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

in

,

r'l

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

I

,I ,

K = 8gR 2S ! .V'

(1-9)

I YI

I

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

2

1110BASIC PRINCrPLES

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

.

(,

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-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!~ ~V~~~,~" V?",,-~ ~ ..... 1,;:\.~j'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 7-12. Andysis and desigll ?f sto.ble mo~ion over the entire periphery of. FIG. hydraulic secti0!1' (a) ~heoretLcl'l.l secthe channel bed, the two forces tion {or given soil properties. and eh!l.l1ll:l mentioned in the above paragraphs . slope, providing Q - 220 cis; (b) mod~fred section for Q" 400 cfs; (c) modIshould be equal; that is,. fied section for Q' 100 efs. wyS cos 30 11-;-24 8-10 2-6 30 11-24 6-10 2-8 'er .....

,~ ! ~u{j ,~/'

t,\I

~.,

'+OJ U

....

i

,

i51

~

,,'

"\'

O

o. 9O.

0,7

~I,

//"'~

'b'

I[/1/

'/

I

Ir

where Yn is a constant of integration. From Eq. (7-5) and w = pg; it. CD,n be shown thatI'1-

o.61--:- f-.0.5 f--.0.4

{j-----iY' it variesBOO

=

vgRS

Vr

(8-8)

v:/

V,.

/"

V,

i-

0.3 c,. 40 50 60 70 80 100

-L-J

I

200 DistallCe x, f j

300

400

I

600

I

The quantity represented by V r has the dimensions of a velocity. Since wi~h the boundary friction To, it is known as the ~n.i]i;;jii1i or shea;' velocity. Thus, Eq. (8-7) may be written . vr'

P,y./('UViJ.f-

L~~:::) .~

0:.

tVl ~

(8-9)

FIG.

8-6. Solution of Example 8-1 for the growth of a boundary layer.

about 30 ft, or a variation of less than 10%. This shows that an increase in rough,ness has a tendency to speed up the boundary-layer growth or to',educe the developmenL l e n g t h . ' ,

. 8-~., Veloc!ty Distribution in Turbulent Flow. The velooity distributiOn In a ulllform channel flow will become stable when the turbulent ",-_Q.9un~ar~ la!el' 1l3i'!!!.v ,develoE~d. In the turbulent boundary layer, the, clistnbu:lOn can be shown to be approximately logarithmic. "," The shearing stress at any point in a turbulent flow moving over a solid surface has been given by Prandt.l [7] as, T

= pl2

(::)2':

Thili equation indicates ,that t~l:9,ci.~.J~he turbulel:t re~~~ logarithmic funotion of the distance .y. It is commonly known as the -. - . -~~-~ .. ~--~~ Prandtl-von Il..(irmdn univeJ'sal-velocity-dist1-ibution Zaw. I This la.w has been VCl,jti.ed by several experiments [10J.' The results indica.te a striking i'imilarity betvveen observed and computed distributions and, therefore, offer reasonable justification for use of this log;arith' mic law in practical problems., When the sul'face is smooth, the constant yo in Eq. (8-9) has been found to depend solely ,on the friction velocity and the kinematic viscosity; that is, ' my (8-10) Yo = Vr where m is a constant equ~l to about

(8-5)

where

mass ~el1sity = wig, wh~re w i~ the unit weight of the fi.tiid and g IS the gravitational acceleration ': l ':'" a characteristic length known as; ~he mixing length ,........ , dvldy velocity gradient at a normal distance y from the solid surfp,cep

% for

smooth surfaces. 2

,

For wavy

1 Von Karman .[9J also proved this law by II similarity hypothesis which assumes a linear shearing-stress distribution, the mixing length being proportiona.l to (dv /dy) /

(d~/dy).

!011

Thill value is derived from Nikuradse's experimental data.

smooth pipes [111.

I

20~

UNIFORM FLOW

!. 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

.. "--

_--"1 -=-..--T,'- __ _1 -.. - __ 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 1 + G:) N Fev,J). + 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".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_

~

~

~!:::, ~~ ~ ~ MO'~ ~~ ~ ~ ~ ,. C'I') M::'t;l '~r ~~ I 000000000000000 _~ ~

0

~

0

.qt

0

1""""1

t-' to to~

f.O

CO

Q ...-I

(10-47).."

(;

and1100 "

'0 .,>0

Missnuri

Ri\~3r

-"~

E2

"E

~ ~c:

.!!~

"

~

'"

\:.)

e ...cor

~,~

..

"

80

0

"

--",-

t"; '" '" ... ..

~

Ii

M '"

'"~ ~

.;::gj .~0

iii

'"

~~

"

... '"~

X

0>h'

xx X1

(10-59)

effect are also n where the velocity- h ea d ch n.nges' d ue t 0 back"'ater neglected. From Eqs. (10-58) u.nd (10-59) I'1." L'r

= Q/v'F

(~)Z

(10-60)

10-7. The Stage-fall-discharge Method for Natural Channels. When flow profiles of a stream in its natural state, without backwater effect, are available for a number of discharges, thestage-fall-discharge method ~ay be used; this method has the advantages of simplicity and economy [29.]. Similar rnethods'have been 'n.lso developed' by others [30-34].11 Reference (30) describes the so-called Grimm Tllethod. It requires a trial computation which, however, can be avoided by using nomQgraphs, !lIl suggested by Stron. be.rg [311. '

, ~vhere Q/ y'F is called the discharge for 1-ft fal!.1 !his equationC8,l: be used ill the flow-profile eomputntion if the stage-fall-discharge relatlOnshtp for uniform flow in the reach is known. The stLlge-fall-discharge relationship f.or 11 selected reach may be determined froni. records of observed stages and discharges Cfable 10-9). The stages or water-surface elevations at the beginning section _of the reach are plotted as ol'dinates, and corresponding values of Q/y'F are plotted as ~bscissas, resulting in a sta.(je-ve1"!:;us~Q/....IF C'urve (Fig. 10-13). When any water-surface elevation ~\t the_beginning section of the reach fS given, the corresPQnding value of Q/ v'F can be read from the curve, and the fall for a dischrLrge Q. be computed by Eq. (10-60). The computed fall, when added to the water-surface elevation at the beginning section of the -reach gives the water-surface elevation at the end section of the reac.h, whicl~ is also the water-surfaee elevation at the beginning section of the next reach. The procedure is repeated for each reach until the complete ,required flow profile is ob,tained. : The stage-versus-Q/v'F curve is generally cOl~structed as an average 'curve for varying river conditions; such as rising and falling of stage,,lIn a similar method devel~ped by Rakhmanoff [341 a term F /Q' is. us 7 d in lieu of Q/ ~F. This term has the nature of /J. resistance factor and therdore IS given !I IIame of resistance'modulus by PavlovskiI [21, p. 1151.

282

GRADUALLY VARIED FLOW

METHODS OF COMPUTATION

283

fluctuating stream bed, and effects of wind, aquatic growth, ice, .and overbank flow .. Owing to these varying conditions, the plotted points 20re often scattered; and a smooth line, giving consideration to the varying conditions, should be drawn through the points, representing the average condition of the-channel. Where sufficient measurements are available, data of doubtful accuracy should be rejected. In general, the more recent meaSurements should be given greate: weight, as reflecting recent channel changes. Other factors that should be considered in constructing the curve are the relative accuracy of individual discharge measurements; the ,flow condition during the measurements, whether rising, falling, or stationary; conditions affecting the stage-fall-discharge relationship, such as the changes in channel roughness, levee bteaks, and shifts of cha,nnel controls; and the existence of substantial local inflow between the stations. The stage-yersus-Q/VF curve may qe extrapolated !].bove or below t.he range of the observed data by extending the curve at its ends in accordance with the general trend of the curvature. Howe-.:er, any abrupt change in hydraulic elements of the channel section will produce a, eorresponding change in the curvature of the curve. In this case, a correc.tion for the change, if known, should be made in extrapolating the curve. This method is used most advantageously when a number of discharges corresponding to known stages, or vice versa, a.re desired in a stream. By making proper allowance for variable conditions, satisfactory results ca.n be obtained for reaches of large rivers 50 to 100 miles from the measuring station. The data required by the methOd are often .les;:! expensive the,n those required by the standard step method. However, this advantage.is usually.offset by the inaccuracy of the-results, because the effecl of the change in velocity head 1s ignored in the present method. For this' reason, the stage-fall discharge method is more satisfactory for problems . iriwhich the velocity is well below critical end decreases in the downstream direction.E:x:amplelO-ll. Compute the water-sudace elevation at section 1 of the Missouri River problem in Example 10-10 by the sto.ge-fall-discharga method. The reach from section 1 to section 5 istakea as the first reach. The water-surface elevations are available from stage records for gages located at sections 1 and 5. The discharges have been observed at the A.S.B. Bridge located about 3,000 ft downstream from section 1. These data are tabulated in Table 10-9.' . Solution. The data arid cornputa tions for the sta.ge versus discharge for a I-it-fall! curve are given in Table 10-9, which contains the following headings: Col. 1. Recorded water-surface elevations at section 1 Col. 2. Recorded wo.ter-surface elevations at section 5 Col. 3-. Fa.ll in ft, which is equal to the differ~nce between elevations entered in cols. 2 and 11

Col. 4.0bserveddischarges,at.the, A ..8J3. Btidgej i!lds Col. 5. Discharge per I-ft faIl, or Q/ y'F, where Q:is,th'e discha.rge in col. 4 and F is tilE; fa.ll in col. 3 . Using water-surfa.ce cle\'ations at section 1 as listed in col. 1 of the ta.ble and the CQl:responding values of Q/ ..\/ifi col. 5, construct 0. sta.ge-versus-Q; vF curve (~'ig. 10-13).

in

755 752,25

\

~ 750:

---I-~-lf--, -~,

1- 'i

crt

745

I ~

~i----~

!~;c

I740r--r-

'0

..,>

I I

I

'0 735.12 0;~

'" 730

"":~0

'" ." .g 725

3;

720

715 0

1

-r~!50 100

o

I

-[- wI

L~I

I

I 1

I

I

ISO

I

200

I

250

I

300

h_ _I".,

,II1

.,,350".,

Values OfQ/IFinl.o00

lJ~its

FIG. 10-13. The stage-vs.-Q(V'F curve for Example 10-11.

This example is taken from [291 with modifications.

For a we.ter-surfaceelevation of 752.25, a value of Q/ -../F =33~,OOO i5 obtained by extrapolation. By (10-60), the fall between sections 1 ancH is equal to (431,000/ 355,000)' = 1.65 ft. Ad'ding- this va.lue. to the elevation at section 1, th~ required water-sufface elevation at section 5 is 753.90. This is about half a foot lower than the elevation. compu~ed by the standa.rd step method; the difference results primarily from the neglect of velocity-head changes in the present-method. The computation may be continued for subsequent reaches. A tabulation,. as 5hown in Table 10-10, is suggested for the computation if a. complete flow profile is requiled. .

Eq.

"

\

e.;.-,

284

GRADUALLY VARIED FLOW

METHODS OF COMPUTATION

285

If .d~ired, the ,,:,ater-surface elevati~m; at the intermediate sections 2 to 4 rna be obtaln~d by breakmg up the reach 1-5 Into four short reaches The profile 1 .~ at th . t d' . e eva,.lons e III, erme la/,,", sectlOns may be obtained by in'erpel-t' " Th.e stage-versus, 1, } Ion. Q/ . v F curves can be drawn for each section and the cbmputat'o b . d fol' the subdivided reaches. ' . I n can e carne outB

effects of velocity-head changes and eddy losses, is required, the Ezra method described in Art. 10-5 should provide more satisfactory results.Example 1012. Determine the w!),ter-surface elevations a.t sections 1 to 5 of the Missouri River at lCa-nsas City, M.a., as described in Example 10-10. The da.ta required for the computation by the Ezra method are given in Table 10-7. The discharge is 431,000 cis. The initia.l water-surface elevation at section 1 is 752,25. It is assumed that eddy losses are included in the friction losses. S:)/u.t-ior.. The first step is to compute value of Z + F(Z) from the given data. The computation'is tabulated in Table 10-11 with the following column headings: Col. 1.' Channel-section number Col. 2. River mileage CoL 3. Length of reach in ft. The upper value f>:Cd is the length of the downstream reach from the selected section, and the lower va.lue Llx. is the length of the upstream reach. ,' CoL 4. '\Vater-surfa.ce elevations. Three elevations are given for each section. Generally, at least, three elevations are seiected for eaah section to provide at least three points for plotting each Z F(Z) curve. Cols.5 to 14. These columns correspond exactly to those in Table 10-8 for the standard step method. The values in the top row for each elevlltion al'e for the mllio cha!lnel, and those in the bottom row arc for the left-overbank section. CoL,15. }'rictioii slope, which is equal to (Q/[()~, where Q = 431,000 cis and lC is from col. 10 Col. 16. Value of .-8 , ClXd/2, where S. is the 'value from col. 15 and ClXd is the upper value in col. 3 Col. 17. Value of S, t..x./2, where S. is the value from col. 15 and C.x" is the, lower value in col. 3 ' Col. 18. Value of F(Z,), which is equal to the sum of the value in coL 14. and the value in col. 16 _ ('..01. 19. Valu~ of F(Z2), which is equal to the sum of the va.hl"- in col. 14 and the value in col. 17 Col. 20. Sum of the values of Z in col. 1 and F(Z,) in coL 18 Col. 21. Sum of t.he values of Z in col. 4 and F(Z,) in col. Ii) The second stP.p is to plot curves of Z F (Z) against Z for each oross section, using , values from cols. 4, 20, and 21 of Tl1ble lCl-11. The resultiug curves are shown in Fig. 10-14. . The third step is to determine the water-surface elevations from the Z F(Z) curves. At section I, for a.n initial water-surfaae elevation of 752:25, the value of ~. F(Z~) is found from t.he appropriate curve (Fig. 10-14) to be 754.14. Taking this value to the Z, + F(Z ,) curve for the next upstream section 2, the corresponding water-slUiace elevation is found to be 752.72. Contiuuing the procedure for other sections, the values are trD.Ced in the direction shown by thc dashed line in Fig, 1014. The rcsults of the IV ater-surfac,e-elevation determinat.ion are tabulated in Table 10-12. They are in very close agreement with those obtained by the standard step method, Example 10-13.. Solve the problent in Example 10-12 for a discharge of 500,000 ds, The corresponding initial wuter-surface elevatior. &t section 1 was estimated from the rating CUrve to be 752.30, Solution. The values of F(Z ,) and F(Z2) for Q = 500,000 cfs may be obtailled by mUltip1ying the corresponding values in Table 10-11 by (500,000/431,000)' = 1.34.' T~,1e values thus obt~jned are tabulated in.cols. 3 and 4 of Table 10-13, respectivel}', and the values of ZI + F(Z.) and Z, + F(Z.) in cols. 5 and 6, t:,espectively.

TAllLE 10-9., DATA, AND COMPUTATIONS FOR SrAGE-vs.-Q/YF CURVE USEDIN EXA~IPLE 10-11 (Missouri River at E:ansas City, Mo., sections 1 to 5)

Water-surface elevation, m,s.l.11

,

FII, 11 ,ft (3)

Section 1 (1) 724.8 725',3 729.13 727.4 727.8 730.2 730.8 731.3 734.6 735.8 736,6 745,0 722.2 724.6 725.0 725.3

Section 5 (2) 725,7 726.2 730.2 728.3 728.8 731.2 731.7 732.3 735.6 736.7 737.7 746.6 723.1 725 6 726.0 726.4

D' ISCh al'ge, cfsI

Q

I

VF!~5)

(4)

I

0.9 0.0 O.G0.9 1.0 1.0 0.9 1.0 1 0 0,9 1.1 1.6 0.8 1.0 1.0 1.1

I

33,600 36,100 66,100 69,500 76,000 97,200 105,000 113,000 141,000 157,000 104,000 326,000 22,900 45,400 49,900 52,300

35,40',) 38,000 85,30U 7'6,200 76,UOO 97,200 111,000 113,000 141,000 165,000 156,000 ' 258,000 25,600 45,100 49,9UO '19,800 10_11BY

+

TA.BLE

10-10. ConiPUTA'I'I~N

OF

THE.'

FLOW P'ROFILE FOil. EXAMPLE

(Missouri River Sea. no. 1 5 River mile 377.58 378.65

THE ST,lGE-FA.LL-nrSCHARGE II-fETHOD at Kansas City Mo' ~ections 1 to 5 f)

+

".'

, '., =

431,000 ds)

I

Length of reach

~rater-silrfaceelevation

Ii

+

Q/-./F

F. =~-2 (Q/ -./F)

+

.....5,655

.

"

......

. ....

752.25 1.65 r 335,000 1.65 753,90 .... ' .. To be continued if desired' ...

I

10.,8. The ~zra Method for Natural Channels.' If flow profiles for a numbel' of discharges or stages are desired, the stage-fall.discharge metho~ can be use.d most advantageously for a simple and economical but approXlrnate solutlOn. However) if a precise computation, including the

TA.BLE

10-11.

COMPllT.AT10N OF Zl + F(Z!) ANn Z2 +F(Z,) FOR E:u.MPLE 10-12 (Missouri River at Kl\llsas City. Mo.; Q - 431,000 cfs)

A

p

R I R%

".

K

A'

Ie'

+F(Z.)

0.12

.... 11,&3.... 11.80I'

753.93

0.11~

754.80

0>

coI~i!;;;:~ . 3:~c-.-_-.t

I.

f

0.10

...

~

1.71

7.55.71

0.13

1.46 1.70

753.46

753.7il

J-0.151(l.l~

1.39 1.65

154..39

754.66

1.3111.55

755.31

755.M

-~'.~,--

;; O. The symbol F l'epresents the Froude number of flow at a. section a distance x from the upstream end of the channeL The value of F can be computed from Eq. (12-13) by replacing F 0 with F and L with x. Since the flow is subcritical; the depth of flow at any section is greater than the critical depth, as shown, L The profile can a.lso be computed bra gra.phica.l method developed by Ca.mp ;4], which requires trial adjustments~

Region B. This region: represents the condition' where the flow is subcritical throughout the channel but where the value of F will first increase as the flow proceeds downstream, reaching a maximum 'value . less than unity, an,d then decrease. It been found that the line di:vidregions Band C can be represented approximately by G = 1 + F . This line indicates all the cases in which the maximum value of F reaches unity. ' Regicrn C. This region represents the condition at which there is supercritical flow in the downstream portion of the channel and a hydraulic i:ump in the channel. It h(l.s been found that super critical flow occurs when G is greater than approximately 1 + F.. The hydraulic jump will fo:-m only if the outlet is sufficiently submerged. As the jump occurs, the control section will be shifted into, the channel, and the elevation

I~

336

GRADUAIJLY VARIED FLOW

SPA.TIALLY VA.RIED FLOW

337

,(\

of the \,rater surface at the outlet will not affect the entire flow profile. 'The flow profile upstream from the . jump cannot be determin!ld froni the : value of YO) but it can be determined from the critical depth yaand the position of the critical section Xc. A dimensionless flQ~" in the sllpercritical reach ('[;~ig. 12-5, ill which A" and A .!we water areas, respectively, of the critical section and the section a distance x from the upstream end of the ch~nnel) has been computed by numei'ica[ inte~ gratioll. This curve can be used to FlO. 1,?-5. Dimensionless flow profile in compute the flow profile below the Btipercriti!\a~ reaches in a sp .. til1!lyvaried-liow cht,nnel of sloping bed and critical section and. above the hyparallel walls, (After W. H. Li [5),) , draulic jump. . The positi'on of a critical section in the lateral-spillway channel can be determined by the method of singular point (Art. 9-6), Regio-n D. This region represents the flow,condition Fo.t which there is supercritical flow throughout the downstream portion of the channel but where the depth of submergence at the ouMet is not enough to create a hydraulic jump in the channeL Thus, the value of Fa is not determined by the depth of su'bmergence. The dividing line between regions C and D (Figs. 12-3. and 12-4) was obtained by numerical integration on the condit,ion of a millimum depth of required to prOdtlCe a hydraulic jump at the outlet. This Jnllllmum depth 'is . equruto the depth neeeSS1J.ryto set the downstream pool level at a seq).lent depth. A depth of submerge'nee greater than this. minimutn depth wlll force the jump to move upstream into the channel, and the oondition of flow Ifill be represented by region C. When the slope of the channel is extremely steep or when the value of G is very large, the flow ,vill become unsteady. The limiting value of G that will keep the flow in a steady condition has not determined. In the above a.nalysis, the' effect of friction has This has been verified as justifiable for the design of wash-)vater troughs and side-channel spillways. For effluent channels around 6ewnge-treatment tanks. however, the effect of friction may increase t~e upskeam depth Yu ~s :much as 10%.* '$LJ 15] has computed curves repre:;enting the increaSe of !/Y horizontal channels. .relmitof friction in

'For an advanced theoretical analysis of spatially varied flow, the method of singular po