Hydraulic Model of Dam Break Using Navier Stokes Equation with Arbitrary Lagrangian-Eulerian Approach
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1 IACIT Ieraoal Joral of Egeerg ad Tecolog ol. 8 No. 4 Ags 6 dralc odel of Dam Break Usg Naer okes Eqao Arbrar Lagraga-Elera Aroac Alreza Lorasb oarram Dolasa Prooz ad Alreza Laae Absrac Te lqd flo ad e free srface sae drg e al sage of dam breakg are esgaed. A mercal sceme s deeloed o redc e ae of a sead comressble scos flo free srface. Te meod oles a o dmesoal fe eleme D a ercal la. Te Naer-okes eqaos for coserao of momem ad mass for Neoa flds co eqao ad fll olear kemac free-srface eqao ere sed as e goerg eqaos. Te mag deeloed o sole gl deformed free srface roblems commo aes formed drg ae roagao rasforms e r model from e scal doma o a comaoal doma Arbrar Lagraga Elera ALE fe eleme modelg ecqe. Ide Terms Dam break aer-sokes eqaos freesrface flos arbrar lagraga elera. I. INTRODUCTION Gra-dre flos de o dam breakg sded b Zekecz [] Pole [] ad oker [] sg e Lagraga descro. Pole reseed a ssemac rocedre for e deermao of e sccesse erms ese easos. oeer ol e leadg-order erms ere cosrced ad aalzed. I bo Lagraga ad Elera descros e easos of e solo me oer seres sold be cosdered as oer solos c are eeded o be correced er solos ear e erseco o. oeer sc a er solo as sccessfll dered a relea roblem cocerg a forml accelerag ae maker b Kg ad Needam [4]. eeral mercal sdes erformed drg e as fe ears ere based o e solo of olear sallo-aer eqaos sg dffere meods sc as e fe-olme meod e fe-dfferece meod ad so o see [5] []. Tere are er aalses of e dam-break roblem. [] sed a kemac ae aromao o oba a closedform solo for a slog cael ad meoed a s solo s ald for large mes. Te dam-break roblem ca be erreed e coe of a lqd colm collasg der gra []. Pee ad Torll [] sded e collase of a fld colm c s srroded b a lger fld. Te aalss as erformed Elera arables bo for small ad moderae mes. Te dered e al asmoc of e solo for ascr receed Ocober 4; resed Febrar 7 5. Alreza Lorasb ad Alreza Laae are Dearme of Cl egeerg College of egeerg Borojerd Brac Islamc Azad Uers Ira e-mal: Ar_lorasb@aoo.com seab@gmal.com. oarram Dolasa Prooz s cool of Cl Egeerg Uers College of Egeerg Uers of Tera Ira e-mal: mdola@.ac.r. fld colms of sem-cldrcal ad emsercal saes ad soed a ese asmoc are o ald close o e base ere e fld eloc s mc ger a e res of e colm. Te mercal solo for free srfaces as bee deeloed sg ree defed eores: Lagraga; Elera; Arbrar Lagraga-Elera descro. For aes arclar e Arbrar Lagraga-Elera s seror erms of adlg g dsoro e grds. To ae a ersale descro of e fld doma s ecessar o ae a meod e beefs of bo Lagraga ad Elera descros o er defceces. c a meod deeloed e las o decades s e "Arbrar Lagraga- Elera formlao" c grd os ma be moed e fld ormal Lagraga descro. Ts meod allos e grds moe deede of e fld moo. II. PROBLE FORULATION We cosder e lae roblem of gra-dre flo c s geeraed e a ercal dam fro of a lqd rego s sddel remoed. Iall codo of e lqd s. Fg.. Flo rego a al me sa =. A e al me sa = e dam s sal remoed ad e gra-dre flo sars. Te reslg flo s oeal ad o-dmesoal. We sall deerme e lqd flo ad e sae of s free srface drg e earl sages of e rocess. Te scal doma srroded b a ecese smoo bodar s so Ts doma s occed b a scos comressble fld e coeffce of cosa kemac scos of ν ad e secfc mass of ρ. Te roblem der cosderao s e sead moo of a srface ae der gra. To-dmesoal sead comressble scos flo s cosdered. Te goerg eqaos are eressed b e sead Naer-okes eqao ad e eqao of co. Le e recaglar coordaes be deoed b ad e corresodg eloc comoes be deoed b. As a resl e eqaos of coserao of momem ad mass for comressble Neoa flds e arbrar DOI:.776/IJET
2 Lagraga-Elera form are ge as follos: g ere ad are e mes eloces ad drecos. Te bodar cosss of o es of bodares: oe s e o c eloc s ge; e oer s e free srface bodar o c e srface force s secfed. Te bodar codos ca be eressed as e follogs.... o c o c o o ere e serscr care deoes a fco c s ge o e bodar ad ad smbolze e dreco coses of e oard ormal o e bodar resec o co-ordae ad. To Eqaos ca be redered dmesoless b rodcg e follog arables: / / / g d d d 4 Usg ese rasformaos e Eq. ad Eq. are modfed as follos: III. NUERICAL ANALYI Te mercal model s based o a fe eleme meod for e saal dscrezao of aral dffereal eqaos. Ts meod s mlemeed sg eged resdal araoal ecqe for e solo aroac eac eleme. A. Basc Coce I e emoral dscrezao e oal me s dded o a mber of sor me cremes. Eac me o s deoed b. eloc ad ressre a e me o ca be defed as: 5 ere ad deoe e coordae a e me o e scal doma. Te arameers ad are e fed coordae a e me o e referece doma. eloc ad ressre a me o ca be defed sbseqel as: 6 I e Elera reame e saal dffereao ca be aromaed e form: IACIT Ieraoal Joral of Egeerg ad Tecolog ol. 8 No. 4 Ags 6
3 IACIT Ieraoal Joral of Egeerg ad Tecolog ol. 8 No. 4 Ags 6 W sbsg Eqaos e eqaos of moo co ad kemac bodar codo ca be dscreded o Te bodar codos corresodg are descrbed b:.. c.. c Te aalss rocedre reseed ere oles comg e ko arable ad e bodar codos sarg from e ko arable. 9 To sole Eqaos e fracoal meod s emloed. Ts meod s oe of e earles ad e mos del sed meod for solg fld damc roblems. I s meod b dscrezg e eqaos of moo e ermedae eloc ca be obaed. oeer s eloc ma o sasf e eqao of co. To correc e obaed ermedae eloc a correco oeal sold be rodced. Te Posso eqao for e correco oeal ca be dered b rg o sasf e eqao of co. B solg e resla Posso eqao e correco eloc ecor ca be obaed. ~ ~ ~ Ts eqao mles a ~ ~ ~ ere s a scalar c s referred o as e correco oeal. B akg e aral derao o bo sdes regard o ad resecel ad addg em ogeer e ae ~ ~ W sbsge eqao of co e eqao for ca be dered as ~ ~ W sbsg e eqao of corollg e eolo of ressre ca be eressed 4 I order o mleme a mercal solo rocedre for e arbrar Lagraga Elera formlao e momem eqao ad e comressbl cosra of e Naer-okes roblem are aalzed sg a rocedre c cosss of s searae ases. Le be e eloces ressre ad ae eg felds a me ere. From bodar secfcaos e felds are calclaed. ad e I. TRANFORATION OF TE BAIC EQUATION INTO TE APPED COORDINATE YTE Te comao of e roagao of free srface aes oles comaoal bodares a do o cocde coordae les scal sace. For e fe eleme meod sc roblem reqres a comlcaed erolao fco o e local grd les c resls e local loss of accrac e comaoal solo. c dffcles reqre a mag or rasformao from scal sace o a geeralzed sace. Ts rasformao smlfes e roblem of gl deformed ar-fld erface a arses e aalss of ae breakg. Ts mag rasforms e ae roagao model from e scal doma o a comaoal doma. Te se of geeralzed coordaes mles a a dsored rego scal sace sc as breakg ae s maed o a recaglar rego e geeralzed coordae sace ere e ko erface cocdes a coordae le as ce e eror os e comaoal doma form a reglar grd ad e bodares cocde coordae les e deermao of s easer a 5
4 IACIT Ieraoal Joral of Egeerg ad Tecolog ol. 8 No. 4 Ags 6 orkg e rreglar scal doma. W ferrg o e scal ad comaoal meses cre e follog mag ca be esablsed. Fg.. Te comaoal grd s so maed back o e scal sace. Te fco F F 5 s erolao fco. W Emlog ree o erolaos as so e ae: F F F F F 4 4 F F Fg.. Tree o erolao fco. 6 A. Elera Descro To ae a Elera descro ere e scal coordae ssem cocde e geeralzed coordae ssem s ecessar o se. B. Elera Descro Dreco ad Lagraga Descro Dreco Elera Descro dreco ad Lagraga Descro dreco ca be aled for obreakg aes. I s case s ecessar o se. Te rasformao s Lagraga dreco ad Elera dreco ad e roblems assocaed s rasformao sold ae sgle ale rofle. C. Arbrar Lagraga-Elera Descro Te Arbrar Lagraga-Elera algorm s emloed modellg ae roagao bo oer slog beaces ere e eolo occrs oer bamer oogra ad oer cosa de regos. Dffere es of ales are roded ad deedg o e are of e roblem. To cosde scal ad comaoal bodar e ales are cosdered o be a ff order olomal fco of as folloed: ad l 4 l l 5 b 5 [ 5 l 5 ] C /5 8 C Defos of b l ad are llsraed. Parameer C s a cosa coeffce ad s ale s obaed b ral ad error o sablze e roblem. Fg. 4. Defo of arameers fco. D. araoal Eqaos e Trasformed Doma aal dscrezao of aral dffereal eqaos e mercal model s based o a Galerk fe eleme meod. Ts meod s mlemeed sg e eged resdal araoal meod for solo eac eleme. Usg sadard lear sae fcos for a recaglar eleme aral coordae ssem e eloc ressre ad correco oeal felds e eleme are erolaed erms of er odal ales as follos: ere J s j k jl j kl J j l j jl lj J l j j l j ~ J j j J J 9 s e erolao fco ad rerese e odal ales a e ode of e eleme. s a scalar c s referred o as e correco oeal base o e Fracoal se meod reseed b aas ad aaaka 99. B ddg e oal me j 5
5 IACIT Ieraoal Joral of Egeerg ad Tecolog ol. 8 No. 4 Ags 6 o a mber of sor me cremes e eqaos of moo co ad kemac bodar codo ca be dscrezed o: ere d d d s d d I sold be oed a all of e deraos are resec o. Fg. 5. Dam break model rese mercal model a =. /=.. Fg. 6. Dam break model rese mercal model a =.5 /=.. Fg. 7. Dam break model rese mercal model a =5. /=... CONCLUION Te roagao ad deformao of free srface dam break oer fla bamer s esgaed. ere a dam searag o saoar aer leels s sddel remoed ad e almos ercal aes rogress o e o domas. Illsraes 'dam break' roblem dagrammacall /=.. ad so e dam break a =.5 sec ad =5 sec resecel. Ts model as e caabl of dam break model l /=4.5. Te arbrar Lagraga-Elera descro s eamed ere e saal coordaes are mog e eloc ad e comao s doe e referece coordae ssem ad. Te reaso for e selecg of arbrar Lagraga-Elera descro for modelg of dam breakg ae s force e model o coe seeral of ae rofles ad e model ca be emloed a geomer der comlcaed bodar codos ad arbrar bamer o a addoal comaoal effor. Te model s aldaed b comarg mercal resls eorecal solos ad resls obaed mercall. Oerall e coform beee e aalable daa ad e comaos s ell ad mos cases e mercal model ges ecelle resls. Te meod s esed o a free sead ae of fe amlde ad e aled o sead aes ad s fod o ge ecelle agreeme deede calclaos based o e oer esg eores. REFERENCE [] R. L. Zekecz J. Talor ad Z. Z Te Fe Eleme eod: Is Bass ad Fdameals Edo 5. [] F.. Pole Te Lagraga eqaos of drodamcs: solos c are aalc fcos of me PD dsserao Ne York Uers UA 95. [] J. J. oker Waer Waes Ierscece Pblsers Ic Ne York 957. [4] A. C. Kg ad D. J. Needam Te al deelome of a je cased b fld bod ad free srface eraco Joral Fld ec ol [5] C. Zoo ad. Robers Elc scemes for dam-break smlaos Joral dral Eg ol [6] P. Glaser olos of a o dmesoal dam break roblem I J Eg c ol [7] P. Glaser Aromae ma solos of e sallo aer eqaos Joral dral s ol [8] F. Lg ad E. F. Toro Eermeal ad mercal assessme of e sallo aer model for o dmesoal dam-break roblems Joral dral s ol [9] C. Zoo ad. Robers Nmercal solo of e odmesoal sead dam break Al a Flds s ol. 4 o [] P. Brfa ad P. Garca-Naarro To dmesoal dam break flo smlao I J Nmer eods Flds s ol [] B. Asmoc solo for dam break roblem J dral D ACE ol [] W. G. Pee ad C. K. Torll Te dserso der gra of a colm of fld sored o a r orzoal lae Pl Tras Ro oc Lodo A A. Lorasb s a academc member Dearme of Cl Egeerg College of Egeerg Borojerd Brac Islamc Azad Uers Ira. e sared eacg cl corse Azad Islamc Uers sce 6. s researc eress clde dao feld sres mercal ad aalcal modelg ad azard assessme mgao ad lag Breakg aes. e s a member of Amerca Cocree Ise ACI ad member of Cosrco Egeerg Dsclar Orgazao Leel Desg ad Corol Ira. 5
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