Key words Non-uniform; specific energy; critical; gradually-varied steady flow; water surface profiles

Transcription

1 Chaptr NON-UNIFORM FLOW 4.. Itroductio 4.. Gradually-varid stady 4.3. Typs of watr surfac profils 4.4. Drawig watr surfac profils Summary Likig up with Chaptr, dalig with uiform i op chals, it may b otd that ay chag i th phomo (i.. rat, vlocity, dpth, ara, bd slop do ot rmai costat) causs th to b o-uiform. This chaptr will discuss th ffct of chag i ay o of th abov quatitis, icludig spcific rgy, critical dpth ad slop, ad typs. How to draw watr surfac profils will also b itroducd. Ky words No-uiform; spcific rgy; critical; gradually-varid stady ; watr surfac profils 4.. INTRODUCTION 4... Gral I th prvious Chaptr, th was uiform udr all circumstacs udr cosidratio. I may situatios th i a op chal is ot of uiform dpth alog thal. I this chaptr th coditios studid rlat to stady, but o-uiform,. This typ of is cratd by, amog othr thigs, th followig major causs: Chags i thal cross-sctio. Chags i thal slop. Crtai obstructios, such as dams or gats, i th stram s path. Chags i th discharg such as i a rivr, whr tributaris tr th mai stram. A o-uiform is charactrizd by a varid dpth ad a varid ma vlocity: V h 0 or 0 (4-) s s If th bottom slop ad th rgy li slop ar ot qual, th dpth will vary alog thal, ithr icrasig or dcrasig i th dirctio. Physically, th diffrc btw tompot of wight ad th shar forcs i th dirctio of producs a chag i th fluid momtum which rquirs a chag i vlocity ad, from cotiuity cosidratios, a chag i dpth. Whthr th dpth icrass or dcrass dpds o various paramtrs of th, with may typs of surfac profil cofiguratios possibl. Fig. 4.. illustrats som typical logitudial fr-surfac profils. Upstram ad dowstram cotrols ca iduc various pattrs. I som cass, a hydraulic jump might tak plac. A jump is a rapid-varid phomo; calculatios wr dvlopd i Chaptr 3. Howvr, it is also a cotrol sctio ad it affcts th fr surfac profils upstram ad dowstram. Chaptr 4: NON-UNIFORM FLOW 70

2 upstram cotrol sluic gat cotrol hydraulic jump dowstram cotrol sharp-crstd wir rapid varid gradually varid rapid varid gradually varid rapid varid upstram cotrol critical dpth suprcritical cotrol hydraulic jump subcritical dowstram cotrol ovr (critical dpth) rapid varid gradually varid rapid varid gradually varid rapid varid 4... Acclratd ad Rtardd Fig. 4.. Exampls of o-uiform A idalizd sctio of a rach of a chal with acclratd ad rtardd coditios is show i Fig. 4.a ad Fig. 4.b, rspctivly. As acclrats, with th rat of costat, th dpth h must dcras form poit to poit, ad a watr surfac profil as show i Fig. 4.a rsults. Rtardd will produc watr surfac profils as show i Fig. 4.b. Sigificat i ach o of th abov cass is th fact that ow th watr surfac is a curvd li ad ot logr paralll to thal bottom ad th rgy li, as was tas for uiform. Th followig poits ar mad i coctio with th abov obsrvatios. Chaptr 4: NON-UNIFORM FLOW 7

3 Th watr surfac, as will b show latr, ca hav a cocav or a covx shap. Th rgy li is ot cssarily a straight li; howvr, it is assumd that th rgy gradit is costat alog th lgth of a rach ad th rgy li will b rprstd ad cosidrd to hav a slop i = H L /L. As was do i tas of uiform, it is hr also accptd that th dpth of, h, is qual to th prssur had i th rgy quatio. Obviously, this applis oly wh th slop of thal bottom is small. For vry stp slops, allowacs for this discrpacy must b mad. H L g watr surfac V g V p h rgy-had li hydraulic grad li z L z i p h Fig. 4.a. Acclratd datum rgy-had li V g p h watr surfac hydraulic grad li i H L V g p h z L z Fig. 4.b. Rtardd datum Chaptr 4: NON-UNIFORM FLOW 7

4 4..3. Equatio of o-uiform i V g h watr surfac h (V+dV) g h+dh z b i b dl Fig No-uiform Cosidr a o-uiform i a op chal btw sctio - ad sctio -, i which th watr surfac has a risig trd (i.. th rgy-had gradit is lss tha th bd slop) as show i Fig Lt V = vlocity of watr at sctio -; h = dpth of watr at sctio -; V+dV = vlocity of watr at sctio -; h+dh = dpth of watr at sctio -; i b = slop of chal bd; i = slop of th rgy grad li; dl = distac btw sctio - ad sctio -; b Q z b = avrag width of thal, = discharg through thal, = chag of bottom lvatio btw sctio - ad sctio -, ad h = H L, chag of rgy grad li btw sctio - ad sctio -. Sic th dpth of watr at sctio - is largr tha at sctio -, th vlocity of watr at sctio - will b smallr tha that at sctio -. Applyig Broulli s quatio at sctio - ad sctio -: V (V dv) zb h (h dh) h (4-) g g V V dv i b.dl h h dh i.dl (4-3) g g V.dV i.dl dh i.dl, glctig g b i or b (dv) g (small of scod ordr) (4-4) dh V.dV i dl g.dl (dividig by dl) (4-5) Chaptr 4: NON-UNIFORM FLOW 73

5 dh V.dV ib i (4-6) dl g.dl W kow that th quatity of watr ig pr uit width is costat, thrfor or q = V.h = costat (4-7) dq 0 dl (4-8) d(vh) 0 (4-9) dl Diffrtiatig th abov quatio (tratig both V ad h as variabls), V.dh h.dv 0 (4-0) dl dl dv V dh (4-) dl h dl Substitutig th abov valu of dv dl dh V dh ib i Eq. (4-6), yilds i (4-) dl gh dl dh V ib i dl gh dh i i dl V gh b (4-3) (4-4) Nots: Th abov rlatio givs th slop of th watr surfac with rspct to th bottom of thal. Or i othr words, it givs th variatio of watr dpth with rspct to th distac alog th bottom of thal. Th valu of dh/dl (i.. zro, positiv or gativ) givs th followig importat iformatio: If dh/dl is qual to zro, it idicats that th slop of th watr surfac is qual to th bottom slop. Or i othr words, th watr surfac is paralll to thal bd. If dh/dl is positiv, it idicats that th watr surfac riss i th dirctio of. Th profil of watr, so obtaid, is calld backwatr curv. If dh/dl is gativ, it idicats that th watr surfac falls i th dirctio of. Th profil of watr, so obtaid, is calld dowward curv. Chaptr 4: NON-UNIFORM FLOW 74

6 Exampl 4.: A rctagular chal, 0 m wid ad havig a bd slop of 0.006, is dischargig watr with a vlocity of.5 m/s. Th is rgulatd i such a way that th slop of th watr rgy gradit is Fid th rat at which th dpth of watr will b chagig at a poit whr th watr is ig m dp. Solutio: Giv: width of thal: b = 0 m bd slop: i b = Lt dh dl vlocity of watr: V =.5 m/s slop of rgy li: i = dpth of watr: h = m b th rat of chag of watr dpth. Usig quatio i (4-4): dh dl ib i V gh = As. 4.. GRADUALLY-VARIED STEADY FLOW 4... Backwatr calculatio cocpt Gradually varid is a stady, o-uiform i which th dpth variatio i th dirctio of motio is gradual ough to cosidr th trasvrs prssur distributio as big hydrostatic. This allows th to b tratd as o-dimsioal with o trasvrs prssur gradits othr tha thos du to gravity. For subcritical s th situatio is cotrolld by th dowstram coditios. A dowstram hydraulic structur (.g. bridg pirs, gat) will icras th upstram dpth ad crat a backwatr ffct. This cocpt has b itroducd shortly i sctio Th trm backwatr calculatio rfrs mor grally to talculatio of th logitudial fr-surfac profil for both subcritical ad suprcritical s. Th backwatr calculatio is dvlopd assumig: a o-uiform a stady that th is gradually varid, ad that, at a giv sctio, th rsistac is th sam as for a uiform with th sam dpth ad discharg, rgardlss of trds of th dpth Equatio of gradually-varid I additio to th basic gradually-varid assumptio, w furthr assum that th occurs i a prismatic chal, or o that is approximatly so, ad that th slop of th rgy grad li ca b valuatd from uiform formulas with uiform rsistac cofficits, usig th local dpth as though th wr locally uiform. Rfrrig to Fig. 4.4., th total rgy had at ay cross-sctio is Chaptr 4: NON-UNIFORM FLOW 75

7 V H z h (4-5) g i which z = chal bd lvatio; h = watr dpth, = kitic-rgy corrctio cofficit as itroducd i Chaptr, ad V = ma vlocity. V g slop of rgy grad li, i dh H h bd bd slop i b z dx datum Fig 4.4. Dfiitio sktch for gradually-varid If this xprssio for H is diffrtiatd with rspct to x, toordiat i th dirctio, th followig quatio is obtaid: dh de V i i b with E h (4-6) dx dx g i which i is dfid as th slop of th rgy grad li; i b is th bd slop (= - dz/dx); ad E is th spcific-rgy had (i.. th rgy had with rspct to th bottom). Solvig for de/dx givs th first form of th quatio of gradually varid : de i b i (4-7) dx It appars from this quatio that th spcific-rgy had ca ithr icras or dcras i th dowstram dirctio, dpdig o th rlativ magituds of th bd slop ad th slop of th rgy grad li. Y (973) showd that, i th gral cas, i is ot th sam as th frictio slop i f (= 0 /R, this quatio will b itroducd agai i Chaptr 7) or th rgy dissipatio gradit. Nthrlss, w hav o bttr way of valuatig this slop tha applyig uiform- formulas such as thos of Maig or Chzy. It is icorrct, howvr, to mix th frictio slop, whiclarly coms from a momtum aalysis, with trms ivolvig, th kitic-rgy corrctio (Marti ad Wiggrt, 975). Not: Th bd slop i ad th frictio slop i f ar dfid as: z H o i = si ta ad if x x R rspctivly, whr H is th ma total rgy-had, z is th bd lvatio, is thal slop ad o is th bottom shar strss. Chaptr 4: NON-UNIFORM FLOW 76

8 Th scod form of th quatio of gradually-varid ca b drivd if it is rcogizd de de dh de that ad that, applyig quatio (4-), Fr dx dh dx dh, providd that th Froud umbr is proprly dfid. Th, quatio (4-7) bcoms: dh i i dx Fr b (4-8) Th dfiitio of th Froud umbr i quatio (4-8) dpds o thal gomtry. For a compoud chal, it should b tompoud-chal Froud-umbr, whil for a rgular, prismatic chal, i which d/dh is gligibl, it assums tovtioal rgy dfiitio giv by Q B/gA 3. Th ratio dh/dx i Eq. (4-8) rprsts th slop or th tagt to th watr surfac at ay poit alog thal. This rlatioship thrfor idicats whthr at ay poit alog thal th watr surfac is risig (backwatr coditio) or droppig (drawdow coditio). Immdiatly th followig dductios ca b mad: Wh dh 0 dx, th slop of th watr surfac is droppig i th dowstram dirctio ad th dpth dcrass dowstram. Wh dh 0 dx, th slop of watr surfac is paralll to thal bottom ad uiform xists. This ca b radily s from Eq. (4-8) sic, for this coditio, i b = i must qual zro. Wh dh 0 dx, th slop of watr surfac riss i th dowstram dirctio ad th dpth h icrass dowstram. Wh dh dx, which rquirs that Fr = 0 or Fr =, th slop of th watr surfac must thortically b vrtical. This occurs wh th chags from subcritical to suprcritical, or vic vrsa, as idicatd by th valu of th Froud umbr. Th formulas drivd do ot actually apply ay logr du to th assumptios mad. A vrtical watr surfac also dos ot occur i rality; howvr, a vry oticabl chag i th watr surfac taks plac. This is spcially so wh th chags from blow to abov. I such istac a phomo kow as th hydraulic jump occurs TYPES OF WATER SURFACE PROFILES Classificatio of profils From th forgoig, it is vidt that th rlatioship xprssd i Eq. (4-8) provids a cosidrabl amout of iformatio as to th shap of th watr surfac profil i a op chal. Ivstigatio of this formula yilds th followig rsults: Chaptr 4: NON-UNIFORM FLOW 77

9 . Th rlatioship btw th slop of thal bottom ad th slop of th rgy grad li dtrmis whthr th umrator of th quatio is positiv or gativ.. Th domiator of th quatio is positiv if Fr <.0 ad vic vrsa. I othr words, if th is subcritical (Fr smallr tha ) th domiator is positiv, ad if th is suprcritical (Fr gratr tha ) th domiator is gativ. Toditios at which i a op chal ca tak plac ad th possibl rlatioships btw th obsrvd dpth h o, th ormal dpth at which is uiform h, ad tritical dpth ar illustratd i Fig It is vidt from this figur that thr ar thr zos of chal dpths at which ca b obsrvd: Zo, with h o gratr tha h ad (i.. h o > h > ) Zo, with h o btw h ad (i.. h > h o > ) Zo 3, with h o lss tha h ad (i.. h > > h o ) h o h h h o h h o h o > h > h > h o > h > > h o Fig.4.5. Thr zos of chal dpths Th rlativ bottom slop dfis whthr uiform is subcritical or suprcritical. Dtrmi th associatd Froud-umbr Fr. Fr V R V gh h gr whr R is th hydraulic radius of th op chal. Subcript dots th quilibrium. Th bottom/wall shar strss is dfid as: c V gr i o f f V i i gr c c Fr f b (th frictio slop i f = th bd slop i b ) f f R i b h cf A P.R W hav: A = B.h h, whr P is th quilibrium wttd primtr. B B Chaptr 4: NON-UNIFORM FLOW 78

10 Fr B i b P cf I cas of turbult :. R B For two-dimsioal : h P So, a propr approximatio for Fr is: Fr i b cf If i b < c f, w hav a mild slop (M typ) Th uiform is subcritical: Fr <, h >. If i b > c f, w hav a stp slop (S typ) Th uiform is suprcritical: Fr >, h <. If i b = c f, w hav a critical slop (C typ) Fr =, h = Not: It ca asily b drivd that is Maig s. c g 3 gr, whr C is Chzy cofficit ad C f Two coditioal chal bottom coditios or slops xist. Ths do ot rally costitut op chal, but gravity ca tak plac alog thm. Thy ar as follows: If i b < 0, w hav a advrs slop (A typ) If i b = 0, w hav a horizotal slop (H typ) It should b oticd that h = h. Not: Th actual dpds o th boudary coditio, i.. mild, stp, tc. dos ot tll us aythig about th actual Sktchig profils I thory, for ach of th fiv slop dscriptios abov thr ar thr zos i which ca b obsrvd. It follows th that a total of 5 thortical watr surfac profils ar possibl, prstd i Tabl 4.. Ths profils, togthr with illustratios of practical applicatios, ar show i Fig Whil this figur is for th most part slf-xplaatory, th followig obsrvatios ad xplaatios ar prstd for furthr clarificatio. Mild slop (i b < c f ). Th M curv is grally vry log ad asymptotic to th horizotal ad th li rprstig h o. Th M- ad M3-curvs d i a sudd drop through th li rprstig ad a hydraulic jump, rspctivly. Critical slop (i b = c f ). Sic = h i this cas, thr is o zo, ad oly two watr surfac profils xist, C ad C3. Th C-curv coicids with th watr surfac that corrspods to uiform at critical dpth. Chaptr 4: NON-UNIFORM FLOW 79

11 Stp slop (i b > c f ). All curvs ar rlativly short. S is asymptotic to th horizotal, whras S ad S3 approach h o. Horizotal slop ad Advrs slop chals. I this cas, h is ifiitly larg ad uiform caot tak plac. Hc thr ar o H- or A-profils. Chal slop Zo Tabl 4.. Typs of profils i prismatic chals Dsigatio Zo Zo 3 Zo Rlatio of h o to h ad Zo Zo 3 Gral typ of curv Typ of Mild M h o > h > Backwatr Subcritical Fr <, M h > h o > Drawdow Subcritical h > M3 h > > h o Backwatr Suprcritical C h o > = h Backwatr Subcritical Critical Fr =, C = h o = h Paralll to chal bottom Uiform critical h = C3 = h > h o Backwatr Suprcritical Stp S h o > > h Backwatr Subcritical Fr >, S > h o > h Drawdow Suprcritical h < S3 > h > h o Backwatr Suprcritical Horizotal i b = 0 Advrs i b < 0 No h o > h > No No H h > h o > Drawdow Subcritical H3 h > > h o Backwatr Suprcritical No h o > (h * ) > No No A (h * ) > h o > Drawdow Subcritical h * i parthss is assumd a possitiv valu. A3 (h * ) > > h o Backwatr Suprcritical S Fig. 4.6.a 4.6.b 4.6.c 4.6.d 4.6. M horizotal M M h M M sctio of largmt M Mild slop M3 CDL M3 M3 CDL = critical-dpth li; Fig.4.6.a. Mild slop (0 < i b < c f ) ad xampls of profils = ormal-dpth li Chaptr 4: NON-UNIFORM FLOW 80

12 horizotal C CDL = C = h C3 critical slop C3 Fig.4.6.b. Critical slop (i b = c f > 0) ad xampls of horizotal S S h S CDL S stp slop S3 sctio of largmt S S S3 Fig.4.6.c. Stp slop (i b > c f > 0) ad xampls of profils S3 horizotal H H H3 CDL H3 horizotal slop Fig.4.6.d. Horizotal slop (i b = 0) ad xampls of profils A A CDL A3 A3 advrs slop Fig Advrs slop (i b < 0) ad xampls of Chaptr 4: NON-UNIFORM FLOW 8

13 Prismatic chal with a chag i slop This chal is quivalt to a pair of coctd prismatic chals of th sam cross sctio but with diffrt slop. Svral typical pattrs alog a prismatic chal with a brak or discotiuity i slop ar show i Fig mild M h h mildr (vry log) M M CDL S S3 h stp (vry log) h S stpr CDL M rsrvoir h mild S CDL H S CDL stp h stp rsrvoir H h M mild (vry log) CDL fr ovrfall rsrvoir H h M mild (short) tailwatr CDL Fig Flow profils with a chag i slop Th profils i Fig. 4.7 ar slf-xplaatory. Howvr, som spcial faturs should b mtiod: Th profil ar or at tritical dptaot b prdictd prcisly by th thory of gradually varid, sic th is grally rapidly varid thr. I passig a critical li, th profil should, thortically, hav a vrtical slop. Sic th is usually rapidly varid wh passig tritical li, th actual slop of th profil caot b prdictd prcisly by th thory. For th sam raso, tritical dpth may ot occur xactly abov th brak of thal bottom ad may b diffrt from th dpth show i th figur. Chaptr 4: NON-UNIFORM FLOW 8

14 Composit profils with various cotrols Chals with a umbr of cotrols will hav profils that ca b composd from th diffrt typs of profils prstd i th prvious sctio. Th ability to sktch tomposit profils is i may cass cssary for udrstadig th i th chal or for dtrmiig th discharg. I all cass it is cssary to idtify firstly th cotrols opratig i thal, ad th to trac th profils upstram ad dowstram of ths cotrols. Two simpl cass ar show i Fig. 4.8; i th first cas th slop is mild, i th scod cas stp. Turvs for th mild-slop situatio ar slf-xplaatory, sic thy icorporat may of th faturs alrady discussd. For th stp-slop situatio w hav alrady s that tritical must occur at th had of th slop i.. at th out from th rsrvoir; thraftr thr must b a S-curv tdig towards th uiform-dpth li. Thr must b a S-curv bhid th gat, ad th trasitio from th S- to th S- curv must b via a hydraulic jump. Dowstram of th sluic gat, th will td to th uiform coditio via a S- or S3-curv; thc it procds ovr th fall at th d of th slop. I this cas thr is othig that impls th to sk tritical coditio. I Fig. 4.8 two profils ar draw i dashd lis abov th M3- ad th S-curv. Ths ar loci of dpths cojugatd to torrspodig dpths o th udrlyig ral surfac profils, ad ar thrfor kow as cojugat curvs. Obviously a hydraulic jump will occur whr such a curv itrscts th ral (subcritical) surfac profil dowstram; th cojugat curv thrfor provids a covit mas of dtrmiig th locatio of a hydraulic jump. M cojugat curv M rsrvoir mild slop h M3 jump ovrfall cojugat curv S jump rsrvoir S S3 ovrfall stp slop h Fig Exampls of composit logitudial profils Chaptr 4: NON-UNIFORM FLOW 83

15 4.4. DRAWING WATER SURFACE PROFILES Dirct-stp mthod Tomputatio of a profil by a stp mthod cosists of dividig thal ito short rachs ad dtrmiig rach by rach thag i dpth for a giv lgth of a rach. I pricipl, th dirct-stp mthod could b applid to ithr Eq.(4-7) or Eq. )4-8), but usually is associatd with th formr. Eq. (4-7) is put ito fiit-diffrc form by approximatig th drivativ de/dx with a forward diffrc ad by takig th ma valu of th slop of th rgy grad li ovr th stp siz x = (x i+ x i ) i which th distac x ad th subscript i icras i th dowstram dirctio. Th rsult is: Ei Ei xi xi (4-9) i i b whr i is th arithmtic ma slop of th rgy grad li btw sctio i ad sctio i +, with th slop valuatd idividually from Maig s quatio at across sctio. Th variabls E i+, E i ad i o th right had sid of Eq. (4-9) all ar fuctios of th dpth h. Th solutio procds i a stpwis fashio i x by assumig valus of dpth h ad thrfor valus of th spcific-rgy had, E. As Eq. (4-9) is writt, x icrass i th dowstram dirctio. I gral, upstram computatios utiliz Eq. (4-9) multiplid by ( ), so that turrt valu of th spcific-rgy had is subtractd from th assumd valu i th upstram dirctio ad x bcoms (x i+ x i ), which is gativ. Thrfor, if th quatio is solvd i upstram dirctio for a M-profil, for xampl, tomputd valus of x should b gativ for icrasig valus of h. Dcrasig valus of h should rsult also i gativ valus of x for a M-profil. For a M3- profil, which is suprcritical, icrasig valus of dpth i th dowstram dirctio corrspod to dcrasig valus of th spcific-rgy had, ad Eq. (4-9) idicats positiv valus of x, sic i > i b. Although th dirct-stp mthod is th asist approach, it rquirs itrpolatio to fid th fial dpth at th d of th profil i a chal of spcifid lgth. Som car must b tak i spcifyig startig dpths ad chckig for dpth limits i a computr program. I a M-profil, for xampl, th startig dpth should b tak slightly gratr tha th computd critical dpth, if it is a cotrol, bcaus of th slight iaccuracy ihrt i th umrical valuatio of critical dpth. I additio, th M-profil approachs th ormal dpth asymptotically i th upstram dirctio, so that som arbitrary stoppig limit must b st, such as 99% of th ormal dpth. Exampl 4.: A trapzoidal chal has a bottom width b of 8.0 m ad a sid slop ratio of :. Th Maig s of thal is 0.05 m -/3 s, ad thal is laid o a slop of If thal ds i a fr ovrfall, comput th watr surfac profil for a discharg of 30 m 3 /s. Solutio: Giv: bottom width: b = 8.0 m sid slop ratio: m: = : Maig s : = 0.05 m -/3 s bd slop: i b = 0.00 discharg: Q = 30 m 3 /s Comput th watr surfac profil. m = h b B Chaptr 4: NON-UNIFORM FLOW 84

16 First, th ormal dpth ad tritical dpth must b dtrmid. Maig s quatio rads as: 3 Q.A.R.i b with trapzoidal chal cross-sctio: A y (b mh ) ad hydraulic radius: A h (b mh ) R P b h m So, th Maig quatio ca b rwritt as: ib A A [h (b mh )] Q. 3 A.R A. P P b h m 5 3 h (8.0 y or, m 3.7 m 8 h From th Froud formula: V Q QB Fr 3 gd A A A g. g B whr B = b+ mh; D = A/B, th hydraulic dpth. I cas of critical : or 3 3 c A c c c 3 c Bc c QB h b h Fr A g b mh 3 h c 8 hc 5 30 m 8 h 9.8 c Chaptr 4: NON-UNIFORM FLOW 85 Q g h =.754 m =.03 m Du to h >, this is a mild slop (i b = 0.00): w hav a M-profil that has a critical dpth at th fr ovrfall as boudary coditio. Th dirct-stp mthod, as applid to Exampl 4., ca b solvd i a spradsht (Microsoft Excl) as formattd i Tabl 4.. Th valus of h ar slctd i th first colum (). Th formulas for dtrmiig th spcific-rgy had E, colum (5), ad th slop of th rgy grad li i, colum (6), for a giv dpth, ar prstd blow. Th arithmtic ma of i (i bar = i ) is computd i colum (7), ad thag i spcificrgy had E, DlE, i th upstram dirctio is show i colum (8). Formulas applid i th spradsht: V E h A y(b mh) g E Dl.E E E (V) Dl.E P b h m i x Dl.x 4 3 Q Q R (ib i bar ) R ; V (i i ) x Dl.E /(i i ) P A i bar (i i ) /

17 Th, th quatio of gradually varid i fiit diffrc form is solvd for th E distac stp x, as x = m i th first stp. i b i Tabl 4.. Watr surfac profil computatio by th dirct-stp mthod. h A R V E i i bar Dl.E Dl.x Sum Dl.x () () (3) (4) (5) (6) (7) (8) (9) (0) E E E-03.6E E E-03.3E E E-03.35E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-03.05E E-03.89E-03.0E E-03.74E-03.4E E-03.60E-03.8E E-03.46E-03.E E-03.34E-03.6E E-03.E-03.30E E-03.E-03.33E E-03.0E-03.36E E-03.9E-03.39E E-03.8E-03.4E E-03.73E-03.44E E-03.65E-03.47E E-03.57E-03.49E E-03.50E-03.5E E-03.43E-03.53E E-03.37E-03.55E E-03.3E-03.57E E-03.5E-03.59E E-03.0E-03.60E E-03.5E-03.6E E-03.0E-03.63E E-03.05E-03.65E E-03.0E E Not that at last thr sigificat figurs should b rtaid i E to avoid larg roud-off rrors wh th diffrcs ar small i compariso to th valus of E. I th last colum, tumulativ valus of x ar giv, ad ths rprst th distac from th startig poit whr th spcifid dpth h is rachd. Aftr th first stp, uiform icrmts i dpth h, with h icrasig i th upstram dirctio, ar utilizd. Th valus of h ar stoppd at th fiit limit of.745 m, which is 99.5% of th ormal dpth. Th lgth rquird to rach this poit is 7 m, which is th lgth rquird for this chal to b Chaptr 4: NON-UNIFORM FLOW 86

18 cosidrd hydraulically log, but that lgth varis, i gral. Th dpth icrmts ca b halvd util thag i profil lgth bcoms accptably small. Altrativly, smallr icrmts i dpta b usd i rgios of rapidly chagig dpth, ad largr icrmts may b appropriat i rgios of vry gradual dpth-chags. A portio of th computd M-profil is show i Fig M watr surfac-profil computd by th dirct-stp mthod,4,8 dpth (m), 0, distac upstram (m) Fig M-curv draw i xampl 4.. Exampl 4.3: A trapzoidal chal with a bottom width of 5 m, a sid slop of :, ad a Maig of 0.03 m -/3 s carris a discharg of 50 m 3 /s at a bd slop of Comput by th dirct-stp mthod th backwatr profil cratd by a dam that backs up th watr to a dpth of 6 m immdiatly i fot of th dam. Th upstram d of th profil is assumd at a dpth qual to % gratr tha th ormal dpth. Solutio: Giv: bottom width: b = 5.0 m sid slop ratio: m: = : Maig s : = 0.03 m -/3 s bd slop: i b = discharg: Q = 50 m 3 /s watr dpth: h = 6.0 m (i frot of dam) Comput th watr surfac profil. A M Dam 6 m CDL h i b = Similar to Exampl 4.: th ormal dpth ad tritical dpth ar: 5 3 [h (b mh )] Q. 3 i b h b m h =.87 m Chaptr 4: NON-UNIFORM FLOW 87

19 c 3 hc b hc Q b mh g =.57 m Bcaus h > thal slop is mild. Th profil lis i zo ad thrfor it is a M curv. Th rag of dpth is 6m at th dowstram d ad (0% x,87) =.90 m at th upstram d. Studts should try to mak a tabl computatio, which is slfxplaatory ad draw a M curv as Fig. 4.0 blow: M watr surfac-profil computd by th dirct-stp mthod watr dpth (m) i frot of dam distac upstram (m) Fig M-curv draw i xampl Dirct umrical itgratio mthod Th dirct itgratio mthod is applicabl to prismatic chals oly. This mthod uss Eq. 4-8 as govrig quatio: dh ib i Fr dx dh (4-0) dx Fr i i I its itgratd form, Eq. 4-0 bcoms: xi hi hi Fr dx xi xi dh g(h)dh i i xi hi b hi b (4-) Th itgrad o th right had sid of Eq. (4-) is a fuctio of h, g(h), whica b itgratd umrically to obtai a solutio for x, as show as i Fig. 4.. g(h) ara = x i+ - x i h o h i h i+ h h Fig. 4.. Watr surfac-profil computatio by dirct umrical itgratio Chaptr 4: NON-UNIFORM FLOW 88

20 A varity of umrical itgratio tchiqus ar availabl, such as th trapzoidal rul ad Simpso s rul, which ar commoly usd to fid tross-sctioal ara of a atural chal, for xampl. Simpso s rul is of highr ordr i accuracy tha th trapzoidal rul, which simply mas that th sam umrical accuracy ca b achivd with fwr itgratio stps. Applicatio of th trapzoidal rul to th right-had sid of Eq. (4-) for a sigl stp producs: g(h i) g(h i) xi xi hi hi (4-) To dtrmi th full lgth of a profil, (x l x o ), multipl applicatio of th trapzoidal rul rsults i l g(h o) g(h l) g(h i) i L x l xo h (4-3) whr L is th profil lgth ad h = (h i+ h i ) is th uiform dpth-icrmt. Bcaus th global trucatio rror i th multipl applicatio of th trapzoidal rul is of ordr (h), halvig th dpth icrmt will rduc th rror i th profil lgth by a factor ¼. By succssivly halvig th dpth itrval, th rlativ chag i th profil lgta b calculatd with th procss cotiuig util th rlativ rror is lss tha som accptabl valu. Chaptr 4: NON-UNIFORM FLOW 89