Chow - OPEN CHANNEL HYDRAULICS

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OPEN-CHANNEL HYDR~AULICS 
VEN TE CHOW, Ph.D. 
Proiessot of Hydtaulic En(finl!erin(f 
University of Illinois 
INTERNATIONAL STUDENT EDITION 
. '. 'T'1::' 
KHAL1D PHOTO STAlb 
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OPEN-CHANNEL HYDRAULICS 
INTERNATIONAL STUDENT EDlTION 
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VIII 
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In recent years water-resources projects and hydraulic engineering 
works' have been developing rapidly throughout the world. The knowl­
edge 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 perma­
nent" (" 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<lh of Stelldy Flows ill Ca.nals and Rivers It), Dunod, 
Pa.ris, 1954; Martin Schmidt, "Gerinnehyqraulik" ("Open-cha.nnel Hydraulics"), 
VE'B Verlag Teehn:ik-Bauverlag GMBH, B~rlin and Wiesbaden, 1957; N. N. PllV-
10vskiT, "Otkrytye rusla i sopdazhenie biefov sooruzhenil" (Open channels and adiust­
ment of water l~veh), in the "Sobranie sochinonil" ("Collected Works"), voL 1, 
pp. 309-543, Academy of Scieooes of U.S.S,R., Moscow and Leningrad, :1955; and 
the new edition of M, D. Chertousov, "Qidraivlika" ("Hydraulias':), Gosenergoizdat, 
Moscow and Leningra.d, 1957. 
• Boris A. Bakhmeteff," Hydraulics of Open Channels," McGraw-Hill IIook Com­
pa.ny, Too" New York, 1932; ao.d Sherman M. Woodward a.ud Chesley J. Posey, 
"Hydraulic.s of Steady Flow ill Open Channels," John Wiley and SOIlS, New' 
York,' 1941. ' 
yii 
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viii PREFACE 
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 to the reader, but they are liste:d 
for academicaud historical intercst. 
In essence, the book is the outgrowth of the author's 20 year,,' experi­
ence 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 insti­
tutions and laboratories in other countries and exchanged knowledge with 
their staff members..In 1956 he visited England, France, Belgium, the 
Netherlands,.Germany, Italy, <I.lld Switzerland. In 1958 he visited 
Austria, Turkey, India, and Japan, and again England, France, and 
Belgium, The information obtained from these countries and from other 
countries through publications and correspondenc'e was eventually added 
. to the final draft of the manus!}ript as supplements to the American 
practice. . 
The text.is Ol:ganitlec,i into five parts-namely, Basic Principles, Uni­
form Flow, Gradually Varied Flow, Rapidly Varied Flow, and Unstea,dy 
Flow. The fin;t three parts cover the material which would ordinarily be 
treatedin a one-semester .course 011 open-channel ;hydraulics. For a one~ 
semester course on the! design of hydraulic structures, Chaps. 7 and 11 
and part IV should suJ:iply most of the material fpr .. the t'eaching purpose. 
Part V on unsteady fldw may' be used either fOl' advanced studies or as 
supplemental material ;to the one-semester course, depending If),rgeiy 011 
the discretion of the irlstructor with .reference to the time available and 
the interest 'ahown by the students. . 
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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 insignifi­
cant 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 suit­
ahlf?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 interpre­
tation 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. intro­
duced. ,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 sedi­
ments, As will be stated later, the sediment transportation il'; consid~red 
as a subject in the domain of river hydraulics, Therefore, only the chan­
nels: of the first type, which :carry relatively clear wa~er in stable con-
dition, are treated in this bodk ,. 
Iii Part III ongl'adually va-ried flow, se,,-eral methods for the compu­
tation 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 
I its original size. This extended table \1nd a table for negative slopes were i • i' 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) J'" . conduits, a varied-ft.ow function table is also· provided in Appendix E. : 
l 
. The method of singular point is a powerful tool for the analysis of flow 
, • profiles. Since this method requires the use of advancedmatheml1tics} 
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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. . 
In Part IV on rapidly varied fiow, the treatmellt of the problems is 
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 
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£ 
advanced Tiiathematics beyond the scope of this wbrk. 
In Pnrt V on llnsteady:fiow, the keatment is general but practical. It 
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. . 
It should be noted that the subject matter of this book dwells mainly 
on the flow of water in channels where water contains little foreign mate­
rial.· Consequently, problems related to sediment trAnsportation and 
air entnl.inment are not fully discussed. In recent years, sedirrient tr!lns­
portatim in channels has become abroad subject that is generally COvered 
in the study of river hydraulics, which is often treated iI).depc:mdently.1 
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" ("Calcu­
la.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 agencieshl1ve ~!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 Labora­
tory, 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, Bel­
gium; 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, Gel'many.. , 
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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xii 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. 
Ye.nTe Chow 
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CONTENTS 
Prefacs . 
PART I, BASIC PRINCIPLES 
Chapter 1. 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. Open Channels and Their·Properties 
2-1. Kinds of Open bl~annel 
2-2, Channel Geometr)~ . 
2-3, Geometric Elements of Cha.nnel Section, 
2-4, Velocity Distribution in a. Channel Section. 
2-5. Wide Open Channel 
2-6. Measurement of Velocity , 
2·7. Velodty-dbtribution Coeflicienu, . 
2·8, DeterminaHon of Velocity-distribution Coefficients 
2-9. PrllS<!ure Dh;tribution in a Channel Section. 
2·10. Effect of Slope on Prcssui'e Distribution. 
Chapter 3. 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. Critical Flow: Its CQl~nputation and Appllcations . 
4-1. Critical Flow 
4-2. The8ection Factor for Crit.ical-fiow Computa.tion . 
_ 4-3. The HydrILulic' for Criticru-fiow Computation 
4-4. Computation of Flow. 
4-5. Control of Flow. 
4-6; Fl9W Measurement. 
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xiv CO!-T'l'ENTS 
PART 11. UNIFOR.M FLOW 
Chapter 5. Development ()f Uniform Flow and Its Founulas 
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. Computation of Uniform Flow . 
6-1. The Conveyance of a Channel Section 
6-2., The Section Factor [or Uniform-fio;; Computation . . 
6-3. The Hydraulic EXjlQnent for Uniform,.flow COmputation. . 
6-4. Flow Chua.cteristics in Jl. Closed Conduit with Of/en-channel Flow 
6-5. Flow in a Channel Section' with Composite Roughness 
6-6. Determination of the Normal Depth and Velocity. 
6-7. 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. D<"sign of Channels for Uniform Flow 
A. NONERODIBLE CHANNELS 
7-1. The Nonel"Ddlble Channel. 
. 7-2. Nonerodible r-,·laterilll and Lining. 
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, ERODIBLE CHANNELS WHICH SeOUR BU'l' Do NOT SILT 
7-8. MethOds of Approach . 
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. 
C. GRA8SEDCHANNELS 
7-16. The Gra.ssed. Channel . 
7-17. The Retardance Coefficient 
7-18. The Permissible Velocity 
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7-19. Selection of Gr8.53. . 
. 7~20. Procedure of Design 
CONTENTS 
Chaptllr 8. Theoretical Concepts of Boundary Layer, Surface Roughness,. 
Velocity Distribution, and Instability of Uniform Flow. 
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. 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. The GrD.phical-int!lgration Method 
10-2. Method of Direct Integration. 
10-3. The Direct Step Method 
10-4, The Standard St.ep Method 
10-5. 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.icalFlow 
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 V flried Flow 
12-1. Basic Principles and' Assumptions 
12-2. Dynamic Equa.tion for Spa.tia.lly Varied Flow 
. ~ -i : 
xv 
184 
184 
.-192 
192 
194 
198 
200 
202 
205 
206 
210 
217 
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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xvi CONTENTS 
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. RAPIDLY VARIED FLOW 
Chapter 13. 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. Flow in Channels of Nonlinear Alignment 
16-L Nature of the Flow . 
16.2. Spiral Flow . 
16-3; Energy Loss. 
16-4. Super elevation 
16-5; Cross Waves. . . . .:. . . . . 
16-6: Design Considerations for Subcritical Flow. 
16.7. Design Considerations for S~percritical Flow 
333 
341 
346 
347 
357 
3.57 
'357 
360 
360 
362 
363 
365 
368 
~70 
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 
l..i 
CONTENTS xvii J( 
Chapter 17. Flow through Nonprismatic Channel Sections 461 
17-1. Sudden Transitions,. 
'--:,.... 
461 
17-2. Subcritica.l Flmv through Sudden Transitions 464 
17-3. Contractions in Supercritical Flo''''' 468 \...-
17-4. Expansions in Supercritical Flow. 470 
1t-5. Constrictions 475 .', 
17-6. Sl!bcritical Flow through Constrictions 476 ''''-
17-7. Backwater Effect due to Constriction 490 
17-8. Flow through Culverts. 493 L .. 
17-9. Obotructions. 499 
17-10. ro'loW' between Bridge Piers 501 , 
17-11. Flow through Pile Trestles 506 '-...-
17 .. 12. Flow through Trash Racks 506 
17-13. Underflow Gates 507 \., 
17-14. Channel Junctions 512 
(-
PART V. UNSTEADY FLOW 
l~. 
Chapter 18. Gradually Varied Unsteady Flow 525 
18-1. Continuity of Unsteady Fiow 525 ,~ 
18-2. Dynamic Equation 'for Unsteady Flow,. 626 
. ( 18-3. Monoclinal Rising Wave . 528 
18-4. Dynamic Equa.tion for Uniformly Progressive Flow 631 
18-5. Wave Profile of UnIformly Progressive Flow 533 
18-6. Wave Prupagation . 537 
( 
'--' 
18-7. Solution of the UnsLeady-flow Equations 640 
18-8. Spatially Varied Unsteady Surface Flolv 543 l_ 
Chapter 19. Rapidly Varied UnsteadY Flow 554 
-.' 
19-1. Uniformly Progressive Flow 554 
19-2. The Moving Hydraulic Jump, 557 C 
19-3. Positive Surges . 559 
19-4. Negative Surges. 566 Ii 19,5. Surge in Power Canals. 568 '-...,' 
19-6 .. Surge in Naviga~ion Canals 572 l. 
19-7. Surge through Channel Transitio!IS 575 I.. 
19-8. Surge at Channel.Junctions 578 
19-9. Pulsating Flow. .. 580 
i, __ 
Chapter 20. Flood Routing 586 
( 
'-20-1. Routing of Flood ,. 586 
20-2. Method of Characteristics. 587 
20-3. Method of Diffusion Analogy. 601 \ 
~, 
20-4. Principle of Hydrologic Routing 604 
\ 20-5. Methods of Hydrologic Routing 607 i 
20-6. A Simple HydrologIC Method of Routing 609 
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CHAPTER ,1 
OPEN~C:HANNEL FLOW AND ITS CLASSIFICATIONS 
1-1. Description. The flow of water in a conduit may be either open­
channel flotu or 'Pip~ flow. The 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 h~@t.2 '., . . 
~ 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 
: 1 It is here assumed that the velocity is uniformly distrib\.ited across the conduit 
s~ction; othenvise a correc,tion ,,",ould have to be made, such Ilf is desaribed in Art. 2-7 
fbr open chanrrels.! , 
: 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<ilts. 2-9 and 2-10). As 
a result, the hydraulic grade Une would not coincide exactly with the water surface. 
3 
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. '-. ... 
) 
" 
xviii CONTENTS 
Appendix A. Geometric Elements for Circular Channel Sections 
Appendix B. Geometric Elements for. Trapezoidal, Triangular, and Parabolic 
Cb-annel Sections 
Appendix C. NQIDographic Solution of the Manning Formula 
, Appendix D. Table of the Varied-flow FuncH,ms 
. r " dlt r" dtl 
F(u.,N) = Jo I-UN and F(u,N)_s, = )0 l+'uN 
Appendix E. Table of the Varied-flow Functions for Circular Sections 
Name Index. 
. Subiect Index 
.~ 
625 
, , 
629 ! 
640 
'~ 
·PART I 
BASIC PRINCIPLES 
641 
657 
663 
669 
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4 BASIC PRINCIPLES 
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. 
Reliableexperimental 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 
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 open­
channel 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. 
, ... 
OPEN-CHANNEL FLOW AND ITS, cL'ASSlPICATI0NS 
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 consider­
ation. 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) 
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<!,.ntinuo,,!-s A.!?llI,_.i~.J91!gQ.._tt.:l._!()a-!si~~e gutters., side-channel 
spillways,' the wa.shwa.te.r troughs in filters, the efHuent channels around 
sewage-treatment tanka, and the mrun drainage c~annels and feeding 
channels in irrigation 8YSt~lDS:-------'-'----- ----~--.-.-. --_ .. 
--The la\v~ntiriufty-or unsteady. flow requires consideration of thEf,,, 
--,--~time effect. Hence, tJ?'~. ~<?f:1tjn~i~ ~~~a~i..Ol~J.oE-~(.:mtiIll!qu.s_~.~tead"y 
flow should include the time element as a variable (Art. 18-1). 
Uniform Floi»-'anil VariedJji;;;;aHu;;;;;'s thecrit;iQ;!:-:.Opell-channel 
flow is said to be uniform if the depth of flow is the same at every section 
of the channel. A uniform flow may be steady or unsteady, depending 
on whether or not the depth changes with time. . 
.P. • .... [Jteady uniform flow is. the fundg,mental type of flow treated in open­
channel hydraulics. The depth of the flow does· not change during the 
time interval undercons'ideration. The establishment of unsteady 'Uni~ 
form flow would require that the water ~urface fluctuate from time to 
·time w.!!ile reE}aininlLIW:!eJlel.!Q.t~nl1el.tott;om: Obvio~his 
I 
6 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. 
Uniform 'flow - Flow in a 
laboratory' channel 
Sluice 
Confraction 
below the 
sluice 
G.v. F. - Flood wove 
Chong'e of depth from 
time to time 
Unsteady unifarm flow - Race 
RV.F. G.V.F. RV.F. 
Varied flow 
----:---~-
~- -'. 
' , 
// 
R.V.F. - Bore 
lJnsteqdy flow • 
FlG. 1-2. Various types of open-channel Bow. G.V.F. = gradually varied flow; 
RV.F. ~mpidly varied flow. 
I, 
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j. 
OPEN-CHANNEL FLOW AND ITS CLASSIFICATIONS 7 
For clarity, the classification of open-channel flow is summarized as 
follows: 
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 signifi­
cant 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~: .-.-.---.--.----.-.- --
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 tran­
sitional, state. 
The effect .of viscosity relative to inertia can be represented by the 
Reynolds number, defined as 
R == VL 
JJ 
(1-3) 
w here V is the velocity of flow in fps; L is a characteristiclength 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 
t ~'. 
http:equal.to
\..J 
I , 
"-
, , 
8 . "BASIC PRINCIPLES ' 
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) 
f.I = 2.09 X 10-5 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. ' - --'--.• -.-.--~--'-''''------
~Th~-i~mi~~:, turbul~ntJ 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.<!~veloped 
f?r flow 11l plP£lS- The Darcy-WfJisbach formula, 1 also developedpJ:i~;' 
nly for flow in pipes,is 
(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 :t, V IS the velocIty of flow in fps, and g is the acceleration due to gravity 
In ftfsec~. ' 
Since da = 48 and the ~ne~gy gradie.u,t S hJlL, the above equation 
may be rewritten for the friction factor 
1. .. " (1-5) 
I 
:'hls equation may a.lso be applied to uniform ahd nearly uniform flows 
~/-m...0pen channels. ' , ---________ _ 
. Th
t
? f-R[IiJrelati~l1S~pJ.~smo~t~J;_~~~ can ~~~e,!2I'.!lssed bv th, e Blas:ius 
equa ton '. '. - ----"-,---- , .. 
. ---' -- ., 0.223 
f = RO.~' (1-6) t 
;. It snould be noted that there is actually no definite upper limit .. 
,! .1 t8 a result ~f Darc(s. stud.: [21: ~n flo~ in pipes, his name is commonly associated 
wlthithat ~ WelS~o.ch [31 In deslgna,tJng~hlS. equation which Weisbach firstformulated.. 
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. 
I 
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OPEN-CHANNEL FLOW AND ITS CLASSIFICATIONS 9 
v..'ith the data obtained b Nikuradse 8 ~ The resulting Prandil-:von 
d1'man equ~..!4sm Is 
1 
2 log (R ...;'1) + 0.4 ' (1-7) 
-'--~Eciuations (1-6) and (1-7) will be used i'n the follo\ving discussion as 
I), basis for comparing flow conditions in open channel.s. It may be noted 
that corresponding equations forflO'wi'n open channels have been derived 
by Jf~~~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 in open-chann.erfloware described below. ,t 
Experimental data available, for the .determinationof the f-R relation­
ship in open-channel flow can be found in various publications on hydmu-' 
lics.1 Figure which plots the relationship for flow iIi 
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) 
, r'l 
From Eq;S. (1-3) and (1-5) it can be shown that 
i • ~ I 
, K = 8gR2S 
I, ! .V' 
(1-9) 
1 Se~ [l(~1 to [23].' ! ':. 
2 The da.;ta. for the rectangular channel; were furnished through the cdurtesy of Pro­
fessor W. :/;1. lAnsford ann processedJoi the present purpose by the author. 
, 
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I 
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> ree­
tanguJar channels and 14 fol' the triangular channel under consideration. 
f 
to 
eta 
0 •• 
., .. 
0.1 
om 
0.06 
().04 
0.02 
0.01 
0.000 
1 
-i.lli:vtRS;'TY n,= fLl!NOlS DATA 
G fI£CTAlfGlA.AR c.HAflNa~,1.5Ff \\'10£. wlTK GL,us WALlS 
&. POI.JSH(O 9Fl4SS PLATE 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; or Sl'Flt;CTtiRAL srUL 
o nU4NCAA...A.R CHA'HN£l.,l'd" TO fsd''V~Rrt)t ,t~I.ES,WlrH 
O.OO41--+-+++4-+-~-+-'f+-+-'-JHh-+-'-I--+-l-+I--!·-++H 
4 • $1'0' 
R 
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 obtainedl at the University of Minnesota [20J 
and the da~ for turbulent flow cdllected individually by Kirschmer 
f 
\ 
\ 
I, 
0.2 
0.08 
O.Oi 
O.qz 
Q k 100.0rt9fM 
.. k. 0.02.831" 
e R .. ',{I td 
o F: '. a.o AI 
(C R • 3,0* 
$ R ., 4,01;:11 
• a .. 4, a til 
~ R .. ~4 tK 
e R • 0.0 tloll 
(I A ... 1.:2 C!A 
11 
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0.008 
10 
.' . 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.
t
;n sp.~in'" of 50 mm; No. 24, cem, ent 
.. 7 Ne 14. except WI .1 a .Q " 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 . " 
. . . (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 
sen Y k whic IS a SIze m . ft' s' repre , . 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 er. . 
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12 BASIC PRINCIPLES 
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 rough­
ness 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, tri­
angular 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 
c haI!!l E11.J:{:!§.i..frul.nce. .... 
3. In the turbl,1lent region most plots appear parallel to the Prandtl­
von Karman curve, This curve serves as an approximate limiting posi­
tion 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 experi­
mental studies .seem necessary to substantiate it. 
~-' -~It should be noted that the ab'Jve descriptions are limited to low­
velocity, 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, 
1 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 Rey­
nolds nu~ber increases, the fio'IV may be cha.nging from quasi-smooth flow to wake­
interference flow, and then tomolatecl·rou.ghness flow (~t. 8-2). 
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[ OPEN-CHANNEL FLOW. AND ITS Ci..AftSIFICATIONS 13 
j) -1. 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 repre­
sented 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/secz) 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 rec­
tangular 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 > v'{jD; 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.<n'§..Jl.nd.. 
below the mean surface level and thus create waves th~t...!~~~..2'::~g.l.!-l 
.or gravity force. It should be noted that a gravity wave can be propa-
... gated upstream in water of subcritical flow but· not in water of super-
.S!"'1 G.€: critical flow, .since the celerity is~ater tha11 the velocity of flQ}! in th.e 
E,.!,. > IL .1.Q!m~t.~~e and less in the latter. T)terefore,~the possibilitY_2I'il!!PQ§.sJ:-
<dl.:(',( f ~ of propagating a gravi~y wa'::~)J.:2stream can be used as a criterion 
.0(. for aistinguiShing between subcritical and supercritical flow. n -2 Since the flow in most channels is controlled by the gravity effect, a 
;::..-~ model used to simulate it prototype channel for testing purposes must be 
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 kinel­
icity or velocitY-head ratio 11; = V'/2gL, proposed by Stevens [28] alld Posey [29J 
respecti vaLr. . • 
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14 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 
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 open­
channel 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 
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OPEN-CHANNEL FLOW AND ITS CLASSIFICATIONS 15 
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 uni­
form 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 super­
c:'1.tlCal-turbulent flow changing to varied subcritical-turbulent. In both 
cases, the diffusion of dye is the evidence. of turbulence. 
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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<l!:...~ 
~Y~.lI!:_~!"~~~(L1:yj.ili increas~de n1Enbt:~. Generally, I;}le effect 
of gravity is practically negligible where the Froude numbeds small, say, 
less than 3. A further irivestigation by Iwagaki [34J indicates that, with 
'increasing Froude number) the friction factor of turbulent flow in both 
smooth and rough open channels becomes lal'ger than that .in pipes. 
is possible that the presence of the free surface in (J~n~channel flow makes 
. the channelrougw- than the pipe.--Whellmore data -and --:evidell(;e 
oecome avaiTab-le, the'-Froude number, ,representing the gravity effect, 
may have to be' considered as an additional faotor in defining the J-R 
relationshipfm' supercritical-turbulent flow. ' 
PROE\LEMS 
, 1-1. 'With reference to Fig. 1-1, show that the theoretical dischl.i:ge of the open­
channel flow may be expressed by 
0-12) 
where A 1 and A. are the cross-sectional areas of the flow at sections' 1 and 2, respec­
tively, 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 
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; 199-
224, 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" (,IText-
book 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. ' 
~l 
I 
I 
! 
! 
I 
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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 decou­
VllJ'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-
18 BASIC PRINCIPLES : 
b!lrt C. Nels01l, Trqnsaction8, American So.ciety o.J Civil Engineers, vol.. 123, pp. 
707-712, 1958.. . 
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 constric-
tions), 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: 1125-
1135, 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. 1135-
1138, 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'<!rcritical flow in open channel with smoDth 
wa.lls) , WGSS/!iTkraJt unci Wasserwirf.8chaJt, Munich, ·vol. 35, no.. 3, pp. 55-59, 1940. 
32. Yuichi Iwagaki; On the laws of resistance to turbulent flow in .open smooth 
channels, llfemoirs of: tJte Faculty oj Engin.eerill,f/, Kyo.to University, Jilpan, vol. 15,· 
no. 1, pp. 27-40, Janua.ry, 1953. 
33. Masashi Hom-rna: Fluid resistance in water flow of high .Froude number, Pro­
ceedings oj the ed Japan National Congress Jor Applied Mechanics, pp. 251-254, 
1952. 
34. Yuichi Iwagaki: On the laws of resistarrce to turbulent flow in open rough channelS, 
Proceedings oj the 4th Japan N a./irma!. C(mgres8 Jor Applied Mechanics, pp. 229-233, 
1954. . 
.j 
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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 chan­
nel may be either natural 01' artificial. 
Natural channels include all watercoUl:~ that exist naturally on the 
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 irregu­
lar. 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 
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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 com­
paratively 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 geo­
metric 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 
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 .. 
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22 BASIC PRINCIPLES 
egg-shaped, ovoid, semielliptical, U-shaped, catenary, horseshoe, basket­
handle, '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 supportedat 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 geome­
try 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 dimen­
sions. 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 chan­
nels, 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 direc-
tion bf flow. ' 
I Many typica.! seWer sections arEi described in [2J nnd [3]. 
I 
.! 
OPEN CHANNELS AND THEIR PROPERTIES 23,\ 
The hydr.aulic radiu,$ R is the ratio of the water area to its wetted 
perimet.er, or. 
R A 
(2-1) 
The hydr~ulic de1Jth D is the ratio of ~le wa.ter area to the top width~ or '1 
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) 
. 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. . 
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Tabie 2-1 furnishes 11 list of formulas for six basic geometric ~lements 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. . ' 
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24 BASIC PIIINCIPLES 
, 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. 
, ' ' \ 
~~' , ' T:44" ....&.. 
~~ .~' 
C~b=20':, .1' \ 
'~//l7~~0~ 
P=46,8 - , 
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,·,: 
2-4. Velocity Distribution in a 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; 
FIG. 2-3. Velocity distrib,ution in'·a. rectangula; 'channeL 
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 
OPEN CHANNELS ,AND 'TIIEIR PROPERTIES 
~',.# 
~' Q,?!/' , 1.5 
. 
t~.5' '. 
, . I 
Tropezuidal chonnel 
Triangular channel 
Sholtow ditch 
Pipe 
Natural irregular channel 
25 
. NarroW' 
(@c!anglJlar 
5~ction 
FIG. 2-4. Typical curvea of equal velocity in various channel sections. 
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 veloc­
ity 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 
on velocity distribution. 
As revealed by cv.refullaboratory investigations, 
the flow in a straight prismatic channel is in fact 
three-dimensiqnal, manifesting a spiral motion, 
i 
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bed 
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 unavoid­
able, 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. 
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26 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) 
(d) ConI our lines 01 
. equal component (vyl Jr 
Co I Con lou, lines of 
equal componenHv,1 
(e) Dire clio n Jines and 
ma9niludes of the 
IOlerol currenlslvor' 
FIG. 2-6. Distribution of the velocity components, fa.cing downstream a.t the mid­
section 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 a rectangular channel of 
infinite width. In other words, under this condition, the sides of the 
!iliJ!imel h~i practicallyDO 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 
~ I . 
OPEN CHAN)lELS AND THEllR PROPERTIES 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 proce­
dure 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 coef­
ficient 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. 
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28 BASIC PRINCIPLES 
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 com­
putation. Therefore, the coefficients are often assumed to be unity. In 
channels of complex cross section, the coefficients for energy and momen­
tum Can easily be as 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 irregu­
larities in alignment, values of a greater than 2.0 have been observed. 1 , 
Precise studies or analyses of flow in such channels will require measure­
ment 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. 
Vo.lue of", ' Value of {j 
Channels 
Min Av 'Max lVIin Av Ma.x 
'-"0-
Regular channels, flumes, spillway:;: .... , 1.10 1.15 1.20 1.03 1.05 1.07 
N o.tural streams and torrents ..... ~ . ~ . 1.15 1. 30 1.50 L05 1.10 1.17 
Rivers.under ice cover., ..... ....... 1.20 1.50 2,00 1.07 '1.17 1.33 
River vlllleys" ove,flooded ..... ...... 1.50 1.75 2,00 L 17 1.25 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:wvll 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' measure­
ments 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 
OPEN CHANNELS AND THEIR PROPERTIES 29 
corrected velocity head for the whole area as a V 2/2g, the total kinetic 
energy 'is ,aw va A/2g. Equating this quantity with 2:wv 3 AA/2g and 
reducing, 
J 1]3 dA L;'~! LlA 
V3A ,'"'" (2-4) 
The momentum of water passing AA, per unit time is the product of 
the mass W<I AA/g and the velocity v, or wv2 AA/g. The total momentum 
is :Z1l!V2 AA/g. Equating this quantity with the corrected momentum 
for the w'hole area, or {3wA V2 / (I, and reciuping, 
, Jy2 dA 2:v2 LlA 
{J = V2 A "". 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 planim­
etering again. Similai:ly, integrals 1:v2 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 = 1 + 3.% - 2t3 
'p 1 + f2 
(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 
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30 BASIC PRINCIPLES 
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 
B 
(al 
, 8 s' 
(c 1 
FIG. 2-7. Pressure distribution in 
straight and curved channels of 
small or horizontal slope at the 
section under considera.tion. h = 
piez~metric head; h. = hydrosta.tic 
head; and c "" pressure-bead cor, 
rection for curvature. (a) Par~ 
allel flow; (b) convex flow; (c) 
concave flow. 
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 sur­
face. Therefore, the pressure at any 
point on the section is directly propor­
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 dis­
tribution in the ,crQ§s se~tiQn of a flQwing 
channel is valid only if th&.!o\v filaments 
have no accele~~n coinponertts in the 
·plane·gf cross sectioE' This type of flow 
~oretically known as parallel flow, l 
that is, such th'at the streamlines ha VI! 
neither substantial curvature nor diver­
gence. ~~uently, there are n.o ag­
preciablc acceleration component.§..JlQ!:= 
mal to the direction of flo'\' that \VO~ 
disturb the hydrostatic-pressure d~J;.iQ.u in thfLcros~ctien of-flt-­
, ~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~, 
TI " t ec't V'Nhether the energy coefficient or the momentum coeffiCient ,IS 
1IS IS no carr . ., .. 1 d 
to be used depends on whether the energy or the moment?m prw::lple IS Ill:ro -:,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]. ' 
OPEN CHAN,lIjELS AND THEIR PROFER'I'IES 31 
_----7 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 appre­
ciable 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 curvi­
iinear 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:l­
ward 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 hydro­
static pressure of a parallel flow. Similarly, when divergence of stream­
lines is great enough to developappl'eciable acceleration components 
normal to the flow; thehydrostati'c pressure distribution will be disturbed 
accQrdingly. 
J, 
Let the deviation from an otherwise hydrostatic pressure h. in a curvi­
linear 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 ' 
~.-~--- . 
p = 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'esstire­
head correction is, therefore, 
d v2 
c=-­gr 
(2-9) 
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 
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32 BASIC PRINCIPLES 
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' =~ r A hI! dA 1 + 1 r A cv dA (2-10) 
~y}o . }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<L§trea!!J.1ines l,:!9ssess 8ubstfilliia.UIJD!J)J,m1Ul.nd di;v~nce. ~ 
seguently, the hy~ro8tatic law of pressure ,distribut~s not hold strictlJl. 
i!!!.IJ!1riifl:U...Y!1:riedJto£ . 
- It should be noted at the outset .that throughout this boqk flow is 
treated in general as either parallel or gradually varied. Ther~ore, the 
effect of the curvature of streamlines, will not be considered (~hat is, it 
will be assumed that a' = 1) unless the flow is specifically described as 
either curvilinear or rapidly varied. 
2-10. Effect of Slope on Pressure Distribution. With fPjerence to a 
straightsloping channel of unit width and slope angle 6 (Fig. 2-8), the 
weight of the shaded :"vater element of lengt;h dL is equaltq..:l:!:'lUlQ~. 
The pressure due to this weight is wy cos2 fJ dL. The unit pressure is, 
therefore,equal to wy cos~ e, and thehead 1 is 
h = Y cos 2, e 
or h = d 90S e 
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 pres-
or--, ' , 
1 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]. . 
i 
I 
I 
. i 
OPEN CHANNELS AND 'l'HEIR PROPERTIES 33 
sure head at any vertical depth is equal to this iiepth multiplied by a cor­
rection fact·or cos2 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 
'on. ~ertical section A'C 
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 appreci­
able. 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 cos2 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~ 
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 or discharge, etc •• all have some; 
bearing on a.ir entrainment. ' . ; 
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34 BASIC PRINCIPLES 
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 surfafaT or the floV\L For practical purpo;es, however, the density 
may be assumed constant; this assumption of I!!llform air distributi_o_I?:_i~_ 
Convex, flow 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 /a) con· 
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 
. [ 
1 
f 
OPEN CHANNELS AND 'rHEIR PROPERTIES 35 
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. 
Distance f,om Distance fro~ 
a reference point Stage, a referenc(J point Stage, 
neClr left bank, ft ft near .left bank, ft ft 
Left bank: -5 5.6 7 -0.1 
-4 4.6 9 -0.1 
-2 4.0 11 -0.4 
0 1.9 13 -0.1 
1 0.8 15 0.7 
2 0.2 17 2.6 
3 0.3 19 3.2 
5 0.2 Right bank: 20 4.1 
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/> ~ %56k~ sin 4.j.] 
YI = if cos 10 
(2-13) 
(2-14) 
where :1'1 and YI, respectively, are the ordinate and abscissa measured from t,h.e mid­
point 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 sec­
tion 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 of 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 pres­
sur~ 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 
j 
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, .36 BASIC PRINCIPLES 
Douma. formula,' that is, 
'U - 10 ~0.2V: - 1 
. gR (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. It is assumed that the depth of tlow section is the same!l.S 
that at section DB. 
;:; 
C 
.2 
::l 
:-
OJ 
t;j 
2 
/ 
4 ,a 
Un;l pressare, II 01 woler 
\«$PilIW1!Y 
Iraining wall, 
eo II cpO!1 
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, 
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. 
2. Leonard Metcalf and H. P. Eddy: "American Sewerage Pra.ctice," McGraw-Hm 
Book Company, 1M., New York, 3d ed., ,1935, vo!. 1. 
3. Harold E. Babbitt: "Sewerage and Sewage Treatment," John Wiley &: Sons, Inc., 
. New York, 7th ed., 1952, pp. 60-:.66. 
4. 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%. ' 
OPEN CHANNELS. AND THEIR PRO'PERT1ES 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 Gewful­
sern 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.ns­
ac:I.ior/.$, American. Socie4/ of Civil Enginee7's, voL 11:0, pp. 633-644, 1945. Dis­
cussions, pp. 645-668. 
23. J. B. Bela.nger: "Essai sur la solution numeriqne de ql,lelques problemes relatifs 
au mou.-ement permanent des eaux courantes" ("Essa.y on tIle Numerica.l Solu­
tion of Some Problems Relative to Steady Flow of Wa.ter"), Carilian-Goeury, 
Paris, 1828, pp. 10-24. 
\ , 
38 BASIC PRINCIPI..ES 
24. It Ehrenberger: Versuche Iiber die Verteilung der Drucke an Wehrriicken infolge 
des I1bsturzcnden '.Vassers (Experiments on the distribution 'of pressuresa\ong the 
f~~e of w(d .. ;; resulting from the impact of the fa.lling water), Die W IMJserwirtschaft, 
Vienna, vol. 22, no. 5, pp. 65-72, 1929. 
25. 'H&rald Lauffer: Druck, Energie und Fliesszustand in Gerinnen mit grossem 
Gefiille (Pressure, energy, and flow type in channels with high gradients), Was­
serkrafl, und Wasserwirtschaft, Munich, vol. 30, no. 7, pp. 78--82, 1935. 
26. J. H. Douma: Discussion on Open channel flow at high velocities, by L. Standish 
Hall, in Entra.inment of atr in flowing water: a symposium,T1'ansactions, American 
Society of Civil Engineers, vol. 108, pp. 1462-1473, 1943. 
LeI!.' foI.l- Pit' 67/\1 c: ,Il.1 ,f;.-N eN 
.2. . h}j c/rct,;.d.i :/'t'--10 c: c ,'" Y.5. , . (, < 
W/ibn .ftu;o 4,4ff luitf&Yl7 tUi,rv:5 (id .. t 
f? . d tU. ,lillJ-' foe-t ~ Y-'.c,tt/).Je1' 
. .k.": In a CCt·n~l.pdjW 4t- /~1{."fp"U.1/f'A(iCL 
I _ P ,~,> J: . . . c. "e J' .,t:; 'I 
HI]~II tt'l"1 ::>..;t;.1".!/f"''' V"'r1.tl.-7 VW' _..:!.~,~" .it.-.:.~(j 
~'i._ .:di.l:£nnd~~L4;'1~ -t&.4i-:5 p,ttt'a . 
e>J rret ,/·1/0 fall . 
.. ,he. ellA-of /:J~"m ;$ dtre,·r/6..,;tLul 
vV (I! ·'n'YII!'. .s~t:rt, 
Vr';,t.. cd cbf #.. is /l. J'Vi4 v..u ¢ ,pver.( e @'we <5 
~y.& o,C;...l."< ~ 
iJ d ( .§ / /J / '. I IS r1.!1 /!.~Ci/ bkrl~)~""'p1) 
/,1 (f?)e. '!, 6 rtf..~' • "'"'t CttYD.iJ,.u.:; r;~yo;o (}v'Iv: vI< /. 
/u..-",,' fYr..f t! Ci.~"Vl/ r1 f~J t· '1' . 
. '7.... ('/. " .) • ( ftvtAP. t'k,~e l',., ctylv) 
!/YiM..ltU ... (l!':71tdVt) ~-;P. f ' 
T>/~~cf Jt#rry.: (IU9A .~) . 
~it..d-(c4blh < .5q,-'Ui4'V"t d.c;ot/.., 
( . bq: ...... e It:;rY:J 
CHAPTER 3 
ENERGY AND MOMENTUM PRINCIPLES 
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 
@ , 
1 
d 
Saelioil 0 ~ I 
I I 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.in­
ing point A on a streamline of flow in a channel of large slope (Fig. 3-1) 
may be written 
V 2 
H = ZA + dA, cos 8 + 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 
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40 
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BASIC PRINCIPLES 
~ different velocity head, owing to· the nonuniform vel~ity distribution 
U1 ~ctual fl~w, . Onl~ in un ideal pal'a:lel fl~w of uniform velocity distri­
butl.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 
V2 
H = z + rl cos () + a --.:. 
. 2g (3-2) 
For channels of small slope, 0 = O. Thus, the tots'! energy at the chan .. 
nel section is 
P H = z +d + c;.-
2g (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 bottoml by So = sin () In uniform £10'" S - S - S - . () . . ", J- w- 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 ... 
V;2 . V" . 
Zl + d l cos /1 + al ~2· = Z2 + dz COs /1 + az ---.l: .L h . g 2g , 'f (3-4) 
This equation applies to parallel or gradualJy varied flow. For a channel 
of small slope, it becomes . 
. V; V~ . 
ZI + YI + al -2 = Z2 + Y2 + a2 - +hf . g 2g . (3-5) 
Either of these tW? equations is known as the energy equation. Wh~n 
al = .a2 = 1 and hr = 0, Eq. (3-5) becomes 
+. V I
2 .·. V 22 
Zl Yl+ -2 = Z2 +y~ + - const .. g 2g (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 ?y lllf~l'ence, to gjve recognition to his pioneer achievement in 
'JI 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. .. . 
, . 
) 
ENERGY AND MOMENTUM PRINCIPLES 41 
-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 . 
V2 
S. = d cos 8 + a 2g (3-7) 
or, for 8: channel of small slope and a = 1, 
V! 
E =y+-
.' . 2g 
(3-8) 
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 ordi­
nate 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· aucord­
ingly. The 1wo curves A' B' and A" B." (Fig. 3-2) represent positions of 
1 The concept of specific energy was first ir~troduced by Bakhrneteff [21 id 1912. 
http:totalheruLQf.J!.ow
42 BASIC PRINCIPLES 
,the specific-energy curve when the discharge is less 'and greater, respec­
tively, 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---- y -----I 
j 
Supercritica I 
,Fw. 3-2. Specific-energy curve. 
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 = 1 J may be written 
Q2 
E=y+--, 
217A -
Diffe~entiatiilg with respect to y and noting that Q is a constant, 
dE = 1 _ ~ dA = I _ VI dA 
dy gA3 dy gAdy 
. ." 
(3-9) 
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 Wall first 
introdl.).ced by BOss [3J. ' 
) 
, 
,< 
ENERGY AND MOMENTUM PRINCIPLES 43· 
At the critical state of ftowthe specific energy is a min~mum, or 
dE/dy =0. The above equation, therefore, gives 
N\':;" 
VI D 4 ( 
2g - 2 ,~ 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 con­
ditions 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 
(3-11) 
For a channel of large slope angle 8 and energy coefficient ct, the criterion 
for critical flow can easily be proved to be 
Vl D cos e 
ct 2g = 2 (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 
(3-13) 
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 or vice versa occurs frequently in open 
channels. Suchchange 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 
varied and is known as a local phenomenon. ' The hydraulic drop and 
~ , , . 
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 
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44 BASIC PRINCIPLES 
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 
/Theore.licOi water surfoce 
---"""",-""-,,,,- ==:::t' ~s~~nLP~~e.!...! I~ ___ -'-
Actual WQter surface - -- - -'\. ____ _____ ~ 
E 
~ 
FIG. 3-3. Free overfa.ll interpreted by specific-energy c~rve. 
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 COl1-
ti~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<!: .The 
theoretIcal water-surface curve of an overfall is shown with a dashed line 
in Fig. 3-3. . 
It ~hould be reme~bered that the detel;mination of critical depth by 
Eq. ,c3-1O) or !3.11) IS based on the assumption of parallel flow a.rid is 
ap.phc~ble only approximately to gradually varied flow. 1'he flQW at the 
brink E' actually ..llllt:llilinear, fru:...:the curYatl..lli} of .flow is -P..'CQllQl!nclill; 
.hen~, the method is invalid 'for determining the critical depth asJ.~ 
ENERGY AND 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 com­
jnite(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 over­
fall 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 Y 
o " 
" "" o.tl: 
-, 
Specific-ene'g)' CUrye Hydraulic jump 
"" c. 
<U 
'0 Y 
Ir.ilial 
deplh 
Specific-·force Culye 
FIG. 3-4. Hydro.ulic ju~np interpreted by.specific-energy and specific-force curves. 
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 
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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 chan­
nels, 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 a gradually 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 sey-
tion can also be obtained from the energy curve at the point of minimum 
energy. At &ectionC 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 
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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 alter­
nate-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 supercritical­
flow 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 
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 two­
dimensional 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 con­
trMtion 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 con-
traction is negligible. - . 
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48 BASIC 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 hori­
zontal energy line showing the elevation'of the tobl head is, therefore, drawn on the 
channel· profile (Fig. 3-6). . 
FIG. 3-6. Energy principle applied to a channel contraction (a) without gr:adual 
hydraulic drop; (b) with gradual hydraulic .(hop. ' 
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'> " + 155.25 = 0 
y . ~Y b'. 
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 pro-
cedure 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<?n, the 
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ENERGY AND MOMENTUM PRINCIPLES 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 sub­
Cl'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,), y, 
2g = '2 
or . y = {/lO,OOO 
, "gb% 
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 exagger­
ated. 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 
weight of water in lb/ft i , Q is the discharge in cfs, and V is the mean 
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 
'~l . 0( 
. resultant of all the external forces that are acting on the body,Applying 
this principletO a 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: 
J(.. 1 
• .;.. "'>l·~)(qt 
(3-14) 
- cit! ,. , 
where Q, w, and 1:::' are :as .. previously defined, with subscl'ipts refe1'l'in'g to ' 
)>1' C( 
_, Yn 
sectionr-l~nd 2; P! and P2 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· , 
1 The application of the inomentum principle was first suggested by Belanger [5J. , 
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50 BASIC PRINCIPLES 
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,' how­
ever, 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 curva­
ture effect of the streamlines of the flow. For simplicity, P1 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 
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 hdA= 1 + ~ fA cdA (3-15) 
AZ)o AzJ~ 
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:allelflow: 
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 
= j.Ad4 1 (cd A 
" . <.l .4 
A >(:i:.;. ! c dA 
/::}. 
j'(,h .{- c)id JI. 
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ENERGY AND )';IOMENTUJI{ PRINCIPLES 
rectangular channel of small slope and wi~lth b (Fig. 3-7), 
PI ;'~Wb1l1:l. 
. and Pz = 7§.wbyz2 
Assume F I = wh/by 
51 
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 
Q ~ ~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, . 
L 
I 
(3-16)1 
I • 
Co t1r IC!cr-'T(~ 0 'tl. 
1. t'Yltkfr; ~t·-Z,a:k.~ 
1.iJ" ~ e"" t Jt-l'!' 
~.~~?} 
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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 dif­
ferent 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 internal­
ellergy 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 differ­
ences in their constitution is important, de.'lpite the fact that m many 
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 
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~. 52 BASIC 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 internal­
energy 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 hydraulic­
jump 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. 
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 fric­
tional 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 ,rgf1lJ,o/,J<. (){;.:aty/'b .d:./n Ct"7fd-t. 'c.~ t5 If) f3 L 
:: 
, . '- 'Y) c'Af:i 
j .. /.:: .r-? 
/, .ve-.le, i:-~ 'r· »( , I <Y'£. 
.1: /)~;..<_. r .. ' _ .. ,:Ad .rI.r...,~;,;,;",,! f. . ...u,:u., 
ENERGY AND MOMENTUM PRINCIPLES 53 
approach section 1 and the downstream section 2 at themininium depth OIl the top 
of the weir, the following equation may be written: 
qw (!1. _ !J_) = HWYI' - MWy2 2 - Hwh(2y, - h) 
g Y2 YI . . 
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 (3-17) 
. ·Y'T //. 
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 -- 171) = PI - P2 
g 
The hydrostatic forces PI and 'P 2 may be expressed as 
PI = wi1A. 1 a.nd P 2 = wi2A 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 (Qr //.,-c 1-1,) (3":18) 
gAl g.fi 2 
'The v:llue of the coefficient actually depends on many factors: mainly, the round­
ing 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 pre­
pared 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. Geo­
logical 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) Minne­
Bola and· Washington tests performed,'respectively, at the university of Minnesota and 
Washington State University [6]. For some formulas and coefficientsof discharge 
developed in the U.S.S.R., see [12]. For an analytical treatment of the problem, lIee· 
'[13]. 
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54 BASIC PRINCIPLES 
The two sides of Eq. (3-18) are analogous and, hence, may be expressed 
for ahy channel section by a general function . " 
Qz' 
F -it zA "(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 
B 
c· 
I 
~=-:-_. I ( 
o 45° for () Ez '£lEiE:, 
channel of ,......, 
zero or small 
slope (0] l.bl 
3-9. Specific-fofce curve supplemented \~ith specific-energy curve. (a) Specific-
energy curve; (b) channel section; (e) specific-force curve. ' 
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 
i 
! 
i 
ENERGY AND MOMENTUM PRINCIPLES 55 
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), ' 
dF Q2 dA ,d(zA) 0 
gP dy T --elY 
For a change dy in the depth, the correspo,nding change 'd(iA) in the 
static'moment of 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 
dF _ Q! dA + A = 0 
gA2 dy 
Since dA/dy = T, Q/ A 
reduced to 
V,and AfT = D, the above equation may be 
(3-10) 
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 indi­
cates 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 F 1, 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 F 1, 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, 
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 exam­
ple,' 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, 
(3-18) 
For ll. recLanguinr channel of width b, Q = V,A, = V.A.., Al = by" A, = bYt, 
2, = yi/2, and ii' - y~/2. Substituting these relations IlJld F, = V d v'Uih in the 
above equation and simplifying, 
(:1!!)' - (2F,' + 1) (J!J)+ 2F,' = 0 y,. ~ 
(3-20) 
Factoring, [(1Lt)' + ~ - 2F,'] (~ - 1) - 0 
YI . YI YI ' 
Then, let (~ V + 1l! _ 2F • = 0 Y.:/ y, 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.ch­
mgflow. 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 
Pa and intercepting the specific-energy curve a.t point P.", which: gives the energy con­
tent E, after the j';lmp. The energy loss in the jump is then equal to El - E" 
represented by /lE.: ' 
a-8.Momentum priIicipl~ Applied to Nonprisrnati~ Channels. The 
sp~cifj.c force, like the specific' energy, varies with the snape ofthechannel 
' .. ,I 
"'i 
f 
i 
\ 
I 
ENERGY AND MOMENTUM PRINCIPLES 57 
section.' In applying the mornentur~l principle to nonprismatic channels, 
therefore,a three-dimensional plot similarto that shown for the applica­
tion 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 
hydraulic jump is involve(i. 
following example shows how the momentum 
design of a channel transition in which a 
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. 
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 
, , . 
http:with,.llJ.gh.mt;mlaldIDe:t,:gy.--illss.es
I \' 
58 'BA.SIC PRINCIPLES 
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. . 
TABLE 3-1. CObfPUTATION FOR A CHAI'1N1:lL EXF.Ai'/SlON DESCRIBED IN 
EXAMPLE 3-4 ., 
Section Low stage High stage 
width 'II" ft PI 1/2, ft F, 
b, ft for E = 10.207 for E = 4.097 
8.00 0.500 78.6 3.940 71.9 
8,50 o 470 78.7 3.960 75.9 
9.00 0.4'.1:3 78.8 ,3.979 79.9 
9.50 0.419 78.8 3.987 83.6 
10,00 0.398 78.8 4.000 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 com­
puted by means of Eq. (3-8) or (3-9), or 
"':"""-1::--=::'- + VI,:"" 10.207 
where b is the width of the section. Similarly, the high st,age 1/2 for a total ene1'gy of 
4.097 can be computed from 
+ 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 <laction the water surface at low ,stage will jump to the high stage, as indicated 
by a vertic,al line. (Fig. 3-10a.). Actuall:)" however, the jump will take place over a 
'short distance, as shown by the dotted line.' The energy loss in the jump is repre­
sented bj' the vertical intercept between the upstr.9a.m and downstream energy lines, 
which is equal to 6.110 ftl covering the energy; difference between the flows in the con­
ne¢ting channels. By varying tlie shape of the cross sections of the connecting chan: 
nels the location of the intersection of theF lines, or the position of the jump, can be 
altered. Changing the depth of flow in the downstrea.m channel will also change the 
po~ition of tile jump. Generally, an increase in the downst~eam depth will mon 
the. jump upstream, and a decreaSe in the depth will move the jump do:wnstream. 
The hydraulic jump (lan be eliminated if the energy loss can be dissipated gradually 
And ·smoothly. This can. be done by introducing proper roughness in the transition, 
r 
~ 
'1 
.1 
ENERGY A.ND MOMENTUM PRINClPLES 59 
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 proportion­
ing 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 sec­
tion, where the :flow cha.nges from low to high stage, that is, from 8upercritical to sub­
critical 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 criti­
ca.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. 
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 300 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 specific-
energy 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 . 
(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. ' 
1 The .concepG of critical depth based. on the theorem of maximum diacharge was 
first introduced by Belanger [20J. 
\ 
". 
! 
60 !!ABIC 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 inflectionat !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 chan­
nel (Fig. 3-11), prove that 
where y. is thesubme!"ged depth; y, is the height of sluice-gate opening; y, is th~ tail­
water 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 
()J2 -YI)' ./' 
4j/lY2 
AE _(3-24)* 
3-12, If !I. hydrAulic jump is forllled on the ,horizontal floor at the toe of the spill­
way 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 
(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. 
ENERGY A.ND MOMENTUM PRINCIPLES 61 
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 chan­
nels 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 V 9.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"), Springer­
Verlag, 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 experi­
inents 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:lu­
lies), llHmoire, gmt" Na.tionale de$ Pont8 e~ ChaU$88I3$, Paris, 1849-1850, pp~ 32-33. 
I~-
CHApTER 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 impor­
tant 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 uni­
form 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 resultin 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 
. curve (Fig. As the curve is almost verticalliear the critical depth, 
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 . 
( '. 
I 
, \ 
I 
64 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, 
. Q 
z=-v'g 
When the energy coefficient is nO& assumed tobe unity, 
. Q 
z=-­
v'g!cY. 
(4-1) 
('1-2) 
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 a. 
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: 
Q =Z Vg 
I 
or by Eq. (4-2) in the following form: . 
Q=z:(2 N;' 
(4-3) 
(4-4) 
.... 
o 
ci 
Op I ~ puo' q/,( lO SGnlOI\ 
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. II . 
~~,.. 
.., 
c 
--:.~ 
0,-, 
'" !" ... 
Q 
:5 . .., ... 
'w ... 
..':! .., 
"'0 '0 
"-
N 
1> 
'" OJ a dt5 
(5 
g 
;;:.. 
.<i ... 
g. 
-0 
";l 
'-' 
:3 
't 
" ..c:: .., 
tj) 
.5 
.~ 
... 
OJ ... 
" '0 ... .s 
fIl 
~ ... 
::> 
0 
-< 
.j, 
ci 
~ 
66 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 will ·help to determine the depth 
y for a given section factor Z, and vIce versa. 
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) 
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 
(4-6) 
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 
dif" "'" 2y 
(4-7) 
Now, taking logarithms on both sides of Eq. (2-3), or Z = 
then differentiating with respect to 1/, . 
A vi A/T, and 
dOn Z) 3 l' 1 dT 
--ay- = 2 J.- 2T (4-8) 
Equating the r~ght sides of Eqs. (4-7) an~ (4-8) and solving for M, 
M = JL (31' -'- .:1- dT). (4-9) 
. A Tdy 
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 SUbsti­
tuted in Eq. (4-9); the resulting equation [2] is simplified arid Mcomes 
M = 3[1 + 2z(y/b)P - :2z(y/b)[l + z(y/b)J . (4-10)* 
. [1 + 2z(y/b>:J[1 + z(yjb)] 
• 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 
" 
I. 
~ 
I 
. . 
CRITICAL :FLOW: ITS C.OMPUTATION AND. APPLICATIONS 67 
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 
5.0 
2.0 
1,0 
o.a 
0 
~ 0.6 ,.. 
'0 0.5 c 
0 0,4 ..a .... ,.. 
0.3 ... 
0 
. ." ., 
.2 0.2 
~ 
0.1 
o,oe 
O.os 
0.05 
0.04 
0.03 
0.02
2
.
5 3.0 3.5 4.0 4.5 5.0 5.5 
Volues of M 
FIG. 4-2. Curve:; of Mvalues . 
A curve for a circular section with M plotted against 1Ildo, where du 
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 yjdD less than 0.7 or so, but increases rapidly 
as the value of y/do becomes greater than 0.7. The significance of this 
z(1/(b) may be constructed. It is obvious that this curve: be identical with the 
eurve for z 1 in Fiilj. 4-2. Fo, convenience in application, however, a. family of 
ourves of M versusy(b an shown, using If. as & pammeter. 
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58 BASIC,P~INClPLES 
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 
N 
'" ~ 
ne pial bee om .. curved' 
"~.n Ihe depth approach.. I 
the Q1oduo II}' converging 
I! 
and phenomenon occur also in other 
crown of a closed conduit P 
types of closed conduit with gradu­
ally closing crown, when the water 
surface approaches the crown of the 
1.1=2. lon9 
i:~g<~~~;o conduit. 
ot P For channel sections of other than 
trapezoidal or circular shape, exact 
values of M can be computed di~ 
rectly by Eq. (4-9), pi'ovided that' 
the derivative dT jdy can be evalu­
ated. Approximate values of Mfor 
any channel section, however,may 
,be obtained . hom the following 
equation 
M (4-11) 
log y' where Z 1 and are section factors 
FIG. 4-3. Graphical determination of the for any two depths Yl and Yz of the 
M value. 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 ordi­
nate 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 straight­
line fOl'm. The hydraulic exponent is 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), 
f, 
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CRITICA.L FLOW: ITS COMPUTATION .AND APPLICATIONS 69 
4 ... 4. Computation of Critical mow. Computation of critical flow' 
involves the determination of critical depth and velocity when the dis­
charge 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 deter­
mined 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 
The velocity is 
D = 1[(10 + 1/) 
10 + 2y 
and A = V(20 + 2y) 
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. 
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. 
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. 
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 0.0394-
1:<'01' this value, the chart gives '!I/b = 0.108 or '!I. ":' 2.Hi ft. 
70 BASIC PRINCIPLES 
FlO: 4-4. Curve Df y versus Z for a. cir(lular 5C)ction. 
In Example 4-3, Zld~'" = 0.226. For this value the clHLrt. gives yld = 0.48 or 
y. = 1.44 ft. 
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 down­
stream 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 con­
ditions 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. 
, 
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. 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 
" I 
Flo'..,. oon<lilian controlled 01 the downstream e~ 
j
BOCkWO!il( 
curve 
rr---_ ~ 
, ---~ 
l.critIC<J~..s..2~..!.e_ctions; 
I 
I 
Row condition contrnl1ed Qt lhe 
upstream enc 
Add~d depth due I 
h:I backwater effect 1 Drowdawn 
Depth of subcriticol 
flow without dam 
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 
~ ---. r 
1 
L 
L 
I 
i 
72 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 
N-N 
SECTION L-L 
FIG, 4-6, Plan, elevation, ana dimensions o.f ,he Parsholl flume. (U.s. SlIil Conser­
vation 8811>ilie [26J.} Plan and elevation of a concrete Parshall measuring fIl\me show­
ing 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.mend of flume; D = width of upstrea.m end of flume; 
E = depth of flume; F length ofthl'oat; G "" length of diverging section; K = differ­
ence 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." 
effect originlj.tillg from ~hepool will not extend far upstream. Instead, 
the flow in the upstream channel will eontinue in:the downstz-eam direc­
tion 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-
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). 
·{l 
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><: * ''.N ' 
S;:-MMc<:)M:r.I~MP)(o)Mp') 
r:OOOMp')~¢I'lO~tO<a\O ,... 
j.:! '<J' "\!1 "¢I ae Q) co QO Q 000'0 
r:- ..... ----roI.<;q~:1N~ 
..100C::OOOOQOOOOO .. 
~_-:-l_."C"')C'}(t)~.:";ImMI.'I? 
;; ""0 0 0 ~ 0 <:;I 0 0: 0 0- <:) 
t: ::::;) ""'" - C'l N ~ C'1 ~'~ ("1 cq ~ 
~oo(Oooooooooo 
a:-t;~~ (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 <qO U1 t.e ~ CC ..... 1:--
I~""~~~------------------I 
.:~i:f 
~ ~ .=~M.~O .• O. 
'" l~ .... ·,......-tto':l(#)MM~"~"'')LI':l 
ii~~~ 
~tO :::fOOlQIt)OIOOf;!)Q 
:~ ..:. Col ,,' 'ttl ~ l.O·tn 0: 10 ..,;. l'-< co 
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74 BASIC PRINCIPLES 
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 condi­
tion 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 measure­
ment 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 critical­
flow 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 
I 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. 
,. 
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" ·r 
CRITICAL FLOW: ITS COMPUTATION AND APPLICATIONS 75 
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: ' 
ThrQat width 
3" 
6" 
9" 
12" to 8' 
10' to 50' 
Equ.a.lion 
Q = 0.9921I.u4T 
Q = 2.06H.ua 
Q = 3.07 H.l.63 
Q ~ 4WH.uuw'·'" 
Q = (3.6875W -I- 2.5)H.1.G 
(4-12) 
{4-13} 
(4-14) 
(4-15) 
(4-16) 
In the above equations Q is the free discharge in cfs Wis the width of 
.thro~t in 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 dis­
charge. 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. . 
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 y. M. Cone at,the hydraulic laboratory of the Colorado Agricultural Experi­
ment 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, dur­
mg 1~.Dec~mber meetIng of 1929. Further developments on the Parshall flume arc 
d~scflbed by R. L. Parshall in [22] to [26J. : 
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76 
b 
'" aa 
f c. 
BASIC PRINCIPLES 
U 
Discharge, cIs 
(OJ 
Discharge, cIs 
( b) 
Oischarge, cIs 
(~) 
FIG. 4~7 
i 
1.9 ZD I 
4.0 4.5 5.0 5.5 6.0 
GRITICAL FLOW: ITS OOMI'iJTATIONAND APPLICA.TIONS 77 
Correction, cIs 
(tt) 
Correction. cis 
(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 cubicfeet 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 subDlerged­
flow discharge. 
http:Parsha.ll
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78 BASIC PRINCIPLES 
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/, 
10 
12 
15 
20 
25 
30 
40 
50 
CorrectiOlt factor 
LO 
1.2 
1.5 
2.0 
2.5 
3.0 
4.0 
5.0 
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 dis­
advantages. The flume CD.nnot be used directly with or, combined with 
a head gate. "It is more expensive to build and requires more accurate 
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CRITICAL FLOW: ITS COMPUTATION AND APPLICATIONS 79 
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~~:s' 4-8;cD()lio!l.Tgar;~sAf{)O~ del~ermalirunE' 'g th.s los,s of h~!l.d t(hrough Parshall flumes of various -\ 
Seroi~e ~61.) lW • • ell .ur ,:z:pmmen Slat~{)'11 251 and U.S. Sml Ccm.ssnJl1tion 
, { 
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80 BASIC PRINCl;l'LES 
workmanship in its construction than i 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. 
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 = V.y, =%H. V%i'if. = 3.09H.1.S (4--17) , 
This is a theoreticru discharge equation i1) which H. is uncertain since the critical sec­
tion 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 equa­
tion 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 = 9.38yol.l (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 ft.; , ' 
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. ' 
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CRITICAL FLOW: ITS (TO¥PUTATION AND APPLrcATIONS 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 
~+~=~~. , 
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. 
FIG. 4:-9. 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. 
PROBLEMS 
. 4-1. Prove the following critical-discharge equations for the triangular trapezoidal ' 
aud circular' sections: I ' , , 
Channel 8e£!lion 
Triangular 
Trapezoidal 
Circular 
Parabolic 
Equation 
Q, = 2.295zH .. ·• 
Q _ 5.671{(b + zy)yJu 
, - -(I) + 2zy)',$ -
Q.= 0.251(0' - sin OJ''' d U 
(sin Yz 9) 0,6 n 
Q. = 2.005TH.L5 
(4-19) 
(4-20) 
(4-21) 
(4-22) 
, In theabove 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 
expressed by 
(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 sec­
tion of <iiameter do, derived by BrILine [40] from ~.n equaticm equ.ivalent to Eq. (4.-21), 
is 
y. = 0.325 (£t + [).083d. (4:-25) 
which is accurate only when 0.3 < y./dQ < 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 
(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 possi­
ble? 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 Exam­
ple 4-5. 
4-12. Determine the dischltrge through the 4i-ft Parsha.ll flume described in Exam­
ple 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.d-
ing 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 chan­
nel ,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 
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CRITICAL FLOW; ITS COMPUTATION AND ,APPLICATIONS 83 
FIG. 4-10. Rating curves of ;l. critical-flow flume proposed for a clos"d conduit. (After; 
J. C. Stevens [32J.) d. ""diameter of the cor.duit. ' 
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. 
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84 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 Engineer­
ing, 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 Jour­
nal, Americar;. Waler Works Association, vol. 34, pp. 1473-1475, September, 1942: 
9. E. S. Crump: Moduling of irrigatioIl channels, Pt"dab Irrigation Brar4ch Publi-
cations, 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: Pre­
liminary 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'agri­
coltura, 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. 
, 
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CRITICAL FLOW: ITS COMPUTATION AND APPLICA~ONS 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 critical­
depth 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 
a;:d ~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. 
33. 
34. 
!. 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. 
Herbert Addison: "Hydmulic Measurements" JohnWilev & Sons In N 
York, 1941. . ' J . ' c., ew 
"Wate'l' 'tvIell.sunnnent Manual," U.S. Bureau of Il.eclamo.tioll May W53 
43-58. ' , , pp. 
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 News-
Record, 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, Agri­
C1!/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 storm­
w2.ter outflows on sewers, Journal, Institution of CiviL Engineers, London vol. 28 
no. 6, pp. 136-163, April, 1947. . '" 
http:Februa.ry
http:gener.al
http:ressa.ut
.PART II 
UNIFORM FLOW 
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CHAFTER 5 
DEVELOPMENT OF UNIFORM· FLOW 
AND ITS FORMULAS 
. .) 
15-1. Qualifications for Uniform Flow. The uniform flow to be con­
sidert: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 allparal­
lei; that their slopes are all equal, or Sj = S'" = Sn = S. For practica.l 
pm-poses, the requirement of constant velocity may be liberally inter­
preted 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 uni­
form-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 experi­
ence 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 assump­
tion 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 engi­
neering 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. I 
" It should be noted that unifor:m flow cannot occur at very high veloci­
ties, 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. 
89 
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90 UNIFORM FLOW 
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. 
Ivorled~w un[f~rm flow on the ove''::Qe ,J 
Tronsitor yt--I 
I zone I . . ' . ' I 
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t;"ifOftl1 fl 0 "' ___ -I. 1 
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FIG. 5-1. :&!ts.blishment of uniform flow in Il long ohe.nne1. 
If the water enters the channel slowly, the velocity and hence th(;l resist­
a~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 establish­
ment 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 transi­
tory length required by the given conditions,. unIform flow cannot be.' 
. attained. Toward the downstream end of the channel the resil5tance 
t 
. .l·. 
DEVELOPMENT OF 1}NIFORl'rl FLOW AND ITS FORM,'()'LAS 91 
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 dif­
ferent slopes: sub critical, critical, and supercritical (Fig. 5-1). At the 
subcritical slope (top sketch in Fig. 5-1) the water surface in the transi­
tory 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 approach­
ing 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 com­
putations 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: 
(5-1) 
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 rough­
ness, 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 
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. . 
1 
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92 UNlF'ORM FLOW 
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; 
A the water area 
. V the mean velocity 
V ma the maximum surface velocity' 
P the wetted perimeter 
R the hydraulic radius 
y . the maximum depth of w-ater area 
SUI the slope of the water surface 
n a coefficient representing the channel roughness, known as the 
coefficient of 7'oughness1 
Q. the sllspended sediment charge 
Qb the bed load 
'IL the dynamic viscosity of the water' 
T the temperature of the water 
There have been dev,eloped and published a large number of practical 
uniform-flow formulas, t but none of these formulas meets the qualifica­
tions 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 amultiple­
correlation 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. . 
1 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. . 
DEVELOPMF.NT OF UNIFORM FLOW AND 1'1'13 FORMULAS 93 
. 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: 
(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 assump­
tions. 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 flow2 
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 oylinder94 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. . 
M~my 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 
.41.65 + 8 -n° 
C - (5-3) 
- ( 0.00281) n 
1 + 41.65 + S vB, 
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 
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DEVELOPMENT OF UNIFORM FLOW AND ITS FORMULAS 95 
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 = __ 157.t'_·_ 
1 + m/vR (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 
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 ................ . 
Bazin's.1n 
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 experi­
mental data and several naturaL streams. The results based on this 
study are showll in Table 5-2. The values of the average variation indi­
cate 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. This formula, an 
implicit function of C, is 
C = -42 log (4i + It) (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). 
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fL~.tG,~I~,LIW.!~I~!~I'L1UT~.ll~ttLI~I~.llk'~~~~~~WW~~WU~~~~o 
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DEVELOPMENT-OF UNIFORM FLOW AND ITS FORMULAS 97 
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 
T.tUILE 5-2, COMPARISON OF VARIATH)NS IN CHi:z.y's C, BAZIN'g m, 
AND KTJT'J'ER'S 11 
Measurements 
Average vaiue3 
varia.tions, % 
C m C n 
Bazin's Series 6 .. , .......... " .. "'... . ... 0.185 O. 0121 .. 1 1 
7 .... , .................. , ... , .. , 0,156'0.0120 .. . 1.0 
8 ........ " ........ " ...... , ..... 0,1420.0116 .... . 2,5 
9 .................. " .... ' ..... 0.1990.0130 ... .. 1.2 
10 ............................... 0.1440.0117 .... , 1.4 
11., ............................ 0.129 O.OU3 .... , 3.8 
12 ............ , ................. 0.324,0.0151. 1.6 1.0 
1.2 
1.8 
1.2 
1.5 
Ill ........................... OJHI0.0148 .... :j2.7 
14., ............... , ........ , .. 0.3210.0150 .... 4.4 
15 .............................. 0.'7150.0209 .... 4,2 
16 .............. ·~ ............... 0,7110.0212 ..... ' 5.7 
17 ....... :: ............... 1 .... 0.7210.0215 16.1 2.2 
0.4 
L2 
8,8 
5.7 
4.9 
22.2 
32 .......................... j .... 0.4240.0' 1.8 
33 , ., ..••.......... ,. . .... , ..... 0.444 0.0 3 . 1 
. 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 
Bogue Phalia River, Miss .• 1914 .... " ....... 63.3· 4.09 0.070424.20.35.1 
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.. .. . " .:, ... , .. , .. , 
• Values a..veraged by the author. 
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 a 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-; 
. ; 
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98 UNIFORM, FLOW 
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 
New Old 
Neat cement surface ............... . 
Unplaned-plank flmlles .... ; ............ . 
Concrete-lined channels. . . . . . . .. . ..... . 
Earth, straight and uniform ............ . 
Dredged ear~li channels ................. i 
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" 
41 65 + 0.00281 + 1.811 
. . 0.005 0.015 = 1242 
+ ('41 6- + 0.00281) 0.015 . 
• 0 0.005 V4J.O 
0= 
Then, by the Chc~y 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 pub­
lished 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 sup­
pose 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 
DEVELOPMENT OF UNIFORM FLOW AND ITS' FORMULAS 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 111 anning' 8 n. This formula was developed from seven differ­
ent formulas, based 011 Bazin's experimental data, and further verified 
by 170 observatipns} Owing to its simplicity of form and to the satis-
a linear meas,ure of roughness and </>(Rlk) is a iunction of Rlk. If <p(Rlk) is considered 
climensionless, It ,\yil! have the same dimensions as those of kH, that is, ~H. 
On the ot.her hand, of course, it is equally possible to assume tha~ the numerator 
of 1.486/n can a.bsorb the dimen'sions of Lwr--I , or that ",(Rlk) involves a dimensional 
factor, thus leaving no dimen~ions for n. Some authors, therefore, preferring the 
simpler choice, consider n it dimensionless coefficien t. ' 
It is inte~esting to note that the conversion of the units for the l\'Ianning formul!l. 
is independent of the dirn~nsions of n, as long as the same value of n is used in both 
systems of units. If n is .assumed dimensionle~s; then the formula in Engliah units 
.gives the numeric."l Gonstallt 3.2B08~ = 1.486 since 1 meter = 3.2808 ft: NfliV" if n 
is assumed to have the dimensions of LhI, it" nllmerical value in English units must be 
different from its valuo in metric units, unless a nnme.rical ~orrection factor is intro­
duced for compensation. Let n be the vallie in metric units and II' the value in Eng­
lish units. Then,n' = (3.2B08H)n = 1.2190n. When the formul(l. is converted from 
metric to English lin its, the resulting form takes the numerical constant 3.2808'~+~~ = 
3.2BOB~~ = 1.811, since. n has the dimensions of Lli. Thus, the reSUlting equation 
should be writtan V = 1.811R~~S}i/n' Since the same. value of n is used in both sys­
tems, the practical form of the formula in the English .system is Y = 1.8UR~S!h1 
1.2HlOn = 1.48I}R~S"~/n, whiGh is identical with the formula derived on the assump-
. tiori that II has no dimensi.olls. 
In a search of early Ii~erature on hydmulics, the author has failed to find an y 
significant discussion regarding ~he dimensions of n. It seems that this was not ~ 
problem of concern to the forefathers of hydraulics. It is most likely, however,. that 
11: was unconsciously taken 2-S dimensionl¢Ss in the conversion of the Manning formula, 
becau,se such a conversion, as shown above, is more direct a.nd simpler. 
Now, considering the approximations involved in the derivation of the formula 
and the uncertainty in the value of n, it seems unjustifiable to carry the numerical 
constant to more than three significant figures. For practical purposes, a value of 
1.49 is believed to be sufficiently ac.;urate [161. 
Manning mentioned that the simplified form of the forl11ulahad been suggested 
independently by G. H. L. Hagen prior to Mo.nning's own work, ,according to a state.­
ment by Major Cunninghllm (171. Hagen's formula was believed to have appeared 
first in 1876 [7J. It is also known that Pllilippe-Gaspl1rd Gauckler [18J had all early 
proposal of the simplified form of Manning's formula in 1868 and that Strickler [191 
presented independentiy the Same form of the formula ill: 1923. ' 
I For the derlva.tion of the exponent of Il, use was made of Bazin's experimental 
do.ta on artificial channels [12J. For different shaj:les and rOllghnesses, the average 
yalue of the el{ponent was found to vary from 0.6499 to: 0.8395. ConsiderinC' these 
variations, Mll,nning adopted an approximate v[l.lue of % for the exponent. 'On the 
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100 UNIFQRMFLOW I 
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factory results it lends to practical applications, the Manning formula has 
become the most widdy used'of all uniform-flow formulas for 6pen-chan­
nel 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 (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 butvaries 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 
. where 
(5-8) 
(5-9) 
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 sug­
gested 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.nd­
book 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." 
DEVELOPj\HlNT OF UNIFORM FLOW A.ND ITS FORMULA.S 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 vrn 
Y"" 1.3 vn 
for R < 1.0 m 
for R > 1.0 m 
15-7. Determination of Manning's Roughness Coefficient, In applying 
t~e 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 
method of f'electing the n value. At the present of knowledge, to 
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 theoreti­
cal 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 there­
fore described below. It should be noted that these factors are to a Cer­
tain 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, 
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102 UNIFORM FLOW 
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 rough­
ness 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, dlstn­
butiol1, 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 deter­
mine 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 bushes.. . 
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, . 
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DEVELOPMENT OF UNIFORM FLOW AND ITS FORMULA.S 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 ofaffected 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 depres­
sions, 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 meander­
ing 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 
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104 UNIFORM FLOW 
, 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. 
When the discharge is too high, the stream may overflow its and 
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. VALUES OF n FOR VARIOUS' STAGES m THE NI5HNJ..BOTNA RIVEI!., 
IOWA, FOR THE AVlllRAca; GROWING SEl.SON. 
Flood-pilli~ cover 
Depth of Channel 
water, it section 
Corn Pasture Meadow Small Brush and 
i 
gt'ains waste 
Under 1 0.03 0,06 0,10 0,10 0,12 
1 to 2 0.03 0,06 O,O~ 0.09 0.11 
2 to 3 0,03 0.07 0.04 0.07 0,08 0,10 
3 to 4 0,03 I 0,07 0.04 0,06 0,07 0.09 
Over 4 0,03 0,05 0,04 0.05 0,06 O,Oi 
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;DEVELOPMENT OF UNIFORM FLOW AND ITS FORMULAS 105 
of .inundfltion, as obseryed in the Nishnabotna River, low-a,fof the aver­
age gfowing season [24]. It should be noted, however, that vegetation 
has a marked effect only up to a certain stage and that the roughness 
coefficient can be considered to remain constant for pl'actical purposes in 
determining overbank flood discharges. 
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Mississippi River oetweenMemphis 
and Fulton, Tennessee . 
o U,S, Geologicol Survey doto 
.. U.S, Corps of Engineers dato 
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. 
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. 
:106 UNlFORM FLOW 
;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 andthe 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 judg­
ment 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!!, . 
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DEVELOPMJilNT OF UNIFORM FLOW AND ITS. FORMULAS 107 
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 character­
istics 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, cotton­
wood, 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, 
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108 UNIFORM: FLOW 
(b) dormant season-willow or cottonwood trees 8 to 10 years old, inter­
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 season­
bushy 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, inter­
grown 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 nor­
mal 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 mainte­
nance 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 examina­
tion of the best available experiments at his time.2 Table 5-6 is compiled 
L The minimum value lor Lucite was observed in the Hydra.ulic Engineel'ing La.bol'a­
Lory 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. 
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DEVELOPMENT OF UNIFORM FLOW AND ITS FORMULAS 109 
'fAllLl!l 5-5. VALUES FOR THE CO]>tl'UTATlON OF THE ROTlGHNESS COE·FFICI.!!lNT 
:BY EQ. (5-12) 
Channel conditionB Values' 
Earth 0.020 
M.ateriul Rock cut 0.025 
in vo I .... ed nQ 
Fine gravel 0.024 
Coarse gravel 0.028 
Smooth 0.000 
Degree of Minor 0.005 
irregularity r., 
Modernte 0.010 
Severe 0.020 
Gradual 0.000 
Va.ria.tions. of 
channel cross Alternating oceaSionally n2 . 0.005 
section 
Alternating irequently O.010~.015 
Negligible 0.000 
Relative Minor O. 01O-{}. 015 
effect of '11.. 
obstruutions 1\. ... " .. ~ .~ 0.030-{}.030 
0.040-0.080 
Low 0.005-0.010 
Medium 0.010-0.025 
VegeLfLtion 
High 0.02S-{}.050 
Very high 0.050-0.100 
Minor 1.000 
Degree of 
Appreciable 1.150 meandering m. 
Severe 1.300 
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110 UNIFORM FLO>'i 
. TABLE 5-6. VALUES OF THE ROUGHNESS COEFFICIENT n 
(Boldface figures are values generally recommended in design) 
Typ~ of channel and description I Minimum I Normal Maximum 
----~~~~~~~~~-I 
A. 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 
:.::." ,.,,,""::".: 
0.009 
O.OlD 
0.013 
0.010 
0.011 
0.012 
0.013 
0.017 
0.021 
0.003 
0.009 
O.OlD 
0.011 
0.010 
0.011 
0.011 
0.013 
0.012 
0.012 
0.015 
O.OlD 
0.015 
0.011 
0.011 
0.013 
0.014 
0.011 
0.012 
0.012 
0.016 
0.018 
0.010 
0.012 
0.016 
0.013 
0.014 
0.014 
0.016 
0.019 
0.024 
0.009 
0.010 
0.011 
0.013 
0.011 
0.013 
0.012 
0.015 
0.013 
0.014 
0.017 
0.012 
0.017 
0.013, 
0.014 
Q.015 
0.016 
0.013 
0.015 
0.013 
0.019 
0.025 
0.013 
0.014 
. 0.017 . 
0.014 
0.016 
0,015 
0.017 
0.021 
0.030 
O.OlD 
0.Of3 . 
0.013 
0.015 
0.013 
0.014 
0 .. 014 
1).017 
0.014 
0.016 
0.020 
0.014 
0.020 
0.017 
0.017 
0.017 
0.018 
0.015 
0.017 
0.016 
0.020 
0.030 
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DEVELOPMENT OF UNIFORM FLOW AND ITS FORMULAS 111 
. TABLE 5-6. VALUES OF THE ROUGHNESS COEFFICIENT n (continued) 
Type of channel and description 
------
D. 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 
Minimum Normal Maximum 
0.011 
0.012 
0.021 
0.010 
0.011 
O.OlD 
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..015 
0.017 
0.015 
0.020 
0.020 
0.017 
0.020 
0.023 
0.011 
0.012 
0.017 
0.023 
0.013 
0.013 
0.015 
0.030 
0.012 
0,013 
0.025 
0.011 
0.013 
0.012 
0.012 
0.013 
0.015 
0.014 
0.013 
0.015 
0.017 
0.017 
0.019 
0.022 
O.OZI) 
0.027 
0.017 
0.020 
0.020 
0.025 
0.030 
0.020 
0.023 
0.033 
0.013 
0.016 
0.025 
0.032 
0.015 
0.013 
0.016 
0.014 
0.017 
0.030 
0.013 
0.015 
0.014 
0.015 
0.015 
0.018 
0.017 
0.015 
0.016 
0.020 
0.020 
0.023 
0.025 
0.020 
0.024 
0.024 
0.030 
0.035 
0.025 
0.026 
0.036 
0.015 
0.018 
0'.030 
0.03.5 
0.017 
0.500 
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112 UNIFORM FLOW 
'l"\13LE. 5-6: VALUES OF TliE Ro'l1GllNESS COEFFlCllilNT n (continued) 
Type 01 channel and description ·Minimum Normal Maximum 
C. Exe.WATIilD OR DREDGED 
a. Ea.rth, straight and uniform 
.1. Clean, recently completed 0.016 0.018 0.020 
2. Clean, e.fter weathering 0.018 0.022 0.025 
3. Gravel, uniform section, clean 0022 0.025 0.030 
4. With short gr!l.8S, f.ew weeds 0.022 0.027 0.033 
i' b. Earth, winding and sluggish 
I 1. No vegetation 0.023 0,025 0.030 
2. Grass, some wGeds 0.025 0.030 0.033 
3. Dense weeds or aquatic plants in 0.030 0.035 0.040 . 
deep channels 
4. Earth bottom lind rubble sides 0.028 0.030 0.035 
5. Stony bottom and weedy banks 0.025 0.035 0.040 
6. Cobble bottom nnd clea.n sides 0.030 0.040 0~050 
c. Dritgline-eltcavated or dredged 
1. No vegetation 0.025 0.028 0.033 
2. Light brush on banks 0.035 0.050 0.060 
d. Rock cuts . I 1. Smooth and uniform 0.025 0.035 0.040 
2. Jagged and irregular 0.035 0.040 0.050 
e. Channels not maintained, wc.eds and' 
brush uncut 
. l.Dense weeds, high as flow dept.~ 0.050 0.080 0,120 
: 2. Clean bottom, brush on sides 0.040 0.050 0.080 
3. Same, highest stage of flow 0.045 0.071iJ 0.110 
4. Dense brush, high stage 0.080 0.100 0.1<10 
D. NATuML S'IREAMS' 
D-l. l\llnnr streams (top width at flood stage 
<100 it) 
a. Streams on plain 
I: Clean, straight, fun stage, no rifts or 0.025 0.030 0.033 
deep pools I 
2. Same as .. bove, but more stones and I 0'.030 0.035 0.040 
weeds I 
3. Clean, winding, some pools and I 0.033 0.040 0;04:> 
4. 
shoals . I 
Same ail above, but llome weeds a.nd 0.035 . 0.045 0.050 
stones . , 1 
5. Same as above, lower stages, more I 0.040 0.0'48 0.055 . 
ineffective slopes and sections I ! 6. Same as 4, but more stones 0.045 0.050 0.060 
1· 7. Siuggish reaches, weedy, deep pools I 0.050 
I 
0.010 0.080 
I 13. Very weedy rea.ches, deep pools, or 0.075 0,100 0.150 
J ftoodways with heavy stand of tim- 1 
I' ber a.nd underbrush 
DEVELOPMENT OF UNIFORM FLOW' AND ITS FORMULAS ll3 
TABLE5c6. V,u,UES OF THE ROUCnl'NESS COE)"FICIENT 7i (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 
boulders 
D-2. 
2. Bottom: cobbles with large boulders 
Flood plains 
D-3. 
<l. Pasture, DD brUsh 
1. Short grass 
,2: High grass 
b. Cultivated areas 
1. No crop 
2. Mature row crops 
g. !vIa ture field crops 
c. Brush 
1. Scattered brush, heavy weeds 
2. LIght brush and trees, in winter 
3. Light brmlh and trees, in sumInar 
,4. Medium to densl'l brush, in winter 
5. Medium to dense brush, in summer 
d. Trees 
1. Dense willows, summer, straight 
2. Cleared land with .tree stumps, no 
sprouts 
3. Same ¥ above, but with heavy, i 
growth of sprouts I ' 
4. Heavy stand of timber, f.l few down I 
trees, little undergrowth, flood stage 
below branches 
6. Same as aboye, but with flood stag" 
reaching bra.nohes 
Major streams (top width at flood stage 
>1.00 it). The 11. vallie is Jess than that 
for minor streams of similar ,description, 
becll.use banksoffer less effective resistance. 
0.030 
0.040 
0.025 
0.030 
0.020 
0.025 
0.030 
0.035 
0,035 
0.040 
0.045 
0.070 
0.110 
0030 
0.050 
0.080 
0.100 
0.040 0.050 
0.050 0.070 
0.030 0.035 
0,035 0.050 
0.030 0.040 
0.035 0.045 
0.040 0.050 
0,050 0.070 
0.050 0~060 
0.060 0.080 
0.070 0.110 
0.100 0.160 
0.150 0200 
0.040 0.050 
0.060 0.080 
0.100 0.120 
0.120 0.160 
0.060 a. Regular section with no boulders or /1 0.0215 
brush 
____ 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, 
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114 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 con­
dition. 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 rough­
ness in streams, mostly in the northwes~ern United S~ates. These in­
clude 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 evalu­
ating n. 
l 
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FIG. 5-5. Typical channels ~howing different n values. (These photo h ' 
duced 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 
~ ,,,. • . ,.am.!er or l!e 01 !Jrs.) , 
115 
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1 
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,-1 
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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 . 
~l' 
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i 
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) 
1 
i 
I 
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t 
( 
; 
I , 
. \ 
! 
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(4) 
~ii) 
(6) 
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 
-
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(8) 
(9) 
. . 1"10. 5-5 (1:"'9) . 
7. 1'1 = .0.020. Irrigatiollcanal, straight, in hard-packed smooth sand. of' 
8. 1'1 - .0.022 .. ~ent-plaster lining applied directly tp the trimmed suria.ce 
the earth channel. With weeds in broken places and looStl ssnd on bottom. 
9. 1'1 = 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 . 
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(14) 
(15) 
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 
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(16) 
(17) 
(18) 
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 slopes.: :' , 
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 
I 
I 
~ 
.1 
'{ 
(19) 
(20) 
(21) 
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 
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(22) 
(23) 
(24) 
. FIG. 5-5 (22--241 . 
.22. n = 0.110. 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 
./ 
I 
f 
124 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 of1.486 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 or 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. ' 
1 It should be noted that the synthetio rating curve thus obtained is very approxi­
ma.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). ' 
.;;." 
Jt 
DEVELOP!'.{ENT OF UNIFORM FLOW AND ITS FORMULAS 125 
Stage, ft Discharge, cis Stage, It Discharge, cfs 
0.3 1.'0 1.50 50.0 
0.4 2.3 1. 75 62.0 
0.5 4,.6 2.00 75.0 
'0.6 7.8 2.25 88.0 
0.7 1l.0 2.50 10'2.0 
0.8 15.0 3.00 132.0 
0.9 20.0 3.50 164.0 
1.'0 25.0 4.00 199.0 
1.25 38.0 
6-1S: By the Cowan method, estimate the n value for !l. slightly curved rellch ill 
channel 21 of FIg. 5-5. ' 
REFERENCES 
L E. C. Schnack~nberg: Slope discharge formuiae for alluvial streams and rivers, 
Proceedings, New Zea.land·lnsWt!lion of Enginesrs, vol. 37, pp. 340-409, Welling­
ton, 1951. Discussions, pp. 4.10-4.4.9. 
2. IVllJl E. Houk: Calculation of flow in {)pen channels, Miami Conservancy District, 
Technical Report, ·Pl. IV, Dayton. Ohio, 191B. ' 
3. 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. ' 
4. Philipp FOfchheimer, "Hydraulik" (" Hydraulics JI), Teubner Verlagsgesellscha.ft, 
Leipzig e.nd Berlin, pp. 139-163, 1930. 
5. 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. 
6. 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. , 
7. G. H. L. He.gen: "Untersuchungen fiber die gleichformige Bewegung des Wassers" 
("Research'!S on. Uniform Flow of Water"), Berlin, l~7f.L , ' 
8. 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, Jalluary­
June, '1897. 
9. A. Brahms: "Anfangsgriinde dcr Deioh- und Wasserbaukunst" ("Elements of 
Dam and Hyd!,aulic Engineering"), Aurich, Germa.ny, 1754 and 1757, vol. I, 
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. 
http:Germa.ny
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 Topo­
graphical 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 deooll­
verts (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 Geo­
physical 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,g­
keitsia.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 <A1.tnp; Disoussion Oil Dete.rmination of Kutter's 11 for sewers pal.t!y 
filled, by C. FI ank Johnson, Trans(lcl~ons, American Societ!! of Civil Engineers 
V<ll. 109, pp. 240-243, 1944. • 
29. ~, ~, Wilcox: A compara.tive test of the ftow Gf water in 8~inch concrete and 
vltTl.Hed clar .sewer pipes, Ur;.ilJersill1 ,of W'ashil1(Jlon, ElI(Ji'l'tee:ring Expariment 
Slatton, Bulwt.ll 27, Mar. 1, 1924. 
30. D. L. Yarn~ll and 8. M. Woodward: The flow of water in drain tile, U.S. Depart-
menl of A.gnc!tll.u.re, Bullelin No. 854,1920. ' 
31. C. }'rank Johnson: Determination of Kutter's n for sewers partly Hl'led T 
ti A'S' , . , ransa;c· ons, 'mcnoon oClely of Ct~ll vol. 109, pp. 223-239 1944. 
32. Woody ~. Cow~n: Estimating hydraulic roughne;;s coeffici:nts, Agricultural 
EngmeerL1l{/, vol. .3(, no. 7, pp. July, 1956. ' 
33, Donal~ Schnepper B,nd Ven Te Chow: Full scale toe-of-slope gutter model 
unp~bI1SI:ed rep~'rt ~f an ~n",:,es~ig~t,ion conducted by the Department of Civii 
Engmeenn.g, ,?nlverSlty of ~l1mols, In coopera.tion with the Division of Highways, 
State of IllinOIS,. and the Bureau of Public Roads, U.S. Departn\ent of Commerce 
May, 1954 (ava.tlable at the University of Illinois library). ' 
34. Robert E, Horton: Some better Kutter's formula coefficients Eng' : N 
1 75"" (, tneer·"ng ews1 
'1'0, " no. 8, pp. 3r3-374, Feb. 24, 1916. Discussions. by Fred C. Scobey and 
Robert E .. ~orton., vol. 75, no. IS, pp, 862-863, May 4, 1916. 
35 . .Romce Wilham I{i?g, "Handbook of Hydraulics," 4th ed., revised by Ernest F. 
~rate~, M?Gra.w-Htll Book Company, Iuc., New York, 1954, pp. 7-102 to 1-111 
3a. 'Engmeer,mg ~Handbook: Hydra.ulics," U.S. Department of Agriculture, Soii' 
Oonservation uervice, 1955, sec. Ii. 
37. F. ~. Scobey; Flow of water ip irrigation and similar canals, U.S. Department of 
Agr::cu1tw'~, Technical BuUeiin No. 852, February, 1939. 
38. C;.t.. Ramser; Flow of water in drainage channels U.S. DSDartmeni 01 Anr' It 
T 11 • I B '1 . N '.. ten u.re, ec .n~ca UL etm o. 129, November, 1929, 
1 
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CHAPTER· 6 
COMPUTATION PF.UNIFORM FLOW 
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, repre­
sented by EC[. (5-1), and t.he water area, or 
Q = VA = CAR-S· = KS" (6-1) 
where 
(6-2) 
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 
and the conveyance is 
Q = K',IS 
l( = Q­.yS 
(6-3) 
(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) 
where C' is Chezy's resistance facto::. Similarly, when the Manning 
formula is used, 
K = 1.49 AR~ 
n 
(6-6) 
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 coef­
ficient 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. 
. 1~8 
.. ~ . r 
COMP!UTATION OF UNIFORM FLOW 129 
(6-6), this factof may be expressed as 
~ 
and, from Eq. (6-4), (6-8) 
Primarily, Eq. (1)-8) applies to a channel s~ction when the flow is uni­
form. 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 be omyr!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: . 
Q = 1.49 AR% v'S 
. n (6-9) 
. ' 
Thisis essentially thc product of the water area and the velocity defined 
by the Manning formula. The).'UP-.?cript ~_g,--.§ometill!!lL1L~.d t<L§Qecify 
ihe_c..Qpdition of uniform ·fl9,F-. 
In order to simplify the computation, dimensionless curves showing 
the relation between depth and section fuctor ARH (Fig. 6-1) have been 
prepared for rectangular,. trapezoidal, and cirpular channel' sections. 
These self-ex:planatory curves will help to determine ,the depth for a 
given section factor A.8~, and vice versa. .Ihe A R~ val ue~ for a circu­
lar section can also· be 'found from the table in AppeEi!ix A. 
I . 
1 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. 
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'" tt: 
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'A 
COMPUTATION OF UNIFORM FLOW 131 
6-3. The Hydraulic Exponent for UniformAiow Computation. Since 
the conveyance !{ is a function of the depth of flow y, it may be assumed 
that 
(6-10Jwhere 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() 
dy 
N 
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, . 
d(1nI{) = .!. dA: + ~ ! dR 
dy A dy 3 R dy 
Since dAjd'b' = T and R . A/ P, the above equation becomes 
~(111 K) = J.- (5T _ 2R d!.) 
dy 3A dy 
(6-12) 
(6-13) 
Equating the right sides of Eqs. (6-11) and (0-13) and solving for N, 
N .:~ (5T - 2R ~:) (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 
N 10 1 + 2z(y/b) 8 (y/b) 
'3 1 + z(y/b) --3' 1 + 2 ~1:=::+=z=~-(y-/-b) (6-15) 
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/dQ, where do 
is the diameter, is also shown in Fig. 6-2. This curve shows that the 
. 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). 
---~-·--:---~~~----~-·-'------~~"--., ... -.. -,-"..---.c~---.~= ... ~-=-,_ ....-: .... =_ .. _ ... '-" .-.~~--- .""-~-.-"--" 
~ -,.~-.,-.~.' ':" ---. 
I 
.1 , 
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 I~+-'-----J 
2.0 
1,0 
0.8 
.0.6 i 
0.5 
0.<1 
0.3 
0.2 
0,; 
0.06 
0.06 
0,q5 
0.04 
0.0 3 
0,02 
2.0 2.5 4,0 
VOlues of N 
FIG. 6-2. Curves of N v.alues. 
4.5 5.0 5.5 
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 
COMPUTATION OF UNIFORM FLOW 133 
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 straight­
line 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 (K2;Y2) are 150r, --,---",-,-, .,--.,--.,-.,-" 
taken' from the straight line, the ap- 100 
proximata value or N may be com-
puted by the following equation: 50 
+ I 
I 
~I 
N' -- 'J log (!(dK~) (6-16) 
1 .- ~ log (Yl/V2) 
/ 
III 
./ The cutved' I plot when 'h. 
{)' •• Kll 1 depth opprooches 
I Ille crOllln ot 0 
I l closed conduit 
log Y 
Fw. 8-3. Graphica! determination of N by 
logarithmic plotting. 
K 
':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 expo­
nents 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, 
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i34 UNIFORM FLOW 
the curve may' be divided into 11 number of short segments', arid eachseg­
ment may be considered a straight line having,a constant slope orhydrau­
lie 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 dQ = 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 'lildo = 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 R3il 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 Y ITT 0)' Both the 
discharge and velocity curves show maximum values, which occur at 
about Q,938do and 0.81do 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 pro­
portional to R*, the depth for the maximum velocity, or 0.81do, 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.938do. 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, 
COMPUTA.:!'ION 'OF UNIFORM FLOW 135 
for example, hilS been shown to increase by as much as 28% from 1.00dn 
to O.25dc, where it appears to be a maximum (see Fig.6-5 and the dis­
cussion in Art, 5-8 regarding t.he as a factor affectiIlg n value). 
This effect callses the actual maximum discharge and velocity to occur 
. at depths of about O.97do and O.94do, respectively. The corresponding. 
curves of Q/Qo and ViTrQ 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 0/00 , vivo • AR<iYAoRJI~, cnd R 1fYR~iJ 
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 C011-
duit 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 maxi­
mum 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 
, I 
J 
136 UNIF,ORM 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] <Lnd Einstein [7,8] assumed thateachpal't of the area has the 
same mean velocity, which at the same time i15 equal to the mean velocity 
of the whole section i that is, VI = V ~ = . . . = V N = V. On the 
basis of this assumption, the equivalent. coefficient of roughness may be 
obtained by the following equation:' .' . 
(6-17) 
There are many other assumptions for the determination of all equiva­
lent 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 
n 
(Plnl 2 + 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~ 
i 
I 
I 
! 
i 
COMPU'l'ATION OF UNIFORM FLOW 137 
where ~l' Hi, ... ,R/'f are hydraulic radii of the subdivided areas. For 
simple c~!\.nnel sections, it may be assumed that 
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 
Smooth iee: 
Withol\t drifting iee blookB 1.3-2.0 0.01D-0.012 
>2.0 0.014-0.017 
With drifting ice blocks 1.3-2.0 0.OI6-0.01E 
>2.0 .0.017-0.020 
Rough ice with drifting ice blocks 0.023-0.025 
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) 
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 
P P1' P! 
C1 = C
1
2 + Czl (6-21) 
Let: the wetted perimeter P z = aPi or P = PI + P2 (1 + a)P1i 
then 
Sin(:e,: by Eq. (5-7), C 
1 a 1 a 
= C
1
2 + C2 2 
1.49Rt'/n, , 
(1 + a),n2 
RH! 
(6-22) 
(6-23) 
138 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 = 11,22 (1 + ~)H (E22 + aEif,) (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), Ezz =aEI)', a.nd 
Tt2 ('(' 'S)2~ n = - a' T 0Z' 7 
VT+li 
(6-25)* 
For wide channels, it may be a,ssumed that 
a = 1. Thus, E22 = ~l~', and 
PI = P z, that is, that 
Tt = ..72 (1 + ~2~l)% 
The roughness coeffiCient for the ice cover is, therefore, 
n~ = (1.68nn - nl~I)~l 
(6-27) 
(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 
nl 
Q = 0.63- Ql 
11, 
(6-29) 
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 = R~/n instead of Eq. (5-7), obtaining 
(6-26) 
It was Belokon [lSj whQ us~d Eq. (5-7). 
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COMPUTATION OF UNIFORM FLOW 139 :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 sub­
sections, 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 thevelocity-distribution coef­
ficients for the corresponding subsections; let .6.A 1 , .6. A 2, •• " .6.AN be 
Fw. 6-6. A channel con8isting of one main section and two' side sedions. 
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: 
VI = [(~ Sf. - 1(2 Sf~ _ 1(/'1 SH 
.6.A. I V2 - Ll.,A z VN - .6.AN 
Q = V A = VI .6..4. 1 + Vz dA 2 + + VN dAN 
N , , 
= {Kl + ](z + .. + ](N)S~~ 0:: IC.r) Sfi 
I 
and 
Incorporating the above expressions with Eqs. (2-4) and (2-5) and 
simplifying, the velocity-distribution coefficients of the entire section are 
and 
N 
I (aNKNJjdA N') 
1 a = ~-N~--~--~-
0; [(NY/ A' 
1 
N l (f3N[(N 2jdAN ) 
(J = -J-'N'-~--~--
(Il(NY/ A 
I 
(6-30) 
" 
(6-31) 
1. 
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140 UNI~ORM 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: 
{J Subsection A, ft' J P, ft I n value I 
Main section. .-.--I·-
S
-, 3-6-0-
1
1\. ~5 11~0-.-O-3-5~JII-1-.-1-0-'--1-' 0-4-
Side section. .. .. .. .. 5,710 405 0.040 1. 11 1. 04 
Sol'u/'i01l:. The computations are given below. 
S"b~"tio" AA I-~ R .. J. " K I /3K2/I!A I ",K'/I!.tl' 
-I---! 
I. 892 X 10' 16-::;4 X 10'1".93 X 10" Main section. 5,360\22523.88.290.035 
Si~::~~i~~.,:: I ~:~~~I~~y~·15:~~ ~:~'~~ 1.244 X 10' 12.82 X 10' B,56 X 1O" 
3.136 X 10' i g. 76 X 108 I 32.49 X 10" 
I I 
By Eqs. (6-30) a:nd (6-31), the.coefficients are 
32.49 X 10'0, 
0'= (3.136 X 10')'/11,070' = 1.29 
and 
9.76 X 10' 
{J = (3.136 X 10°)'/11,070 =1.10 
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 
The velocity is 
R = y(lO + y)_ 
10 + y vS 
and A = y(20 + 2y) 
'100 ./ 
y(20 + 2y) 
./ ." 
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]. 
:' 'I of ~ /71 
/r 
I . 
[md simplifying, 
COMPUTA.TION OF UNIFORM FLOW 
V.:=: LJ"'L ~ J../l '.rT" ,', 
200 = 1.49 [Y(lO + Y)-J~~ O.0016H 
y(10 + y) 0.025 10 + y V5 
or 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 J A -~I RH Am~ Remarks 
-----1 
3.00 
I 
78.0 2.34 I 1.762 137.4 Y too sll~all 
3.50 94.5 2.65 1.915 181.0 Y too large 
i I 
I 
I 3.30 87.7 2.53 1.852 162.6 I 3.35 89.5 2.56 
i 
J .870 167.2 I 
, 3.36 I 89.8 I 2.56 I 1.870 I 158.0 The closest 
The normal depth is,therefore" Yn = 3.36 ft. 
B. Graphical 111 ethod. 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). Com­
pute 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. 
-_., 
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142 UNIFORM FLOW 
C. Method of Design Chal·t. The design. chart for determining the 
normal depth (Fig. 6-1) ~all be used with great expediency ... 
In Exalllple 6-2, ARn =< '167.7. The v",[ue of AR~fi/M~ is 0.0569. I For this value, 
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 yla., = il.72, 
or 11 = 0.72 X 3 = 2.16 ft. 
6-7. Determination of the Normal and Critical When the dis-
charge 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. 
o 
FlO. 6-7. A curve of !I vs. ARJ~ for n. circula.r section. 
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 uni-
form 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. 
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COMPUTATiON OF 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.) From the given data it is found that R = 2.56 it and V '= 446 f . 
S b ' . h ' ps. 
u stltutmg t ese va.lues; in the Manning formula and .~ol-dng for Sn, 
or 
A, Ad - 1.49 256"'8 I" 
•••• C' - 0.025. ~.,,' 
s. 0.0016 
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:nding. 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, ' 
7.66 1. 97HB.l." 
or S. 0.(1067 
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 S.n = 0.0070 
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. 
Example 6-6. petermine the limit slope of !I. rectangular channel (Fig. 6-8) with 
b = 10 ft and n = Q.0l5. 
.S~ll'tion. .Since the limit slope is the :;mall~.st critical slope, its value' maybe deter­
mmed 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 
.Q = K vB. (6-32) 
or, 'when the Manning formula is used, 
(6-33) 
or, for the rectangular channel, 
(f}-34) 
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144 UN)FORM FLOW 
:2. The second condition, from Eq. (4-3), is 
Q =Z,...;g 
or, for the .ectangulul' chmnel, 
Q = 1O....;g y'" 
(6-35) 
(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. :3 4 5 6 7 
Slope in 10-3 
(al 
Slape In 10-3 
. (b) 
F!G~ 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 dlS­
cha.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, super­
critical. 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· con­
tinuity 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:' ." 
COMPUTATION OF UNJl<'ORM FLOW 145 
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 com­
putation 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 indi­
cated by a check mark (v') and the unknowns required in the problem 
by a question mark (?). The unknmm.val'iables that can be determined 
from .the known variables are indicated by a dash The last 
TABLE t>-2. SOME TYPES OF PROBLEMS OF UNn'ORM-FLOW COMPUTATiON 
Type of Dis- VeIoe-
I Geo-
De th Rough- Slope metric 
prob- cha.rge ity p ness 
S ele-
. Exa.mple 
lem Q V 11 n ments : 
A ? - v v v I v Prob. 5-5, (Ex, 5-0 
B - ? v v v v Prob. 5-5, (Ex. 5-1-) 
c v , - 1 v v v Example 6-2 
D v v ? v 'I. Prob.5-6 
E v' .- v v ? , v Example 6-4a 
F v .- v v v; ? Example 7-2 
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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. Fur­
thermore, flood flow is in fact varied and unstean'y, and use of a uniform­
flow 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 contracted­
opening method, in which the principle of energy is applied directly to a 
contractedopening 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 con-
I 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 
i 
j 
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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 uniform­
flow 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 
F' 
S =£ (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 Zj2g. The energy slope is, therefore, equal to 
s=!!:!. 
L (6-39) 
where (6-40) 
and kis a factor.' When the reach is (;ontracting (Vu < Vd ), 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 UNIFORM 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. 
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 = 11,070 
Ad = 10,990 
K, = 3.034 X 106 
K. = 3.103 X 10 6 
au = 1:13" 
O'd = 1.177 
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(/ - ."" 2(/ 1.354 
V d'_ 1177 (97,000/10,990): = 1,42,1 
ad 2g - . 29' 
-0.070 . 
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 6-6 
Appr.oxi- I As~umed Flo. ~ a~ ~l h, I S v'S 1 Computed 
matlon Q I 2g 2U Q 
I-~----I-!-----1st 
2d 
3d 
4th 
5th 
97,000 
90,000 
91,200 
91,000 
0.500 ...... , .. \°.50°
1
°.0010000.0316 97,000 
0.500 1.354 1.424 0.430 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 
, , 
.~ 
j ,. , 
COMPU'l'ATlON OF UNIFORlI'I FLOW 149 
surface flo~ occurs mostly as a result of natural runoff, and is called oo'er­
land 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 
.J,~,~Ofti~T!~ ~V~~~,~" 
V?",,-~ ~ ......... 1,;:\.~j·· 
~" 'r- . ~,' '\\ 
.~ f' "'"(i"', r .' 1 '.. •• ., .~~ 
w(Y"l~'Y)S (_ • 'flop'li' 't~o .... \ .. , 
" Ii! . t~F"'~~101l ~ ;.:.; fr. 
.( . _,(. J,;.iii:. 
i n7 .... , ~, ) ' .. ,~ 
.' . ··'·1:':··~7 
,/ ,., AifC 
~~1:;;t· 
FHl. fr9. Uniform laminar open-channel flow. 
dynamic viscm:ity i" and the shear stress r at a dist.ance y from the 
bOlmdary surface (Fig. 6-9)" as follows: 
du 
r = I> 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 _ t) 
)I • . 2 
(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-
http:6-9),.as
150 UNIFORM FLOW 
grate Eq. (6-42) from Y = O. to Y = Ym and divide the result by Yrn; the 
average velocity is 
1 1011
°' V = - vdy 
Y •• 0 
(6-43) 
and the di:;;charge per unit width is 
(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<i [22], from R = 300 to 330 by Hopf [23}, 
and from R = 548 to 773 by Horton [24J. However, Horton believedthat the Reynolds criterion is not satisfn.ctory for sheet flow over relatively 
rough surfaces. He reasoned that, at the transition point, the velocities 
for laminar and turbulent· flow are nearly equn.l, because this condition of 
equal velocities represents the minimum amount of energy capable of 
. maintaining turbulent £low.. Thus, the flow cannot be turbulent if the 
velocity is less than 
(6-46) 
where Y", is the avewge depth of flow. 
As the natural ground surface is rarely even and uniform in slope, over­
land flow is apt to cha!lgefrom lamim\'r to turbulent, and vice versa, within 
a short di.'lta.rice. Consequently, the flow is mixed between the laminar 
and turbulent. For very rough surfaces or areas densely covered with 
vegetation, the flow ill general is highly turbulent. Experiments have 
indica.ted tha.t the discharge of overland flow per unit width of flow varies 
with the aver~ge depth o~ flow as follows: 
(6-47) 
'where C is a coefficient a~d where the exponent :j: varies between 1.0 for 
highly turbulent. flow and 3.0 for mixed flow. . 
:r ." ..... -, ,,-'~ 
I 
I 
I 
\ 
COMPUTATION OF UNIFORM FLOW 151 
PROBLEMS 
6-1. Detern:ine the normal discharges in channels ha.ving the following sections for 
'y = 6 ft, n = 0.015, and S = 0.0020: . 
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 . 
6-2. 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 
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 pre­
cedmg 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 [01-
lowing 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:.. (3 T _ U dP) 
A 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 , 
( 
-\ 
http:incres.se
'152 . UNIFORM FLOW 
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 maxi-
mum 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') - In R] (6-50) 
where x = 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 
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 sur-
face 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) .. Com­
pute 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: ' 
;;1 
r 
I 
I 
i 
I 
l 
C01-IPUTATION OF UNIFORM FLOW 
Q.. The cross sect·ion of the stream at 11 typical upstrel1m station in the uniform 
Tea<:h is given hy the following cQordinates: 
StrLUr:f(1. Elev. m.s.!. Sta/·ian Elev. m.s.l. 
Left b8lnk: 0 + 00 590.0 6. + 00 543.7 
1 + 00 580.7 8 + 00 540.0 
1 + 50 578.2 10 + 00 572.2 
3 +00 582.0 11 + 00 573.2 
4 +00 581.0 12 + 00 568.5 
5 + 00 580.0 14 + 00 590.0 
b. The value of n for the main channel is estimated as 0.035, for the side channels 
1l.S 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 
ch~.nnBl. 
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 con­
prete \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.ve­
ment 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 anuncovered 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 
[ (n.)% ] . I:>y = 1.31! n - 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 
-----------
Main section ........ 5,320 205 0.035 1.12 1.05 
Side section .... "':1 5,670 408, 0.040 1.10 1.0-1 
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 
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 
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 dis­
charge 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. lIH (6-53) 
·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<L~ a fall of 0.35 ft in wa~er surfa.ce during It 
flood. Compute ~he flood discharge through this reach, using the following data: 
Subsection A, ft' 11. Of ~ 
Upstream: 
Main channel. , ~ . ~ ....... 4,250 0.038 1.10 1.0'1 
Side channel .... .. , ....... . 25,620 2,050 0.038 1.20 L08 
Downstream: 
channel ... , . ...... .. 5,760 320 0.042 1.10 1 .. 04 
Side channeL. ....... ... 25,610 l,gOo 0.038 
i 
1.18 1.06 
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 = 177. 
I ,. 
i 
I 
! 
I 
I 
I" . . '! 
"> ,,/.-:' 
COMPUTA'l'lON OF UNIFOR1I1 FLOW 155 
REFERENCES 
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 opell­
cha.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-mat­
icheskikh i tsilindriche.~kikh ruslakh pri usta.novivshemsia dyil!henii (On the con­
struction 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 sec­
tion 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.lIo­
I 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 deriva­
tsionnykh 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 
http:8urfa.ce
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.ns­
actiO'flS, All-UnionScientific 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 For-
mula"), 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, TTans­
actions, 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 dis­
cussion 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 exca­
vated 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'actica­
bility, 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 
./ 
http:lining.of
UNIFOR\I1 FLOW" 
158 '~. 
. water to pick up lining blocks and push them out .~~ ??SlulOn. 
movmg .. h Id be designed against such POSSibilItIes. 
Accordingly, the Immg SOli, 'bi VelO~l'ty The mim:mum pel'missible h M' , urn PerrnlSSI e '-. . 
7 -3, T e mlm "t'n v~locit'lJ 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 ' 'It load or for desiltecl flow, t IS . d For water carrymg no Sl , , 
determme . 'fi ,t for its effert onplant growth. Gen-f t . h s little si"'nI canee excep . - h 'h 
. ac OJ a '. t> l't' f 2 t 'Hps may be used safely w en" e II kino' a mean ve OCI y 0 0 • . 't f 
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 ::allihe longitudinal bottom Glope of a 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 0 w , 1 F'" lple channels' used for W:Lter-dlstrl-f the channe 'or exan , h d r 
' pur~ose o. " ' b se used in irrigation, water supply, y :au I~ 
butlOn plll poses, such as to, . h' , level at the pomto! 
. . , d h droDower proJects reqUIre a Ign . 
mll1Jl1g, ~J.l1 y. 1 -I' '~d 'rable in order to keep the los3 III delivery; therefore, a smaL b ope I::> eSI 
elevation to a minimfum'h 1 deIJend mainly on the kind of materiaL, The side slopes 0 a e anne . h . 
" 1 . d f the slopes suitable for use Wit 'l'fLrJOUS Table 7-1 give:, a genera I ea o. ' 
's R CHANNELS BUILT IN V,UllOU.S I{INDS TABLE 7-1. SUl'rABLE SID;;; LOPES FO 
OF MATElRIALS 
. 111 aterial 
Rock .... ,.,.. . .. , ........ ', ... , * •••••• ~ ••• w ••••• 
Muck and .peaL soils . .. ", .. . .... : .' ........ , ... ~ ........ . 
Stiff clay or earth with, concrete Immg ... , , . ' , .......... . 
" I" or earth for large eila,nnels ....... . Etlft.h ,Vlth Btone Wing, . 
Firm clay or earth for small ditches ........... , , .......• 
Loose sa.ndy earth. , ....... , ... , .... , .... , , .. , ... , : . , .• 
Sandy I06.m 'or porous clay .. , . : .. : ... , ...... , . , .. , .... , 
Side slope 
Nearly vertic al 
~:i: 1 
H':1 to 1:1 
1:1 
1%:1 
2:1 
3:1 
. . erodible material, however, a more accurate 
kinds of m~LterJal. For h'd b checked [wainst the criterion of . , f th lopes SOUl e " "0 
determll1atwn 0. . e Slit (Art 7-10) or by the principle of tractive 
maximum permli5s1bb~~e. °f:c~ors to' be considered in determining slopes 
force (Art. 7-14). ,~1. ~dition of seepage loss, climatic change, 
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 h 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 , 
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DESIGN OF CHANNELS FOR UNIFORM FLOW 
159 
a.now the pradicable Use of just about any type of lining or lining tl'eat­
ment 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 substruc­
ture may be endangered by any overflow. 
There is no universally accepted rule for the'determination of free­
board, since wave action or wat~t-8urface fluctuation in a channel ma.y 
bi;: created by many uncontrollablecauses. 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 slop­
ping over, 
Freeboll.rd in 'an unlined cann.'l OJ' lateral will nornu;.lly be governed by 
considerations of canal size and locatiOJ1, storm-water inflow, and water­
table 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 
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160 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. 
Capacity. cIs 
FIG. 7-1. Retommended freeboard and lleight of bank of lined channels. (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, there­
fore, 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 
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DESIGN OF CH .... NNELS FOR UNIFORM FLOW 161 
be noted that a best hydraulic section is the section that gives the mini­
mum area for a given discharge but not necessarily the minimum exca­
v!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. BEST HYDRAULIC SECTIONS. 
I AArea Wetted /' Hydraulic' Top Hydrau- Section 
Cross se<;tion perimeter radius width lie depth . factor 
I_~ __ ·. P il, __ R-I __ T_I_D~ __ Z __ 
I' I 
Tmpezoid, half I .: . 
of a hexagon I ,13 yl: 2 ... /3 11 %,y %' v3" 11 '%y %y'" 
Rectangle, half I 
of a square I 2y l ,II 4y ! 
Triangle, half of I 
a :;quare y' I 2 V2 11 I 
Semicircle .) , ~Y' i Try I 
" I P,arabola, 
T=2V2y 
Hydro~tatic 
catenary 
1% V'2 y' )%'V2 Y 
! 1. 39.58Sy' r 2, 9836y 
~YI 2YI Y 2:\1',6 
"~YI "I ~' ~y" 
72YI 2111 -y - y'" 4 4 
HYt 20YI %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). . • 
Example 7-1. Show that the best hydraulic trapezoidal 8ecti~n is one-half of a 
hexagon, 
Solutioll . 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 - z) 
I 
For a minim~m wetted perimeter, dP Idy ~ 0, or 
b = 2Y(VI ,+ ZZ - 2) 
b 
y 
Substitl!ting this equation for b/n the previ~us two equations for A ELnd Pand solving 
http:decreD.se
1,62 . UNl.F'ORMFLD'W 
simu1t,a.neDusiy for: P, 
p = z 
Now, find tile va.lue of z tha.t makes P th~ least. 
z, equcting dP Ida to zero, and sol viug for z, 
z = ~3 ~ ta.n30· 
This means that the section is a half he"agon, 
Diifer:entiating P with respect to 
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, 
(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 
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 
practicability. 
5 .. For the design of irrigation channels; the channel section is some~ 
times proportioned by empirical rules such as thesimple rule given by·the 
early U.S. Reolamatiau Service [51 for the full supply depth of water in 
feet .. 
y= 0.5 -vA (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) 
DESIGN OF CHANNELS FOR UNIFORM FLOW 163 
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 have uS,ed an enlpirical formula V = VA 13 '= 0577 : r7A h'ch . . I : I' V .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, 
35 
30 
'-
.= 25 
'" <>. .. 
~ 20 
'0 
<: 
" 
== ,,,! 15 
" 
E 
2 
0 
OJ 
:> 
Q! 
10 100 ZOO 300400 600 IPOQ '1,000 '!,pOD 
capodry ill cis 
(
FUIG
S
' 7B-2. Experience cur:re.sshowing bottom, width and depth of lined· channels 
. . .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", Zlavmg aBslope of 0.0016 and n =- 0.025. Proportion the section dimensions 
,,0 !!t,on. Y' Eq. (6-8), . 
AR~1 = _0:025_ ~ '" 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) in the ~ove 
-::::---'-":......:..~~;.--' == 167.7 
y)% . 
Assuming b "'" 20 it and z = 2 and simplifying, 
7,680 + 1,720y = (y(1O + y)Ju 
Y,= 3.36 It 
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164 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 free­
board 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 va/a. 
B. ERODIBLE CHANNELS WHICII SCOUR :BUT DO NOT SILT 
7-8. Methods of Approach. The behavior of flow in an erodibl~ chan­
nel 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 chan­
nel. 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 
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. 
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DESIGN OJ!' CHANNELS FOR UNIFORM FLOW 165 
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 engi­
neering 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 veloci-
T.~llLE· 7-3. MAXIMUM PERMISSIBLE VELOCITIES RECOMMENDED' BY FOR'rIEll 
,AND SCOBEY ANi. THE CORRESPONDlNG UNIT-TIIAC"l'IVE-FORCE VALUES 
CONVERTED BY THE U.S. BUREAU OF RECLAMATION" 
(For straight channels of small slope, after aging) 
Material '7\ 
, ! Water trans-
Clear water _I porting col­
loidal silts 
v, I rD, V, TO, 
fps ilb/ft2 fps Ib/ft' 
--'1---------
Fine sand, colloidal ... , .. , . , .. ,. " ""'1 0.020 1. 50 i O. (127 2,50 0.075 
Sandy loam, non colloidal. . .,. . . , , . . .. 0,020 1. 75 . 0,037 I 2.50 0,075 
Silt loalll, nOllColloidal .. " ", ... .", ''', 0.020 2.00 0,048 3.00 0, n 
Alluvial silts, noncolloidal. ......... 0.020 /2.00 0,048 3.50 0.15 
Ordinary firm loim,., ..... , ....... , .. 0.020 2.50 0.075 3.50 0.15 
Voleanicash" ...... " ........ " .... 0.02012.50 0.075
1
3.50 0,15 
Stiff clay, very colloidal. , ...... "" , .... " 0.025 I 3.75 0.26 I 5.00 0.413 
Alluvial silts, colloidal.. """'''' "''-'1 0,025 3.75 0.26 5.00 0.46 
ShalesandhardPl1ns .... : ......... ".: ... j 0.025 6.00 0.67 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. , , . ,.1 0.0;10 4.00 0.43 5.50 i 0.80 
Coa-rse gmvel, ~0IlC0I10idlll .......... , .. ; ... / 0.025 4.00 0.30 .16,00 I 0.67 
CObblesandshmglE'3, ... , ................ 0.035 5,00 0,91 5.50 1.10 
, • 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. 
ties tha.n new ones, becau$e the old cha.nnel bed is usually better stabi­
lized, 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. 
LG6 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] how­
ever, it is dOHbtful whether such a velocity Mtually exists. In 1915, 
~ 
'" .~ 
" "iii ,. 
'" :is 
j o~~, -=l==I=t:I=l= 
0,.8 
()'5~H-++--t 
O.4!-H-++-fflt-f--+ 
o.:;f--f-+-++-+H-H--t 
Q.~ !-,-,-.---l...--'---r",-,--,--W.-..c 
Cloy 
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 theqhan­
nel 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. Com­
prehensive bibliographies can be found in [19] to [25]. 
~, : 
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DESIGN OF CHA.NNELS FOR UNIFORM FLOW 167 
2 
,g 0.7 
2 .. 
'0 g 0.5f----j--·,--+-.. ·---/---';-',,\},c-:-+---f-1--i 
0.4 
:3 
Per"r.issible va loci!i es. fps 
FIG. 7-4, Curves showing' U.S.S,u, da.ta on permissible velocities for cohesive soils. 
.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.xi­
mum permissible velocities (Figs. 7-3 and 7-4) 
ahove which scour would be produced in noncohe-
sive 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 ap­
proximate, since no accurate data are available 
at the. present time. 
7-10. Method of Permissible Velocity. Using 
the maximum permissible velocity as a criterion] 
the design procedure for a channel section, as­
silmed to be trapezoidal, consists of the following 
steps: 
Carreclion foctor 
FIG. 7-5. Curves showillg 
U.S.S.R. corrections of 
permissible velocity fo,r 
depth for both collesive 
and noncohesive mate­
rials. 
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168 UNIFORM FLOW 
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 per-
missible 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 simul-
taneously 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. . ' 
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. 
1.49 --
4.5 = 0.025 R~ ";0.0016 
or R ." 2.60 ft 
ThenA = 400/4.5 = 88.8 ft', a.nd P = AIR 88.8/2.60 = 34.2 ft. Now 
A = (b + zy)y = (b + l!y)y ~ 88.8 ft' 
and P = b + 2 v'l + ii Y = (b + 2 y'S y) )\4.2 it 
Solving the above two equations simultaneously, b 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 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 
channel resistance in 11 uniform flow was sta.ted 1;Iy Brahms earIy in 1754 (see Art. 5-4). 
I 
.Ii". 
DESIGN o.F CHANNELS FOR UNIFORM l"L~W 169 
It should be noted that the unit .tractiv~ force in channels, except for 
wide open channels, is not uniformly distributed along the wetted perim­
et.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 
i 
1.5 
O:97QwyS 
:FIG. 7·6. Distrihution of tra.ctive force in !1 trapezoidai channel section. 
123'156789 
bh 
On sid s S 01 channels 
. Ii, 
l-fl--+--!--+---l- - -I I rmm] 
!4-+-1-+--+=1t++i~ 
o :-.-JI:~ 
o I 2 3 4 5 6 7 B 9 10 
bh 
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!1gular 
channels. A typical distribution .of tractive force in a trapezoidal chan­
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 
http:88.8/2.60
170 UNIFORM FLdw 
. . 
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-forceRatio. 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 gnlvitycfol'ce component 'YV; sin ¢, 
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 inde­
pendently by' the U.S. Bur~a.u of Reclamation under the direction of E, W. Lane 
[29,35]. 
,. 
J 
DESIGN OF QHANNELS FOR UNIFORM ;FLOW 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 
T, = ,c! cos rb tan II '\;1- tan' (1 (7-7) 
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 <to = 0: 
}v, tan fI = an (7-8) 
Solving for the unit tractive force n that causes impending motion on a 
level surface, . 
lV, . 
TL = - tan e 
a 
(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 
T f tan2 ¢ 
K = ~ = COli '" ~ 1 - tan 2 0 (7-10) 
Simplifying, ) J( = /1 _ Si.ll ~ '" 
'\; Sill" ~ 
(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 com­
paratively clear .vater, become so great in propprtion to the gravity­
force component causing t.he particle to roll down that the gravity force 
wn safely be neglected. Therefore, the angle of repose need be con­
sidered 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 
1 . 
http:equal.to
I 
! , 
172 UNIFORM FLOW 
Panic 1 e size in in clles 
0.1 . 0.2 O.l 004 0.6 o.e 1.0 
L-----.rL--; I , I I I I I I I I 2.0 3.0 4.0 
42 3/16 1/4 lIB 112 3/4 I 
~-•..... l ~j -T--[I 
I 
H/2 
-;:; 
" " N 
o 
4 0 
8 
,6 
.., :3 4 
2 
." 
'" .., 0 
.; 
'" e 
Il c. 
'" '-
'0 
.!? 
'" 
'2 6 
c: .. f--- ... 
2 4 
2. 2 
20 
I 
r r ... 
---~. I // V ' 
// / 
I V V// V/ 
-1-- // , V V// V 
~~i/I / VI/';I/I 
~~"'/ V L/ 
Ii ; V L .---;----
~ ·Z .... " t7-.'" 
~'" 
~o $O~ V 
1 
~/ " 
~ .If': b'" if t1 ! 
I.;;:,~ 'ti ~<~I I 
J~Ci ;;"'/ / 
V."O- 'b V i---
~ 0° ~~ ; 
~ .) r j , L ,'" 
V:o."':;" 
I) 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 sub­
stantially 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 
i , . 
I 
-\ 
I 
I 
A 
DESIGN OF CHANNELS FOR UNIFORM FLOW 173 
particle size for noncohesive material and upon compactness or v:o~ds 
ratio for cohesive materiaL Other' soil properties such as the plastIcIty 
index1 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 
\ I . I I 
/ .<1r-+~I+H Recommeoded value for 
"-:; canals in coarse l 
noncohe~ive material~ 
si.e 25% lorg er 
50 100 
FIG. 7-10. Recommended permissible unit tr.active forces for canals in noncohesive 
Ino.terial. (U.S. Burea:!' of Rec!amalit.m.) 
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: 
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 
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, <lnd (3) for canals with clear water. 
For cohesive materials,the data based on convei·sion. of permissible 
velocities to UIJit tractive forces and given in Table 7-3 and Fig. 7-11 are 
recommended as design references. ' 
The pel'missible tTactive forces mentioned above refer to straight chan­
nels. For sinuous channels, the values should be lowered in order to 
reduce. scour. Approximate percentages of reduction, suggested by 
Lane [29J, are 10% for slightly sinuous canals; 25% for moderately sInu­
ous canals, and 40% for very sinuous .canals. 
DESIGN OF CHANNELS FOR UNIFORM FLOW 175 
7 -14. Method of Tractive Force. The first step in the design of erodi­
ble channels by the method of tractive force consists in selecting a11 
approximate channel section by experiene.e or from design tables,l collect­
ing 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: 
Example 7-4. Design a trapezoidal channel laid on a slope of 0.0016 a.nd carrying 
a discharge of 400 cfs. The cha.nnel is to be excavated in earth containing noncol­
lnidal coarse gravels and pebb['~s, 25 % of which is 1.25 in. or over in diameter, Man-
ning's n. = 0.025. . 
SoLution. For trapezoidal channels, .the maximum unit tracth'e force OIl the slop­
illg 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 permis­
sible 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 b Iy = 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. 
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 UNIFORM: FLOW 
7 -15. The Stable Hydraulic Section. The section of an erodible chan­
nel 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 to· 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 sec­
tion-a situation, however, that never actually occurs. Fortunately, 
the mathematical analysis made by the Bureau I has shoWn that the actual 
I 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 
·1 
I 
i 
DESIGN OF CHANNELS FOR:UNIFORM FLOW 
177 
transfer (if tractive force has little effect on the results and can safely be 
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 
where rP is the slope angle of the 
tangent at AB. . 
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 corre­
sponding unit tractive force on the 
sloping area ~B is,therefore, equal