ESTIMATION OF HYDRAULIC JUMP LOCATION USING NUMERICAL SIMULATION

Transcription

1 Twel Inernaonal Waer Tecnology Conerence, IWTC 8, Aleandra, gyp STIMATION OF HYDRAULIC JUMP LOCATION USING NUMRICAL SIMULATION M. T. Samaa,, M. Hasem and H. M. Karr, Cvl ngneerng Deparmen, Facly o ngneerng, Mansora Unversy, l-mansora, gyp Correspondng Aor: -mal: aresamaa@yaoo.com Cvl ngneerng Deparmen, Facly o ngneerng, Ass Unversy, Ass, gyp Cvl ngneerng Deparmen, Hger Inse o ngneerng, Hoon, Lbya ABSTRACT Te Bossnesq eqaons are nmercally solved o smlae e ormaon o ydralc jmp n a recanglar cannel avng a small bed slope. Te MacCormac sceme s sed or er solon by applyng speced nal and bondary condons nl a seady sae low s reaced. Te locaon o e ydralc jmp s deermned as a par o ese compaons. Te arcal vscosy ecnqe sold be sed n e compaons o dampen e speror oscllaons near e seep graden o e smlaed ydralc jmp. Twelve laboraory epermens were carred o or vercaon o e nmercal model. Upsream Frode nmber or ese epermens ranged rom.7 o.8. Te smlaed ydralc jmp proles sng e MacCormac sceme sowed good agreemen w e epermenal daa. Applcaon o e model was eended beyond e lms o e daa sed n e vercaon process. A power emprcal eqaon was developed o deermne e locaon o ydralc jmp sng regresson analyss based on smlaed daa. INTRODUCTION Hydralc jmp s e ranson penomenon rom spercrcal low o sbcrcal low, were waer srace rses abrply, accompaned by consderable rblence and energy dsspaon. Te applcaons o ydralc jmp n open cannel low are val as energy dsspaon devce over ydralc srcres, mng o cemcals sed or waer prcaon, and aeraon o lows. Several laboraory and eld nvesgaons ave been carred o o deermne e varables nclded n e ydralc jmp penomena sc as s leng, locaon, energy dsspaon, and ecency. Cow [4] comped e waer srace proles or bo e spercrcal low sarng rom e psream bondary and e sbcrcal low sarng rom e downsream bondary o deermne e ydralc jmp locaon n a cannel. Te jmp locaon s esablsed a e poson were e specc orces on bo sdes o e jmp are eqal. McCorqodale and Kala [] sed a srp

2 Twel Inernaonal Waer Tecnology Conerence, IWTC 8, Aleandra, gyp negral meod o predc e jmp leng, velocy dsrbon, waer srace prole and pressre a bed. Kaopodes [8] sed e ne elemen meod o solve e S. Venan eqaons nmercally nl a seady sae was reaced. Te locaon o ydralc jmp was aomacally comped as a par o e solon. Te S. Venan eqaons are based on e assmpon o norm velocy dsrbon and ydrosac pressre dsrbon [,,9]. Tese eqaons sold be replaced by e eqaons o e Bossnesq ype wen e assmpon o a ydrosac pressre dsrbon s ncorrec. In a rapdly vared low avng seep waer srace graden, McCowan and Basco [5,6] spposed a e vercal velocy dsrbon ncreases rom zero a e cannel bed o s mamm vale a e ree srace o derve Bossnesq eqaons. Te Bossnesq eqaons can be redced o e S. Venan eqaons, by neglecng e Bossnesq erms, wc accon or e non-ydrosac pressre dsrbon. Te se o an eplc ne derence sceme s oen avorable wen e governng eqaons are complcaed and e solon as a bg space me cange rae or even dsconnes [6,]. Te Bossnesq eqaons, or wc closed-orm solons are no avalable, can be solved nmercally. Order- and ger order scemes oen generae spros oscllaon and Gbbs errors n e vcny o dsconnes, so an arcal vscosy sold be added o preven srong spros oscllaons [6,,,4]. In s paper, e MacCormac ne derence eplc sceme [,,5,6,,4,5] s sed o solve e Bossnesq eqaons. Te governng eqaons are rs presened, en deals o e MacCormac ne derence eplc sceme are eplaned. Sably condons and e nclson o nal and bondary condons or e MacCormac sceme are dscssed. Usng e me dervave erm as an erave parameer w e speced bondary condons, e nseady low compaons are conned n s sceme nl ey converged o a seady sae low o deermne e locaon o e ydralc jmp. Te comped resls were vered by comparng em w e epermenal laboraory daa. Wen a g level o vercaon can be aceved, en may be possble o een e applcaon o e model beyond e lms o e daa sed n e vercaon process. A power emprcal eqaon was developed o deermne e locaon o ydralc jmp Usng regresson analyss based on smlaed daa obaned rom e maemacal model. GOVRNING QUATIONS McCowan and Basco [5,6] assmed a e ld s ncompressble, e cannel s prsmac, recanglar n cross secon, and as small boom slope o derve e Bossnesq eqaons. Also, ey assmed a e low velocy n e laeral drecon s zero and e vercal velocy dsrbon ncreases rom zero a e

3 Twel Inernaonal Waer Tecnology Conerence, IWTC 8, Aleandra, gyp cannel bed o s mamm vale a e ree srace. Te Bossnesq eqaons or one-dmensonal low can be rewren as: = () g g( S S ) = () n wc = () were = dsance along e cannel bed; = me; = waer dep; = low velocy n e drecon; g = gravaonal acceleraon; S = cannel bed slope; S = rcon slope; and = Bossnesq erm. Te rcon slope can be epressed sng e Mannng's eqaon as: n S = (4) 4 / R n wc n = Mannng's rogness coecen; and R = ydralc rads. Te Bossnesq eqaons () and () may be rewren n a vecor orm as: U F S = n wc U = F g S ; = ; and = g( S S ) (5) (6) NUMRICAL SOLUTION MTHOD plc derence scemes a a ne nmber o grd pons n e recanglar spaalme grd can be sed o oban a nmercal solon o e Bossnesq eqaons. Te nnown varables n ese eplc scemes are comped a a recanglar grd pon on an advanced me lne sng e nown vales and condons a grd pons on e presen me lne. Te MacCormac eplc sceme was appled or e solon o e Bossnesq eqaons sng speced nal and bondary condons. Te locaon o ydralc jmp was deermned by connng e solon nl a seady sae was aceved. Te MacCormac sceme s a predcor correcor sceme, n wc e solon s obaned sng orward derence appromaon or e space dervave n e predcor sep and bacward derence appromaon o e predced vales n e correcor sep. Te compaonal grd or e MacCormac sceme s sown n Fgre.

4 Twel Inernaonal Waer Tecnology Conerence, IWTC 8, Aleandra, gyp Predcor S F F U U = (7) Correcor S F F U U = (8) U U U = (9) Fgre. Compaonal Grd or e MacCormac Sceme Te sbson o eqaons (7) and (8) no eqaon (5) and smplcaon o e resled eqaons yeld e ollowng: Predcor = () () S S g g g = = () = () Upsream bondary condon Inal condon Q, y Known Q, y Unnown Downsream bondary condons

5 Twel Inernaonal Waer Tecnology Conerence, IWTC 8, Aleandra, gyp Correcor = (4) (5) S S g g g = n wc = (6) = (7) Te advanages o e MacCormac sceme nclde smplcy and g accracy; owever, as a drawbac a n e compaon o soc waves speror oscllaons near e seep graden o e comped solon may appear, so an arcal vscosy meod sold be sed [,,6,,]. ARTIFICIAL VISCOSITY Arcal vscosy meod s a soc-caprng meod. I s very smple, and as smoong or lerng eec n resrcng e developmen o parase oscllaons. Te soc-caprng meod plays an mporan role only n e rapdly varyng area o e solon, wle as very small eec on e res par o e solon. Te arcal vscosy sold cange a dsconny crve (soc wave) no a narrow layer were e solon as a seep b smoo ranson. Also, sold only ave a slg mpac on e accracy o e smoo par o e solon. Arcal vscosy orm sed by Jameson e al. [,6,] s nrodced as ollows. A parameer λ I s rs comped rom e nally comped low deps as: = λ (8) λ λ κ λ, ma / = (9) n wc κ s a dsspaon coecen sed o reglae e amon o dsspaon. Ten, e comped veloces and waer deps are moded as: / / = λ λ ()

6 Twel Inernaonal Waer Tecnology Conerence, IWTC 8, Aleandra, gyp INITIAL AND BOUNDARY CONDITIONS Inal and bondary condons are e essenal reqremens or nang e smlaon o ydralc jmp. Te nal condon descrbes e velocy and e dep o low a all compaonal nodes along e -drecon o e cannel a e nal me =. Te low n e enre cannel a nal me s assmed o be seady spercrcal. Te low dep and velocy a all e compaonal nodes a e nal me are deermned by e nmercal negraon o e ollowng derenal eqaon descrbng e gradally vared low [,6,]: d S S = () d g Te nal low dep, and velocy descrbng e psream bondary condons a e enrance o e cannel are ep ncanged drng e compaons. A consan low dep s speced a e downsream bondary, wle e low velocy s comped a advanced me level sng e caracersc orm o eqaons () and () [,6] as: ( ) g( S S ) g = c () n wc c = g s e celery o a gravy wave n a recanglar cannel. STABILITY CRITRION For eplc meods, e vale o ms be less an some mamm vale allowable or sably. Te MacCormac sceme s sable sased e ollowng Coran-Fredrcs-Lewy (CFL) creron: = Cn () ma g n wc C n = Coran nmber wc ms be less an or eqal o one [,,6,,,4]. qaon () sows a e eplc me sep msn be greaer an e me reqred or a wave o propagae rom one grd pon o e ne one. DSCRIPTION OF LABORATORY XPRIMNTS Te laboraory epermens were carred o o ge e reqred daa or e vercaon o e MacCormac sceme. A recanglar recrclaon perspe lme 7.5 cm wd, 5 cm eg, and 4.8 m leng, locaed a e ld mecancs laboraory, Hger Inse o ngneerng, Hon, Lbya, was sed o ge e reqred daa, Fgre. Te waer s sppled rom a consan ead an rog a sarp edged slce gae. Te lme s eqpped w a algae o conrol e al waer dep and a

7 Twel Inernaonal Waer Tecnology Conerence, IWTC 8, Aleandra, gyp conrollng screw o conrol e slope o e lme. A e end o e lme, e waer dscarges no e smp rog a wegng an. Te wegng an was sed o measre e dscarge and a pon gage operang along e lme was sed o measre e waer deps. A ral and error procedre was sed o deermne e Mannng's coecen or e lme. Drng e nal seady spercrcal low, e Mannng's coecen n =. was deermned by macng e comped waer srace prole w e measred waer deps n e lme. Twelve laboraory epermens were carred o or vercaon o e MacCormac sceme. Te deals o e laboraory epermens are lsed n Table. Tal an Tal gae Pon gage Sarp-edged Gae Head an Cloc Flow (4 Wegs Tlng screw ( Pmp Spply ppe Conrol valve Upsream carrer Smp Measrng an Downsream Carrer Fgre.. Layo o permenal Insallaon permen nmber Dscarge (m /sec) Table. Deals o permens Waer dep a vena conraca (m) F r a vena conraca Bed slope Tal waer dep (m)

8 Twel Inernaonal Waer Tecnology Conerence, IWTC 8, Aleandra, gyp MODL APPLICATION Nmercal model were consrced o solve e Bossnesq eqaons sng e second-order MacCormac sceme. Te Coran sably creron and e space grd sze resrced e me sep sze. Te Coran nmber was sed eqal o.65. Followng e ncrease o e downsream waer dep a me =, e nseady low compaons were conned nl e seady sae was reaced sng e me dervave erm as an erave parameer. Te jmp raveled rom e downsream o e psream and en moved lle bac and or nl was sablzed n one locaon as sown n Fgre ().. Waer dep (m) Dsance (m) =7.777sec =5.5sec = sec =5.894sec Fgre.. Waer Srace Prole a Deren Tmes, permen No. () Te nmercal scemes were calbraed sng e epermenal resls. Te smlaed jmp proles sng dsspaon coecen κ =. and =. m gave e mos accrae resls. Te obaned resls sng =. m seems o be more accrae an a obaned sng =.5 m, cold be de o e eec o e sed arcal vscosy ecnqe and e rncaon error []. Te comparson beween e smlaed resls, obaned sng =. m, Coran nmber =.65 and κ =., and e measred daa sows a good agreemen as llsraed rom Fgre (4) o Fgre (8). Fgres (4) and (5) llsrae e eec o al waer dep on e locaon o ydralc jmp. Te ydralc jmp moves owards e psream bondary wen e al waer dep ncreases. Fgre (6) sows a, or e same dscarge, e ydralc jmp moves owards e downsream bondary as psream waer dep decreases de o e ncrease o Frode nmber. Fgre (7) ndcaes also a e ydralc jmp moves owards e downsream bondary as Frode nmber ncreases (dscarge ncreases) or e same psream waer dep. Te ydralc jmp moves also owards e downsream bondary wen e lme slope ncreases as sown n Fgre (8).

9 Twel Inernaonal Waer Tecnology Conerence, IWTC 8, Aleandra, gyp Te specc orce was comped a e begnnng o e jmp and e appromae locaon a e end o e jmp. I may be conclded a e comped resls sasy e eqaly o specc orce a e begnnng and e end o ydralc jmp becase er vales dered rom eac oer by.9% o 4.9%. Waer dep (m) Smlaed ( =.8 m) Smlaed ( =.8 m) Smlaed ( =.745 m) Measred ( =.8 m) Measred ( =.8 m) Measred ( =.745 m) Dsance(m) Fgre 4. Comparson beween Measred and Smlaed Jmp Proles Usng Deren Tal Waer Deps,permens No. (), () and (4). Waer dep (m) Smlaed ( = m) Smlaed ( =.78 m) Measred ( =. m) Measred ( =.78 m) Dsance (m) Fgre 5. Comparson beween Measred and Smlaed Jmp Proles Usng Deren Tal Waer Deps,permens No. (5) and (6).

10 Twel Inernaonal Waer Tecnology Conerence, IWTC 8, Aleandra, gyp Waer dep(m) Smlaed ( =.8 m) Smlaed ( =.8 m) Measred ( =.8 m) Measred ( =.8 m) Dsance (m) Fgre 6. Comparson beween Measred and Smlaed Jmp Proles Usng Deren Upsream Waer Deps,permens No. (4) and (7). Waer dep (m) Smlaed (F r =.8) Smlaed (F r =.6) Measred (F r =.8) Measred (F =.6) Dsance (m) Fgre 7. Comparson beween Measred and Smlaed Jmp Proles Usng Upsream rad nmbers,permens No. (8) and (9). r Waer dep (m) Smlaed (S =.5678) Smlaed (S = ) Smlaed (S =.456) Measred (S =.5678) Measred (S = ) Measred (S =.456) Dsance(m) Fgre 8. Comparson beween Measred and Smlaed Jmp Proles Usng Deren Bed Slopes,permens No. (), () and ().

11 Twel Inernaonal Waer Tecnology Conerence, IWTC 8, Aleandra, gyp GNRAL QUATION USING RGRSSION ANALYSIS I s assmed a e dependen varable (L), wc s e dsance rom e begnnng o e lme o e locaon o ydralc jmp as sown n Fgre (9) s a ncon o e ollowng ndependen varables: densy o low ρ, e psream waer dep ( ) a vena conraca, e al waer dep ( ) a e end o e cannel, e velocy a vena conraca ( v ), e acceleraon o gravy (g), bed slope (S ). L Fgre 9. Denon Sec o Hydralc Jmp Te general ncon relaonsp beween e above varables can be wren as: ( L,,,,, g,s). ρ (4) = Usng e dmensonal analyss, e π erms obaned are, = v / g = Fr π = L /, = / π, π, and π 4 = S. Tese π erms may be arranged n e ollowng nondmensonal orm: L (,, Fr, S ) =. Te general orm o eqaons relang a dependan P-erm w a nmber o ndependen P-erms sng e regresson analyss [] n s wor s n e orm o e prodc o powers o relevan P erms,.e., a a a π C π π π (5) am = π m (6) qaon (6) can be ransormed o a lnear epresson by ang logarms o bo sdes o e eqaon, as ollows: = log C a a a a m m (7) Fnally, e eqaon can be rewren n marces orm as ollows:

12 Twel Inernaonal Waer Tecnology Conerence, IWTC 8, Aleandra, gyp N log π log π m log π log π log π m log π log π log π log π m log π m log C log log log m π a π π log π m log π log log a = π π log a m log log m π m π π m (8) Were, N s e nmber o observaons. Ten, e vales o e parameers C, a, a,..., a m are obaned and can be replaced bac no eqaon (6). I e erm ( L/ ) s aen as a dependen erm, e eqaon orm wll be as ollows: Or L.7 = 65 r S.9 ( F ) r.7.66 (9) 65 F S L = () qaon () was sed o predc e locaon o ydralc jmp. Te predced vales were compared w e smlaed vales o cec e valdy o e above eqaon. Te comparson beween e predced and observed vales sows a good agreemen as llsraed n Fgre (). 4. Predced Vales Smlaed Vales Fgre. Comparson beween Prdeced and Smlaed Vales or Te Locaon o Hydralc Jmp SUMMARY AND CONCLUSION Bossnesq eqaons were solved nmercally, sng e MacCormac Sceme o smlae e ydralc jmp n a recanglar cannel. Te nseady low compaons were conned n s sceme nl ey converged o a seady sae low. Te

13 Twel Inernaonal Waer Tecnology Conerence, IWTC 8, Aleandra, gyp locaon o e ydralc jmp was deermned as a par o ese compaons. Deals o e MacCormac Sceme, sably condons, and e nal and bondary condons were eplaned. Twelve laboraory epermens were carred o o ge e reqred daa necessary or vercaon o e MacCormac sceme. Te comped resls sng e MacCormac scemes were vered by comparng em w e epermenal daa. Te comparson beween e smlaed resls, obaned sng =. m, Coran nmber =.65 and κ =., and e measred daa sows a good agreemen as sown rom Fgre 4 o Fgre 8. Usng regresson analyss e power emprcal eqaon () was developed o predc e locaon o ydralc jmp. Te predced vales were compared w e observed vales and a good agreemen was aceved as sown n Fgre. RFRNCS. Abbo, M. B., Compaonal Hydralcs; lemens o e Teory o Free Srace Flows, Pman Pblsng Lmed, London, Anderson, J. D., Degrez, G., Dc,. and Grndmann, R., "Compaonal Fld Dynamcs; An Inrodcon", A Von Karman Inse Boo, Sprnger-Verlag, USA, 99.. Cadry, M. H., Open-Cannel Flow, Prence Hall, Inc., New Jersey, USA, Cow, V. T., "Open Cannel Hydralcs", McGraw-Hll Boo Co., Inc., New Yor, USA, Fennema, R. J. and Cadry, M.H., plc Nmercal Scemes or Unseady Free-Srace w Socs, Waer Resorces Researc, Vol., No., 9-9, Garang, A. M. and Cadry, M. H., Nmercal Smlaon o Hydralc Jmp", Jornal o Hydralc ngneerng, ASC, Vol. 7, No.9, pp. 95-, Sepember Gnal, M., and Narayanan, R., "Hydralc Jmp n Slopng Cannels", Jornal o Hydralc ngneerng, Vol., No. 8, pp , Ags, Kaopodes, N. D., "A Dsspave Galern Sceme or Open Cannel Flow", Jornal o Hydralc ngneerng, ASC, Vol., No. 4, pp , Mamood, K., and Yevjevc, "Unseady Flow n Open Cannels", Vol. I, Waer Wyle,. Resorces Pblcaons, McCorqodale, J. A., and Kala, A., Inernal Flow n Hydralc Jmps, Jornal o Hydralc ngneerng, ASC, Vol. 9, No. 5, pp , 98.. Molls, T., Zao, G., "Dep-Averaged Smlaon o Spercrcal Flow n Cannel w Wavy Sde Wall", Jornal o Hydralc ngneerng, ASC, Vol. 6, No. 6, pp , Jne,.

14 Twel Inernaonal Waer Tecnology Conerence, IWTC 8, Aleandra, gyp. Samaa, M.T., plc Nmercal Scemes or Te Smlaon o Hydralc Jmp, Mansora ngneerng Jornal, Facly o ngneerng, Mansora Unversy, gyp, Vol. 8, No., pp. C.9-C.5, Sepember.. Terrence, J.T., Appled Nmercal Meods or ngneerng, Jon Wley & Sons, Inc., New Yor, USA, Weyan, T., "Sallow Waer Hydrodynamcs; Maemacal Teory and Nmercal Solon or a Two-dmensonal Sysem o Sallow Waer qaons", lsever Scence Pblsers, Amserdam, Te Neerlands, Zang, W., and Scmd, B.H., "Nmercal Smlaon o Rapdly Varyng Flows n Nonprsmac Cannels", nd Inernaonal Conerence on Rver Flood Hydralcs, Yor, ngland, pp. 7-46, -5 Marc, 994. NOTATION Te ollowng symbols are sed n s paper: C n = Coran nmber; c = wave celery; F = vecor o les n -drecon; F r = Frode nmber a vena conraca; = general ncon; g = acceleraon de o gravy; = low dep; = al waer dep; = psream waer dep a vena conraca; = compaonal nodes; = me level; L = e dsance rom e psream bondary o e locaon o jmp; n = Mannng's rogness coecen; Q = dscarge; R = ydralc rads; S = vecor o sorce erms (rcon and boom slope); S = rcon slope; S = bed slope o e cannel; = me; U = vecor o low varables = (, ) T ; = low velocy n -drecon; v = e velocy a vena conraca ( v ); = dsance along cannel boom posve n downsream drecon. = me nerval; = space nerval; and κ = dsspave coecen.