Chow - Open Channel Hydraulics

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

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OPEN-CHANNEL )

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

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VEN TE CHOW, Ph.D.

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

INTERNATIONAL STUDENT EDITION

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

HYDR~AULICS

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KHAL1D PHOTO STAlb

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McGRAW·HILL BOOK COMPANY; I:r.;G. New York

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URQUHART, O'ROURlIL

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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. . •

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BASIC PRINCIPLES .

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

OPEN-CHANNEL FLOW AND ITS CLASSIFICATIONS

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

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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

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

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(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.

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BAstC ' PRINCIPLES

It is'believed th~t gravity action may have a definitive effect upon the flow resistance in cliurmels at the tut'bulent-flow range~ The experi,mental data studied by Jegorow [311 and Iwagaki [32J for smooth rec'tangular channels. and by Hom-:ma [33J for rough :channels have shown that, ~n the supol'critical-turhulent regil;l1e of flo~, the friflj;ion fact known as curvilinear flow. The effect of the curvature is to produce appreciable ELGceleration components or cell trifugal forces normal to the direction of flow. Thus, the pressure distribution over the section deviates from the h}'drostatic if ourvilinear flow occurs rn the vertical plane:' Such curviiinear flow may be either convex or concave (If;lg~nd c). In both cases the .nonlinear pressure distribution is represented by AB' instead of tlte straight distribution AB that would occur if the flow wel'e parallel. It is assumed that ail streamlines are horizontal at the section under consideration. In concave flow the centrifugal forces are pointing dDW:lward to reinforce the gravity action; so the resulting pressure is greater than the otherwise hydrostatic pressure of a parallel flow. In convex flow the centrifugal forces are acting upward against the gravity action; consequently, the resulting pressure is less than the otherwise hydrostatic pressure of a parallel flow. Similarly, when divergence of streamlines is great enough to developappl'eciable acceleration components normal to the flow; thehydrostati'c pressure distribution will be disturbed accQrdingly. Let the deviation from an otherwise hydrostatic pressure h. in a curvilinear flow be designated by c (Fig. 2-7b and c). Then the true pressure or the piezometric height h = h; + c. ' If the channel has a curved longitudinal profile, the approximate centrifugal pressure may be computed, by Newton's law of acceleration, ~~,~he E~~~~~ oL~L@iWt d and a croSQ..§ection. Qf 1 sq ft, that is, wd/g, and the centrifugal ac~ v2/r; or ' _----7

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

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(2-8)

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

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

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BASIC PRINCIPLES

OPEN CHANNELS AND 'l'HEIR PROPERTIES

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

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

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Pressure di.stribution

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

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

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

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

-

(2-10)

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

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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.

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BASIC PRINCIPLES

OPEN CHANNELS. AND THEIR PRO'PERT1ES

Douma. formula,' that is, 'U -

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(2-15)

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

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

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

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

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

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

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

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

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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.

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ENERGY AND MOMENTUM PRINCIPLES

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

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