Sina Mirzaei Sefat TATALAMENTO E AUTORROTAÇÃO DE PLACA PLANA VERTICAL ARTICULADA INDUZIDOS POR CORRENTE UNIFORME

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1 TATALAMENTO E AUTORROTAÇÃO DE PLACA PLANA VERTICAL ARTICULADA INDUZIDOS POR CORRENTE UNIFORME Sna Mrzae Sefat Tese de Dotorado apresentada ao Programa de Pós-gradação em Engenhara Oceânca, COPPE, da Unversdade Federal do Ro de Janero, como parte dos reqstos necessáros à obtenção do títlo de Dotor em Engenhara Oceânca. Orentador: Antono Carlos Fernandes Ro de Janero Novembro 0

2 TATALAMENTO E AUTORROTAÇÃO DE PLACA PLANA VERTICAL ARTICULADA INDUZIDOS POR CORRENTE UNIFORME Sna Mrzae Sefat TESE SUBMETIDA AO CORPO DOCENTE DO INSTITUTO ALBERTO LUIZ COIMBRA DE PÓS-GRADUAÇÃO E PESQUISA DE ENGENHARIA (COPPE) DA UNIVERSIDADE FEDERAL DO RIO DE JANEIRO COMO PARTE DOS REQUISITOS NECESSÁRIOS PARA A OBTENÇÃO DO GRAU DE DOUTOR EM CIÊNCIAS EM ENGENHARIA OCEÂNICA. Eamnada por: Prof. Antono Carlos Fernandes, Ph.D. Prof. Sergo Hamlton Sphaer, D.Sc. Prof. Jan Batsta Vlla Wanderley Ph.D. Prof. Celso Ppo Pesce, D.Sc. Dr. Rcardo Francss D.Sc. RIO DE JANEIRO, RJ BRASIL NOVEMBRO DE 0

3 Mrzae Sefat, Sna Tatalamento e Atorrotação de Placa Plana Vertcal Artclada Indzdos por Escoamento/ Sna Mrzae Sefat. Ro de Janero: UFRJ/COPPE, 0. XV, 7 p.: l.; 9,7 cm. Orentador: Antono Carlos Fernandes Tese (dotorado) UFRJ/ COPPE/ Programa de Engenhara Oceânca, 0. Referêncas Bblográfcas: p Atorrotação.. Tatalamento. 3. Manfold. I. Fernandes, Antono Carlos. II. Unversdade Federal do Ro de Janero, COPPE, Programa de Engenhara Oceânca. III. Títlo.

4 v To my parents, Mohammad and Zahra, and my ssters Azade and Atefe.

5 Agradecmentos Thanks God for all the blessngs that he brngs nto my lfe. I am n debt to my advser Prof. Antono Carlos Fernandes for hs gdance and spport drng the research, also all of hs favors and spports from the frst day that I came to Brazl p to now. I wold lke to thank Fabo Morera Coelho, Lís Vctor Cascão, Mohammad Mehd Armande and Amanda Slva de Albqerqe for ther techncal assstance n epermental tests and for many dscssons and sggestons. I lke to acknowledge Agênca Naconal de Petróleo ANP, for all of ther fnancal spports drng my PhD perod. I also wold lke to thanks from all staffs of LOC/COPPE/UFRJ (Laboratóro de Ondas e Correntes da COPPE, da Unversdade Federal do Ro de Janero): Ivan Bragança Marnho Falcão, Lz Antôno Ferrera, Anderson Araúo do Santos and Werner de Barros. Fnally, I wold lke to send my especal thanks to my top frends becase of ther frendshp and all of ther spports n my lfe: Saman Karm (Unversty of Illnos, USA), Mesam Zadhossen (Derak Co., Iran), Arash Roshanzamr (ISOIO Co., Iran), Reza Froozkooh (Cesos NTNU, Norway) and Arash Abbasna (Amrkabr Unversty of Technology, Iran). v

6 Resmo da Tese apresentada à COPPE/UFRJ como parte dos reqstos necessáros para a obtenção do gra de Dotor em Cêncas (D. Sc.) TATALAMENTO E AUTORROTAÇÃO DE PLACA PLANA VERTICAL ARTICULADA INDUZIDOS POR CORRENTE UNIFORME Sna Mrzae Sefat Novembro/0 Orentador: Antono Carlos Fernandes Programa: Engenhara Oceânca O obetvo deste estdo é entender o problema de osclação qe pode ocorrer com o lançamento lvre de dspostvos oceânco (offshore) na ága drante sa a nstalação no fndo do mar. Este estdo aborda a rotação de ma placa plana vertcal artclada torno de se eo vertcal ectada apenas por m escoamento de ága horzontalmente ncdente. No caso de placa fa, as cargas agndo sobre ela foram meddas para os ânglos de ataqe de 0 a 90, para dferentes números de Reynolds. Para o caso dnâmco, ma análse dmensonal mostra qe o movmento de rotação ndzda é governado essencalmente pelo número de Reynolds, pelo momento admensonal de nérca. Um modelo qase-permanente desenvolvdo pela tese é capaz de reprodzr tanto o tatalamento (fltterng) qanto a atorotação obtdas no LOC. Confrmo-se a estênca da freqênca natral de tatalamento (fltterng) pleno qe é lnearmente proporconal à velocdade do escoamento ncdente e nversamente proporconal à largra da placa. v

7 Abstract of Thess presented to COPPE/UFRJ as a partal flfllment of the reqrements for the degree of Doctor of Scence (D. Sc.) FLUTTERING AND AUTOROTATION OF A HINGED VERTICAL FLAT PLAT INDUCED By UNIFORM CURRENT Sna Mrzae Sefat November/0 Advsor: Antono Carlos Fernandes Department: Ocean Engneerng The obectve of ths stdy s to nderstand the oscllaton motons that may occr when an offshore devce s released n the water drng the nstallaton n seabed. Ths stdy addresses the rotaton of a hnged flat plate allowed to rotate abot a vertcal as nder the nflence of a nform crrent. In the statc case, the loads and moments actng on the flat plate were measred for angles of attack 0 90 and for dfferent Reynolds nmbers. For the dynamc case, the dmensonal analyss proves that the moton n flow ndced rotaton s governed essentally by dmensonless moment of nerta and Reynolds nmber. A qas-steady model s sggested to model the fltterng moton as obtaned n LOC s flme. It s confrmed that the natral and response freqences of fltterng moton are lnearly proportonal to the ncomng flow velocty and nversely proportonal to the flat plate wdth. v

8 Smáro Lsta de Fgras Lsta de Tabelas v Capítlo : Introdção Capítlo : Epermentos 4 Capítlo 3: Análse Teórca da Placa Estátca 6 Capítlo 4: Modelagem Teórca Qase-Permanente de Rotação Indzda de Placa Plana 8 Capítlo 5: Análse de Establdade da Rotação Indzda pelo Escoamento 9 Capítlo 6: Conclsões e Sgestões para Trabalhos Ftros 0 Apêndce : Introdcton Apêndce : Eperments n LOC s Crrent Flme 9. Epermental Setp 9. Fed Plate 3.. Hydrodynamc Forces and Moment on a Fed Flat Plate 3.. Hydrodynamc Loads and Moment Reslts 4... Lft Moment verss Mnk Moment 7..3 Strohal Nmber (St) 3.3 Flow Indced Rotaton Dmensonal Analyss Eqvalent Harmonc Angle of Fltterng Response Freqency of Atorotaton Natral Freqency of Fltterng Phase dagram of Fltterng and atorotaton 4 Apêndce 3: Theoretcal Analyss of Statc Plates Introdcton Krchhoff s Model Roshko s Model 48 v

9 3.4 W s Model Ktta s Model for Moment Coeffcent 58 Apêndce 4: Theoretcal Modelng of Flow Indced Fltterng and Atorotaton 6 4. Fltterng Natral Freqency of Fltterng Moton Atorotaton (Tmblng) Eqaton of Moton of Atorotaton W s Streamlne Theory Moment coeffcent model Ktta s Moment Coeffcent Model LOC Moment Coeffcent Model 75 Apêndce 5: Stablty Analyss of Flow Indced Rotaton Fed Ponts Conservatve System and Phase Dagrams Ktta s Moment Coeffcent Model W Streamlne Theory Moment Coeffcent Model LOC Moment Coeffcent Model Epermental Phase Dagram 84 Apêndce 6: Conclsons 87 Apêndce A: Comptatonal Fld Analyss 90 A. Trblence Modelng 93 A. ANSYS CFX Modelng 96 Apêndce B: Free Streamlne Theory 00 B. Krchhoff s Model 00 B. Roshko s Model 06 B.3 W s Model 0 References 3

10 Lsta de Fgras Fg. - Schematc of flow ndced rotaton of a flat plate hnged at the center and s free to rotate n nform crrent 5 Fg. - Pendlm Installaton Method (PIM) of a manfold, () frst phase: almost a free fall n water snce the nstallaton cable s slack, () second phase: s characterzed by the crvlnear moton wth the nstallaton cable becomng tat, () s st the smooth vertcal traectory down to the bottom 5 Fg. -3 Drng several model testng by Fernandes et al. [,], Oscllatory behavor of a manfold drng the frst phase of the PIM (Pendlos Installaton Method); (a) frst model of a comple obect (manfold), (b) second model showng clearly the fltterng behavor 6 Fg. -4 The vertcally hnged flat plate n the crrent flme of LOC/COPPE/UFRJ (Laboratory of Waves and Crrent of COPPE, Federal Unversty of Ro de Janero) 6 Fg. - (a) Schematc llstraton of reference as, (b) Pctre of the epermental setp n the LOC/COPPE/UFRJ (Laboratory of Waves and Crrent of COPPE, Federal Unversty of Ro de Janero) 9 Fg. - Assembly detals of the torqe sensor and load cell 0 Fg. -3 Data Acqston System for: (a) the load cell and (b) the torqe sensor Fg. -4 Dagram of forces and center of pressre setp arond the flat plate; note that for 0 the plate s transversal to the flow whle for 90 t s algned to the flow 4 Fg. -5 Drag coeffcent obtaned by eperments at LOC and calclated wth ANSYS CFX 5

11 Fg. -6 Lft coeffcent obtaned by eperments at LOC and calclated wth ANSYS CFX 5 Fg. -7 Comparson of normal coeffcent of epermental reslts at LOC and Hoerner [5] and also nmercal reslts 6 Fg. -8 Poston of the center of pressre of nmercal modelng compared wth eperments at LOC and two other epermental reslts of Flachsbart [4] and Tachkawa [37] and lnear segments as proposed by (-6-a) and (-6-b) 7 Fg -9 Schematc of the nteracton of nform crrent wth a D slender body 8 Fg. -0 Moment coeffcent obtaned nmercal modelng and eperments at LOC and also the Mnk moment coeffcent gven by (-5), and moment coeffcent segments, proposed n Table - 30 Fg. - Strohal nmbers of epermental reslts at LOC epermental and the nmercal smlaton reslts for dfferent block coeffcents 3 Fg. - (a) Tme hstory of atorotaton moton for I*=0.5 and for dfferent Reynolds nmbers, (b) Tme hstory of atorotaton moton for Re=9945 and for dfferent dmensonless moment of nerta 36 Fg. -3 Eqvalent harmonc angle of fltterng moton verss crrent velocty for flat plates wth dfferent dmensonless moment of nerta 37 Fg. -4 The epermental reslts of response freqency of atorotaton moton as a fncton of dfferent veloctes and for dfferent plates 39 Fg. -5 The epermental reslts of natral freqency as a fncton of dfferent veloctes and for dfferent plates for fltterng moton 40 Fg. -6 Natral freqency of flow ndced fltterng verss vorte sheddng freqency 4 Fg. -7 Phase dagram to classfy dfferent motons of flow ndced rotaton based on Reynolds nmber and dmensonless moment of nerta, obtaned by eperments at LOC. Each pont means a

12 dfferent test; prelmnary phase dagrams wll be dscssed later n Apêndce 5 4 Fg. 3- Schematc of free streamlne theory, for flow past a flat plate 45 Fg. 3- The z-plane of the actal flow based on Krchhoff s theory and other conformal mappngs planes [4]. 46 Fg. 3-3 Flow assmpton of Roshko s modelng [3,33] 48 Fg. 3-4 The flow n the physcal space (z-plan, where z = + y) 5 Fg. 3-5 Comparson of normal coeffcent of streamlne theory wth σ.5, and epermental reslts at LOC and Hoerner [5], and also ANSYS CFX nmercal reslts 53 Fg. 3-6 Locaton of stagnaton pont and center of pressre of flat plate [43]; µ s poston of stagnaton pont [43] 54 Fg. 3-7 Comparson of poston of the center of pressre wth streamlne theory (Eq. (3-)) and ANSYS CFX and epermental reslts of LOC, Flachsbart [4] and Tachkawa [38] 55 Fg. 3-8 The comparson of the lnear segmented reslts of poston of the center of pressre fond by streamlne theory and nmercal ANSYS CFX reslts 56 Fg. 3-9 Comparson of moment coeffcent of streamlne theory (Eq. 3-3) wth σ.0 and also σ.5 and nmercal reslts of ANSYS CFX and eperments at LOC 57 Fg. 3-0 Comparson of the moment coeffcents proposed by epermental reslts of LOC, streamlne theory, Mnk and Ktta 60 Fg. 4- (a) Tme seres of fltterng moton of the hnged flat plate abot the vertcal as sbmtted to a nform crrent (Re= ), (b) The respectve freqency doman response 6 Fg. 4- Undamped oscllaton when the dfference of the natral and ectng force freqency s small, and t reslts beatng moton, and are the frst and second waves freqency 63 Fg. 4-3 Comparson between epermental reslts from LOC and Eq. (4-5), I*=0.00, C B =0.4; Segment.: C M = -0.45, Segment.: C M =

13 -0.605, Segment.3: C M = Fg. 4-4 Comparson between epermental reslts from LOC and Eq. (4-5), I*=0.0038, C B =0.4; Segment.: C M = -0.45, Segment.: C M = , Segment.3: C M = Fg. 4-5 Comparson between epermental reslts from LOC and Eq. (4-5), I*=0.0047, C B =0.4; Segment.: C M = -0.45, Segment.: C M = , Segment.3: C M = Fg. 4-6 Comparson between epermental reslts from LOC and Eq. (4-5), I*=0.0378, C B =0.48; Segment.: C M = -0.45, Segment.: C M = , Segment.3: C M = Fg. 4-7 The epermental setp for atorotaton moton 69 Fg. 4-8 The phase dagram as a fncton of angle and anglar velocty for for dfferent Reynolds nmbers, I*= Fg. 4-9 The atorotaton moton of plate wth I * =0.5 and Re= Fg. 4-0 The atorotaton moton of plate wth I * =0.5 and Re= Fg. 4- The atorotaton moton of plate wth I * =0.5 and Re= Fg. 4- The atorotaton moton of plate wth I * =0.5 and Re= Fg. 4-3 The atorotaton moton of plate wth I * =0.508 and Re= Fg. 4-4 The atorotaton moton of plate wth I * =0.5 and Re= Fg. 5- The phase dagram of Ktta s moment coeffcent model 8 Fg. 5- The phase dagram of Streamlne theory moment coeffcent model 8 Fg. 5-3 The phase dagram of LOC adsted moment coeffcent model 83 Fg. 5-4 Typcal phase dagram of the small fltterng state, obtaned by eperments at LOC, Re= , I * = and = π/ 84 Fg. 5-5 Typcal phase dagram of fltterng moton, obtaned by eperments at LOC, Re= , I * = and = π/ 85 Fg. 5-6 Typcal phase dagram of atorotaton moton, obtaned by eperments at LOC, Re= , I* = 0.5 and θ = π/; note that the mnmm s close to zero (arond rad/s) 85 Fg. A- The velocty n RANS eqatons represent as tme average and flctaton velocty terms 9

14 Fg. A- Mesh generaton model 97 Fg. A-3 CFX pre model 98 Fg. A-4 Tme hstory of drag and lft coeffcents for dfferent angles of 99 attack Fg. B- Plans nder the assmpton of Krchhoff s modelng [0] 0 Fg. B- Poston of the center of pressre base on Krchhoff model (Eq. (B- 30)) 05 Fg. B-3 The hodograph of Roshko and W modelng 06 Fg. B-4 Plans nder the assmpton of Roshko s modelng [33,34] 06 Fg. B-5 Plans nder the assmpton of W s modelng [43] 0 Fg. B-6 Lft coeffcents as fncton of angle of attack for dfferent wake nder-pressre coeffcents [43] Fg. B-7 Drag coeffcents as fncton of angle of attack for dfferent nderpressre wake coeffcents [43] v

15 Lsta de Tabelas Table - The eperment setp propertes Table - Lst of moment coeffcent segments 3 Table -3 Lst of dmensons of the varables 34 Table 3- Drag coeffcent vales for dfferent vales of k [33] 49 Table 3- The moment coeffcents proposed by eperments, streamlne theory, Mnk and Ktta 59 Table 4- Lnear segments of moment coeffcent 64 Table 4- The root mean sqare of angle of rotaton for dfferent Reynolds nmbers 7 Table 4-3 Eqatons of moton based on moment coeffcent proposed by epermental reslts of LOC 75 Table 5- Fed ponts for dfferental eqaton models of moton 77 Table 5- Stablty analyss of the lnearzed eqatons n the neghborhood of the fed pnts 78 Table 5-3 The conservatve energy systems for dfferent dfferental eqaton models 80 Table 5-4 The vale of energy level for separaton traectory of fltterng to atorotaton 84 Table A- RANS eqaton terms 9 Table A- Mesh propertes 96 Table A-3 Propertes of fld doman and bondary condton for k-epslon modelng n ANSYS CFX 97 v

16 Capítlo Introdção A nteração de m corpo em movmento com corrente é m dos prmeros problemas a ter recebdo atenção. Entretanto, a maora das pesqsas se concentra nos movmentos de translação, pelo menos em Engenhara Naval e Oceânca. Apenas algns têm realzado nvestgações sobre o movmento de rotação. Conseqentemente, algns aspectos mportantes da rotação ndzda por escoamento anda não são bem compreenddos. Este estdo aborda a rotação de ma placa plana vertcal artclada em torno de se eo vertcal, ectada apenas por m escoamento de ága horzontalmente ncdente. O obetvo deste estdo é entender o problema de osclação (qe pode ser ecessva) qe pode ocorrer com o lançamento lvre de dspostvos oceâncos (offshore) na ága, drante sa a nstalação no fndo do mar. O comportamento da placa plana é estdado epermentalmente e teorcamente e o estdo evdenca a ocorrênca de tatalamento (fltterng) e de atorrotação conforme fatores como a ntensdade da corrente, momento de nérca da placa, condções ncas, entre otros, como eplcado na tese. A tese está organzada do segnte modo: ela tem ses capítlos e a cada capítlo corresponde m apêndce de mesmo número onde o assnto é apresentado pelo ator O Dconáro Mchaells Portgês Inglês (Edção ) assm se refere ao verbo Tatalar: sm bras whr as a wng stroke vnt (...). to whr, to fltter

17 qe é mas flente em nglês e onde, portanto, pode epressar-se melhor. A tese tem também mas dos apêndces, o A e B, onde aspectos teórcos são anda mas aprofndados. Estes últmos apêndces são ctados pelos capítlos e apêndces nmércos. Assm sendo, o Capítlo descreve a confgração epermental para medções no canal de correntes do LOC/COPPE/UFRJ (Laboratóro de Ondas e Correntes do Programa de Engenhara Naval e Oceânca da COPPE, Unversdade Federal do Ro de Janero). No caso de placa fa (sem movmento), as cargas agndo sobre ela foram meddas para os ânglos de ataqe de 0 a 90, para dferentes números de Reynolds. Para o caso dnâmco (placa lvre para osclar apenas em rotação em torno de eo vertcal), ma análse dmensonal mostra qe o movmento de rotação ndzda é governado essencalmente pelo número de Reynolds, pelo momento admensonal de nérca, pela largra da placa e pelas condções ncas. O Capítlo 3 é alocado para eplcar ma teora qe permte a modelagem analítca pela Teora da Lnha de Corrente (Streamlne Theory). Esta teora fornece resltados qe podem ser sados para calclar as cargas (forças e momentos) agndo na placa plana. Além dsso, a tese desenvolve ma modelagem nmérca sando o sstema ANSYS CFX. O Capítlo apresenta comparação dos resltados das três metodologas ctadas: epermentos, ANSYS CFX e Teora da Lnha de Corrente (Streamlne Theory). No Capítlo 4, o trabalho sgere m modelo qas-permanente analítco tanto para o tatalamento (fltterng), como para a atorrotação. Novamente, segem comparações com resltados dos epermentos no canal de correntes do LOC. Neste capítlo, dentfca-se e estda-se a freqüênca natral de vbração do movmento de tatalamento (fltterng) bem como a freqênca de atorrotação.

18 No Capítlo 5, o trabalho realza ma análse de establdade através da confecção de dagrama de fase. Aborda o comportamento da rotação ndzda pelo flo dentfcando a natreza da transção do tatalamento (fltterng) para a atorrotação. Evdenca-se também a dependênca das condções ncas. Fnalmente, o Capítlo 6 fecha a tese, apresentando Conclsões e Sgestões de Trabalhos para o Ftro. Ver mas detalhes no Apêndce. 3

19 Capítlo Epermentos Este capítlo apresenta os epermentos realzados, começando com a confgração epermental para medções no LOC/COPPE/UFRJ. No caso da placa fa, as forças e momentos agndo sobre a placa plana foram meddos para os ânglos de ataqe θ=0-90 e para dferentes números de Reynolds. Os resltados estátcos são útes para se compor as propredades mportantes para o caso dnâmco nos movmentos de tatalamento (fltterng) e no de atorrotação ndzdos apenas pelo flo. Da mesma forma, os resltados para o número de Strohal, sto é, dentfcando a freqüênca de geração de vórtces, são também apresentados. Trata-se de mas ma contrbção da tese, ma vez qe o número de Strohal é bem conhecdo apenas para corpos clíndrcos. Por otro lado, para o caso dnâmco, a análse dmensonal mostra qe o movmento de rotação ndzda é governado essencalmente por admensonal do momento de nérca, número de Reynolds, largra da placa e condções ncas. Com base em város ensaos eastvamente realzados, foram possíves defnr os lmtes de bfrcação de establdade qando o tatalamento (fltterng) mda para a atorrotação. Para tal, m dagrama de fase fo preparado baseado nos epermentos. Observaram-se três tpos de estados: o estado de peqeno tatalamento (fltterg), o 4

20 tatalamento (fltterng) pleno e o estado de atorrotação. A dependênca é clara do número de Reynolds, momento de nérca admensonal e condções ncas. Por otro lado, os epermentos e a análse dmensonal confrmaram qe a freqüênca natral é lnearmente proporconal à velocdade de entrada e nversamente proporconal à largra da placa. Também com base na análse dmensonal e epermentos, fo confrmado qe o ânglo de rotação eqvalente harmônco de rotação é apromadamente constante em dferentes velocdades. Ver mas detalhes no Apêndce. 5

21 Capítlo 3 Análse Teórca da Placa Estátca Neste Capítlo, a tese sstematzo modelagem teórca para o caos da placa estátca sbmetda a escoamento nforme ncdndo por város ânglos de ataqe. Uso a chamada Teora da Lnha de Corrente (Streamlne Theory). Esta teora bascamente modela, com bons resltados, o escoamento a sante. O prncípo básco da Teora da Lnha de Corrente (Streamlne Theory) assme qe o flo bdmensonal permanente e espacalmente constante à sante para cada ânglo de ataqe, caracterzando ma stação plenamente desenvolvda. As bordas da placa são assmdas cantos vvos e consdera-se o escoamento precocemente separado. Mostra-se qe a dependênca do número de Reynolds é nsgnfcante. Além dsso, neste capítlo, a espessra da placa plana é assmda nla. Assmr qe a sante a dstrbção da pressão é constante sgnfca qe a velocdade perto dos dos pontos de separação é qase a mesma. Vórtces se formam alternadamente em cada lado. Esta formação de vórtces ocorre à sante de qalqer corpo rmbdos e a freqüênca de desprendmento de vórtces é ma característca da forma do corpo. São três as dferentes Teoras da Lnha de Corrente (Streamlne Theores) descrtas neste capítlo (Krchoff, Roshko e W), além de ma teora herístca (Ktta). Para cada ma, o coefcente admensonal de momento fo calclado para dferentes 6

22 ânglos de ataqe, e a aderênca com resltados epermentas é notável para a Teora de W. Isto permte a tlzacão dos sempre útes resltados analítcos. Ver mas detalhes no Apêndce 3. 7

23 Capítlo 4 Modelagem Teórca Qase-Permanente de Rotação Indzda de Placa Plana O capítlo apresenta ma apromação qas-permanente com o obetvo da obtenção de mas conhecmentos sobre a natreza do fenômeno de tatalamento (fltterng), bem como para a atorrotação da placa plana ndzda pelo escoamento. Mostra-se ser possível modelar a dnâmca da placa através de ma eqação dferencal ordnára com contrbções de momento de nérca adconal e coefcentes de momento obtdos dos capítlos anterores. Baseado nesta modelagem qas-permanente do movmento, confrma-se qe para o caso de tatalamento (fltterng), a freqüênca natral do movmento de osclação é lnearmente proporconal à velocdade do escoamento de entrada e nversamente proporconal à largra da placa. A modelagem matemátca mostra qe a freqüênca natral é qase gal ao resltado epermental da freqênca de desprendmento de vórtces o qe leva a m movmento com característcas de batmento no caso de tatalamento (fltterng). Por otro lado, o método de Rnge-Ktta fo sado para estmar as traetóras de movmento de atorrotação com m modelo qas-permanente semelhante. Estas estmatvas teórcas têm boa aderênca com as traetóras epermentalmente obtdas nos ensaos do LOC. Ver mas detalhes no Apêndce 4. 8

24 Capítlo 5 Análse de Establdade da Rotação Indzda pelo Escoamento Este capítlo desenvolve ma análse de establdade sobre a rotação ndzda pelo escoamento nforme e aprofnda-se na bfrcação entre o fenômeno de tatalamento (fltterng) para atorrotação. Prmeramente, os pontos fos para os dferentes modelos são obtdos. Cada ponto é classfcado por meo da eqação lnearzada. Encontraram-se apenas centros e pontos de sela. Em segda, os dagramas de fase são montados em fnção da velocdade anglar e ânglo de rotação. Os níves de energa ncas são dentfcados e mostram-se essencas para entender a bfrcação ctada. Mostram-se os resltados para os modelos ctados da Teora da lnha de corrente (Streamlne Theory), Ktta e m advndo dos epermentos, chamado de LOC. Como era de se esperar, o dagrama de fase com base no modelo do LOC se porta melhor qando comparado com os epermentos. Na verdade, neste modelo de apromação do coefcente de momento é mas prómo da realdade em todos os ânglos. 9

25 Capítlo 6 Conclsões e Sgestões para Trabalhos Ftros A tese apresento o estdo epermental combnado com sstematzação teórca sobre a hdrodnâmca nstável de ma placa plana vertcal artclada nm eo fo qando sbmetda apenas a m escoamento nforme. Identfco-se a estênca de três tpos de comportamento: o peqeno tatalamento (fltterng), o tatalamento (fltterng) pleno e a atorrotação. As cargas hdrodnâmcas estátcas foram obtdas epermentalmente no canal de correntes do LOC. Hove também o processamento nmérco através do ANSYS CFX. Fnalmente m método analítco, com base na Teora da Lnha de Corrente (Streamlne Theory), corroboro os resltados anterores. Estes resltados estátcos são evdentemente útes para a obtenção de resltados dnâmcos de rotação ndzda pelo escoamento. Um modelo qase-permanente desenvolvdo pela tese é capaz de reprodzr tanto o tatalamento (fltterng) qanto a atorotação obtdos no LOC. Para qe este modelo tvesse mas consstênca, a tese recorre com scesso à ma análse dmensonal. Confrmo-se a estênca da freqênca natral de tatalamento (fltterng) pleno qe é lnearmente proporconal à velocdade do escoamento ncdente e nversamente proporconal à largra da placa. Como era de se esperar, devdo à mesma dmensão, o mesmo tpo de dependênca se aplca à rotação no fenômeno de atorrotação. 0

26 Fnalmente, o dagrama de fase da análse de establdade corrobora a estênca dos três estados de peqeno tatalamento (fltterng), o tatalamento (fltterng) pleno e a atorrotação, mas mostra o papel da energa ncal no lmte de transção entre eles. Em partclar, eplcto-se a dependênca das condções ncas. Com estes resltados, nem qe sea qaltatvamente, espera-se contrbr para a melhora do Método de Lançamento Pendlar de Manfold (PIM - Pendolos Installaton Method) qe fo a motvação ncal da presente nvestgação. No ftro, sgere-se contnar as nvestgações, sempre sando ma abordagem combnando teora e epermentos, como a tlzada na tese. Como o obetvo é dmnr o tatalamento (fltterng) e a atorrotação, sgere-se estdar corpos com farngs e bordo de ataqe, conforme á ncado por (Fernandes et al -- OMAE0). Fnalmente, sgere-se estdar corpos com espessra.

27 Apêndce Introdcton The nteracton of a movng body wth crrent s one of the earlest hydrodynamc problems to receved attenton. However, most of the researches concentrate on translatonal motons, and st few nvestgatons have been performed on rotatonal moton. Hence, some mportant aspects of the flow ndced rotaton are stll not well nderstood, althogh ths phenomenon has ganed nterest n varos areas sch as morphology of plants and anmals, meteorology and aeroballstcs, aerospace engneerng, wndborne debrs occrrng n wndstorms [,7,39] and the fallng obect [,4,3,3,6,37]. Ths eplans the choce of ths thess abot stdyng the behavor of a hnged flat plate allowed to rotate abot a vertcal as nder the nflence of a nform crrent (see Fg. -). Ths s ndeed a fndamental problem wth only one degree of freedom that, as shown net, reqres a vast knowledge of several hydrodynamc sses as lft, drag, vorte sheddng, self ectaton, separaton, etc. The fltterng and atorotaton are two dfferent phenomena whch may occr n flow ndced rotaton of a plate abot a fed as, whch s free to rotate n the crrent. In the atorotaton, the plate rotates contnosly arond a vertcal as and t never damps ot. On the other hand, the fltterng moton s another nepected perodc oscllaton of the plate arond a stable poston whch the plate s normal to the flow. For a free fallng obect, the fltterng moton s called the same and the body oscllates

28 ether perodcally or chaotcally from sde to sde as t descends alternatng gldng at low angle of attack and fast rotatonal moton [6]. Bt the atorotaton moton for the fallng obect s called tmblng whch s characterzed by the end-over-end rotatonal moton of body. Mawell [3] was the frst who tred to eplan tmblng moton of a flat-plate. Later, several epermental researches nder both free-fall and fed-as condtons were made n order to classfy and qantfy the types of these rotatonal motons. The theoretcal and epermental reslts before 979, sch as Dplech [4], Smth [34], and etc, have been evalated and smmarzed by Iversen [8]. Accordng Iversen [8], the moton of a freely fallng and also fed as rectanglar plate s governed by the Reynolds nmber (Re=Ub/v), the aspect rato (c/b), thckness rato (t/b) and the non-dmensonal moment of nerta of the flat plate (I * =I/ ρ f cb 4 ), where b, c, t and I are wdth, span, thckness and moment of nerta of plate respectvely. Also U, ρ f and v are the velocty, densty and knematc vscosty of fld, respectvely. Iversen shows that, the transton from fltterng to tmblng shold be completely governed by these parameters. Tachkawa [39] frstly, presented a two-dmensonal eqaton of moton to eplan the traectores of free fallng of sqare and rectanglar plates n a nform flow. Secondly, he obtaned by eperments, the aerodynamc coeffcents of atorotaton n dfferent sqare and rectanglar flat plates, whch are reqred for solvng the eqatons of moton. The reslts show that the traectores of a flat plate released nto a flow from rest s closely related to the ntal mode of moton and s dstrbted over a wde range. Lgt [,] arged that, for a vscos fld, at least qaltatvely, a ffth-order polynomal for the dampng terms n the pendlm eqaton s necessary to smlate the 3

29 self-ected oscllaton (fltterng) and atorotaton for bodes wth as fed n a parallel flow. Feld et al. [3] prepared a bfrcaton dagram based on epermental data, showng the dynamcal behavor of fallng dsks as a fncton of the two parameters: the dmensonless moment of nerta and Reynolds nmber. Accordng to ths bfrcaton dagram, dependng on the dmensonless moment of nerta, the moton cold be fltterng for smaller dmensonless moment of nerta, or tmblng for larger dmensonless moment of nerta. Based on these measrements, the transton from fltterng to tmblng s nearly ndependent of Reynolds nmber and t appears at I * Tanabe et al. [37] gave a phenomenologcal model for free fall of a paper, assmng zero thckness and ncompressble deal fld. They dscovered fve dfferent patterns of: perodc rotaton, chaotc rotaton, chaotc fltterng, perodc fltterng and smple perpendclar fall. Mttal et al. [5] show a comptatonal stdy on flow ndced moton of a hnged plate pnned at ts center. Ther focs s on the effect of Reynolds nmber and plate thckness rato and non-dmensonal moment of nerta on vorte ndced rotaton of plates. Based on ther nmercal reslts, they sggest that the fltter and tmble freqences of large aspect-rato plates are governed by the Von-Kármán vorte sheddng process. Anderen et al. [] nvestgated the dynamcs of freely fallng plates epermentally, nmercally and by qas-steady modelng, at Reynolds nmber of 0 3, whch descrbes the moton of freely fallng rgd plates based on detaled measrement of the plate traectores and from ths to assess the nstantaneos fld forces. 4

30 Fg. - Schematc of flow ndced rotaton of a flat plate hnged at the center and s free to rotate n nform crrent. In the present work, controllng the oscllatory behavor of manfolds n the pendlos nstallaton method (PIM) [8,0,,,35] s the frst motvaton to stdy the flow ndced rotaton. Fg - descrbes the PIM and Fg -3 shows some selected pctres wth fltterng from other epermental tests [9]. Thogh, the stdy on flow ndced rotaton of a vertcal hnged flat plate nder the nflence of nform crrent s a -DOF problem, bt t ams at the nderstandng of the fltterng problem of fallng obects n ar or water. The reslts cold be generalzed for some other phenomena whch are 3-DOF, lke as PIM method, fallng obects, wndborne debrs, etc. Fg. - Pendlm Installaton Method (PIM) of a manfold, () frst phase: almost a free fall n water snce the nstallaton cable s slack, () second phase: s characterzed by the crvlnear moton wth the nstallaton cable becomng tat, () s st the smooth vertcal traectory down to the bottom. 5

31 (a) (b) Fg. -3 Drng several model testng by Fernandes et al. [,], Oscllatory behavor of a manfold drng the frst phase of the PIM (Pendlos Installaton Method); (a) frst model of a comple obect (manfold), (b) second model showng clearly the fltterng behavor. In ths work, the flow ndced rotaton of a fed as flat plate s stded epermentally and theoretcally and both the fltterng and the atorotaton have been clearly dentfed. The eperments were condcted n a flme (m.4m 0.5m) at the LOC/COPPE/UFRJ (Laboratory of Waves and Crrent of COPPE, Federal Unversty of Ro de Janero), see Fg. -4. Fg. -4 The vertcally hnged flat plate n the crrent flme of LOC/COPPE/UFRJ (Laboratory of Waves and Crrent of COPPE, Federal Unversty of Ro de Janero). 6

32 Apêndce descrbes the epermental setp for measrements at LOC/COPPE/UFRJ. In the statc case, the loads and moments actng on the flat plate were measred for angles of attack θ0 90 and for dfferent Reynolds nmbers. On the other hand, for the dynamc case, the dmensonal analyss proves that the moton n flow ndced rotaton moton s governed essentally by dmensonless moment of nerta and Reynolds nmber. Certan combnatons defne the stablty bondares between fltterng and atorotaton. Hence n ths apêndce, a bfrcaton dagram prepared from the eperments to classfy dfferent states, observed small fltterng, fltterng and atorotaton based on dfferent Reynolds nmber and dmensonless moment of nerta. For the analytcal modelng, the so-called Streamlne theory has been sed to appromate the wake far downstream and Apêndce 3 s allocated to eplan ths theory. A comparson of the reslts among the three methodologes of eperments, ANSYS CFX nmercal modelng and Streamlne theory s dscssed. In Apêndce 4, a qas-steady model s sggested to model the traectory of atorotaton moton va analytcal modelng. The comparson of analytcal modelng wth eperments at water flme of LOC was made. It shold be mentoned that most of the prevos researches on flow ndced rotaton, condcted epermentally and rarely nmercally. In ths apêndce, the natral freqency of fltterng moton has also been sggested. In Apêndce 5, a stablty analyss s performed on flow ndced rotaton phenomenon to gan nsght nto the natre of transton from fltterng to atorotaton. At frst, the fed ponts for dfferent models of moton are obtaned and each pont analyzed sng the lnearzed eqaton. Secondly, the phase dagrams as a fncton of anglar velocty and angle of rotaton have been presented for dfferent dynamc models 7

33 from Apêndce 3. Fnally, phase dagrams obtaned by eperments at LOC for small fltterng state, fltterng and atorotaton are presented. Apêndce 6 shows the conclsons and sggestons of ftre works. 8

34 Apêndce Eperments n LOC s Crrent Flme. Epermental Setp The eperments were condcted n a crrent flme wth m n length,.4 m n wdth and 0.5 m n depth at LOC/COPPE/UFRJ (Laboratory of Waves and Crrent of COPPE, Federal Unversty of Ro de Janero), shown n Fg. -. The epermental set p was assembled to measre forces, moment and the center of pressre for dfferent angles of flow ncdence on the fed flat plate and for dfferent Reynolds nmber. Sbseqently, free rotaton of the flat plate was also assessed. (a) (b) Fg. - (a) Schematc llstraton of reference as, (b) Pctre of the epermental setp n the LOC/COPPE/UFRJ (Laboratory of Waves and Crrent of COPPE, Federal Unversty of Ro de Janero). 9

35 The epermental apparats was monted n accordance wth the detals shown n Fg. -. A load cell and a torqe sensor were postoned at the top of the plate. These sensors were calbrated already postoned n the apparats. Fg. - Assembly detals of the torqe sensor and load cell. The vorte sheddng freqences for the statc flat plate were obtaned throgh spectral analyss of the sgnal acqred from an Acostc Doppler Velocmeter (ADV) whch was postoned at a pont n the wake of the plate. The velocty of the crrent flowng n the channel was obtaned from a trbne type flow-meter. The rotaton was measred by two dfferent systems. The frst one s a Qalsys system and the other s potentometer. The Qalsys system spples a range of hardware and software prodcts for moton captre and analyss of movement data. The key components of the system are the cameras and the Qalsys Track Manager (QTM) software to measre the rotaton angle and also freqency. 0

36 The potentometer placed on the top of the flat plate was powered by a voltage sorce. The otpt voltage of the potentometer was obtaned wth the same acqston system sed for the load cells. The oscllaton freqency was obtaned by spectral analyss of the rotaton sgnal acqred for each velocty of ncdent crrent. The data acqston from the load cell was performed sng a system (NI - 97 modle n conncton wth the nversal NI Daq - 99) whch has A/D converters and cstomzed connectons for the stran-gages brdge complement reqred for the cell. The dgtal sgnal was recorded by software blt n LabVew 8.. The acqston of data from the torqe sensor was performed smlarly, bt sng an analog sgnal condtoner that amplfes and flters the sgnal before beng dgtalzed. Ths confgraton was reqred de to the sensor fll scale compared to the low magntde of the torqe sgnal for some angles of flow ncdence. Fg. -3 (a) and (b) present the mentoned data acqston systems. (a) (b) Fg. -3 Data Acqston System for: (a) the load cell and (b) the torqe sensor.

37 Intally an almnm plate wth wdth of 0.6 m, draft of 0.5 m, and thckness of 5 mm was tested. Ths means that the blockage coeffcent (C B =plate wdth/flme wdth) s In ths block coeffcent, some wall blockage effect was observed. To decrease ths effect a smlar almnm plate 0.3 m n wdth was tested (C B =0.4) nder the same epermental condtons. Fnally, to observe the effect of the nerta varaton n the behavor of the oscllatng flat plate, two PVC plates wth wdth 0.3 m (C B =0.4) and 0. m (C B =0.4) were tested. Ther thcknesses were 5 and 8 mm, respectvely. The water depth was kept constant at 0.5m. The characterstcs of the flat plates tested are smmarzed n Table -: Plate Propertes Wdth Materals 0., 0.3 and 0.6 m Almnm, PVC Fld Parameters Fld sed Water Velocty range m/s Table - The eperment setp propertes. Based on tests n wnd channels, by prevos researchers [3,7] and also the eperments n LOC, t s proved that the atorotaton (tmblng) occrs at large dmensonless moment of nerta. In the water flme case, becase of large densty of water n comparson wth the ar, ncreasng of the dmensonless moment of nerta s not straght forward. For ths prpose a nqe apparats was constrcted to model the atorotaton whch s dscssed n Apêndce 4.

38 . Fed Plate.. Hydrodynamc Forces and Moment on a Fed Flat Plate The loads actng on a flat plate mmersed n a flow stream may be decomposed n normal pressres and tangental stresses on ts srface. When ntegrated, these pressres and stresses gve rse to the resltant forces and moment. The component of the load whch s n the same drecton of the flow s the drag and the lft force s perpendclar to the flow drecton. There s also a bnary gven by the moment of the force abot an as. If ths as s taken at the center of pressre, by defnton, the bnary s zero. The dmensonless lft C L, drag C D, normal C N and moment C M coeffcents are defned as below. C C C L (-) 0.5 U S L f D 0.5 U S (-) D f N 0.5 U S (-3) N f C M z 0.5 U M f Sb (-4) where L s the lft, D s the drag, N s the normal force, M Z s the moment abot z as (the flow s n,y plane), s the fld densty, U s the pstream flow velocty and S s a reference srface. Here S=bh, where b s the flat plate wdth and s the depth of the water n the flme. The dstance from the centre of pressre to the centre of the plate s c. The relatonshp between the moment coeffcent, the normal force coeffcent (C N ) and the poston of the centre of pressre s: C ( c / b) (-5) M C N 3

39 The dmensons of the flat plate and the dagram of drag and lft forces setp arond the flat plate, as well as the angle of attack are shown n Fg. -4. Fg. -4 Dagram of forces and center of pressre setp arond the flat plate; note that for 0 the plate s transversal to the flow whle for 90 t s algned to the flow... Hydrodynamc Loads and Moment Reslts The loads and moment coeffcents were measred as a fncton of angle of attack by eperments at LOC. Smlar measrements were carred ot wth a CFD smlaton sng the ANSYS CFX code (See Apêndce A). Fgs -5 and -6 show respectvely the drag and lft coeffcents as a fncton of angle of attack for both the eperment reslts and the nmercal smlaton wth ANSYS CFX. The Reynolds Nmber was kept arond In both fgres, the nmercal smlaton of lft and drag coeffcents show good agreement wth epermental reslts. 4

40 ..8 CFX - CB=0.06 CFX - CB=0.4 LOC - CB=0.4.5 C D θ[deg.] Fg. -5 Drag coeffcent obtaned by eperments at LOC and calclated wth ANSYS CFX CFX - CB=0.06 CFX - CB=0.4 LOC - CB=0.4 C L =πα C L θ[deg.] Fg. -6 Lft coeffcent obtaned by eperments at LOC and calclated wth ANSYS CFX. A classcal reslt of potental theory for flat plate s that for a small angle of attack, [0,4] (Note that here ). Fg. -6 also shows the 5

41 comparson of ths reslt wth epermental and nmercal reslts. It s clear that, for the cases wth smaller blockage effect the reslts match well wth the theoretcal small ncdent angle lmt. Flachsbart [4] measred the normal coeffcent of a two dmensonal flat plate. These data are reported by Hoerner [5]. Fg. -7 shows the comparson of normal coeffcent of nmercal smlaton wth ANSYS CFX and the epermental reslts n LOC and Hoerner [5] reslts LOC CFX - CB 0.06 HOERNER C N θ [deg.] Fg. -7 Comparson of normal coeffcent of epermental reslts at LOC and Hoerner [5] and also nmercal reslts. Fg. -8 shows a good agreement between the poston of the center of pressre fond by nmercal modelng and eperments at LOC and two other epermental reslts of Flachsbart [4] and Tachkawa [40]. Consderng these reslts, two lnear regons may be devsed as show n Fg. -8. These two lnear segmented models adsted to ft the poston of the center of pressre are gven by Eqs (-6-a) and (-6-b). 6

42 Center of pressre (c/b) LOC CFX - CB=0.06 TACHIKAWA FLACHSBART assmed C P = 0.38θ C P = 0.0θ θ[deg.] Fg. -8 Poston of the center of pressre of nmercal modelng compared wth eperments at LOC and two other epermental reslts of Flachsbart [4] and Tachkawa [40] and lnear segments as proposed by (-6-a) and (-6-b). b 0.0 c for angle of attack 55 (-6-a) b c for angle of attack (-6-b)... Lft Moment verss Mnk Moment Based on the D'alembert's parado the net force on a body whch s mmersed n an nvscd flow s zero, bt not necessarly a zero moment and any shape other than a sphere generates a moment. Ths moment may be called Mnk moment n the elongated body contet [4]. The Mnk moment arses from the asymmetrc locaton of the stagnaton ponts, where the pressre s hghest. The Mnk moment sally arses for the case of a long slender body nclned at angle to the drecton of moton n an nbonded fld (see Fg. -9). 7

43 Fg -9 Schematc of the nteracton of nform crrent wth a D slender body. Resolvng the free stream velocty U nto components parallel and normal to the as of the body, the cross flow velocty (v) and the added mass ( m d ) gves s an elemental momentm ( m vd ) and f ths changes wth tme, Newton s second law gves s the relaton d df mdv (-7) dt Or df d d dm dv m v v m (-8) dt dt dt Mnk omtted the second term becase he was not consderng accelerated flow. By consderng v U sn and d dt U cos, (-8) may be wrtten as (-9) df d dm dm d dm v v U sn (-9) dt d dt d The added mass s [7] m y S (-0) where S y (-) By consderng (-0) and (-), the reslt s 8

44 df ds U sn (-) d d The ntegral over the entre length of the body s [7] F U sn S B (-3) where S B s the base area, f the body s trncated. The Mnk moment abot the vertcal as of the body can be obtaned n a manner smlar to that sed to obtan the lateral force. For steady moton, we obtan the followng epresson [7] M m U m m U sn b dm sn yy (-4) where m s added mass along the body -as (forward), and m yy s along the body lateral y-as, and also shold be noted that. Ths moment may be appromated for small angles of ncdence. Accordng to the above statements and etendng them, for the D case, the Mnk moment coeffcent for a flat plate can be epress as C M ( )sn (-5) 4 On the other hand, based on the potental theory for a flat plate n small angles of attack, the lft coeffcent s [4]: C sn (-6) L The locaton of the center of pressre, for small α vales s located at one forth of plate wdth (b/4) from the nose. Hence, the center of pressre cold be appromated as [30,3]: ( c / b) (/4) cos (-7) Therefore the lft moment coeffcent can be epress as 9

45 C M sn ( ) cos ( )sn cos ( )sn 4 4 (-8) whch t s eactly matched by the formlaton of the moment coeffcent of Mnk moment for flat plates (Eq. -5). The comparson of the moment coeffcent calclated by nmercal modelng and eperments at LOC and also the Mnk moment coeffcent gven by (-5) s presented n Fg. -0. Ths comparson shows that the range of ts valdty s (75 90) θ[deg.] C M = -0.45θ C M = -(π/4)snθ C M LOC CFX - CB=0.06 Mnk Fg. -0 Moment coeffcent obtaned nmercal modelng and eperments at LOC and also the Mnk moment coeffcent gven by (-5), and moment coeffcent segments, proposed n Table -. Table - shows a model adsted to ft the avalable epermental and nmercal data for moment coeffcent. For angles of attack less than 30 (30 ), the normal coeffcent s almost constant; hence t s obvos that by mltplyng wth the lnear term of center of pressre, the resltant moment coeffcent s lnear too. Bt the normal coeffcent n angles of attack between 30 to 75 (30 75 ), s not constant and cold be epressed as a polynomal fncton. So t epected by 30

46 mltplyng the normal coeffcent wth lnear term of center pressre, the resltant moment coeffcent also be a hgher order polynomal fncton. Bt becase of small vales of the nonlnear terms, a srprsng lnear behavor for moment coeffcent shows p. Ths s shown n Fg -0 and Table -. So ndeed we have a Hydrodynamc Torsonal Sprng for angles of attack less than 75. Moment Coeffcent Angle C C M M 4sn Table - Lst of moment coeffcent segments The reason for sch a srprsngly hgh range of valdty of the lnear behavor may be nderstood that the pressre dstrbton n the wake part of the flat plate s almost constant (ths wll be addressed n Apêndce 3). It means that ths regon do not contrbte to the moment. In fact the man contrbton to the moment comes from the front part of the plate n a flow that may receve a potental theory treatment that leads to the lnear behavor, provded a snglarty n the corners are takng off as wll show n Apêndce 3 by the Streamlne theory. 3

47 ..3 Strohal Nmber (St) The Strohal nmber s a dmensonless nmber characterzng the vorte sheddng freqency and s gven by (-9) St f sb (-9) U where f s s the freqency of vorte sheddng, b s the flat plate wdth and U s the velocty of the fld. The epermental reslts of Strohal nmber for the dfferent flat plates n terms of Reynolds nmber and also the nmercal smlaton reslts are presented n Fg. -. All reslts are consstent ecept the flat plat wth C B =0.48 whch t s de to the large blockage effect. Fg. - sggests that the Strohal nmber wthot the blockage effect for the flat plate s n the range of 0.9 to Strohal Nmber LOC - CB=0.4 LOC - CB=0.4 LOC - CB=0.48 CFX - CB= Reynolds Nmber Fg. - Strohal nmbers of epermental reslts at LOC epermental and the nmercal smlaton reslts for dfferent block coeffcents. 3

48 .3 Flow Indced Rotaton Wth the nderstandng of the statc behavor n one hand, the work went on releasng the flat plate to rotate. Three dfferent behavors cold be devsed epermentally: small fltterng state, fltterng and atorotaton. Several tests were performed at the LOC to nvestgate these motons of plates wth dfferent dmensonless moment of nerta and dfferent ranges of velocty. An mportant observaton drng the tests was that the atorotaton moton st occr nder two condtons. The frst s ntal condton: the plate mst be released wth an angle less then stall angle (-5 5 ), then the moton s affected by the large lft force. The other condton s that the dmensonless moment of nerta shold be hgh enogh to store sffcent anglar momentm to overcome the hydrodynamc torsonal sprng. The reslt wll be shown net, bt before ths a dmensonal analyss wll be performed..3. Dmensonal Analyss In order to mprove the nderstandng of flow ndced rotaton of flat plates, a smple dmensonal analyss s performed. The flow ndced rotaton moton of a flat plate may be characterzed by eght dmensonal parameters: b the wdth of the plate, the thckness of the plate, the densty of the plate, the densty of the fld, I the moment of nerta of the plate, A 66 added moment of nerta, v the knematc vscosty of the fld, and U the nform flow velocty. The scheme that has been sed for redcng these dmensonal varables nto dmensonless grops s the Bckngham-p theorem, whch has s steps [5]: 33

49 . Lst and nmber the varables nvolved n the problem. As mentoned before, the nmber of varables s 9 (n=9) as lsted below: f b, t,,, I, A66, U (-0) R s f,. Lst the dmensons of each varable accordng to {MLT} or {FLT}. The lst s gven n Table -3. b t R s f I A U 66 M L T Table -3 Lst of dmensons of the varables. 3. Fnd. In ths stdy the s eqal to 3, ths the nmber of dmensonless nmbers wll be 6 (n=6). 4. Select scalng parameters whch do not form a p prodct. Hereby we pcked p the nform flow velocty, plate wdth, and fld densty. 5. Algebracally fnd the eponents whch make the prodct dmensonless. In or case the reslts are: t s I A66 R f,,,, 5 5 (-) b f f b f b U b 6. Wrte the fnal dmensonless fncton. From the fve parameters of step 5, we form fve non-dmensonal nmbers, the thckness rato, t/b, the dmensonless densty,, the dmensonless added moment of nerta, Iˆ I / A66, and Reynolds nmber, Re=U.b/v 34

50 , * R f, Iˆ, (-) Re Therefore the flow ndced rotaton moton of a flat plate s governed by the Reynolds nmber, thckness rato, dmensonless densty and dmensonless moment of nerta of the flat plate. It shold be noted that the dmensonless moment of nerta can be rewrtten as a fncton of dmensonless densty and thckness rato. Actally the moton n flow ndced rotaton moton and n partclar, the transton from fltterng to atorotaton, shold be completely governed by dmensonless moment of nerta and Reynolds nmber. It shold be mentoned that for the comparson prpose, nstead of Î, n ths * research, we wll se I 8 I ˆ as sed by Iversen [8] and Feld et al. [3]. Fg. -(a) and -(b) show the tme hstory of atorotaton moton for dfferent Reynolds nmber and dmensonless moment of nerta Re=9945 Re=7800 Re=66800 Re=58360 Re=4775 Re=37575 θ R [rad.] (a) Tme [sec.] 35