I 1

MeGRA W-HItr, CIVIL ENGLNEERING SERIESHARMER E. DAVIS, Consulting Editor."

OPEN-CHANNEL)

HYDR~AULICS

I)

\

BABBI'IT '. Engineering.in Public Health BENJAMIN' Statically Indeterminate St~uctures Cnow . Open,-cha,nnel Hyqraulics DAVIS, TROXELL,'AND WrsKoCIL . Tl1e Testing and Inspection of ' Engineering Materials DUNl'iAM . Foundations of Structures DUNHAM' The Theory and Practice of Reinforced Concrete DUNHAM AND YOUNG.' Contracts, Specifications, and Law for Engineers GAYLORD AND GAYLORD' Structural Design HALLERT 'Photogrammetry HENNES AND EKSE . Fundamentals of Transportation Engineering KRYNINE AND JUDD' Principles of Engineering Geology and Geot.echnics LINSLEY AND FRANZINI . Elements of Hydraulic Engineering LmsLIDY, KOHLER, AND I'A ULHUB ' Applied Hydrology LINSLEY, KOHLER, AND PAULHUS' Hydrology f9r Engineers LU:8DER . Aerial Photographic Interpretation MA'l'SON, SMITH, AND HURD' Traffic Engineering MEAD, MEAD, AND AKERMAN' Contracts, Specifications, and Engineering Relations NORRIS, HANSEN, HOLLEY, BIGGS, NAMYET, AND 1fINAMI . :Structural Design for Dyiramic Loads PEURIFOY' Construction Planning, Equipment, and Methods' PEURIFOY' gstimating Constructi()u Costs TROXELL AND DAVIS' Composition and Properties of Concrete TSCHEBOTARIOFF . Soil Mechanics, Foundations, and Earth Structures URQUHART, O'ROURl reetanguJar channels and 14 fol' the triangular channel under consideration.to

e

R

..

o(C

F: '.

R 3,0*

a.o AI4,

',{I td

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

$

~

a ..R

R ., 4,01;:11..

~4 tK

a til

e(I

R 0.0

tloll

A ... 1.:2 C!A

0

G

fICTAlfGlA.AR c.HAflNa~,1.5Ff \\'10. wlTK GL,us WALlS&. POI.JSH(O 9Fl4SS PLATElhjrro~,

0.2

!~

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

.,..0.1

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

FACE;o

or

Sl'Flt;CTtiRAL

srULfsd''V~Rrt)t ,t~I.ES,WlrH0.08

nU4NCAA...A.R CHA'HNl.,l'd" TO

f

om

O.Oi

f

0.06

().04Q

k

100.0rt9fM

.. k.O.qz0.02

0.02.831"

,

\1O.OIl-----4----+-....:..--t---~'r-~0.008

0.01

0.000

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

10

"--''-.---

.J ~

i

4

$1'0'

R;

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

I

\

.' . 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, andlO mm In roughened by tra.nsverse wooden stripS 27. sp.~in'" of mm; No. 24, cem, ent .. 7 Ne 14. except WIt .1 a te : spacmg; No.1, same B.S. ' d K' her's cha.nnel: smooth concre . lining; and No. 26, u~pIL

n-2

"

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

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

-I

'J14.

itJOPEN-CHANNEL FLOW AND ITS CLASSIFICATIONS

BASIC PRINCIPLES .

rI I

15

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

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

1I

1

~I,_I

1

'1,-,'

1I

(

'-J

jf,I

~\

i

)

~J\

iI

C:--'\\..-

I!

Velocily, Ips

~.~(Afler

FIG.

1~5.

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

Roberlson and Rouse [3D].)

,_-

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

'~

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

L

i

.

.

..

(Courtesy{_.

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

,

,

16

BAstC ' PRINCIPLES

~lI ! ! II ,

(

I

\.

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

~ %56k~ sin 4.j.]

. I(2-13)l

I

YI = if cos 10

(2-14)

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.PROBLEMS2-1. Verify the formulas far geometrio elements of the seven channel sections given in Tahle 2-1. $-2. Verify the cUrY'es shown in Fig. 2-1. 2-3. Construct curves siinilar tQ those shown in Fig. 2-1, for a square channel section. 2-4. Construct cu'rY'es similar to those shown in Fig. 2-1 for an equilateral triangle with one side as the channel bottom. 9.-5. From the data g(ven below on the cross section 1 of a natural stream con

.[

/a)

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

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

, i

"

( ,

of

1

f

)

j)

.1

!36Douma. formula,' that is,'U -

,.BASIC PRINCIPLES

OPEN CHANNELS. AND THEIR PRO'PERT1ES

37

10

~0.2V: . gR

- 1

(2-15)

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

R Is the hydraulic radIus. . 2-12. Compute the wall pressure on the section OA (Fig. 2-10) of the spillwaydescribed in Prob. 2-11. that at section DB.It is assumed that the depth of tlow section is the same!l.S

/

\

$PilIW1!Y

Iraining wall,

eo II

cpO!1

;:;

C .2

.)i.~I

:OJ

::l

t;j

2

/,i 4

,aUn;l pressare, II 01 woler

\I

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

REFERENCES1. 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. Leonard Metcalf and H. P. Eddy: "American Sewerage Pra.ctice," McGraw-Hm Book Company, 1M., New York, 3d ed., ,1935, vo!. 1. Harold E. Babbitt: "Sewerage and Sewage Treatment," John Wiley &: Sons, Inc., New York, 7th ed., 1952, pp. 60-:.66. H. M. Gibb: Curves for solving the hydrostatic oatenary, Engineering News, vol. 73, no .. 14, pp. 668-670, Apr, 8, 1915.

2.

3..

4.

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

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: "Hydra.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 &; 80ns, 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 WilliamKing; i'Handbook of Hydraulics," 4th ed., l'evised by Ernest F. Brater, McGraw-Hill Book Company, Inc., New York, 1954, p. '7-12. 19. Morrough P. O'Brien and Joe W. Johnson: Velocity-head correction for hydrau1ia flow, Engineering News-Record, vol. 113, 0.0.7, pp. 214-216, Aug. 16, 1934. . 20. Th. P..ehbQck': Die Bestimmung der I,age der Energielinie bei ftiessenden Gewfulsern mit HilIe des GeschwindigkeitshOhen-Ausgleichwertes (The determina.tion of the position of the energy line in flowing water with the o.id of velocity-head a.djustment), Der Bau.ingenieuT, Berlin, vol.. 3, no. 15, pp. 453-455, Aug. 15, 11122. 21. Boris A. Bak&meteff: CorioIis and the energy principle in hydraulics, in "Theodore von !Urman Anniversary Volume," California. Institute of Teohnology, Pasadena, 1941, pp. 59-65. 22. W. S. Eisenlohr: Coefficient's for velocity distribution in open-Channel flow, Tra.nsac:I.ior/.$, American. Socie4/ of Civil Enginee7's, voL 11:0, pp. 633-644, 1945. Discussions, pp. 645-668. 23. J. B. Bela.nger: "Essai sur la solution numeriqne de ql,lelques problemes relatifs au mou.-ement permanent des eaux courantes" ("Essa.y on tIle Numerica.l Solution of Some Problems Relative to Steady Flow of Wa.ter"), Carilian-Goeury, Paris, 1828, pp. 10-24.

\

,

38

BASIC PRINCIPI..ES

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

CHAPTER

3

ENERGY AND MOMENTUM PRINCIPLES

1

LeI!.'foI.l-

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

eN

.2.

.

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

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

,Yl

.

(,

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

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

.

c. "e J' .,t:;

'I

1d

~'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 'n'YII!'. .s~t:rt, Vr';,t.. cd cbf #.. is /l. J'Vi4 v..u ,pver.( e @'we - According to Newton's second law of motion, the change of momentum per unit of time in the body of water in a flowing channel is equal to the . resultant of all the external forces that are acting on the body,Applying this principletO channel of large slope (Fig. 3-7), the fplfo~~ing expl:ession. for the momentum change per unit tirnein;the body of water enclosed betv.:een sections 1 and 2 may be written:

cfs,

a

-

J(.. 1 .;.. "'>l~)(qt

'~l 0(

.

(3-14)

cit!C(

)>1'

_, Yn

,I

II !

! ,

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

2;

1

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

,

50

BASIC PRINCIPLES

ENERGY AND )';IOMENTUJI{ PRINCIPLES

51

r-

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

rectangular channel of small slope and wi~lth b (Fig. 3-7), . and AssumePI ;'~Wb1l1:l. P z = 7.wb yz2 FI = wh/by

L

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

I.\

I

Q

~ ~i(Vl

+ V~)bii

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

and

sin

{j

=

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

(3-16)1

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

1.

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

I Co t1rI

IC!cr-'T(~ 0

'tl.

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' =

i

1-:: fA AZ)o

hdA= 1

+ ~ fA AzJ~

cdA

(3-15)

where z is the depth of the centroid of the wa.ter area A below the free surface, h is the pressure head on the elementary area dA, andc is the pressure-head correction [Eq. (2-9)1, It can easily be seen th'1t pI is !.Treater than 1.0 for ooncave flow, less than 1.0 for and equai =~~ to 1.0 f ;;:..fIl

.s0

" ...

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

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 t "d' indicolas the lull tlow condition

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

II0.6 0.70,8

0.9

Votues of 0/0 0 , vivo AR-8. Curves of critica.! slope vs. discharge.

curve MLN indicates a minimum valueof S, O.004Il.t L, which IS the required Iimii slope. ' , . : ' . Assumil1g that the maximum expected depth of flow in the ch$.IlneLis 5 ft, a dlScha.rge cUl:veOM (Fig. can be cOIl1itrueted according to Eq. (6-9). It becomes evident tha.t within the sha.ded area. betweEn the curves OM and MLN, a.ll expected flows wHi b~ subcritical. On the right side of the Cllrves, the flows will be, supercritical. Since t1:,e point L is below the cUr'le 'OM, the limit slope is possible in the exj:lected flLnge of flow. . . . Similll.dy, the maximum expected depth of flow is !\.Ssumed to be 1.5 ft, and the curves ar~shown in Fig. 6-8b. In this case, the point L is above the Cl.lCVe01l1; there-. fore, the limit slope cannot be expected to occur in the realm under cQnsideratioll.

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

t>-2.

SOME TYPES OF PROBLEMS OF UNn'ORM-FLOW COMPUTATiON

Type of problem

Discha.rge Q

VeIoeity V

IDe th p11

Roughness

SlopeS

n

Geometric elements

. Exa.mple

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

AB

?

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

c

D EF

v v v' v

,?

-

v v v v v1

v v v?

.-

.-

v v

v v v v v;? ,

I

v v v 'I. v?

145

UNIFORM FLOW

COMPUTATION OF UNIFORM

:now

147

l

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

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

i

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, orS

=

F'

(6-38)

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

vB.

(6-3)

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

s=!!:!. Lwhere

(6-39)(6-40)

jIi

i

and kis a factor.' When the reach is (;ontracting (Vu < V d ), k = 1.0. When the reach is expanding (V" > V.), k = 0.5. The 50% decreru;e in the value of Ie for an expanding reach is customarily assumed for the recovery of the velocity head due to the expansion of the flow. The corresponding discharge is then computed byEq, (6-3) u.'ling the revisedflow, 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.

~

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 !

I

J 1,

,

.~

,COMPU'l'ATlON OF UNIFORlI'I FLOW

148

UNIFORM FLOW

149

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 Ad= =

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

.J,~,~Ofti~T!~ ~ . . . . . 1,;:\.~j ~V~~~,~" V?",,-~'r-

" Ii!

w(Y"l~'Y)S ( _ . t~F"'~~101l

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

.' 'flop'li' 't~o1

.

~,'

'\\;.:.;

~

....

' . . . ,

\

.~~ ..,

.(

.

11,070 10,990

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

au = 1:13"O'd

i n7 .... , ~,.' .

_,(.

=

1.177

,/

'1:':~7~~1:;;t

) ' ..,~,., AifC

J,;.iii:.

fr.

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: .. 2(/ ad

V,,._ 1'13 A (97,000/11,070)" =."" 2(/

1.3541,42,1 FHl. fr9. Uniform laminar open-channel flow.

V d ' _ 1177 (97,000/10,990): 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

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

r

at a dist.ance y from the

=

I>

du dy

(6-41)

6-3.

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

6-6

Appr.oximatlon 1st2d 3d

IAs~umedQ

Flo. ~

4th 5th

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

0.500 . . . . . . , .. \.50 .0010000.0316 97,000 0.500 1.354 1.424 0.430 1 0.000860 0.0293 90,000 10.500 1.165 1.~25I0.44010.0d0880 0.0297 .91,200 10.500 1.H.l5 1.258 0,437i 0.000874 0.0296\ 91,000 1 0 . 500 1.190 11.253 1.437\ 0.000874) 0.0296 91,000

I

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

I

2g

a~ ~l2U

h,

I

S

v'S 1ComputedQj ,. ,

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) dyJ'

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

6-10. Uniform Surface Flow.

When water flows across a broad sur-

g8 (YYm _)I .

face, so-called S1J.1"!ace flow is produced. The depth of the flow may be sothin 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

t)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-

150

UNIFORM FLOW

COMPUTATION OF UNIFORM FLOW

151

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

=

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

PROBLEMS

V = -1 Y

10 ' vdy11

(6-43)

'y

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

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

0

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 JeffreYmpute 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)

REFERENCES1. 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 elements of opellcha.nnel cro~s-sections, Civil Engineering, vol. 18, no, 10, p. 47, October, 1948. . 4. N. N. PavlovskiI: "Gidra.vlicheskii Spravochnik" ("Handbook of Hydmulics"), Onti, Leningra.d and Moscow, 1937, p. 515. 5. A. N. Rakhmanoff: 0 post.roenii krivykh svobodnolpoverlthnosti V. prir-maticheskikh i tsilindriche.~kikh ruslakh pri usta.novivshemsia dyil!henii (On the construction of curves 01 free surfaces in prismatic nnd cylindrical channels with established flow), hvestiia V sesoiuznoyo Naudmo-Is:sledouaf.eI'.kO'lo bl.lititula Gidrotekhllim (1'ranS(lctions, All-Union Scienli.fi.c ResC(Lrcfl InstituteDf Hydraulic Engineering), Leningrad, vol. 3, pp. 75-114, 1ll31. 6. Robert E. Horton; Separate roughness coefficients for channel bottom and sides, Jj7'dl""'.""di News-ltccord, vol. 111, no. 22, pp. 052-653, Nov. 30, 1933. 7, H. A. Einstein; Del' hydraulische odsr ProJil-Radius (The hydraulic (11' cross section radius), Sch",ei$~rische Ba.uzeilv,1!g, ZUrich, vo!' 103, no. 8, pp. 89-91, Feb. 24, 1934. 8. Ahmed M, Yassin: Mean roughness coefficient in open cho.nnels with different, roughness of bed and side walls, Eid(jelll!,~.rische technische H oCMchllle Ziirich, Mitieilungen ruts der Vcnu.chsa.nst(lltflIT Tfr(lsse!'oau una Erd:bau, No. 27, Verlag Leemann J ZUrich, 1954. 9. N.N . .Pavlovskii: K voprOStl 0 raschetnoI formule dlia ravnomernogo dvizheniia y yociotoka.hk s llcodnorodllymi stenkami (On a design formula. for uniform move men~ in channels with nonhomogeneous walls), Izul1siia VsesoiUZllogo Nau.cir.lIoI 8sledovatel' skolJo Instiluta. Gidratekhnikt (T"(LMrtClions, ,111- Unio;~ Scienlifu; Resear~h Inslill1le of Hyd1"altiic Erl9ineeril~g), Leningrad, vol. 3, pp. 157-164, 1931. 10. L. MUhlhofer: Rauhigiteitsuntersuclmngen in einem Stollen mit betonierter Soble und unveddeidete~ Wand en (R(Jughne5S investigations in a shaft with concrete bOUom and unlined wo.lis), Wass8rlcra!t u.na Wasserwir/,sdtafl, Munich, vol. 28, no. 8, pp. 85-83, 1933. . ll. H. A. Einstein and R. B. Banks: Fluid resistance of composite roughness, Trans- . actions, Am.erican Geophysical Union, vol. 31, no. 4, pp. 603-6jO, August, 1950, 12. G. l{. Lott.er: SOQbrazheniia k gidravlicheokomu raschetu cusel s l'!>tlichnoI sherokhovatosliiu stenoI!: (Collsidera.tionson hydraulic design of channels wi~h different roughness of \ir8.1I8), /zu.estiia. V~e~oiu2nogo Nauchno-!ssledvvateJ,'skof/Q Instiluta. Gid1'otekhniki (Transa.ctiqna, .till-Union Sde1\tijic Reseprch In.stilutc of Hyd~at;lic En(Jineering), Leningrad, voL 9, pp. 238-241, 1933. . 13. G. It Lotter: Vliianie uslovii ledoobrazovaniia. i tolshchinY l'da naraschct derivatsionnykh ka.nruov (Influence of condi~ions of ice formation and thickness on the design of derivation ca-nlLls), IZTleiltiia. Vsesoiu.znogo Nav.chno-I:ssled()va.t~l'slco(jo. Ins/itu/a Gidrolekhltiki (Tra.nsactions, All-Union Scimtiji.c ReseO;rch 17Ultitute of H1!d~G7j.lic Engineering), Leningrad, vol. 7, pp. 5&-80, 1932. 14. G. Ie LoLter; Metod akademilca N. N. Pavlovskogo dUo. ojJcedeleniia koeflitsienta

I

i

I

a.nd that

thi~

slope for a wide cluumel isScrt.

= 14.5n>

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 critical-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 = 26.5", t,lle tractive-fol'ce ratio by Eq. (7-11) is K = 0.587. For a size of 1.25 in., the permissible tractive force on a level bottom is TL = 0.1 X 1.25 = 0.5 lb/ft' (sam.e from Fig. 7-10), and the permissible tractive force on the sides iST. ~- 0.587 X 0.5 = 0.294 lb 1ft'. For a state of impending motion of the particles. on side slopes, 0.078y = 0.294, or y = 3.77 ft. Accordingly; the bottom width is b = 3.77 X 5 = 18.85 ft. For this trapezoidal section, A = 99.5 ft' and R = 2.79 ft. With n = 0.025 aud S = 0.00113, the discharge by the Manning formula is 470 crs. Further computation will show that, for z = 2 and bIy = 4.1, th" ~ection dimensions are y = 3.82 ft and b "" 15.66 ft and that the discharge is 41"4 cfs, which is close to thO} design discharge .. Alternative section dimensions may be obtained by as~uming other '(alues oI z or side slopes. . b. Checking the Proportjalled Dimensions. With z = 2 and bly = 4.1, the maximum unit tra,ctive force on the channel bottom (Fig. 7-7) is 0.97wyS = 0.97 X 62.4 X 3.82 X 0.0016 = 0.370 [b/ft', less than 0.5 Ib/ft', which is the pel'mu;sible tractive . force on the level boLtom. Example 7-4.

!

a discharge of 400 cfs.

Voids ratio

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

are tentatively recommended (1) for canals with high content of fine sediment in the water, (2) for canals with 10',11 content affine sediment ill the water, 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 maihematicalderivation of this section by the Bureau, the follow'ing assumptions are made: .. . L The soil particle is held against thechanllel bed by the component of r,he submerged weight of the particle acting normal to the bed. 2. At ,and above the water, surface the side slope is. at the of repose of the material under the action of gravity. 3. At the center of the channel the side slope is zero and the, force alone is sufficient to hold the .particles at the point of incipient instability. 4. At points between the center and edge of the channel the particles are kept in a state ,of incipient motion by the resultant of the gravity component of the particle's submerged weight acting on the side slope and the tractive force of the flowing water. 5. Thc tractive force act.ing on an area of the channel bed is equal to the weight component of the water d,irectly above the area acting in the direction of fiow. This weight component is equal to the weight times the longitudinal slope of the channel. If assumption 15 is to hold there. can be no lateral transfer of tra.ctive force between adjacent currents moving at different velocities in the section-a situation, however, that never actually occurs. Fortunately, the mathematical analysis made by the Bureau I has shoWn that the actualI

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

ttal

V

where rP is the slope angle of the \ tangent at AB. . 29.5'-------1 The other assumpti(;ms stated above have been used previously to develop the equation for the trac. tive-forc~.ratio K (Art. 7-12). The unit tractive force all the level bottom at the channel center is 1'L wYuS, where yo is the depth of flow at the center. The corresponding unit tractive force on the sloping area ~B is,therefore, equal to wyoSIL , . III o~der to achieve impending Andysis desigll ?f sto.ble mo~ion over the entire periphery of. FIG. 7-12. secti0!1' and ~heoretLcl'l.l sechydraulic (a) the channel bed, the two forces tion {or given soil properties. and eh!l.l1ll:l mentioned in the above paragraphs . slope, providing Q - 220 cis; (b) mod~fred section for Q" 400 cfs; (c) modIshould be equal; that is,. fied section for Q' 100 efs. wyS cos 30 11-;-24 8-10 2-6 30 11-24 6-10 2-8 ,nsions, but, owing to the difference in shape and orientation, they may produce identical roughness effect and, thlls, their roughncsscs will be clesignated by the same roughness height. 2 The position from which the rOllghness hcight should he measured is IL disputable mli\tter. It is a.''I'h~I,

UNIFORM FLOW

average roughness height for a given surface can be determined by expenment. Table g1 gives averaged from manv The con~ept, of l"~ughness in conduits was further advanced by' Morris [5]. Morns assumed that the flow over. a rough surface is due largely tl) roughness element. The f o.

I;

THEORETICAL CONCEPTS RELATED TO UNIFORM. FLOW

i97

II

.i

of abnormal turbulence. 1 In 'such a flow,therefore, the ratio yl).will be an importn.nt correlating parameter.. . ~si-~m.ooth ~o~..2.ccurs when_ the roughness elements. are.~.Q.~... together tb.!--'l!i..~' ski.ms the _cre5t...2l..the_.e!e~nt, . The grooves between the elements will be filled with dead water contall1mg

! i

!.~etermiM,

andTA1l!.E

8-1.

ApPROXIMATE VALUES

OF

ROUG.lNESS HE1(lHT

It

. Mc.lerial

k,

it

B:ra.ss, copper, lead, glass ... ". , , , ,. Wrought iron, steel. ...... ' ....... Asphalted east iron, . . . . . . .. . Galvanized iron ... ,., ........ '. Cast iron ............ , .... " ..... Wood stave ....... , ............. Cement. . . . . . . . . . . . . . . . . . . . . . . . .. Concrete., ........... , ...... , .... Dra.in tile .......... ,.............. Riveted steel. ..... , .............. Natural river bed. . . . . . . . . .. . . . . ..

0.0001:"'0,0030 0.0002-0.0080 0.0004-0.0070 0.0005-0.0150 0.0008-0.0180 0.0006-0.0030 0.0013-0.0040 0.OU15-Q.0l00 0.0020-0.0100 0.0030-0.0300 0.1000-3.0000

tudinal ~f the r~~ss el~.mentsis the roughness dimension of paramount Importance in rough-conduit flow. Under this concept, flow ~~vel' IOtlgh surfa.ces can be classified into three basic types (Fig. : I 'L~olC!!:e.?::..~:01J.2hJ1.es.s flow, wake-interference flow, and quasi-smooth (01'~

(0)

---------

I

slcimmt'ng) flow.~ solated-roughne~~Jow.

'

.-'--;:;,

. , 1".

prevails when the roughness elements are so far apart that the wake and VOl'tex a.t each element are completely developed and dissipated before the next element is reached. The apparent roughness would thereforere8ult from the form on the roughness elements, represented primarily by the height of projection k of the element, in addition to the friction drag 011 the waH surface between elements, ,which depends on the spacing of the elements. In such a flow, the. ratIO kl}.. may be taken as a significant correlating parameter influencmg the apparent friction factor in the flow. . Walee-interference flow :esults when the roughness elements are placed ,s~ clo~together tha.t the ~nd .YQrJj)x at each element wiUjnterfere WIth those developed !l:L~tJ resulting in intense and c.!?~pl~x."yor~i~ and tu~bulen~g. hi such ~ flow, the height of the e~em~nt is. rehlthr ely unimportant, but the spacing is obviously of major ImpOltance. Also, the average depth y of flow above the crests of the elements will control in part. the vertical ex tent of the surface region

( c1

FiG. 8-4. Sket.ches showing concept of three basic typ;;s of How: (a) iso~ lll.ted-roughness. !low; (b) walJ:e-interf~rence flow; (c) quasi-slnooth tlcw.

_~ stable eddies, creating a:?~1}. Large projections Illoe ~tisent from this pseudo wall, and the surfuce !'cts hydrauli~l\.ll-y smooth.In such a flow, the ratio k/A (or .ill., where j is the groove width) will again be a significant parameter. Quasi-smo.Q.thJiy.:.:Y.....ba~ factor than flow ovel' a true smootll surfaft because the eddies in the groove~me a certain. am~f~gy. The nbove concept ap'pears to be substantiated adequately by experi1 Morris llsed the pipe radius instea.d of the depth in defining the parameter because he was ',;oncerned primarily with pipes instead of channels;

198

UNIFORM FLOW

THEORETICAL CONCEPTS RELATED TO UNIFORM

now

.j

mental 'data from many different sources. The concept can also be extended to surfacs of variable roughness by u5ing average values of the roughness dimensions that vary or by combining the friction factors for each flow type to give all over-all apparent friction factor of the flow. 83. Computation of Boundary Layer. For the development of a turbulent bound~ry layer in wide channels, an-;:pprQximate but practical meth~2!L h~_~.E!:Q12Q~_Qy_Jl~lJ&t.f.1, This method waS' developed primarily for flo\v i.,1l; ~.han~els.oI large ~l'?.P.e, but it has been found applicable also to channels of small slope; provided the flow is . -.::- - ,-...... -- .--:' ...."

. computing the varied-flow surface is used. In the case of tIle boundarylayer development of a uniform flow, the application of Bau61:'s method is just the same except that the water surface requires 110 computation, for it is simply parallel to the ehanneLbottom.Example 8-1. Aooncret.e overflow spillway of indefinite length has a. surface slope angle 11 = 538' (Fig. 8-5) and a roughness snch that k 0.005 ft. When the discharge is 3130 cfs per foot of spillway width, compute the length for boundary-layer dev!:')opment, the profile of the boundary layer, and the water surface, Solution.. The computation is shown in Table 8-2; the headings are explained as follows: Col. 1. Arbitrarily assigned length of x in ft, !DeMured from 0 Col. 2.. Values of xlk, where k 0.005 ft Col. 3. V",lues of lJ/:z, computed by Eq. (8-4) Col. 4. Values of ~ in HI obtained by multiplying x by 8/x CoL 5. Velocity head vo'/2g in ft, equal to x sine = 0.80x Col. G. Velocity in fps corresponding to the velocity head in the preceding column CoL 7. Potentia.l thickness of Row in it, equal to the dischal'f!;e 1160 cis/it divided by the yelocity fl. . CoL 8. Actual thickness of Lhe flow in ft, equal to the potent.ia.l thickness plus the displacemce is 0,00215 ft. Plot the profiie of the turbulent boundary layer, 1).0(\ est.im!l.te the length -of channeL required for a fun development of the boundary layer \vhich begins tobe turbulent u.t the entrance. 8-S. Determine whether the c.hannel'described in the preceding problem is hydrauli' cally smooth or rough. 8-4, A trapezoidal channel (Fig. 2-2) with b ,.. 20 H, z 2, and S = 0.001 ca.rries a uniform Row at a depth of 6 ft. Compute the unit tractive force :Lnd the fric~icm velocit,y developed in the channel, Wha.t are maximum friction velocities on the sides and bottom of the dUlIInel? 8-li. Snow t.1U1.t(8-36)

I

i,Il

V

=

xyV V",- V

where x = exponent of the hydraulic radius in the general uniform-flow formula Eq. (5-1). Thus, x = 2 for laminn.r flow (Art. 0-10), x OJ) 'for turbulent flow if the Chezy formuln. is and x = H for turbulent flow if the Manning formula is used. V = mell.n velocity , V", = absolute velocity of disturbance waves in channel ~ = a shape factor of channel section, defined bydP 'Y=l-RdA.(8-34)

where R is the hydraulic radius) P is the wetted perimeter, and A is the water area: Thus,"( -= 1 for very wide channels, and 'Y 0 for very narrow channels. It will be shown that V", - V is equal to the celerity c of the waves (Art. 18-6) or to the critical velocity V.. Since ,the Froude number F =- VIV., Eq. (8-33) may be reduced toV

where f is Darcy's friction factor and 11 is Mll.rming's roughness coefficient. 8-6: Using Eq, (8-13) for expressing the theoretical velocity distributi.on in wide rough cha.nnels, (a) show that the I:iverag8 of the velocity of 0.2 depth alid the velocity at 0.8 depth gives the velocity at 0.6 depth, and (b) compute the position of the mean velocity below the free surface) and compare the result with that determined by the rule of the U,S. Geological Survey (Art.. 2-6)., 8-1. A wide channel carries a uniform .flow a~ a depth of 5 in. on a slope of O.OOOL The channel surface is rough, having a value ork = 0.25 in. Compute and construct a curve showing ~he theoretical velocity distribution in the channel section. What are the mean velodty and its position? . 8-8. Determine the values of k in Probs. 5-9 and 5-10. 8-9. Eq, (8-13), show that the theoretico.l velocitJ:"-distribution coefficients in wide channels can be expressed as;

212

UNIFORM FLOW THEORETICAL CONCEPTS RELATED TO 'UNIFORM j,'LOW,

A wide st.ream carries approximately uniform flow- at a depth of 12 ft Th ve OCI les at 0.2 and 0.8 depths are found to be 1.85 o'nd 1 32 f ...e mate (a) the roughness coefficient (b) th I . . ps, respectIvely. Estl' ' n, e mean ve oClty (e) the slope f tJ h I an d ~d) tn8 dis,:harge PCI' unit width of the channel. ' , o l e C anne . 8-LS In a wIde' t ' . . . s ream IlaVing moving sediment bed the f I' " 1 able: S = 0,003 k, = 050' d 1. , 0 . 0 " mg (ata are aVO,IJJ,"' mm, an ,,,~ = 0040 mm TI e I d r d' al'ea al'e found to be It = 0.85! _ 1 8, d A _ _. I ,Iy lau lC ra IllS and li'ater , J ' . an - 2/3y - 797 where R d ' ano, A IS in [('.' Construct tl. e disc lIar t" ' an yare m ft " . , ge-ra mg curve of the strean' , b-14. Show that the Vedernilcov' number in wide channels is V ~ 2F '. Row, V = O,67F fo), turbulent flol'v: 'f th "T " f ' for lammal' 1 e ,',-annmg orm'll- IS us d dV 0 turbulent flolV if the Che~y formula is used. ,~e ,an = ,SF for ' 8-11i. Using t.he Vedernikov' criterion expla'n tl ' . I"" . . fYI ,~ti;_n7

CHAPTER

.)J

!n~ for '/, Jir""tJy nt~;tv' ai"r- c;(,,"" nel. . 1 .

9

THEORY AND ANALYSIS

, l/}~ctr~ a:lt/ -~~ ~~"7~7~~~1/2-prn#.J 11-.I~ [af,;d~..td on -Ilu hMCd'9- tJ7Ur/&-'n1 t!c# t.".-.ru;dfL-!

9-1. Basic Assumptions. The gradually varied flow to be discussed in Part III of this book is the steady flow whose depth varies gradually dong the length of the channel (Art. 1-2), This definition signifies two conditions: (1) that the flow is stea,dy; t.hat is, that the hydraulic characteristics of flow remain constant fqr the time interval under consideration; and (2) that the streamlines are practicaily parallel; that is, that hydrostatic distribution of pressLlre prevails ovel' the channel section, The development of the theory of gradually varied flow dates back to the eighteenth .century. Mn.ny early hydraulicians L have contributed to this development. The theories thus developedprac.tically all hinge on the following basic assumption: -----p A. The head loss at a section-is the same as for a Hnijo'(m flow having the velocity and hydra111'icraciius of the section. . According to this assumption, the u:r.iform-flow fonmila may be used to evaZ1/ate the enngy slope of a gracl1wlly va-riecl flow at a giveTI. channel sect'ion, and the corresponding coefficient of roughness developed primarily f01' wd.form flow is applicable to the va'ned flow. This as~umption has never been precisely confirmed by either experiment 2 or theory, bnt errors arising . from it are believed to be small compared with those ordinarily incurred in the use of a u;1iform-flow formula and in the selection of the Toughness coefficient. Over years of use this assumption has proved to be a reliable basis for design. The assumption is undoubtedly more correct for varied flow where the velocity increases than where _the velocity decreases, . because in a flow of increasing velocity the head loss is caused almost. entirely by frictional eff~g~, whereas in a flo,"v of decreasmg -velocity-~here may.E(;l!~t.];~:s.c::~l~_e~_dy_.lo..~~I Bela.nger ll] is believed to be the outstanding contriblltor. Also El.mong early contributors are Bernoulli, Bresse, Poncelet, Saint-Yenant, BOllssiuesq, El.lld others [2] to [5]. _ . 2 Using the experimentlll data from the .Sunderland Technical College lind ICing's College in England and from the University of lllinois, Bettes [G] has dClL'ived an j-R relationship (Art. 1-3) for gradually variedftow in smooth open cblHlels, which WIIS found to agree very closely with the relationship for uniform ftow obtained by Allen [7]. Also, the computlltion of backwater curves based on this assumption ho.s been verified satisfactorily by many experiments. These experimental verifications, though not VEry rigorous, indicate the validity of the assumption for practical purposes. 217 .

~l'il1i;;~~ &~"Rtff

lV1301HOd qn'gt.:;~

218

GRADUALLY VARIED FLOW

'l'HEORY AND ANALYSlS '

In addition to the above basiQ assumption, the following aSEum:ptions will also be. used where further simplification is necessary in subsequent discussions: . B. The slope of the channel is small; so that: 1. The depth of flow is the same whether the vertical or normal (to . the channel bottom) direction is used. 2. The pressure-correction fact,or cos IJ [applied to the depth of. the flow section" Eq. (2-12)] is equal to unity . . 3. No air entrainment occurs. 11'1 case of notable air entrainment, the computation may be carried out assuming no entrainment and then corrected approximately, at the end, using Eq. (2-15). C. The channel is prismatic; that is, the channel has constant alignment and shape. D. The velocity distribution in the channel section is fixed. Thus, the velocity.-distribution coofficients are constant,. E. The conveyance K (Art. 6-3) and section factor Z (Art. 4-3) are expOl18ntial fUllctions of the depth of flow. . F. The roughness coefficient is independent of the depth of flow and constant throughout the channel reach under consideration. 9-2. Dynamic Equation of Gradually Varied Flow. Consider the profile of gradually varied flow in the elementary length dx of an open channel (fig. 9-1). The total head above the datum at the upstream section 1 is

IIi

:)

!

. H

= z + d cos e.+ a 2g

V2

(3-2)

T ,,4FIG. 9-1. Derha.tion of the gradually-varied-fiaw equation.. " t

where Ii is the total hea,d in ft i z is the vertical distance of the channel bottom above the datum in ft; d is the depth of flow section in ft; (J is the bottom-slope angle; ex is the energy coefficient; and V is the mean .. velocity of flow through the section in fps, It is aSllUmed that 8 and ex are\ constant throughout the channel reach under consideration. Taking the bottom of the channel as th!3 x axis and differentiating Eq. (3-2) with respect to the length :1.: of the w[.;ter-sUl'face profile, which is measured along the x axis, the following equation is obtained: . dH = -dz + cos edd (9-1) Ct -d -dx cl.t _ ..:.::.Jb:. dx 2g

and the slope of th~ ch~,l1nel bottom So = sin 0 = -dz/dx. t,hese slopes in Eq. (9-1) and solving for ddjdx,dd

Substituting

dx = cos ()

+ Ctd(V~j2g)jdd

So - Sf

(9-2)

'((

+

(V2)

It shouldJ?e n(jte3'"

. :";

".,

.

i

i,I

.:; i .220GRADUALLY VARIED. FLOW.

~ .

\

,

THEORY AND ANALYSIS

221

the cbannel bottom when. ddjdx. = 0, rising when dd/ dx is positive, and . loweling when del/dx is negative. .. In the abovH equation, the slope angle 0 has been assumed constant or independent of x. Otherwise, a term -d sin (J(d8/dd), where (J is a function of X, wonh:l have been added to the denominator. For small .OJ cos 8"" 1, d "" y, and cld/dx ~ dy/dx. Thus Eq: (9-2) becomes

. velocity and hydra1,llic radius of the section. is used, the energy slope is

When the Mann~ng formula(9-8)

When the Chezy formula is tiseo,Sf=

ViGZR

(9-9)

.;:rJ,x - 1 + ad(Tn/2Q)/dy

dy _

So - Sf

(9-3)

In: a

In most problems, the channel slope is smaH; accordingly, Eq. (9-3) will be used in subsequent discussions. The term ex d(P/2g}/dy ill the varied-fiow equation represents the change in velocity head. The coefficient a has been [;,ssumed to be constant from section to section of the channel reach under consicielp.tioll. Ot,hel'wise, the change in velopity head would have been expressed as (I(Cl V2/2g)/dy, where IX is a function of x. Since V =Q/ il, Q is constant, and dA/dy '1', the velo~ity-hf.)ad term may be developed as follows:1

form, expressed in terms of the conveyance [(, the energy (6-4), may be written(9-l0)

Suppose that a uniform flow of a discharge equal to Q occurs in the section. ._'!h.~.}~!!~~~~()jJe ,vo.u1d ~~t)9.ual to the bot_tom slope, an~q .. (9-10) may be written (9-11) where K" is the conveYf-llCe for uniform flO'w at a depth y". This Kn should be distinguishedfro}n K in Eq. (9-10). The notation K represents simply the numerical v!),lue of the conveyance at a depth yof the gradually varied flow. The value [(" is t~e conveyance computed for Q at the depth yAas if the flow were r.onsideted uniform. Dividing (0-10) by Eq. (9~1l),SfSo =Kn~

d (V~)a dy

\Since

21}

=

aQ2 dfi- 22g--dy= -

aQ2 dAgA3 dy'= -

aQ2T gfP

(9-4)

Z = VJP/T,

the above may be written

d (V2)a

aQl= -

ely \ 2y

gZ2

(9-5)

Suppose that a critical flow of discharge equal to Q occurs at the section; Eq. (4.4:) gives(9-6)

](2

(9-12)

Substituting Eqs. (9-7) and (9-12) in Eq. (9-3),., ~Y = So 1 - (Kn/IC)2 dx . 1 - (Z,/Z)2(9-13)

where Z, is the section factor for cl',i~ical-fiow computation for discharge Q at depth Ye. The symbol Z, used herein should be carefully distinguished from the Z in Eq. (9-5). The symbol Z simply represents the numerical value of VA3/T, which is computed for the discharge Q at a depth equ~l to y of the gradually vUl'iedfiow. The value of Z. is the section factor.; which is computed for Q at the depth v, as if the flow were considered critical. Now, substituting Eq. (9-6)' for Q in Eq. (9-5) and simplifying,(9-7)

This is anotber form of the gradually-varied-flow equation. There are other popular forms of the gradually-varied-flow equation that can readily be derivea, such asdy dx

=

. 1 - (Kn/KF So 1 - r(K,,/J()~

(9-14)

where r So/ S,n, or the ratio of the channel slope to the critical slope at the normal depth of discharge Q (Art. 6-7) ;(9-15)

'\

The term Sf inEq. (9-3) represents the energy slope. According to..the :first assumption in Art. 9-1; this slope at a channel section of thegr!ldually ~al'ied flow is equal to the' energy slope of the uniform fiow that has t),e

222

GRADUA.I,LY VARIED FLOW

THEORY AND ANALYSIS

223

where Q is the given discharge of the gradually varied flow at the actual depth y; Qn is I;he nQrma.1discharge at ~. depth equal to y; and Q. is the cntic!1r discharg;;:t a depth equal to y; [\nd ';'

dy (IX

= ,1

So - Q~/C2A2R - aQ2/ gA2D

(9-16)

where D i~ the hydraulic depth, C is Chezy's resist~nce factor, and the rest of the notation is as defined in this article. For wide rectangular channels, 1. When the Ma11n.ing formula is used, (9-17) 2. When the Chezy formula is used,dy dx

Since the values of K and Z increase 01' decrease continuously with the depth y, the first case indicates y > y" and Y > Y.. As y > y" the flo'N must he subcritical. If Y > Yn > y" the sUDcritlcal flow must occur in a mild channl!l (i.e., a channel of subcritical slope). On the other hand, if y > y, > Yn, the subcritical flow must oceur in a steep channel (i.e., a channel of supercritical slope). Similarly, the second case indicates Y < Yn and y < yo. The corresponding flow must be supercritical; and it occurs i:n a mild channel if Yn > y, > !I and in a steep channel if y, > Yn > y. For a drawdowll curve, dy/d:!: is negative aild Eq. (9-13) gives two possible cases: 1. 1 - (K n /K)2 2. 1 .- (Kn/K)"

> 0 and 1

(Zc/Z)2(Z./Z)"

0

.!

Yn. and, thus, that the flow is superCi:itical in a steep channeL 'Slffiiiarly, the second case indici,tes that ~ V > Yo, 9E.. that the flow is subcritical in a rild channel. vVhen the water surface is parallel to the bottom of the channel, dy/dx = 0, and Eq. (9-13) gives 1 - (1(',/ K)2 = 0, or Y == yn, which indicates a uniform now. Tbe flow is a unjform critical flow if y = Yn = Yo, 8, Ulliform subcritjcal flow if y = y ... > Yet and a uniform sllpercritieaIflow if Yo > Yn = y. 70-r purposes of discussion, cha.nnel slope may be classified as sv.stair.ing and nonsusta.ining. A sustaining slope is a chai1l1el slope that falls in the direction of flow. ,Hence, a sustaining slope is always positive and' may also be ,called a positive slope.' A sustaining or positive slope may be critical, mild (sllbcritical), or steep (supel'critical). A nonsustaining slope may be either horizontal or adverse. A horizontal slope is a zero slope. . An adverse slope is a negative slope that rises in the direction of flow. . In a channel of horizontal slope, Qt" So = ~ (9-11) gives Kn = 00 . or ~ince K ... ~ = Q, Eq. (9-13) fo~ hOrizo\ltal channels may be writtendy _ -(Q/K)~ dx - 1 - (Zo/Z)2

l

,

(9-19)

Considering Un1. Yn

=

00,

> 0 and 1 - (Z,/Z)2 > 0

2.1- (Kn/KP

< 0 and

1-

(Z./Z)~

Y>,'J> >y

Dra.wdown1:1

'I Subcritieali SupercrhJcal! SubcrlU",,1Suberitir.t'l

fill

!In.

'1>l)

ynlin

>>u>">

y.

I B ...kwnt;,r

!II/n

Bllckwn.terBa.ckwa~er

SupererltjoalSub.ducalUniform~

11>

y,

Critical8,-8,>0

C2C3

j]

:=

Parallel to1),:;

11-

'lJ~

channel bQttom

en tlcnl

II,

> >!I

y. > 11

Bll.ckwJlter

Super-critical

Steep

81 182

I133 .

'11>

1I.

Yn

Backwater~

Subcrit.icul

8.

> 8, > 0None

y,

>y.'lin> !III'

.-

Sup.relitlcal 8 -lpexcri,ti.en.l1

11

> >

I"

Adver

A2 A3

J.!!

1i>

(lIA)*

None8ubcriticnl

(lin)' >!I > Y,11,,~)

S, Se > S3). In the spiral type, the flow profile that pa.sseg through P and is asymptotic to Fl ;"" 0 indicates a discontinuous flow 1 changing from supercritical to subcritical in a channel with sligMly concave bed (8 1 > Se > 8 a). In the vortex type, the, flow profile that passes through the singular poiut is the point itself and has no hydmulic significance. A general solution for the transitional profile in all four typ~ h SL, the condition depends as follows on the relation of Bo to lh.

I-' ((oj

.. "--

_----=-..--T,'- __ _ "11 -.. - _ _ a

I

~

\ )

Let K" = Q/V8o,K = L49AR%/n, Z, Then, the above equation becomes2.22RY.iSo

=

Q/VU, and"

Z

=

A,fIT. '(9-28)

s

S

UncriticO/.S/o pe

//,

= n~gD

This is a theoretical condition for the establishment of the transitional depth.: It indicates that the transitional depth depends only 011 the channel geometry, roughness, and slope. This equation contains no discharge; therefore, t.he tranllitiOlial depth is independent of the actual discharge. It is logical to say that there is a certain discharge Qt that occurs at the transitional depth Yl./ This discharge may be called the transitional discharge. According to the definition of the tranl?itional depth, the transitional discharge shoilld be a normal discharge and also a critical discharge. Referring to the critical-slope curve discussed in Example 5-5 (Figs. 5-8 and 9-12), the transitional discharge can be represented by a point on the curve. It is evideilt from the curve that, for a given slope B~, whit::h is greater than the limit slope BLI there are two possible critical discharges, say, Q., and Qb, both of which are transitional discharges. The actUal discharge is designated by Q and the corresponding critical slope by Be. 'I

, (d)

fi---'A. When the slope is close to the limit slope.

PAl

FIG. 9-12. Flow pro.files expla.ined by a critical-slope curve.

In this case, the condition

Here .the 'upstream normal flow changes to tbe downstream normal flow at an abrupt tr'ansition formed by a hydraul~{l jump. , See [f2], [II)], [28] to [30]. and [32] ., I

will depend further on the magnitude of the actual discharge Q with1:

respect to the smaller and larger transitional discharges Qo and Qb, respectively. '

.1,

I

0,,)

244

GRADUALLY VARIED FLOW

.1 , '",THEORY AND ANALYSIS9-7. f?ketch the possible flow profiles in the channels sbown in Fig. 9-13.

\"."j

245

I"

i

t'i. I

J

(Fig. 9-12b), tIlen So < Se and Ye < y" < Ya < Vb. Since So < Sc, the flow is sub critical, and the profile should be of the 1VI1 typ~ . However, the profile will contain two points Ta. and Tb at which the slope is horizont.aL Bet-';veen these two points a p(')int of inflection apparently exists. The depths at the two poihts are transitional depths Ya. and y~. If Qa < Q < Qb (Fig. 9-12c), then So > S,and y. < Yn -< y. < Yb. Since So > Sc, the flow profiles are of the S type. However, there will be a point Tb where the slope is horizontal on the Sl profile and a point T a where the slope is horizontal on the 83 profile. If Q. < Qb '< Q (Fig. 9-12d), then So < S, and Y. Qb, the flow will be sub critical and the profile will be of the M 1 type. If Q < Qb, the flow will be supercritical and the profile .will be of the S1 type. C. When Ihe. slope is very large. In this case, the large transitional discharge Qb is considered to exceed the maximum expected discharge (see Fig. 6-8). Thus, the flow is supercritical and the profile is of the S1 type. The highest point of theS1 profile is very close to the downstream end. The above discussion was developed for the case in which the point L of the limit slope is below the curve of maximum expected discharge (Fig. 6-8a) and in which the channel sections !1re rectangulal' or trapezoidal or similar to such forms. If the point L is above tbe curve of maximum expected discharge (Fig. 6-8b), the larger l;ransitional depth of the flow will be greater than the maximum expected depth, or Yb > Ym, and the larger transi~ional discharge will be greater than the maximum expeeted discharge, or Qb > Qm. The foregoing discussion, however, remains valid as long as the actual discharge Q does not exceed Qm. If Q exceeds Qm, the discussion has no practical meaning. Similariy, the flow profiles remain t,he same, but the useful part of the profiles will be where the depths al'e less than y",.If QPROBLEMS9-1. Show that the wnter-surface slope S" of a gradually varied flow is equaJ to the slim of the energy slope S and the slope d~e to velocity change d(", V'/2g)/dx. 9 c 2. Show that the gradually-varied-fiow equation is'reduced to a uniform-flow formula if du/dx ... 0, 9-3. Prove Eq.' (914). 9-4. Prove Eq.: (9-15), 9-5. Prove Eq. (9-16). 9-6. Prove Eqs. (9-17) and (9-18).

< Q.

LEGEND:

-------~-

Criticol-c\eplh line

- - - -Noqnal-deplh line

FIG. 9-13. Channels for Prob. 9-7.EI.1274 EIJ272 :s2 EI.f270

The vertical scale is exaggerated.

..

--x-jEU266

[=I

v

.-+----j.500'----~.tl proliles .. d. Construct the renl and some other possible flow pIoliles. 9-11. Show' that the gradually-varied_flow cq\!ation for flow in a rechar.gular channel of va.riabli!. width b may be (lli:pressed as=

S. - Sf

1 - ",Q'b/gAJ

+ (",Q'ylgA~) (dbldz)

...._........

(9-29)

All notation has been previously defined.

REFERENCES1. J. B. Belanger: sur Ill. solutionnumiirique de quelques problemes relatifs au mouvement. permanent des eallli: courantes" ("Essay on the Numerical Solution of Some Problems Reletiv~ to the Steady Flow of Water"), Cn.rilian-Goe\lry, Pa.ris, 1828. . 2. J. A. Ch. Bresse: "'Cours de mecJl.nique appliquee," 2e po.rtie, Hydraulique ("Course in Applied l\'Iechanics," pt. 2, Hydraulics), Ma.1let-Bachelier, Paris, 1860. 3. Boris A. Bikhmeteff: "Hydraulics 'of Open Channels," appendix I, Historicnl and bibliographical notes, McGraw-Hill Book Cr)mpany, Inc., 'New York, 1932, pp.299-301. 4~ Charles Jaeger: Steady flow ill open channels: The problem of Boussinesq, J Ot,rnal, lnetitution of Civil Engineers,. London, vol. 29-S0,.pp. 338-348, November, . 1947-0ctober, 1948. 5. Charles Jaeger: "Engineering Fluid M.echanics," translated from ~he Germa.n lly P. O. Wolf, Blackie & Son, Ltd., Glasgow, H,56, pp. 93-97. 6. F. Bettes: Non-uniform flow in channels, Civil Engineering and Public Works Review, London, vol. 52, no. 609, pp. 323-324, March, 1957; no. 610, pp. 434-436, April, 1957. . 7.(-Allen: Streamline and turbulent flow in open channels, The.Lond(}tl, Edinburgh and Dublin Philosophical Magazine and Journal Science, ser. 7, vol. 17, pp. 1081-1112, June, 1934. S. Hunter Rouse and Merit P. White: Discussion on Varied flow in open channels of adverse slope, by Arthur E. Ma.tzke,. Trallsadion.s, American Society of Civil Engineers, voL 102, pp. 671-676, 1937. 9. ShermanM. Woodwa.rd and Chesley J. Posey: "Hydraulics of Steady Flow in Open Channels," John Wiley & Sons, Inc., New York, 1941, p. 70. 10. Ivan M. Nelidov; Discussion on S~face curves for steady nonuniform flow, by Robert B. Jansen, Tra.nsa.ctions, American Society of Civil Engineers, vol. 117, pp. 1098~1l02, 1952. .

0;

11. Dwight F. Gunder: Profile curves for open-chruloel flow, Transadions, American Socieey IJj Ci.-il Engineer", vol: lOB, pp. 481-488, 1943. 12. G. Mouret: "Hydraulique: Cours de rneca.nique appliquee" (It HydJ:aulies: Course tn Applied Mechanics "), L'Ecole Nationalc des Ponts et Chaussees, Paris, 1922-1923, pp. 447-458; revised lIB "Hydraulique giinerale" ("General Hydraulics"), cours de 1'Ecole NatiQnii.le des Ponts et CtwLusseesi Paris, 1927-1928. 13. A. Merten: Recherches 'sur Ill. forme des axes hydra.uHques dans un !it prisma.tique (Studies on the form of fiow profiles ill. a. prismatic channel), Anna!e.~ de jl.:l18socic:Ucm de. In(16nieuTS Bortis des ~cole8 8pec.iales de Gand, Ghent, Belgium, '101.5, ser. 3, 1906. 14. M. 'Boudin: De I'axe hydraulique des coms d'eau contenus dans un lit prisma.tique et des dispositifs reaiisll.nt, en p~!l.tique, Bes formes diverses (The flow profiles of w.ater ill a prisma.tic ch!1nnel' and actuI'1.1 "dispositions ill various forms), A nnales des trcwaux pu.bliqucs de Belgique, Bru~s!'f -.80:::

[IOJ

1111(12]

N=2+2m.~1-3

P

1

UI;OD"\ A.!c:rv~

tt

tfM:t

K:

tt

AIZ2

cc lI CO I.l,at1%

lIS1[14.15] [16](17)

K.:

ct lI'~1

1JJl

None

Table_~~~2.~iv:~s_::\...~~.~~~X ..~xi~Hrlg methods of dii'eJ:;i.integ.I:aJ:.i.c>n, .. /

arranged chronologically.'" Although the list is incomplete, it provides !l generarJile}iofUie deveTopment of the dil'ect-in tegration method. Note that most of the early methods were developed fOl~ channels of a specific cross section but that later solutions, since Bakhmeteff, were designed for channels of all shapes, Most early methods use Chazy's formula, whereas later .methods use Manning's formula. . In .the Bakhmeteff me~hod [8] the channel length under consideration is divided into short reaches, The change in the critical slope within the small; l'o.nge of the va.rying depth in each reach is assumed constant,1 and,

the integration is carried out by short-range steps and with the aid of a varied-flow function. In an attempt to improve Bakhmeteff's method, Mononobe [13] introduced two asllumptions for hydraulic exponents. By these assumptions the effects 9f velocity change and friction head are taken into account' integrally without the necessity of dividing the channel length 'into short reaches. Thus, the Mononobe method affords a more direct and accurate computation procedure wherebyresuH.s, can be obtained without recourse.in kinetic energy the friction slope, or r in Eq. (9-14), i3 a.ssUI;ned cOMbnt in each rea.ch. Since an increa.se or decrease in depth will cha.nge both these fa.ctors in the sa.me direction, their ratio is relatively stable a.nd can-be IloBsumed constant for pralltical purpose!!.

to

1S

I In the Bakhme(eff method, Eq. (9-14.) is used. The coefficient r in this equa.tion a.ssumed cOlUltllnt in the reach. Thus, it ca.n be shown tha.t the ra.tio of the change

254

GRADUALLY VARIED FLO","

METHODS OF COMPUTATION

255(10-7)

to successive st.eps. In applying this method to practical problems, it has been found !.ha,t the first assumption (see Table 10-2) is not very satisfactory in many cases. Another drawback of this method perhaps lies in the difficulty of using the accompanying. charts, ."hich are not sufficiently accurate for pl'actical purposes. Later, Lee Tl4] and Von Seggern [16] suggested new assumptions which result in more satisfactory solutions. Von Seggern introduced a new varied-flow hmction in [t,dclition to the fUllction used by Bakhmeteff j hence, aD additional table for the new functicn is necessary in his method. In Lee's method, however, no new. function is required. The method [18] described here IS the outcome of a study of many existing methods.' By this method, the hydraulic exponents arE; expressed in terms of the depth of flow. From Eqs. (6-10) and (4-6), /{,,2 = Gly"N, /(2 = G,yN, Z.2= C2y,M, and Z2 = C2 y M, where G, and C2 are coefficient,s. If the;3e expressions are substituted in Eq. (9-13), thegradually-v.aried- .' flow. equation becomes(10-2)

where

fCv,J) ';", . f" 1 dv

)0

-

J

V

This is a varied-flow function like F(u,N), except that the variables u and N are ;'eplaced by v and J, respectively.l . Using the notation for va,ried-flow functions, Eq. (10-'1) may be writt.en

\

I.

xor wherex

==

t

[ 11 -

F(u,N)

, 'M J + G:) N Fev,J).1+ canst

(l0-8)( 10-9)

A[u - F(u,N)

+ BF(u,J)] +1t =

constJ=

A

=~:,

B -

_ (y,)M -, J y"N

Yn.

't,

N

N - M+ 1

. .-

and where F(u,N) and F(v,J) are varied-flow functions. By Eq. (10-9), the length of flow profile between tVTO consecutive sections 1 and 2 is equai to

. 'j-I\

L

= x~=

-

Xl

A! (u, - u,) - [F(u2,N) - F(1I:"N)]

Let u

=

yly,,; the above'equation may be expressed fOl' dx asdx

+ B[F(U2,J)

- FCli"J)] I (10-10)

~ ~: [ 1 -

1

~

ltN

+ (~:)'{ 1U~-:;N ] dtt

(10-3)

This equation can be integrated for the length ,; of the fio,v profile. Since the change in depth of a. gradually varied flow is generaIJy small, the hydraulic exponents may be assumed constant within the range of the limits of integration. II:_. ?-~I1~e ..U!.e _hY'dra~lic eXQonentL1l.

log y

!2E

"''''''' 0>".0>

.>..>.'

. ..2

01

2. .2 52

CJIo OJ. OJ.

~~~'r;u~i~'e~;~~::~~~~iC plots of depth against Z and lIf, respectively, for vari[,ble)

sion ~s sa~isfactory in most rectangular and trapezoidal channels. -As ~escnbe~ mArts. 4-3. and 5-3, the hydraulic exponents may vary appreciably with res~ect to the depth of flow ~vhen the channel section has abr~pt changes m cross-sectional geometry or is topped with a gradually closmg crown. In s~ch cases, the channelhmgth should be divided "into a number of reaches III each of which the hydraulic exponents appenr to be constant. ' ' Referring to Fig. 10-5, it is assu~ed th:;Lt the hydro.ulic exponents in the range o~ depth from YI to Y2 of a reach ~re practicaJlyconstant. Let N n be the N value at the normal depth y,,;'let N be the average N value

Thus,

(10-30)

For a uuiform flow in the circular conduit with a discharge equal to Q of the actual flow, Eq.(9-11) gives (10-3'1) Q = Knv'So From the above two equations the following may be developed:

i 1

\

262

GRADUA"LI,Y VARIED FLOW

METHODS OF COMPUTATION

,

263

where (ICo/KF is evidently a function of y/d o and, hence, can be represented by fl(u/do). Ji'r"omEqs. (\)-4) and (9-7), the following may be written:0:(.221' ZC)2 = 'qA3 (Z

=""Ci7 q(A/d o2)l

aQ2

T /d o "=

aQ! (Y) -([;Sf. C4

(10-3.'3)

applications. The direct step method I is a simple step method applicable to prio:matic channels. Figure 1O-6illustrates a short" channel reach of length I1x. Equating the total heads at the two end sections 1 and 2, the following may be written:

where (1'/ d o)/ g(A/d 0 2 ) Sis appa,rently afunction of y/d o and, hence, can be represented by hey/do). Substituting Eqs. (10-32) and (10-33) in Eq. (9-13) and simplifying,(10-34)

(10-40)

(10-41)x _ do [ (v/do

- So}o

dey/do) 1 - (Q/QoFh(y/d o)

-~)oor

aQ2 (v/e.,

Ia(y/do) d(u/d o) ] 1 - (Q/QoFfl(y/d o)

+ const

(10-35)

where E is the specific energy or, ass1_unillg cr.l = a2 = a,

where and

.(x _aQ:~ y.) +"const do' (y Q) .(Ii/d. -dey/do) X FI do' Qo =}o 1 - (Q/QoFh(y/d o) Y = F? (X, R) = -f.(u/do) dey/do} - do Qo )0 1 - (Q/Qo)2!t(y/uo)x = - do SoT ."

V2 E=y+cr.2q

(10-42)

(10-36)

FIG. 10-6. A channel reach vation of step methods.

fOl"

the deri-

=

. (10-37)(10-38)

(v/d

o

In the above equations, y is the depth " of flow, 'V is the mean velocity, cr. is the energy coefficient, So is the bottom slope, and S,is the friction slope. The average value ofB! is denoted oy Sr. When the Manning formula is used, the friction slope is expressed by(9-8)

These are the varied-flow functions for circular conduits, depending all y/d o and Q/Qo. They can be evaluated by a procedure of numerical integration, say Simpson's rule. A table of these functions for positive slopes, I prepared by Keifer and Chu, is given in Appelldix E. The length of flow profile between two consecutive section3. of depth "Vl and Ya, respectively, in a circular conduit may be expressed as (10-39) where A = -do/So and B = cr.Q2/d o 10-3. The Direct Step Method. In general, a step method is chal"acterized by dividing the channel into short reaches:and carrying the computation step by step from one end of the reach to the other. There is a great variety of step methods. Som.e methods o..ppear superior to others in certain n~spects, but no one method has been found to be the best in all5

The direct step method is based on Eq. (10-41), as may be illustrated by the IDliowingexo..mple. .Example 10-7. Compute the flow profile required in Example 10-1 by the direct step method. . Solution. With the data given in Example 10-1, the step computations are carried out as shown in Table 10-4. The ,,&lues in each column of the table are expln.ined as foll is negat,ive. Since the actual discharge Q must be positive, (Q/Q.)2 bee"omes" negative. Thus, the integration procedure must be done for negative values of (Q/Qo)l in the two varied-flow functions.1

"

, t;

!,., >, f",,~-

i

.: -,

.. --/

\

METHODS OF COMPUTATION

265

1 (

f

~

~

21

1

~'""""!

~~~~~~~~~~~~~~~J

NMl!"Jt-...~;:S~~~~8~g~c::. ~ r--: u:cr: ~ c ~ 00 00 00 J:-..l"-..;! ~

;g ~ ;g

I.(')

-..:tt 0

~

e,p 00

r-...

...-I

C"I .D C'l

Col. 8. Change of sp.ecifi~ energy in ft, equal to the difference between the E value in col. 7 and that of the previous step CoL 9. Friction slope computed by Eq. (9-8) with n "" 0.025 and with V as given in c'o1. 5 and R~i.in col. 4 Cbl. 10. Aver~ge friction slope between the steps, equal to t.he arithmetic mean of the friction slope just computed ill col. 9 and tha.t of the previous step Col. 11. Differe,nce between the b0ttom slope 0.0016 and the average friction slope CoL 12, Length of the reach in ft between the consecutive steps, computed by Eq. (10-41) or by dividing the value of b.E ic col. 8 by the vahle in col. 11 . Col. 13. Distance from the section under consideration to the dam site. This is equal to the cUlmllativ~ sum of th~ values in coL 12 computed for pr\lvious steps. Th~ fiow profile thus computed is practically identical with that obtained by graphi~l integration (Fig. 10-3). . .,.,... 'Example 10-8. A 72'-in. reinforced-concrete pipe culvert, 250 ft long, is raid. on a slope of 0.02 with a free outlet. Comput,e the flow profile if. the. culv;;l't dischs.rges 232 ds, n = 0.012, and" = LO. Sol'utian . . ,Fronl tl:e data, V' = 4.35 it and Yn = 2.60 ft .. Since y, > y", the chr.nnel slope i~ steep. As shown in Fig. 10-7, the control section is at the e.ntrance; water will enter the culvert at the critical depth !lnd .thereafter flow: at a depth less than 'Y, but grer.ter than y . The flow profile is of the 82 type. Table 10-5 shows the computat,ion of. the fio\v profile, which is self-explanatory. The computed profile is plotted Il.S shown in Fig. 10-7. Plotted also in the figure is the energy line indicating the variation of energ:' along the culvert. The comp\,tll.tion has be~.n car~ied to exceed the length of the eulvert, so. that the depth of flow at the outlet can be interpolated. This depth is fOHnd to be 2.81 ft, and the corresponding outlet velocity is 19,4 fp3. It should be noted that, if the. pipe were fiowing full at the outl'et, the outlet veloCity would be only 10 fps.

! ./1_

~

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0

~

0

.qt

0

1""""1

t-' to to~

f.O

CO

Q ...-I

(10-47).."

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'0> .,0

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Ri\~3r

-"~

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~ ~c:

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