Linear and Weakly Nonlinear Instability of Slightly Curved Shallow Mixing Layers

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1 WEA TAACTIO on FLUID MECHAIC Irn Ele Andre Kolshn Lner nd Wel onlner Insbl o lhl Crved hllow Mn Lers IIA EGLITE ADEI KOLYHKI Deprmen o Enneern Mhems Tehnl Unvers Mez sr blo 4 LATVIA rnele@mlom olsns@rbslv Absr: - The pper s devoed o lner nd wel nonlner sbl nlss o shllow mn lers The rds o rvre s ssmed o be lre Lner sbl problem s solved nmerll sn olloon mehod bsed on Chebshev polnomls I s shown h or sbl rved mn lers rvre hs sblzn ee on he low Wel nonlner heor s sed o derve n mplde evolon eqon or he mos nsble mode I s shown h he evolon eqon n hs se s he Gnzbr-Lnd eqon wh omple oeens Epl ormls or he llon o he oeens o he Gnzbr-Lnd eqon re derved merl lorhm or he ompon o he oeens s desrbed n del Ke-Words: - Lner sbl wel nonlner heor mehod o mlple sles Gnzbr-Lnd eqon olloon mehod Inrodon hllow mn lers or n mn enneern pplons Flows n ompond nd ompose hnnels nd lows rver nons re pl emples o shllow mn lers Mehods o nlss o shllow mn lers nlde epermenl nveson nmerl modeln nd sbl nlss [] Epermenl nveson o shllow mn lers s onded n mn ppers []-[4] I s shown n []-[4] h boom ron pls n mporn role n sppressn perrbons In ddon he re o rowh o he mn ler s lso reded n omprson wh ree mn lers Lner sbl nlss o shllow lows s perormed n [5]-[] d-ld ssmpon s sed n [5] o deermne he rl vles o he bed ron nmber or we lows nd mn lers The pplbl o he rd-ld ssmpon o he sbl nlses o shllow lows s nlzed n [6] where s shown h or smll Frode nmbers he error n sn he rd-ld ssmpon s qe smll The ee o Frode nmber o he sbl o shllow mn lers n ompond nd ompose hnnels s sded n [8] The resls presened n [5]-[] show h he bed ron nmber sblzes he low nd redes he rowh o mn ler Cenrl nsbl n lso or n shllow mn lers The ee o smll rvre o he sbl o ree mn lers s nvesed n [] I s shown n [] h rvre hs sblzn ee on sbl rved mn ler nd desblzn ee on nsbl rved mn ler Lner sbl nlss n be sed o deermne how prlr low beomes nsble Crl vles o he prmeers or emple rl bed ron nmber rl wve nmber nd so on re lso esmed rom he lner sbl heor Developmen o nsbl bove he hreshold nno be nlzed b lner heor Wel nonlner heores [] [] re sed n order o onsr n mplde evolon eqon or he mos nsble mode These heores re bsed on he mehod o mlple sles [4] nd re pplble he low s nsble b he vle o he prmeer or emple enolds nmber or bed ron nmber or shllow lows s lose o he rl vle In hs se he rowh re o nsble perrbon s smll nd one n hope o nlze he developmen o nsbl b mens o relvel smple evolon eqons h n pproh s sed n [] or plne Poselle low n [5] nd [6] n order o nlze nsbl o wves enered b wnd nd n [] [7]-[9] or shllow we lows In mplde eqons re sed n he lerre n wo ws Frs prlr orm o he evolon eqon s seleed pror nd he oeens o he eqon re esmed rom epermenl d Then he eqon wh esmed oeens s sed o model he phenomenon o neres eond one n ll derve n I: Isse Volme 6 Aprl

2 WEA TAACTIO on FLUID MECHAIC Irn Ele Andre Kolshn evolon eqon rom he eqons o moon Ths pproh s sed n [] [] [5] [7] nd [9]where s shown h or wo-dmensonl ses he evolon eqon s he omple Gnzbr-Lnd eqon In he presen pper we perorm lner sbl nlss o slhl rved shllow mn lers The rl vles o he sbl prmeers re evled Wel nonlner heor s sed ler o derve n mplde evolon eqon or he mos nsble mode I s shown h he evolon eqon s he Gnzbr-Lnd eqon wh omple oeens Epl ormls or he llon o he oeens o he Gnzbr- Lnd eqon re presened Dels o he nmerl lorhm re dsssed Lner sbl problem We onsder shllow wer eqons o he ollown orm v p v h v v v p v v h v v where nd v re he deph-vered velo omponens n he nd -dreons respevel s he dreon lon he sremlne s perpendlr o p s he pressre h s wer deph s he ron oeen nd s he rds o rvre I s ssmed here h / << Inrodn he srem non b he relons v 4 nd elmnn he pressre we rewre he ssem - n he orm h h 5 where he sbsrps nde he dervves wh respe o he vrbles nd nd Consder perrbed solon o 5 n he orm ε ε ε 6 where he role o he prmeer ε wll be lred ler The bse low s reled o b he orml In lssl heor o hdrodnm sbl [] he bse low s sll smple solon o he eqons o moon As n emple we onsder he ver-oes eqons where he velo veor hs onl one nonzero omponen whh s non o rdl oordne onl olvn he ver- oes eqons we obn prbol velo dsrbon he Poselle low Ths pproh does no wor or shllow wer eqons: s no possble o nd smple nll solon o - Bse lows n he se o shllow wer eqons re sll hosen n he orm o relvel smple model velo proles sh s hperbol nen prole or shllow mn lers or hperbol sen prole or shllow we lows These proles re hosen on he bss o rel nlss o vlble epermenl d The ollown wo bse low proles wll be sed below: nh 7 nh 8 Prole 7 orresponds o sbl rved mn ler he hh-speed srem s on he osde o he low-speed srem whle prole 8 represens nsbl rved mn ler he hh-speed srem n on he nsde o he low-speed srem I: Isse Volme 6 Aprl

3 WEA TAACTIO on FLUID MECHAIC Irn Ele Andre Kolshn bsn 6 no 5 nd lnerzn he resln eqon n he nehborhood o he bse low we obn L 9 where L h o o Usn he mehod o norml modes we see he solon o 9 n he orm ep[ ] where s he wve nmber s he phse speed nd s he mplde o he norml perrbon bsn no 9 we obn L where '' L [ nd ' / / / / b / h s he sbl prmeer b s he hrers lenh sle The bondr ondons re ± 4 Problem -4 s n eenvle problem The omple eenvles r deermne he lner sbl o he bse low whh s sd o be sble ll < nd nsble les one > The ondon 5 orresponds o nerll sble perrbons merl mehod The psedosperl olloon mehod bsed on Chebshev polnomls s sed o solve eenvle problem -4 nmerll The nervl < < s rnsormed no he nervl b mens o he rnsormon r rn The solon o s hen soh π n he orm r r T r 6 where T r os rosr s he Chebshev polnoml o he rs nd o deree nd re nnown oeens The ollown se o olloon pons s sed o solve 4: πm r m os m 7 bsn 6 no nd evln he non r nd s dervves p o order wo nlsve he olloon pons 7 we obn he enerlzed eenvle problem o he orm B D 8 where B nd D re omple-vles mres nd T More deled desrpon o he nmerl lorhm s ven n bseon n he one o wel nonlner llons The or r rnees h he bondr ondons 4 n erms o he new vrble r re ssed omll r ± There re les wo resons wh solons o he orm 6 re more onvenen hn hose obned b lssl olloon mehods []: he se o he bse nons h ss he ven zero bondr ondons onsderbl redes he ondon nmber [] nd b he mr D n 8 s no snlr Problem 8 s solved nmerll b mens o he IML rone DGVCCG The resls o nmerl llons re presened n Fs nd Three nerl sbl rves or he se o sbl rved mn ler bse low prole 7 re shown n F or he ollown vles o he prmeer / : nd 4 rom op o boom The low beomes more sble s rvre nreses I: Isse Volme 6 Aprl

4 WEA TAACTIO on FLUID MECHAIC Irn Ele Andre Kolshn F erl sbl rves or bse low prole 7 The vles o he prmeer / re nd 4 rom op o boom erl sbl rves or he se o nsbl rved mn ler bse low prole 8 re shown n F As n be seen rom he re rvre hs desblzn nlnee on he low he rl vles o he sbl prmeer nrese s / nreses domnn nd here s lle hope o nlze he developmen o nsbl nlll However he rowh re o he nsble mode s relvel smll hen wel nonlner heores n be sed n order o develop n mplde evolon eqon or he mos nsble mode h eqons re obned n he ps or he se o plne Poselle low shllow wer lows wves on he sre enered b wnd nd n some oher sons see [] [] [5]-[9] ppose h nd re he rl vles o he sbl prmeer wve nmber nd wve speed respevel Then he mos nsble mode n ordne wh he lner heor s ven b wh nd where he eennon n be repled b C The onsn C nno be deermned rom he lner sbl heor In order o nlze he developmen o nsbl nlll n he rmewor o wel nonlner heor we onsder smll nehborhood o he rl pon n he -plne where prmeer s ssmed o be slhl below he rl vle: ε The onsn C n hs se wll be repled b slowl vrn mplde non A Follown he pper b ewrson nd r [] we nrode he slow me nd londnl oordnes nd b he relons F erl sbl rves or bse low prole 8 The vles o he prmeer / re nd 4 rom boom o op Wel nonlner nlss Usn lner sbl heor one n deermne he ondons nder whh prlr low beomes nsble merl solon o he orrespondn eenvle problem llows one o obn he rl vles o he prmeers o he problem nd deermne he srre o he nsble mode However lner heor nno be sed o pred he evolon o he mos nsble mode bove he hreshold In he nsble reon perrbon rows eponenll wh me see I he rowh re s lre hen nonlner ees ql beome ε ε where s he rop velo Ths A A nd he non n now hs he orm A ep[ ] 9 where he bbrevon mens he omple one The srem non n 6 n be represened s ollows: Usn he hn rle we n rewre he dervves o wh respe o nd n he orm I: Isse Volme 6 Aprl

5 ε ε ε In oher words he derenl operors nd re repled b ε ε ε bsn 6 no 5 sn nd ollen he erms o order ε we obn he ollown eqon or he non : ] [ ] [ L oe h he operor L on he le-hnd sde o s he sme s n 9 nd wll be he sme or ll orders n ε mlrl ollen he erms o order ε we obn ] [ ] / 5 [ L e we onsder he solon o I n be shown h sbsn 9 no he rh-hnd sde o he ollown hree rops o erms wll emere: he erms h re ndependen on me b he erms proporonl o he rs hrmon ] ep[ nd he erms proporonl o he seond hrmon ] ep[ here nd n seqel we drop he sbsrps nd se he noon nd or onvenene Ths he non shold lso onn he sme hree rops o erms More presel we see he solon o n he orm ] ep[ ] ep[ A A AA 4 where nd re nnown nons o A denoes he omple one o A he spersrp reles he nde o he hrmon omponen nd he sbsrp represens he order o ppromon bsn 9 nd 4 no nd ollen he erms proporonl o AA elds WEA TAACTIO on FLUID MECHAIC Irn Ele Andre Kolshn I: Isse Volme 6 Aprl

6 WEA TAACTIO on FLUID MECHAIC Irn Ele Andre Kolshn [ [ The bondr ondons hve he orm ] ] 5 ± 6 mlrl sbsn 9 nd 4 no nd ollen he erms proporonl o ep[ ] we obn he ollown eqon or he non wh he bondr ondons ± 8 7 Comprn 7 nd one n see h he rh-hnd sde o 7 s el he sme s he rh-hnd sde o s repled b Ths 7 s resonnl ored nd solvbl ondon shold be ppled hs se o rnee he esene o he solon Usn he Fredholm s lernve [] we onlde h eqon 7 hs solon nd onl he rh-hnd sde o 7 s orhoonl o ll eennons o he orrespondn homoeneos don problem The don operor L nd don eennon re dened b he relon The le-hnd sde o 9 s eql o zero sne s he solon o Ths he don eqon s dened b he orml L Inern he le-hnd sde o 9 b prs nd sn he bondr ondons 4 we obn he don operor n he orm L The bondr ondons re ± The don eennon s he solon o he problem Appln he solvbl ondon o 7 we obn [ ] d Eqon denes he rop velo η η where η d nd 4 Ld L d 9 I: Isse Volme 6 Aprl

7 WEA TAACTIO on FLUID MECHAIC Irn Ele Andre Kolshn η [ 5 ] d Fnll sbsn 9 nd 4 no nd ollen he erms proporonl o ep[ ] we obn 8 [ 4 ] 4 8 / wh he bondr ondons / 6 ± 7 olvn hree bondr vle problems nmerll we obn he nons nd The non he seond order orreon s hen ven b 4 Le s onsder he solon he hrd order n ε Eqon lso hs solon nd onl he rh-hnd sde o s orhoonl o ll eennons o he orrespondn homoeneos don problem bsn 9 nd 4 no he rh-hnd sde o nd ppln he solvbl ondon we obn he mplde evolon eqon or slowl vrn mplde non A o he orm A A σa δ µ A A 8 Eqon 8 s he omple Gnzbr-Lnd eqon wh omple oeens σ δ nd µ where σ δ µ σ δ µ 9 η η η The oeens σ δ nd µ re ven b σ d δ [ µ {6 } d ] d ] [ 4 The oeen η n 9 s ven b 4 Formls 9 represen he oeens o eqon 8 n erms o he hrerss o he lner sbl o he low More presel n order o obn σ δ nd µ we need o perorm he ollown llons: solve he lner sbl problem 4 nd deermne he rl vles o he prmeers nd he orrespondn eennon ; solve he homoeneos don problem nd deermne he don eennon ; solve hree bondr vle problems nd deermne he nons nd ; evle he nerls n 9 merl mehod In hs sbseon we propose nmerl mehod or he llon o he oeens o he Gnzbr- Lnd eqon The solons o lner sbl problem - 4 don problem bondr vle problems re soh n he orm r r T r 4 I: Isse Volme 6 Aprl

8 WEA TAACTIO on FLUID MECHAIC Irn Ele Andre Kolshn where r represens n o he nons r r r r r rell h r rn Usn he hn rle we ompe π he dervves o he rs seond nd hrd order o wh respe o : d d πr d os π dr d 4πr d 4 πr os sn os d π dr π πr 4 n d 6πr d os d π dr 4 πr sn os π 4 5 πr d 8 os dr π 6 πr d dr πr d dr 4 The dervves o wh respe o r re evled sn 4: d dr d dr d dr r [ rt T [ T ''' r] r r ' [ 6T r 4rT ' r 6rT T ' r] r r '' r T '' r] 4 In order o evle he non r nd s dervves p o he hrd order we need o ompe he vles o he Chebshev polnoml r nd s dervves he olloon pons 7: mπ T rm os mπ sn ' T rm πm sn πm mπ os sn '' T rm πm sn T mπ os πm sn πm os ''' mπ T rm sn πm 5πm sn sn 4 πm os πm os 4πm sn The vles o r s dervves p o order wo nlsve nd he oeens o eqon he olloon pons 7 n be evled sn ormls 4-4 so h he elemens o he mres B nd D see 8 n be omped nd he enerlzed eenvle problem 8 n be solved nmerll mlr pproh n be sed n order o solve bondr vle problems 5 6 nd 6 7 sem o lner lebr eqons o he orm F G 44 s obned n eh se er dsrezon where T The mr F s no snlr or problems 5 6 nd 6 7 Thereore n lner eqon solver n be sed n order o nd Ths he nons nd n be evled b mens o he epnsons o he orm 4 The sme orm o he epnson 4 s sed o solve bondr vle problem 7 8 Eqon o he orm 44 s lso obned er dsrezon n hs se b he mr F s snlr sne he orrespondn homoeneos pr o 7 hs nonrvl solon nd Eqon 44 s solved n hs se b mens o he snlr vle deomposon mehod [4] I s nown h F s omple mr hen here es orhoonl mres U nd V sh h U H FV Σ 45 where Σ d γ γ γ Eqon 45 s lled he snlr vle deomposon o he mr F nd γ γ γ re he snlr vles o F In or se onl he ls o he snlr vles wll be eql o zero I: Isse Volme 6 Aprl

9 WEA TAACTIO on FLUID MECHAIC Irn Ele Andre Kolshn γ > γ > > γ > γ Hene he solon o 44 n hs se n be wren n he orm H VΣ U G 46 H where he ls olmn o V he ls row o U he ls olmn nd he ls row o Σ re deleed In omponen orm he solon o 46 s H U DV γ 47 H where U nd V re veors olmns o he H mres U nd V respevel Hene he vles o he non n be omped sn orml 4 where he oeens re he omponens o he veor n 47 The nl sep o he omponl proedre nvolves he llon o nerls n 9 Adpve qdrre orml desrbed n [] n be sed o ompe he nerls n 9 Gnzbr-Lnd model po-emporl dnms o omple lows s oen nlzed b relvel smple evolon eqons sh s Lnd or Gnzbr-Lnd eqons [5]- [6] The models re relvel smple so h phsss r o se hem n order o desrbe omple phenomen In mn ses he orm o he eqon s ssmed b he oeens o he eqon re lled rom epermenl d n oher words phenomenolol model s sed Emples n be ond n [7]-[9] where s shown h he Lnd nd Gnzbr-Lnd eqons n be sessll sed n order o desrbe epermenl observons o lows behnd bl bodes n wde rne o enolds nmbers In he presen pper we show h he Gnzbr- Lnd eqon does no need o be ssmed or slhl rved shllow mn lers s ll derved rom he shllow wer eqons nder he rd-ld ssmpon 4 Conlson In he presen pper sbl nlss o slhl rved shllow mn lers s perormed Lner sbl hrerss o hperbol nen velo prole re lled I s shown h sbl rved mn ler s even more sblzed b he nresn rvre whle or nsbl rved mn lers rvre hs desblzn ee on he low Mehod o mlple sles s sed n he pper o derve n mplde evolon eqon or he mos nsble mode n he rmewor o wel nonlner heor I s shown h he evolon eqon s he omple Gnzbr-Lnd eqon Epl ormls or he llon o he oeens o he Gnzbr-Lnd eqon re presened merl lorhm or he llon o he oeens s proposed nd nlzed n del eerenes: [] GH Jr Lre sle low srres nd mn proesses n shllow lows Jornl o Hdrl eserh Vol9 pp [] VH Ch nd Bbrs Connemen nd bed-ron ees n shllow rblen mn lers Jornl o Hdrl Enneern ACE Vol4 988 pp [] WJ Uewl nd Boo Ee o shllowness on he developmen o ree-sre mn lers Phss o Flds Vol pp 9-4 [4] WJ Uewl nd J Ter Developmen o qs wo-dmensonl srres n shllow ree-sre mn ler Epermens n Flds Vol4 998 pp 9- [5] VH Ch JH W nd E Kh bl o rnsverse sher lows n shllow open hnnels Jornl o Hdrl Enneern ACE Vol7 99 pp 7-88 [6] D Chen nd GH Jr Lner sbl nlss o rblen mn lers nd es n shllow wer lers Jornl o Hdrl eserh Vol6 998 pp 85-8 [7] M Ghdo nd AA Kolshn Lner sbl nlss o lerl moons n ompond open hnnels Jornl o Hdrl Enneern ACE Vol5 999 pp [8] AA Kolshn nd M Ghdo Grvonl nd sher nsbles n ompond nd ompose hnnels Jornl o Hdrl Enneern ACE Vol8 pp [9] BC Prooen nd WJ Uewl A lner pproh or he evolon o oheren srres n shllow mn lers Phss o Flds Vol4 pp I: Isse Volme 6 Aprl

10 WEA TAACTIO on FLUID MECHAIC Irn Ele Andre Kolshn [] AA Kolshn nd M Ghdo bl nlss o shllow we lows Jornl o Fld Mehns Vol494 pp [] WW Lo Lner nsbl o ree rved sher lers Phss o Flds Vol6 99 pp [] K ewrson nd JT r A non-lner nsbl heor or wve ssem n plne Poselle low Jornl o Fld Mehns Vol48 97 pp [] C Godréhe nd P Mnnevlle Hdrodnms nd nonlner nsbles Cmbrde Unvers Press 998 [4] J Kevorn nd JD Cole Mlple sle nd snlr perrbon mehods prner 996 [5] PJ Blennerhsse On he eneron o wves b wnd Phlosophl Trnson o he ol oe o London er A Vol98 98 pp [6] Lebln Amplon o nonlner sre wves b wnd Phss o Flds Vol9 7 pp [7] F Feddersen Wel nonlner sher wves Jornl o Fld Mehns Vol7 998 pp 7-9 [8] M Ghdo AA Kolshn JH Ln FC Chn Q L nd K X Lner nd nonlner nlss o shllow wes Jornl o Fld Mehns Vol548 6 pp 9-4 [9] AA Kolshn nd zrovs Lner nd wel nonlner nlss o wo-phse shllow we lows WEA Trnsons on Mhems Vol6 o 7 pp -8 [] PG Drzn nd WH ed Hdrodnm sbl Cmbrde Unvers Press 98 [] C Cno MY Hssn A Qreron nd TA Zn perl mehods: ndmenls n snle domn prner 6 [] W Henrhs Improved ondon nmber or sperl mehods Mhems o Compon Vol5 o pp -9 [] D Zwllner Hndboo o derenl eqons Adem Press 998 [4] GH Golb nd CF vn Lon Mr ompons The Johns Hopns Unvers Press 98 [5] L Arnson nd L Krmer The world o he omple Gnzbr-Lnd eqon evew o Modern Phss Vol74 pp 99-4 [6] MC Cross nd PC Honenber Pern ormon osde o eqlbrm evew o Modern Phss Vol65 99 pp 85- [7] M Provnsl C Mhs nd L Boer The Bénrd-von Kármán nsbl: rnsen nd ored remes Jornl o Fld Mehns Vol8 987 pp - [8] T Lewee nd M Provnsl The low behnd rns: bl bod wes who end ees Jornl o Fld Mehns Vol pp 65- [9] M hmm E Berer nd PA Monewz el-eed osllons n he we o wodmensonl bl bodes Jornl o Fld Mehns Vol7 994 pp 7-5 [] GE Forshe MA Mlolm nd CB Moler Comper mehods or mheml ompons Prene Hll 977 I: Isse Volme 6 Aprl