Chapter 3. Design Philosophy

Transcription

1 Chapter 3 Desgn Phlosophy 3.1 Introducton In chapter, process of energy dsspaton and use of hydraulc jump as energy dsspator are dscussed. The mportance of the ssue of approprate locaton of jump n stllng basn over a wde range of dscharge condtons and lmtatons of exstng practce of stllng basn desgn are addressed. Present chapter proposes a mathematcal procedure to address the ssue of stllng basn desgns to overcome ther lmtatons. Prmarly the procedure s developed for horzontal apron. It s further modfed emprcally and appled to hydraulc jumps on slopng aprons. It s observed that exstng stllng basn desgns try to match the jump heght curve (JHC) wth the tal water ratng curve (TWRC). In these desgns, there s full dependency on TWRC. The efforts are made to match these levels so as to form hydraulc jump wthn basn. In case of tal water defcency, normally ether basn floor s depressed or broad crested wer s provded. Sometmes floor s depressed partly and partly wer s provded. As these adjustments are based on desgn dscharge, JHC may ntersect TWRC at desgn dscharge condton only and at lower dscharges, t remans below TWRC. Ths produces drowned jumps at lower dscharges. When wer s provded, TWRC s dsconnected from JHC and agan drowned jumps are produced at lower dscharges. Therefore, n the present study new stllng basn desgn s proposed for tal water defcency. Ths can largely reduce dependency on TWRC wthout dsconnectng t from JHC. Efforts are made to produce requred JHC at all dscharges to form clear jumps even at lower dscharges. 3. Scope of the Chapter Ths chapter proposes new desgn of hydraulc jump type stllng basn for tal water defcency. A mathematcal procedure s developed to desgn rectangular stllng basn wth wer at ts end. The plan sze of basn s governed by the sze of spllway and length of hydraulc jump at desgn dscharge condton. Producton of requred JHC s manly governed by the end wer characterstcs. Therefore a focus s made on end wer desgn. A new parameter submerged flow coeffcent (k s ) for sharp crested 38

2 stepped wer s ntroduced. A relaton between average submergence rato (S r ) and k s s establshed for horzontal apron. 3.3 Proposed Methodology To acheve approprate locaton of hydraulc jump n rectangular channel of wdth B, t s essental to consder nput parameters whch govern locaton of jump. Fg 3.1(a) demonstrates defnton sketch of hydraulc jump n rectangular channel wth horzontal slope. Sluce Gate H Clear hydraulc jump Proposed wer (crest) y y 1 h y' Rectangular Channel Secton A y t Flow drecton Test Secton B Plan A Fg. 3.1(a) Defnton sketch of hydraulc jump n rectangular channel Factors affectng hydraulc jump are total head on upstream of sluce gate (H), prejump depth (y 1 ), post jump depth (y ), crest heght of wer (y ' ), head over wer crest (h) and tal water depth (y t ). Another nondmensonal mportant parameter that governs hydraulc jump phenomenon s supercrtcal Froude number F r1. Fg. 3.1(b) shows dscharge Q on X-axs and post jump depth (y ) and tal water depth (y t ) on Y-axs to form two curves - jump heght curve (JHC) and tal water ratng curve (TWRC) respectvely. 39

3 Case 1 Case y y t Stages y and yt JHC=TWRC Stages y and yt JHC TWRC y 1 Dscharge Q Dscharge Q Case 3 Case 4 Case 5 Stages y and yt TWRC JHC Stages y and yt TWRC JHC Stages y and yt JHC TWRC Dscharge Q Dscharge Q Dscharge Q Fg. 3.1(b) Classfcaton of tal water condtons for the desgn of energy dsspator Dependng upon the relatve magntudes of y and y t at varous dscharges, fve dfferent cases would arse (Chow 1959). The vertcal dstance between JHC and TWRC represents the dfference between y and y t for a partcular dscharge. When ths dfference s zero,.e. y t = y, the hydraulc jump forms at vena contracta of the supercrtcal flow. If ths condton s acheved for all the dscharges, t becomes an deal condton (Case -1) that ensures formaton of clear hydraulc jump at vena contracta for all the dscharges. Ths stuaton does not requre any appurtenances n the stllng basn and only horzontal apron wth ts top at rver bed elevaton would suffce the requrement. But such condtons are of rare occurrence n practce. In most of the cases on feld, the tal water depths are ether on hgher sde or lower sde. Sometmes these are partly on hgher sde and partly on lower sde and vce versa over the whole range of dscharge. Hence t drectly affects the locaton of hydraulc jump on apron. If y t < y (Case -), then hydraulc jump sweeps out of the basn and the correspondng jump s referred to as swept up jump. In ths case the 40

4 jump may form far away from toe of spllway but reman partly on apron or t may completely sweep out and form n the downstream channel entrely away from the stllng basn. If y t > y (Case -3), then the hydraulc jump rdes on slopng spllway surface and the correspondng jump s referred to as submerged or drowned jump. In ths case a hgh velocty flow extends on apron upto a large dstance and t may cause scourng of the apron. Sometmes t may extend even upto the end sll. Secondly, n the drowned jump, large scale pressure fluctuatons are observed (Bower and Toso, 1988). In case-4, y t < y for lower dscharges and for the remanng dscharges y t > y, whereas n case-5 y t > y for lower dscharges and for the remanng dscharges y t < y. Present study focuses attenton on the tal water defcency condton (Case-). It s also partly applcable to case-4 where there s tal water defcency at lower dscharges. Case- shows that over a complete range of dscharge, the tal water depth s lower than the post jump depth. The dfference between these two depths goes on ncreasng wth ncreasng dscharge. Accordngly the locaton of jump shfts on apron and the toe of jump moves farther and farther away from the desred locaton.e. toe of spllway or sluce gate. As the jump moves away from ts desred locaton, there s proportonate reducton n the dsspaton of energy. The tal water defcency condtons encounter manly on spllways located on flanks. As the tal channel s ntended to convey flood water to the orgnal rver channel, t has large elevaton dfference n short dstance, hence t s relatvely steeper. Ths tends to produce hgher veloctes and subsequent lower flow depths n the tal channel. Hence due to non avalablty of suffcent tal water depths for formaton of jump, the jump has a tendency to sweep out. As the swept up jump s rather harmful than the drowned jump, t s preferred to have a hydraulc jump type stllng basn n the form of depressed horzontal apron and an end sll whch produces drowned jumps at low dscharges. Another soluton s to have horzontal apron may be at rver bed level or slghtly depressed wth a rectangular broad crested wer located at the end of the apron, generally referred as end wer. As drowned jumps dsspate less energy than the clear jumps, the resdual energy carred by the water n the tal channel after t enters the tal channel s comparatvely large. In ths study t s presumed that the apron elevaton on just upstream of the end wer 41

5 and elevaton of the channel floor just downstream of the end wer are same. In some of the cases only the apron elevaton s lowered to match the y levels wth that of y t levels. But an addtonal excavaton cost s also nvolved n lowerng the apron elevaton. Wth an am to provde a generc soluton to the problem of tal water defcency, t s proposed to desgn an end wer geometry whch wll cater to wde range of dscharge by takng the cognzance of correspondng range of tal water submergence. By way of the proposed soluton an attempt would be made to overcome tal water defcency condton (Case-) to attan deal condton (Case-1). It s known that a sngle rectangular suppressed wer of partcular heght s able to cater to only sngle partcular dscharge. If a wer desgned for low dscharge s exposed to hgher dscharge, the jump wll sweep out. If a wer desgned for hgh dscharge s exposed to lower dscharge, a drowned jump wll be formed as shown n Fg. 3.1(c). It means practcally, under constant head, every partcular dscharge demands, a partcular heght of end wer whch wll be proportonal to that dscharge. On the other hand, f a contracted wer s desgned, t results nto lowerng of crest heght and for every partcular dscharge a partcular wdth of wer s obtaned whch s nversely proportonal to dscharge. In short, f wdth s kept constant, crest heghts vary and f crest heght s kept constant then wdths vary. To acheve the am of the study, t s necessary to desgn sngle end wer geometry n such a fashon that t wll provde the effect of all such ndvdual wer heghts or wdths. It s therefore thought that a rectangular stepped wer may serve ths purpose. On felds, normally end wers are provded n the form of broad crested wers whch are gravty structures. In other small projects they can be provded n the form of sharp crested wers. As broad crested wers have hgh modular lmt ( as per USBR Water Measurement Manual), they are less senstve to submergence effects. Hence t s requred to study the effect of submergence wth respect to sharp crested wers whch are relatvely more senstve to submergence effect. Hence the study s ntated wth sharp crested wers. Broad crested wers are used for case studes. 4

6 Desgn dscharge condton Dscharge less than desgn dscharge condton Fg. 3.1 (c) Locaton of hydraulc jump at desgn and lower dscharge for the end wer desgned for desgn dscharge condton 3.4 Governng Factors The factors whch govern desgn of stepped wer geometry are as follows 1. Head on upstream = H. Wdth of channel = B 3. Maxmum dscharge n the range = Q max 4. Mnmum dscharge n the range = Q mn 5. Coeffcent of dscharge = C d 6. Submerged flow coeffcent = k 7. Startng heght of wer crest = y' The head on upstream H s the head of water on upstream of sluce gate or t s the heght of spllway plus half of the head over spllway crest above stllng basn level for mnmum dscharge n the range. The head over spllway can be neglected f t s smaller n magntude because some frctonal head loss s occurrng on spllway surface whch may compensate t. 3.5 Desgn Phlosophy of Stepped Wer Secton A stepped wer s to be desgned for varyng dscharges and a specfc range of tal water submergence. Varaton of tal water depth wth dscharge s not lnear. Dependng upon the magntude of the dscharge and the geometry of the downstream channel, a specfc tal water depth s formed n the channel. When the free flow over wer exsts, the tal water level does not affect the depth of flow on upstream of wer. But when the crest of wer undergoes submergence, consderaton of only dscharge 43

7 varaton wll not suffce requrement of wer desgn; the tal water depths for the correspondng dscharges are also needed to be consdered. Ths can be acheved by desgnng a rectangular stepped wer n such a manner that, every step of t wll fulfll followng two condtons. Condton 1 - The vertcal dstance between the channel bottom and any partcular step top, on upstream of a wer, should be equal to the post jump depth (y ) for that partcular dscharge. Condton The wdths of ndvdual steps should be such that the cumulatve dscharge passng through the wer, wth the water level upto any partcular step top, should be equal to that partcular dscharge mentoned n condton 1. Assumptons 1) Stllng basn s rectangular wth horzontal slope. ) Flow s steady and head on upstream of spllway (H) s constant. 3) Spllway s ether ungated or the gate openng s unform. 4) Dscharge condtons are varyng. (Varaton upto 0% of desgn dscharge). 5) Coeffcent of dscharge (C d ) s constant Development of mathematcal procedure The proposed stepped wer should cater to wde range of dscharge from desgn dscharge (Q max ) to mnmum dscharge equal to 0% of the desgn dscharge (Q mn ). N ntermedate dscharges between Q mn and Q max wth an nterval of (Q max - Q mn ) / (N+1) are consdered resultng n (N+) dscharges correspondng to whch there would be (N+) steps n a stepped wer. A stepped wer s consdered to be made up of number of rectangular wers. The equaton for dscharge Q over a rectangular sharp crested wer (free flow) s gven by 3 3/ Q = Cd b g h (1) As h = y y, the requred wdth of wer can be expressed as (refer Fg.3.) 3 Q b = () C 3/ d g ( y y' ) 44

8 y s calculated from Belanger momentum equaton (Subramanya, 1986). As llustrated n Fg. 3., wdth of frst step s gven by b = (3) C d g Q [ (y ) y' ] 3/ 1 Where, y = (y ) 1 / 4 s desgned for Q 1 (.e. Q mn ). (y ) 1 s the post jump depth correspondng to Q 1 and s calculated by Belanger momentum equaton n the followng manner. (y (y 1 ) = ( ( F r1 ) ). 1 ) + Q Where, (y 1 ) =, v = g H, B v (F ) = r1 g( v 1 y 1 ) a 3 / a 3 / (y ) 3 (y ) (y ) 1 a / crest y' B b 3 a / b b 1 tread rse Fg. 3. Desgn of rectangular stepped wer (Secton A-A taken from Fg. 3.1(a)) (frst 3-steps shown) Usng equaton (), for any dscharge Q n, the wdth of the correspondng step b n can be calculated as follows, b = b + a.. n > 1 (4) n n 1 n Where, a n represents the ncremental wdth at n th step. Where, a n = 3 C d g Q n [ = n -1 [ ( y ) ( y ) ] 3/ = 1 n (Q n ) ] n 1 = [for =1] (6) 3 and, (Q ) C g b [(y ) (y' ] 3/ n d 1 n ) = 3 [(y ) (y ] 3/ [for = to (n-1)] (7) 45 (Q n ) Cd g a n ) 1 (5)

9 The mathematcal procedure s based on the free flow rectangular wer formulae. Here t s presumed that stepped wer s made up of number of rectangular wers. As gudelnes for stepped wer s C d are not avalable, t s necessary to evolve C d for free flow stepped wer. In practce, mostly the crest of stepped wer s submerged, therefore t becomes essental to evolve modfed dscharge coeffcent whch would be applcable to submerged condtons. More lght s thrown on these aspects n the subsequent sectons. Another aspect regardng mathematcal procedure s about assumptons and 5. In ths desgn, supercrtcal velocty v 1 s calculated by the method suggested by Peterka (1984). C d s calculated usng formula gven by Subramanya (1986). But t has generated a zg-zag geometry of stepped wer as shown n Fg. 3.(a). Though mathematcally correct, t s not practcable. For each dscharge, for calculaton of y, t s necessary to calculate correspondng prejump depth y 1 and thus correspondng prejump velocty v 1. v 1 can be calculated by the procedure gven by Peterka n USBR Monograph 5 ( that consders varaton of head whch s proportonal to dscharge. On the contrary, the assumpton of H to be constant, whch n turn mples pre jump velocty also to be constant, gves a practcal soluton. As, fnally the desgn of stepped wer s based on the magntudes of post jump depths (y ) for (N+) dscharges, these depths are calculated by both the methods dscussed above. For ths purpose data of an exstng dam s consdered. B = 55m, H = 1.m, Q max = 1736 m 3 /s, Q mn = 347. m 3 /s. It s found that the values of y farly agree wth each other as correlaton coeffcent s found to be The comparson of y values s presented herewth for ready reference. 46

10 Table 3.1 Calculaton of y by two methods Step No. y m - based on monograph method y m - based on proposed method Fg. 3.3 shows a pattern of stepped wer whch s mathematcally correct as well as practcable. As an addtonal exercse, C d values are calculated by the formula proposed by Swamee (1988). But t has also resulted nto a smlar zg-zag geometry of stepped wer. After extensve trals on mathematcal model wth dfferent nput data, t s found that even mnute varaton of H and C d s resultng nto an mpractcable geometry of stepped wer. b 11 b 3 b b 1 B y Fg. 3. (a). Schematc of zg-zag geometry of stepped wer 47

11 b 11 b 3 b (y ) 11 b 1 y' B Fg. 3.3 Defnton sketch of geometry of rectangular stepped wer No lterature s avalable on rectangular sharp crested stepped wer. Recently some work on compound broad crested wers s reported by Jan et al (006) and by Gogus et al (009). Jan has nvestgated the rectangular-rectangular compound broad crested wer and derved the dscharge equatons for 0. h/b w Gogus has nvestgated dfferent compound broad crested wer geometres for 0.1 h/b w 0.7. Gogus further concluded that, for h/b w 0.5, the C d values are almost remanng constant. In the case study dscussed n chapter 6, the h/b w rato for the mddle wer of step wdth b 1 (whch s contrbutng more than 80% of the dscharge for any y on ts upstream) ranges from 0.5 to 1.7. Thus, wth the reasonable approxmaton, t s assumed to treat both H and C d as constant throughout the desgn. It s found that for F r1 > 4.5, the mathematcal procedure gves practcable geometry of stepped wer whch s also mathematcally correct. (The detaled calculatons by consderng varaton of H and C d s dscussed n Appendx D) Determnaton of Approprate C d for Free Flow Condton on Horzontal Apron In the mathematcal desgn of stepped wer, for free flow condton, a constant value of C d = 0.63 s adopted accordng to Francs formula. The valdty of the same has been emprcally determned wth the help of laboratory experments. As there are dfferent knds of uncertantes nvolved n the flow condtons, the analytcal determnaton of C d s hardly possble. To menton a few, there s presence of 48

12 hydraulc jump and assocated turbulence on upstream of stepped wer. The upstream reach for the stepped wer, beng equal to length of stllng basn, s small. Hence t s decded to emprcally judge the approprateness of Cd by the approprateness of hydraulc jump locaton (whch s vulnerable to uncertantes due to some assumptons nvolved n the mathematcal desgn of wer) whch s the man am of study. Beng an mplct functon, Cd needs to be assumed at ntal stage. If the wer performance s not satsfactory the step wdths can be modfed emprcally. It s proposed to conduct tests n a laboratory flume under free flow condton. For creatng free flow condtons over the stepped wer, the bottom of channel n the frst half (m long) s rased. Ths was done by layng a wooden platform of 5cm heght nsde a flume. Fg. 3.4 shows vews of the actual setup n laboratory. (a) Vew from downstream (b) Rght hand sde vew (c) Vew from downstream wth free flow from the wer Fg. 3.4 Photographs showng rased platform n the laboratory tltng flume and free flow condtons created over the stepped wer 49

13 Common parameters for whch wers are desgned are H=0.4m, B=0.3m, Q max =0.01 m 3 /s, Q mn =0.00 m 3 /s, y'=0.015m. Three wer models, each for C d =0.6, 0.65 and 0.63 are desgned. As per present practce, energy dsspators are desgned for the desgn dscharge condtons and ther performance s tested at lower dscharges (Vttal and Al-Garn 199). Generally apart from desgn dscharge, the performance s tested at 3 lower dscharges whch are 75%, 50% and 5% of desgn dscharge. To ncrease the accuracy, t s proposed to consder 10 lower dscharges and reduce the lowest dscharge to 0% of desgn dscharge. Thus, ncludng desgn dscharge, there would be total 11 dscharges. Thus n the proposed stepped wer, there are 11 steps correspondng to 11 dscharges. Tables 3.1(a, b and c) gve the geometry of these wers and other parameters related to hydraulc jump. Table 3.1 (a) Output of Mathematcal Procedure for Laboratory Data (Horzontal apron and C d = 0.6) Sr.No. Q m 3 /s y 1 y h F r1 m m m m b Wth C d = 0.6, the performance of wer s tested for boundary condtons of dscharge, that are Q mn and Q max to check the possblty of jump formaton and the adequacy of ts locaton. It s presumed that f the wer performance t unsatsfactory over these extreme dscharges, then t would probably be unsatsfactory for the ntermedate dscharges also. The jumps are found to be shfted n the downstream drecton for both the dscharges. Ths shows that the area of flow secton of stepped 50

14 wer s larger and need to be reduced. Ths can be done by reducng the step wdths (b) whch requres ncrease of C d as b vares nversely wth C d. Table 3.1 (b) Output of Mathematcal Procedure for Laboratory Data (Horzontal apron and C d = 0.65) Sr.No. Q m 3 /s y 1 y h F r1 m m m m b In the second tral another stepped wer s desgned wth C d = 0.65 and tested n a smlar manner. Durng ths, the jumps are found to be drowned. Hence for the thrd tral C d = 0.63 s adopted and accordngly, tests are taken. For stepped wer wth C d = 0.63, the hydraulc jumps have formed nsde the basn and the fronts of jumps n all the cases were found to be located near the sluce gate. Thus C d = 0.63 s confrmed emprcally for the condton of free flow over the wer. For hgher submergences (S r > 0), the coeffcent of dscharge wll decrease and takes the form of modfed coeffcent of dscharge (C dm = k s C d, where k s <1). In ths case, k s s a submerged flow coeffcent for stepped wer whch s newly ntroduced. Determnaton of C dm based on K s s explaned below. 51

15 Table 3.1 (c) Output of Mathematcal Procedure for Laboratory Data (Horzontal apron and C d = 0.63) Sr.No. Q m 3 /s y 1 y h F r1 m m m m b Determnaton of Approprate C dm for Submerged Flow Condton on Horzontal Apron When tal water depth (y t ) rses above crest, wer s sad to be submerged. For determnaton of C dm, t s necessary to determne k s. The effect of tal water submergence s dscussed n USBR publcatons lke Monograph 5 (1984), Manual of Water Measurement (001) and Desgn of Small Dams (1974). In 1947 Vllemonte (Subramanya 1986) has studed effect of submergence on dscharge over rectangular sharp crested suppressed wer and has gven a relaton between k and S r. But a smlar detaled analyss s not avalable for rectangular stepped wer. Thus for stepped wer, k s (for gven S r ) s determned emprcally and s compared wth k for rectangular wer gven by followng Vllemonte s equaton. k 3 ( 1 (S ) Q = (3.9) Q s = r ) f Where, Q s and Q f are the submerged and free flow dscharges respectvely and 5

16 S r ( y t y ' ) = (3.10) ( y y ' ) To fnd k s emprcally, 6 stepped wer sectons wth dfferent k s values rangng from 1 to 0.8 at an nterval of 0.04 are desgned. The other desgn parameters are same as before. The wer sectons are tested n flume for correspondng ranges of S r. Submergence s created by puttng specfc number of vertcal stop logs at the end of the channel nstead of operatng tal gate. Use of vertcal stop logs ensures contnuaton of stream lnes. The performance of wer secton s judged from the ablty of wer secton to form clear hydraulc jumps at approprate locaton for boundary dscharges. The detals of the trals are gven below. In order to obtan k s, seres of laboratory experments are carred out. The expermentaton s carred out for three mean submergence ratos vz. 0.45, 0.65 and 0.75 and accordngly three k s values are determned. Once relaton between S r and k s s establshed, a relaton between S r and C dm can be obtaned and t can be used for practcal applcatons. In the frst tral approprateness of C d for free flow condton of stepped wer s determned emprcally. In the second tral, the rased platform s removed from the flume and tal water depths are rased to create submergence condtons on downstream of stepped wer. The submergence so created s kept proportonal to dscharge. Over a range of dscharge and a correspondng range of S r (0.4 to 0.5), agan a stepped wer wth C d =0.63 (.e. k=1) s tested. It s observed that for both the dscharges drowned jumps are formed. Ths ndcates that the area of flow at the stepped wer secton s not adequate to form clear jumps near sluce gate. Then next wer secton wth C dm = 0.6 (.e. k s =0.96 and havng relatvely large area of flow than the prevous case) s tested and t has formed clear jumps at approprate locaton. Ths has confrmed that the stepped wer under consderaton s operatng satsfactorly. Further to reconfrm the adequacy of ths stepped wer, for the same submergence condtons a wer secton wth C dm = 0.57 (.e. k s =0.9) s tested and t s observed that the front of jump moved away from the approprate locaton (.e. swept up jump). Thus the value of C dm = 0.6 for the average S r = 0.45 s fnally confrmed. Smlar tests are carred out wth average submergence ratos 0.65 and 0.75 and the 53

17 correspondng values of C dm are found to be 0.55 and 0.5 (.e. k s = 0.88 and 0.8) respectvely. Table 3. gves comparson of emprcally determned k s values wth k values gven by Vllemonte s equaton. Sr. No. Table 3. Comparson of submerged flow coeffcents k and k s Range of Average k k s C dm = submergence submergence k s x C d rato (S r ) rato (S r ) to to to The values of k s are slghtly greater than k. Ths s because of followng approxmatons nvolved n the process. ) Vllemonte s equaton (whch s applcable to rectangular wer) s appled to stepped wer. ) ) In conventonal rectangular wers the whole crest of wer s under submergence. But n case of stepped wer, crest and few steps are under submergence. In conventonal wers the flow on upstream s gentle. Whereas n present case there s presence of hydraulc jump on upstream of stepped wer. 3.6 Closure A new stllng basn desgn s proposed n the form of horzontal apron wth a stepped wer at ts end. A mathematcal procedure s developed to desgn geometry of rectangular sharp crested stepped wer for varous operatng condtons. Wer geometres are desgned for laboratory condtons. In case of horzontal apron, a new parameter - submerged flow coeffcents (k s ) s ntroduced and s determned emprcally for dfferent average submergence ratos (S r ). A relatonshp between S r and C dm s establshed for horzontal apron. The equaton for close startng heght of wer (y') s gven. To optmze geometry of stepped wer, few trals wth mathematcal procedure are requred to be taken wth values around y'. 54