NUMERICAL ANALYSIS OF FLOW-INDUCED VIBRATION OF TWO CIRCULAR CYLINDERS IN TANDEM AT LOW REYNOLDS NUMBERS

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1 11th World Cogress o Computatoal Mechacs (WCCM XI) 5th Europea Coferece o Computatoal Mechacs (ECCM V) 6th Europea Coferece o Computatoal Flud Dyamcs (ECFD VI) E. Oñate, J. Olver ad A. Huerta (Eds) NUMERICAL ANALYSIS OF FLOW-INDUCED VIBRATION OF TWO CIRCULAR CYLINDERS IN TANDEM AT LOW REYNOLDS NUMBERS PAULO R. F. TEIXEIRA * AND ERIC DIDIER * Uversdade Federal do Ro Grade Aveda Itála km8, Campus Carreros, Ro Grade, RS, , Brazl e-mal: paulotexera@furg.br, Departameto de Hdráulca e Ambete Laboratóro Nacoal de Egehara Cvl Av. do Brasl, 101, , Lsboa, Portugal emal: edder@lec.pt, Key Words: Fte Elemet Method, Flow-duced Vbrato, Crcular Cylders, Tadem Arragemet, Wake Iterferece Abstract. Uform flows over crcular cylders dfferet arragemets appear may practcal stuatos. Aalyses of sde-by-sde, tadem ad staggered arragemets have show sgfcat dffereces amog flow parameters ad the teracto of the flow ad the cylder, by comparso wth sgle cylder parameters. Ths paper descrbes the study of two crcular cylders tadem arragemet subject to b-dmesoal uform lamar flows at low Reyolds umbers from 90 to 140. Cases wth dstace betwee cylders equal to 5.25 dameter for both fxed cylders ad for a fxed upstream cylder ad a dowstream oe elastcally mouted trasversal drecto are aalysed. The umercal model Ifeco, whch s based o the fte elemet method ad uses a parttoed scheme that cosders two-way teracto of flud flow ad structure, s employed the aalyss. The flud flow model uses a sem-mplct two-step Taylor-Galerk method to dscretze the Naver-Stokes equatos ad the arbtrary Lagragea-Eulera formulato to follow the cylder moto. The aalyss of the cylder movemet s carred out by usg oe DOF dyamc equato for the trasverse drecto dscretzed tme by the mplct Newmark method. For both fxed cylders, dffereces terms of lft ad drag coeffcets ad Strouhal umber are foud by comparso wth the oes of the sgle cylder. Whe the dowstream cylder s elastcally mouted, a lock- pheomeo s observed a rage of Reyolds umbers, characterzed by the dscotuty of the lft ad drag coeffcets, vortex vbrato ad atural frequeces approxmato ad crease vbrato ampltude. 1 INTRODUCTION The vortex-duced vbrato pheomeo ca occur as a result of the acto of wd o brdges, sleder buldgs, chmeys ad eergy trasmsso cables besdes the acto of water flow o ppeles ad rsers. The wake aroud crcular cylders due to a uform flow leads to

2 several complex pheomea. Despte the smplcty of geometry, the flow aroud cylders requres deep studes, sce t may duce usteady forces o structures assocated wth vortex sheddg. Ths pheomeo, whch appears may practcal stuatos of dfferet bluff body arragemets, has ts complexty creased due to the wake terferece amog crcular cylders. Sde-by-sde, tadem ad staggered arragemets betwee two equal dameter cylders are specfc cases that are curretly used to expla ths terferece. Aalyses of these types of arragemets have show sgfcat dffereces amog flow parameters ad the teracto of the flow ad the cylder, by comparso wth sgle cylder parameters. Several terferece regmes have bee classfed, accordg to the Reyolds umber ad dstace betwee cylders, based o expermetal studes, such as Igarash s [1], Zdravkovch s [2,3] ad Sumer et al. s [4]. Cosderg tadem arragemet of crcular cylders [1,5], a crtcal dstace, Lc, betwee both cylders s detfed (L/D 4, where L s the ceter to ceter dstace betwee cylders ad D s the dameter) whch dscotuty of the flow behavor ad vortex sheddg occur. For shorter dstaces (L < Lc), the mea drag of the dowstream cylder s small ad egatve. For loger dstaces (L > Lc), fluctuatg lft ad drag forces of the dowstream cylder become hgher tha the upstream oes. Researchers, such as Zdravkovch [6], Brka ad Laevlle [7], vestgated expermetally the flow duced oscllatos of two terferg crcular cylders cosderg dfferet arragemets ad dstaces betwee them. The authors observed that the vortex duced oscllatos are strogly depedet o the arragemet of cylders ad the gap betwee cylders. Ass et al. [8], Okajma et al. [9] ad Huera-Huarte ad Gharb [10] aalyzed expermetally the flow duced oscllatos specfcally for tadem arragemets. Other researchers, such as L et al. [11], Slaout ad Staby [12], Mttal et al. [13], Meegh et al. [14], Sharma et al. [15], Carmo [16], Carmo ad Meegh [17], studed several parameters of the flow for dfferet arragemets betwee cylders based o umercal smulatos. Some umercal aalyses (Carmo et al. [18], Mttal ad Kumar [19], Papaoaou et al. [20]) have bee developed for the study of flud-structure teracto tadem arragemets of crcular cylders cosderg two-dmesoal cases at dfferet Reyolds umbers, by usg vortex dscrete method, fte volume method ad spectral elemet method. Carmo et al. [18] coddered the fxed upstream cylder ad the free dowstream oe to oscllate trasverse drecto subjected to flows at Reyolds umbers equal to 150 (bdmesoal smulato) ad 300 (three-dmesoal smulato) ad vared the reduced velocty by chagg the structural stffess. They cocluded that there are sgfcat chages the dyamc behavor of the cylders by comparso wth a sgle cylder. Ths paper descrbes the study of two crcular cylders tadem arragemet subject to b-dmesoal uform lamar flows at low Reyolds umbers (from 90 to 140). Cases wth ceter to ceter dstace betwee cylders equal to 5.25D for both fxed cylders ad for fxed upstream cylder ad a dowstream oe elastcally mouted trasversal drecto are aalysed. The umercal model Ifeco [21] whch s based o the fte elemet method ad uses a parttoed scheme that cosders two-way teracto of flud flow ad structure s employed for the aalyss. The flud flow model uses a sem-mplct two-step Taylor-Galerk method to dscretze the Naver-Stokes equatos ad the arbtrary Lagragea-Eulera formulato to follow the cylder moto. The classcal Galerk weghted resdual method s appled to the space dscretzato ad a tragular elemet s employed. The cylder 2

3 movemet s carred out by usg the oe DOF dyamc equato for the trasverse drecto dscretzed tme by the mplct Newmark method. 2 IFEINCO MODEL The umercal model Ifeco s based o a parttoed scheme, whch the flud flow ad the structure are solved two-way teracto. Essetally, the flud-structure teracto adopted by the code cossts the followg steps: (a) update the varables of the flow from stat t to t+ t; (b) mpose pressure ad vscous stress as a load to the structure; (c) update the varables of the structure from stat t to t+ t; (d) mpose the body moto to the flow terms of the updated velocty vector ad boudary posto. Bascally, updatg the varables of the flow cossts of followg steps [22]: ~ a) Calculate o-corrected mometum per volume U at t+ t/2, where the pressure term s at t stat, accordg to Eq. (1). ~ + 1 / 2 = U U t f 2 x j j τ j p + x j x where p s the pressure, w are the velocty compoets of the referece system, τj s the vscous stress tesor, U = ρ v, f j = v j ( ρ v) = v ju (, j = 1, 2), ρ s the specfc mass ad v are the velocty compoets. w j U x b) Update the pressure p at t+ t, gve by the Posso equato: where p = p +1 p ~ + 1/ 2 1 = U t p p t 2 (2) c x 4 x x ad = 1, 2. c) Correct the velocty at t+ t/2, addg the pressure varato term from t to t+ t/2, accordg to the equato: U + 1 / 2 t p = ~ + 1 / 2 U (3) 4 x d) Calculate the velocty at t+ t usg varables updated the prevous steps as follows: U + 1 = U + f j t x 1/ 2 j + 1/ 2 τ j x j + x + 1/ 2 + 1/ 2 p + 1/ 2 w j U x The classcal Galerk weghted resdual method s appled to the space dscretzato of Eq. (1), (2), (3) ad (4), ad a tragular elemet s employed. I the varables at t+ t/2 stat, a costat shape fucto s used, ad the varables at t ad t+ t, a lear shape fucto s employed [21]. The Posso equato, whch s a result of spatal dscretzato of Eq. (2), s solved by employg the cojugate gradet method wth a dagoal precodtog [23]. Spatally dscretzed Equato (4) s explctly solved by the teratve process usg the lumped matrx (1) (4) 3

4 [24]. The scheme s codtoally stable ad the local stablty codto for elemet E s gve by t β h v (5) E where he s the characterstc sze of the elemet (lowest elemet edge), β s the safety factor (0.25 has bee adopted ths study) ad v s the flud velocty. The mesh velocty trasversal compoet w2 s computed to dmsh elemet dstortos, keepg prescrbed veloctes o movg ad statoary boudary surfaces. The mesh movemet algorthm adopted ths paper uses a smoothg procedure for the veloctes based o these boudary les. The updatg of the mesh velocty at ode of the fte elemet doma s based o the mesh velocty of the odes j that belog to the boudary les [21]. I order to update the rgd body moto of the structure, t s ecessary to calculate dsplacemets ad rotatos of a hypothetcal cocetrated mass at ts gravty ceter. I ths case study, there s oly movemet trasverse drecto (oe degree of freedom DOF) ad, cosequetly, dsplacemet, velocty ad accelerato ths drecto are the varables to be determed at each tme step. To update the varables of the structure, the rgd moto of the cylder s calculated at each stat, after the varables of the flow (pressure ad vscous stress) are kow. For ths case study, oe DOF dyamc equato s cosdered for the trasverse drecto, as follows: E m & y + cy& + ky = F (8) where & y&, y& ad y are the trasverse accelerato, velocty ad dsplacemet, respectvely; m s the mass; c s the dampg coeffcet; k s the stffess; ad F s the dyamc force. I Ifeco code, Eq. (8) s dscretzed tme by usg the mplct Newmark method [25] ad the accelerato, the velocty ad the dsplacemet trasverse drecto are calculated at each tme step accordg to the followg algorthm: a) Italze 0 & y&, 0 y& e 0 y; b) Calculate the tegrato costats; a 0 1 = ; a α 2 t δ = α t 1 ; 1 1 δ t δ a2 = ; a 3 = 1; a 4 = 1; a 5 = 2 ; (9) α t 2 α α 2 α ( δ ) a = 1 ; a = δ t 6 t c) Determe the effectve stffess coeffcet: Ad for each tme step: a) Calculate the effectve loads at stat t+ t, as follows: t + t t t Fe = + 7 k e = k + a 0.m + a 1.c (10) F + m a b) Solve the dsplacemets at t+ t: t+ t t t t t t t ( y + a y + a && y) + c( a y + a y& a & y ) 0 2 & t+ t c) Calculate the acceleratos ad the veloctes at t+ t: e e (11) y = F / k (12) 4

5 t + t t+ t t t t ( y y) a y& a & y & y = a0 2 3 (13) t + t y& t t t+ t = 7 where δ ad α are 0.5 ad 0.25, respectvely, ths study. 3 NUMERICAL SIMULATION y& + a6 && y + a & y (14) Both cylders, whch are mmersed water, have dameters D = m (Fg. 1). The dowstream cylder (mass m = kg) s elastcally mouted trasversal drecto. The sprg stffess, k, s equal to 579 N/m, resultg a atural frequecy of ths system of f = Hz. The dampg coeffcet, c, s equal to kg/s ad the correspodet damper rato s ζ = c / 2mω = , where ω s the agular frequecy. The Reyolds umber (Re = ρ U D/µ, where ρ s the specfc mass, U s the free stream velocty, D s the flow cylder dameter ad µ s the vscosty) raged betwee 90 ad 140, thus, characterzg a regme whch the vortex street s fully lamar ad b-dmesoal. Costat velocty (U ) s mposed o the let boudary whereas, o the lateral boudares (far from cylders), a slde codto s mposed; the outlet boudary s free ext, but ull pressure s mposed o ts ceter. The ma dmesoless parameters that fluece the VIV behavor are the reduced velocty (VR = U /D f), ragg from 5 to 7.8, ad the mass rato (M = m/ρd 2 ) that s equal to 166 [26]. I ths case, whch the ceter to ceter dstace betwee cylders s 5.25D, vortces shed off the upstream cylder ad roll up before strkg the dowstream cylder; the, they teract strogly wth t. The computatoal doma cossts of a rectagle whose sdes have the mmum dstace from the cylders of 100D. The cylder boudary s dscretzed 200 segmets ad the sze of the frst elemet aroud the cylders s 0.016D, totalzg odes ad tragular elemets [27]. The tme step s betwee 6x10-5 s ad 8.5x10-5 s, accordg to the Reyolds umber. Fgure 1: Sketch of the case study 3.1 Statoary upstream ad dowstream cylders These results are compared wth prevous oes of a statoary sgle cylder obtaed by Goçalves et al. [27]. Results of statoary cylders tadem arragemet showed that the Strouhal umbers follow the same tred observed for the sgle cylder ad are approxmately 10% lower tha results of a sgle cylder alog the Reyolds umber rage (Fg. 2). The sheddg frequecy happes to be the same for both cylders for all Reyolds umbers. The root mea square of the lft coeffcet (CLrms) for the dowstream cylder s fourfold the oe 5

6 of the sgle cylder, whch s smlar to the upstream result (Fg. 3). I terms of mea drag coeffcets (CDmea), the upstream cylder presets smlar results to the oes of the sgle cylder, but the value of the dowstream cylder s approxmately half of that of the sgle oe (Fg. 4). These results show that dfferet postos of resoace for tadem arragemet are expected by comparso wth the sgle cylder case whe the dowstream cylder s free, maly because of dffereces Strouhal umbers. Moreover, hgher lft coeffcets of the dowstream cylder may provde hgher vbrato ampltudes for tadem arragemet cases by comparso wth the sgle cylder case, due to the strog terferece betwee both cylders sce the dowstream oe s placed sde the wake of the upstream oe. Fgure 2: Strouhal umber versus Reyolds umber for statoary cylder cases Fgure 3: Root mea square of lft coeffcet versus Reyolds umber for statoary cylders case 6

7 Fgure 4: Mea drag coeffcet versus Reyolds umber for statoary cylder cases 3.2 Statoary upstream ad oscllatg dowstream cylders Whe the dowstream cylder s elastcally mouted, a sgfcat crease CLrms from to at Reyolds umber from 110 (VR = 6.12) to 119 (VR = 6.63) ad a abrupt decrease at Re = 120 (VR = 6.68) are observed. Ths pheomeo s characterstc of the resoace (lock-) ad also occurs the sgle cylder, but wth some dffereces coeffcet magtude ad Reyolds umber rage, as show Fg. 5. The Reyolds umber rage whch ths pheomeo occurs for sgle cylder s wder tha the rage the tadem arragemet case. Besdes, these regos, the mea drag coeffcet for oscllatg cylder gets hgher values abruptly, reachg 0.85 for Reyolds umber of 117 (VR = 6.51), as show Fg. 6. Smlar behavor occurs the sgle cylder case, but wth a wder Reyolds umber rage. Fgure 7 shows the root mea square of lft coeffcet ad the mea drag coeffcet versus Reyolds umber for statoary upstream ad oscllatg dowstream cylders. It may be otced that whle the dowstream cylder expermets hgh varatos of coeffcets the lock- rego, these coeffcets decrease smoothly for the upstream cylder. I ths case, flow terferece betwee both cylders s dfferet from the statoary dowstream cylder due to the moto of the dowstream cylder ad chages wake patter ad pressure feld. Fgure 5: Root mea square of the lft coeffcet versus Reyolds umber for statoary ad oscllatg dowstream cylders tadem arragemet ad for fxed ad oscllatg sgle cylders 7

8 Fgure 6: Mea drag coeffcet versus Reyolds umber for statoary ad oscllatg dowstream cylders tadem arragemet ad for fxed ad oscllatg sgle cylders Fgure 7: Root mea square of lft ad mea drag coeffcets versus Reyolds umber for statoary upstream ad oscllatg dowstream cylders tadem arragemet Fgure 8 shows the dmesoless ampltude (Y/D) ad the vortex duced vbrato frequecy (f/f) for dowstream cylder ad a comparso wth the sgle oe. It ca be otced that the resoace, for the sgle cylder, occurred for Reyolds umbers betwee 102 (VR = 5.68) ad 113 (VR = 6.29) case whch the vbrato ampltude s hgher ad the vortex frequecy s approxmately equal to the atural frequecy (f) of the dyamc system. I tadem arragemet wth L/D = 5.25, the dowstream cylder expermets the resoace a shorter Reyolds umber rage, betwee 115 (VR = 6.40) ad 120 (VR = 6.68). Furthermore, the maxmum dmesoless ampltude s for Re = 118 ad ths s much hgher (more tha 70%) tha the oe for the sgle cylder (0.422 for Re = 103). The terferece effects o tadem arragemet chage the Strouhal umber relato to the sgle cylder oe ad to the mode of terferece betwee both cylders, sce the dowstream cylder s mmersed the perodc wake of the upstream oe. Ths s why there are dffereces betwee both lock- regos. 8

9 Fgure 9 shows the temporal seres of lft coeffcet (CL) ad dmesoless vbrato ampltude (Y/D) for Re = 118. Vortcty dstrbutos ad streamles are show Fg. 10 where tme postos of each pcture are dcated Fg. 9. It may be otced that the lft coeffcet presets hgher o learty by comparso wth vbrato ampltude whch shows harmoc behavour. The phase agle betwee them s aroud 42 o ths case. Fgure 10 shows that the vortex sheddg occurs for both cylders, as expected for L/D > 4, ad the wake behd the dowstream cylder s formed by the combato of vortex shed from both cylders. The maxmum vbrato ampltude (Y/D = 0.721) occurs at stats show Fg. 10a ad 10b. (a) (b) Fgure 8: Dmesoless vbrato ampltude (Y/D) (a) ad vortex sheddg frequecy (f./f ) (b) versus Reyolds umbers for sgle cylder ad cylders tadem arragemet 9

10 Fgure 9: Temporal seres of lft coeffcet (C L) ad dmesoless vbrato ampltude (Y/D) for Re = 118 (letters represet tme postos of the vortcty dstrbutos, show Fg. 10) (a) (b) (c) (d) (e) (f) (g) (h) () Vortcty (s -1 ) Fgure 10: Vortcty ad streamles aroud cylders for Re = 118 at e stats (show the graph Fg. 9) durg oe cycle 4 CONCLUSION Ths paper preseted a umercal aalyss of a uform flow over crcular cylders tadem arragemet wth ceter to ceter dstace equal to 5.25D. Smulatos were carred 10

11 out by usg Ifeco model whch s based o the fte elemet method ad employs the semmplct two-step Taylor-Galerk method to dscretze the Naver-Stokes equatos ad the Newmark method for the dyamc equato of the structure. The root mea square of lft coeffcet, the mea drag coeffcet ad the Strouhal umber at Reyolds umber ragg from 90 to 140 for both statoary cylders were compared wth those obtaed for a sgle cylder. Importat dffereces were observed: the lft coeffcets of the dowstream cylder tadem arragemet were much hgher tha those of the sgle cylder, whereas ts drag coeffcets were aroud half of the value; Strouhal umbers are about 10% lower. Cosderg the dowstream cylder elastcally mouted trasversal drecto, the lock rego was captured by Reyolds umber from 115 to 120; ths rage was dfferet from ad shorter tha the oe for the sgle cylder (102 to 113). The maxmum vbrato ampltude, whch was 0.721D, was hgher tha the oe for the sgle cylder (0.422D). At Re = 120 ( the lock- rego), the drag coeffcet ad the lft coeffcet showed abrupt varatos; the latter was much more tese. Complex terferece occurs betwee both cylders sce the dowstream cylder s mmersed the perodc wake of the upstream oe. REFERENCES [1] Igarash, T. Characterstcs of the flow aroud two crcular cylders arraged tadem. Bullet of JSME (1977) 24(188): [2] Zdravkovch, M.M. Revew of flow terferece betwee two crcular cylders varous arragemets. ASME Joural of Fluds Egeerg (1977) 99: [3] Zdravkovch, M.M. The effects of terferece betwee crcular cylders cross flow. Joural of Fluds ad Structures (1987) 1: [4] Sumer, D., Prce, S.J. ad Païdousss, M.P. Flow-patter detfcato for two-staggered crcular cylders cross-flow. Joural of Flud Mechacs (2000) 411: [5] Dder, E. Numercal smulato of low Reyolds umber flows over two crcular cylders tadem. Coferece o Modellg Flud Flow, Budapest, September 9-12, (2009) [6] Zdravkovch, M.M. Flow duced oscllatos of two terferg crcular cylders. Joural of Soud ad Vbrato (1985) 101(4): [7] Brka, D. ad Laevlle, A. The flow teracto betwee a statoary cylder ad a dowstream flexble cylder. Joural of Fluds ad Structures (1999) 13: [8] Ass, G.R.S., Meegh, J.R., Araha. J.A.P., Bearma, P.W. ad Casaprma, E. Expermetal vestgato of flow-duced vbrato terferece betwee two crcular cylders. Joural of Fluds ad Structures (2006) 22: 819:827. [9] Okajma, A., Yasu, A., Kwata, T. ad Kmura, S. Flow-duced treamwse oscllato of two crcular cylders tadem arragemet. Iteratoal Joural of Heat ad Flud Flow (2007) 28: [10] Huera-Huarte, F.J. ad Gharb, M. Vortex- ad wake-duced vbratos of a tadem arragemet of two flexble crcular cylders wth far wake terferece. Joural of Fluds ad Structures (2011) 27: [11] L, J., Chambarel, A., Doeaud, M. ad Mart, R. Numercal study of lamar flow past oe ad two crcular cylders. Computers & Fluds (1991) 19:

12 [12] Slaout, A. ad Staby, P.K. Flow aroud two crcular cylders by the radom-vortex method. Joural of Fluds ad Structures (1992) 6: [13] Mttal, S., Kumar, V. ad Raghuvash A. Usteady compressble flows past two cylders tadem ad staggered arragemets. Iteratoal Joural for Numercal Methods Fluds (1997) 25: [14] Meegh, J.R., Saltara, F., Squera, C.L. ad Ferrar, J.A. Numercal smulato of flow terferece betwee two crcular cylders tadem ad sde by sde arragemets. Joural of Fluds ad Structures (2001) 15: [15] Sharma, B.. Le, F.S., Davdso, L. Ad Norberg, C. Numercal predctos of low Reyolds umber flows over two tadem crcular cylders. Iteratoal Joural for Numercal Methods Fluds (2005) 47: [16] Carmo, B.S. Estudo umérco do escoameto ao redor de cldros alhados. Master Thess, Escola Poltécca da Uversdade de São Paulo (2005). [17] Carmo, B.S. ad Meegh, J.R. Numercal vestgato of the flow aroud two crcular cylders tadem. Joural of Fluds ad Structures (2006) 22: [18] Carmo, B.S., Sherw, S.J., Bearma, P.W. ad Wllde, R.H.J. Flow-duced vbrato of a crcular cylder subjected to wake tereferece at low Reyolds umber. Joural of Fluds ad Structures (2011) 27: [19] Mttal, S. ad Kumar, V. Flow-duced oscllatos of two cylders tadem ad staggered arragemets. Joural of Fluds ad Structures (2001) 15: [20] Papaoaou, G.V., Yue, D.K.P., Tratafyllou, M.S. ad Karadaks, G.E. O the effect of spacg o the vortex-duced vbratos of two tadem cylders. Joural of Fluds ad Structures (2008) 24: [21] Texera, P.R.F. ad Awruch, A.M. Numercal smulato of flud-structure teracto usg the fte elemet method. Computers & Fluds (2005) 34: [22] Texera, P.R.F. ad Awruch, A.M. Three dmesoal smulato of hgh compressble flows usg a mult-tme-step tegrato techque wth subcycles. Appled Mathematcal Modellg (2001) 25: [23] Argyrs, J., Doltss, J.St., Wuesteberg, H. ad Pmeta, P.M. Fte elemet soluto of vscous flow problems. Fte Elemets Fluds. Wley, New York, Vol. 6, , (1985). [24] Doea J., Gula, S., Laval ad H., Quartapelle, L. Fte elemet soluto of the usteady Naver-Stokes equatos by a fractoal step method. Computer Methods Appled Mechacs ad Egeerg (1982) 33: [25] Bathe, K.J. Fte elemet procedures. Pretce-Hall, (1996). [26] Fredsφe, J. ad Sumer, B.M. Hydrodyamcs aroud cyldrcal structures. World Scetfc Publshg Co. Pte. Ltd., Sgapore, Vol. 12, (1997). [27] Goçalves, R.A., Texera P.R.F. ad Dder, E. Numercal smulatos of low Reyolds umber flows past elastcally mouted cylder. Thermal Egeerg (2012) 11(1-2):

NUMERICAL SIMULATIONS OF LOW REYNOLDS NUMBER FLOWS PAST ELASTICALLY MOUNTED CYLINDER

NUMERICAL SIMULATIONS OF LOW REYNOLDS NUMBER FLOWS PAST ELASTICALLY MOUNTED CYLINDER R. A. Goçalves a, P. R. F. Texera b, ad E. Dder c, d a, b Uversdade Federal do Ro Grade Escola de Egehara Campus Carreros