VORTEX-INDUCED INSTABILITIES IN OFFSHORE CATENARY RISERS: A NON-LINEAR NORMAL MODE APPROACH

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1 Proceedgs o PACAM I Corgh 9 ABCM h P-Aerc Cogress o Aled Mechcs Jr -8 Foz do Igç PR Brzl VORTE-INDUCED INSTABILITIES IN OFFSHORE CATENAR RISERS: A NON-LINEAR NORMAL MODE APPROACH Crlos E.N. Mzzll cezz@s.r Césr T. Sches cesr.sches@gl.co Cool Mechcs Loror Srcrl d Geoechcl Egeerg Dere Escol Poléc Uers o São Plo A. Pro. Aled Prdo r.. 8 São Plo Brsl. Asrc. Oshore cer rsers re sed he eloo o dee-wer ol d gs elds sch s hose log he Brzl Sohes cos. The re seced o seere dcl lods sch s hose reled o sde low o lds der hgh ressre eerl low o se crres d osed oo he log rodco lor de o se wes. I hs er rodcor sd s reseed h ddresses he dc sl de o VIV oredced ros csed se crres. The oo o he TDP och-dow o where he rser oches he seed d he oorg oo he rser o edwhere ees he log re egleced here or he ske o slc lhogh s kow h he l or role he dc resose o cer rsers o e ddressed re work. The o-ler orl odes o e-elee odel o cer rser wh ed eds re ll deered sg cool ool [8] sed o he r old rocedre roosed Shw d Perre [9]. The ode h s rclrl roe o e eced VIV s seleced d redced hecl odel h coles he srcrl odl resose d he ld dcs o he VIV s descred der Pol oscllor [] s sed o oresee he os-crcl ro ldes. Resls re cored wh hose cog o ro ler lss. Kewords: Rsers Cer VIV No-Ler Modes Fe Elee Mehod Oshore. INTRODUCTION The ol d gs dsr s cg eer-cresg chlleges s seerl oshore d oshore eloro elds he red he glol ded or eerg hs grow d he ol rces he show srog oll rece es. Ths scero hs led he oerg coes o ocs o dee d lr-dee wer eloro rgg orh ew cocers o cos redco hrogh dced echologcl deeloes. Coseqel he oshore eloro dsr hs cresed s eo o ew rels o reserch sch s o-ler dcs o srcres or dee d lr-dee wer d cool ld dcs secll regrdg ore-dced ro VIV lss. I s-se eloro hs ee cole d dedg c ro ow o ew o he cog deeloes deewer d lr deewer wll e ee ore chllegg. The oshore dsr hs lred eg o elore wer dehs he ls o he crre echolog d hs ls o ccess dehs oer 5. I s deelog s-se rodco sses rero or lr deewer rodco whch clde ore lele rsers d hrsher eroel codos. Uder hese crcsces o-ler eecs wll crese her lece oer he rser glol dcs. Moreoer he edec owrds ore lele d lgher s-se srcres wll oos he oerll o-ler ehor cldg he oe seed ro VIV. The oece o hs er s o rese ercl sd o he srcre-ld ercos d he srcrl oler dc ehor o deewer rser wh VIV oo resorg o he o-ler odl lss echqe d he der Pol oscllor roosed Bles [] o rerese he ld dcs. A cool odel s sogh sg he e-elee ehod. The eloed e elees cosder geoercl o-leres d he o-ler eqos o oo ke o cosdero o hrd-order o-leres whch led o hrd- d orh-order erl d sess esors. These o-ler eqos llow or eher cle or e ehor. I hs work he oo o he log rodco FPU s well s he ochdow zoe TDZ coc hs ee egleced. Ised he cer rser hs ee odelled wh ed eds s llowg or roo. Frher dces regrdg ercl roch or he TDZ re der deeloe sce cosderle rogrg eor s reqred o ke o cosdero he coc role he TDZ. B he w lcl sd o he TDP oo d he TDZ lerl coc lece o he o-ler orl odes o cer rsers hs ee ddressed []. I s eleed h o-ler odes l or role he ccre srcrl rereseo o dee-wer rsers odels wh sll er o degrees o reedo. A ree-hgg cer s ddressed cosderg s 6 h oler orl ode o ro or whch he rl reqec s close o h o ore sheddg. The le gol s o deelo ore corehese low-desol odel o he dcs o rser rodcg o-ler odels o cre he coled dcs o he ld-srcre sse.

2 Proceedgs o PACAM I Corgh 9 ABCM h P-Aerc Cogress o Aled Mechcs Jr -8 Foz do Igç PR Brzl. NONLINEAR MODES Ths er shold e regrded s l reserch o ele he o-lerl coled ld-srcre dcs o rsers whch wll e odelled s le res wh geoerc o-leres de o he colg o gel d rsersl dslcees. For he e eg erl ler wll e ssed. Torso d D eecs s well s FPU oo d o-leres de o he lerl coc he TDZ wll e egleced. Alhogh erl resoce coe o l hs reqrg he cosdero o he so-clled o-ler lodes he er wll cocere o he o-ler orl odes. I s eleed h he o-ler odes eher orl or l l or role o geere relle odels wh ew degrees o reedo sll keeg he essel ehor o rsers der dere se-lodg codos sch s he cses o hgh-reqec VIV d low-reqec drg... FEM Forlo Tcll he eqos o oo o geerl degree-o-reedo e-elee odel o elsc le re wh geoerc o-leres der ree ros red: M rs s Drs s U r where re he geerlzed coordes d he rces o ss M rs eqle dg D rs d elsc orce ecor U deed o he geerlzed coordes d eloces s ollows: r M rs M rs M rs M rs Drs Drs Drs Drs U r K rs s K rs s K rs s rs where M rs M rs M rs Drs Drs Drs K rs K d K rs r s re coss h dee secod- hrd- d orh-order esors. Ese s coeo or so hs ee eloed... Ler d No-ler Modes Drg odl oo he hse recores o dscresed ler sse re coed o wo-desol egeles ch he se w s he hse recor o -do sse wh geerlzed coorde res coed o he le. De o hs rce roer sch egeles re r olds o he dcl sse. I o-ler sses he r olds re o loger les d he oos whose recores re coed o he re clled o-ler orl odes. I geerl or -do sse here re r olds ech oe corresodg o dere ode; hese olds co he eqlr o d hs o re ge o he corresodg egesces o he lersed sse. Sch geoerc chrcerzo o odl oo sggess he so-clled r-old rocedre o deere orl odes whch Shw d Perre [9] roosed d led o sses o ew degrees o reedo. Sores d Mzzll [8] eeded he rocedre o ll e-elee odels o le res. A lere echqe o ele o-ler orl odes o e-elee odels ws roosed Mzzll d Brcho Neo [6]. Sch lere echqe s sed o he ehod o lle scles. To hdle cses o coled odl oos o o-ler sses de o erl resoce he lle-scles rocedre hs ee sccessll eeded Brcho Neo d Mzzll []. Here he esg ro kes lce r old eedded he hse sce whose deso s wce he er o he orl odes h erc. Ths old cos sle eqlr o d s ge here o he s-egesce o he lersed sse whch chrcerzes he corresodg ler odes. The l-ode c e locll descred ler coo o he ler odes. O hs old he sse ehes lke M-do oscllor where M s he er o coled orl odes... Ir Mold Procedre Here he del ses o he r-old rocedre re ollowed hg d s lco o e-elee odels o rsers. Irodcg he oo d sse c e wre rs-order or s

3 Proceedgs o PACAM I h P-Aerc Cogress o Aled Mechcs Corgh 9 ABCM Jr -8 Foz do Igç PR Brzl. Seres esos or he cos he eghorhood o he eqlr o re rodced Eq.: R N L H G F E C B where N L H G F E C B d R re kow coss. I drg odl oo he recor o he solo he hse-sce s resrced o wo-desol srce he s e ossle o eress ech geerlzed dslcee or eloc s co o wo o he or sce k d k les he eghorhood o he eqlr o. B ssg he eressos 5 we rre k k 6 whch s o-ler sse o rl derel eqos hg he cos s kows h e s dcl o sole s he orgl eqos. Howeer we look or roe solo hese cos c e wre s where 9 re coss o e deered. Now we sse 7 d 6 sse o o-ler olol eqos hg he s d s s kows s ored. I geerl here re solos o hs sse ech oe corresodg o dere se o odl relos 5.e. dere r old. Moreoer ssg o hese solos 7 d he reslg eressos 5 he k-h eqo - clled he odl oscllor eqo chrcerzes he dcs o he corresodg ode. Dels o he rocedre s oled re oded here or re c e od Sores d Mzzll [8] where s lso show h he solo o he o-ler olol eqo eoed oe c e oded roded he egeles d egeecors o he lersed sse re kow.

4 Proceedgs o PACAM I Corgh 9 ABCM h P-Aerc Cogress o Aled Mechcs Jr -8 Foz do Igç PR Brzl. FLUID EQUATIONS Cosder he l od show Fg. s clder o deer D rereseg he cross seco o rser. The eloc U s he se-crre eloc. Fro hs o o wll e sl sed hs e U or he se-crre eloc sre eloc sed o U. Fgre reses o Kárá Vore Sree creed whe cldrcl l od ddes sre o eloc U. The Srohl er S s he redo reqec o ore sheddg s lled he clder deer D d dded he ree sre eloc U s ollows: s D S. 8 U For s-crcl rege d drg coece eql o. he Srohl er c e od o e.. A low rege s cosdered s-crcl whe Reolds er rges ro o 5. S-crcl rege Fgre. Vo Kárá ore sree see [7]. I order o dec he dcs o he coled ld-srcre odel he ollowg sse o eqos ws ssed Iw d Bles []: d d ρd U dz ω ς s ς F ω h. o.. 9 d d D d d z U dz dz U d ω s z d D d UD d D d where s geerlzed dslcee chose s odl rle he ω s he rl reqec o he chose ro ode ς s s he srcrl dg ro ς F s he ld dg ro ρ s he sewer secc wegh s he srcre ss ls dded ss U s he sre eloc D s he clder deer z s he hdde rle see [] ω s s he ore sheddg reqec d re kow coss see []. I hs er he solo o eqo 9 s sogh sg he r old roch. Thereore secod- d hrd-order o-ler ers wll rse s coseqece o he o-ler srcrl orlo.. CASE STUD Fgre shows he rser e-elee odel wh 77 degrees o reedo. The low er o elees sed s de o he cosderle cool eor reqred o work o he hrd- d orh-order esors h led hs FEM odel o lloce roel GB o RAM eor. The lss ook 7 hors o rocessg e sg.6 GHz rocessor. Tle reses he rser d sed o odel he srcrl sse. Clclos were erored ssg Srohl er eql o. d he sre eloc U.5 /s orl o he oo le here ssed o e he cer le. Tkg o cco hese ssos he ore sheddg reqec roches he rl reqec o he 6 h ro ode. Thereore s ssed he lock- occrs or hs ode. As resl he ollowg sse o eqos s sogh o rerese he dcs o he coled ldsrcre sse here ddressed:

5 Proceedgs o PACAM I Corgh 9 ABCM h P-Aerc Cogress o Aled Mechcs Jr -8 Foz do Igç PR Brzl z z z 7 z 5 z 98 where d re resecel he odl dslcee d he odl eloc so dslcee s he 5 h degree o reedo s show Fg... The chose odl Fgre. Rser e-elee odel d he chose odl rle 5. Tle. Tcl seel rser d. og s dls E. N/ Rser legh 8 Cross-seco re A. Cross-seco oe o er I.7 5 Rser eerl deer D. Rser hckess e 9.5 Il eso he o 6 T N Il eso he 5 T 6.9 N oo Rser ss er legh wer sde dded ss 8kg/ Rser wegh er legh 77N/ I Fg. hse orr or oh ler le d o-ler red resoses c e osered. The ler resose s regrded o e h o he coled ld-srcre sse whe ol he srcrl sse s lersed. Ths he dcs o he ld res o-ler ee whe he ler srcrl sse s cosdered. The -o dslcee lde sed-se os o.565 he o-ler solo d o.567 he ler oe. Alhogh here s o oeworh derece regrdg he ol ldes s eresg o sress h he lde erees re dere wh he o.57 d he o or he o-ler srcrl sse. O he oher hd he ler srcrl sse ges he se sole les or he erees.8 d -.8. Hece he o-ler sse s le o cre he serc sess whch ws orged ro he rser geoerc ser.

6 Proceedgs o PACAM I Corgh 9 ABCM h P-Aerc Cogress o Aled Mechcs Jr -8 Foz do Igç PR Brzl Fgre. Phse Porr ler s le; o-ler s red. 5. ACKNOWLEDGEMENTS The hors ckowledge cl sor ro he Rol Soce der Jo Proec schee. The rs hor lso ckowledges he sor o CNPq der he Gr 587/ REFERENCES [] Brcho Neo O.G.P. Mzzll C.E.N. Elo o l-odes or e-elee odels: sses ed o : erl resoce Ierol Jorl o Solds d Srcres Vol [] Bles R.D. Flow-Idced Vro Secod Edo Kreger. [5] Mzzll C.E.N Lec S. Norl ro odes o sleder e o elsc odo wh lerl coc II ICTAM Adelde 8. [6] Mzzll C.E.N. Brcho Neo O.G.P. Elo o o-ler orl odes or e-elee odels Coers & Srcres Vol [7] Meegh J.R. Proeos de Pesqs o Tóco Gerção e Desredeo de Vórces o Escoeo o Redor de Cldros Tese de Lre Docêc Escol Polécc d Uersdde de São Plo São Plo. [8] Sores M.E.S. Mzzll C.E.N. Noler orl odes o lr res dscresed he e elee ehod Coers & Srcres [9] Shw S.W. Perre C. Norl odes or oler ror sses Jorl o Sod d Vro RESPONSIBILIT NOTICE The hors re he ol resosle or he red erl clded hs er.