform flow. From Eq.

This is true for channels in which the v'alue of AR3, al'ways increasllII. with increase of depth. since Eq. (6-8) will give one value of AR~,. which in turn gives only one depth. In the CMe of a closed conduit having a gradua.ily closing top, the va.lue of Ami will first increase wit~ depth and then decrease with depth when the full depth is approached, becaUse a n;taximum value of AR3i usually occurs in such a conduit at a depth slightly less than :the full depth. Consequentlj, it is possible to have two depths for the 'lame value pf ARH, one greater and the either le5J3 than the depth for the maximum value of Ami. For further discussion onthie subject see Art. 6-4.

This equation can Qe used to compute the convey!mce when the discharge and slope of the channel are given. . When the Chezy formula is used, Eq. (6-2) becomes K

=

CAR~~

(5-5)

I

.

1~8

1

.

COMPUTATION OF UNIFORM FLOW

131

6-3. The Hydraulic Exponent for UniformAiow Computation. Since the conveyance !{ is function of the depth of flow y, it may be assumed that

a

(6-10J

where C is a coefficient and lV is a pn.rameter called the hydra.ulic l3:cponent for uniform~jlowcompulalton. Ta.king logari~hlTIs on both side.$ of Eq. (6-10) and then differentiating with respect to V, d(ln l()

N

dy

2y

(6-11)

Now, taking logarithms oil both sides of Eq. (6-6), K lA9AR~~/n, alld then differentiating this equation ,... ith respect toy under the assumption that n is independent of y, .

..; ...

..s... . Cl,

~

d(1nI{) dy

= .!. dA: A dy

+~!

dR 3 R dy

(6-12)

0

d '0 c; Q

~

co

.0

Since dAjd'b' = T and R

....'"

~

~(111 K) =

hO

~=

dy

.S

?- s

-

N .:~ (5T - 2R ~:)

...

,..; I

<0

ci

I!i;

8

This equation indicntes that the value of N for the.tmpezoidal section is a function of z and y/b. For values of z = 0, 0.5, 1.0, 1.5, 2.0,.2.5, 3.0, and 4.0, a family of curves for N versus y/b can be constructed (Fig. 0,,2). 2 These curves indic;lte that the value of N varies within a range of 2.0 to 5.3. The curve for a circula.r section with N plotted against y/d Q, where do is the diameter, is also shown in Fig. 6-2. This curve shows that the

o

j

lI

I

I This equation (IJ was "Iso developed independently by Chuga.ev [2) through the use of the Chez.y lormula. . S Similar curves to tho'le in Fig. 6"2 for trapezoidal channels were constructed by .Kirpich [3J and al~o prepared iudependently by Pa.vlovskil [4] alJd R~khml\noff (5).

.

N.

d

130

1 l>.

'A

- - -~-·--:---~~~----~-·-'------~~"--., ...-..-,-"..---.c~---.~=...~-=-,_....-: ....=_ .. _... '-" .-.~~-----. ~

-,.~-.,-.~.'

':"

(6-14)

This is the general equntion for the hydraulic exponent N. For a . trapezoidal channel section having a bottom width b (l.lld side slopes 1 on z, the expressions for A., T, P,. and R may be obf,aind frorn Table 2-1. Substituting them in Eq. (6-14) and simplifying, the resulting equatioll1 is 10 1 + 2z(y/b) 8 (y/b) N (6-15) '3 1 + z(y/b) --3' 1 + 2 ~1:=::+=z=~-(y-/-b)

!:l

<3

0p/" puo q/" }OsanlOll

(6-13)

Equating the right sides of Eqs. (6-11) and (0-13) and solving for N,

:>

0

2R

'0

l<J

0

0

(5T _ dyd!.)

.,.

"CJ

.,'"

d

J.-

3A

'"tt:
I

. A/ P, the above equation becomes

.""-~-.-"--"

132

UNIFORM FLOW

value of N. decreases rapidly as the depth of flow approaches the top' of the channel. Further mathematical analysis has reveo.led that the value of N will be equal t~ zem at Vida = 0.938 and will then become negative at greater depths. The significance of this fact will be discussed \ later. in this article and the next. 6.0

5.0 >.4--1-_-1---1_-'- -.'---'-4.0 I-+--/----J 3,0

and abscissa are interchanged. If a constant n value is assumed, Eq. (6-6) indicat~ that K IX he~ce, the~e curves for AR~~ should show the same characteristics as if the curves were plotted for Ie From (6-10), it can. be seen that the hydraulic exponent for the straightline range of the plot is equal to twice the slope of the plotted straight line. Thus, if any' two points with coordinates (K1,Yl) and (K 2;Y2) are 150r,--,---",-,-,. , - - . , - - . , - . , - " taken' from the straight line, the ap100 proximata value N may be computed by the following equation: 50

or

I~+-'-----J

2.0

N' -1

§

0.3

::: "

0.2

~

'J

~

log (!(dK~) log (Yl/V2)

i

III

0.<1

<;

.-

(6-16)

+ I

1,0 0.8

.0.6 0.5

133

COMPUTATION OF UNIFORM FLOW

I

{)'•• Kll

I

./ The cutved' plot when 'h. depth opprooches

1 Ille

I I

~I

/

l

crOllln

K

ot 0

closed conduit

0,;

0.06 0.06

0,q5 0.04

log Y

0.0 3

I .1,

0,02 2.0

Fw. 8-3. Graphica! determination of N by

logarithmic plotting. 2.5

4,0

4.5

5.0

5.5

VOlues of N

FIG. 6-2. Curves of N v.alues.

Fot channel sections other than the rectangular, tnpezoidal, nnd 'circular shapes, exact values of N filly be computed directly by Eq. (6-14), provided that the derivative dP/dy can be eVllluated. For most chan-. 'nels, except for channels with abrupt changes in cross-sectional form and for closed conduits with gradually closing top, a logarithmic plot of K as ordinate o,gainst tIle depth as abscissaJFig, 6-3) willappear approximately as a straight line. This can also be seen from tll!') dimensionless curves for ARH in Fig. 6-1, which are plotted similarly except that the ordinate

':I

Fro. 6-4. Typicltl channel sections having appreciable variation ill N value with respect ro depth. (After R. R.

C}~ugaev

[2].)

When the cross section of a channel changes abruptly with respect to depth, the hydraulic exponent will ohange accordingly .. Several typical sections are shown in 6-4. In such cases the logarithmic plot of N against y rna.y appear as a broken line or an evident curve, For the nearly straight portions of the broken line or curve) the hydraulic exponents may be D.Ssumed constant. When the depth of flow approaches the gradually closing crOWl1 of a closed conduit, the logarithmic plot will appear as a curve. The hydraulic. exponent in the range of the curved plot is equa,l to twice the slope of the tangen't; to the curve at the given depth (Fig. 6-3). For practiaalpurposes,

i34

I

! i

i,

J.

i

I I· (

UNIFORM FLOW

the curve may' be divided into 11 number of short segments', arid eachsegment may be considered a straight line having,a constant slope orhydraulie exponent. N ow take the circular section as an example, The dimensionl€&s log:l.rithniie plot of ARH against depth is shown in Fig. 6-1. Assuming a const,ant '('Llue of n, this curve will show the same characteristics as if the depth were plotted against K. As the depth' increases, the curve deviates gradually from a straight line and finally reaches a prononnced curvature at Yldo = 0.938, where the value of A RH/dij~'J is a maximum. Since the n value is assumed constant, t,his ratio yl d Q = 0.938' also corresponds to the maximum value of the conveyance K. The slope of the tangent to the curve at this depth, according to the gl'aph in which the ordinate and abscissa a,re interchanged, is hQrizontal, and thus the hydmulic exponent N is equal to zero. For depths with ratio greater than 'lild o = 0.938, the curve shows a decrease in t.he value 'Of A RH!do'~ and, hence, a decrease in the conveyance K if r:. is ai$Bumed constant. The slope of the tangent to the. curve and with it the hydraulic exponent will thus beconie negative. 6-4. Flow Characteristics in a Closed Conduit with Ope-.l~channel Flow. Taking the circular section as an example, the dhnenslol1less curves for ARHI AoRoH and R 3i l Roy" are shown by the full lines in Fig. 6-5. The subscript zero indicates the full-flow condition. 'If the n value is assumed constant or independent of the .depth variation, these" two curves will represent the variation of the ratios of the discharge and velocity to their corresponding full-flow values (Le., QI Qoand YITT 0)' Both the discharge and velocity curves show maximum values, which occur at about Q,938d o and 0.81d o respectively. Mathematically, the depth for the maximum discharge, or O.93ado) can be obtained oimply by equat.ing to zero the first derivative' of ARH with respect to y, since the discharge computed by the Manning formula is proportional to ARH for consta,nt nand S. Similarly, as the velocity by t,he Manning formula is proportional to R*, the depth for the maximum velocity, or 0.81d o, can be obtained by equating the first oerivative of R"" to zero. Flu;thermore, the dimensionless curve of QIQo shows t.hat, when the depth is greater than about 0.82do, it is possible to have t'l'IO different depths for the same discharge, one !J.bove and one below the value of O.938d o. Similarly, the cut've of V /1'0 shmvs that, when the depth is gn:&ter than the half-depth, it is possible to have two different depths for the same velocity, one above . and one below the value of O.81do. The above discussion is based on the assumption that the roughness . coefficient remains constant as the depth che.nges. Actually, the value of n for average clean sewer· pipes and drain tiles, both. clay and concrete,

135

COMPUTA.:!'ION 'OF UNIFORM FLOW

for example, hilS been shown to increase by as much as 28% from 1.00d n to O.25dc, where it appears to be a maximum (see Fig. 6-5 and the disas a factor affectiIlg n value). cussion in Art, 5-8 regarding t.he This effect callses the actual maximum discharge and velocity to occur . at depths of about O.97d o and O.94do, respectively. The corresponding. curves of Q/Qo and ViTr Q are shown by the dashed lines in 6-5. According to the assumption of constant n value, the velocity would be the same for a half-full pipe ru; for a full pipe; whereas, if the n value is . Subscri:>t "d' indicolas the lull tlow condition

r----:r:....-.-,---,--.-----r-- . . r--r~-.-__.~._-r--~ I

II 0.6

0.7

0,8

0.9

Votues of 0/0 0 , vivo • AR
FIG. 6-5. P'low characteristics of a circula.r section.

(Alter T. R. Camp, [27J 0/ Chap. 5.)

taken to vary with the depth, ru; shown, the veloc.ity at the half-depth is only 0.8 the full velocity. The discussion for the circular conduit applies also to any l:losed C011duit with a gradually closing top.. The exact depths for maximum discharge and velocity, however, will depend 011 the shape and roughness variation of the specific conduit section. Since the ma.x;imurn discharge and velocity of a closed conduit of gradually closing top do not occur at the full depth, this means that the conduit WillllOt. flow full at the maximum capacity as long as it maintaimr an open-challlHil flow 9n a uniform grade free from obstructions. For praetical purposes, however, it may sometimes be assumed that the maximum discharge of a cir('.ular cOllduit or similar ,closed conduit with gradually closing top does occur at the full I

I

136

UNIF,ORM FLOW

137

COMPU'l'ATION OF UNIFORM FLOW

depth, because the depth for maximum discharge is so close to the top . that there is always.a possibility of slight backwater to increase this depth closer to and eventtu:IUy equal to the full depth. ' . 6-0. Flow in a Channel Section with Composite Roughness. In simple chann81s, the roughness along the wetted perimeter may be distinctly different fl'OJU part to PE1l't of the perimeter,but the mean velocity can stiil be computed by a uniform-fl,uw formula without actually subdividing the section.. For example, a rectangular channel built with a wooden bottom and glass walls must have different n values for the bottom and the walls. In applying the Manning formula to ~uch channels, it is sometimes' necessary to compute an equivalent n value for the entire perimeter and use this equivalent value for the computation of the flow in the whole section. . . For the determination of the equivalent roughness, the water area is divided imnginatively into N parts of which the wetted pel'imeters P 2, ... , PN and the coefficients of roughness nl, n2, • . . ) nN are known. Horton [6]
where ~l' Hi, ... ,R/'f are hydraulic radii of the subdivided areas. simple c~!\.nnel sections, it may be assumed that

For

RI = R2 = . . . = RN = H . Raugnne88 of Ice-covered Channels. When a. channel is covered with the wetted perimeter of the flcrw is greatly increased. . The bottom surface of the ice cover may be either as smooth as a finished concrete sUl1fa:ce or as rough as the natur&'l channel bed v,rhenclrifting ice blooks .exist. Tabb 6-1 gives the 11, values for drcdg;ed channels covered with ice, as proposed by Lotter [13]. .

.lee condition

Velooity of flaw, fps

n Value

1.3-2.0 >2.0

0.01D-0.012 0.014-0.017

Smooth iee: Withol\t drifting iee blookB

With drifting ice blocks

1.3-2.0

0.OI6-0.01E

>2.0

.0.017-0.020 0.023-0.025

Rough ice with drifting ice blocks

Let nand n1 be the roughness coefficients for channds with and without ice cover, respectively. By means of Eqs. (6-17) to (6-19)' it is possible to compute the roughness coefficient 7l-1 of the ke cover. However, the coefficient thus computed may sometimes be a negative ,falue . , which is, of course, unrealistic, . In order t.o develop a realistic approach to the problem, Pavlovskil [14) assumed that the total force resisting the noW' is equal to the sum of the resisting forces due.to the channel bed and' the ice cover. Thus, from Art. 5-4, (5-20)

(6-17)

There are many other assumptions for the determination of all equivalent l'Oughness.Pavlovski! (9] and also Muhlhofer [10] and Einstein and Banks [111 assumed that the total force resisting the flow (that is JCV'"PL; see Art. 5-4) :is equal to the sum of the forces the flo~ developed in the subdivided 3reas.By this assumption, the equivalent roughness coefficient L'3

where the subscript 1 refers to the channel bed and 2 to the ice cover. Since Chezy's C = -vu;jR OJ' J( = W/C2, the above equation b~comes

(Plnl 2

n

,I J

+ P2n2~ + ... + PN'1i~!)H P}i

(6-18)

Lotter [12] assumed that the total discharge 'of the flow is equal to thei sHm of the discharges of the subdivided areas. Thus, the equivalent roughness coefficient .is

PRH'

n = ...-----'-.-

(lH9~

P

C1

i

II

! i

=

P1' P! C1 2 Czl

+

Let: the wetted perimeter P z = aPi or P = PI then 1 a 1 a = C12 C2 2

+

Sin(:e,: by Eq. (5-7), C

+ P2

(1

+ a)P1i (6-22)

t'/n,

1. 49R , (1 a),n2

+

(6-21)

RH!

(6-23)

138

UNIFORM FLOW

COMPUTATION OF UNIFORM FLOW

It is further assumed that the total hydraulic radius R is made up of . two parts: t.he hydraulic radius Rl due to the channel bed and the hydraulic l':1,dius R~ due to the ice cove'r; that is, that R '= Rl + R 2 • Now, let El = Rl/R2 a,nd E2 = nt/nz. Then Eq. (6-23) is reduced to

(1 + a)~2

=

11,2

(1 + ~)H

2

(E22

+ aEif,)

Tt2

VT+li

(a'' (T'

0Z''S)2~ 7

For wide channels, it may be a,ssumed that PI Thus, E22 = ~l~', and

:i

channel section is equal to the total discbirge divided by the total water "1 area. Owing to the differenr.es that exist among the velocities of the subsections, the velocity-distribution coefficients of the whole section are different from those of the subsections. The values of these coefficients may be computed as follows: Let Vt, V2, : • • , vI! be the mean velocities in the subsect,ions; let al .. a2, . . . 1 aN and fJI I fJ~, ... ,fJN be the velocity-distribution coefficients for the corresponding subsections; let .6.A 1 , .6. A 2, • • " .6.AN be

(6-24)

For the condition of maximum discharge, Pavlovskil. postulated that the relation !.LlUOJ;lg R I , R 2, and n is such that dnjd~i = O. Thus, from Eq. (6-24), Ez z =aEI)', a.nd n = -

139

(6-25)* =

P z, that is, that

a = 1.

Tt =

..72 (1 + ~2~l)%

Fw. 6-6. A channel con8isting of one main section and two' side sedions.

(6-27)

the water areas of the corresponding subsections; let K l , [(2, . • '. , KN be the con veyances of the corresponding subsections; let V be the mean velocity of the to~al section; and let A be the total water area. From the continuity equation Emd Eq. (6-3), the following can be written:

The roughness coeffiCient for the ice cover is, therefore, n~ =

(1.68n n -

nl~I)~l

(6-28)

Now let the discharges wit-hand without ice covel' be Q and Ql1 respectively. Then, using the Manning formula and assuming R = RI/2, where Rand R 1' are the hydraulic radii with and without ice cover, respectively, the discharge of an ice-covered channel is Q

nl

= 0.63Ql 11,

VI =

Q = VA

=

(6-29)

=

Eq. (5-7).

Sf~

_ 1(/'1 SH VN -

.6.AN

+ VN dAN

+ ](z + .. + ](N)S~~

0:: IC.r) N ,

,

Sfi

and Incorporating the above expressions with Eqs. (2-4) and (2-5) and simplifying, the velocity-distribution coefficients of the entire section are N

I a

R~/n instead of Eq. (5-7), obtaining

,~ I

f ~

i !

X.I,

. .t .

=

(aNKNJjdA N')

(J

(6-30)

0; [(NY/ A'

l and

1.

1

~-N~--~--~-

1

N

i

"

us~d

1(2

Ll.,A z

= VI .6..4. 1 + Vz dA 2 +

;

(6-26)

It was Belokon [lSj whQ

{Kl

V2 -

I

Channels of Compound Section. The cross section of a channel may be composed of several distinct subsect.ions with each subsection different in roughness from th~ others. 'For example, an alluvial channel subject to seasonal floods generally consists of a main cha.nnel and two side channels (Fig. 6-6). The side channels a.re. usually found to be rougher than the main channel i so the mean velocity in the main channel ia greater than the mean velocities in the side channels. In such a case, the Manriing formula may be applied separately toearih subsection in determining the mean velocity of the subsection. Theni the discharges in the subsections can ,be computed. The totaL discharge is, therefore, equal to the sum of th~se discharges. ' The mean velocity for the whole .. PavlovskiI [11] used the relation. C

[(~ Sf.

.6.A. I

(f3N[(N

2

jdA N )

= -J-'N'-~--~--

(Il(NY/ A I

"

(6-31)

:' ' I

140

UNI~ORM

I

FLOW

Example 6-1. ' Compute the velocity-distribution coefficients at a peak flow in a natural stream channel consisting of a main sectioll and an overflow side section. The data obtained at the peak flow stage are:

A, ft'

Subsection Main section.

. - . - - I · --,3-6-0-1\.

S 5,710

I

{J I ~5 11~0-.-O-3-5~JII-1-.-1-0-'--1-'0-4-

P, ft

1

Side section. .. .. .. .. Sol'u/'i01l:.

J

405

0.040

1. 11

1. 04

The computations are given below.

S"b~"tio"

AA

I-~

R

J. "

I /3K2/I!A

K

..

-I---!

I

",K'/I!.tl'

10' 16-::;4 10'1".93

10"

Main section. 5,360\22523.88.290.035 I. 892 X X X 1.244 X 10' 12.82 X 10' B,56 X 1O" I 3.136 X 10' i g. 76 X 10 8 I 32.49 X 10"

Si~::~~i~~.,:: ~:~~~I~~y~·15:~~ ~:~'~~

I

32.49 X 10'0, (3.136 X 10')'/11,070' = 1.29 9.76 X 10' {J = (3.136 X 10°)'/11,070 =1.10

and

6-6. Determination of the Normal Depth and Velocity. The normal depth and velocity may be computed by a uniform-flow formula. In the following COlnplltations, the M.anning formula is used with threedi.fferent methods of solution.· ./ A. Algebraic Method. For geometl'ically simple channel sections, the un'Hol'm-ftow conditIon may be determined by an algebraic solution, as ' i!lustrated by the following example: Example 6-2. A trapezoidal channel (Fig, 2-2), with b = 20 ft, Z = 2,S. = 0.0016, Ik"ld n = 0.025, carries a discharge of 400 cfs. ComIJ.!!te t~rmal 'depth and velocity. ... Sol,lt/ion 1: The Analytical Approach. The hydraulic raclius and water area of the given section are expressed in terms of the depth y as

, !

R

=

+ y)_ + y vS

y(lO

10

and

A = y(20

+ 2y)

The velocity is

I

, II

'100 ./ y(20 + 2y) ./

200 y(10 y)

+

."

Substituting the given quantities and the ILbove expressions in the Manning formula • Besides the methods described here, there are other methods for the computation of uniform flo~, suc.h as the use of hydraulic tables., Popular tables for this purpose can be found in [16] to [20].

=

1.49 0.025

[Y(lO 10

+ Y)-J~~ O.0016H

+ y V5

7,680 +:1,720y = [y(lO

+ !:::)J2.5

Solving this equation for Y by trial and errol', V. = 3.36 ft. This is the normal depth. The corresponding arelL is An = 89.S ft' alld the normal velocity is V. = 400/89.8 = 4.46 fps. From EXlLmple 4-2, it is known that the critical depth for the same discharge in the channel is 2.15' ft. ,Since the normal depth is greater than the critica! depth, , the flow is subcritical. Soh;/ion 2: The Trial~and-e,.~or Approach. Some engineel's prefer to solve this type of problem by trial and error. Using the given data, the right side of Eq. (6~S) is 1IQ/1.49 ...;s = 167.7. Then, assume a value of y and compute the section fnctor ''AR%: Make several such trials until. the computed value of AR3; is very' closely equal ......t.u-liiJ....7.; Lhen the a.ssumed ii-i~;~ th~-~l~sest trial :e. the.llormaCdepth:-·This tl'ial-and= error computation is shown as follows:

y

0'=

.

'.rT"

,',

[md simplifying,

I

By Eqs. (6-30) a:nd (6-31), the.coefficients are

/71

COMPUTA.TION OF UNIFORM FLOW V.:=: LJ"'L ~ J../l

or

n value

~

of

/r

J

-----1 3.00 3.50 3.30 I 3.35 , 3.36

I

I

A

78.0 94.5 87.7 89.5 89.8

-~II II

2.34 2.65 2.53 2.56 2.56

Ii I

I

RH

Am~

Remarks

1.762 1.915 1.852 J .870 1.870

137.4 181.0 162.6 167.2 158.0

Y too sll~all Y too large

i

I

I

The closest

The normal depth is,therefore" Yn = 3.36 ft.

B. Graphical 111ethod. For channels of complicated cross sel}tion and variable flow conditions, a graphical solution of the problem is found to be convenient.. By this procedure, a curve of y against the section factor A RYJ is first constructed and the value of nQ/IA9 VS is computed, According to Eq. (6-8), it is evident that the normal depth may be found from the V-A R~1 curve where the coordinate of A RH equals the computed value ~f nQ/l:49 VS. When the discharge changes, new values of nQ/1.49 VS are then computed and the eorresponding new normal depths can be found from the same curve. ' Example 6-3. Determine the normal depth of flow in a 36-in. culvert (Example 4-3) laid 011 a slope of 0.0016, having n = '0.,015, and carrying a discharge of 20 cfs. Sollllion. Construct a curve of y vs, Am, for the given culvert (Fig. 6-7). Compute nQ/1.49 V8= O.DlS X 20/1.49 YO.0016 '" 5.04. From the y-Am; curve, find the depth corresponding to the value of 5.04 for AR3!. This depth i~ the required norma.l depth, or y" = 2.16 ft. Since this depth is greater than tlie critical depth determined in Example 4-3 under the same condition, the flow is subcritical. The table in Appendix A for the geometric elements of a circular sect,ion may also be used for the solution of this problem. Since do = 3.0 ft and do" = 18.75, AR,;/ d.}; = 5.04/1S.75 = 0.269. From the table, lI/do = 0.72, or V = 0.72 X 3 = 2.Hl ft.

-_.,

142

UNIFORM FLOW

COMPUTATiON

The design. chart for determining the normal depth (Fig. 6-1) ~all be used with great expediency ... C. Method of Design Chal·t.

In Exalllple 6-2, ARn =< '167.7. The v",[ue of AR~fi/M~ is 0.0569. the uhnrt gives yllJ = 0.168, or 11" = 3.36 ft. In Example 6-3, AR15ldo~1 - 0.269. For this va.lue, the chart or 11 = 0.72 X 3 = 2.16 ft.

I

UNIFORl\I FLOW

143

b. Determine the critic.!tl slope a.nd the corresponding norma.l depth when the dis" charge is 400 cis. ' '

c.D~term!ne the critir.;a.1 slope at the n01'l:11al depth of 3.35 ft, a.nd compute the ~or respondmg dlscharge. So:1ttion. ' . (v.) h From the given data it is found that R = 2.56 it and V '= 446 ' f ps.. Su bstltutmg t ese va.lues; in the Manning formula and .~ol-dng for Sn,

For this value,

yla., =

OF

il.72,

A,

Ad -

1.49 256"'8 I" 0.025 . ~.,,'

s.

0.0016

•••• C'

6-7. Determination of the Normal and Critical When the discharge and roughness are given, the Manning formula can be used to determine the slope of Ll. 'prismatic channel in which the flow is uniform at a given normal depth Yn; The slope thus determined is sometimes culled specifically the normal slope Sn.

or

-

This is the slo~e that will maintain a unifO~'m flow i~ the gh'en channel at a depth oI 3.36 ft and a discharge of 400 cfa (see Example 5.2), b. From the given data t.he critl!:al depth is found to be 2.15 ft (see Example 4-2) The corresp:ndin g. v:J.lues of R ~nd V are R 1.97 f~ and V 7.60 Ips. Substitutin~ the known ~ A.lues m the MnnnlIlg formula and solvmg for Sa, '

1. 97HB.l."

7.66 or

S.

J I ·'l.

!

,I I

0.(1067

t

Th~s is the slope that will maintain a. unifornl and critical flow in the givell cha.nnel for

a dis~harge of 400 cfs. The depth of flow is 2.1& ft. c. For the given normal depth of 3 .•36 it, it is found that R = 2.68 ft, Ii = 89.8 fV D :" 2.~8 ft, and, by Eq. ~I-ll), the critical velocity V. 9.3'fps. Substi~ tutmg tne known values m the Manning formula and solving for 13 m , 1.19' 9.3 = 0.025 2,56~~S",,~ or

o

This is ~he slope that will maintain a uniform and critical flow in the given channel at the glvp,n normal clepth of 3.36 ft. The disoharge is equal to 9.3 X 89.8 = 835 ds.

FlO. 6-7. A curve of !I vs. ARJ~ for n. circula.r section.

Example 6-6. petermine the limit slope of !I. rectangular channel (Fig. 6-8) with 10 ft and n = Q.0l5. .S~ll'tion. .Since the limit slope is the :;mall~.st critical slope, its value' maybe determmed gra.phlc.\J.lly.rro~ a Cll::ve ~f.the critical slope plotted aga.inst discharge. ~or the determmatJOn of a cntlcal slope, the following two cOlldition~ should be satisfied: L The first condition, from Eq. (6-3), is

By varyillg the slope of the c:hannel to a certail1 value, it is to change the normal depth and make the uniform flow occur in a cl'it,ical state for the ghren dischatge and roughness. The slope thus obtained is the critical slope 8., and the corresponding normal depth is equI11 to the critical depth. The smallest critical slope for a channel of shape and roughness is called the l~:1nit slope SL. Furthermore, by adjusting the slope and the discharge, a critical uniform flow may be obtained at the given llorinal depth. ·slope thus obtained is known as the critical slope at the given normal depth So... ,The folloH.'ing examples will illustraie the above discussion. Example 6-4. A trapezoidal channel has a bO,ttoro w'idthof 20 H, side slopes of 2:1, and u = 0.025. a. Determine the normal slope at a normal depth of 3.S6 it whf:)n the discharge is 400 cfs.

S.n = 0.0070

b

=

.Q = K

vB.

(6-32)

or, 'when the Manning formula is used,

(6-33)

or, for the rectangular channel,

I

1

i I.

,t .

(f}-34)

i

'I

:2. The second condition, from Eq. (4-3), is (6-35)

Q =Z,...;g or, for the .ectangulul' chmnel,

Q=

1O....;g y'"

(6-36)

By using Eqs. (il-34) and (6-36) .iLnd eliminating 11, the relation between Q and S, can be established. This relation is expr;lssed, however, as an implicit function, and

a direct solution is mathematically complic!l.ted, A practical solution of the problem i~ to assume different values of ii, substitute 11 in Eq. (6-36) and solve for Q, and then substitute lIll.!ld Q in Eq. (6-34). anti solve for So. Following this procedure, the rel!l.tion between Q and S< was compute.d ..nd plotted as shown ilt Fig, 6-8. The plotted.

Supercrltlcal

flow

2.

Slope in 10-3 (al F!G~

145

COMPUTATION OF UNJl<'ORM FLOW

UN)FORM FLOW

144

:3 4 5 Slape In 10- 3

6

7

. (b)

t>-8. Curves of critica.! slope vs. discharge.

curve MLN indicates a minimum value·of S, O.004Il.t L, which IS the required Iimii slope. ' , . : ' . Assumil1g that the maximum expected depth of flow in the ch$.IlneLis 5 ft, a dlScha.rge cUl:veOM (Fig. can be cOIl1itrueted according to Eq. (6-9). It becomes evident tha.t within the sha.ded area. betweEn the curves OM and MLN, a.ll expected flows wHi b~ subcritical. On the right side of the Cllrves, the flows will be, supercritical. Since t1:,e point L is below the cUr'le 'OM, the limit slope is possible in the exj:lected flLnge of flow. . . . Similll.dy, the maximum expected depth of flow is !\.Ssumed to be 1.5 ft, and the curves ar~shown in Fig. 6-8b. In this case, the point L is above the Cl.lCVe·01l1; there-. fore, the limit slope cannot be expected to occur in the realm under cQnsideratioll.

6-8. Problems of Uniform-flolw COII}putation. The computation of uniform flow may be performed: by the use of two equations; the· continuity i equation and a unifodn-flow. formula. When :the Manning formulli. is used as the uniform-fl6w formula, the computati/;m will involve ." the following six variables:'

1. The normal discharge Q 2. The mean velocity of flow V 3. The normal depth 'Ii 4. The coefficient of roughness n , 5. The channel slope S " B. The geometric elements that depend on the shape of the channel section,such as A, N, etc.

When any four of the above six variab'lesare given, the remaining two unknowns can be determined by the two equations. The following are some types of problems of uniform-ftow computa~ioll: A. To compute the normal discharge. Inpractical applieD.tlons, this comput.ation is required.for the determination of the capacity of a given channel or for the construction of a synthetic rating curve of the channel. B. To: determine the velocity of. flow. This computation has many appli.cations. For example, .itis often required for the study of scouring and silting effects in a given channel. C. To compute the normal depth. This computation is required for the determination of the of flow in a given channel. D. To determine t.he channel roughness. This computation is used to ascerta.in the roughness coefficient in a given channel; the .coefficient thus determined may be used in other similar channels. E. Ta compute the channel slope. This computation is required for adjusting the slope 0(0, given channel. F. To determine the dimensions of the channel section. This computation is required mainly for design purposes . Table 6-2 lists the known and unknown variables involved in each of the six types of problems mentioned above. The kllown variables are indicated by a check mark (v') and the unknowns required in the problem by a question mark (?). The unknmm.val'iables that can be determined The last from .the known variables are indicated by a dash TABLE

Type of problem

t>-2.

Discha.rge Q

A

?

B

v v v' v

c

D E F

SOME TYPES OF PROBLEMS OF UNn'ORM-FLOW COMPUTATiON

VeIoeity V

,

?

-

.-

.-

I De th p 11

v v 1

v v v

Roughness

n

v v v ?

v v

Geometric elements

Slope S

v v v v ? ,

v;

I

v v v 'I. v ?

. Exa.mple

: Prob. 5-5, (Ex, 5-0 Prob. 5-5, (Ex. 5-1-) Example 6-2 Prob.5-6 Example 6-4a Example 7-2

145

UNIFORM FLOW

column of the tabie shows the example given in this book for each type of problem. The examples shown. in parentheses are solved by the. use of the Chl:z;y formula. It should be noted, however, that Table 0-2 does not include all types of problems. By varying combinations of various known and unknown variables, more types of problems can be formed. In design problems, the use of the best hydraulic· section and of empirica.l rules is generally introduced (Art. 7-7) and thUl'J new types of problems are created . . 6l.9. Computation of Flood Discharge. In uniform-flow computation it is understood, theoretically, that the energy slope Sf in the uniform-flow formula is equal to the slope of the longitudinal water-surface profile and also to the slope of the channel bottom (Art. 5-1). In natural streams, however, these three slopes are only approximateiy equal. Owing to inegular channel conditions, the energy line, wB,tel' surfa.ee, and channel bottom cannot be strictly parallel to one another. If ·1;he change in velocity within the channel reach is not appredable, the energy· slope may be taken roughly equal to the bottom 01' t.he surfs.ce slope. On the other hand, if the velocity varies appreciably from one end of the reach. to the other, the ·energy slope should be taken as the difference between the total heads nt the ends of the reach divided by the length of the reach. Since the total head includes the velocity head, which is unknown, a solution by successive approximation is necessary in the discharge computa~ion.· . During flood stages, the velocity var~eS greatly,. and the velocity head should be included in the total head for defining the energy slope. Furthermore, flood flow is in fact varied and unstean'y, and use of a uniformflow formula for discharge computation is acceptable only when the changes in flood stage and dischurge are relatively gradual. The direct use of a uniform-fl.ow formula for the determination of flood discharges is known as the slope-aJ'ea method. The flood discharge may also be determined by another well-kno".,n method called the contractedopening method, in which the principle of energy is applied directly to a contracted opening in the stream. Both methods l require information about the high water marks that are detectable in the flomded reach. Good locations for coll~cting such information may be found not only on main streams but also on smaller tributaries; but they ml.\st be either comparDtjvely regular valley channels free from bends and thus well suited to the slope-area method or else contrll-cted openings wiyh sufficient constriction to produce definite indrease in head and velocity and thus suited t.o the contracted-opening method. The foilowingis a description of the slope-area method. 2 The conI

~

For a comprehensive description of the metholis, see [21]. : It shoul.d be noted tha.t the slope-area method actually deals with gradually var.ied

COMPUTATION OF UNIFORM

:now

147

tracted-opening method is related to rapidly varied flow and, therefore, will be descri I:>ed later, in Art. 17-6. . The Slope·-area Method. The following inform:ttion is necessary for the slope-area methqd: the determination of the energy slope in the channel reach; the measurement of the a vel'age cross-sectional area, and the le.ngth of the reach; and the est.imation of the roughness coefficient applicable to the channel reach, so that frictional losses can be calcu!il.ted. When this information is obtained, the discharge can be computed by a uniformflow formula, such as l'vIa.nlling's. The procedure of computation is as follows: 1. From the known values of A, R, and n, compute the conveyances Ie, andKd,respeetively, of the Epstl'eam and dO\\'llstrealll sec~ion8 of the reach. 2. Compute the average conveyance K of the reach as the geometric mean of 1(u and F:.o,' or (6-37)

3. Assuming zero velocity head, the energy slope is equal to the fall F of water surface in the reach divided by the length L of the reach, or S

F'

=£

(6-38)

The corresponding 'discharge may, therefore, be computed by Eq. (6-3), or Q= K

vB.

(6-3)

which gives the first approximation of the discluirge. 4. Assuming ~he disclul.rge p.qltal to thefil'st approximation, compute the velocity heads at the upstream [1,nd downstream sections, or au V,,2j2g and ad V d Z j2g. The energy slope is, therefore, equal to

s=!!:!. L where

i

j I

I !

I i

J1,

(6-39) (6-40)

and kis a factor.' When the reach is (;ontracting (Vu < V d ), k = 1.0. • When the reach is expanding (V" > V.), k = 0.5. The 50% decreru;e in the value of Ie for an expanding reach is customarily assumed for the recovery of the velocity head due to the expansion of the flow. The corresponding discharge is then computed byEq, (6-3) u.'ling the revised flow, but it is believed that at this stage of reading the rcad~r should be able to follow the procedure.described here. This method shows how the uniform-flow formula can be applied to gradually vllded flow and thus paves the way for a more cOinprehensive treatment on the subject of gradually varied flow in Part Ill.

·l

i

,

,

.~

148

149

UNIFORM FLOW

COMPU'l'ATlON OF UNIFORlI'I FLOW

slope obtained by Eq. (6-39). This gives the secolid approximation of the discharge. 5. Repeat step 4 for ~he third and fourth approximatiOJls, and so on until the assumed and computed discharges agree. 6. Average t.he discharges computed for several reaches, weighting them equally or as circumstances indicaie.

surface flo~ occurs mostly as a result of natural runoff, and is called oo'erland flow. Uniform flow may be turbulent or laminar, depending upon such factors 11..'> discharge, slope, viscosity, and degree of surface roughness. If velocities and depths of flow are relatively small, the viscosity becomes a domina.tiQg factor and the flow is laminar. In this case the Newton's lD.w of viscosity applies. This law expresses the ~~at~.9l1_p~tween the

Example 6-6. Compute the flood discharge through a riv.er reich of 500 it ha.ving· known values of the ws.t"r areas, conveyances, and energy e,oelficient8 of the upstream and downstream end sections. The fall of water surface in the reach was found to be 0.50 ft. Solution. ';rI.e. wl).ter areas, conveyances, and energy coefficients for the two elld sections of the' reach are: ," Au Ad

11,070 10,990

= =

K, = 3.034 X 10 6 K. = 3.103 X 10 6

.J,~,~Ofti~T!~ ~V~~~,~" V?",,-~ ~ ..... 1,;:\.~j··

~"

'r-

.~ f' "'"(i"', r

w(Y"l~'Y)S ( _ • . t~F"'~~101l

=

~

.

au = 1:13" O'd

1

" Ii!

.(

....

.~~ ..,

\

;.:.;

) ' ..,~

,/

··'·1:':··~7

,., AifC

~~1:;;t·

The a.verage K = Y3.034 X 10' X 3.103 X 10' = 3.070 X 10'. FOI' the first approximl,tion, assume h, = 0.50 ft. Then S = 0.50/E,00 = 0.0010, v'S = 0.0310, and Q = K VB = 3.070 X 10' X 0,0316 = 97,000 ds. For the second approximation, assume Q = 97,000 cfs. Then the velocity heads at the t\~O end sections are: .

V,,·._ 1'13 A (97,000/11,070)" =

«. 2(/ ad

.""

2(/

V d ' _ 1177 (97,000/10,990): 2g - . 29'

=

1.354 FHl. fr9. Uniform laminar open-channel flow.

1,42,1

-0.070 .

dynamic viscm:ity i" and the shear stress bOlmdary surface (Fig. 6-9)" as follows:

Since l'u is less than V d, the flo," is contracting, and k = 1.0. Hence, h, = 0.500 0.070 = 0.430, S = 0.430/500 = 0.00086, "IS = 0.0~93, and Q = 3.070 X 10' X 0.0293 = 90,000 ds. Similarly, other approximations are made, as shown in Table 6-3. The estimated discharge is found to bl! 91,000 cfs. TABLE

6-3.

COMPUTATION OF F1.OOD DISCHARGE BY THE SLOPE-AREA METHOD FOP. EXAMPLE

Appr.oximatlon

IAs~umed

1st 2d 3d

4th 5th

Flo. ~

Q

97,000 90,000 91,200 91,000

I

I

2g

a~ ~l 2U

6-6

h,

I

S

v'S 1Computed Q

-~----I-!-----

0.500 . . . . . . , .. \°.50° °.0010000.0316 97,000 0.500 1.354 1.424 0.430 1 0.000860 0.0293 90,000 10.500 1.165 1.~25I0.44010.0d0880 0.0297 .91,200 10.500 1.H.l5 1.258 0,437i 0.000874 0.0296\ 91,000 1 0 . 500 1.190 11.253 1°.437\ 0.000874) 0.0296 91,000

6-10. Uniform Surface Flow.

When water flows across a broad sur-

face, so-called S1J.1"!ace flow is produced. The depth of the flow may be so thin in comparison with the width of flow that the flow becomes a wide-' open-channel flow, known specifically as sheet flow. In a drainage basin

=

r

,.,j

I>

r

at a dist.ance y from the

du dy

(6-41)

For uniform laminar flow, the component of the gravitational force parallel to the flow in any laminar layer is balanced by the frictional force. In other words, thc shear stress T per unit arel;\ of the flow along 'the laminar layer PP (Fig. 6-9) is equal to the effective component of the gravitational force, that is, r = tV(Ym - y),c;. Since the unit weight tV = pg and 1>/ P = v (Art. 1-3), T = gl>(Ym - y)S/v. Thus, from Eq. (6-41), .

dv

=

g8 (Ym - y) dy J'

Integrating and noting that v = 0 w;hen Y = 0, v

=

g8 (YYm _ )I

•

.

t) 2

fr.

J,;.iii:.

i n7 .... , ~, .

'\\

' . . •• . ,

_,(.

.'

1.177

~,'

.

.' 'flop'li' 't~o

(8-42)

This is a quadratic equation indicating that' the velocity of uniform, ·laminar flow in a wide open channel has a parabolic distribution. Inte-

COMPUTATION OF UNIFORM FLOW

UNIFORM FLOW

150

grate Eq. (6-42) from Y average velocity is

=

O. to Y = Ym and divide the result by Yrn; the

V = -1 Y••

10 °' vdy 11

PROBLEMS

(6-43)

'y

a. A rectangular section 20 ft wide b. A triangular section with a. bottom angle equal to GO" c. A trapezoidal sectiC'u with a bottom width of 20 It and side slopes of 1 on 2 d. A cil'cular section 15 ft in diameter . e. A parabolic section having a width of 16 ft at the depth of 4 ft .

(6-44)

where CL = gSj3v, a coefficient involving slope and viscosity. Uniform surface flow becomes turbulent if the surface is rough and if the depth of ftmv is sufficiently large to produce persisting' eddies. In this case the sui'face roughness is a dominating factor, and the velocity can readily be expressed by the Manning formula. Thus, the discharge per unit width is (6-45) where y", is the average depth of flow and where CT = 1.49So·~/n, a coefficient involving slope and roughness, The change of state of sheet flow from laminar to turbulent hn.s been studied by ffi8.ny hydraulicians. The transitional region was found variously at R = 310 by JeffreY
'where C is a coefficient a~d where the exponent highly turbulent. flow and 3.0 for mixed flow.

,,-'~

6-1. Detern:ine the normal discharges in channels ha.ving the following sections for .

= 6 ft, n = 0.015, and S = 0.0020:

0

and the di:;;charge per unit width is

:r ." ..... -,

151

:j:

varies between 1.0 for .

6-2. Prove the following equ1Ltion for the dischuge ina. tl'iangular highway gutter (Fig. 6-10) having one side vertical, one 3ide sloped at 1 pn z'. MallTlilig's n, depth of

I

I I \

FIG. 6-10. A highway gut:ter section. flow y, and longitudinal slope S: (G-48)

where . 6~3. Compute the discharge in the triangular highway gutte!' described in the precedmg problem when z = 24, n = 0.017, Y = 0.22.ft, and S = 0.03. 6-~. U~ing the Manning formula, determine the hydmulic exponent N for the [01lowing channel sections: Ca) a very narl'OW rectan(!;ls, Cb) a very wide rectangle, (c) a Y~ry wide panbola for which the wetted perimeter i~ practically equal to the top Width, and (d) an equilateral triangle with a vertex at the bottom. . 6-5. UBing the Ch6zy formula,' ~how that the general equation for the hydmulic exponent N is .

N

=

1:..

A

(3 T _ U dP) dy

(/H9) .

6-6. Solve Prob. G-4 if the determination of the hydl'!l.)llic exponent is based on the Chezy formula. Compare the results with those obtained in Frob. 614. I The G. K. formula shows that Chezy's C is a f~nction of the hydrauiic radius and hence of the depth ii. Thus, the Chezy formula has hot been found very convenient for determination of the N value. Forcanal~ in earth and gravelly soil the N value is generally found to havBan increase of 0.30 to 0.50 due to the v!l.ri!l.ti~n in Chezy's C with respect to the depth. Thi~ i~crease, however, brings the N valuc closlJr to that bMcd on the Manl1ing formula.

1, (

-\

;;1

'152 .

C01-IPUTATION OF UNIFORM FLOW

UNIFORM FLOW

r

6-7. Compute the l)ydraulic exponent N of the trapezoidal channel section (Fig. 2-2) having a normaldep.h of 6 ft, using (al Eq.(6-15), (b) Fig. 6-2, and (el the graphical method based on Eq. (6-16). 6-11. Compute the hydraulic exponent N of a 36-in. circular conduit having a n~r mal depth of 24 in. above the·invert, usi,ng (al Fig. B-2 and (b) the graphical method based on. Eq. (6-16). ' 6-9. Using the Manning formula, show that the depths for a maXimum discharge and velocity in a aircular conduit are, respectively, 0.938d. and O.Sld..· . 6-10_ On the basis of the Chezy formula, determine the respective depths for maximum discharge and maximum velocity in a circular conduit. . 6-1~_ A~ what depths will the maximl.lin discharge and velocity occur in a square cOlldUlt laId flat on one side? . ' . 6-12. Prepare the curves of dis~harge: andveloeity variations with respect to the depth in ll. square conduit laid on one side. 6-1S. A channel is assumed to have a constant hydraulic radius R for any dep'th of flow. .Prove that the CrOss section of this channel can be defined by y = R[ln (x

+

vx' - R')

II

i

Q.. The cross sect·ion of the stream at 11 typical upstrel1m station in the uniform Tea<:h is given hy the following cQordinates:

I

StrLUr:f(1.

l

Elev. m.s.!.

Sta/·ian

Elev. m.s.l.

590.0 580.7 578.2 582.0 581.0 580.0

6. + 00 8 + 00 10 + 00 11 + 00 12 + 00 14 + 00

543.7 540.0 572.2 573.2 568.5 590.0

Left b8lnk: 0 1 1 3

+ 00 + 00 + 50 +00 4 +00 5 + 00

b. The value of n for the main channel is estimated as 0.035, for the side channels 0.050.. e. The nl1~ural slope of the stream is about 1 it/mile. Construct a synthetilJ re..ting curve. It is suggested that th", wl1ter areas of the '. main channel and the side channels be seplJ.l·ated by the extended sides of the main 1l.S

ch~.nnBl.

- In R]

(6-50)

= R when y = o. Draw the sketch of this section and discuss its properties. (~IN;~: Frrm~ the giv~n condition, R = A(P =dA/dP "'" "dY/Vd:!"' + dy'. Solve

where x

thIS mfferentlill equatlOn, and evaluate the mtegration constant by the condition that x = R when 11 = O. MathematicallY, the section is formed by two catenaries as sides. For practical purposes, an. artificial bottom should he provided ~iilce the .theo~eti.cal section is bottomless. A.uniform-flow fOl'mulfl, such as the Manning formula, lIlfiIcates that the hydrfLlrlic radius is the sole shape parameter for the velocity. The adequacy of this indication can bo verified experimentally hy testin'l' a channel built of tha section oi constant hydraulic radius. If the indication is tru:, then, once this cl~annel is designed for a safe velocity; it should I;>e nonscouring and nonsiit,ing bver a WIde mnge of stages. In .earthen canals, however, the large variation in water surface during the change of stage wouldetode the sides very easily.) . . 6-14. Verify Eqs. (6-17)to (8-19). 6~16. A rectangular testing channel is 2 ft, wid'e and laid on a slope of 0.1035%. When the channel bed and walls were made smooth· by neat cement, the measured normal depth of flow was 1.36 ft fOT a discharge of 8.9 cfs. The same' channel was then roughened by cemented sand grains, and thus the measured norrne,! depth became 1.31 ft for a discharge of 5.2 cfs. a. Determine tile discharge for a normal d~pth of 1.31 it if the bed were roughened and the walls were kept smooth. , . ' . b. Detel'mine the discharge for a normal depth of 1.31 it if the walIs were rOl.]ghened and the bed were smooth. _ .' c. The discll!ir~es for the conditions described-in a and b were actually measured and found to be' G.50 'and 6.20 cis, respectively'. Determine the corresponding n; values, and compare these values with those computed by Eqs. (6-17) to (6-19). ' . 6-16. A chann~l consists of a main section and two side section.s (Fig. 6-6) .. Compute the total disbhfLrge, assuming that the main section and the two side sections are separated Ca) by vertical division lines and (bl by extended sides of the main channel Given: n· = 0.025 for the main channel. n = O.OBOfor the side channels, andS i= 0.001: 6-17. The hydrographic survey of a stream jndicates tha.t the hydraulic p¥operties of the stream llre relatively unifOl'm for a length of over 2' miles. The data 0btained by the survey" are: '

6-18. Compute the discharge in I1n overfliJ~ved highway gutter (Fig. 5-10) having Il. depth of flow of 3 in. and a longitudinRl slope of 0.03. The gutter is made of conprete \vith n = 0.017 and has a t~iangular section with a vertical curb side, a sloped side of z = 12, and a top width of T = 2 ft. The overflowed soil-aggregate pD.vement has a crosg slope of z. = 24 and n, = 0.020. 6-19~ For all equal amount of discharge, an ice-covered .channel should have greater depth of fl(HV than an uncovered channel, lor two reasons: (1) thc wetted perimeter is greater in an ice-covered channel and thus results in greater resistance or less velocity. and (2) the thickness of the ice cover is greater than a depth of wa~er oi equal weight, since the spGCific gra\'ity Qf i~e is about 0.917. Sbow that the increase . in depth due to resistanee in' an iue-covered wide open channel may be expressed by . I:>y

=

[1.31! (nn.)% - 1 ] Y

(6-51 )

where n, is the roughness coefficient of the channel with' ice cover. n is the roughness coeflicient of the channel without ice cover, and y is the depth of flow in the channel carrying the same discharge but without ice cover. . 6-20: Compute the conveyauce and velocity-distribution coefficients of a channel ser.tion 500 ft downstream f!"om the section described in Example !l-l. The survey data at the secti.on 'fol' the same flood are:

Subsection

A., ft'

P, ft

n

'"

(3

5,320

205 408,

0.035 0.040

1.12 1.10

1.05 1.0-1

----------Main section ........ Side section ....

"':1

5,670

6-21. Solve Example 6-2 by the G. K Form.ula, 6-22. A rectangular channel with 20 ft width, S = 0.006, and n = 0.015, carries a discharge of 200 cfs. Compute the normal depth and velocity. 6-23. Using the l\'Ianning formula, determine the norm!,1 depths in channels having the following sections when Q = 100 cfs, n = 0.015, and ,8 =' 0.0020:

UNIFOll.iVI FL01-V

COMPUTA'l'lON OF UNIFOR1I1 FLOW

155

a. A rectangular section 20 H wide D, A tril1ngulltr section with the bottom angle equal to 60~ c, A ~rapezoidal section "'ith a bottom width of 20 ft and side slopes of 1 on Z d, A circuliI' ~cction 15 H in diameter e. A parabolic section having a wicl'th of 16 ft at the depth of 4 ft

REFERENCES

6-24. Solve Example 6-2hy the gr'aphkal method. 6-26. A re.ct'o.ngular channel 20 it wide has a. l'oughnells coefficient n = 0.015. a. Determine the normal slope at a normal "depth of 1.23 i~ when the discharge is 200 cfs. . D. Determine the critical slope o.lIa the corresponding nonna.l depth ",hen the discharge is 200 cis. c. Determine' the critic!,l slope at tbe normal depth of 1.23 ft, and 'lr>mpute the corresponding disch'al'ge. 6-26. Show that the critic!!.l slope at n given norma.l depth !i4 may be expressed by (6-52)

a.nd that

thi~

slope for a wide cluumel is S

= 14.5n> crt.

(6-53)

lIH

·6-27. Deterrnhie the limit slope of the channel described in Exa.mple 6-4, 6-28. Construct the critical-slope curves of the cha.nnel described in Exampie 6-5 for bottom widths b == 1 ft, 4 It, 2() ft, and "'. 6-29. Determine the criti·cal-slope. curvl',s oi the channel desi::ibed in Example 6-4 for side slopes z €I, 0.2, 0.5, 1, 2, 5, and "'. 6-30. A cha.nnel reac)11,OOO ft loug h
Subsection Upstream: Main channel. , . ....... Side channel .... .. , ....... . Downstream: channel ... , . ...... Side channeL. ....... ... ~

~

..

4,250 25,620

2,050

5,760 25,610

l,gOo

320

11.

Of

~

0.038 0.038

1.10

1.0'1

1.20

L08

0.042 0.038

1.10 1.18

1 ..04 1.06

i

I,.

6-:11. Prove Eq. (6-46), 6-32. Using Eqs. (1-5) and (6-43), determine the value of ]( in Eq. (1-8}. 6-33. Compu~e the discharges ·per unit width of a ghect flow on a. surflLc€ with 11 0.01 and AS = 0.035 when the depth of flow is (a) 0.01 ft and (b) 0.004 ft. The temperature of water is 6soF. . .' 6-34. Compare Horton's criteria. for sheet flow in Prob. 6-33 with those shOW\l by . ~he cha'tt of Fig. 1-3. " .... . 6-35. Show that the velocity-distribution coeffici(mts for la.minar uniform flow in wide open channels are a "" 1,.54 and ~ 1.20. . 6-36. Using the Blasius equation (1-6) for turbulent flow in open cha.nnels, ;ahow that the cOI'nsponding exponent, in Eq. (6-47) is:c = 1 77.

i

I !I

I

I"

.

.'!

"> ,,/.-:'

1. Yen Te Chow: Integrating the eqltation of gradually varied flol\', paper 838, ProceediniJs, American ,"!ocietl! of Civii Engineers, voL 81, pp. 1-32, November, . 1955. 2. R. R. Chugaev: NekotorY(l voprosy neravnomemogo dvizheniia vody v otkrytykh prizmaticheskikh ruslakh (Abollt some questions concerning nonuniform flow of w!).ter in open channels), /zv6stiia. l'Besoiuznogo Nauchno-Issierlov(Ltel'skoIlO [nstitllta. aidrotekhniki (Tri.lnsa:clioM, AU-Union Scientific Re6eard~ institl,/e of Hydraulic Enyincerinp), L~ningl'!ld,·vol. 1, pp. 157-227, 1931. 3. Phillip Z. Kirpich: Dimeru;ionless cons~ants for hydr>l.ulic element·s of opellcha.nnel cro~s-sections, Civil Engineering, vol. 18, no, 10, p. 47, October, 1948. . 4. N. N. PavlovskiI: "Gidra.vlicheskii Spravochnik" ("Handbook of Hydmulics"), Onti, Leningra.d and Moscow, 1937, p. 515. 5. A. N. Rakhmanoff: 0 post.roenii krivykh svobodnolpoverlthnosti V. prir-maticheskikh i tsilindriche.~kikh ruslakh pri usta.novivshemsia dyil!henii (On the construction of curves 01 free surfaces in prismatic nnd cylindrical channels with established flow), hvestiia V sesoiuznoyo Naudmo-Is:sledouaf.eI'.kO'lo bl.lititula Gidrotekhllim (1'ranS(lctions, All-Union Scienli.fi.c ResC(Lrcfl InstituteDf Hydraulic Engineering), Leningrad, vol. 3, pp. 75-114, 1ll31. 6. Robert E. Horton; Separate roughness coefficients for channel bottom and sides, Jj7'dl""'.""di News-ltccord, vol. 111, no. 22, pp. 052-653, Nov. 30, 1933. 7, H. A. Einstein; Del' hydraulische odsr ProJil-Radius (The hydraulic (11' cross section radius), Sch",ei$~rische Ba.uzeilv,1!g, ZUrich, vo!' 103, no. 8, pp. 89-91, Feb. 24, 1934. 8. Ahmed M, Yassin: Mean roughness coefficient in open cho.nnels with different, roughness of bed and side walls, Eid(jelll!,~.rische technische H oCMchllle Ziirich, Mitieilungen ruts der Vcnu.chsa.nst(lltflIT Tfr(lsse!'oau una Erd:bau, No. 27, Verlag Leemann J ZUrich, 1954. 9. N.N . .Pavlovskii: K voprOStl 0 raschetnoI formule dlia ravnomernogo dvizheniia y yociotoka.hk s llcodnorodllymi stenkami (On a design formula. for uniform move· men~ in channels with nonhomogeneous walls), Izul1s€iia VsesoiUZllogo Nau.cir.lIoI 8sledovatel' skolJo Instiluta. Gidratekhnikt (T"(LMrtClions, ,111- Unio;~ Scienlifu; Resear~h Inslill1le of Hyd1"altiic Erl9ineeril~g), Leningrad, vol. 3, pp. 157-164, 1931. 10. L. MUhlhofer: Rauhigiteitsuntersuclmngen in einem Stollen mit betonierter Soble und unveddeidete~ Wand en (R(Jughne5S investigations in a shaft with concrete bOUom and unlined wo.lis), Wass8rlcra!t u.na Wasserwir/,sdtafl, Munich, vol. 28, no. 8, pp. 85-83, 1933. . ll. H. A. Einstein and R. B. Banks: Fluid resistance of composite roughness, Trans- . actions, Am.erican Geophysical Union, vol. 31, no. 4, pp. 603-6jO, August, 1950, 12. G. l{. Lott.er: SOQbrazheniia k gidravlicheokomu raschetu cusel s l'!>tlichnoI sherokhovatosliiu stenoI!: (Collsidera.tionson hydraulic design of channels wi~h different roughness of \ir8.1I8), /zu.estiia. V~e~oiu2nogo Nauchno-!ssledvvateJ,'skof/Q Instiluta. ·Gid1'otekhniki (Transa.ctiqna, .till-Union Sde1\tijic Reseprch In.stilutc of Hyd~at;lic En(Jineering), Leningrad, voL 9, pp. 238-241, 1933. . 13. G. It Lotter: Vliianie uslovii ledoobrazovaniia. i tolshchinY l'da naraschct derivatsionnykh ka.nruov (Influence of condi~ions of ice formation and thickness on the design of derivation ca-nlLls), IZTleiltiia. Vsesoiu.znogo Nav.chno-I:ssled()va.t~l'slco(jo. Ins/itu/a Gidrolekhltiki (Tra.nsactions, All-Union Scimtiji.c ReseO;rch 17Ultitute of H1!d~G7j.lic Engineering), Leningrad, vol. 7, pp. 5&-80, 1932. 14. G. Ie LoLter; Metod akademilca N. N. Pavlovskogo dUo. ojJcedeleniia koeflitsienta

I·

i

I

i

156

. UNIFORM FLOW

sherokhovl1tostl rusel·, pokrytyl,h I'dam (Method by Academy Member N. N. Pavlovskil for der.crmination Df roughness coefficients of ice-{lovered channels), I TllVest{ia V seso;wm<Jljo N auchno,J ssledava te/' skOlia I nstitttUJ. Gidr612khniki (T1'Q.nsactiO'flS, All-Union Scientific Research Institute of Hyd1'auiic E1I{)ineering), Lenin.gro.d, no. 29, I94L 15. P. N. Belokon: "Inzhenernaia gidravlika patoka pod i<;dia.nym pokrovom /I ("Engineering Hydraulics of a Current under Ice Cover "), Gosenergoizdat, Moscow IlIld Leningrad, 1940. IG. ·Horace William King, "Manning Formula Tables," vol. II, "Flow in Open Channels," McGraw-Hill Book Compa!lY, Inc., New York, 1039. 17. "Hydraulic and Excavo.tion Tables," U.S. Burea.u of R.eclam:l.tiou, 10th s.d., 1950. 18. "Hydraulic Tables," U.S. Corps of Engineers, U.S. Government Printing Offiee, Washington, D.C., 2d ed., 1944•. 19. Horaoe William King: .. Handbook of Hydraulics," 4th ed., revised by Ernest F, Bra.ter, McGra.w-HiII Book Company, Inc., New York,.1954, sec, 7, table \lD. 20. P. A. Arghyropoulos: "Ce]cu\ de I'ecoulement en conduites sous pression ou a sUl'fll.ce libre, d'upres Ill. fOl'mule de Manning-Strickler" ("Computation of Flow in Conduits under Pressure or with Free Surface, Usiug Ml:lIlning-Strickler Formula"), Dunod, Paris, HISS. ' 21. IViLn E. HOllk: Calculation of flow in open channels, Miami Conserl.'ancy Districl, Technical Report, Pt. I.V, Do.yton, Ohio, H118. 22. H. Jeffreys: Flow of water in an inclined channel of rectangular section, London, Edinburgh and D"blin PIIi!osophical Magazine and Jou.rnal of Science, vol. 49, no. . 293, pp. 793-807, May, 1925, 23. L. Hopf: Turbnlen:1; bei einem Flusse (Turbulence in a flow), Annalen der Physik., Halle aml Leipzig, vo!. 32, sec. 4, pp. 777-808, April-July, 1925. 24. Robert E. Hortun, H. R. Leach, a.nd R. Van Vliet: Laminar sheet ftow, TTansactions, A:merican GeophysicaJ Union, vol. 15, pt. 2, pp, 393-404, 1934.

CHAPTER

7

DESIGN OF CHANNELS FOR UNIFORM FLOW

Channels to be discussed in this chapter include nonerodible channels erodible chann.els, and grassed channels. For erodible channels, the discussion will be limited mostly to those· which scour but do not silt (see Preface). A. NONERODIBLE CHANNELS

7-1. The Nonerodible Channel.' Most Hned channels a!ldbuilt-up channels can withstand eroslonsatisfactorily and are therefore· considered nonerodibZe. Unlined channels are generally erodible, except those excavated in firm foundations, such as rock .bed. . In designing nonerodihle channels, such' factors f:t:;, the maximum permissible velocity (Art. 7-9) and the permissible tractive force (Art. 7-13) are not the criteria to be considered, The designer simply computes t.hedimensions of the channel by a uniform-flow formula and then decides the final dimensions on the basis of hydraulic efficiency, or empirical rule of best pi'acticability, and economy [1,21. The factors .to be considered in the design are: the: kind of mat.erial forming the channel body,which determines the roughness coefficient; the minimum permissible velocity, to avoid deposition if the water carries silt or debris; the channel bottom slope and side slopes; the freeboard i and the most efficient section, either . hydraulically or empil'ically determined. 7-2. Nonerodible Material and Lining. l The nonel'odible materials used to form the lining.of a channel and the body of a built-up channel include concrete, stone masonry, steel, cast iron, timber, glass, plastic, etc. The selection of the material depends mainly on the availability and cost of thf; material, the method of construction, and the purpose for which the channel is to be used. The purpose of lining a channel is in most cases to prevent erosion, but oacasionally it may be' to check seepage losses. In lined channels, the ma:r;imum permissible velocity, i.e" the maximum that will not cause erosion, can be ignored; provided that the water does not catty sand, gravel, or stones. If there are to be very high velocities over a lining, however, it should be remembered that there is a tendency for the rapidly 1

For detailed information on channel lining, aee [3). .

157

./

'~.

UNIFOR\I1 FLOW"

158

DESIGN OF CHANNELS FOR UNIFORM FLOW

159 a.now the pradicable Use of just about any type of lining or lining tl'eatment now or in tho future ant.icipatcd by the Blireau. 7-6. Freeboard. The freeboard of a channel is the vertical distance fi'Oill the top of the channel to the water surface nt the design condition. This distance shouid be sufficient to prevent wnves or fluctuations in water surface from overflowing the sides. This ftwtor becomes imp or. tant particularly in~he design of elevated flumes, for the Rume substructure may be endangered by any overflow. There is no universally accepted rule for the'determination of freeboard, since wave action or wat~t-8urface fluctuation in a channel ma.y bi;: created by many uncontrollable causes. Pronounced Waves and fiu.ctuatioll of water surface are generally expected in channels where the velocity is so 11igh and the slope so steep th~t the flow becomes very unstable, or on curves where high velocity and large deflection angle may cause appreciable superelevuted water surface on the convex side of a curve, or in chnrmels where the velocity of flow appt'oaches the critical state at which the water may flow at alternate depths and thus jump from the low stage to I;hc high stage at the least obstruction. Other natural causes such as wind movement und tidal adion may also induce high waves and Tequire special considemtion in design. Free boards va.rying from less than 5 % to grea,ter tha.n 30 % of the depth of flow are commonly used in design. For smooth, il1terior, semicircular metal flumes on tangents, carrying water at velocities not, greater than . 80 % of the criticnl velocity with a maximum of 8 fp.'l, experience has indicated that a freeboard of 6 %. of the flume diameter shouid be used, For flumes on curves with high velocity or deflections, wave Ilction will be produced; so freeboard must he increll.sed to prevent water from slopping over,

. water to pick up lining blocks and push them out .~~ ??SlulOn. movmg .. h Id be designed against such POSSibilItIes. Accordingly, the, Immg SOli, 'bi e VelO~l'ty h M' urn PerrnlSSI ' - . The mim:mum pel'missible . 7 -3, T e mlm "t'n is the lowest velocity that Will not" velocity, ?L' I'.he ~onS'O" ~d ~ duc~ the growth of aquatic plant and mo~s. start sedlmentatlOn an IJ1 t ' . d its e"act value cannot be er.sIly 1'1 . I ' ty is very uncer aJl1 :Ln l\. h' liS ve. OCI ' ' no ISlt load d For water carrymg , or for desiltecl flow,, t IS . , t for .its effert fdetermme t . h s little si"'nI'fi canee excep - onplant growth.h Gen'h . ac IIOJ a kino'' .a mean t> l ' t ' f 2 t 'Hps may be used safely w en" ve OCI y 0 0 • . 't ef era y spea "". , t' the channel is small, and a mean veloci y 0 percentage of sIlt preseDillln. t a growth of vegetation that would not less than 2.5 fps W pi even,. ' th ,. i'lg capaCity of the channel. , sedously decrease longitudinal bottom Glope of cha.nnel is 7,.4. Channel Slop , ra hv and the energy head required for generally governed by the topog , P', h slope may ,depend also on the h f! f ater In many cases, t e . , t e . ow 0f the w channe , 1 F'" 'or exanlple , channels' used for W:Lter-dlstrlh d r ' pur~ose o. " ' b se used in irrigation, water supply, y :au I~ butlOn plll poses, such as t o , . h' , level at the pomto! . . , d h droDower proJects reqUIre a Ign . mll1Jl1g, y . a smaL1 b-I' '~d eSI'rable in order to keep the los3 III delivery;~J.l1 therefore, ope I::>

v~locit'lJ

::allihe

a

elevation to a minimfum' h 1 deIJend mainly on the kind of materiaL, The side slopes 0 a e anne . h . " 7-1 give:, a genera1 I.d ea o f the slopes Table . suitable for use Wit 'l'fLrJOUS ' TABLE

7-1.

' sLOPES

SUl'rABLE SID;;;

FO R CHANNELS BUILT IN V,UllOU.S I{INDS

OF MATElRIALS

. 111 aterial Rock .... ,.,.. . .. , ........ ', ... , * • • • • • • • • • Muck and .peaL soils . .. ", .. . .... : .'........ , ... ~ ........ . Stiff clay or earth with, concrete Immg ... , , . ' , .......... . " Etlft.h ,Vlth Btone I"Wing, or earth . for large eila,nnels ....... . Firm clay or earth for small ditches ........... , , .......• Loose sa.ndy earth. , ....... , ... , .... , .... , , .. , ... , : . , .• Sandy I06.m 'or porous clay .. , . : .. : ... , ...... , . , .. , .... , ~

w

•••••

Side slope Nearly vertic al ~:i: 1

H':1 to 1:1

'an

1:1

1%:1 2:1 3:1

. . erodible material, however, a more accurate kinds of h'd b e checked [wainst the criterion of . m~LterJal. , f th For lopes SOUl " "0 determll1atwn 0. . e S l i t (Art 7-10) or by the principle of tractive maximum permli5s1bb~~e. °f:c~ors to' be considered in determining slopes ,~1. ~dition of seepage loss, climatic change, force (Art. 7-14). are method of constru<..; lon, COl . . 'd 1 hould be made as . t Generally speakmgj SI e s opes s channel Size, .e c. d Id·b designed for high hydraulic efficiency steep as practICable a~ s ou 1 e th US Bureau of Reclamation [4] and st:Lbility. For 11l1ed can~~, e'~'5'1 slope for the usual sizes has been considering standarfdlzhl~lg ~n a .. that it is sufficiently flat to. of canals. One advantage a t IS S ope IS ,

~.

h

I I

1;,

Freeboll.rd in unlined cann.'l OJ' lateral will nornu;.lly be governed by considerations of canal size and locatiOJ1, storm-water inflow, and watertable fluctuations eaWled by checks, wiud action, soH characteristics, percolation gradients, operating road requirements, and availability of excavated material. According to the U,S. Bureau of Reelamation [4J, the approximate range of freeboard frequently used extends from 1 ft for smalliateral"/ with shallow depl;hs to 4 it in canals of 3,000 cfs or more capacity with reheively large \vater depths. The Bureau recommends that preliminary estimates of the freeboard required under ordinary conditions be made according to the following formula: (7-1)

where F is the freeboard in ft, y is the depth of water in the canal in ft, and C is ll. coefficient varying from 1.5 for a canal capacity of 20 cfs to 2.5 for a canal capacity of 3,000 cfs or more~ This approximation is

.\

II

160

I

I,

I

UNIFORM FLOW

be noted that a best hydraulic section is the section that gives the minimum area for a given discharge but not necessarily the minimum excav!1tion. The section of minimum excavation 'occurs only if 'the water surface is at. the level of the bank tops. Where the water surface is below the bank tops, as frequently occurs, channels narrower than those of the best hydraulic section will give minimum excavation. If the' water surface overtops the banks and these il.l'e even,with the ground level, wider channels will provide minimum excavation, TABI,E 7-2. AArea

Cross se<;tion

II_~__·. I'

Tmpezoid, half I of a hexagon I Rectangle, half of a square I Triangle, half of a :;quare Semicircle

T=2V2y Hydro~tatic

catenary Capacity. cIs

T_I_D~ _ _Z_ _

il, _ _

.:

.

4y

,II

y'

I 2 V2 11

,

Section . factor

R-I_ _

P

,13 yl: 2 .../3 11 I 2y l

Hydraulie depth

%'

%,y

!

~YI

Y

1. 39.58Sy'

2, 9836y

'%y

2YI

Y

%y'" 2:\1',6

"I ~' ~y"

I

Try

y'

v3" 11

2111

20YI

4-y

-4 y'"

%YI % y'3y u

0.46784y 1. 917532yIo, 72795y 1.19093y ' ·&

.' The pri~ciple of the best hydraulic section applies only to the desig.n of nonerodlble channels. For erodible channels, the principle of tractive force must be used to determine an efficient section (Art, 7-15). . •

(U.S: Bureau

of Reclamation.)

7-6. The Best Hydraulic Section. It is knO\vn that the conveyance of ar chn.nnel section increases with increase in the hydraulic radius or wit.h .decreD,se in the wetted perimeter, ]'rom a hydr~ulic viewpoint, therefore, the channel section having the least wetted peri!lleter for a given area has the maximum conveyance; such a section is known as the best hydrauNc section, The s~micircle hrLs the least perimeter among all sec,tions with the same area; hence it is the most hydraulically efficient. of . all sections, • The geometric elements of six .best hydraulic sections are listed in iTable 7-2, but these seytions may not always qe practical owing to 'difficulties in constructioJ;! and in use of material. : In general, a channel :section should be designed for the best hydraulic. efficiency but should be : modifi~d for practicability. From a practical point of view, it should

Wetted /' Hydraulic' Top perimeter radius width

~Y' i I "~YI 72YI " I 1% V'2 )%'V2 HYt ! r

.)

P,arabola,

BEST HYDRAULIC SECTIONS.

I

I

FIG. 7-1. Retommended freeboard and lleight of bank of lined channels.

161

DESIGN OF CH .... NNELS FOR UNIFORM FLOW

based upon average Bureau practice; it will not, however, serve for all conditions. . For lined canals or h1.teralS, the height of lining abo"';'e the water surface will depend upon a number of factors: size of canal, velocity of water, curvature of alignment, condition of st0rm- and drain-water inflow, fluctuations in water level due to operation of flow-regulating structures, . and wind action. In a somewhat similar mannel', the height of bank. above the water surfitce will vary with size and location of canal, type. of soil, amount of intercepted storm or drai.n water, etc. As a guide for. lined-canal design, the U.S. Bureau of Replaination [3] has prepared curves (Fi~. 7-1) for average freeboard and bank heights in .relation to capacit.ies.

Example 7-1. Show that the best hydraulic trapezoidal 8ecti~n is one-half of a hexagon, Solutio ll . Table 2-1 gives the water area. and wetted perimeter of a tl'apezoid as A = (b

+ zy)y

and

P = b

+ 2 VI + .' y

whe~e y is th~ depth, b is the bottom width, and .: 1 is the side slope,'

.. Flrst, conslder A and z, to b,e constant. : Differentiating the above two equations with respect to y and solvmg Simultaneously for dP Idy, . . dP dy = 2( v'f+Zi

-

b y

z)

I

f t I

I

1

For a minim~m wetted perimeter, dP Idy ~ 0, or b = 2Y(VI

,+ ZZ

-

2)

Substitl!ting this equation for b/n the previ~us two equations for A ELnd Pand solving

11

simu1t,a.neDusiy for: P, p

=z

Now, find tile va.lue z tha.t makes P th~ least. z, equcting dP Ida to zero, and sol viug for z,

of

z =

163

DESIGN OF CHANNELS FOR UNIFORM FLOW

UNl.F'ORMFLD'W

1,62 .

Diifer:entiating P with respect to

where x. is the width-depth .ratio b/y and z is the horizontal projectkm of t~e SIde slope corresponding to 1 ft vertical. Similarly, engineers in IndIa [61 uS,ed an enlpirical formula: V = VA I 13 '=' 0577 V: r7A h'ch . . have I .ti, 'IV 1 IS equlva ent to x = 3 - z for trapezoidal sections' alld Ph'II'lP' , . . . f7] El' . . , pme el1gJ.-: ne61S t use q. (,7-3) WIth z 1.5, or z ~ 2.5, far earth canals,

~3 ~ ta.n30·

35

This means that the section is a half he"agon,

7 .7. Determination of Section Dimensions. The determin8,tion of section dimensions for nonerodible ch:u1l1eis includes the followin.g steps: 1. Collect all necessary information, estimate n, and select S. 2. Comput~ the sectioll factor ARH by Eq. (6-8), 01'

AR'i=~ 1.49 vIS,

practicability. 5 .. For the design of irrigation channels; the channel section is some~ times proportioned by empirical rules such as the simple rule given by·the early U.S. Reolamatiau Service [51 for the full supply depth of water in

-vA

25

'"<>.

..

'0 <:

20

"

of Art. 6-6. . 4. If the best hydn.ulic section is required directly, substitute in Eq. (6-8) the expressions for A and R obtained from Table 7-2 and sol verOr the depth. This best hydraulic section may be modified for

y= 0.5

'-

.= ~

(6-8)

3. Substitute in Eq. (6-8) the expressions for A B,ud R obtained from Table 2-1, and solve for the depth. If there are oth('J' unknowns, such as band z of a trapezoidal section, then assume the values for these unknowns and solve Eq. (6-8) for the depth. By assuming several values of the unknowns, a number of coltlbinations of section dimensions can be obtained. The finaldimellsions are'decided on the basis of hydraulic efficiency and practicability. For lined canals, the trapezoidal section is 'Commonly a.dop~ed, and the p.S. Bureau of Reclamation [3J has developed experience curves (Fig. 7-2) showing the ayerage relation of bottom. widths and water depths to canal ca.pacities. These curves can be used as a guide in selecting proper section dimensions. • The determination of the depth for the computed value of AR7~ can be simplified by use of the design chart (Fig. 6-1). Some engineers prefer a. solution by trial and erro!", similar to Solution 2 for Example 0-2

feet ..

30

== 15 ,,,!

"E 2 0

OJ

:>

Q! 10

100

IPOQ

'1,000

'!,pOD

FUIG ' 7B-2. Experience cur:re.sshowing bottom, width and depth of lined· channels ( .S . .ureau of RG~aJltatum,,) . '

6. Check the minimum permissible velocity if the watel' carries silt. 7. Add a proper freeboard to the depth of the channel .section. b :~ample 7-Jl. A trf,pez~idal channel carrying 400 ds is ,built with nonerodible e", Zlavm g aBslope of 0.0016 and n =- 0.025. Proportion the section dimensions ,,0 !!t,on. Y' Eq. (6-8), . 025 AR~1 = _0: _ ~ '" 167.7 1.19 VO,OOIS

ex~~!~I~t~ting

A, = (b

+ zy)y

a,nd R

=

(t,

+ zV)y!CIJ + 2 Vi + z' y)

-::::---'-":......:..~~;.--'

y)%

(7-2)

where A is the water area in ft2. For a trapezoidal section it can be shown that this rule may alsabe expressed by a simple formula x= 4- 2 (7-3)

ZOO 300400 600

capodry ill cis

Assuming b "'" 20 it and z

== 167.7

in the

~ove

[

.

= 2 and simplifying, 7,680

+ 1,720y = (y(1O + y)Ju Y,= 3.36 It

\

I

164

i'

UNIFORM FLOW

It should be noted that thia solution is exactly the same as the computation of the normal depth given in Solution 1 of Example 6-2. Accordingly, the solutions by trial il.nd error and by the graphical method described in Exo.mple 6-2 can also be applied to the present prublem. Similarly, assume other suitable values of band z, and compute the corresponding depths. The final' decision on dimensions will depend on practical considerations. If the 'values of band z are decided o.t the 'beginning of the compuio.tion, the depth will be c,)mputed only once. Suppose that b =20 ft, 2 = 2, and y = 3.36 ft are the final vo.lues. Assign a freeboard of 2 ft; the total depth of the channel is, therefore, 5.36 ft and the top width of the channel (not the width of the ,vater smface) is 41.4 ft: The water area is 89.S ft', and the velocity is 4.46 fps, which is greater than the minimum permissible velocity ' for inducing silt, if any. When the best hydraulic section is required, substitute A, = vl3y' and R = 0.5y, obtained' frem Table 7-2, in AR~~ = Hi7.7 !!p..d simplify; the depth is found to be y = lUI it. Add 3 ft freeboard; the total depth is 9.1) ft. The corresponding: bottom width is 7.6ft, the top width of the dUlllnel is 18.7 ft, the water area is 75.2 ft', afld the velocity is 5.32 fps. Since the, best hydraulic trapezoidal section is the half hexagon, t,he side slopes are 1 on

r

L

of Reclamation and is tentatively recommended for design of erodible channels. It should be noted that either method at the present stage will serve only as a guide and -\vill not supplant experience and sound engineering judgment. 7-9. The Maxim'urn Permissible Velocity. The max-im1tm permissible velocity, or the nonerodible velocity, is the greatest mean velocity that. will not cause- erosion of the channel body. This velocity is very uncertain and variable, and ,can be estimated only with experience and judgment. In gencml, aId and well-seasoned channels will stand much higher velociT.~llLE·

7-3.

MAXIMUM PERMISSIBLE VELOCITIES RECOMMENDED' BY FOR'rIEll

,AND SCOBEY ANi. THE CORRESPONDlNG UNIT-TIIAC"l'IVE-FORCE VALUES

U.S. BUREAU OF RECLAMATION" (For straight channels of small slope, after aging)

CONVERTED BY THE

,

va/a.

Material

I

)

I It has been noticed that certain channels are erodible whereas others v:ary similar in cha,nnel geometry, hydraulics, and soil physical properties are not. As a further step in investig~tioIl, the chemical properties of the material forming the channel body should be explored, It may be that an ion exchange between water and soil or hydration of the, material is providing a binder in some places and thus affecting the erosion. For a general discussion of the compiexity of this problem, see [81 and [91.

!

Clear water

_I

'7\

B. ERODIBLE CHANNELS WHICII SCOUR :BUT DO NOT SILT

7-8. Methods of Approach. The behavior of flow in an erodibl~ channel is influenced by so many physical factors and by field conditions so complex and uncertain that precise design of such channels at the present· stage of Imowledge is beyond the realm of theory.! The uniform-flow formula, which is suitable fOr the design of stable nonerodible channels; ,provides an insufficient condition for the design of erodi.ble channels. This is because the stability of erodible channels, which governs the design, is dependent mainly on the properties of the material forming the channel body, rat.her than only on the hydral..lllcs of the flow in the channel. Only after a stable section of the erodible channel is obt,ained can the uniform-flow formula be llsed for ,computing the velocity of flow and discharge. Two methods of approach to the proper design of erodible channels are described here: the method oj perrn1;sSible velocity and the method of tractive f01·ce. The method of permissible velocity has been used exteu- . sively for the design of. earth canals in the United States to ensure freedom from scour. The method of tractive force hOes sometimes been used in Europe; it is now under comprehensive investigation by the U.S. Bureau

165

DESIGN OJ!' CHANNELS FOR UNIFORM FLOW

v,

I

fps

ilb/ft2

Water transporting colloidal silts

V, fps

rD,

TO,

Ib/ft'

--'1---------

'1

I

iI i

J

Fine sand, colloidal ... , .. , . , .. ,. " ""'1 Sandy loam, non colloidal. . .,. . . , , . . .. Silt loalll, nOllColloidal .. " ", ... .", ''', Alluvial silts, noncolloidal. ......... Ordinary firm loim,., ..... , ....... , .. Voleanicash" ...... " . . . . . . . . " .... Stiff clay, very colloidal. , ...... " " , .... " Alluvial silts, colloidal.. " " " ' ' ' ' " ' ' - ' 1 ShalesandhardPl1ns .... : ......... ".: ... j

i

0.020 1. 50 O. (127 0,020 1. 75 . 0,037 0.020 2.00 0,048 0.020 /2.00 0,048 0.020 2.50 0.075 0.02012.50 0.075 0.025 I 3.75 0.26 0,025 3.75 0.26 0.025 6.00 0.67

I

I

1

2,50 0.075 2.50 0,075 3.00 0, n 3.50 0.15 3.50 0.15 3.50 0,15 5.00 0.413 5.00 0.46 6.00 10.137

~~:~:tl:~m' ~~. ~~b'b'l~~ ";h~~ :~~~~~li~i~l~l: :~:~~~

i: ~~

~:~~5 ~:~~ I g:~~

Graded siits to cobbles when colloida.l. , , . 0.0;10 Coa-rse gmvel, ~0IlC0I10idlll .......... , .. ; ... / 0.025 CObblesandshmglE'3, ... , ................ 0.035

4.00 4.00 5,00

0.43 5.50 0.30 .16,00 0,91 5.50

,.1

,

•

i

0.80

I 0.67 1.10 I

I!

• The Fortier and Scobey values were recommended for use in 1926 by the Special Committee on Irrigation Research of the American Society of Civil Engineers.

f f

ties tha.n new ones, becau$e the old cha.nnel bed is usually better stabilized, particularly with the deposition of colloidal matter. When other conditions are the same, a deeper channel will convey water at a higher mean velocity without erosion than a shallower one. This is probably because the scouring is caused primarily by the bottom velocities and for the same mean velocity, the bottom velocities are greater in .the shal~ lower channel.

i' i

f J

t

IL !~

, I

UNIFORM FLOW

LG6

DESIGN OF CHA.NNELS FOR UNIFORM FLOW

Attemp~1 were made early -to define a mean velocity that would cause neither silting nor Scoill:ing. From the present-day viewpoint] however, it is dOHbtful whether such a velocity Mtually exists. In 1915,

[

167

2

J _Ir j

! ,g 0.7 :

~, ~

'"

.~

.2

'0

g 0.5f----j--·,--+-..·---/---';-',,\},c-:-+---f-1--i

.)

0.4

,." :is '" "iii

I

I

j o~~,-=l==I=t:I=l=

·t

0,.8

()'5~H-++--t

-{

O.4!-H-++-fflt-f--+

:3

o.:;f--f-+-++-+H-H--t Q.~

Per"r.issible va loci!i es. fps

!-,-,-.---l...--'---r",-,--,--W.-..c Cloy

!

FIG. 7-4, Curves showing' U.S.S,u, da.ta on permissible velocities for cohesive soils.

FIG. 7-3. U.S. and U.S.S.R. data on ]lermissible veloCities for no,nco,hesive soila,

Etcheverry [2Gl published probably the first table of maximum meIl.n velocities that are safe il:gaillllt e1'Osion. . In 1925, Fortier and Scobey [27J pli bUshed the well-known to ble of "Permissible Canal Velocities 11 shown ill Table 7-3. The values in thiE table are for lVell-seasoned channels of small sfopes and for depths of flow less .than 3 ft. The table also shows 1 The first famous fo,rmula for this nonsiiting and no,neroding vciocit,y fo,r silt-laden water was published in 1895 by Kennedy [lOj. From a stl\dy o,f tne dischargo and depth of 22 canals of the Upper Bari Doab irrig.!l.tion system in Punjab, India, the Kennedy fOrImll", WD-S developed as .

(7-4)

where V, is the nonsilting and noneroding mean velocity in fps; y is the depth of flow in ft; C - 0.84, depending primarily or. the firmness of t,he material forming theqhannel body; and z "" 0.64, an exponent which varies only slightly. Ba.sed on later studies by other engineers, the valu~s of C generally recommended are 0.,55 for extremely fille soils such a.s tho,se found.in Egypt; 0.84 for fine light sand soils such as those found in the Punjab, India; 0.92 for coarse light sandy soils; 1.01 for sandY loamy silts; and 1.09 for coarse silt or hard-soil debris. For clear water, a "alue of z = 0.,5 has been suggested. For the design of canals carrying sediinent-laden water, the Kennedy formula is now practically obsolete and is being replaced by methods based on Lacey's regime theory [11-16], Einstein's bed-load function [17), a.nd Maddock-Leopold's principle of channel geometry [18J. There are voluminous writings on these methods. Comprehensive bibliographies can be found in [19] to [25].

,, l I~ l~

. I

.suitable n values for various materials and the converted values for the corresponding permissible tractive force, which will be discussed later (Art. 7-13). In 1036, a .u.~l'''''.<1iH magazine [28] published values of m[l.ximum permissible velocities (Figs. 7-3 and 7-4) ahove which scour would be produced in noncohesive materiai of a wide range of pa.rticle sizes and various kinds of cohesiVe soil It also gave the variation of these velocities with channel dep!;h ~ (Fig. 7-5). The maximum permissible velocities mentioned abo Ire are with reference to straight channels. For sinuous channels, the velocities should be lowered in order to 'reduce scour. Percentages of reduction suggested by Lane [29] are 5% for slightly sinuous canals, 13 % for moderately sinuous canals, and 22 % for very sinuom; canals. percentage val tieS, however, are veJ;'Y approximate, since no accurate data are available Carreclion foctor at the. present time. FIG. 7-5. Curves showillg 7-10. Method of Permissible Velocity. Using U.S.S.R. corrections of the maximum permissible velocity as a criterion] permissible velocity fo,r the design procedure for a channel section, as- depth for both collesive silmed to be trapezoidal, consists of the following and noncohesive materials. steps:

I

I

r i

-j

168 I, l

It should be noted that the unit .tractiv~ force in channels, except for wide open channels, is not uniformly distributed along the wetted perimet.er .. Many attempts have been made to determine the distribution of the tractive force in a chann~l. Leighly [31] attempted to determine this ,distribution in many trapezoidal and several rectangular and tl'iangular . channels from: the published data on the velocity distribution in the

1. Fot' the given kind of material forming the channel body, estimate the roughness coefficir.nt n (Art. 5-7), side slope z (Table 1-1), and the maximum permissible velocity V (Table 7-3 and Figs. 7-3 to 7-5). ' 2. Compute the hydraulic radius R by the Manning formula. 3. Compute the water area required by the given discharge and permissible velocity, or A = Q/V. ' 4. Compute the wetted perimeter, or P = A/R. 5. Using the expressions for A and P from Table 2-1, solve simultaneously for band y. The solution may be expedi~ed by the charts given in Appendix B. ' 6. Add a pl'0per freeboard, and modify the section for practicability.

.

i 1.5

'

Example 7-3. ,Compute the bottom width and the 'depth of fiow ,of !l. trapezoidal cha.nnel laid on a slope of 0.0016 and carrying 8. design dischfl.rge of 4{)O cfs. 'rhe channel is to be excavated in earth containing noncolioidal coarse i1;rllvels and pebbles. Solution. For the given conditions, the following rm!estimated: n 0.025, z = 2, , and nlll:llili1um pe;l'missible velocity = 4.5 fps. Using the Manning formula, !wlve for R. 4.5

=

O:97QwyS

:FIG. 7·6. Distrihution of tra.ctive force in

!1

trapezoidai channel section.

1.49 -0.025 R~ ";0.0016

R ." 2.60 ft

or ThenA = 400/4.5

=

88.8 ft', a.nd P

A = (b

and

169

DESIGN o.F CHANNELS FOR UNIFORM l"L~W

UNIFORM FLOW

P

=

b

+ zy)y =

=

(b

AIR

+ l!y)y

+ 2 v'l+ ii Y =

(b

88.8/2.60 = 34.2 ft. ~

Now

88.8 ft'

+ 2 y'S y)

Solving the above two equations simultaneously, b

)\4.2 it

18.7 ft and y

3.46,ft.

.

'I -11. The Tractive Force. When water 'flows in' a channel, a. force is developed that acts in the direction of flow on the channal oea-:-"This fm'cb, which is simply the pull of wilter on the wetted area, is known as the tract~'ve force. l In It uniform flow the tra()tive forceis apparently equal to the effective component of the gravity force acting on the body afwater, parallel to the channel bottom and equal to wALS, where w is the unit weight of wa.ter, A is the wetted area,' L is the length of the channel reach, and S is the slope (Art; 5-4). Thus, the a.verage value of the tractive fOl'ce pel' unit wetted area, or the so-called tmit tractive f~rce TO, is equal to wALS/PL = wRS, where P is the wetted perimeter and R is the hydraulic radius; that is .

.

. [TV

=

w~·

V

Ii,

l- fl--+--!--+---l- -

123'156789

-I

I

rmm]

7

B

9 10

!4-+-1-+--+=1t++i~ :-.-JI:~

o o

I

2

3

4

5

6

bh

bh

On sid s S 01 channels

On bcttom of cl1annels

FIG. 7-7, Maximum unit tractive forces in terms of wyS. '

channels. U nfoi·tunately,. to deficiency of data, the results of his study were not very conclusive. In the CS. Bureau of Reclamation, . Olsen and Flol'ey[32] and other engineers ha,ve used the membrane analogy and analytical and finite-difference methods for determining the distribution of tractive fOl:ce in trapezoidal, rectangular, and tria!1g ular a trapezoidal chanchannels. A typical distribution .of tractive force Itel reSUlting from the m~mbrane-analogy study is shown in Fig. 7-6. The pattern of distribution varies with the shape pf the section but is

q-5)

In a wide open channel, the hydraulic radius is equal to the depth of flow y; hence TO '" wyS. 1 'l"his is also known as the shear force or the drab fOl·ce. The idea of tractive force is gellorally believed 1.0 h!we been first :introdue~d intohydraulia litera.ture by du Boys in 1879 [po 149; of 30]. Howelrer, the principle of balancing this foroe wit~ the

in

channel resistance in 11 uniform flow was sta.ted 1;Iy Brahms earIy in 1754 (see Art. 5-4).

I .Ii".

170

.

UNIFORM FLdw

DESIGN OF QHANNELS FOR UNIFORM ;FLOW

.

practically unaffected by the size of the section. Based on such studies, curves (Fig. 7-7) showing the maximum unit tractive forces on the sides and bottom of various channel sections have been prepared for use in cu,ilal design. Generally speaking, for trapezoidal c.hannels of .the ilhapell ordinarily used ill canals, the maximum tractive force on the bottom is close to the va.lue wyS, .and on the sides close to 0.76 wyS. 7 -12. Tractive-force Ratio. On a soil particle resting sm the sloping side of a channel sect.ion (Fig. 7-8) in which water is flowing, two forces are acting: the tractive force aT, and the gnlvity cfol'ce component 'YV; sin ¢,

171

is equal to the force tending to cause the motioil. The resistance to motion of the particle is equal.to the normal force W, cos ¢ mUltiplied by the coefficient of frictioil, or tan (J, where e is the angle of repose. Hence, (7 -(3)

Solving for theullit tractive force T, tlmt causes impending motion on a sloping surface, . .. W . / tan" tP (7-7) T, = cos rb tan II '\;1- tan' (1

,c!

,.

J

Similo,rly, when motion of :1 pa.rticie on the level surface is impending owing to t.he iractive force an, the following is obtained fromEq. (7-6) with
-

a

tan

.

e

(7-9)

The ratio of T, to TL is called the tractive-force ratio; this is an important ratio for design purposes. From Eqs. (7-7) and (7:'9), the ratio is

Simplifying, ) PIG. 7-8 .. Analysis of forces ac.ting on a pa.rticle resting on the surface of a chn.nnel bed.

which tends to cause the particle to roll down the side slope. l The symbols used are a = effective area of the particle, T, = u\lit tractive force on the side of the channel, W, = submerged weight of the particle, and", = angle of the side: slope. The resultant of these two fO;'ces, which are at right angles to each other, is

VlV,2 Si112

'"

+ oh,2

When this force is large enough, the particle ~vill move. By the principle of frictional motion in mechanics, it may be assumed that, when mot.ion is impending, the resistance to mqtion of t.he particle 1 The concept of the three-dimensional analysis of the 'gravity and tractive forces acti]lg on a particle resting on a slope at the Btate of impending motion was first given by F'orchheimcr [33J. A complete analysis of a chann"l section using this .concept Wa.'l first developed by Chia.-Hwa Fan [34J. The a.nalysis ,vas also developed independently by' the U.S. Bur~a.u of Reclamation under the direction of E, W. Lane [29,35].

=

~

tan 2 ¢

f

T

K

= COli '"

J( =

~ 1 - tan 2 0

/1 _

'\;

~

Si.ll '" Sill" ~

(7-10)

1

(7-11)

It can be seen that this ratio is a fUllctioll only ~f the inclination of the sloping side", and of the angle of repose of the material O. Fot cohesive and fine noncohesive materials, the cohesi vc fOI'ces,even with comparatively clear .vater, become so great in propprtion to the gravityforce component causing t.he particle to roll down that the gravity force wn safely be neglected. Therefore, the angle of repose need be considered only for coarse noncohesive materials. According to the U.S. Bureau of Reclamation's investigation, it was found in general t.hat the' angle of repose increases with both size and anguiarity df the material. For use in design, curves (Fig. 7-9) were prepnred by the Bureau, showing values of the angle of repose for nOli cohesive material above 0.2 in. in . diameter for various degrees of roughness. The diameter referred to is the diameter of a particle than which 25% (by weight) of the material is larger. 1 Equation (7-10) was presented by the U.S. Bureau of Reclamation [35,361 and Eq. (7-11) by Fan [341. The two equations are mathematically identical.

.

1

172

UNIFORM FLOW

0.1

. 0.2

L-----.rL--;

42

~-

I

I

, I

I

3/16 1/4 lIB 112 ~j

• .....

-T--[I

l

I

I I I I

I

3/4

I

8

""

,6

-1--

:3 4

2

..,'"

I !,

.

'"c:

~<~I ;;"'/ /

.If': ~/ ~

b'"

Il

I.;;:,~

'0 .!?

t1

$O~

'2 6

f--- ... 24

2. 2

'ti

V if

J~Ci

V."O- V ~ ~0° .~~ L ,'") r

I

---V~.' // /// V / / V/ V//V L/

V

~ .'"...."

3.0 4.0

173

DESIGN OF CHANNELS FOR UNIFORM FLOW

particle size for noncohesive material and upon compactness or v:o~ds ratio for cohesive materiaL Other' soil properties such as the plastIcIty index 1 or the chemical action may probably also be taken as indexes for defining permissible tractive force more precisely. However, sufficient data and information on these indexes are lacking_ The U.S. Bureau of Reclsmation has made 'a comprehensive study of the problem, using data fol' coarse noneohesive material obtained from the Sall Luis Valley

VI/';I/I

/

Ii t7-

~o

0

.;

'"c.e ''"

~~"'/ ~'"

."

//

~~i/I

N

o ..,

V , V

I

2.0 I

H/2

I...

r

r

40

-;:;

,i .

Panic 1e size in in clles O.l 004 0.6 o.e 1.0

;VL

.---;----

·Z

1 "

!

\

I

I

.I

/ .<1r-+~I+H Recommeoded value for

"-:;

i---

'b

canals in coarse l

I

noncohe~ive material~

si.e 25% lorg er

;

j

,

V:o."':;" 20

I)

I

.FIG. 7-9. An .. Jes of repose of nonconesive material.

(U.S. Bureau of Rec/a.1l1alion.)

7-13. Permissible Tractive Force. The pe?'m-issible tract1've'01"ce is the • • J maXImum Ulllt trnctive force that will not cauge seriOlls erosion of the material forming the channel bed 011 a level surface: This unit tractive fol'c~ c"n. be determined by laboratory experiments, and the value thus obtamed IS known as the critical t1'uctiz,c JOT'ce. However, experience has shown that actual canals L.'1 coarse nUl1Cohasive material C[l.n st!].nd substantially higher va.lues than the critical tractive forces measured in the laboratory. This is prohab!y because the water {md soil in £l.ctUtll canals (J?nt~in slight amounts of conoidl1.1 and organic matter whiGh provide a bmdlllg power and also because slight movement of soil particle" can be t.?lerated in P:a.~tical designs. without endangering channel stability. B.mce the permISSIble tractive force is the design criterion fOl' field. condi- ' tlOl1S, the permissible value may be taken less than the critical value. Tb,e determination of permissible. tractive force is now based upon

50

100

FIG. 7-10. Recommended permissible unit tr.active forces for canals in noncohesive (U.S. Burea:!' of Rec!amalit.m.)

Ino.terial.

canals [37), values converted from permissible velocities, given by Etche-, verry and by Fortier and Scobey,' the U.S,S.R. values, etc. (Alt.7c9). As a result, values of permissible tractive force recommended for canal design were developed as follows:

I

-\

I

I

A

1 The pla~ticity index is 'the difference in .percent of moisture between plas~ic limit and liq:uid llmit in Atterberg soil tests. This index has been investigo,ted by the U.8. Bureau of Reclamaiioll as a soil characteristic that can be lL~ed to indics.te resistance to SCour for cohesive materials. For cs.nal deSign. a plasticity illdex of 7 may be. taken tentatively as the criticnl vll-Iue, with SCOill' OCCUrring for moder:.te 'tractive forces below this value. However, acours a.re still observed in many cases where the index is above 7. Research'shows that determinat.ion of the plasticity index in (:00" junction with consolidated-shear testa may possibly be necessary.

174

UNIFORM FLOW

For coarse noncohesive matel'ial, with sufficient factor of safety, the Bureau recoIilluends tenta.tively a value of permissibletracti Ire force in pounds per square foot equal to 0.4 times the diameter in iilches of a particle than which 25 % (by weight) of the material is larger: This .recommendation is shown by the straight line in the design c'hart (Fig. 7-10). For .fine nOll cohesive material, the size specified is t.he median size, or size sma.ller than 30% of the weight. Three design curves (Fig. 7-10)

\'0

., E==E=E~j=R+q:B=t==+=a r\

,.....

Sondy cloys (sand < 50 %) _~

Llilil! I I

DESIGN OF CHANNELS FOR UNIFORM FLOW

175

7 -14. Method of Tractive Force. The first step in the design of erodible channels by the method of tractive force consists in selecting a11 approximate channel section by experiene.e or from design tables,l collecting samples of the material fo;:ming the channel bed, and determining the'required properties of the samples .. With these data, the designer investigates the section by applying tractive-force analysis to ascertain probable stability by reaches and to determine the minimum section that appears stable. For channels in noncohesive materials the rolling-down effect shouid be considered in addition to the effect. of the distribution of tracth-e'forces; for channels in cohesive material the rolling-down effect is negligible, and the effect of t.he distribution of tractive force alon:e is a criterion sufficient for design. The final proportioning of the channel section, ho\-vever, will depend on other nonhydl;aulic practical considera!;iops.· The analysis for tractive force is best described by the following example: Design a trapezoidal channel laid on a slope of 0.0016 a.nd carrying The cha.nnel is to be excavated in earth containing noncollnidal coarse gravels and pebb['~s, 25 % of which is 1.25 in. or over in diameter, Manning's n. = 0.025. . SoLution. For trapezoidal channels, .the maximum unit tracth'e force OIl the slopillg sides is usually less th!>ll that Oil the (,ottom (Fig. 7-7); hence, the side force i~ the controUing value .in the analysis. The design of the cha.nnel should therefore include (It) the proportioning of the section dirnensioIlll for the maximum unit tractive force on the sides. and (b) c.heckiilg the proportioned dimensions for the maximum unit tJ:active force on the bottom, . a. PTopol'lioning the. Section D!:men,~ions. Assuming side slopes of 2: I, or z = 2, 3,nd a base-depth ratio bly ",. 5, the maximum unit tractive force on the sloping sides (Fig. 7-7) is 0.775wi;s = 0.775 X 62.4 X 0.0016Y = 0.078y lb/ft'. Considering a very rounded ma.terial 1.25 in. in diameter, the angle of repose (Fig. 7-9) is 8 = 33.5". 'Nith fJ = 33,5' and z ~ 2, or cJ> = 26.5", t,lle tractive-fol'ce ratio by Eq. (7-11) is K = 0.587. For a size of 1.25 in., the permissible tractive force on a level bottom is TL = 0.·1 X 1.25 = 0.5 lb/ft' (sam.e from Fig. 7-10), and the permissible tractive force on the sides iST. ~- 0.587 X 0.5 = 0.294 lb 1ft'. For a state of impending motion of the particles. on side slopes, 0.078y = 0.294, or y = 3.77 ft. Accordingly; the bottom width is b = 3.77 X 5 = 18.85 ft. For this trapezoidal section, A = 99.5 ft' and R = 2.79 ft. With n = 0.025 aud S = 0.00113, the discharge by the Manning formula is 470 crs. Further computation will show that, for z = 2 and bIy = 4.1, th" ~ection dimensions are y = 3.82 ft and b "" 15.66 ft and that the discharge is 41"4 cfs, which is close to thO} design discharge .. Alternative section dimensions may be obtained by as~uming other '(alues oI z or side slopes. . b. Checking the Proportjalled Dimensions. With z = 2 and bly = 4.1, the maximum unit tra,ctive force on the channel bottom (Fig. 7-7) is 0.97wyS = 0.97 X 62.4 X 3.82 X 0.0016 = 0.370 [b/ft', less than 0.5 Ib/ft', which is the pel'mu;sible tractive . force on the level boLtom. Example 7-4.

a discharge of 400 cfs.

Voids ratio

FIG. 7-11. Permissible unit tl'l1ctive forces for canals in cohesive material as converted from the U.S.S.It. data on permissible velocities. .

are tentatively recommended (1) for canals with high content of fine sediment in the water, (2) for canals with 10',11 content affine sediment ill the water,
I Typica.l average earth sections of irrigation ca.nals and laterals, constructed or proPQsed by the U.S. Bureau of Reclamatiotl. and selected for the flows required on the basis of econom.v and stability, are given in Fig. 5, paragraph 1.120, of [4].

!

176

DESIGN OF CHANNELS FOR:UNIFORM FLOW

UNIFORM: FLOW

transfer (if tractive force has little effect on the results and can safely be

7-15. The Stable Hydraulic Section. The section of an erodible channel in which no erosion will occur, at a minimum water area for a given dischilrge is called the stable hydraulic 8ection. Empirical pi'oJiles, such as the ellipse and the parabola, have been suggested as stable hydrauUc sections by many hydraulicians.. The U.S. Bureau of Reclama.tion [38] has employed the principle of tractive force t o · a theoretically stable section for erodible channels carrying clear water in noncohGS'ive .materials.· . In designing trapezoidal sections, as described in.the preceding article, the tractive force is made eqnal to the permissible value over only a part pf the perimeter of the section, where the forces are close to maximum; on most of the perimeter forces are less than the permissible value. rn other words, the impending instability occurs only over a small pr.rt of the perimeter. In developing a stable hydraulic section· for maximum efficiency, it is necessary to satisfy the condition that impending motion shall prevail everywhere on the channel bed. For material with a given angle of repOSE> and for r~ given discharge, this optimal section will provide noi only the chl1Ilnel of minimum water area, but also the channel of minimum top width, maximum mean velocity, and minimum excavation. In the maihematical·derivation of this section by the Bureau, the follow'ing assumptions are made: .. . L The soil particle is held against thechanllel bed by the component of r,he submerged weight of the particle acting normal to the bed. 2. At ,and above the water, surface the side slope is. at the of repose of the material under the action of gravity. 3. At the center of the channel the side slope is zero and the, force alone is sufficient to hold the .particles at the point of incipient instability. 4. ·At points between the center and edge of the channel the particles are kept in a state ,of incipient motion by the resultant of the gravity component of the particle's submerged weight acting on the side slope and the tractive force of the flowing water. 5. Thc tractive force act.ing on an area of the channel bed is equal to the weight component of the water d,irectly above the area acting in the direction of fiow. This weight component is equal to the weight times the longitudinal slope of the channel. If assumption 15 is to hold there. can be no lateral transfer of tra.ctive force between adjacent currents moving at different velocities in the section-a situation, however, that never actually occurs. Fortunately, the mathematical analysis made by the Bureau I has shoWn that the actual I

ignored. . . '. It· y . According to assumption 5, the tractlve force actmg on any e em en ar. n.rea AB OIl the sloping side (Fig. 7-12a) per unit length o~ the channell~ equal to wyS dx, where w is the unit weight of water, Y is the depth 0 'water· above AB, and S is. the longitudinal slope. , . Since the area AB is VCdX)2 + (dy)2, the unit tmctive force is equal to

t

V

tal

where rP is the slope angle of the \ tangent at AB. . • 29.5'--·-----1 The other assumpti(;ms stated above have been used previously to develop the equation for the trac. tive-forc~.ratio K (Art. 7-12). The unit tractive force all the level bottom at the channel center is 1'L wYuS, where yo is the depth of flow at the center. The corresponding unit tractive force on the sloping area ~B is,therefore, equal to wyoSIL , . III o~der to achieve impending 7-12. Andysis and desigll ?f sto.ble mo~ion ·over the entire periphery of. FIG. hydraulic secti0!1' (a) ~heoretLcl'l.l secthe channel bed, the two forces tion {or given soil properties. and eh!l.l1ll:l mentioned in the above paragraphs . slope, providing Q - 220 cis; (b) mod~fred section for Q" 400 cfs; (c) modIshould be equal; that is,. fied section for Q' 100 efs. wyS cos

+ (lL)£ tan (dy)t dx Yo •

2

. At the center of the channel, y

Takiflg the effect of lateral tnl.litive force into accoUnt, an a.!ternative assumptioll

was ina.de by the' BUreau, which states that the tractive force acting on a particle is prop~rtiona.1 to the square of the ll1ean velocity in the channel at 'the point where the

177

·1

I i

(I

=

ya' and

:t

tan~

fJ

= O.

With this condition

pa.rticle is loea~d. This assumption gives ~ solution .that agrees very clqselY w~th the solution bailed on assumption 5. Therefo.re, neglect of the trB?"fer 0,1 .tractwEl force in the a.~!l.lysis will give equally sn.tisfactory results, Bad wll:.h cOJ;lslderably less ,,'ork.· .

.. ~ .178

1

the solution of the above differential equation is tan (7-12) Y = Yo cos ( - -8 x) yo , This equation shows that the shape of a stable hydra,ulic section under the specified assumptions is a simple cosine curve. From the results of the Bureau's mathematical anaiysis, the following properties of this stable section can be stated ~ TO

Yo

o.97wS

l' A

=

1.35 - 1.19 tan 8 yo%Sl! n 2.04Yo 2 tan e

I

(7-14)

·11

(7-15)

a

= 0.99

(1 - ~ T

Example 7-6. Determine the profile of the stable hydro.ulic section to replo.ce the tra.pezoidal section of the channel described in Example 7-1. SolUlian. For the given conditions, TO = 0.5 lb/ft l , S = 0.0016,. e.= 33.5°, a'J.d n = 0.025. By Eq. (7-13), the center depth is Yo = 0.5/(0.97 X 62.'1 X 0.0016) 5.16 ft. From Eq. (7-12), the shape of the theoretical section is

which

is

y = 5.16 cos 0.128z

plotted as shown in Fig. 7-12a..

II l I

I I

\

(7-16)

On the other hand, if the channel is to can'y more than the theoretical section will carry, it is necessary to add a rectangular section at the center. Suppose the. discharge to be carried is Q", which is greater than Q, and the top width of the added rectangular area is Til. The value of T)) may be computed by /I _ n(QI! - Q) (7-17) T - 1.49Yo~~SI~

.

I

(7 -13)

where 70 is the permissible t.ractive force in IbjfV, V is the mean velocity in. the section in fps, A is the water area in ft~, 8 is the angle of repose for the material or the slope angle of the section at the water edge of the channel, T is the top width, and the rest of the symbols are hS previously defined. The discharge of the theoret,ical section is equal to Q '= FA. If the channel lS to carry a discharge less than Q, it is necefisary to remove vertical portion of the section at the channel center. S'upposc the dis- . charge to be carried is Q', which is less than Q, and the top widths of the designed section and the removed area are T and T f , respectively. The value oLT' may be computed by

T'

179

. DESIGN OF 'CHANNELSFOR UNIFORM: FLOW

UNIFORM .FLOW

It should be noted that the angle. of ~he

~o~inc function is expressed inradiansj it may be converted to degrees: by multLplymg

I

I

I

l [ I!

.

I

.

it by 180/11" or 57.3. The top width mllY be computed by Eq. (7-12) with if = 0, or cos 0.128::; = O. Thus, 0.128::; = 11"/2 and x = 12.3 Tli.e top width is T = 2::; = 24.6 ft. By Eq. (7-14), the mean velocity is V= (1.35 - 1.19 tau 33.5') 5.151' X 0.0016}'/ 0.025 = 2.69 fps. By Eq. (7-15), the water Mea is A = 2.04 X 5.16'/tan 33.5° = 82.2 fV. Hence the discharge is 220 cfs. Since the design discharge is 400 cfs, it is necessary to add a rectangular area at the middle (Fig. 7-12b). The width of the rectangle may be computed by Eq. (7-17) as

n.

. ,,_ ·0.025{400 - 220) _ T - 1.49 X5.l6~~ X 0.00W1 - 4.9 ft TherefCJre the top width is 24.6 + 4.9 = 29.5 ft. If the channel is 'designed to carry 100 cis, it isnecesso.ry to remove !l. vertical area ir.:im the middl~ (Fig. 7-12c). The top width of the removed a.rea may be computed by Eq. (7-1il). T' = 0.96(1 - V 1 0,%20)X 21.6 = 7.7 ft Therefore the top width is 24.6 - 7,7

=

lB.\) ft.

C. GRASSED CHANNELS

7-16. The' Grassed Channel. Presence of grass or vegetation in channels will result in considerable turbulence, which means loss of energy and retardance of flow, For earLh channels used for carryin~ water on farm lands, however, a lining of grass is often found to ndvantageous and desirable. The grass will stabilize the body of the channel, consolidate the soil mass of the bed} and check .the erosion on the channel su1'face and the movement of soil particles along the channel bottom. The U.S. Soil Conservation Service (39-411 has conducted a series of experiments on channels lined with various kinds of grass (Fig. 7-13) .. The . results thus obtained under different testing conditions and the procedure suggested for the design of grassed channels will be described in. the following articles. 7-17. The Retardance Coefficient. The Manning coefficient of roughness for grassed channels is specifically known as the relardance coefficunt. According to the investigation by the Soil Conservation Service, it was found that Manning's n for just one kind of grass varied over a wide range depending on the depth of flow and the shape and slope of the channel. Thus, the selection of a desJgn value for n would be nearly· impossible. Fortunately, it was discovered that the retardance coefficient n holds a.certain relationship with the product of the mean velocity or flow V and the hydraulic radius R. This relationship is charact~ristic of the vegetation and practically independent of channel slope .and shape. As a result, therefore, a number of experimental curves for n vers4S' VR (Fig. 7-14) were developed for five different degrees of retardance: very high, high, moderate} low, and very low. For very low retardance

be

\

-1

1

. . ---'i

180

UNIFORM FLOW DESIGN OF CHANNELS FOR UNIFORM FLOW

lSI

TAllLE 7-4. CLASSIFICATION OF DEGRE1!l OF RE'l'ARDANClll FOR VAIUOUS KINDS OF GRAS/;*

. Cover

Condit.ion

A Very high

Weeprng love grl\SS ...•.•....• Yellow bluestero ischa.emuln ...

Excellent stand, tllll (av 30 in.) Excellent sta.nd, trtll (av 36 in.)

B. High

Very den'sE. growth, unout :Kudzu ............. , ..... . Bermuda: gra.ss ......... , .... . Good sta.nd, taU (av 12 in.) Native grass mixtur~ (little bluestem, blue gramat a.nd othr:r lon.g .l!.nd short Midwest grtUlSes) . . . . . . . • .. . ., ... . Good stand, unmowed Weeping love gra.sS ........... . Good stand, ta.ll (av 24 in.) Lespedeza ~ericea., ........... . Good sta.nd, not woody, tall (av. 1 \J in.) Good st!tnd, uncut (av 11 in.) AHal£a ........ . Weeping l.)ve grass ...... ' ... . Good stll.nd. mowed (av 13 in.) Dense growth, unout Kudzu...... . .......•.. Illue granla ............•...... Good stand, uncut (av 13 in.)

Reta.rdo.nce

Moderr.te

C

\

.l

Good stand, uncut (6 to 8 in.) dense cover (av 6 in.) Good stand. headed (6 to 12 in.)

D

I.ow

E

Very low

Bermuda gras.s....... . ....... Good stand,. cut to 1.5in.heighj. Bermuda grass. . . . . . . . . . . . . .. Burned stubble

;.

!

Fair stand, uncut (10 to 48 in.) Good stand, mowed (~W 6 in.) Good sta.nd, uncut (!loV 11 in.)

Bermuda grass .........•...... Good stand, cut ~o 2.5 in. height Comm·:lo let;pedeza ......... . Excellent stand, uncut (av 4.5 in.) Buff.Jo grass.... . ........... . Goodstand, uncut (3 to 6 in.) Grass-legume wixture-fall, spring . (orchard grass, redtop, ltaliar, rye grass, and common le5pe·iez.ll.).. ..• . . . . . .. . . .... . Good stand, uncut (4.to .5 in:) Lespedeza. sericea, .... After cutting to 2 in. height., very good 'stand before clltling

Fm. 7-1~. CenLipede grassed channel. (Courtesy of W. O. Ree, U.S . .'igricultura! Research.Smn:ce.) .t A ) Before experiment; (B) after test a.t t1 flow equal to 15 cfs for 4~ mm; (C) dunng test at a flow equal to 30 cf.5; (D) at completion of the whole experiment.

only the average ctll've is shown, together with the curves for low retardance. The classification of degree' of l'etardance is based on the kind of vegetation and the condition of growth, as described in Table 7-4. The term" stand" used'in the tabie refers to the density of grass, or the count of vegetation, which is sometimes expressed as the 'number of stems per square foot. The n- VR curves thus developed may also be applied to other kinds of gmss, provided that their characteristics and degree of I'etardance can be identlfied. For this purpose, Table 7-5 is provided as a guide in the selection of the vegetal retu.rdance for diffetent conditions of stand and average length of the grass. .

Crab grass. . .. . .. .. .Bermuda grass ............... . Common Jespedezo. ... , ....... . Grthss-leg'.1me mixture_ummer (orah1lrd grass, redtop, Italian rye grass, and common le8pede:z.a) ..•....... , ....... . Centipede gr3.5s . .' ........... . Kentucky bluegrass .......... .

U.~.

Soil Conservation

[41J.

182

UNIFORill FLOW·

DESIGN OF CHANNELS FOR UNIFORM FLOW

183

.sl---l--t--........f.--+-++-+-!-l-f---+ ~1---4--4--.--+---+--~-~+++----r--+--~~~-+-+~~+---+--+---~

H

r:: .1 : .09 .OS .O~

.00

.O~

!

...

-.I

VR

VR

(0) Curves for A or very high vegetal retardance.

(e) CurYe.s for C or modera~e vegetal retardancB.

i

.\,

VR . (b) Curves for B or high vegetal reta.rdllnce. FIG.

7-14. Experimental n-VR curves.

(d) Curves for· D or low vegetal reta.rda.nce, a.nd a.n .e.vernge curve for E or very low vegetal reta.rdll.nce.

(U,S. Soil Conservation Service.)

l

'f I

i .1

I

~I

184

UNIFORM FLOW

TABLE

7-5.

GUIDE I,:! SELECTION OF VEGE'L'AL RE'l'ARDANCE*

Stand Avcrage length of grass, in.

Good

Foil' .

Degree of retarciance

>30 11-;-24 8-10 2-6 <2

A . Very high B High C Moderete D Low E Very low

>30 11-24 6-10 2-8 <2

B C

D D E

High Moderate Low Low Very low

* U.S. Soil Conserv!!.tion Service [411. 7-18. The Permissible.Velocity. The permissible velocity of Row in a gl'assed channel is the velocity that will prevent severe erosion in the channel for a reasonable length of time.' Permissible velocities for different veg.etal ?overs! ch~nnel slopes, and soil conditions, recommended on the baSIS of mvestlgatlOn by the Soil Conservation Service, are shown in Table 7-6.

, I

!

DESIGN OF

1

7-19. Sel~ction of Gra~s. The selection of grass for the channel lining depe.nds mawly on the clImate and soil in which the plant will grow and surv.I~~ under ,the given conditions. From the hydraulic viewpoint, stablhu y . and other f~ctors should also. be considered. In general, a higher dlsch:.l.l'ge requu'cs a stronger or better lining. On steep slopes ?unch grasses; such as alfalfa, lespedeza,and kudzu, will develop channel~ mg of the flow and" hence, are unsatisfactory for lining. For slopes greater than 5 %, only fine and uniformly distributed sod-forming grasses sueh as Bermuda grass,. Kentucky bluegrass, and smooth brome ar~ re~om~nended fOl' lining· where the main flow occurs. Because' of the obJec.tlOnable spreading nature of sod-forming grasses, the top portion of the. Sides and the berm may be planted with gras.:;es that do Ii.ot.spread easliy, such as weeping love gl'B,SS. 11'01' fust establishment of the lining, Bermudagl'assand weeping love, grass fLl'e recommended. Sometirne~ an~uals are used as temporary protectibn until permanent coveb by native gras~e~ are ~stablished. Silt deposition in channels may be con~rolled by Iml?g With bunch grasses, which will develop channeled flow lnCl'ease velOCity, and thus reduce silting.: , ' . 7-20. Proced?re of ~esign., After the !kind of grass for channel ]ining IS selected, the qegree of retardrmce can be determined from the condition of the stem length and the density of growth. During the period of

I !

CHANN~LS

~85

FOR UNIFORM FLOW

establishment, tht; grass will grow and the channel will be stabilized under a condition of low degree of retardance. The channel will not reach its maximilm capacity until' the. grass 'cover is fully q,eveloped and well established. Therefore, it is suggested that the hydrll:ulic design of a gras:;ed channel ,consist of two stages. The first stage (A) is to design the channel for stability, that is, to determine the channel dimensions under the condition, of a lower degree of retardance. The second stage , TABLE

7-0.

PERMISSIBLE VELOCITIES FOI! CHANNEI.S LINED WITH GRASS*

,

Slope IMlge,

Cover

%

----

Bermuda grass

0-5 5-10 >10

Buffalo g'rass, Kentucky bluegrass, smooth brome, blue grama

GrllSs mixture

,

I. I I

Permissible velocity, fps Erosion-resistant soils

Easily eroded' soils

8

6 5 4

7 8

0-5 5-10 >10

7 5

5 4 3

(}-5

5 4

3

5-10 II Do not use

6

all

4

slopes steeper ~han _10~%_ _ __

I

I

0-5 ,3.5 2.5. Lesp,edeza sericea, -w-e-ep-i-~~g-lo-v-e grass, ischaemuffi (yello.v blue- I Do not use on slopes steeper than 5 %, except lOr stem), kudzu, aHalf!\., crabgrass side slopes in a combination channel

I

------

Annuals---used on mild slopes or as~ --~-=5---1 3 ,5 2.5 temporary pr"tection until per- Use on slopes steeper than 5 % is not reCOffimanent CQyel'S al'e established, mended common lespedeza, Sudan grass

I

REMARKS. The values apply to average, uniform stands of each type of covel'. lIse velocities exceeding 5 fps onlr where good covers and proper maintenll.nce cll,n be obtairied.' , . * If.s. Soil Conse.rv!l.tion Service 1411.

(B) is to review the design for maximum capacity, that is', to determine the increase in depth of flow necessary to maintain a maximum capacity under the conditiun of' a higher degree of retardance. For instance, if. common lespedeza is selected as the grass for lining, the comn-lOn lespedeza of low vegetal ret!l.rdar\.ce (greell~ average length 4.5 in.) is used for the first stage in design. Then, in the second stag~, the cummon lespedeza of moderate vegetal retal'dance' (green, uncut,: average length 11 in.) shou1.d be used. Finally, a proper freeboard is added to the computed'

186

I

UNIFORM

depth of the channel, which will further increase the maximum design capacity. The design procedure I is described as follows: A. Design Jor Stabilt·ty. Given the discharge, channel slope, and kind of grass, the Ilrst stage of design may proceed in the following stiOlPs: 1. Assume a value. of n, and determine the corresponding values of VR from the n- VR curve (Fig. 7-14). 2. Seler:t the permissible velocity from Table .7-6, and c()mpute the value of R. 3. Using the Manning formula, compute the value of L49R%SH

n

and check this value against the value of V R obtained in step 1. 4. Make other trials until the computed value of V R is equal to the IT R value obtained from the n- V R curve. 5. Compute the water area, or A Q/Y. 6. Since the correct values of.A and R have been obtained, the section dimensions may be determined the procedure described in Art. 7-7, The sections generally used for grassed channels are the trapezoid,' po,rabola, and triangle, named in order of increasing depth required in excavation. Owing to the norl11al action of channel deposition and erosion, trapezoidal and tLiallgular sections, if selected but not well maintained, will generally become para.bolic after a long period of service. Example 7-6. Determine the section of a coannellined with grltSS miX~ure, laid on erosion-resistant soil at a slope of 0,04, and carrying a discharge 0; 50 ds. Solution. In designing for stability, the grass mixture that offers a low vegetal retardance; j,e., that of the dormaut ~eaaon, is considered. Therefo~e, the corre~pond ·jng n- 'V.n curve should be used in the c.omputation. . From TablCl 7-5, the permissible velocity for design is taken as 5 fps. Using the n~l'R curve (Fig 7-14) for grn.s8 mixtute for fall, winter, and spring, the trial computations involved in tho design procedure !lore ¥ follows:

Trial no.

R

1.49R~5SH

Side slope z:1

Bottom width b, ft

Depth

17.0 12.5 0

0.53 0.62 1.00

Top width T,ft

!I, it

....

3;1 6:1 Triangle ........... Para.bola ..........

10: 1.

1

187

DESIGN OF CHANNELS FOR UNIFORl!vl FLOW

1

---~-

20.18 19 94 20.00 2().00

o.n

B. Design j,'}r lI1a:rim~tm Capacity. The second sta,ge in design. is to determine the additional dept.h necessary to sustam the maXImum capacity of (1, fully developed lining. The procedure is as follows:. . 1. Assume Lne depth y, and compute the water area A and the hydraulIc radius R. . 2. Comput.(J the velocity Y by Y = Q/ A and the value of V R. 3. From the n-V R curve of a higher degree of retardance for the selected lining, detcrmiac the value of n. 4. Compute the velocity by the Manning formula, and check this value of V against the value obtai!led in step 2. 5. Make trial computations unt.il the computed V in step 4 is equal to the computed IT in step 2. ·It should be note~ that this velocity is a.lw.''\ys les5 than the permissible velocH,y assumed 111 the first stage of deSign, that is, in design for stability, since the cross section has heen enlarged in the second of the design. 6. Add proper freeboard to the computed depth. Example 7-7. Modify. the selected sections in Example. 7~6 for maxi,mum ca,pacity. Soh,tion. For determining the proper depth for maXlnlum. capacIty of t.he proposed sections, the grass mixture of the growing. season, whIch offers a X?oderate vegetalreta.rdallce, is considered in the computatlOn. TnI' n-VR curve (FIg. 7-14) for grass mixhure in summer is therefore used. Other known da.ta are Q = 50 cfs and S 0.04.. . For the tl'apezoidal section with 3:1 side slope and b = 17.0 ft, the trial computations are given below:

11.

'Trial 1

2

' L35 0.34

3

4.72

4

2,50

The correct va-lues for the determination of sections o.re.R = 0.50 It and A = 3% = 10 ft'. Several cha.nnel sections meeting these requirements are proposed as follows (using charts in App~ridb:: D ) : ' , I

For an eXl).mple .of the pra.ctical

of a. grassed channel, see £42J.

noQ.

!I'

A

R

V

l'R

n

I

V

1.491mSH 11.

-

1

2 3

0.70 0.60 0.65

13,4 11.3 l2.3.

0.G3 0.54 0.58

3.73 4,42 4.07

2.35 2.39 2'.36

0.051 0.050 0.051

1

4.2.2

3,96 4.07

1 1

The correct depth is 0.55 ft. Adding a freeboa.rd of 0.2 tt,.the total depth is 0.85.ft. . For the trapezoidal ::;ection with a side slope of 6: 1 and b 12.5 ft, t~12 computat~on results in a total depth of 0.94 ft. Simila.rly, the tot!).l depth of the trlangula.r sectIOn with a side slope of 10: 1 is found to be 1.33 ft,.

.1 ..;....

-!~

" •

188 For the parn.bolic :3ection of y as follows: Trial no. ---

y

0.80 0.90 0.85

I

2 3 ·1

DESIGN OF CHANNELS FOR UNIFORM FLOW

UNIFORM FLOW

I

0.85

I I

=- 0.73 ft and T = 20 ft, the trial computn.tions are

T

A

21.0 22.2 21.6 21. 7

11.2 13.3 12.2 12.4

I

I

R

V

0.52 0.60 0.57 0.58

4.406 3.76 4.10 4.03

I- - -I- - VR

n

2.32 2.25 2.34 2.34

0.051 0.051 0.051 0.051

I

I

1.49m1S~~

n

3.78 4.15 4,03 4.07

It should be noted that the computation for the parabolic section is simplifi
, i

7 -1. Show t~at the most efficient rectangular or triangular section is one-half of a square. . 7-2. Explain (a) that any section formed by a p'olygon which can be inscribed by !\ semicircle with the ccnter in the water surface will have its hydro,ulic radius equal to one-half the radins of th~ inscribed circle, and (b) that such section will have the best hydl'll.ulic effiuiency. . 7 -3: Detenuine the best hydraulic section of the channel in Example 7-2 if the section is (n) rectangular, (b) triangular, (e) cirl!ular, (d) parabolic, and (e) in the form of It hydraulic catenary. . . 7-4. Solve Example 7-2 by the empirical rule of Eqs. (7-2) and (7-3). 7 -6. Design a nonerodible channel carrying 200 cfs with n = 0.020 and S = 0,0020. Use your own judgment and assumptions. 7-6. Based on the p.ractice of t1u~ U.S. Bureau or'Reclamation, det.ermine (0) the freeboard of the channel designed in Example 'r-2 when the channel is unlined and (b) the heights of the lining and benk if the channel is lined. ' 7-7. Solve Example 7-3 if the material forming the channel body is fine silt having an average particle size of 0.006 mm. Estimate the permissible velo,~ity with the aid of (a) Fortier and Scobey's table, (b) U.S.S.R. data, and (e) the Kennedy formul~. modified for clear ,vater. . . 7-8. SOlVE Example 7-3 if the mn.terial forming the channel body is fairly compact heavy clayey soil with a voids ratio of l.0. 7 -9. Solve Example 7-3 if the channel has !l. pa.rabolic section. 7 -10. Design the section of a canal to carry a discharge of 200 cfs throllgh a land of erodible soils with n = 0.020 and S = 0.0020. Assume other necessary data and use your o~vn jlldgment. . . 7-lt. 111e All-American Canal is desigr:ed to divert 15,155 cfs of desilted water from the Colorado River to irrigate the Imperial Valley in southern Califpmia. This canal is 80 millls long. . The typical maximum section has a bottom width of 160 ft, width at wo.t¢r surface of 232 it, water depth of 20.6 ft, minimum freeboard of 6 ft and bank width of 27 to 30 ft. The terminal capacity is 2,600 cfs. The canal was 'excavated

189

mostly in n.Uuvi!ll soil, ra~ging from light sand~ or silty [oams to adobe and having an average particle size of 0.002b in. Review the hydraulic design of the channel section. 7-12. Review the stability of the section dimensions obtaincd in Example 7-3 by the method of trn.ctive force, !l.Ssuming that 25 % of the material forming the channel bed is L2b in. or over in diameter. . 7-13. Solve Example 7-4 for the following conditions, respectively: (a) If the side slopes are. assumed as 1 on 1.5. . (b) If the mateJin.1 forming the channel bed contains fine noncohesive particles, 50% of which are larger than 1 mmin diameter. The water is clear. (e) If the m~.terial forming the channel bed is cohesive compact clay, having !L voids ratio eqllal to 0.5 .. (d) Ii the channel is moderately sinuous. 7-14. The conversion from the maximum permissible velocity to permissible tractive force (Table 7-3) is based on 0. flow depth 0'£ 3 ft _and an avernge channe.l section ha.ving a bottom width of 10 it and side slopes of L5: 1. For a.llllvial noncolloidal silts and. clear-water floW, the m!lximum permissible velocity recommended by Fortier and Scobey is 2.00 fps r.nd the n value is taken as 0.020. Compute the corresponding permissible tractive foree. 7-15. Compute the maximum tractive force per unit area on the section of the AllAmerican Canal described in Prob. 7-11. . 7-16. Determine the cross ser.I,ion and discharge of the st~ble hydraulic section of 0 .channel excavated in a' noncohesive IDr.terial ho.ving To = 0.1 lblft', S = 0.00040, o "" .31", and n = 0.020. . 7-17. Determine the modffied profile for the channel section obtained in the preceding problem if the channel is tOCllUY (a) 75 cfs ILnd (b) 300 efg. 7-18. Design a waterway lined with Bermudn. grass on- erosion-resistant soil and cn.rryillg a discharge of 200 cfs .. The n.vernge slope of the channel is 3 %. Use the curve for modemte vegetn.l rel,ardance. 7 -19. Determine the total depth for maximnm capacity of the channel section propo,,,ed for the preceding problem. Allow 11 fre,eboard equal to 20% of the computed depth. .

REFERENCES 1. Victor L. Streetel': Economical conal cross section~, Transactions, American Society of Civil Engineers, vol.. 110, pp. 421-430, 1045. 2. I van E. Houle" Irrigation Engineering," vol. 2, "Projects, Condllits, und Structures," John Wiley & Sons, Ill~., New York, 1956. 3. ",Linings for lrrigrttion Canals," U.S. Bureau of Reclamn.tion, Jl1ly, Hl52. 4. Cll-nals !lnd related structures, U.S. Bureau of Reclamation, Design and Constnlclion Manila!, Design Supplement No.3, Apr. 17, 1952; vol. X, 'pt, 2,. ghop. I, paragraphs 1.15, 1.8, and 1.18. . . 5. B. A. Etcheverry: "Irrigation Practice and Engineering." vol. II, "Conveyance of Water," iVIcGmw-HiIl Book Company,Jnc., New York, 1st ed., 1915, p. 122. 6. G\lilford L. Molesworth: "Pocket-book of Engineering Formulae (Useful Formulae and Memoranda) for Civil·and Mechanical Engineers," E. & F. N.Spon, London, 7th ed., 1871, p. 175. ' 7. Isidro D. Carino: A graphical solution for flow in earth channels, paper 1360, Proceedings, American Society of Civil Engineers, J oumal, Irrigation and Dra.ina.ge Division, voL 83, no. IR2, pp. 1-9, September, 1957. .

190

UNIFORM FLOW

DESIGN OF CHANNELS FOR UNIFORM FLOW

8. 'Pete W. Terrell [LIld Whitney M. Borland: Design of stable canals and channels in erodible ma.terial, Transactions, AnLerican Society of Civa Engineers, vo!' 123, pp. lOl-U5, 1D£i8. . 9. E. W. Lane: Stable channals in erodible material, Tmnsact·irms, American Society of Cilia Engineel's, vol. lO2, pp. 123-142, 1937; and discussions. 10. Robert G. Kennedy: The prevention oI silting in irrigation canals, Proceedings,. Inslit:l.tiolL c,f Civil Engineers, London, vol. 119, pp. 281-290, 1895. 11. Edward S. Lindley: Regime channels, Minutes of Proce,ed£ngs, Pun}ao Engtneering Congress, Lah01'e, India, vol. 7, pp. 63-74, 1919. 12. Gerald Lacey: Stable channels in alluvium, Proceedings, Inst·itution of Civil Engineers, London, vol. 229; pp. 2.'i9-384, HJ30.' 13. Gerald L"c"y: Regime flow in incoherent alluvium, Cenl,.al Board of h1'1:galion, . PlLblication No. 20, Simla, Iudia, 1940. 14. Gerald Lacey: A general theory of flow in alluvium, Journal, Instit'ution of Civil Engineers, Londan, vol. 27, pp. 16-47, 1946. '. 15. Thoinas Blench: "Bydrs-ulies of 'Sediment-bearing Cano.ls and Eivers," Evan~ 'Industrie$, Ltd., Vancl)uver, B.C., Canada, 1951. 16. Thomas Blench: Regime theory for oclf-formeci sediment bearing'channels, Tmnsactions, American Bociety oj Ciliil En.gineers, vol. 117, pp. 38.1-400, Hi52. 17. H. A. Einstein: The bed-load fUllction for sediment tran~Jlorta~ion in open channet flows, U.S. Depa"bILent of Agriculture, Technical BuUelin No. 1025, 1950. 18. L. B. Leopold and Thomas Maddock, Jr.: The hy.draulic g~ometry of stream cha.nnels and some physiographic implictttions, U.B:G'eological BUTVP,Y, Professional Paper 252, 1953. 19. ,K W. Lane: Sta.ble channels in erodilJle material, Tra1tSdctio,is, American Society of Civil Engineers, vol. 102,pp. 123~142, 1937. .' 20. Serge Leliavsky: "An Introductiqn to F1uvial Hydraulics," Constable & Co., Ltd., London, 1955.' 21. Instit\ltion Research Committee: Recent developments hydraulics, Proceedings, In.stilution of Civil Enginp,cr.s, London, pt. III, vol. 4, pp. 990-1049, December, 1955. 22. Ning Chien :'The present status of research on sediment transport., Tl'anoactions, A lILerican Society of Civil Engineers, va!. 121, pp. 833-868, 1956. 23. Ning Chien: Graphic design of olluvial channels, Transactions, Amer':can Society of Cilia Enuineers, vol. 121, pp. 1267-1280, 1956. ' 24~ Ning Chien: A concept of the regime theory, Transactions, American Society of Civil Engineers, vol. 122, pp. 785-793, 1957. 25. Eno~ J. Carlson and Carl R. Miller; Research needs'in sediment hydraulics, paper 953, Procced£ngs, American Society of Civil Engineers, J oumal, Hydraulics D£visio1];, vo!' 82, no. 11 Y2, pp. 1-33, April, 1956. 26. D. A. Etcheverry: "Irrigation Practice and Engineering," va!. II, McGraw-Hill Book Company, Inc., New York, 1915. 27. S. Fortier and Jf. C. Scobey: Permissible cana.l ;relocities, Tran8~tions, AlILerican Society of Cillil Engineers, vol. 89, pp. 940-956, 1926. , 28. The maximum' permissible ,mean 'velocity in open channels, Gidrotekhnichesko·ie Stroitel'stvo, (Hydrotechnical Constmction),I\{oscow, no. 5, pp.5-f, May, 1936. 29. Emory W. Lane: Design of stable channels, Transactions, Am.erican Society of Civil' Engineers, vol. '120, pp. 1234-1260, 1955. . 30. P. du Boys: Etudes du regime du B,h6ne et l'action exercee par les eaux su~ un lit ~ fond de graviers indetiniment affouillable '(The Rhone and streams with movable •bedsj, Annales rles ponts et Cha1L8S~eS, ser. 5, vo!. 18, pp. 141-195, 1879.

31. J. B. Leighly; Toward a thaory of the morphologic significance of turbulence in the flow of water in stream, Uni~ersily of Ca.17/orlliCL, Publications in GelJgraphy, vol. 6, no. 1, pp. 1-22, Berkeley, 1 9 3 2 . , . 32. O. J. Olsen and Q. L. Florey (compilers): Sedimentation studies in open channels: Bo.und~y shear and velocity distribution by membrane 'ano,logy, ana.lytical and fil11te-dlfference methods, reviewed by D. McHenry and It. E. Glovel' U.S. Bu,rea.u of Reci.rultation, LabOl·atory. RepOl·t, No. Sp-34, Aug. 5, 1952. ' 33. Ph.ili~p Forchhei~eT: "Hydraulik" ("Hydraulics "), Teubner Verlag:sgesellschaft, Lelp,ag and Berlin, 1st ed., 1924, p. 495; 3d ed., 1930, p. 551. 34. Chi~-Hw~ Fan:.A study.oI stD,blc channel cross section (in Chinese), Hydraulic En(]~neer.~1!!l, Chtne.se SoCtely of Hydraulic Enginse'rs vol. 15 no 1 pp 71-79 Nanking, 1947. ' ",. " 35. E. W. Lane': Progrl:'ss report on results of studies on design of stable channels U.S. Bureau of [leclamation, Hyd'·(L!J.lic Laboraiot']J Repol·t, No. lIyrl-352 , J\1l1e' 1952. ' ,

in

191

36. A. C. ~~rter: Critica~ tractive fones on channel side ~lo]Jes, U.S. Bgrea.l! of Rcclmrt(lMTc, Hydrav.hc LaboratoJ'Y Report No. Hyd-3G6 (supersedes Hyd-295) Feb. 18, 1953. '. . . , 37. E. W. Lane ~tl1d E. J. CarIllOn: Some factors affecting the stn.bility of canals coustructed in coarse granular materials, Pr'oceedings of the Minnesota [nlernalional IIydmulics Convention, Sept. 1-4, 1953, Joint Meeling of Inte1'1wtional As.sociation for H ydt·l}.ulic Research and HydmldiC5 Divi.sion, American Bociety of Civil Eng,;rwers, pp. 37-48, August, 1953. . 38. R. E. Gloyer and Q. L. Florey: Sta.ble channel profiles, U.S. But'eau of Re,clama,ti01l, HydTflHl'ic Laboratory Report No. Hyd-325, Sept. 27, 1951. The work was started by R. G. Conard and revie.wed by E. \Y, Lane. 39. W ..0. Ree: Hy~raul.ic characteristics of veget.ation for vegetated waterways, Agnculttlral .Enqmeenng, vol. 30, 110. 4, pp. 184-187 and 189, April, 1949. ' 40. ~'. O. Ree and V. J. Palmer: Flow of water ill channels protected by vegetat.ive lmmg, U.S. Boil Co,~seTVation. Service, Technical Bulletin No. 9lH February 1949. 41. Stillwater Outaool' Hydraulic Laboratory: Handbook of chandel desig;ri' for soil an~ water conservation, U.S. Soil Conservation Service, SCS-TP-Gl, Marcll., 1947; reVised, ,June, 1954. . . 42. V. D. Fredcnhagen and E. H. Doll: Gmssed waterways, Agricullural Enuineming, vo!. 35, no. 6, pp. 417-419, June, 1954. .

.,\

.j

I ·1

-\ 1, (

·1

I

'l'HEORET1CAL CONCEPTS RELATED' TO 'UNIFORM FLOW

. CHAPTER I

lI

8

THEORETICAL CONCEPTS OF BOUNDARY LAYER, SURFACE ROUGHNESS, VELOCITY DISTRIBUTION, AND INSTABILITY OF UNIFORM FLOW

This chapter presents assorted theoretical concepts which have been developed in the mechanics of open-channel flow. These concepts, though not to be thoroughly discussed, may shed some light upon the solution of many practical problems. 8-1. The Boundary Layer. When watel' enters a channel, the velocity' distributionncross the channel section, owing to the presence of boundary

193

bution; (3) the depth of Row is indefmitely 1111'ge,so the depth of ftow cun be considered constant as the water enters the channeL 1n th!'i,~ha!!neJl the effect on the velocity di!J.trib!Jti®~t9. boun.1.::I£Y-XQ.hlghnes~ is indicatedbythe- ii~~" IlHf,- Outside the surface represented by ABC, --7'- th~ ;~I~city7 d'i;tribution is practically uniform. Near the channel surface and within the region ABC, v~locjty"'y£l.ries l\.ccording_tQ_ 4isww.e iromthe channel surface. . The region inside ABC, though not distinc'ti;;;,i;k~~~~ boundary layer 1 and its thickness is deSignated by o. Since the bouhdary layer is not distinctive, jiB. thi A common definition is that the lrei&-C,;11 _ ' ._~ thickne;:;s 6 is the magnitude of the ':((5 w/,bu l( ~n normal distance from the boundary ;ll~ft1,ceat ,~hi()h th;Vclocity ;; is ' equitnQ99%Of theli:rrd~~: -n:y Vo, ''I'hTcli.the velocity:'distribu1ion-:-b~ur:ve1n the boundary Tayer

1

approaches

asy!llptoticall~(Fig.

. g:<~---~-~ ....--_~

The effect of the bounda.ry layer on the flow is equivale!lt to a fictitious upward displacement of the channel bottom toa virtual position by an amOl1!l,t equal to the so-called displllcementthiclmess 0* (Fig. 8-2), which is defined as '

The e,1lranCE condition is hypolhelicol

\

r-

~

FlO. ~L Development of the boundary layer in an open chl1nnel with an idea.l entrance condltlOn. . .

i

i

roughness, will vary with the distance over wllich the water t!'avels in the;. channel. (Fig. 8-1).1 If· the flow is uniform and stable and if' the channel is prismatic and of constant roughness, the veiocity rustdbution will eventually reach a definite pnttern. For simplicity of discussion the following m'e assumed; (1) th~ fiow enteX'ing the channel is lamil1al' and of uniform velocity distribution; (2) no restriction exists at the entrance that will cause abrupt disturbanc~ of the water surface and the velocity distri. ,1 For· the sake of simplicity, the two-dimensiona.l profile of:a. wide open channel With exagger~ted verticll.l sc.ale isahown.

192

r",-

. '

(8-1)

"-

FlO. 8-2 .. Distribution of velocitv over a smooth channel Sur!l1c9{not in sC!lle). / '

where l' ill the velocity at any elis" / £-,.. . .. _--"-" tance'li from the channel surfnce ,!!-nAvl-is the 'yJ;!lGtci~t~Jit~!toftE.E!. _~9.1!.~l.g3lry.l~r.. The value of t.he displacement tl1ickness generally varies . from one-eighth to. one-tenth of the thickness of the boundary layer, depending on the magnitude of the Reynolds number. At the beginning of the now in the channel (Fig. 8-1) the ftow iii entirely laminar and a laminar boundary layer is developed along the channel surface, .as shown by the curve AB.· The velocity distribution in th~ layer is. approximately .parabolic. As. the water travels farther along in the channel; the ftow in the boundary byer will eventtmlly change to turbulent. (rhe point where the challge tn,kes place is indicated by B. Downstre~m from B a tl1rb1dent boundary layer is developed, 'as shown I

For a. compl'ehensive treatment of this shbject see !ll to [4].

; . I-I/:. .,. .If {J. ,(.r C> -=- vI(.')j"~oJ.fCL 1'r"G<:.-' .tId .;)' l.tfol V{~U_1. .d~ ~/..

_'

~-"

.t:d _ . i

/'/-t . . . ~7;1'.,A;..

<¥ /i

194

THEORETICAL CONCEPTS RELATJi!D TO UNIFORM FLOW·

by BC. The velQcity distribution in this layer can be shown analytically to be approximately log:1l'ithmic (Art. 8-4). If the channel suds.ce is relatively smooth, the velocity close to the channel surface is low; thus, a very thin stable film offiow known as the la:minar sublayer will be develope-d on the surface. Within the laminar sublayer the flow is kept laminar.l The top surface of th~ laminar sub~ ___ '-_~ layer corresponds to the transitional zone of flow from laminar to t.urbu-' lent (Art. 1-3) and, hence, cannot be precisely defined.

195

irregularities forming the roughness elements is called the 1'oilgl!1i68.s height! k. The ratio kl R of the roughlless height to the hydraulic radius known as tha relative roughne.ss. --_~""''''' If the roughness height is less t,han a certain fraction of the thickness j( -< So of the laminar sublayer, the s~rfa.ce irregularities will be so small that all ronghness elements will be entIrely submerged in the laminar sublayer (Fig. 8-3a).' Under this condition the roughness has no effect upon the fio'w ou.tside the lam.inar sublayer, and the surface is said to be hydrau_._.;;;> liGally sn;ooth. A hydraulically smooth surface is said to be wuu!LiUh.e aVer~iW.8Jl.....l1UrY:e (Fig. 8-3b). ---. . In connection with flow in pipes or on flat plates at zero incidence, Schlichting (see p. 454 of [ID . the following condition for a surface to be hydraulically smooth:

is

V,k

<5

or

k

<

51'

1

(8-2)

)I

where V f a term known as the friction lJe/ocitu (Art. 8-4). Using the Gh6zy formula. it can be shQ\rn~ above condition that, for a surface to be hydraulically smooth, the roughness height must be less than a critical rotlghne.ss expressed by

k.

'~-----Y-

5C ;

(8-3)

where C is CheZ~I's C, I' is th.e kinem(\tic viscosity, and l' is the mean velocity. The above conditionis·giveri for r0l1ghi1ess obtained with sand . having \'alue;s of C greater than 100, prob"bi,r. For the avcri1ge COlldi~ion, Schlichtillggives kc = 100"/1', which corresponds to C '" 113.5. ail approximD.tion, (8-3) U1:;ty be applied to channels. If the roughness height is gl"eater tha~l th'l critical va111e..d~y EJi: . (8-3) (Fig. 8-3c), the roughness elemeuts will have sufficient magnitude ~ angu!a.ritY..to extend...tht'lis~ffects beJ:::9l!.c!. the !l).mil~[1,r subl~r..i1ncl. thus to disturb the flow in the chi1nne1. The surface is therefore said to bC"ro1Lg/i:--in rough ch~i;,nJS, ~ the-~eI;;~ity distribution will depend 011 the form and ~ize of the roughness projections, .!1l1d a stable laminar subli1yer can no longer be formed.

If the conditions for ul1iformflow exist throughout the channel, the turbulen t boundary layer will be fully developed at section CD; thereafte! the velocity distribution w~e patt.er.!!,:. In a laporutory ·cha.nnel, the laminar boundary layer AB _gan be elh?inated easily by placing a..l."ou~hness el_e!l~e!l~ el1t.!~lce: Thus, the turbulent bounC!=ary layer will be developed very beginning of the channel, an~ the totallen&h of the :/lone for the full development of boundary la~l' c!tllbe slloi'·tel1~~ SinGe the flow in ordinary chanl1els is usually turbui~nt:the f9UO;:;;g articles will deal only with the turbulent boundary layer. . 8-2. Concept of Surface Rough~ The concept of the existence ofa laminar sublayer in the turbulent· boundary layer offers a picturesque explanation of the behavior of surface roughness. When the surface profile of a channel is enlarged (Fig. 8-3), it can be seen that the surface is composed of irregular peaks ~l1d valleys. The ejJectiue height of the i A refined concept of the Ia.minar subla.yer will consider that there exi~ts in the s\lblayer a small a.mount of eddy which dec~eases very ra.pidly to zero· at the boundary surfll.ce. .

=

,

, i

lI

I It should be noted that the roughness height is mel'eLy a measure of the linear dimension of the rQughness elcmeuts but is not necessarily eqoaL to t.he ac~ual, or ev~n an average, height. For example. two roughness elements ll13.Y have diffctcllt linear dirnr>,nsions, but, owing to the difference in shape and orientation, they may produce identical roughness effect and, thlls, their roughncsscs will be clesignated by the same roughness height. 2 The position from which the rOllghness hcight should he measured is IL disputable mli\tter. It is a.'<StlInJad here tha.t k is measured from 1\ datUllI that li,,~ at. n distance of O.5k below the average bottom of the channel.

I

~I 196

UNIFORM FLOW

r->'I'h~

average roughness height for a given surface can be determined by expenment. Table g··1 gives averaged from manv The con~ept, of l"~ughness in conduits was further advanced by' Morris [5]. Morns assumed that the flow over. a rough surface is due largely tl) roughness element. The o.f

I,

I ;

I I

.i

THEORETICAL CONCEPTS RELATED TO ·UNIFORM. FLOW

i97

of abnormal turbulence. 1 °In 'such a flow,therefore, the ratio yl).will be an importn.nt correlating parameter.. . ~si-~m.ooth ~o~..2.ccurs when_ the roughness elements. are.~.Q.~... together tb.!--'l!i..~' ski.ms the _cre5t..§.2l..the_.e!e~nt£, . The grooves between the elements will be filled with dead water contall1mg

!i

!.~etermiM,

and TA1l!.E

8-1.

ApPROXIMATE VALUES

OF

ROUG.lNESS HE1(lHT

. Mc.lerial

B:ra.ss, copper, lead, glass ... ". , , , ,. Wrought iron, steel. ...... ' ....... Asphalted east iron, . . . . . . .. . Galvanized iron ... ,., ........ '. Cast iron ............ , .... " ...•.. Wood stave ....... , ............. Cement. . . . . . . . . . . . . . . . . . . . . . . . .. Concrete., ........... , ...... , .... Dra.in tile .......... ,.............. Riveted steel. ..... , .............. Natural river bed. . . . . . . . . .. . . . . ..

k,

It

it

0.0001:"'0,0030 0.0002-0.0080 0.0004-0.0070 0.0005-0.0150 0.0008-0.0180 0.0006-0.0030 0.0013-0.0040 0.OU15-Q.0l00 0.0020-0.0100 0.0030-0.0300 0.1000-3.0000

tudinal ~f the r~~ss el~.ments·is the roughness dimension of paramount Importance in rough-conduit flow. Under this concept, flow ~~vel' IOtlgh surfa.ces can be classified into three basic types (Fig. : I 'L~olC!!:e.?::..~:01J.2hJ1.es.s flow, wake-interference flow, and quasi-smooth (01'

(0)

~

I

.-'--;:;,

. , 1" .

slcimmt'ng) flow.

---------

'

~ solated-roughne~~Jow.

prevails when the roughness elements are so far apart that the wake and VOl'tex a.t each element are completely developed and dissipated before the next element is reached. The apparent roughness would thereforere8ult from the form on the roughness elements, represented primarily by the height of projection k of the element, in addition to the friction drag 011 the waH surface between elements, ,which depends on the spacing of the elements. In such a flow, the. ratIO kl}.. may be taken as a significant correlating parameter influencmg the apparent friction factor in the flow. . Walee-interference flow :esults when the roughness elements are placed ,s~ clo~together tha.t the ~nd .YQrJj)x at each element wiUjnterfere WIth those developed !l:L~tJ resulting in intense and c.!?~pl~x."yor~i~ and tu~·bulen~g. hi such ~ flow, the height of the e~em~nt is. rehlthr ely unimportant, but the spacing is obviously of major ImpOl·tance. Also, the average depth y of flow above the crests of the elements will control in part. the vertical ex tent of the surface region

( c1

FiG. 8-4. Sket.ches showing concept of three basic typ;;s of How: (a) iso~ lll.ted-roughness. !low; (b) walJ:e-interf~rence flow; (c) quasi-slnooth tlcw.

_~ stable eddies, creating a£:?~1}. Large projections Illoe ~tisent from this pseudo wall, and the surfuce £!'cts hydrauli~l\.ll-y smooth. In such a flow, the ratio k/A (or .ill., where j is the groove width) will again be a significant parameter. Quasi-smo.Q.thJiy.:.:Y.....ba~ factor than flow ovel' a true smootll surfa£ft because the eddies in the groove~me a certain. am~f~gy. The nbove concept ap'pears to be substantiated adequately by experi1 Morris llsed the pipe radius instea.d of the depth in defining the parameter because he was ',;oncerned primarily· with pipes instead of channels;

198

UNIFORM FLOW

THEORETICAL CONCEPTS RELATED TO UNIFORM

mental 'data from many· different sources. The· concept can also be extended to surfac€s of variable roughness by u5ing average values of the roughness dimensions that vary or by combining the friction factors for· each flow type to give all over-all apparent friction factor of the flow. 8··3. Computation of Boundary Layer. For the development of a turbulent bound~ry layer in wide channels, ·an-;:pprQximate but practical meth~2!L h~_~.E!:Q12Q~_Qy_Jl~lJ&t.f.§1, This method waS' developed primarily for flo\v i.,1l; ~.han~els.oI large ~l'?.P.e, but it has been found applicable also to channels of small slope; provided the flow is . -.::- -.... ,-...... -- .--:'

. computing the varied-flow surface is used. In the case of tIle boundarylayer development of a uniform flow, the application of Bau61:'s method is just the same except that the water surface requires 110 computation, for it is simply parallel to the ehanneLbottom. Example 8-1. Aooncret.e overflow spillway of indefinite length has a. surface slope angle 11 = 53°8' (Fig. 8-5) and a roughness snch that k 0.005 ft. When the discharge is 3130 cfs per foot of spillway width, compute the length for boundary-layer dev!:')opment, the profile of the boundary layer, and the water surface, Solution.. The computation is shown in Table 8-2; the headings are explained as follows:

~

"

.j

now

\

,j

Col. 1. Arbitrarily assigned length of x in ft, !DeMured from 0 Col. 2.. Values of xlk, where k 0.005 ft Col. 3. V",lues of lJ/:z, computed by Eq. (8-4) Col. 4. Values of ~ in HI obtained by multiplying x by 8/x CoL 5. Velocity head vo'/2g in ft, equal to x sin·e = 0.80x Col. G. Velocity in fps corresponding to the velocity head in the preceding column CoL 7. Potentia.l thickness of Row in it, equal to the dischal'f!;e 1160 cis/it divided by the yelocity fl. . CoL 8. Actual thickness of Lhe flow in ft, equal to the potent.ia.l thickness plus the displacem<:lntthickness, which is assumed w be 10 % of the thickness 0 of the boundary la.ye!'

Nomin.o\ limit of boundory'loye.r

TA.BLE

k

Dam

8-2.

= 0.005 it

x/k

BOtiNDARY-LAYER COMPUTATION (J

53°8'

q

=

u. i /2g

li/x

360

cfs/n v.

glv,

y

I

( 4)

F" •.

50

Fm. 8-5. Growth of bounda.ry layer over the surfac<:l of an overflow Rpillway.

either accelerating or. uniform, and, if accelerating,

~~M t;-~l1!use boulldary~layer S~~~llS. -

100 200 400

th.~l£~~~idlr

600

.

The st,udy of boundary-layer de.velopment by Bauer was made on concrete overflow spillways (Ii'ig. 8-5). In this case the transition from the la~illar to the turbulent boundary layer usually occurs far upstream from the zone under consideration; hence the laminar boundary layer can be ignored, fo'r it is all insignificant part of the .problem. From the results of Bauer's investigation, the following equation may be written:' l) _

1 X 10; 2 X 10·

4 X 104 8 X 10' L2 X (05

0.0073 0.001313 0.0060 0.0055 0.0052

0.36

40

50.S

0.136 1.20 2.20 3.12

SO

71.8

7.09 5.00

7.13 5.07

160

101.5

3.~

3.65

320,

113.5 175.8

250

2.72 2.35

480

2.04

The computed water surfa.ce a.nd t,he bounda.ry layer are plotted on logarithmic paper (Fig. 8-6). The point of intersection of the two profiles indica.tes the location of the section where the bOlmdary layel: reache.s its maximum, or where it is fully developed. The lengtb of the development is shown to be approximately 460 ft. It shOUld be noted that the simplified method of 'computing the water-surface profile of a varied flow is justified in this problem because the flow is relatively thin and the slope is very steep.. As the frictional loss in this case is :negligibly small, the potential energy of water is practics.lly entirely converted into the lunetic energy expressed by the velocity head. For the computation of surfl:\Ce proj\Ies of varied flow in general, various methods will be described in Part Ill. To illustrate the effect of change in roughness upon the growth of a boundary layer, the computation has been repea.ted using ~he8amc da.ta but with a rOl\gbness twice as large, The line marked with It 11 = 8, k = 0.01 itO (Fig. 8-6) is the result of this computation. The change in water surface ispractic.!Llly negligible. It can be seen tha.t the difference between the values of the development lengtll for the two cases is

0.024.

x - (x/k)o.13 where 8 is the thickness of the turbulent boundary layer at distance x from 0 in the diiectionof flow (Fig, 8-5) and where k is the roughness height, Bauer has shown the application of this method by the 'following ·example. It should be noted that the flow on the spillway surface to be described in this example is varied and that a sirtiplified method of

1 I

! I

I I

.. I

i

J

200

~I',

UNIFORM FLOW

10

:FEfIi ! I t

9

,~ ! ~u{j ,

s

i'.r-.,

5

I,

t-

~>'er.....

~.,

~

I

"

"/.bf:

I

JiV

....

'+-

i

,

OJ U

,,'

~

i5

"..'"

't>'

"\<-

1

~I

O.

,

0.4

v: /

I r

[/1/

/" V ,

i-

{j

I

I

200 DistallCe x, f j

300

I

400

600

I

BOO

P,y./('·UViJ.f-

vgRS

(8-8)

Vr

. 8-~., Veloc!ty Distribution in Turbulent Flow. The velooity distributiOn In a ulllform channel flow will become stable when the turbulent ",-_Q.9un~ar~ la!el' 1l3i'!!!.v ,develoE~d. In the turbulent boundary layer, the, clistnbu:lOn can be shown to be approximately logarithmic. "," The shearing stress at any point in a turbulent flow moving over a solid surface has been given by Prandt.l [7] as T

= pl2

(::)2':

L~~:::) .~

0:.

tVl ~

(8-9)

Thili equation indicates ,that t~l:9,ci.~.J~he turbulel:t re~~~ logarithmic funotion of the distance .y. It is commonly known as the -. - . -~~-~ .. ~--~~ Prandtl-von Il..(irmdn· univeJ'sal-velocity-dist1-ibution Zaw. I This la.w has been VCl,jti.ed by several experiments [10J.' The results indica.te a striking i'imilarity betvveen observed and computed distributions and, therefore, offer reasonable justification for use of this log;arith' mic law in practical problems., When the sul'face is smooth, the constant yo in Eq. (8-9) has been found to depend solely ,on the friction velocity and the kinematic viscosity; that is, ' my (8-10) Yo = Vr

8-6. Solution of Example 8-1 for the growth of a boundary layer.

about 30 ft, or a variation of less than 10%. This shows that an increase in rough,ness has a tendency to speed up the boundary-layer growth or to',educe the developmenL l e n g t h . ' ,

(8-5)

where m is a constant equ~l to about

mass ~el1sity = wig, wh~re w i~ the unit weight of the fi.tiid and g IS the gravitational acceleration ': l ':'" a characteristic length known as; ~he mixing length ,........ , dvldy velocity gradient at a normal distance y from the solid surfp,ce

where

=

The quantity represented by V r has the dimensions of a velocity. Since wi~h the boundary friction To, it is known as the ~n.i]i;;jii1i or shea;' velocity. Thus, Eq. (8-7) may be written . vr'

'1-

.

-L-J

,

(8~7)

yo

-----iY' it varies

0.3 c,. 40 50 60 70 80 100

FIG.

I

I~ In JL

Vp

·where Yn is a constant of integration. From Eq. (7-5) and w = pg; it. CD,n be shown that

I .~

V,

v = 2.5

O·

'/

.,~

I

//"'~

I

o.61--:- f-. 0.5 f--.

'++

Lbo..:\) . '

t

.. I

oP~c:7 epV:l

J.QLth~n near the acEd surfac\l, ,Prandtl introduced two assumptions:' (1) that themixh}g 'length is proportional. to 'Ii, and (2) that the shearing stress is constant; Since the shearing stress at the, channel surface is equal to the unit tractive force (Art. 7-11), the second assumption 'gives T = TO.:, From these two assumptions,Rq. (8-5) may be . . wrItten (8-6)

where K is a constant for the proportionality between L~nd }I:, The value of K has been ·determined py many experiments [8] to be about 0,40. 11ltegr'ating li<J. (8-6), ' c.( '1 .=::> I '" ~ "3'

I

'b'

o. 9 0,7

r

/'

,\

201.

THEORETICAL" COl-iCEPTS RELATBD TO 'UNIFORM FLOW

r

p

% for

smooth surfaces. 2

,

For wavy

1 Von Karman .[9J also proved this law by II similarity hypothesis which assumes a linear shearing-stress distribution, the mixing length being proportiona.l to (dv /dy) /

(d~/dy·).

!

• Thill value is derived from Nikuradse's experimental data.

I

011

smooth pipes [111.

20~

! i

UNIFORM FLOW

surfaces, the' value of m wHI depend on the shape of the wa.ve profiles. Substituting Eq. (8-10) for Yo in Eq. (8-9). and simplifying, v

=

9yVI

575V . r log - v -

for smooth surfaces .

if:

= 5.75YI

log

30 -!Co

~Q'

for rough surfaces

=

P - 'YY

(8-15)

where P is the wetted perimeter and l' is a function depending on the shape of the section. Thus, the water area is equal to

(8-11)

'k

11

=

(

}o

.

B d.y = Ph - 2: 11,2 2

(8-16)

Substituting in Eq. (8-14) the value of v from Eq. (8-9), that of B from Eq. (8-15), n.nd that fO!' A from .Eq. (8-16) and then integra ling and

(8-12)

where the const.ant l m is equal to approximately Eq. (8-12) for YQ in Eq. (8-9) and simplifying,

203

proportional to its vertica.l distarice Y from the boundary i that is,

B

This equation gives the velocity distribution in turbulent flOlv OVer smooth surfaces. When the surface is rough, the constant Yo is found to depend on the roughness height ; that is,

v

THEORETICAL CONCEPTS RELATED TO UNIFORM FLOW

.<

Substituting

(8-13)

lli

equation' gives th~ velocit.y difitribution in tru:l:mlellt flow ml£,r surfaces. It should be noted. that thexoughness height k in this equation is the mean dilolmeter of the sand grains used by.NikIJm.d.®; ---~ . it is Imow!LSpecifically as the Nikurad136 8and rough!.!§M. It is to be noted that the constant.s nsed in the above velocity-distribution e@ations for smooth ®d rm~'faces were derived from data on pipes. It is assumed, however, tha.t it is permissible to apply the llniversal-Velocity-distribution h~w to other cases of turbulent flow, using the constants determined from tests on circular pigcs. • -. 8-0. Theoretical UnIfOrm-flow Equations. Using' the Prandtl-voll Karman universal-velocity-distribution law, KeulElgan (13)2 has derived equations for mean velocity of turbulent flow in open channels. . :I;<"o.r the sake of clarity l;1.nd simplicity, Keulegan's derivation will be modified below. By the cQntiuuit.y equation, the. total discharge through an ordinary channel section (Fig. 8-7) may be written

~h

Q

=

FA

t V=k v dA

=.

1.= O .

=

lch vB dy , 0

.

i'LG. 8·7. Channel

simplifying, the following equat.ion is

v=

.

or

-V = VI {5.75 log

1

[m~ exp ( -1 -~~) + 5.75 lug n~~}

=

(8-17)

0

+ 5.75 log mR) Yo

r

(8-18)

This is the general theoretical equation for the mean velocity of uniform . flow in open channels. For. smooth channels, Yo mn:y be represented by Eq. (8-10). Also,

1

[n

i

I

lf ...... .

obtl~ined:

[li!!:. exp (-1 _1'h~)l Yo R . 4A J

V Yr (A

where h is the depth of water, A is the vlater area, B is the length of the curve of equal velocity, and y is the vertical depth measured from the boundary to the curve of equal velocity. Since the laminar sublayer is relatively very thin, 00 can be assumed to he zero. It is further assumed that the maximum velocity is at the free surface and that the length B is

",,,',

2.5Vr In

to illustrate notation.

. In thp, above equation t.he quantity represented by the first term on the right-hand side is a function of· the shape of t.he chn.nnel section. However, the variation of this quantity with different shapes of the section is relatively small. F~r the sake of simlJlicity, the quantity may be represflnted by an ov~r-all constant" 11 0• This constn.nt will include not only the shupe function but also other uncertain factors, such as the effect of free ~urfn.ce and the effe.ct of nonuniform distribution. of the tract,ive force at the boundary. Accordingly, Eq. (8-17) may be wribten

(8-14)

This' value is derived from Nikuradse's experimental data on Tough pipes [121. Nikuradse's experjments, sand grains were cemented to the inner walls of pipes to . simulate roughness. Such roughness is known as artificial T"Ougimess. 2 Other references on this subject fLre [141 to r21J~

~ec~iol)

I t

* It is entirely feasible to simplify the derivation by assuming this constant at the very beginning. However, the procedure given here is aimed to give a logical demon·stration of the basic principles involved.

f

.J

-I

204

THEORETICAL CONCEPTS RELATED TO UNIFORM FLOW

UNIFORM FLOW

from ICeulegan's study "of Nikuradse's data [111,' Ao is found to be 3.25. The theol'eticaluniform-fiow equation for smooth channels is, therefore,

Y VJ (3.25 + 5.75 log R:/) " =

for smooth channels

"d' ber functions 0 f t h e F. rou e num . as follows: IT = VI

(8-19)

l'

Umng Bazin's data [22] for wavy surfaces, Keulegan obtained a value of Ao = 1.3 for smallwood channels and -3.0 for large wood channe1s. For rough channels, yo may be expressed by Eq. (8-12). From Keulegan's study of Bazin's data [22], the valu~ of Ao was found to have a wide range, varying hom 3.23 to 18.92. Thus; a menu value of 6.25 for Ao may be used. The theoretical uniform-fiow equation for Tough channels is, therefore,

v=

V, (6.25

+ 5.75JOg~)

for rough channels

=

205

Let" E<,qs. (8-1 g) and (8-20) be expressed

I _ _ RY /) "A, + ~.7u log--;-

for smooth channels

"I (AT + 5.7510g ¥ ) r o r rough channels

(8-2·1) (8-25)

·t'"

.. , .' f' ns of the }'roude number .. In the above eouatlOns, A. and £l~ ale .mc 10 . ft. The 1· d t' f" ct~ of other mlllOI' ac OIS. These constaill.s may al,;o ll,/; u e ne e Ie·" .' L

•

(8-20) 7

From the CMzy formula F = C ,IRS and from the definition' of friction velocity 1', = ',fiRS, it cltn be shown that

v VI

C

=

Yo

6

5

(8-21)

.;

4

"c:c

Substituting this equation ahd the Reynolds Il'lmber R = RV / ~ in Eqs. (8-19) and (8-20), the following express:olls fo\' Chez}"s C are obtained:

20.8R C -- 32"61 " ogc;-

for smooth channels

(8-22)

C

for rough (!hannels

(8-2:3)

3

4:

2

f

0

= 32.6 log 12}!!:

-I

After Keulegan's analytical study on laws of turbulent flow in channels, Powell [16-18J experimented with small rect~tnglllar sills used as artificial roughness and arrived at Eq. (5-5) for Chezy's C. Similarly, Robinson and Albertson [19] used baffle plates as artificial roughness and developed another empirical formula of C for rough chaunels. :Moore, Rand, and Hama [3] £\lso conducted similar studie:3, using transverse-bar roughness of various sizes. A further" study was made by Iwagaki [21] on experimental data obtained from many sources. The results of the study have disclosed that resistance to turbulent flow i;l open channels becomes obviously ·f--.»arger than that in pipes with increase in the Froude number. Iwagaki ~easonedthat this is due to the increa:led instability of the hee surface at high Froude numbers. Using the data presented by Iwagaki, therefore, it is possible to introduce. the effect of free-surface instability into Keulegan's equations by assuming that the constants in the equations are

j

-2 -3 0.2

0.3

0.4

F FIG.

8-8. Rel~tions amoog A"" rL, and F.

. . . F' 88 by the lotted data are found to be much scnttered(as shown III 'Ig" I - I "1 . . t " l ' ffects "When t 1e c lu,nne shaded boundaries) probably OWlIlgo t le:;e e . I h slope becomes too large, the flow will be unstab~e (Al:t. 8-8).. l~ suc cases, the \u,ws of turbulent flow described in tIllS o.rtlCle are no onger

P

valid." .. , R hess Coefficient. S-S. Theoretical Interpretation of M.an?m g s oug. n .. he theoIt should be interesting to study Mannmg s n by relo.tl~g l~ tfn\he preretical channel roughness for rough channels, as ~e(v8e ~~e Ma ning's n ceding article. Eliminating C from Eqs. (5-7) [1.n -~, n

-="

206

THEOBETICAL CONCEPTS RELATED TO UNIFORM FLOW

UNIFORM FLOW

may be expressed as follows:

n where

~ (~)k;6

(8-26)

(R/k)'A 21.9 log (12.2R/k)

(8-27)

The plot. of t.his equation, (Fig. 8-9) indicates that, for a wide range of R/ k, the variation in
A. M etlwd of Velocity M easuremellt. With reference to the logarithmic law of velocity distribution expressed ih Eq. (8-13), it can be seen that

the velocity distribution depends on the roughness height,- which may be related to Manning's n by Eq. (8-26). : In other words, the roughne8s in terms of Manning's n can be taken as a dominating fa,etor that affects the velocity distribution. If the distribution is known, the value of Manning's nean, therefore, be determined, This concept has been used by Boyer (24} and others [25-28] in estimating the value of 11 from the vertical velocit,y distribution ill a stream, Let VQ.2 be the velocity at two-tenths the depth, that at a distance O.8y from the bottom of a wide rough channel, where y is the depth of flmv, By Eq. (8-13), the velocity may be expl'essed a.s .

,

...---o~':-

S J feems in No,lhweslern U.S. (I

~ ..;,n.. _ _ _ _ II

"

'\.

Sfricklerts __ _ ~

~

=

,'"

Similarly, let

UO.s

be the veloeity at eight-tenths the depth; t.hen Yo.s =

R/l<.

5.75Yr log

Eliminating VI from the above two equations,

FIG. 8-9. Functiqn ",(Rlk)

I

median sieve size of the material. Data [24J· collected from sever",] streams in the l1orth\ycstcl'll United States and from the 'Mississippi River are also shown in. Fig. 8-9 for purposes of comparison, If
----

1 This constant has been converted to ft-Ib-sec units. in ft.

The value of k is expressed

Y

og'k =

O._.~77~8~x~ __.1....,. 3~8_1

x

(8-28)

!la.dvo.s.· Substi.tuting Eq. (8-28) ill Eq. (8-20), with R ~ y, "there x and simplifying, V 1.7&(:1: 0.95)

+

f='-~x-l

(8-29)

From Eqs. (5-7) and (8-21), with R = .1(,

V Equating the right-hand siues of

J

)

24y

5.75V r logT

Uo.t .

;;.

207

(8-30) (8-29) and (8-30) and solving for 1'1., (8-31)

This equation gives the va.lne for 1'1. for a wide rough channel with logarithmIc velocity distribution. When this equation is applied to a.ctual streams, the value of it may be taken as the mean depth. A pli1t of this equ:1tion (n/y;f, vs. x) has been made to compare with a similar curve developed by Boyer [24Jand also with actual observations taken . from several streams in the northwestern United States and the Mississippi Rivet;. The. comparison leads to the belief that a general