NUMERICAL SIMULATIONS OF LOW REYNOLDS NUMBER FLOWS PAST ELASTICALLY MOUNTED CYLINDER

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1 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 CP , Ro Grade, Ro Grade do Sul, Brazl a rafa.a.g@best.com.br b paulotexera@furg.br c Laboratóro Nacoal de Egehara Cvl Departameto de Hdráulca e Ambete Av. do Brasl, 101, , Lsboa, Portugal edder@lec.pt d Uversdade Nova de Lsboa Faculdade de Cêcas e Tecologa Campus de Caparca , Mote de Caparca, Portugal ABSTRACT The vortex-duced vbrato (VIV) pheomeo has draw the atteto of researchers Egeerg for several decades. A example s the rser used for petroleum explorato, whch t s subected to mare flows that may cause oscllatos due to vortex sheddg. I ths paper, umercal aalyses of the pheomea that occur the teracto amog flows at low Reyolds umbers ad elastcally mouted cylders are preseted. The smulato s carred out by usg the umercal model Ifeco that 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 rgd body moto descrpto s calculated by usg the Newmark method. Frstly, the characterstcs of the vortex geerato process for the fxed cylder are aalyzed. I ths case, the Strouhal umber, the mea drag ad the RMS lft coeffcets for Reyolds umbers ragg from 90 to 140 are show. Afterwards, a aalyss of a flexble supported cylder (wth a sprg ad a damper) trasverse drecto subect to flows wth Reyolds umbers ragg from 90 to 140 s carred out. The cylder dsplacemet ad the vbrato frequeces are studed; the sychrozato betwee the vortex sheddg ad the vbrato frequecy (lock-) s aalyzed. Smlar results to the expermetal oes developed by Aagostopoulos ad Bearma (1992) were obtaed ths study. Keywords: fte elemet method, oscllatg cylder, flud-structure teracto. NOMENCLATURE c Dampg coeffcet, kg/s C D Drag coeffcet C L Lft coeffcet D Dameter, m F Force, N f Frequecy, Hz F D Drag force, N F L Lft force, N f Natural frequecy, Hz g Gravty accelerato, m/s² m mass, kg p Pressure, Pa Re Reyolds umber St Strouhal umber U Mometum per volume, kg m/s /m 3 U Free stream velocty, m/s v Flud velocty, m/s w Mesh velocty, m/s Y Ampltude of cylder oscllato, m y Cylder dsplacemet, m y& Cylder Velocty, m/s & y& Cylder accelerato, m/s 2 Greek symbols τ Vscous stress tesor, Pa ρ Specfc mass, kg/m³ k μ Sprg stffess, N/m Vscosty, kg/(ms) INTRODUCTION Vortex-duced vbrato (VIV) s a pheomeo that s foud several egeerg felds. Some examples are the followg: wd ca cause oscllatos o brdges, sleder buldgs, chmeys ad eergy trasmsso cables; flows wth hgh veloctes ca duce orbtal movemets teral tubes of a heat exchager; ad currets ad waves ca cause vbrato o ppeles. The wake aroud a crcular cylder due to a uform flow leads to varety of complex pheomea. Despte the smplcty of geometry, the flow aroud a cylder s very complcated ad of partcular mportace, sce t may duce usteady forces o structures assocated wth vortex sheddg. Therefore, ths case has bee studed for several decades, ad, owadays, the behavor of the flow s kow. For Reyolds umber (Re) up to 49, two symmetrc statoary recrculato zoes attached to the rear part of the cylder wall are observed. From 49 to 190, the wake s stll lamar ad twodmesoal ad t s composed by two perodc staggered rows of alteratg vortces (vo Kármá vortex sheddg). For hgher Reyolds umbers (from 190 to 260), the wake becomes threedmesoal ad progressvely turbulet. Ths regme Egehara Térmca (Thermal Egeerg), Vol. 11 No. 1-2 Jue ad December 2012 p

2 s followed by a shear layer trasto (up to 1200), whch separatg shear layers become ustable, ad, fally, by the boudary layer trasto (aroud 10 5 ) assocated wth fast decrease of the drag coeffcet. For these regmes, the flow exhbts a perodcty whch s kow as Strouhal frequecy. Whe a perodc vortex street s well establshed, ths frequecy correspods to that of the vortex sheddg frequecy; other cases, whch the vo Kármá streets are ot clearly vsble, the frequecy ca be defed as the oe of the fluctuatos of the streamwse velocty compoet, for example (Placzek et al., 2009). I may applcatos, the cylder oscllates ad teracts wth the vortex sheddg process. For forced oscllatos a rage of frequecy ad ampltude, the cylder moto s able to cotrol the stablty mechasm geerated by vortex sheddg. Oe of the most terestg characterstcs of ths flud-structure teracto s the sychrozato (lock-) betwee the vortex sheddg ad the vbrato frequecy. Smlar pheomea are observed for VIV, whch the flow causes the oscllato of the cylder at ts atural frequecy. Ths frequecy depeds o the mass, the rgdty ad the dampg of the cylder. I ths pheomeo, whch occurs a rage of flow velocty, the ampltude reaches a maxmum value. Ths complex flud-structure teracto pheomeo s stll a good test case to valdate the umercal models. Several umercal aalyses ca be foud the lterature for a large rage of Reyolds umbers, cludg Reyolds Averaged Naver- Stokes (RANS) methods (Saghafa et al., 2003; Gulmeau ad Queutey, 2004), Large Eddy Smulatos (LES) (Breuer, 2000; Pasquett, 2005; Al-Jamal ad Dalto, 2004), Drect Numercal Smulatos (DNS) ad methods that use fte volume or fte elemet approxmatos to solve the Naver-Stokes equatos (Aagostopoulos ad Bearma, 1992; Nobar ad Nareda, 2006; Mttal ad Kumar, 2001). Ths paper descrbes smulatos whch are carred out by usg the umercal model Ifeco (Texera ad Awruch, 2005) that uses a semmplct two-step Taylor-Galerk method to dscretze the Naver-Stokes equatos ad the arbtrary Lagragea-Eulera formulato to follow the cylder moto. The rgd body moto descrpto s calculated by usg the Newmark method. Frstly, the characterstcs of the vortex geerato process for the fxed cylder are aalyzed. I ths case, the Strouhal umber, the mea drag ad the RMS lft coeffcets for Reyolds umbers ragg from 90 to 140 are show ad compared wth expermetal ad umercal results of the lterature. Afterwards, a aalyss of a flexble supported cylder (wth a sprg ad a damper) trasverse drecto subect to flows wth Reyolds umbers ragg from 90 to 140 s carred out. The cylder dsplacemet ad the vbrato frequeces are studed ad the sychrozato betwee the vortex sheddg ad the vbrato frequecy (lock-) s aalyzed. Numercal results are smlar to expermetal oes obtaed by Aagostopoulos ad Bearma (1992). NUMERICAL MODEL The umercal model Ifeco s based o a parttoed scheme, whch the flud flow ad the structure are solved two-way teracto. Bascally, 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 (Texera ad Awruch, 2005): ~ a) Calculate o-corrected velocty U at t+ t/2, where the pressure term s at t stat, accordg to the equato: ~ + U = U Δ f t τ p + 2 ρg w U (1) where ρ s the specfc mass, p s the pressure, g are the gravty accelerato compoets, v are the velocty compoets, w are the velocty compoets of the referece system ad τ s the vscous stress tesor U = ρ v, f = v ( ρ v) = v U (, = 1, 2). b) Update the pressure p at t+ t, gve by the Posso equato: 1 ~ Δ = Δ U p t 2 c +1 + Δt 4 Δp (2) where Δp = p p 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 Δt Δp = ~ + U (3) 4 d) Calculate the velocty at t+ t usg varables updated the prevous steps as follows: U + 1 = U p f Δt w + U + τ + ρg + (4) 62 Egehara Térmca (Thermal Egeerg), Vol. 11 No. 1-2 Jue ad December 2012 p

3 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 (Texera ad Awruch, 2001). The mesh velocty vertcal compoet w 2 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 that belog to the boudary les. I order to update the rgd-body moto structure, t s ecessary to calculate dsplacemets ad rotatos of a hypothetcal cocetrated mass at ts gravty ceter. I ths study case, 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 study case, oe DOF dyamc equato s cosdered for the trasverse drecto, as follows: m & y + cy& + ky = F (5) 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 ths code, Eq. (5) s dscretzed tme by usg the mplct Newmark method (Bathe, 1996) ad the accelerato, the velocty ad the dsplacemet trasverse drecto are calculated at each tme step. NUMERICAL SIMULATIONS The case study cossts of a cylder (dameter ad mass equal to m ad kg, respectvely) subect to a uform water flow (specfc mass, ρ, ad vscosty, μ, equal to kg/m 3 ad 0.001kg/(ms), respectvely). The cylder s mouted o a sprg ad a damper trasverse drecto ad fxed towards the flow. The sprg stffess, k, s equal to 579N/m ad the dampg coeffcet, c, s equal to kg/s. The atural frequecy of ths system s f=7.016hz (Aagostopoulos ad Bearma, 1992). The fluece of the sze of the computatoal doma was aalyzed for Reyolds umber equal to 135. The best computatoal doma that satsfed both accuracy ad computatoal cost crtera was a rectagle 0.320m wde ad 0.384m log, as show Fg. 1. The cylder ceter s located at the ceter of the doma trasverse drecto to the flow ad 0.160m from ts left sde logtudal drecto. I ths computatoal doma, the smallest dstace from the boudary to the cylder ceter s equal to 100D. After aalyzg the mesh covergece, a fte elemet ustructured mesh composed by tragles wth 200 elemets aroud the cylder was used. Elemet szes crease gradually towards the boudares of the doma. The mesh has odes ad elemets. A costat velocty s mposed o the left sde of the computatoal doma; o the lateral boudares, a slde codto s mposed; ad the rght sde s free ext, but ull pressure s mposed o ts mddle. The tme step used for the smulatos s equal to 5.0 x 10-5 s. Fgure 1. Numercal doma. Frst, the behavor of the flow cosderg the fxed cylder s aalyzed. Specfcally, the flow characterstcs at Re=105 (velocty U equal to m/s) are studed. Fgure 2 shows the velocty vectors ad streamles at eght stats alog oe perod of vortex formato, respectvely. The vortex formato was clearly observed, showg two dfferet regos behd the cylder where the flow separato occurs. Near the cylder surface, whle the larger vortex s oe drecto, the opposg vortex s aother oe. Drag (F D ) ad lft (F L ) forces o the cylder obtaed by umercal smulato for Re=105 are show Fg. 3. The drag force has a perodc behavor wth a lttle varato aroud N, whereas the lft force has a perodc behavor wth ampltude equal to N ad frequecy equal to 6.828Hz. Fgure 4 shows a comparso amog the Strouhal umbers, St=f D/U, obtaed by umercal results (where f s equal to the lft force frequecy) for Re from 90 to 140 ad expermetal results of Wllaso (Wllaso, 1989). Numercal results are very good agreemet wth expermetal oes; the mea dfferece was oly 0.03%. I Fg. 5, a comparso of the mea drag coeffcet (C D ) wth umercal oes obtaed by Poldszech ad Grudma (2007) for the same rage of the Re s show. The authors used a Spectral Egehara Térmca (Thermal Egeerg), Vol. 11 No. 1-2 Jue ad December 2012 p

4 Elemet Method ad preseted two curves wth szes of the computatoal doma of 70D ad 4000D. It may be otced that the curve of ths study s located betwee both Poldszech ad Grudma s curves ad the mea dffereces related to the curves are oly 0.8% (4000D) ad 0.4% (70D). The root mea square (RMS) lft coeffcet (C L ) s show Fg. 6 ad t s compared wth umercal oe obtaed by Baray ad Lews (2006), who used a Grd Based Method ad a sze of the computatoal doma of 40D. I ths case, the mea dfferece betwee both umercal results s oly 0.2%. It s worth metog that the hgh accuracy of the parameters St ad C L shows the capacty of the umercal model Ifeco to reproduce the frequecy ad the magtude of forces that s mposed over the crcular cylder the flud-structure teracto process. (a) t = s t = s (b) t = s t = s Fgure 3. Drag (a) ad lft (b) forces for fxed cylder wth Re=105. t = s t = s t = s t = s Fgure 4. Strouhal umbers versus Reyolds umbers. t = s Fgure 2. Velocty vectors at eght stats alog a perod of vortex formato. Fgure 5. Mea drag coeffcet (C D ) versus Reyolds umbers. 64 Egehara Térmca (Thermal Egeerg), Vol. 11 No. 1-2 Jue ad December 2012 p

5 the vortex sheddg ths rego. Ths fact ad the fluece of the free surface cotrbute to develop the three-dmesoal behavor of the flow, ulke the umercal codtos. Fgure 6. RMS lft coeffcet (C L ) versus Reyolds umbers. Afterwards, the teracto amog a cylder mouted o a elastc fxg trasverse drecto ad flows at Reyolds umbers from 90 to 140 s aalyzed. Fgure 7 shows the relato betwee the ampltude (Y) of the cylder oscllato ad ts dameter (D) fucto of Reyolds umbers (90 to 140). Fgure 8 shows the relato betwee the frequecy of vbrato (obtaed by the tme seres of the C L ) ad the atural frequecy (f./f) fucto of Reyolds umbers. The umercal results show that the lock- pheomeo was captured for Reyolds umbers betwee 102 ad 113. Ths s observed due to the crease of the ampltude of cylder moto ad the equalty of vbrato ad atural frequeces. Out of the lock- rego, the ampltudes of cylder moto are eglgble ad the vbrato frequeces are smlar to that obtaed for a fxed cylder, descrbed by the Wllaso s curve. Fgures 7 ad 8 also show the results obtaed expermetally by Aagostopoulos ad Bearma (1992) ad umercally by Dettmer ad Perć (2006). The latter used a model that employs the stablzed low order velocty pressure fte elemets, a arbtrary Lagraga Eulera formulato ad the dscrete mplct geeralzed-α method for the rgd body moto. Both umercal results are smlar terms of ampltudes ad frequeces alog the Reyolds umber rage. The umercal ampltude values are always smaller tha expermetal oes ad there are lttle dffereces the lock- rage. The frequeces obtaed umercally show the same behavor: the lock- rego, the vbrato frequecy s equal to the atural frequecy, whle, out of the lock-, the vbrato frequecy follows the Strouhal oes. The dffereces amog ampltudes obtaed expermetally ad umercally the Reyolds umber rage the lock- rego were explaed by Dettmer ad Perć (2006) comparg the umercal doma ad boudary codtos ad the real stuato of the experece. The expermet was carred out a 0.70m deep chael where 0.12m of the cylder was submerged. The lack of a horzotal plate the ed of the submerged cylder allowed Fgure 7. Ampltude (Y/D) versus Reyolds umbers. Fgure 8. Frequecy of vbrato (f./f) versus Reyolds umbers. Fgure 9 shows the behavor of the drag ad the lft forces for oscllatg ad fxed cylders wth Re=105. Ulke the drag force for the fxed cylder, whch has a small harmoc varato aroud N, the force oscllates betwee N ad N at a frequecy equal to Hz (almost twofold the frequecy trasverse drecto) for the oscllatg cylder. I the trasverse drecto, the lft forces dffer terms of frequecy ad ampltude. The ampltudes for the fxed ad the oscllatg cylders are N ad N ad ther frequeces are 6.828Hz ad 6.984Hz, respectvely. Both frequeces are related to the vortex sheddg ad the latter s closer to the atural frequecy of the dyamc system of the cylder (f=7.016hz). A represetatve case out of the lock- rego (Re=123) was chose to show ts dfferet force behavor. Fgure 10 shows the tme seres of the drag ad the lft forces for Re=123. The drag forces exhbt lttle varato aroud N for both fxed ad oscllatg cylders, wth more dsturbaces the case of the fxed cylder. The drag force for the fxed cylder has a harmoc behavor wth ampltude ad frequecy equal to N ad 8.313Hz, respectvely. For the oscllatg cylder, ths force oscllates perodcally (frequecy of 8.163Hz) wth a evelope wth mmum ad maxmum ampltudes of N ad N, Egehara Térmca (Thermal Egeerg), Vol. 11 No. 1-2 Jue ad December 2012 p

6 respectvely. Although these ampltudes are hgher tha those the prevous case (Re=105), the dsplacemets are smaller, sce ths Reyolds umber, Re=123, s out of the lock- rego. (a) (b) Fgure 9. Drag (a) ad lft (b) forces for cylder wth Re=105. (a) CONCLUSIONS I ths paper, umercal aalyses of the pheomea that occur the teracto amog flows at low Reyolds umbers ad elastcally mouted cylders were preseted. The smulato was carred out by a umercal model, Ifeco, that uses a semmplct two-step Taylor-Galerk method to dscretze the Naver-Stokes equatos ad the arbtrary Lagragea-Eulera formulato to follow the cylder moto. The rgd body moto descrpto s calculated by usg the Newmark method. For the fxed cylder, the behavor of the vortex formato was correctly reproduced: ear the cylder surface, the larger vortex was oe drecto, whle the opposg vortex was aother oe. The Strouhal umbers were calculated for Reyolds umber rage from 95 to 140. These values were smlar to those obtaed by Wllaso's expermets. The mea drag ad the RMS lft coeffcets were also calculated ad compared wth those obtaed by Poldszech ad Grudma (2007) ad Baray ad Lews (2006), respectvely. Good agreemets were obtaed (0.8% ad 0.2%, respectvely) ad showed the accuracy of the umercal smulato. For teracto amog the cylder mouted o a elastc fxg trasverse drecto ad flows, the lock- pheomeo was captured for Reyolds umbers betwee 105 ad 110, characterzed by the crease of the ampltude ad the equalty of vbrato ad atural frequeces. Comparg the umercal ad the expermetal results (Aagostopoulos ad Bearma, 1992), some dffereces were observed due to the presece of the three-dmesoal effects of the expermet that were ot cosdered these umercal smulatos. Comparg the results obtaed by Ifeco wth umercal oes obtaed by Dettmer ad Perć (2006), t was show that both umercal results are smlar terms of ampltudes of cylder moto ad frequeces for the Reyolds umber rage studed, cofrmg the ablty of the Ifeco code for modelg complex flud-structure teracto pheomea. ACKNOWLEDGEMENTS The frst author wshes to thak CAPES for the post-graduate scholarshp. The secod author ackowledges the support of Coselho Nacoal de Desevolvmeto Cetífco e Tecológco (CNPq - proect /2009-5). (b) Fgure 10. Drag (a) ad lft (b) forces for cylder wth Re=123. REFERENCES Al-Jamal, H., ad Dalto, C., 2004, Vortex Iduced Vbratos Usg Large Eddy Smulato at a Moderate Reyolds Number, Joural of Fluds ad 66 Egehara Térmca (Thermal Egeerg), Vol. 11 No. 1-2 Jue ad December 2012 p

7 Structures, Vol. 19, No. 1, pp Aagostopoulos, P., ad Bearma P. W., 1992, Respose Characterstcs of a Vortex-Excted Cylder at Low Reyolds Numbers, Joural of Fluds ad Structures, Vol. 14, No. 6, pp Baray, L., ad Lews, R. I., 2006, Comparso of Grd-Based ad Vortex Dyamcs Predctos of Low Reyolds Number Cylder Flows, Aeroautcal Joural, No. 2983, pp Bathe, K. J., 1996, Fte Elemet Procedures, Pretce-Hall. Breuer, M., 2000, A Challegg Test Case for Large Eddy Smulato: Hgh Reyolds Number Crcular Cylder Flow. Iteratoal Joural of Heat Flud Flow, Vol. 21, No. 5, pp Dettmer, W., ad Perć, D., 2006, A Computatoal Framework for Flud-Rgd Body Iteracto: Fte Elemet Formulato ad Applcatos, Computer Methods Appled Mechacs ad Egeerg, Vol. 195, pp Gulmeau, E., ad Queutey, P., 2004, Numercal Smulato of Vortex-Iduced Vbrato of a Crcular Cylder wth Low Mass Dampg a Turbulet Flow, Joural of Fluds ad Structures, Vol. 19, pp Mttal, S., ad Kumar, V., 2001, Flow-Iduced Vbratos of a Lght Crcular Cylder at Reyolds Numbers, Joural of Soud ad Vbrato, Vol. 5, No. 245, pp Nobar, M. R. H., ad Nareda, H., 2006, A Numercal Study of Flow Past a Cylder wth Cross Flow ad Ile Oscllato. Computers of Fluds, Vol. 35, No. 4, pp Pasquett, R., 2005, Hgh-Order Methods for the Numercal Smulato of Vertcal ad Turbulet Flows-Hgh-Order LES Modelg of Turbulet Icompressble Flow, Comptes Redus Mécaque, Vol. 333, No. 1, pp Poldszech, O., ad Grudma, R., 2007, A Systematc Approach to the Numercal Calculato of Fudametal Quattes of the Two-Dmesoal Flow Over Crcular Cylder, Joural of Fluds ad Structures, Vol. 23, pp Placzek, A., Sgrst, J., ad Hamdou, A., 2009, Numercal Smulato of a Oscllatg Cylder a Cross-Flow at Low Reyolds Number: Forced ad Free Oscllatos, Computer & Fluds, Vol. 38, No. 1, pp Saghafa M., Stasby, P. K., Sad, M. S., ad Apsley, D. D., 2003, Smulato of Turbulet Flows Aroud a Crcular Cylder Usg Nolear Eddy- Vscosty Modelg: Steady ad Oscllatory Ambet Flows, Joural of Fluds ad Structures, Vol 15, No. 1, pp Texera, P. R. F., ad Awruch, A. M., 2001, Three-Dmesoal Smulato of Hgh Compressble Flows Usg a Mult-Tme-Step Itegrato Techque wth Subcycles, Appled Mathematcal Modellg, Vol. 25, pp Texera, P. R. F., ad Awruch, A. M., 2005, Numercal Smulato of Flud-Structure Iteracto Usg the Fte Elemet Method, Computers & Fluds, Vol. 34, pp Wllaso, C. H. K., 1989, Oblque ad Parallel Modes of Vortex Sheddg the Wake of a Crcular Cylder at Low Reyolds Numbers, Joural of Flud Mechacs, Vol. 206, pp Egehara Térmca (Thermal Egeerg), Vol. 11 No. 1-2 Jue ad December 2012 p