Weak Three Dimensionality of a Flow Around a Slender Cylinder: the Ginzburg-Landau Equation

Transcription

1 Weak Thee Dimeninaliy f a Flw Aund a J.A.P.Aanha DF, Mech. Eng, EPUSP S. P, Bazil apaan@up.b Weak Thee Dimeninaliy f a Flw Aund a Slende Cylinde: he Ginzbug-Landau Equain In hi pape a weak hee-dimeninaliy f he flw aund a lende cylinde i cnideed and he elaed mdel, he -called Ginzbug-Landau equain, i hee bained a an aympic luin f he 3D (dicee) avie-ske equain. The deivain i in line wih exiing lende bdie heie, a he Lifing Line They, f example, whee he baic D flw, leading Landau equain, i influenced nw by a idewah ha mdifie bi-dimeninally he iginal flw hugh ma cnevain. The hey i aympically cnien and e n an aumpin ha hld in he viciniy f he Hpf bifucain (ec 45) fuheme, i lead a well-eablihed way deemine numeically bh he Landau cefficien and Ginzbug cefficien. Agumen ae given uggeing ha hi aumpin huld hld fa beynd Hpf bifucain (e >> ec) and, wih i, exend he Ginzbug-Landau equain alm he bde f he aniin egin e 05. In hi wk nly he heeical develpmen i addeed numeical eul will be peened in a fhcming pape. Keywd:Hyddynamic abiliy, Ginzbug-Landau equain, lende cylinde Inducin Vicu flw aund a D cicula cylinde i knwn pduce pnaneu hamnic cillain f he wake f eynld numbe abve a ciical value e c 45, he cillaing pa f he peue giving ie an hamnic anvee fce n he cylinde ha ha impance in eveal engineeing applicain. I eem be nw well eablihed ha hi i eenially a abiliy pblem, ee Huee & Mnkewiz (990), and ha he cillay wake can be idenified wih Hpf bifucain in he language f he dynamic yem hey: hi idea wa fi advanced, in a me peainal way, by Bihp & Haan (964) and i ha been veified expeimenally, amng he, by Pvanal e. al. (987) in hei udy in he viciniy f he ciical eynld numbe. Empiical evidence hw ha bh he hedding fequency (Suhal numbe) and he whle phenmenn f he vex induced vibain, a lea in i me maccpic appeaance, ae eenially invaian wih eynld numbe up he aniin zne (e 0 5 ). Thi bevain ha led me auh (Iwan & Blevin (975), f example) ppe (heuiic) phenmenlgical mdel, baed n Van de Pl equain, pedic he hyd-elaic ineacin, wih eul ha ae impeive given he mewha le fluid dynamic backgund n which hey ae baed ineeingly enugh, he pedicin fm uch mdel ae, in me apec, in much bee ageemen wih expeimen han he ne bained fm diec CFD cmpuain. Being heuiic a hey ae, hweve, hey can be ued nly a inepla and hadly exaplae eul iuain much beynd he empiical daa n which hey ae baed fuheme, me hyeic behavi beved in he expeimen ae n ecveed by hee mdel and, bviuly, he diec link wih he avie-ske equain i lacking in uch appach. The final pupe f he n-ging eeach i deive a fluidelaic cilla mdel diecly fm avie-ske equain, endeing i n nly whle pedicive bu al making i pible be ued in diffeen iuain fm he ne beved in he exiing expeimenal faciliie in paicula, he cae whee he inciden cuen change bh in diecin and ineniy alng he cylinde pan i paiculaly elevan f ffhe applicain. ice al Peened a ECIT004 0h Bazilian Cnge f Themal Science and Engineeing, v Dec. 03, 004, i de Janei, J, Bazil. Technical Edi: Aila P. Silva Feie. ha he link wih he me fundamenal avie-ske equain ha hee an even geae mivain, ince diec cmpuain wih CFD did n pduce ye a eliable eul. In he peen pape nly he fi ep wad hi final gal i addeed, namely, deive he fluid cilla mdel by cnideing he cylinde fixed in he flw. The mdel i epeened by he -called Ginzbug-Landau Equain, ee (3.a), fi pped by Albaède e al (990) in he cnex f VIV, wih a baic diffeence, hweve, in elain he uual appach: nw hi equain i n fied exenally he pblem bu i eul fm a cnien aympic appximain f he 3D (dicee) avie-ske Equain (SE). In paicula, he cefficien f hi equain he Landau cefficien and he Ginzbug cefficien ae n infeed fm he expeimen bu hey ae diecly cmpued by well eablihed numeical pcedue baed n he Finie Elemen Mehd (FEM) applied he D c-flw pblem: a uual in lende bdie heie, ne ha hu an eenially D eff cmpue a 3D eul. The dicee FEM mdel i deived, a alway, in a finie fluid egin and ne ha ceainly a difficuly define he dicee fluid flw pea in due he lely knwn fm f he ppe bunday cndiin a he ule f. By cnideing he flw equain in he wake i ha been pible expe he eiance ffeed by he wake n he flw wihin, named hee he wake impedance, by an explici expein ha depend lely n he velciy and acceleain f he flw n he bdeline ha define he ineface beween and he wake. Thi deivain i elabaed elewhee and i may have an impance ha ancend he pecific applicain aimed in hi wk. The final dicee SE emulae he cninuum SE wih a lcal ineia, a cnvecive ineia and a vicu diipain ha incpae he cnibuin fm bh he finie fluid egin, ha i acually diceized, and he wake. Thi dicee e f equain ae hu peced in he lenidal and gadien ub-pace and andad eul in Linea Algeba ae ued hw he inne cniency f hee pecin in paicula, he pecin n he lenidal ub-pace, ha deemine he velciy field, lead a nmal quadaic dynamic yem which he uual aympic pcedue can be applied, deemine fi Landau Equain in he D cnex and afe he 3D Ginzbug-Landau Equain. A i i knwn, a myiad f ineeing mall cale feaue, me f hem uncveed by a deailed numeical analyi, appea cncmianly wih he g maccpic deed behavi f he wake ha eally mae in he udy f he hyd-elaic phenmenn (VIV): he pupe in hi wk i i n, hu, J. f he Baz. Sc. f Mech. Sci. & Eng. Cpyigh 004 by ABCM Ocbe-Decembe 004, Vl. XXVI,. 4 / 355

2 J.A.P.Aanha peen a axnmy f he cha bu ahe capue he undelying de. T achieve hi gal a f blindne i needed, avid a deailed picue, and he aympic hey i u a echnical file ha pvide i. Thi hey e n a well defined aumpin, ee (.), ha can be diecly veified by he numeical eul fuheme, alhugh icly uified nly in he viciniy f Hpf bifucain (e e c ), i eem hld in a much bade ange f eynld numbe, wha make pible exend Ginzbug-Landau equain hi ange. Incidenally, hi ye peculaive eul can funih a heeical backgund f he called phenmenlgical mdel ha ae in fac applied, wih a elaive ucce in he pedicin f VIV, in a ange f eynld numbe fa beynd he Hpf bifucain (e >> e c ). The pape i ganized a fllw: in ecin he wdimeninal pblem i addeed, leading Landau equain, and in ecin 3 a weak hee dimeninaliy f he flw i cnideed and Ginzbug-Landau equain i bained. Sme me echnical eul, including he deivain f he wake impedance, ae deived elewhee and numeical eul will be peened in a fhcming pape. Tw-Dimeninal Sluin: Landau Equain In hi ecin he w dimeninal c-flw aund a cylinde i cnideed. Pin in he c ecin plane ae deignaed by he vec x = xi + y, he fluid velciy by he vec field u(x,) = u(x,)i + v(x,), he peue by he funcin p(x,) while he diffeenial pea i defined by he expein = i/x + /y hee nain will be kep all hugh he wk, even in he nex ecin whee he hee dimeninal cecin will be addeed. Le d = be he ypical dimenin f he cylinde c ecin (he cylinde diamee in he cae f a cicula cylinde) and U = be he inciden velciy alng he x-axi a he infinie bviuly, he nn-dimeninal fequency d/u cincide numeically wih and bh fm will be ued hee, depending n he cnvenience. The fluid deniy will be al aumed uniay ( = ) and hu v = v /Ud = /e, whee v i he fluid viciy and e he eynld numbe. I i deiable wk hee wih a velciy field u(x,) ha aifie hmgeneu bunday cndiin bh a he infinie and a he c ecin cnu line B. Wih hi pupe in mind ne induce hee an auxiliay vec field u p (x) uch ha up 0 up( x) 0 lim u ( x) i, x xb p (.a) wih he ubidiay cndiin ha u p (x) appache i limi value i fa enugh, namely: u p (x) i f x > 5d, f example. In he cae f a cicula c ecin hi field can be deemined wih he help f he eam funcin in in ( x) in a a a / / ( ½), p c a c c (.b) whee u p (x) i f x > 5d wih an e malle han 0.05% f = 5. If he c ecin i abiay hi funcin u p (x) can be deemined numeically, f inance, bu nce hi i dne he acual velciy field u T (x,) can be wien a u ( x,) u ( x) u( x,). (.c) T p Inducing nw he (vlume) fce vec fp( x) upup u, (.a) p e he flw pblem i educed deemine he field {u(x,) p(x,)} uch ha u e u 0, p p u u u u u u u p f () x (.b) p ubeced he hmgeneu bunday cndiin ux (,) lim ux (,) 0 x xb x lim p( x,) 0. The dicee vein f (.) will be addeed nex. (.c) Bunday Cndiin and Wake Impedance In de deal wih (.) ne mu pecify, fi f all, a finie fluid egin, a hwn in Fig.(), whee he flw vaiable will be diceized by Finie Elemen. Bunday cndiin mu be imped n he bde f : nly wih hem he fluid flw pea can be ppely defined wihin. Figue. Finie fluid egin ( = Bilw) and wake. (u T (x,) = u p (x) + u(x,) u p (x) i f x b). The bunday i made by he c ecin cnu line B, by he inle i a he veical line x = b, by he laeal ide l a y = b and by he ule w a x = l ha define he ineface beween and he wake egin x > l ( = B i l w ). The velciy field u(x,) i ceainly null a B and if b i lage enugh (b 5d) and l i n much lage han b (l 0d) i eem eanable aume ha u(x,) i al null a i l : he peence f he cylinde huld n peub he incming flw a ufficien diance bh upwind and laeally. The fllwing eenial bunday cndiin i hu aumed n hi pa f : ux (,) 0. (.3a) x B i l The ame hmgeneu bunday cndiin cann be exended he ule w unle he diance l i vey lage (and i mu be b). In fac, he viciy geneaed a he cylinde die u vey lwly dwneam, ypically in a diance f de l e f he lage wavelengh, and ince b inceae ughly wih l / he 356 / Vl. XXVI,. 4, Ocbe-Decembe 004 ABCM

3 Weak Thee Dimeninaliy f a Flw Aund a cndiin (.3a) culd be puhed w nly if he finie egin becme vey lage. The velciy field a w huld hu emain unpecified while he ppe bunday cndiin a he ule f will be defined belw in hi ecin. If nw he dynamic equain in (.b) i muliplied by a viual velciy u(x) ha aifie, a uual, he ame eenial bunday cndiin (.3a), and he cninuiy equain in (.b) i muliplied by p(x) and bh expein ae fuhe inegaed in ne bain, afe paial inegain, ha u u d u p u u u p u u ( u) v ( v) d e u uud p( u)d f ( x) ud I( u (y) u (y)) p w w p( u)d 0. (.3b) In (.3b) he nain {(u w (y,) u w (y)) (u(l,y,) u(l,y)) y b} wa ued define he velciy and he viual velciy ve w and I(u w (y) u w (y)) i he wake impedance, namely, b u v I( uw(y) uw(y)) p u w(y) v w(y) dy e x w e x b w, (.3c) whee, auming cninuiy, () w and f he e field in he wake a x = l: he wake impedance i hu he viual pwe dne by hi e field n he viual velciy u w (y) and i epeen he eiance ffeed by he wake f he flw wihin he finie fluid egin. The wake egin i bunded n he lef by w and by w emiinfinie line, a indicaed in Fig.(). Since u(x,) wa aumed null a he laeal ide l i i ceainly cnien wih hi aumpin ake u(x,) 0 n, ince he peubain caued by he cylinde huld be even malle ve han ve l. I un u hen ha he flw in he wake i fced lely by he field u w (y,) and hu u p F Xuw e x w v F Yuw, e x w (.4a) whee he funcinal X,Y () can be deemined by lving he flw pblem in he wake. If bh he velciy and viual velciy {u w (y,) u w (y)} a he ule ae diceized a w uw (y,) U w,k() U w,k () w i h uw(y) k U w,k Uw,kw k,w (y), (.4b) whee {h k,w (y) k =,,, w } ae he ineplain funcin f he velciy field eiced w and {U w () U w } ae he ndal value vec, i can be hwn ha I( uw(y) uw(y)) Uw I( Uw) IU ( w) Mw U w Kw Uw w( Uw) U w, (.4c) he maice {M w K w w (U w )} being cmpued fm explicily defined Fuie eie 3. Wihin he dicee velciy and peue field can be expeed a ux (,) U () iu () h ( x) k e p( x,) P () ( x), k k k (.5a) wih {h k (x) k =,,...,} and { (x) =,,,e} being, epecively, he ineplaing funcin f he velciy and peue field. The funcin {h k (x)} ae neceaily cninuu bu he { (x)} may may n be in ealiy, he peue field de wk n u and i eem eanable che he { (x)} in cnfmiy wih he dicee field u bained fm he {h k (x)}. Placing (.5a) in he inegal ha appea in (.3b) and defining he maice d ( ) d ( ) d u ( u) v ( v) d e p, p p u u U M U u u u U U U u u u u u U K U p( u)d U P p( u)d P U fp() x ud U Fp, (.5b) he dicee fm f he flw equain in weak fm ead (ee (.3b) and (.4c)) U M U Kp, U ( U) U P Uw Mw Uw Kw Uw w( Uw) Uw U Fp U0. (.5c) Obeving ha {U w U w } ae, in fac, he pa f {U U} defined in w, ne can ake he wake impedance n he lef ide f (.5c) and um bh cnibuin bain U M U Kp U U U P U Fp U0. ( ) (.6) The dynamic pacel f (.6), ppinal {M K p (U)}, cme bh fm he finie fluid egin, whee he flw vaiable ae diceized, and he wake egin dwneam: M i he lcal ineia maix, K p epeen, a indicaed in (.5b), he influence f he vicu e and he cnvecive acceleain due he auxiliay field u p (x) and (U)U i he cnvecive ineia fce. The pacel { U P} ae due ma cnevain and he elaed cnain fce (peue) defined in he finie egin : he maix Thi eem be ue even f a mall eynld numbe: f e = 4 he wake ha aleady begun cillae inuidally fa dwneam, ee Van Dyke (98), plae 46. The exiing numeical eul pedic, a a ule, a ciical eynld numbe abve 40 (ec 45) alhugh he expeced value huld be belw (ec 35). 3 F he Fuie eie expanin ne mu impe a finie beadh W f he wake, wih W being abiaily lage (W >> b). I can be hwn ha he numbe f em nl in hee eie inceae bh wih W and e ypically, nl O(W(e)/). J. f he Baz. Sc. f Mech. Sci. & Eng. Cpyigh 004 by ABCM Ocbe-Decembe 004, Vl. XXVI,. 4 / 357

4 J.A.P.Aanha epeen he (dicee) gadien pea and he (dicee) divegence pea. Obviuly, ma i aleady cneved in he wake luin (.4c) and hi cnain huld n appea again a (.6). The (dicee) avie-ske equain (.6) emulae hu he ucue f he iginal equain (.b) and i will be analyzed nex: in hi dicee fm he mahemaical analyi i much imple, nce i baed nly n me geneal eul in Linea Algeba, and, fuheme, i ucme ha an peainal appeal given i diec link he final numeical eul. The Slenidal and Gadien Sub-Space Le W be he -dimeninal linea pace f he dicee velciy vec U and L e be he e-dimeninal linea pace f he dicee peue vec P in bh pace i will be aumed he andad inne pduc <UV> = U V and he elaed nm U = <UU> = U U. Le al {J G } be he -dimeninal and -dimeninal ubpace f W defined by he elain J VW : V0Le G VW : V L. e (.7a) Elemen f G ae gadien f cala field L e and f hi ean G i called he gadien ub-pace f W nice ha G i geneaed by he linea cmbinain f he clumn vec f. Elemen f J have null divegence and J i called he lenidal ub-pace by definiin, hei elemen ae hgnal he clumn vec f and hu J i he hgnal cmplemen f G ( + = ) ee Ladyzenkaa (969). One induce hee he pea W J, (.7b) G e :L L e :W W, (.8) whee i he (dicee) Laplacian pea and will be named he cnugaed Laplacian. Bh and ae epeened by ymmeic, piive emi-definie pae maice, he paene being a cnequence f he lcal chaace f he Finie Elemen diceizain. By definiin V = 0 if V J and V = 0 J ull (). In ealiy, i can be hwn ha J ull (). Le {T =,,,} be an hnmal bai f ull () J and {G =,,,} be he hnmal eigenvec cepnding he piive pecum { >0 =,,,} f, namely: T 0,,..., G G,,...,. (.9a) Obviuly {G G G } i a bai f G while {T T T } i a bai f J auming ha 0 < cnide he maix. (.9b) I Ceainly i a ymmeic, piive emi-definie pae maix wih a pecum in he ineval [0] fuheme T T, (.9c) a elain ha can be ued deemine an hnmal bai f he lenidal ub-pace 4. The Laplacian pea may have a nn-empy null ub-pace (ull () ) bu i ceainly ha a piive pecum in fac, if ˆ, (.0a) G G hen ne can eaily check ha Gˆ = wih Obeving nw he cnugaed elain Gˆ G ˆ. (.0b) G ˆ G, (.0c) he fllwing eul can be deived: he piive pecum f mu cincide, neceaily, wih he piive pecum { >0 =,,,} f. In fac, if wee a piive eigenvalue f wih eigenvec Ĝ hen GGhuld ˆ be an eigenvec f wih he ame eigenvalue and { >0 =,,,} nce, by definiin, hi i he e f all piive eigenvalue f. The pea {} eablih, hu, a dualiy beween he ub-pace Ĝ Le, geneaed by he vec { Gˆ, and he Gˆ ˆ... G } gadien ub-pace G W : if Ĝ hen G and if V G hen V he ub-pace ull () i he hgnal cmplemen f Ĝ Ĝ and L ˆ e ull( ) G W ull( ) G. (.a) Le {S =,,,e-} be an hnmal bai f ull (), named he puiu peue mde in he pecialized lieaue, ee Gunzbuge (985) hey aify he elain S 0,,...,e (.b) and, a i will be een in he nex iem, hee mde play in he dicee pblem he ame le played by he cnan peue field in he cninuum pblem, namely: hey d n inefee wih he dynamic f he flw. In accdance wih (.9b) ne induce hee he maix I, (.c) whee, again, i a ymmeic, piive emi-definie pae maix wih pecum in he ineval [0] hi maix will be ued in he nex iem in he cnex f he Pin equain f he (dicee) peue field. I eem whwhile finih hi ecin wih a me echnical emak abu he Finie Elemen diceizain, elaed he called div-abiliy cndiin (Ladyzhenkaya Babuka Bezzi 4 The APACK algihm i pecially uied deal wih eigenvalue pblem f a lage pae maice, ee Lehucq & Senen & Yang(997). 358 / Vl. XXVI,. 4, Ocbe-Decembe 004 ABCM

5 Weak Thee Dimeninaliy f a Flw Aund a cndiin). The pin i ha f me clae f Finie Elemen (FE) he malle eigenvalue becme mall a he meh ize h ge ze, indicaing ha elemen f he gadien ubpace G end lip in he lenidal ub-pace J a h 0. In hi cae he lenidal ub-pace J becme aefied nce a lea me f he lenidal field ae, in fac, lipping mde f G hi pblem i paiculaly acue f he imple FE diceizain, whee he velciy field i linea piecewie cninuu and he peue i cnan in each elemen, ee Gunzbuge (985) and Bahe (996), f example. Hweve, a hwn in Aanha (003), if he acual lenidal ub-pace i enlaged by hee lipping mde in a way dicaed by he divabiliy cndiin hi pblem can be vecme wihu impaiing he andad Finie Elemen cnvegence ae. In he peen cnex me f he eigenvalue-eigenmde in G ae naually cmpued in he eff deemine he bai {T =,,,} f J and hi enlaging pce can hen be wked u eaily, in h, quein elaed he div-abiliy cndiin ae f n pecial cncen hee. The Slenidal Velciy Field and Pin Equain The luin f he (dicee) avie-ske equain (.6) will be deal in w age: fi, he pecin f (.6) in J will eul in a andad nnlinea diffeenial equain f he velciy ecnd, Pin equain f he peue will be deived by pecing (.6) in G. The imple ucue f he dynamic equain in J allw ne develp andad aympic analyi f he undelying nnlinea yem and bain, in hi way, Landau equain in he viciniy f he Hpf bifucain in he he ide, i i pible hw, wih he help f iem (.), ha Pin equain ha a luin and ha hi luin i unique. Wih hi pupe in mind ne induce he maice T[ T T... T] M T MT Kp, T Kp T Fp, T Fp, ( q) T ( Tq) T (.a) whee T = 0 ince T J al, given abiay -dimeninal vec {q()q} ne ha F D d, T d, G () f () f () f () F () T f () F () G. d, D d, D (.4b) By placing nw U = G in (.6) ne bain, wih he help f (.8) and (.0a), he fllwing Pin equain f he peue P P F Gˆ D() f d,() ˆ, (.4c) G ince T 0 (T J ). Bu P L e and {S =,,..,e}{ G ˆ } i an hnmal bai f L,,.., e : expeing P in hi bai and uing (.0b) i i eay check ha he geneal luin f (.4c) i given by (ecall ha S = 0) P f () e d, ˆ a S G, P whee {a =,,,e-} ae abiay cefficien. The puiu mde P = a S play hee he ame le played by he cnan peue field in he cninuum pblem: in fac, i (dicee) gadien P and i i in hi way ha he peue appea in he dicee flw equain (.6) i null ince P 0 (ecall ha S = 0) and P P P P P 0. (.5a) The luin P f he Pin equain i hu uniquely defined by he expein f () d, P() ˆ G, (.5b) and i i impan pin u ha P() can be al bained by epeaed muliplicain f he pae maix in fac, fm (.c) i fllw ha G ˆ G ˆ, U() J U() Tq() UJ U Tq. (.b) and fm he cnvegence f he gemeic eie ( ) n = /( ) ne bain The viual velciy U W in (.6) belng eihe J G auming fi U = Tq J and ecalling ha T = ( T) = 0 ne bain, wih he help f (.b), he fllwing equain f he vaiable q() (ee (.a)): M q K q ( q) qf. (.3) p, p, The aympic luin f (.3), leading Landau equain, will be addeed in iem (.5). Inducing he dynamic fce vec (ee (.6)) F () MU K U( U) UF W, (.4a) D p p n P() F D(). (.5c) n0 The eie (.5c) ha an alm gemeic ucue and i cnvegence can be hu acceleaed by Shank Tanfmain, ee Bende & Ozag (978). The Seady Sae Sluin (Ue) Le q e,p be he eady luin f (.3), namely K q ( q ) q F, (.6a) p, e,p e,p e,p p, ne can wie D () in he fm ({T }{G } i an hnmal bai f W ) whee bh K p, and F p, ae funcin f he eynld numbe (K p, = K p, (e) F p, = F p, (e)) and i i he eady luin: q e,p = J. f he Baz. Sc. f Mech. Sci. & Eng. Cpyigh 004 by ABCM Ocbe-Decembe 004, Vl. XXVI,. 4 / 359

6 J.A.P.Aanha q e,p (e). If (.6a) i diffeeniaed wih epec e and he maix K = K (q e,p ) i defined by he ideniy ( f and f he deivaive wih epec e) K ( q ) q K q ( q ) q ( q ) q e,p e,p p, e,p e,p e,p e,p e,p e,p p, e,p p,, (.6b) ne bain f qe,p he fllwing nnlinea (K = K (q e,p )) diffeenial equain: a K q K q F. (.6c) The luin f (.6c) bviuly exi and i i unique a lng de K ( q ) 0, (.6d) and if hi lae cndiin i fulfilled ne i able, by inegaing (.6c), mach he f (.6a) a he eynld numbe inceae: equain (.6c,a) define a pedic-cec mehd deemine he eady luin. In he he hand, if de K (q e,p ) = 0 ne wuld have a claic bifucain f he equilibium: he eady luin culd hen be mached u in e by defining a ppe ( aically able ) banch afe he bifucain. A i will be een in he nex iem, he undelying aumpin behind he aympic hey be develped in hi wk implie in a cndiin nge han (.6d): wihin hi cnex i can be aken hee ha a eady luin exi and i i uniquely defined. If {x k k =,,,} ae he nde f he Finie Elemen meh hen, given any cninuu field u(x,), he ndal ineplae u h (x,) i defined by he expein h k k k k k k k u ( x,) u( x,) i v( x,) h ( x ) U () i U () h ( x ), e,p k (.7a) and, wihin he cnex f he dicee mdel, ne can igne he diffeence beween u(x,) and i ndal ineplae u h (x,): hee i hu a ne--ne elain beween he field u(x,) and he ndal value vec U() (u(x,)u()). If nw U p i he ndal value vec f he auxiliay field u p (x) (u p (x)u p ), he eady luin u e (x)u e i defined by he expein Ue Up Tq e,p, (.7b) and he glbal luin u T (x,)u T () can be hu wien a U () U U () U () Tq (). (.7c) T e Keeping in mind hee definiin and inducing al he field u (x,)u (), i i an eay ak hw ha he maix K can be expeed in he fm (ee (.5b)) u e u u u e u d u ( u) v ( v) d e U K U K T KT. (.7d) The pacel [(u e )u + (u )u e ] f he cnvecive acceleain induce, a uual, a nn-ymmey in K ha play a le in abiliy hey be addeed nex. Hpf Bifucain and Aympic Sluin f Fluid Equain The diffeenial equain f he peubain U () = Tq () n he eady luin U e can be eaily deived by uing he definiin q() = q e,p + q () in (.3) ne bain hen, wih he help f (.6a), he hmgeneu nnlinea equain M q K q ( q ) q 0, (.8a) he nn-ymmeic maix K being defined in iem (.4). The eigenvalue f K ae hu cmplex, in geneal, and hey will be defined a fllw: i i 3 3i 3... i (.8c) The fi mde, he ne ha becme fi unable ince, i f he fm () e q E, (.9a) whee he mde E i uch ha ((*) and f he cmplex cnugae) M K E0 ( E ) M E. * (.9b) ice ha (e) = (e) + i(e) whee, fm (.9b), i fllw ha (e) * ½( E) K K E * ½( TE) K K ( TE) i (e) * ½( E ) K K E * ½( T E ) K K ( T E). (.9c) elain (.9c) can be ued in cnuncin (.7d) pvide me explici expein f {(e) (e)} indeed, if e(x)te ne bain * * e * e e * e * * (e) ½ u( e x) e ev( e x) e e e( x)ee x y ee x y d x x y y b (e) ue ve ½ e x e y d e x e y d x y e * * e * e e * e * * i (e) ½ u e( x) e ev( e x) e e e( x)ee x y ee x yd, x x y y (.0a) whee e (x) = ½(v e /x + u e /y) i he hea ae f defmain f he eady luin u e (x) and e (x) = ½(v e /x u e /y) i viciy. The abve expein ugge wie (e) a (e) = a (e) b (e)/e, wih b (e) > 0. The Suhal numbe S(e) 5 (e)/ i knwn change weakly wih e and fm he ucue f (.0a) ne huld expec ha bh {a (e) b (e)} al d if nw ne wie {a (e) = a(e)s(e) b (e) = b(e)a(e)s(e)} and aume ha {a S b } ae ypical value f he lwing vaying funcin {a(e) S(e) b(e)} ne bain b(e) b (e) a(e) S(e) S e a, (.0b) e 5 The acual Suhal fequency (e) diffe lighly fm (e), ee (.3b). 360 / Vl. XXVI,. 4, Ocbe-Decembe 004 ABCM

7 Weak Thee Dimeninaliy f a Flw Aund a an expein ha ha me empiical upp, a dicued in iem (.6) belw. I fllw ha (e) i (ughly) mnnically inceaing wih e, whee (e) < 0 f e < b = e c and (e) > 0 f e > b = e c. The value f e c infeed fm numeical imulain eem calece aund 45 (e c 45) alhugh hee ae expeimenal evidence hwing ha hi hehld value i a bi malle (e c 35). Obviuly, f e belw e c he eady luin i able ((e) < 0) while i becme unable f e > e c ((e) < 0) fuheme, (e c ) 0 and ne ha hu a ypical Hpf bifucain. F e abve e c bu cle i ne ha 0 < (e) <<, ince (e c ) = 0 fuheme, he expeimen ugge and he numeical eul cnfim ha nly ne mde i unable in hi ange f eynld numbe ( < 0 3). The aympic luin be develped i baed n he fllwing aumpin i)0 (e) (e) O() ii) (e) 0 f 3,...,, (.) ha huld be icly aified in he viciniy f Hpf bifucain nice ha (.) implie, neceaily, ha { (e) 0 (e) 0}, ince {(e) 0 (e) < 0}, and de K (e) 0. The adin eigenvalue pblem (K K ) play a le, a i will be een, in he deivain f he aympic hey. Thi pblem ha he ame eigenvalue (.8c) bu he eigenvec ae diinc in paicula, he unable eigenvalue i i aciaed he adin unable mde E a whee M K E 0. (.a) a Fm (.) i fllw ha he unable eigenvalue = + i i a ingle f he elaed chaaceiic equain: if i wee n, me f he wuld be equal and he cndiin (ii) in (.) wuld be n fulfilled. Unde he cndiin ha i a ingle i i pible hw ha E M E, (.b) a 0 and hu E a can be nmalized by he cndiin E M E, (.c) a a elain ha i will be ued belw in hi ecin. The agumen nw i claic and i will be u keched hee: f 0 < (e) << he ampliude A()e i f he unable mde E = E + ie I inceae (iniially) expnenially wih ime (A() e da/d = A) and he luin f he dynamic yem (.8a) i aaced, ince < 0 f 3, he (unable) w dimeninal manifld angen he plane geneaed by {E E I } he expnenial gwh in hi manifld i haled by he nnlinea em and expanding (q )q in pwe eie in he ampliude nly he cubic em A() A()e i can mach he em A()e i ha caue he expnenial gwh. The equain f he ampliude A() namely, Landau equain i hu given by 6 da A A A0 = i, (.3a) I d he eady luin (limi cycle) A c exp(i ) being given by A c I. (.3b) The fmal aympic luin f he (dicee) avie-ske equain will be deived nex. In fac, wiing he luin f (.3) in he fm q() q q (), (.4a) e,p wih he peubain q () aifying (.8a), ne mu have, leading de, ha q () [A()E + (*)], ince he luin f (.8a) huld fllw, a lea iniially, he unable mde E. The ampliude A(), hweve, i uch ha (ee (.3)) da O( A), d / A() O( ) and expanding q () in he mall paamee A() O( / ) ne bain, wih an e in (.4a) f he fm [ + O( )], ha / O( ) O( ) i 3 3i 3 33 O( ) 3/ (.4b) i i q() A() e (*) A() 0 A () e (*) E + A() A() e A () e (*) O( ). The nnlinea em in (.8a) can, accdingly, be wien a (.5a) i i 3i ( q ) q 0 e (*) 3e 33e (*) O( ) (.5b) wih * * 0 E E ( E ) E ( E) E (.5c) * * 3 ( E) 0 ( 0) E( ) E ( E ) ( E) ( ) E. 33 Placing (.5) in (.8a) ne bain ( = + i) da i A ME A A 3( M K) 3 e d i A K0 0A i M K e (.6a) 3 3i A 3iM K e O( ) 0, 6 I ha been implicily aumed hee ha ne ha a upe-ciical Hpf bifucain wih = eal > 0. Thi aumpin ha an expeimenal upp, ee Pvanal & Mahi & Bye (987) and Leweke & Pvanal (994) amng he, and i ha been al veified numeically by ack & Eckelmann (994). Sme peliminay numeical cmpuain, be publihed n, cbae hi eul. J. f he Baz. Sc. f Mech. Sci. & Eng. Cpyigh 004 by ABCM Ocbe-Decembe 004, Vl. XXVI,. 4 / 36

8 J.A.P.Aanha whee he em A A(M 3 )e i O( 5/ ) ha been added u f cnvenience if nw { 0 33 } ae luin f he (nningula 7 ) linea yem K i M K 0 3i M K 0, (.6a) educe da A ME A A 3( M K) d a (.6b). (.6c) Muliplying (.6c) n he lef by he adin unable mde E a and uing (.a,c) Landau equain (.3a) i bained wih E. (.7) Summaizing: lving he eigenvalue pblem (.9b) and (.a,c) he vec { 0 } can be cmpued fm (.5c) and he luin { 0 } f he linea yem (.6b) can be deemined wih hem he vec 3 can be bained fm (.5c) and Landau cefficien i hu given by (.7). Wiing nw 3 in he fm M E E (a) 3 3, (a) a 3, 0, and placing (.8a) in (.6c) ne bain, wih he help f (.3a), he equaliy (a) 3 3, (.8a) q M K 0 all q J. (.8b) Inducing he ( )-dimeninal ub-pace {J J a, } f J by he definiin J q : q E 0 q S x J q : q E 0 q S x, a, a, a, a a, a, wih x being an abiay ( )-dimeninal vec, and he maice M S M S K S K S, a,, a,, he vec 3 can be bained fm he luin f he (nn ingula 8 ) linea yem (a) M, K, x3sa, 3, 0 S x. 3 3 (.8c) (.8d) (.8e) eveing he ndal value vec U() = T(q e,p + q ()), ee (.c), (.7), ne ha 7 ice ha (.) ule u {0 i 3i} a pible eigenvalue f K nce < 0 f 3 and 0 he maice {K (im + K) (3iM + K)}in (.6b) ae hu nn-ingula. 8 ecall ha (.) implie ha i a ingle f he chaaceiic plynmial f K and he maix M + K, eiced he ub-pace hgnal he eigenvec {E Ea}, mu be nn-ingula. U() U A() E e (*) A() A () e (*) wih e,p U i i 0,U,U / O( ) O( ) i 3 3i 3,U 33,U ( ), 3/ O( ) + A() A() e A () e (*) O (.9a) Ue,p EU 0,U,U 3,U 33,U T qe,p E , (.9b) and placing (.9a) in (.4a) ne bain F () F A() e (*) A() A () e (*) F F F i i D 00 0 / O( ) O( ) i 3 3i + A() A() F 3 e A () F 33 e (*) O( ), O( ) 3/ wih (ee (.7d) and (.3a) wih = + i) F 00 Kp Ue,p ( Ue,p) Ue,p Fp F MKEU F 0 K 0,U 0,U F i MK,U,U F MK ME F 3i MK, 3 3,U U 3,U 33 33,U 33,U (.30a) (.30b) whee { 0,U,U 3,U 33,U } ae defined a in (.5c) wih { k k,u EE U 0 0,U,U }. Cnideing nw he luin f he Pin equain (ee (.4c)) P F (k) {(00)()(0)()(3)(33)}, (.30c) k k he (dicee) peue field i given by (P e = P 00 i he peue elaed u e (x)u e ) i i P() Pe A() e (*) A() 0 A () e (*) P P P / O( ) O( ) i 3 3i P3 P 33 O( ) 3/ + A() A() e A () e (*) O( ), (.30d) wih he ame e fac [ + O( )] f he velciy field appximain. Expein (.9a) and (.30d) ynheize he aympic luin f he (dicee) avie-ske equain in he viciniy f Hpf bifucain (e e c ) in he nex iem, he pibiliy exend hi luin he ange e >> e c i dicued. Exenin f Landau Equain Beynd Hpf Bifucain The aympic luin f he (dicee) avie-ske wa deived auming (.), w cndiin ha hld in he viciniy f he Hpf bifucain bu n neceaily nly hee. The pupe hee i give agumen ha ugge ha (.) and, wih i, he given aympic luin can be exended fa beynd Hpf bifucain, ha i, he ange e >> e c. 36 / Vl. XXVI,. 4, Ocbe-Decembe 004 ABCM

9 Weak Thee Dimeninaliy f a Flw Aund a In fac, numeical eul by ack & Eckelmann (994) indicae ha (.) emain eenially cec in he ange e c 45 < e < 300 and Henden (997), afe a deailed numeical wk, affim ha up e = 000 n he bifucain, beide he ne a e c, culd be beved in he w dimeninal mdel, in he wd, ha { < 0 3} in hi ange. Fm he expeimenal ide, Pvanal & Mahi & Bye (987) have beved ha he infeed value f (e) culd be fied he expein (e) d e (e) 0.0 c, (.3) U e while Leweke & Pvanal aumed (.3) in he ange e c 45 < e < 300. A aleady dicued, he empiical elain (.3) eem have a f n he me baic e f equain ha decibe he fluid flw, ince a imila expein can be deived exacly, ee (.0b) fuheme, i indicae ha (e) can be cnideed a mall paamee f de 0.0 ( le). All gehe, hee evidence ugge ha (.) culd be puhed a lea up e 000 ince, fllwing Henden, n he bifucain (in D) culd be fund in hi ange while (.3), gehe wih (.0b), eem indicae ha (e) emain in fac mall a aeed in (.). Expeimenal eul n VIV ae mly in he ange 0 3 < e < 0 4, ee Khalak & Wiliamn (996), and hey d n eem depend vey much n e. The beved hamnic paen i vey nea and hi, undubedly, wa he mivain behind a bld aumpin induced by Bihp & Haan (964) decibe he flw aund a cicula cylinde: hey pped epeen he flw by an ne degee f feedm wake cilla mdel baed n Van de Pl equain ha lead, f a mall enugh, Landau equain (.3a) hi idea wa fuhe develped and i i he bai f he -called phenmenlgical mdel ued pedic VIV, ee Iwan & Blevin (975). In depie f he le link wih he me baic flw equain he pedicin fm hee mdel have me accuacy, hwing ha he deed cillay behavi in he wake can be appaenly decibed by mean f an ne degee f feedm yem elaed he unable mde f he pblem. The mallne f, a cmmn feaue in all wake cilla mdel, cupled he flw epeenain by nly ne unable mde, can be anlaed in he fllwing wd: he baic aumpin (.) i, appaenly, cec in he ange 0 3 < e < 0 4 f eynld numbe f he VIV expeimen. Hweve, fm e 0 4 unil he aniin egin e 0 5 nhing vey much diffeen ccu and ne can pibly puh (.) up he aniin egin e 0 5. F e > 0 6 he bunday laye i fully ubulen bu a (elaively) well defined Suhal fequency can be deeced again: in line wih he veall view aken hee, ne peculae ha he ame cndiin (.) can hld if ne eache f he abiliy f he ime aveaged (ubulen 9 ) ymmeic luin f he flw aund he cicula cylinde. Thee exenin huld be bviuly cnfimed numeically bu ceainly he peen hey ha a ange f applicain much bade han feeen a pii me han ha, (.) can be ued deemine peciely hi ange. Weak Thee-Dimeninaliy: Ginzbug-Landau Equain The peubain u (x,) n he D eady luin u e (x) i, leading de, given by u (x,) A()e(x) whee e(x)te i he 9 Bunday laye ubulence i likely due he cncmian inabiliie f eveal ymmeic mde, a elabaed in a fhcming pape i will be al dicued hee a pible cenai f he aniin egin 05 < e < 06. unable mde and A() i cmplex ampliude: A() = A() exp(()). I i knwn f a lng ime ee, f inance, Tebe (969) ha he vice ae n hed in phae alng he pan f a fixed cylinde, and in fac he celain amng he vice emied in diinc ecin end ze vey fa 0, in a lengh f de f 0d. The phae huld hen change in he lngiudinal z-diecin and i de he ampliude A and he peubain u (x,) {A = A(z,) u = u (x,z,)}, whee x = xi + y cninue epeen he piin vec in he c ecin plane and u (x,z,) = u (x,z,)i + v (x,z,) he peubed velciy field in hi plane a he z-level. The lngiudinal cmpnen f he peubed velciy field will be deignaed by w (x,z,). The vaiain f A(z,) in he lngiudinal z-diecin huld be expeed by an even deivaive wih epec z, ince hee i n pefeed diecin, and beving ha he vicu diffuin in hi diecin, given by (/e) u /z, implie leading de in a em ppinal A/z, he fllwing 3D cecin i pped f Landau equain (.3a): A A z A A A 0. (3.a) Thi i he Ginzbug-Landau Equain (GLE), fi pped by Ginzbug me han fify yea ag in hi udy n upecnduciviy, ee Ginzbug & Landau (950) nice ha, in geneal, bh Landau cefficien and Ginzbug cefficien ae cmplex numbe: { = + i I = + i I }. Auming, a befe, << (ee (.)) and ecalling ha A O( / ), a ppe balance f he em in (3.a) indicae ha he lengh cale l z f he lngiudinal vaiain f A(z,) mu be uch ha d lz O d, (3.b), in h: (3.a) deal wih a weak hee-dimeninal vaiain f he flw field. A i i dicued in he la iem f he peen ecin, GLE can be eaily exended he cae whee bh he gemey and he inciden flw change in he lngiudinal diecin if he ae f change i weake han (3.b). In wha fllw, (3.a) will be bained a a cnien aympic appximain f he SE and, in deiving i, a pcedue deemine Ginzbug cefficien will be defined. Aympic Appximain f 3D Field Equain Le u (3d) (x,z,) = u e (x) + [u (x,z,) + w (x,z,)k] be he 3D velciy field f he flw aund a lende cylinde, wih u e (x) being he D eady luin and [u (x,z,) + w (x,z,)k] he peubain n i if, a defined in ecin (), = i/x + /y, he 3D SE f he peubain [u (x,z,) + w (x,z,)k] i given by u u u ueu uue uu up w e e z z 3/ O( ) w p w w ue w w u w w e z z e z w u, z O(w ) O( ) / O( w ) / O( w ) O( w ) / O( w ) O( w ) (3.a) 0 Vey fa in he elain he lngiudinal lengh f he lende cylinde in fac, vey lwly in he naual lengh cale d f he c flw pblem. J. f he Baz. Sc. f Mech. Sci. & Eng. Cpyigh 004 by ABCM Ocbe-Decembe 004, Vl. XXVI,. 4 / 363

10 J.A.P.Aanha whee (3.b) wa ued eimae he de f magniude f he z- deivaive and {u p } O( / ), ee ecin (). Fm he w - equain ne bain, a nce, ha w O( ), (3.b) hwing ha he lngiudinal peubed velciy w i, a expeced, f malle de han he peubed c-flw u. Fuheme, if em f de ae diegaded, a befe, i i pible check fm (3.a) ha he lngiudinal velciy w de n affec he (dynamic) u -equain: he lngiudinal flw affec he D luin u nly hugh he ma cnevain equain, wih a em f de O( 3/ ), he ame de f magniude f he lngiudinal diffuin in he u -equain. One i lef, hu, wih he equain u ueu uue uu u w p uew w e z u O( ) w, z 3/ O( ) O( ) u p e e z 3/ O( ) (3.c) ha define, wih an e fac [ + O( )], an aympic appximain f u (3d) (x,z,). Equain (3.c) i, in me ene, andad in exiing lende bdie heie : he D ucue, epeened by he dynamic u - equain, i n piled by he lngiudinal velciy w, he influence f hi pacel appeaing nly in an blique way in he pblem. In fac, a pnaneu 3D peubain n he c-flw induce a peue gadien ha fce a lngiudinal flw w and nly hen, hugh ma cnevain, he 3D peubain feed back he D iginal equain: a in he well knwn Lifing Line They, he hee-dimeninaliy, epeened by he idewah w, affec eenially he kinemaic f he D flw. Fuheme, he peue gadien, and he idewah, i ppinal A/z and hen u A/z : hi cecin n he c-flw i added he aigh diffuin em, ppinal u /z, pduce he Ginzbug cefficien = + i I. The influence f he idewah n he final equain i hu wfld: fi, i give ie a lngiudinal diffuin, ppinal w, ha gehe wih he c-flw diffuin u /z deemine ecnd, he lngiudinal flw w induce a kind f cmpeibiliy f he c-flw u (u A/z ) and an acuic wave equain mu be expeced hen, decibed hee by he lngiudinal wave pea A/ i I A/z = 0 wih a dipein elain + I k = 0. If he nnlinea em i I A A i added hi wave pea ne bain he cubic Schödinge equain ia/ + I A/z I A A = 0, a cnpicuu peence in he udy f nnlinea dipeive wave yem, ee Whiham (974). In wha fllw he dicee luin f (3.c) will be defined and dicued. The Sidewah and Ma Cnevain One a by cnideing he w -equain, fced by he em p /z leading de ne ha (ee (.30d) uing p (x)p ) p z x A z, i 3/ p ( ) e (*) O( ) and hu wiing w (x,z,) in he fm A i w(,z,) x w () x e (*), (3.3a) z he fllwing equain f w (x) can be bained : i w ( x) ( ue )w ( x) w ( x) p ( x ). (3.3b) e Taking he ame meh ued diceize bh he velciy and peue field in ecin (), namely, auming x, x W, e x, x P, w ( ) W h ( ) W p ( ) P ( ) P, and inducing he maice {m k w } by he expein (3.3c) w ( x) w( x)d W mw p ( x) w( x)d W P w uew ( x) w( x) w ( x) ( w( x)) d, e W k W (3.3d) he fllwing algebaic equain i bained f he ndal value vec W, i w mk W P, (3.4a) he nn-ingulaiy f (3.4a) being ganed by he fac ha all eigenvalue { =,,..,} f he maix k have neceaily negaive eal pa and { i all }. The cninuiy equain in i weak fm ead A i p( x) u w ( x) e (*) d 0, z and he dicee fm f hi equain i given by A i w e U W (*). (3.4b) z If (3.4b) i muliplied n he lef by and he definiin f he cnugaed Laplacian = i ued, ne bain f U () he equain A i U ( ww) e ( *), (3.5a) z whe geneal luin can be wien a (ecall ha ull() = J ) F impliciy, hmgeneu bunday cndiin i aumed n. Ohe bunday cndiin culd be ued inead bu hi imple ne i eanable and eaie fm a me echnical pin f view. The pacel ppinal ue in (3.3d) lead an ani-ymmeic maix while he ne ppinal /e lead a ymmeic piive definie maix. Fm hi i fllw a nce ha eal < 0 if (m + k)x = / Vl. XXVI,. 4, Ocbe-Decembe 004 ABCM

11 Weak Thee Dimeninaliy f a Flw Aund a A U Tq C z i () (z,) e (*) wih C G being he unique (wihin G ) luin f, (3.5b) C ( w W) G. (3.5c) ice ha C can be defined by he um f he eie (ee (.9b)) C W, (3.5d) n ( w ) n0 whee (3.5d) ha again an alm gemeic ucue and i cnvegence can be hu acceleaed by Shank Tanfmain, ee (.5c). The Ginzbug-Landau Equain (GLE) The dicee fm f he u -equain in (3.c) i given by A U MUKU U UP U MTE i (3.6a) ( ) e (*) e z whee he leading de em U i / (z,) T E A(z,)e (*) O( ) (3.6b) wa ued in he igh ide f (3.c). Placing nw (3.5b) in (3.6a) ne bain, afe pecing in he lenidal ub-pace, he equain A i e M q K q C M E (*) ( q ) q 0 e z C T i MKC, (3.6c) whe aympic luin (.5a) ha an ampliude A(z,) ha aifie he GLE (3.a) wih ii Ea C, (3.7) e ince E a ME =. The emaining em f he velciy and peue field ae given by (.9a)(.30b,c,d), adding he peue pacel ppinal A/z. The aenin will be uned nex a me deailed analyi f GLE. GLE: Bunday Cndiin and Wave-Like Limi Cycle If he cylinde pan i defined in he ineval l z l bunday cndiin mu be imped a he cylinde end z = l, ne in each exemiy. Uing again he nain {e(x)te w (x)w } he velciy field can be wien, leading de, in he fm u A x u x ex xk i (3D) (,z,) e() A(z,) () (z,)w() e (*) z u (,z,) x w(,z,) x, (3.8a) and w cndiin can be naually imped n A(z,), namely: A i) ( l,) 0 w ( x l, ) 0 (3.8b) z ii) A( l, ) 0 u ( x l, ) 0. Bunday cndiin (i) i appaenly me apppiaed f he cae whee he cylinde end eihe a he fee-uface in a wae channel ele if he end-cylinde echnique 3 i ued a i bm end: in hee iuain ne expec ha he peubain n he D eady luin huld be, by fa, dminaed by he cflw u (x,z,). Bunday cndiin (ii) i me awkwad be inepeed hugh i eem be adequae epeen a cylinde ending in he inei f he fluid, whee hen he flw peubain w (x,z,) in he lngiudinal diecin huld be nge han he peubed c-flw u (x,z,) nea hi fee end. Pibly linea cmbinain f (3.8b), including peidic bunday cndiin, ee (3.3b) belw, culd al be imped. The GLE (3.a) depend n hee cefficien, { = + i I = + i I }, and i i impan undeand hw he qualiaive behavi f hem affec he luin. The eal paamee { } huld be all piive: > 0 i a negaive diipain ha caue he inabiliy, > 0 i a nn-linea diffuin due he D c flw and > 0 i eenially a linea vicu diffuin caued bh by he vicu e /e( w ) f he lngiudinal velciy in he c-ecin plane and by he c-flw vicu e /e( u /z ) nice ha he inequaliy > 0 he upeciicaliy f he Hpf bifucain wa dicued in ecin () while he elain > 0 can be infeed fm he Pinciple f he Viual Pwe. The GLE (3.a) ha wave-like luin f he fm i(k 0 z 0 A(z,) ) 0 0e / k0 0 0 k0i I 0, (3.9a) he abiliy cndiin f hee wave-like luin being given by he cndiin { 0 (,, )}. (3.9b) I I 0 0,L ice ha beide a eicin n he cupled effec f dipein ( I I ) and diffuin ( ), cndiin (3.3) ffe al a eicin n he wave ampliude 0 : hi ampliude huld be, in geneal, lage han a lwe bund 0,L = 0,L (,,) f, hewie, he wave luin (3.) i pibly wihin he epulin bain f he unable null luin A(z,) 0. In he he hand, he exience f a cninuum f able luin (3.9c) in he ange 0,L 0 (/ ) / ( in he ange 0 k 0 k 0,L ) eem be in line wih he diveiy f hedding mde (blique, paallel, ec) fund in he expeimen wih fixed cylinde, ee Khalak & Williamn (996). ice, in paicula, ha (3.9c) aifie he fllwing bunday cndiin a he cylinde end z = l, A ( l,) ik 0A( l,) 0, (3.9c) z 3 amely, a lage cylinde i mhly fied he bm end, inceaing lcally he c-flw and ceaing a bm end cndiin imila he ne fund a he fee uface ee Khalak & Williamn (996) J. f he Baz. Sc. f Mech. Sci. & Eng. Cpyigh 004 by ABCM Ocbe-Decembe 004, Vl. XXVI,. 4 / 365

12 J.A.P.Aanha ha educe he cndiin (i) in (3.8b) when k 0 = 0 (D luin): a a mae f fac, he end-cylinde echnique wa induced u ceae cndiin fav he paallel hedding n a cylinde in a wae channel, ee again Khalak & Williamn (996), ince i fce, appaenly, he cndiin A/z 0 a he bm end. Obviuly, me aeive aemen abu me feaue f he luin can nly be dne by diec numeical imulain f (3.a) bu i i fel ha hi imple abiliy analyi help fcu me elevan iue. Spanwie Vaiain f Gemey and Cuen S fa he analyze wa eiced he imple unifm flw alng a cylinde being me pecie, all empiical and numeical evidence cmmened hee ae elaed pecifically he flw aund cicula cylinde alhugh ne huld be cncened, fm a me pacical pin f view, wih pblem whee he c ecin gemey and he inciden cuen change alng he pan, bh in ineniy and in diecin. elevan example ae he flw aund a apeed cylinde, ued emulae a cuen vaiain alng he pan, he flw aund a cicula cylinde wih ake, vey impan fm a pacical pin f view, ele he change f he f he inciden flw diecin alng deph, a iuain uually encuneed in Ocean Engineeing. In all hee example he unable mde E U change in he z-diecin and i de he ecinal peue field P ha fce he lngiudinal flw w (x,z,), ee (.30b,c,d) wiing, a befe, p (x,z)p (z), ne ha ha p /z O(p /l g,c ), whee l g,c i he lengh cale f he lngiudinal vaiain f bh he gemey and he cuen, and he quein ne inend anwe i he fllwing: hw mall can be l g,c in de he GLE (3.a) emain valid, wih he ame e fac f he fm [ + O( )], even in he peence f hee vaiain? Obviuly, he baic paamee huld hen change (weakly) wih z namely, { = (z) = (z) = (z) = (z)} bu he ucue f (3.a) wuld emain he ame and he D expein ued cmpue hem. T anwe uch quein ne mu ecall ha he heedimeninaliy wa fced by he peue gadien ha, leading de, i given by p A p x z z z i 3/ (,z,) p A e (*) O( ), (3.4a) and beving ha eain he final e O( ) nly he em f de O() mu be kep in (3.4a), he vaiain p /z culd be igned if i i f de O() malle: in hi cae he em Ap /z wuld be f de O( 3/ ) malle, ince A O( / ), and he vaiain f he gemey and/ he inciden flw diecin wuld appea a m a he de O( ) and can, hu, be diegaded. I un u ha (3.a) emain cec, wih he ecinal paamee {(z) (z) (z) (z)}, wheneve he lengh cale l g,c i lage ha (p /z O(p /l g,c )) p O( ) g,c O d l. (3.4b) z A een in (.3), empiical evidence ugge ha 0.0 and l g,c O(5d), he fae change in z-diecin wihin he cnex f GLE being defined by he elain l g,c 5d. I i a mae f cuiiy beve ha 5d i a ypical lengh cale f m f he uppein device ued miigae ( eliminae) VIV f inance, hi i a ypical value f he helicidal pich f he ake f he wavelengh f he wavy cylinde analyzed by Beaman (000). Cncluin In he peen pape a cnien aympic appximain f he flw aund a lende cylinde wa develped, leading he Ginzbug-Landau equain. The hey i baed n an aumpin cncening he behavi f he eigenvalue elaed a D peubain n he eady D luin u e (x) i ae ha hee exi nly ne unable mde wih eigenvalue = + i and, fuheme, ha { 0 0 < << }, ee (.). Bh cndiin ae aified in he viciniy f he Hpf bifucain a e e c 45 bu bain he deied appximain me me echnical eul wee needed. Fi, he wake impedance wa induced, by cnideing ppely he flw in he wake and deemining hen hw he wake ei he flw wihin he finie fluid egin ha i acually diceized ecnd, by pecing he dicee flw equain in he lenidal and gadien ub-pace i ha been pible, uing me andad eul in Linea Algeba, hw n nly he inne cniency f he mdel bu al bain he cefficien f he Ginzbug-Landau equain. In paicula, he Ginzbug cefficien = + i I wa analyzed, whee he diffuive feaue f he eal pa wa elabaed and al he wave feaue f I wa eablihed, nce i i elaed he cmpeibiliy f he c-flw, namely, he wk dne by he c-flw peue field n he divegence f he c-flw velciy field. The final gal f he n-ging eeach i adde he VIV pblem, f cnideable impance in me Ocean Engineeing applicain, mainly in he analyi f he ie f a flaing pducin yem. Thi pblem ha been enaively addeed, wih a elaive ucce, by he -called phenmenlgical mdel, whee he flw i imply decibed by a Van de Pl cilla wih cefficien infeed fm me expeimenal eul. The iuain hee i n vey much diffeen, a lea fm he peainal pin f view, he uual appach elaed Ginzbug- Landau equain (GLE), nce hi mdel i fied exenally he pblem and he cefficien ae hen infeed al by me expeimenal ( numeical) eul. Bu hee i a cncepual diffeence, a lea in i igin, in bh appache: GLE wa hugh be valid nly in he viciniy f e c (alhugh i ha been ued fa beynd i, a lea up he ange e 300) while he Van de Pl mdel wa aimed, fm i vey fi mivain, deal wih he expeimen n VIV, whee 0 3 < e < 0 4 ughly peaking. Obeving ha he Van de Pl mdel aume, implicily, ha nly ne mde i unable wih a negaive damping <<, ne may be emped cnclude, baed n he elaively gd pedicive abiliy f hee phenmenlgical mdel and al n he aleady pped exenin f GLE he ange e 300, ha he undelying aumpin f he peen aympic hey hld, in fac, in a much bade ange f eynld numbe han feeen a pii. Thi cnecue can be aied he au f a hyphei in he mahemaical develpmen and, wih i, exend cnienly he GLE he whle ange f e afewad, by lking he acual numeical eul ne can cnfim ( n) hi hyphei. The peen hey de n eem be a dd, hu, wih adiin in phyical cience: indeed, given a e f a mewha dipee eul and bevain, all f hem elaed hweve he cnpicuu cillay behavi f he phenmenn, hey can be gaheed by mean f an aumpin, ynheized by (.), ha place all hem in an unique famewk, namely, he GLE in a wide ange f eynld numbe. Fuheme, a aed abve, hi hey bing in i lay u he pibiliy be efued, nce he baic aumpin can be checked diecly by mean f (numeical) expeimen, fm a me pacical pin f view, i allw ne 366 / Vl. XXVI,. 4, Ocbe-Decembe 004 ABCM

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