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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 '...
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I 1
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
U.E.T (,
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McGRAW·HILL BOOK COMPANY; I:r.;G. New York
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KOGAKUSHA·COMPANY, LTD.
URQUHART, O'ROURl
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CO!-T'l'ENTS
PART 11. Chapter 5.
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1....1
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Development ()f Uniform Flow and Its Founulas
Computation of Uniform Flow .
The Conveyance of a Channel Section The Section Factor [or Uniform-fio;; Computation . . The Hydraulic EXjlQnent for Uniform,.flow COmputation. . Flow Chua.cteristics in Jl. Closed Conduit with Of/en-channel Flow Flow in a Channel Section' with Composite Roughness Determination of the Normal Depth and Velocity. Determination of the Normal a.nd Critice.l Slopes 6-8. Problems of Uuitorm-fiow Computa.tion. 6-9. Computation of Flood Discharge. 6-10. Uniform Surface Flow . Chapter 7.
81l
89 89 91
93 94 98 10l
101 108 114
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It is here assumed that the velocity is uniformly distrib\.ited across the conduit othenvise a correc,tion ,,",ould have to be made, such Ilf is desaribed in Art. 2-7 fbr open c h a n r r e l s . ! , : 2 If the flow were curvilinear or if the slope of the channel w~re large, the piezometric height would be appreciably different from the depth of flow (l
.P. • ....
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6
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BASIC PRINCIPLES
is a practically impossible condition. The term «uniform flow" is , , therefore, used hereafter to refer only to steady uniform flow. Flow is varied if the depth of flow changes along the length of the channel. Varied flow may be either steady or unsteady. ~nce unsteady 2l~fI.~~e, the t~,rm II unsteady flow" is used hereafter to designate unsteady varied flow exclusively:-'-. ._ ... Varied. flow roay be furt~; da.ssified as either rapidly or gradually varied. The flow is rapidly; varied if the depth changes abruptly over a comparatively short distance; otherwise, it is gradually varied. A rapidly varied flow is also known as a local phenomenon,' examples are the hydraulic jump and the hydraulic drop.
OPEN-CHANNEL FLOW AND ITS CLASSIFICATIONS
I
For clarity, the classification of open-channel flow is summarized as follows:
f'
I
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'{'
I r
I Chong'e of depth from time time
to
Uniform 'flow - Flow in a laboratory' channel
G.V.F.
RV.F.
Sluice
Confraction below the sluice
----:---~-
-
~ -'.
,
//
G.v. F. - Flood wove
R.V.F. - Bore
lJnsteqdy flow •
FlG. 1-2. Various types of open-channel Bow. RV.F. ~mpidly varied flow.
G.V.F. = gradually varied flow;
t ~'.
The flow is turbulent if the viscous forces are 'weak relative to the inertial fDrces. In turbulent flDW, the water particles move in irregular paths which are ,neither smooth nor fixed but which in the aggregate still represent the .forward motion of the entire stream. Between the laminar and turbulent ,states there is a mixed, or transitional, state. The effect .of viscosity relative to inertia can be represented by the Reynolds number, defined as
Varied flow
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A. Steady flow L Uniform flow .2. Varied flow c. Gradually varied flow b. Rapidly varied flow B. Unsteady flow L Unsteady uniform flow (rare) 2. Unsteady now (i.e., unsteady varied flow) a.Gradually varied unsteady flow b. Rapidly varied unsteady flow Various types of flow are sketched in Fig, 1-2. For iIlu.strative purposes, these diagrams, as well as other similar sketches of open channels in this book, have been draw.l to a greatly exaggerated vertical scale, since ordinary channels have small bottom slopes. 1-3. State of Flow. The state 01' behavior of open-channel flow is governed basically by the effects of viscosity and gravity relative to the in.ertial forces of the flow. The surface tension of water may affect the behavior of-flow under certain circumstances, but it does not piay a significant role in most open-channel problems encountered in engineering. , Effect of Viscosity, Depeilding on the effect of viscosity relative to inertia, the flow may be laminar, turbulent, or transitional. The flow is laminar if t.he viscous forces are so strong relative to 'the inertial forces that viscosity plays a signiftcant part in determining flow behavior. i~ laminaLfl.mY...Jh~~arti.91esa,ppeal' to IAove in deE!lite smooth paths, or streamlines, and infinitesimally thin layers of. fi»id se~11l To slia:e ove~\:n~Q~!.lt l~ye.r~: . - . - . - - - . - - . - - - - . - . - - -
Unsteady unifarm flow - Race
RV.F.
7
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II
i
j.
R == VL JJ
(1-3)
where V is the velocity of flow in fps; L is a characteristic length in ft, here consider'ed equal ,to the hydraulic radius R of a conduit; and 11 (nu) is the kinematic viscosity of water in fV/sec.
The kinematic viscosity
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"BASIC PRINCIPLES
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OPEN-CHANNEL FLOW AND ITS CLASSIFICATIONS
ill ft2/sec is equal ~o the dynamic viscosity, f.I (mu) in shlg/ft-sec divided by the mass densl(;Y p (rho) in slug/ft,·. For water at 68"F (2000) 5 f.I = 2.09 X 10- and p = 1.937; hence, 11 1.08 X 10-$. " An open~chan?,el flow is laminar if the ReYl10lds number R is small and tUl'bt~lent if. R IS ,large. Numerous experinlents have shown that the flow 10 n. pIpe . c~an~es, from laminar to tul'l)ul~l1t in the range of R betwee~ the cntlCal val.ue~&Q.Q. ~n~ a valuethfl,t inay be as high as 50,000. . I~ th~se exp~nment.s the dIameter of the pipe was taken as the chal'act~rIStlC. le~gth m defining the Reynolds number. When the hYdra~11C radlUS 15 taken as the characteristic length, the corresponding' range IS from ,500 to 11,50q, ,. since th.e~!.fl:,l!l~~~_ of '!..,Eipe is fo~r times its hydraulic mdlUs. ' --'--.•-.-.--~--'-''''------
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r I i
,
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;t
I
(1-4)
,where 14, ~ the fl'1::Lional los; i.n ft. for flow in the pipe, i is the friction fact~~. L L~ the l~ugth of the pIpe III ft, do is the diameter of the pipe in V IS the velocIty of flow in fps, and g is the acceleration due to gravity In ftfsec~. '
:t,
hJlL, the above equation
1. .."
(1-5)
I
. Th f-R[IiJrelati~l1S~pJ.~smo~t~J;_~~~ can ~~~e,!2I'.!lssed bv th, e Blas:ius equatton ' . '. - ----"-,---- , .. .---'- ., 0.223
?
=
RO.~'
(1-6)
t
.1 t8 a result ~f Darc(s. stud.: [21: ~n flo~ in pipes, his name is commonly associated wlthithat ~ WelS~o.ch [31 deslgna,tJng~hlS. equation which Weisbach firstformulated..
in
r'l
From Eq;S. (1-3)• and (1-5) it can be shown that i
In
Act~IlYI dAubulSson [4] presenteji, prIOr to Darcy, a formula. that can be reducEid to the form of Eq. (1-4). i . , t tn. this equation, ~he hydraulic :radius is used as the characteristic lengtll in defin.. -.. mg the Reynolds .number. If the; diameter of pipe were used as ~he characteristic length, the numerlllal constant of ~he numerator in this equationfwould be 0.316.
(1-7)
basis for comparing flow conditions in open channel.s. It may be noted that corresponding equations forflO'wi'n open channels have been derived byJf~~~l~L§>lld ~.epear to be very_similaLtQ_ihrnQe-fiow t:lJl'\l.!:l:jJ.QTI!3~ giYQ!L:;tj;lov:e. It must be remember.ed, however, that, owing to the free "mlrface and to the ~~oUhe }u~draulic ~us. discharge... and slope, the f-R relationship inopen-channel flow does not follow exactlyJ;htl. , sinlpleConcepts thathohlfor pipe flow. Some specific features of the '.f-R relationshiP open-chann.erfloware described below. ,t Experimental data available, for the .determinationof the f-R relationship in open-channel flow can be found in various publications on hydmu-' which plots the relationship for flow iIi lics. 1 Figure is based on data developed at the University of HUn an University of Minnesota [20]. In this plot the following features may be noted: l.The 'plot shows clearly how the state of flow changes from laminar to turbulent. as the Reynolds number, increases. The discontinuity of the plot and the spread of data characterize the transitional regIon, as they do in ,the Stanton for fio'w in pipes. The transitional range, however, so as "pipe flow. The iower critical Reynolds number depends to some extentoa channel shalle. The value :ia~0rQD1 Q.OQJ;9~-(i60;oemgg~i;~rally l~g~;-th-;:;-th-; value 'for pipe flow. For practical purpoSes,thetransitioUal range of:R Iol:-opei1='channel -~.~ flow mfty be a.~sumed to bc..§QQJ.Q..~..QQP. ' It should be noted, however, that the uppe.r value is arbitrary, since there is no definite upper limit ' for all flow conditions. 2. The: data in the laminar regiQu can be defined by a general equation 'I( f F -R • (1-8)
,
;. It snould be noted that there is actually no definite upper limit..
,!
+ 0.4 '
---
:'hls equation may a.lso be applied to uniform ahd nearly uniform flows channels. ' , - - -_________
~/-m...0pen
f
2 log (R ...;'1)
The resulting Prandil-:von
I),
and transitional stat.es ot open-.channel flow c~m be expressed by. a.~lagram thatshows a relation between the Reynolds number and the fnctlOn factor of the Darcy-Weisbach formula. Such 9~ dlUgra~, g~I1el'ally knoWn us the BtanloTL4:f.ggn;,m Ill, has beeg.
(1-9)
Se~ [l(~1 to [23].' ! ' : . The da.;ta. for the rectangular channel; were furnished through the cdurtesy of Professor W. :/;1. lAnsford ann processedJoi the present purpose by the author. 1
2
11 10
BASIC PRINCrPLES
Since V and R have speeific ~'alues for any given chantiel shape, K is a purely numerical factor dependent only on channel shape. For laminar . flow in smooth cha.nllE)ls, the value of J( call be determined theoretically [20} .. The pJot in Fig. 1-3 indica.tes that Kia approximately 24. for tht> reetanguJar channels and 14 fol' the triangular channel under consideration.
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FIG. 1-3. The f-R relationship for flow in smooth channels.
3. The data in the turbulent region correspond closely to the Bllisius:j=lrandtl-vo1n Karman curve. This ~ndicates that the law for turbulent fiowin smaoth pipes may be approximately representative of a.Usmooth channels. The plot also shows that the shape of the channel does not have an important influence 011 friction in turbulent flow, as'it does in ," laminar fl o:W. The dataJor laminar flow obtained at the University of Minnesota [20J and the da~ for turbulent flow cdllected individually by Kirschmer l
.' . h channelS. . Bo.zin's channeis; No.4, FIG. 1-4. The f-R. ~elatlonshlp for Row In rOU~ed wood; No. 14, 'unpolished wo~d gravel embedded In aement; No. Il,. unpo m lon 10 mm high, and·lO mm In roughened by tra.nsverse wooden stripS 27. sp.~in'" of mm; No. 24, cem, ent .. 7 Ne 14. except WIt .1 a te· : spacmg; No.1, same B.S. ' d K' her's cha.nnel: smooth concre . lining; and No. 26, u~p<Jhshed woo. IrSC In . "
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. . . (23] are shown in the diagram for fio~ in (15,16}, Elsner [221, and Kozelnny f the' data channel roughnes.~ is (Fig. 1-4). 13Qme 0 ' • f I _ .r . • h' . e8.SUre of the roughness fll1rttcles or;m k whic IS a SIze m . f t ' s'. Y , . represen d' . illustrates the followmg ea ur.e ing the channel surface. The lsgram d fi d b Eq (1-8) In this • • . . th d til. can be e ne Y . . . L In the lammat reg~on e ~ly h' h than it is for smooth channels region, the value of K l.S genera 19 e r . .
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and ranges bfltween 60 and 33, indicating the pronounced influence of the channell'oughnesson the friction factor. . 2. In the turbulent region the channel shape has a pronounced effect on the friction factor. It is believed that, when the degree of roughness i~ const.ant the fdction. factor decreases roughly in the order of rectangular, t:'iangular, trapezoidal, and circular channels. At the suggestion of Prandtl, Kirschmer [15,lDj explained that the effect of channel shape may be due to the development of secondary flow, which is ~pparently more pronounced in rect(1ngular channels than in,say, triangular channels: The secondary flow is the movement of water particles on /l, cross section normal to the longitudinal direction of the channel. A high secondary flowrinvolves high energy loss andthns r.,ccounts for high chaI!!lE11.J:{:!§.i..frul.nce. . . . . 3. In the turbl,1lent region most plots appear parallel to the Prandtlvon Karman curve, This curve serves as an approximate limiting position toward which a plot moves as the over-all resistance becomes less, Accoi'dingtQ a concept advanced by Morris [241 (Art. 8-2), the rise of the plots above the smooth-conduit curve may be explained as a resul~ of additional energy loss generated by the roughlless elements. When the Reynolds number is very high, some plots become essentially horizont,al, reaching a stllte of SO-Galled complete turbulrnce. At this state the value of fis independent of Reynolds number and dep~nds solely on roughness, hydraulic' radius, .and channel shape . . 4. The' plot of Varwick's data [16J for a give~ roughness,- hydraulic radius, and channel sha~ star~ off from ll. curve parallel to thLPran..Q.tj~ Voill'{arman curve, then rises as the Reypolds n~mber incre~§, and final1y becomes horizo'ntal as a state of complete turbulence is reached. The rise of the plot is a peculiar phenomenon which demands explanation, I and, since this finding has not been verified by other data, mor.e experimental studies .seem necessary to substantiate it. ~-'-~It should be noted that the ab'Jve descriptions are limited to lowvelocity, 01' subcritical, flow (which will be defined later in this.-ar.ti.cl.e)_ and to Row onwhich 8.ll:.rface tension dOEls not have a §1IDlificant influence. .. _._._----;---,. .. - .... -In most open channels laminar flow occurs very rarely. ,Th~!!l;~t tfu;.1 the surface of a streilr!!:J!:eE!~!g_~<JmlOoth and glassY_.JQ..Jtn ob.sru:.v.ru.2...by no mea~__a..lL.i.ll.dicati2.~~inar; }llOID;' prohllJily, it indicates that the sU11aqe vel.QQi!! is lower than that required for capillary wav~Qnn.· :J;.,.aminat open-cha~Jm9,!..n t'O exist, however, ~~::wJ1~~ th.in sheets...2f_ ~t?l' flow over the grq,,';1nd Dr where it is created deliberat~ ~testlng ~~!lnels,
-
---
According to thelloncept of Morris [241. thIS phenomenon probably represents a tJ'ansition of thefiow to a.nother type of flow having higher energy loss. As the Reynolds nu~ber increases, the fio'IV may be cha.nging from quasi-smooth flow to wakeinterference flow, and then tomolatecl·rou.ghness flow (~t. 8-2). 1
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BASIC PRINCIPLES
-
OPEN-CHANNEL FLOW. AND ITS Ci..AftSIFICATIONS
13
As the flow in most channels is turbulent, a model empl.oyed to simulate a prototype channel should be designed so that the Reynolds number of flow of the model channel is in the turbulent range. E.ffect oj Gravity. The effect of gravity upon the .state of flow is represented bt aratio...91 ine~al forces to gravity forc~ This l'atio is,given by the FrQuile numbe1'z' defined» ',.
v
(1-10)
where, V is the mean velocity of fiow in fps, g is the acceleration of gra Yity in ft/sec z) and Lis a.characteristic length in ft. In open-channel fiow the characteristic leng~h is made equal to the lmdraulic deptAJ2., which is defined as the cross-sectional area of the water normal to the direction of. flow in the channel divided by the width of .the fr¢e surface. For rectangular channels this is equal to the depth of the flow' section, When F is equa.l to unity, Eq. (1-10) gives V
=
.../gD
(1-11)
and the flow is said to be in a critical sta.te. If F is less thaJ1 unity. or V < V(jJ5, the flow is s'libcriticaZ. In this state the role played. by gravity forces is mOfe pronounced; so the flow has a low velocity and is often described as tranquil a.nd streaming. If F is greater than unity, or V > the flow is 8upercriticaZ. In this state the inertial forces become domL'lant; so the flow has a high velocity and is usually described as rapid, shooting, and torrentiaL IE the mechanics of water wave~Jhe.critic& velocity v'gD is ideirtified as the celerity of the small gl'avitywaves that~ccu;fns-hallo;v wnterm channels as a result of any momentary change in the local depth of the . _water (Art. 18-6). Such a change may .Q~opecLbjullsJ;urha.ll.c~~u;l.!: obstacl~in...thJL!lb.!IJlp.el that cause a di§placement of waj;.!'lr.J.!.b.
20
(
BA.SIC PRINCIPLES
slopes. The drop is similar to a. chute, but the change in elevation is effected in a shott distance. The culvert flowing partly full is a covered channel of comparfl,tively short length installed to drain water through highway and railroad embankments. The open-flow t'unnel is a comparatively long covered channel used to ca.rry water through a hill or any obstruction on the ground. 2-2. Channel Geometry. A chann\'ll built witlulllY.a:t+ing cross sectiOll ti..'1dc~bottom slope is ~~lled a 'B.risma(ikma1Jnel. Otherwise, the channel is nonprismaticj an example is a trough spillway having variable width and curved alignment. Unl\lss specifically indicated, the channels dGScl:ibed in this book are prismatic. The term channel Bection used in this book refers to the cross section of a channel taken normal to the direction of the flow. A l'ertical channel s8ctio-n, howeve".is the vertical section passing through the lowest or bottom point of·the channel section. For hvl'izontal channels, therefore, the channel section is always a vertical channel section. Natural channel sections are in general very irregular, usually varying from an approximate parabola to an approximate tmpezoid. For streams subject to frequent fioods, the channel may consist of a main channel section carrying normal discharges and one or more side channel sections for accommodating overflows. Artificial channels are USUally designed with sections of regular geometric shapes. Tablc 2-1 lists seven geometric shapes that are in common use. The trapezoid is the commonest shape for channels with unlined earth banks, for it provides side slopes fol' stability. The rectangle and triangle are special cases of the trapezoid. Since the rectangle has vertical sides, it is commonly used for channels built of stable materials, such as lined masonry, rocks, metal, or timber. The triangular section is used only for small ditches, roadside gutters, and laboratory works. The cirde is Hle popular section for sewers and culverts of small and medium sizes. The parabola l is used as an approximation of sections of small and medium-size natural channels. The round-cQrnered rectangle is a modification of the rectEmgle. The round-bottom triangle L':! an . approximation of the parabola; it is a form usually created by excavation with .shovels. Closed geometric sections other than the circle are frequently used in sewerage, particularly for sewers enough for a man to enter. These sections are given various names according to their form; they may be
I
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r 1
i
I !
.i!:
+
I;'.>
+
~
1 The side slope z; 1 of a pa.rl!.boli~ section at the intersection of the sides with the free surface can be computed easily'by the simple formula z = Russian e!lgineers [11 also use semielliptica! and para.bolic of higher order: y azP with 11 .;, 3 Qr 4. The constant CL is computed fl'om the side slope assumed· at the' free surface ..
21
22
BASIC PRINCIPLES
egg-shaped, ovoid, semielliptical, U-shaped, catenary, horseshoe, baskethandle, 'etc. The complete rectangle and square are also common for large sewers. Dimensions and'properties of sewer sections may be found in textbooks on sewerage.! A special geometric section kilOwn as hydrostatic catenary or lintMrw [4,5J is the shape of the cross section of a trough, formed of flexible sheets assumed to be 'Iveightless, filled with water up to the top of the seCtion, . and firmly supported at the upper edges of the sides but with no effects of fixation~ The hydrostatic catenary has been used for the design of the sectiolls of some elevated irrigation flumes. These flumes are constmcted of metal plates so thih that their weight is negligible, and are :firmly attached to beams at the upper 2-3, Geometric Elements of Channel Section.' Geometl"l:c elements are pro'perties of a channel section tha.t can be defined entirely by the geometry of the section B.nd the depth of flow. These elemerits are very importa.nt and are used extensively in flow computations. For simple regular channel 'sections. the geometric element.s can be expressed mathematically ill terms of the depth of flow and other dimensions. of the sectioll. For complicated sections and sections of natural streams, however, ,110 simple formula can be 'written to express these elements, but curves representing the relation -between these elements and the depth of flow can be prepared for use in hydraulic computations. The definitioris of several geometric elements of basic importance are given below. Other geoIlwtric elements used in this book will be defined where they first appear. The depth of flow y is the vertical distance of the lowest point of a /' 'chanilel section from the free surface. This term is often used inter-' changeably with the depth offlow section d. Stridly sp~aking, the depth of flow section is the depth of flow normal to the direction of flow, or the height of the channel section containing the water. For a channel with a longitudinal slop~ angle e, it can be 8een that the depth of flow is equal to the depth of flow section divided by cos e. In the case of steep channels, therefore, the two tei'ms should be used discriminately. The stage is the elevation or vertical distance of the free smfa-ce above 'a' datum. If the lowest point of the channel section is chosen as the datum, the stage is identical with the depth of flow. The top width T is the width of channel section at the free surface. / The wafer al'ea A is the cro~s-sectional area of the fi~w norlD,al to the / direction of flow. The wetted perimeter P is the length of the line of intersection of the channel wetted surface' with a cross-sectional plane normal to the direction bf flow. ' I Many typica.! seWer sections arEi described in [2J nnd [3].
OPEN CHANNELS AND THEIR PROPERTIES
23,\
The hydr.aulic radiu,$ R is the ratio of the water area to its wetted perimet.er, or. A R (2-1)
'1
The hydr~ulic de1Jth D is the ratio of ~le wa.ter area to the top width~ or D
=
T
(2-2)
The slJction factor for critical-flow contputtllion Z is the product of the . water area and the square root of the hydraulic depth, or
Z=A
(2-3)
.j
. The section factor for uniform-flow compulaUon A]z% is the product of the water area and the two-thirds power of the hydraulic radius. .
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I
.!
.
FIG. 2-1. Geometric elements of a. circular section.,
I
~lements
Tabie 2-1 furnishes 11 list of formulas for six basic geometric of' 1 seVf\11 . commonly' used channel sections, For a circular section, the curves in Fig. 2-1 represent the ratios of the geometric elements of the I s~ction to' the corresponding elements when t11e section is fl~wing full.] These curves are prepared from It table. given in Appendix,A. For cer- '. tain trapezoidal, triangular, and parabolic sections commonly found in practical uses, the diag.rams given in Appendix B provide a convenient Ii means of determining the geometric elements. . '
I
'{ [
·1
,\
\ ' \
i
--
24 )
), ,-
"
~',.#
, Example 2-1. Compute the'hydraulic radius, hydraulic dep~h, and section factor.?: Of the~ra.pezoidal chUlluel section in Fig. 2-2. The depth of flow is' 6 ft. ,
'
~~' , '
'
Q,?!/' ~ '
\
~~ .~' C~b=20':, .1' \ '~//l7~~0~
.
....&..
T:44"
P=46,8 -
25
OPEN CHANNELS ,AND 'TIIEIR PROPERTIES
BASIC PIIINCIPLES
,
1.5
t~.5'
,
.
'.
I
Tropezuidal chonnel
Triangular channel
,
FIG. 2-2. A channel crosS'section. $oluti{)ll. By formulas given in Table 2-1, the following are computed: P = 20 +' 2 X 6 ";5 = 46.8 ft; A = 0,.5(20 + 4'!L0 6 = 192.0 ft>; R = Hl2/46.8 = 4.10 ft; D = 19%4 = 4.37 ft; and Z = 192 -./4.37 = 401 ft,·,:
Sholtow ditch
.
NarroW' (@c!anglJlar 5~ction
a
),
\'I
2-4. Velocity Distribution in Channel Section. Owing to the pres~ ence of a free surface and to the friction along the Ghannel wall, the velocities in a channel are not uniformly distributed in the channel section. The measured maximum velocity in ordinary channels usually appears to occur below the free .sUlface at a distance of 0.05 to 0.25 of the depth;
Pipe Natural irregular channel
FIG. 2-4. Typical curvea of equal velocity in various channel sections.
FIG. 2-3. Velocity distrib,ution in'·a. rectangula; 'channeL
I r
r I, .
the closer to the banks, the deeper is the maximum. Figure 2-3 illustrates *e general pattern of veJocity distributio;l over \;al'ious vertical and horizontal sections of a rdctangular channel sectio~ and the curves of equal velocity in the cross section. The general patterns for velocity distribution in several channel sections of other shapes are illustrated in Ji]ig. 2-4. ; , I The velocity distributfoJ in a channel section depends also on other f('l.clors, such as the unusu~l shape of the section, tlle roughness of the
channel~ <md the pJ·esence of bends. In a broad, rapid, !111d, shallow stream,or in a very smooth channel, the maximum velocity may often be found l,Lt the h'ee surface. The ,roughness 6f the channel will cause the curvature of the vertical-velocity-distribution curve to increase (Fig. 2-5). On a bend the velocity increases greatly at the convex side, owing to the centrifug!11 action of the flow. CCfmtrary to the usual belief, a surface wind has very little effect i on velocity distribution. '/ As revealed by cv.refullaboratory investigations, ~ROugh' bed the flow in a straight prismatic channel is in fact three-dimensiqnal, manifesting a spiral motion, although the velocity component in the transvel:se 'FIG., ,2-5. Effect of chann~i section is usu!),lly small, and insignificant roughness on velocity compMecl with the longitudinal velocity com- distribiltion in an open , chann~I. ponents. Shukry [6] found that, in short labora. tory fl~mes, a small disturbance ~t the entrance, which is usually unavoidable, is sufficient to cau()e the zqne of highest water level ito shift to one side, thus giving rise to a single ~piral motion (Fig. 2-6). : In a long and uniforrri reach femotefl'om the entrance, a double spiral mqtion will oc~ur to per¢it equalization of shear stresses on both sides of th~ channel [7,8J.
26
OPEN CHAN)lELS AND THEllR PROPERTIES
BASIC PRINClPLES
The pattern will include. one spira}.on each side of the center line , where . th€ water level is the highest. In practical considerations, it is quite safe to ignore the :;piral motion in straight prismatic channels. Spiral flow in curved channels, however, is an important phenomenon to be considered in design add wHl be discussed later (Art. 16-2).
(cleonlcur Jines
'or
equal componenl lv'l
(o)Con!our lines 01 equal ve c for (v)
Co I Con lou, lines of equal componenHv,1
(d) ConI our lines 01 . equal component (vyl
(e) Dire clio n Jines and
Jr
ma9niludes of the IOlerol currenlslvor '
FIG. 2-6. Distribution of the velocity components, fa.cing downstream a.t the midsection of a. straight flume. Voloeitiestl.re In em/sec (= 0.0328 fps); y/b = 1.0; R "" 73,500; and Q = 701iters/s.ee (= 2.47 cfs). (Afler .A. ShlLkry [6J.)
2-5. Wide Open Channel. Observations in very wide open channels have shown that the velocity disttibution in the central region of the section is essentially the same as it ;would be in rectangular channel of infinite width. In other words, under this condition, the sides of the !iliJ!imel h~i practically DO influence on the velocit.y distribution in the C~gi~l, and. the flow i~_Y;~~.Eehtral region c~n therefore be regarded as two-dimensional in hydraulic ana:1yses. Careful experiments indicate, further, that this central region '~Xi~t8 in rectangular channels only when
a
~
I
27
the width. h,grea~.E 5 to 10 times. the depth f flow ...4~IJe~on the .2_'?~ion of l:lu):fac~ roughness. Thus, a wide Open channel can safely be defined as a rectangular channel whose width is greater than 10 tiri1es the dept.h of flow. For either experimental or anaJytical purposes, the flow in the cen tral region of a wide open channel may be' considered to be the same as the flow in !1 rectangular channel of infinite width. 2-6. Measurement of Velocity. According to the stream-gaging procedure of the U~S. Geological Survey,'lthe channei crosssecticH1 is-ciivid~d -~rti6a1 strips by a number of successive vert~cals, and mean velocities in verticals are determined by measuring t.he velocity at 0.6 of the depth J!-l_~Gh vertical, or, where morc reliable results are required, by taking the average of the velocities at 0.2 and 0.8 of the depth. When the stream i.s covered with ice, the mean velocity is no longer close t.Q 0.6 of the water dept.h, l;mUI!e average at 0.2 and 0.8 of the water depth still gives reliable resul.1§. The average of the ~ean velocities in any two adjacent verticals multiplied by the are:1- between the verticals gives the discharge through this vertical strip of the cross section.: ThG sum of discharges through all strips is t.he total discharge. The mean velocit.y of the who.Ie section is, therefore, eq ua! to the total discharge divided by the whole· area. It should be noted that the above methods are simple and approximate. For precise measurements more elaborate methods must be used, which are beyond the scope of this book. 2-7. Velocity-distribution Coefficients. As a result of nonuniform 9.istributiol1 of velocities over a channel section/the velocity hea.d of an open-channel flow is generally greater than the value computed according to the expression V 2 /2g, where V is the ri;lea.n velocity. When the,energy principJe is used in computation, the true velocity head may be expressed as a V 2/2g, where a is known as the energy coefficient or Co-riolis coe.fficient, in honor of G .. Coriolis [12J who first proposed it. Experimental data indicate that. the value of a varies from about 1.03 to 1.36 for fairly straight pri!:imR.tic channels. The value is generally higher for small channels and lower for large streams of considerable depth. The nOllUniform di.stribution of '/elocities also affects the co.rnputation of momen~umin open-channel flow. From the principle of mechanics, the "momentum of the fluid passing through a channel section per unit time is expressed· by {JwQVj(J, where f3 is known as·the mom.entum coefficient or Boussinesq coe:ffi.cient, after J. Boussinesq [13] who fii'st proposed it; w.is the unit weight of water; Q is the discharge; and V i.'3 the mean velocity. It is ,generally found that the value of (3 for fairly straight prismatic cha.nnels varies approximately from 1.01 to 1.12. The two velocity-distribution coefficients are always slightly larger than the limiting value of unity, a.t which the velocity distribution is 1
For details see
!9J
to [11 J.
.
_,. t:../
_
)
'I
!
) .
· ,
\
28
,
BASIC PRINCIPLES
OPEN CHANNELS AND THEIR PROPERTIES
strictly uniform across the channel section. For channels of regular cross section and fairly straight alignment, the effect of nonuniform velocity distrihution on the computed velocity head and momentum is small, especially in comparison with other uncertainties involved in ~he computation. Therefore, the coefficients are often assumed to be unity. In channels of complex cross section, the coefficients for energy and momentum Can easily be great as 1.6 and 1.2, respectively, and can vary quite rapidly from section to section in case of il'I'egular alignment. Upstream' from weirs, in the vicinity of obstructions, or near pronounced irregularities in alignment, values of a greater than 2.0 have been observed. 1 , Precise studies or analyses of flow in such channels will require measurement of the'actual velocity and accurate determination of the coefficients. In regard to the effect of channel slope, the coefficients are usually higher , in steep channels than in flat channels. For practical purposes, Kolupaila [16J proposed the values shown below for the velocity-distribution coefficients. Actual values of the coefficients for a number of channels may be found in {l7) and [18J.
as
I
"I
I!
Vo.lue of", '
Value of {j
Channels Min
Av
'Max
lVIin
Av
Ma.x '-"0-
Regular channels, flumes, spillway:;: .... , N o.tural streams and torrents ..... . . Rivers.under ice cover., ..... ....... River vlllleys" ove,flooded ..... ...... ~
r
I,
I r I
~
1.10 1.15 1.20 1.50
1.15 1. 30 1.50 1.75
1.20 1.50 2,00 2,00
1.03 1.05 L05 1.10 1.07 '1.17 L 17 1.25
1.07 1.17 1.33 1.33
2-8. Determination of Velocity-distribution Coefficients. Let LlA be an elementary area in the whole water area A, and w the unit weight of water; then the weight of water pasSing LlA per unit ,time with a velocity v is 'wv LlA. The kinetic energy of water passing AA per unit time is wv 3 AA/2g. This is equivalent to the product of the weight wu AA and the velocity head v2/2g. The total kinetic energy for the whole ,WB.ter area is equal to 2:wv ll t1Aj2g. Now, taking the whole area as A, the mean velocity as V, and the 1 A va.lue of", = 2.08 was computed by Lindquist [141 uslng da.~a from Wl}it' measurements made by Ernest W. Schoder and Kenneth B. Turner. In the case;of clolled conduits, much larger values of ,. have been obS,erved [15]. A value of '" = a.87, observed at the outlet section of a. dro.ft tube in the Rublevo power pmnt, is probably the largest known value obtained from actual measurements; the re",l value there must have been still lar!!~r-,-10,2% more, if the effect of a. IS" curvattire of the streamlines is taken into account. The largest known value from laboratory meO:suremen1;$ is believed to be '", = 7.4, which was derived by V. S. Kv:ia.tkovskii in 1940 in the VIGM (All-Union Institute for Hydraulic Machinery, U.S.S.R.) for the spiral flow under !l. model turbine wheeL
29
2
corrected velocity head for the whole area as a V /2g, the total kinetic energy 'is ,aw A/2g. Equating this quantity with 2:wv 3 AA/2g and reducing, J1]3 dA L;'~! LlA (2-4) V3A ,'"'"
va
The momentum of water passing AA, per unit time is the product of the mass W known as curvilinear flow. The effect of the curvature is to produce appreciable ELGceleration components or cell trifugal forces normal to the direction of flow. Thus, the pressure distribution over the section deviates from the h}'drostatic if ourvilinear flow occurs rn the vertical plane:' Such curviiinear flow may be either convex or concave (If;lg~nd c). In both cases the .nonlinear pressure distribution is represented by AB' instead of tlte straight distribution AB that would occur if the flow wel'e parallel. It is assumed that ail streamlines are horizontal at the section under consideration. In concave flow the centrifugal forces are pointing dDW:lward to reinforce the gravity action; so the resulting pressure is greater than the otherwise hydrostatic pressure of a parallel flow. In convex flow the centrifugal forces are acting upward against the gravity action; consequently, the resulting pressure is less than the otherwise hydrostatic pressure of a parallel flow. Similarly, when divergence of streamlines is great enough to developappl'eciable acceleration components normal to the flow; thehydrostati'c pressure distribution will be disturbed accQrdingly. Let the deviation from an otherwise hydrostatic pressure h. in a curvilinear flow be designated by c (Fig. 2-7b and c). Then the true pressure or the piezometric height h = h; + c. ' If the channel has a curved longitudinal profile, the approximate centrifugal pressure may be computed, by Newton's law of acceleration, ~~,~he E~~~~~ oL~L@iWt d and a croSQ..§ection. Qf 1 sq ft, that is, wd/g, and the centrifugal ac~ v2/r; or ' _----7
~.-~---
, ~l"tlow.
",/'
statements. Some authors have proposed the use of the mOluentUnl coeffic~en.tto , r'epJace the energy coefficient even in computations based on the energy PrJ?ClpJ~, " t ec't V'Nhether the energy coefficient or the momentum . , coeffiCient . . o1 ,IS d TI 1IS IS no carr . to be used depends on whether the energy or the moment?m prw::lple -:,e . The two coefficients are derived independently from baslca.lly different principles (Art. 3-6). Neither of them is wrong and neither ca.n be replaced by t~e other; both should be used in th~ correct sense. ' " ' I Specific qualifica.tions for parallel flow were clea:rly stated for the first tlme by Belanger [23]. '
=
wd~~ g
r
)
(2-8)
where w is the unit weight of water, g is the gravitational acceleration, v is the velocity of flow, and'r is the radius of curvature. The pl'esstirehead correction is, therefore, d v2 (2-9) c=-gr
IS Ill:r
;
\
.
p
~
.I
)
,
For computing the value ofc at the channel bottom, r is the radius of cu::',rature of the bottom, d is the depth of flow, and for practical purposes
J,
(
.
. i
32
"
BASIC PRINCIPLES
OPEN CHANNELS AND 'l'HEIR PROPERTIES
v m~y be ass~med equal to the average velocity of the flow. Apparently, c is positive for concave. flow, negative for convex flow, and. zero for parallel flow. ,. In parallel flow the pressure is hydrostatic, and the pressme head may be represe11ted by the depth of flow y. For simplicity, the pressure head of a curvilinear fiow may be represented by City, where a'is a correction ~oefficiimt for the curvature effect. The l:orrecti.on~oeffici~nt is referred to as apressure-dislribtttion coejfiy'ient.. Since this coefficient is applied to a pl'esswe head, it may be specifically SLa~'/m1~:!t!~~!1fficienl. It can be shown that the pressure coefficient is expressed by a'
/., i
I\ I
r.
~y}o
A
1
hI! dA
+
1
r
A
cv dA
sure head at any vertical depth is equal to this iiepth multiplied by a correction fact·or cos 2 e. . Apparently, if the angle (J is small, this factor will not differ apprecill.bly .fl'Om unity. In fact, the correction tends' to decrease the pressure head by an amount less than 1 % until e is nearly , 6" i a slope of about 1 in 10. Since the slope of ordinary channels is far less than 1 in 10; the correction foi: slope effect can usually be safely _ _-:i"" ignored .. However, when the cha~l_slope is large and its eff5l9t becoroe~ appreciable, thecorrection should be made if Mcumte comput~ttion is -'-~-.----.
-'
Pressure di.stribution
FrG. 2-8. Pressure tlistribution in parallel How in cha.nnels of lurge slope.
desired. A channel qf this type, say, with a slope gre~ter than 1 in 10,1.'.3 hereafter called a channel of large slope. Unless specifically mentioned, all chMlnels descrlbeanereafte-r are ~onsidered to be channels of small slope, where the slope effect is negligibie. If a channel of large slope ,has a. 10ngitudin!11 vertical profil£,; of appreciable. curvatul'e~ the pressure head should be cOl;rected fo]' the effect of the curvature of streamlines (Fig. 2-9). In simple notation, the pressure head may be expressed as g'y cos 2 Jt., """~~~~S== In channels of large slope the usu . andhighel' thl1n the critical velobity. When this velocity reaches a certain riHl.gni, "tude, the flowing water ,vill entrain nil', produ'cing a swell in its volume:i .and Ml increase in depth.l For this rel1son the pressUl'C computed uy Eq. {2-11) or (2-12) 4,~n shown in several gases to b.e higher than th~
h = Y cos 2, e h = d 90S e
'
.
'on. ~ertical section A'C
where d = ~ cos e, the depth measured perpendicularly from the water surface. It should be noted from geometry (Fig. 9-1) that Eq. (2-11) does not apply strictly to varied flow) piwticularly when 0 is v~ry large, whereas Eg. (2-12) still applies. Eq~lition ·(2-11) states that ,the presor--,
-
(2-10)
}o '
where Q is the total discharge andy is the depth of flow. It can easily be seen that a' is greater than 1.0 for concave flow, less than 1.0 for convex. flow, and equal to LO for parallel flow. . For complicated curved profiles, the total pressure distribution can be determined approximately by the fjow~net method or DlOre exactly by model testing. , In ra..mr!l.t.Y!1ried flow theghange in depth of fl.oJYJs so rapid and abl'Upt .:that th/2)l!kl; and 11 is the slope angle at the point (XI,l!I), varying from 0 at the bottom of the curve to 9, at the ends. The above equations will define' the. cross section when the flow is at its full dept.h. The slope angle at the ends of a hydrostatic catenary of best hydraulic efficiency is found mathema.tically to be II, = 35'37'7". (a) Plot this section with' a depth y = 10 ft, and (bl determine the values of A, R, D, and Z at the full depth . 2-7. Estimate the Ylllues of momentum coefficient (j for., the- given values of energy c(lefficient ex = 1.00, 1.50, and 2.op. , 2-8. Compute the energy and mo~entum coefficients of the cross st'ction shown in Fig. 2-3 (a) by Eqs. (2-4) and (2-5), and (b) by Eq~; (2-6) and (2-7). The cross section and the curves of equal velocity can be transferred to a piece of drawing paper and enlarged for deSired ll.ccuracy. 2-9. In designing side walls steep chutes and overflow spillways, prove that the overturning moment due to the pressure of the flowing water is equal to Yswy' cos' 9, wherew.is the unit weight of water, y is the vertical depth of the flowing water, and 9 is t,he slope angle of the channel. 2 ..10. Prove Eq. (2-10). 2-11. A high-head overflow spillway (Fig. 2-10) has a 60-ft-radius flip bucket u.t its downstream end. The bucket is not submerged, but acts to change the direction of the flow from the slope of the lipillway face to the horizontal and to discharge the flov1 into the air' between vertical training walls so ft apart. , At: a discharge of 55,100 ds, ;the water surface at the vertical section OB is at El. 8.52. 'Verify t.he curve that represents the computed hydraulic ,pressure acting on the training wall at section DB. The computatiQn is bailed an Eq,' (2-9) and on'the following assumptions: (1) the velqcity is uniformly di~tl'ibuted across the section; (2) the vo.lu,e used for r, fQr pressur~ values near the wall base, is 'equal to the radius of the bucket but, for other pre;isure values, is equal to the radius of the concentric flow lines; and (3) the flow is entto.ined with air, and the density ,of the air-water mixtureca~ be estimated by the
(,
"
, i
of
/a)
1 It is common practioe to show the cross section of a stream in a direction looking , downstream and to prepare the lQngitudinal profile qf a channel so that the wate~ flows from left to right, ;unless this arrangement would bit to show the feature to b~ illustrated by the cross'section and profile. This practice is generally fqllowed bt most 'engineering offices. However, for geographical reasons or in order to depict clearly the location and profile of a stream, the profile may be shown with water ftow~ ing from right to left and the cross section ma.y be shown looking upstream. This happens in ma.ny drawings pre'pared by the TennesseEj Valley Authority, because the Tennessee River and most of its tributaries flow from:east tQ west, and so are shown with the direction of flow from right to left on a, conventional map.
1
f
)
j )
.1
,.
! 36
BASIC PRINCIPLES
OPEN CHANNELS. AND THEIR PRO'PERT1ES
Douma. formula,' that is, 'U -
10
~0.2V: . gR
- 1
(2-15)
where u is the percentage of entrained .air by voiume, V is the velocity of flow, and
R Is the hydraulic radIus. . 2-12. Compute the wall pressure on the section OA (Fig. 2-10) of the spillway described in Prob. 2-11. that at section DB.
It is assumed that the depth of tlow section is the same!l.S
/
\«
$PilIW1!Y
Iraining wall,
eo II
cpO!1
;:;
C .2 ::l
.)
:OJ
t;j
i. 2
~ I
/, i
4
,a Un;l pressare, II 01 woler
\I
FIG, 2-10. Side-wall pressures on the flip bucket of a spillwa.y. 2-13. Compute the wall pressure on the section OA (Fig. 2-10) of the spillway descrIbed in Prob. 2-11 if the bucket is submerged with a tailwater level at EL 75.0. It is !l.SSulned that the pressure resultbg from the centritugal force or the submerged jet need not be considered beca.use the submergence will reault in a severe reduction in velocity.
REFERENCES 1. S. F. Averillnov: 0 gidravlicheskom raschete rusel krivolineinoI formy poperech,
2.
3. .
4.
nogo secheniia (Hydraulic design of channels with curvilinear form oithe crosS section), lzvestiia Akademii Nauk S.S.S.R., Otdelenie ~'ekhnic"eskfk;h Nauk" Moscow, no. 1, pp. 54-58, 1956. Leonard Metcalf and H. P. Eddy: "American Sewerage Pra.ctice," McGraw-Hm Book Company, 1M., New York, 3d ed., ,1935, vo!. 1. Harold E. Babbitt: "Sewerage and Sewage Treatment," John Wiley &: Sons, Inc., New York, 7th ed., 1952, pp. 60-:.66. H. M. Gibb: Curves for solving the hydrostatic oatenary, Engineering News, vol. 73, no .. 14, pp. 668-670, Apr, 8, 1915.
I This iormull!. [26J is based on da.ta obtained from actual· conorete and wooden chutes, involving errOnl of ±10%. '
37
5. George Higgins: "Water Channels," Crosby, Lockwood &: Son Ltd., London, 1927, pp.15-36. . . 6. Ahmed Shukry: Flow around bends in an open flume, Transactions, AmericilTl Society of Civil Engineers, vol. 115, pp. 751-779, 1950.· ' 7. A. II. Gibson: "Hydraulics and Its Applications,'" Constable &: Co., Ltd., London, 4th ed., 1934, p .. 332. , 8.J. R. Freeman: "Hydr·a.ulic Laboratory Practice," Amedcan Society of Mecha.nical , Engineers, New York, 1929, p. 70: ' 9. Don M. Corbett and ot.hers:8trealn-ga.ging procedure, U.S. Geologicnl SlI1vey, Water Supply Paper 888, 1943. 10. N. C. Grover and A. W. Harri'ngtoo.: "S.ream FlOW," John Wiley &; 80·ns, Inc.) New York, Hl43. 11. Standards for methods and records of· hydrologi~ measurements, United Natio7ls Economic Comm.isslcn for Asia: and the Fa:r Ei.I$~, Flood Control Series, No.6, Ba.ngkok, 1954, pp. 26-30. , 12. G. CorioUs: Sur.l'etablissemellt de Ill. formule qui donne la figure des remons, et .sIU· 12. ilorrection tiu'on doH y int,roduire POllr tenir compte des diffel'ences de vitesse dans les diVers points d'une marne section d'un COUl'ant (On the ba.ckwater-curve equation a.tid the corrections to be introduced to !lccount for the difference of the velocitie$ at different points on the same cross section), Ivnmoire No. 268, ..,l,n'nalca du punts et chaw;sees, vol. 11, ser. 1, pp. 314-335, 1836. 13. J. Boussinesq: Esg's'i sur la theorie des eaux courantes (On the theory of flowing waters), M~moire& ]fr/;sentes par diven savants ri l'Academie des Sciences, Paris, 1877. . . 14. Erik G. W. Lindquist: Discussion un Precise. weir measurements, by Ernesf W. Schader andT(ennethB. Turner, 1"7'(tllaac:l.ions, American Society of Civil Engineers, vol. 93, pp. 1163-1176, 1929. 15. N. M. Shcha.pov: H Gidrometriia Gidrotelchnicheskikh SoorllllheniI i Gicir,omashin" (" Hydrometry of Hydrv.lllic Structures and MacJ:Jnery ") I Gosenel'goizciat, . . Moscow, 1957, p. 88. 16. Stcponas Kolupaila: Methods of determin!l.tion of the kinetic energy facto!', The Port Engineer; Calcutta, India., vol. 5, no. I, pp. 12-18, Januo.ry, 1956. 17. M. P. O'Brien and G: H. Hickox: "Applied Fluid Mechallics," McGraw-Hill Book Company, Inc., New York, 1st ed., 1937, p'.272. ' 18, Horace William·King; i'Handbook of Hydraulics," 4th ed., l'evised by Ernest F. Brater, McGraw-Hill Book Company, Inc., New York, 1954, p. '7-12. 19. Morrough P. O'Brien and Joe W. Johnson: Velocity-head correction for hydrau1ia flow, Engineering News-Record, vol. 113, 0.0.7, pp. 214-216, Aug. 16, 1934. . 20. Th. P..ehbQck': Die Bestimmung der I,age der Energielinie bei ftiessenden Gewfulsern mit HilIe des GeschwindigkeitshOhen-Ausgleichwertes (The determina.tion of the position of the energy line in flowing water with the o.id of velocity-head a.djustment), Der Bau.ingenieuT, Berlin, vol.. 3, no. 15, pp. 453-455,·· Aug. 15, 11122. 21. Boris A. Bak&meteff: CorioIis and the energy principle in hydraulics, in "Theodore von !Urman Anniversary Volume," California. Institute of Teohnology, Pasadena, 1941, pp. 59-65. 22. W. S. Eisenlohr: Coefficient's for velocity distribution in open-Channel flow, Tra.nsac:I.ior/.$, American. Socie4/ of Civil Enginee7's, voL 11:0, pp. 633-644, 1945. Discussions, pp. 645-668. 23. J. B. Bela.nger: "Essai sur la solution numeriqne de ql,lelques problemes relatifs au mou.-ement permanent des eaux courantes" ("Essa.y on tIle Numerica.l Solution of Some Problems Relative to Steady Flow of Wa.ter"), Carilian-Goeury, Paris, 1828, pp. 10-24.
\
,
38
BASIC PRINCIPI..ES
24. It Ehrenberger: Versuche Iiber die Verteilung der Drucke an Wehrriicken infolge des I1bsturzcnden '.Vassers (Experiments on the distribution 'of pressuresa\ong the f~~e of w(d ..;; resulting from the impact of the fa.lling water), Die W IMJserwirtschaft, Vienna, vol. 22, no. 5, pp. 65-72, 1929. 25. 'H&rald Lauffer: Druck, Energie und Fliesszustand in Gerinnen mit grossem Gefiille (Pressure, energy, and flow type in channels with high gradients), Wasserkrafl, und Wasserwirtschaft, Munich, vol. 30, no. 7, pp. 78--82, 1935. 26. J. H. Douma: Discussion on Open channel flow at high velocities, by L. Standish Hall, in Entra.inment of atr in flowing water: a symposium,T1'ansactions, American Society of Civil Engineers, vol. 108, pp. 1462-1473, 1943.
LeI!.'foI.l-
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ENERGY AND MOMENTUM PRINCIPLES
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CHAPTER
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3-1. Energy in Open-channel Flow. It is known in elementary hydraulics that the total energy in foot-pounds per ponild of water in any streamline passing through 9, channel section may be expressed as the total head in feet of water, which is equal to the sum of the elevation
J rret ,/·1/0 fall . ..vV ,he. ellA-of /:J~"m ;$ dtre,·r/6..,;tLul ·'n'YII!'. .s~t:rt, Vr';,t.. cd cbf #.. is /l. J'Vi4 v..u ¢ ,pver.( e @'we l~o.!ll.p~nsating external en~,ls sup~lie
having a prolonged reversed curve of water surface; t.his.phcnomenoll may ~Clllled ~ grad'unl hud7'a1t~ic drop and is no longer a local phenomenon~ Hydraulic Jump. When the rapid change in the depth of fiow is from a low stage to a. high stage, the result is usually an abl'llPt rise of water ~e. (Fig;. 3-4, in which the vertjc!l,l scale is exaggerated). This local' phenomenon is known a~J.he hydraulic j1tmp. It occurs frequently in a canal below a regulating sluice, at, the foot of a spillway, or at the place where a steep chann!)l slope suddenly turns jlat.' If thejump is low, that is, if the change in depth is small, the water will not rise obviOtisly and abruptly but will pass from the low to the high stage through a series of undulations gradually diminishing in size. Such a low jump is called an uooular jtimp. . When the jump is high, that is, when, the change in depth is great, the jump is called a direct jump. The direct jump involves a relatively large amount of energy loss thro,ugh dissipation in the turbulent body of water in the jump. Consequently, the energy content in the flow after the jump is appreciably less than that before the jump. . It may be noted that the depth before the jump is always less than the
46
BASIC PRINCIPLES
depth after the jump. The depth before the jump is called the initial .depth y 1 and that after the jump is called .the .sequent depOt Y2; The initial and sequent depths VI and Y2 are shown on the specific-energy curve (Fig. 3-4). They· should he distinguished from the altern!J.te. A~p~hs YI and Y2'! whi(J~~e the two nossible depth~ fo!:....the same specific ener~l' _.~~itial and s~lent depths are actual depths before and after a jump in which ~rgy lOss b.E is invoh-ed. In other words, the specific -energy E 1 at the initial depth Vl is greater tha.n the specific energy at the sequent depth .y~ by an amount equal to the energy loss AE. II there were 110 energy losses, the initial and sequent depths would become identical with the alternate depths in a prismatic channel. 3-5. Energy in NQnprismatic Channels. In preceding discussions the channel has been assumed prismatic so t.hat one specific-el~ergy curve could be applied to evil sections of the channel. For non prismatic channels, however, the channel section varies along the length of the channel and. hence, the specific-energy curve differs from section to section. This cbm'plication can be seen in a three-dimensional plot of the energy curves along the given reach of a nonpl'ismatic channel. For demonstrative purposes, a nOllprismatic channel with variable slope is taken as an example, in which agradually varied flow is carried from a, stibcritical state to a supercritical state: (Fig. 3-5) .. The vertical profile of the channel along its center line is plotted on the Hx plane with the x axis chosen as the datum. For a variable-.slope channel, it is more convenient to plot the total energy head H = z + y + V Z/2g, instead of the specific energy, against the depth of flow on the By plane. For simplicity, the pressure correction due to the slope a.ngle and curvature of flow is ignored in this discussion. An energy line is then plotted on the Hx plane below a line parallel to the x axis and passing through the initial total head at the H axis. The exact position of the energy line depends on the energy losses along the channel. Four channel sections are then selected and four energy curves for these sections are plotted in the Hy planes ~ shown. The initial section 0 is an upstream section in the sUbcritical-flow region. The two depths corresponding to a given total energy H 0 can be obtained from the energy curve. Shice this section is in the subCl'itical-flow region, the high stage yo should be the actual depth of flow, whereas the low stage is the alternate depth. Similarly, the alternate depths in other sections can be obta,ined. In the downstream sections rand 2, the low stages Yl and Y2 are the actual depths of flow singe they are in the supercritical-flow region. The critical depth at each se ytion can also be obtained from the energy curve at the point of minimum energy. At &ection C the critical flow occurs, and the depth y, is the critical depth. On the H x plane, varioUs lines can finally be' plotted; showing the channel bottom, water surface, critical-depth line, and
ENERGY AND MOMENTUM PRINCIPLES
47
alternate-depth line. At the critical section, it is noted that the three lines, namely, the .yater surface, the critical-depth line, and the alternate-depth line, intersect at a single point. It is seen' tho.t, in passing through the critical Reotion, t,he water surface entei'S the supercriticalflow region smoothly. The· three-dimensional plot of energy curves is complicated. The description given here is used only for helping the reader to visualize the problem. In actual applications, the energy C'lrves may be constructed
1 \
II
I I i I !
I
I
! II
X
FIG'. 3-5. Energy in a non prismatic channel of variable-slope, carrying gradually va.ried flow from sllbcritical to supen::ritical state.
separately on a number of tWo-dimensional Hy planes for the chosen sections. The data obtained from these curves are then used to plot the water surface, critical-depth line, and alternate-depth line on a twodimensional Ex plane. For simple channels, the energy curves are not necessary because thi;l critical depth and alternate depths C3.n easily be computed directly, Eumple3-L ·A rectangular cha.nnell0 ft wide is narrowed down to 8 ft by a contrMtion 50 ft long, built of straight wa,lls and a horizontal fiooc. If the discharge is . 100 cfB and the depth of Bow is 5 ft On the upstream side of the transition section, determine the flow-surface profile in the contra.ction (0.) allowing no gradual hydraulic drop in the contraction, Ilnd(b) a.llowing a gradual ,hydraulic drop ha.ving its point of inflection a.t the mid-sectioll of the contra.ction. Th!Jrip.tionalloss through the contraction is negligible. .
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48
BASIC PRINCIPLES
ENERGY AND MOMENTUM PRINCIPLES
. , . Sol1ttiDn. From the given data; the total'energy in the approaching flow meas~red above the channel bottom i~ u: = 5 {100/(5 ?< 1O)1'/2g = 5.062 ft. This energy is kept constant throughout l,lIC contractIon, since energy losses are negligible. A horizontal energy line showing the elevation'of the tobl head is, therefore, drawn on the channel· profile (Fig. 3-6). . "
49
., critical depth at t}Jis section is equal to the total head divided by 1.5 (Prob. 3-3), or 5.062/1.5 = 3.375 ft. By :r::q, (3-10), the critical velocity ts'eqllnl to V, = V3.375g = 10.4.5 fps. Hence, the width of this' critical section should be 100/(10.45 X 3.38) = 2.83 fi. ' With the size of the mid-section determined, the side walls of the contraction can bo drawn in ,with straight lines. The lm~ and high stages at each section are then computed by the equation previously'given, As the flow upstream from the critical section is subcritical, lts water surface should follow the high stage. Downstream from the critical secti6n, the flow is sllpercritical and its Burface profile 'follows the low-stage line. The criti.::al-depth line is shown to sepai'ate the high from the low stage or the subCl'itical from the stipercritical region of flow. On the basis of Eq. (3-10), the critica.l depth can be computed from the equation
+
(lOO/by,),
2g
y,
=
'2
. y = {/lO,OOO , "gb%
or
where' b is the width of the channel, which can be measured from the plan, It should be noted :that the vertical scale of the channel profile is greatly exaggerated. Furthermore, the outline of the gradual hydraulic drop is only theoretical, based on the theory of parallel flow. In reality, the flow near the drop is more or less curvilinear, and the .actual profile would deviate from th/'! theoretical one.. . This example also serves to demonstrate 11 method of designing a channel transition (Arts. ll-5 to 11-7), The designer may fit any type of contraction walls he desires to suit a given flow profile, or vice versa, .
3-6. Momentum in Open-channel Flow. As stated earlier (Art. 2-7), the momentum of. thefiow passing a channel section per 'unit time is expressed by pwQV /y, where p is the momentum coefficient, w is the l.mit and V is the mean weight of water in lb/ft i , Q is the discharge in velocity in fps, ,-----?'?>- According to Newton's second law of motion, the change of momentum per unit of time in the body of water in a flowing channel is equal to the . resultant of all the external forces that are acting on the body,Applying this principletO channel of large slope (Fig. 3-7), the fplfo~~ing expl:ession. for the momentum change per unit tirnein;the body of water enclosed betv.:een sections 1 and 2 may be written:
FIG. 3-6. Energy principle applied to a channel contraction (a) without gr:adual hydraulic drop; (b) with gradual hydraulic .(hop. '
cfs,
The alternate depths for the given tot ..l energy. e..n be computed by Eq. (3-9) as follows: . ' " 100' 5.062 = Y + 2g(by)' or
. , - 5 06'> " y . ~Y
+ 155.25 b'.
=
a
0
This is a cubic equtl.tion in which b is the width of the channel. At the entrance sec-' tion, where b = 10 ft, its s61ution gives two positive roots: a low stage 'YI = 0.589 ft, which is the altemate depth; and ahigh stage Y2 := 5.00 ft, which is the depth of flow. At the exit section ,where 11 = 8 ft, this. equation gives a low ati\ge YI =. 0.750 ftiand a high stage Y. = 4.964 ft. ! When no gr~dual hydraulic drop is allowed in'the contraction (Fig. 3-6a), the1depth of flow at the exit section.should be kept at the high stage, as shown, The high stages for other interm£ldiate sections are then compute~ by the above equation, whicli giveB the flow-surface ptofile. Similarly, the low stages are computed by the aboye procedure and indicated by the alternate-depth line~ ; When a grodua:~ hydrauiic drop is desired in the contraction (Fig. 3-6b), theldepth of flow at the exit ~ection should be at the low stage, Since the point of inAection of the drop or 11. critipal section is maintained at th~ mid-section of the c~ntracti