A STRONG COUPLING SCHEME FOR FLUID-STRUCTURE INTERACTION PROBLEMS WITH DYNAMICALLY MOVING BOUNDARIES IN VISCOUS INCOMPRESSIBLE FLOWS

Transcription

1 A STRONG COUPLING SCHEME FOR FLUID-STRUCTURE INTERACTION PROBLEMS WITH DYNAMICALLY MOVING BOUNDARIES IN VISCOUS INCOMPRESSIBLE FLOWS J. Yang *, S. Predkman, and E. Balaras * * Department of Mechancal Engneerng Unversty of Maryland at College Park College Park, Maryland, 20742, USA e-mal: jmyang@glue.umd.edu, balaras@eng.umd.edu Facultad de Ingenería Unversdad Naconal de Río Cuarto Ruta Naconal 36 Km 601, 5800 Río Cuarto, Argentna e-mal: spredk@vt.edu Key words: Flud Structure Instructons, Strong Couplng, Immersed Boundary Method, Incompressble Flows. Abstract. In the present paper an embedded-boundary formulaton that s applcable to flud structure nteracton problems s presented. The Naver-Stokes equatons for ncompressble flow are solved on a fxed grd whch s not algned wth the body. A corotatonal formulaton s used to descrbe the dynamcs of a body that moves through the fxed grd undergong large-angle/large-dsplacement rgd body motons. A strong couplng scheme s adopted, where the flud and the structure are treated as elements of a sngle dynamcal system, and all of the governng equatons are ntegrated smultaneously and nteractvely n the tmedoman. A demonstraton of the accuracy and effcency of the method wll be gven for a varety of flud/structure nteracton problems. 2131

2 MECOM 2005 VIII Congreso Argentno de Mecánca Computaconal 1 INTRODUCTION Numercal smulatons of flud flows nteractng wth dynamcally movng boundares are amongst the most challengng problems n computatonal mechancs. The man dffculty comes from the fact that the spatal doman occuped by the flud changes wth tme, and the locaton of the boundary s usually an unknown tself that depends on the flud flow and the soluton of addtonal equatons descrbng the moton/deformaton of the body. There s only a lmted number specal cases where establshed boundary-conformng formulatons can be drectly appled to such problems wth a relatvely small overhead. The use of movng reference frames 8, or coordnate transformatons 11,16 are characterstc examples. In more complex confguratons formulatons utlzng movng and/or deformng grds that contnously adapt to the changng locaton of the body have to be adopted (see for example [17, 15]). For problems that nvolve multple bodes undergong large motons and/or deformatons, these algorthms are farly complcated and have an adverse mpact on the accuracy and effcency of the flud solvers. Recently, non-boundary conformng formulatons are ncreasngly ganng attenton as tools to study flud structure nteracton problems (see [9] for a recent revew). A common feature n these methods s the soluton of the Euleran form of the Naver-Stokes equatons on a fxed grd, and trackng of the movng body n a Lagrangan fashon. Consequently, the grd lnes and the surface of the body are almost never algned. Ther advantage compared to boundary-conformng formulatons s that the need for regeneraton and/or deformaton of the grd s elmnated and n most cases effcent Cartesan solvers can be used. On the other hand, the mposton of boundary condtons on the movng body s not trval, and a varety of strateges wth dfferent degrees of accuracy and complexty have been developed over the years. The 'mmersed-boundary' formulaton ntroduced by Peskn 14, for example, uses a smooth forcng functon to enforce velocty boundary condtons on elastc bodes. The forcng functon can be drectly derved from some consttutve law applcable to the elastc body under consderaton. To the lmt of rgd bodes, however, such laws are not well posed and the dervaton of the forcng feld can be problematc. To overcome ths lmtaton, alternatve strateges lke the 'embedded-boundary' method 4 have been developed. In ths case a forcng functon s also used to mpose boundary condtons; t s not, however, specfed n the contnuous space by means of some physcal arguments, but rather n the dscrete space by drectly requrng the soluton to satsfy the boundary condtons. The approach can be farly complex f the spatal dscretzaton utlzes a large stencl, but t enforces boundary condtons wth accuracy that s equvalent to that of boundary-conformng formulatons. Embedded-boundary formulatons have been manly used n smulatons wth statonary rgd bodes 4,7,18,2. Ths s due to complcatons ntroduced to the temporal ntegraton scheme by grd ponts that change phase (for example a pont n the sold body can became a pont n the flud at the next tmestep and vse-versa) as the boundary moves on the Euleran grd. Recently Yang & Balaras 22 proposed feld-extenson strategy that addresses ths ssue. Results on two-dmensonal lamnar flows and three-dmensonal turbulent flows nteractng wth complex boundares undergong prescrbed motons were n excellent agreement wth reference experments and smulatons. In the present study a formulaton that extends ths 2132

3 J. Yang*, S. Predkman, and E. Balaras* method to problems where there s a two way nteracton between the flud and the body, whose moton wll be now determned by structural propertes and the flud forces, wll be presented. Emphass wll be gven on the couplng scheme whch s crtcal for the accuracy and effcency of the overall method. In partcular, we wll present a strong couplng scheme, where the flud and the structure are treated as elements of a sngle dynamcal system, and all of the governng equatons are ntegrated smultaneously and nteractvely n the tmedoman. The detals on the numercal method wll be gven n the next secton. A valdaton of the method wll be gven n the results secton, where vortex nduced vbratons n crcular cylnders wth one and two-degrees of freedom are consdered. Conclusons and future recommendatons wth be gven n the fnal secton. 2 PROBLEM FORMULATION AND NUMERICAL METHODS The governng equatons for unsteady ncompressble vscous flow are: 2 u uu j p 1 u = + + f, t x x Re x x j j j (1) where ( 1, 2, 3) u x = 0, x = are the Cartesan coordnates, u are the velocty components n the correspondng drectons normalzed by a reference velocty U, and p s the pressure 2 normalzed by ρ U wth ρ the densty of the flud, and f represents a external body force feld. The Reynolds number s defned as, Re = UL ν, where s L a reference length scale and ν the knematc vscosty of the flud. The equaton of moton that s generally governng the moton of a two-dmensonal, elastcally mounted body, oscllatng n the ( X Y ) plane has the from: Mx () t + Cx () t + Kx() t = f x( τ), x ( τ), x( τ), τ 0 τ t (3) where M s the mass matrx, C s the dampng matrx, K s the stffness matrx and f s the vector x t = X t + Y t j+θ t k s the vector of generalzed hydrodynamc forces per unt span, and ( ) 0( ) 0( ) ( ) of generalzed structural dsplacements. Here, X 0 ( t) and Y0 ( t ) are the dsplacements of the center of mass of the body n the x and y drectons, respectvely, and Θ ( t) s the rotaton around an axs perpendcular to the ( X Y ) plane. Equatons (1) and (2) governng the dynamcs of the flud, and equaton (3) governng the dynamcs of the body need to be solved n a coupled manner. In the present study a strong couplng scheme s adopted, where the flud and the structure are treated as elements of a sngle dynamcal system, and all governng equatons are ntegrated smultaneously n the tme-doman. A fundamental complcaton wth the applcaton of a tme-doman approach to two-way, flud-structure nteracton problems lke the one descrbed above, s that the predcton of the flow feld and hydrodynamc loads (2) 2133

4 MECOM 2005 VIII Congreso Argentno de Mecánca Computaconal requres knowledge of the moton of the structure and vce-versa. To address ths ssue an teratve predctor-corrector scheme s adopted. It s based on Hammng's 4 th -order, predctor-corrector method 3. Hammng's scheme was adapted to the present problem to avod evaluatng the hydrodynamc loads at fractonal tme steps and to accommodate hydrodynamc loads that are proportonal to the acceleraton (the so called added-mass effect). In the followng paragraphs the flud flow solver, the structural solver and the couplng scheme wll be dscussed n detal. 2.1 Flud flow solver The governng equatons (1) and (2) are solved usng a hghly effcent fractonal step method on a staggered Cartesan grd. All spatal dervatves are approxmated usng secondorder central dfferences. The tme advancement scheme s a second-order Adams-Bashforth scheme, where all terms n the rght-hand-sde (RHS) of momentum equatons are advanced explctly. The overall splttng scheme can be summarzed as follows: uˆ k uˆ Δt k 1 = RHS + f k k p = H( u ) + H( u ) + f 2 2 k 1 k k k x, (4) 2 k k φ 1 uˆ = x x Δt x j, (5) k k k φ u ˆ = u Δt, (6) x k k 1 k p = p + φ (7) where k s the tmestep ndex, u ˆk s the ntermedate velocty, φ s the pressure correcton, H s a spatal operator contanng the convectve and vscous terms, f k s the dscrete forcng functon used to mpose boundary condtons on an arbtrary mmersed boundary that does not concde wth the underlyng grd and Δ t s the tme step. An mmersed body movng on the fxed Cartesan grd s tracked n a Lagrangan manner usng a seres of marker partcles. After establshng the grd-nterface relaton, all Cartesan grd nodes are splt nto the three categores shown n Fg. 1a.: (1): flud-ponts, whch are ponts n the flud phase; (2) forcng ponts, whch are grd ponts n the flud phase wth one or more neghborng ponts n the sold phase; (3) sold-ponts, whch are ponts n the sold phase. The detals on the trackng scheme and taggng procedure can be found n [2]. The Naver-Stokes solver descrbed n the prevous paragraph s appled on all ponts of the Euleran grd as f the flud/sold nterface s not present. The effect a movng sold body on the flow s ntroduced through the dscrete forcng functon, f. It s computed only at the 2134

5 J. Yang*, S. Predkman, and E. Balaras* forcng ponts by substtutng u ˆk wth u Ψ n equaton (4) and solvng for f : f k k 1 uψ u = RHS Δt In the specal case where the nterface and the forcng pont concde, u Ψ s smply the local velocty on the rgd body, and therefore the boundary condton one wshes to enforce. In the general case, however, the ponts on the Euleran grd and the nterface almost never concde and u Ψ has to be computed usng some nterpolaton strategy. An example s shown n Fg. 1b, where the velocty at the forcng pont s computed by means of lnear nterpolaton that nvolves the projecton of the forcng pont on the nterface (pont 1 n Fg. 1b) and two ponts n the flud phase (ponts 2 and 3 n Fg. 1b). In the general case the velocty or any varable,, n the two-dmensonal space can be wrtten as follows: k, (8) φ = b1+ b2x+ b3y (9) The coeffcents b 1, b 2, and b 3 n Eq. (9) can be found by solvng the followng system: where ( x, y ), ( x, y ), and (, ) b1 φ1 1 x1 y1 φ1 1 b2 φ 2 1 x2 y = = 2 φ2 b 3 φ 3 1 x3 y 3 φ A (10) x y n the 3 3 matrx A are the coordnates of the three ponts n the nterpolaton stencl shown n Fg. 1b. The nverson of matrx, A, at every forcng pont s performed every tme the locaton of the nterface s updated. Hgher-order reconstructons for both Drchlet and Newmann boundary condtons can be acheved n a straghtforward manner usng the same overall procedure (see for example [18]). The above solver has been extensvely tested for a varety of lamnar and turbulent flow problems nvolvng statonary and movng mmersed boundares wth prescrbed moton wth results n excellent agreement wth reference computatons and experments 2, ODE Solver and strong couplng scheme Equaton (3) can be rewrtten n non-dmensonal form, as a system of 2n frst-order, nonlnear, ordnary-dfferental equatons ( n s the number of degrees-of-freedom of the structure) as follows: () t ( ) ( ) y = F y τ, y τ, τ 0 τ t, (11) where half of the vector F represents generalzed veloctes and the other half represents generalzed forces dvded by the correspondng nertas. In general the loads depend explctly on y, and mplctly on the hstory of the moton and the acceleraton of the structure. Hammng's 4 th -order, predctor-corrector method s used to ntegrate equaton (11) 2135

6 MECOM 2005 VIII Congreso Argentno de Mecánca Computaconal n the tme doman 3. The detals of the basc numercal procedure used to determne the current value of the vector y are gven next: 1. Let tj = jδ t denote the tme at the j -th tme step, where Δ t s the tme-step sze used to obtan the numercal soluton, and ( tj), ( t ), j ( tj), ( tj) = = = (12) j j j y y y y F F y y 2. Compute the predcted soluton, p y j, and modfy t, 1 y j, usng the local truncaton error, j 1 e, from the prevous tme step: y = y + Δt ( 2F F + 2 F ), y = y + e (13) 3 9 p j j 1 j 1 j 2 j 3 1 j p j j 1 3. Correct the modfed, predcted soluton by: k j k j where = ( ) y 1 = Δ t ( + ) 8 y y F F F (14) k+ 1 j j 1 j 3 k j j 1 j 2 F F y and 1 y j = p y j. k s the teraton ndex. Convergence s acheved when the teraton error, e = y y s less than a prescrbed tolerance ε. j k+ 1 j k j 4. Compute the local truncaton error, to the next tmestep: j e, and the fnal soluton, ( ), j y, at step j and advance j k + 1 j p j j k + 1 j j e = y y y = y e (15) Note that n the above procedure nformaton from four prevous tmesteps s requred. Therefore, startng from the ntal condtons, Euler, Adams-Moulton, and Adams-Bashforth methods were used. The flud solver descrbed n Secton 2.1 s coupled to the structural solver as follows: 1. Fnd the predcted locaton and velocty of the body usng equaton (13). 2. Fnd the predcted flud velocty and pressure felds usng equatons (4)-(7) wth the boundary condtons provded by step 1. Then, compute the resultng loads on the structure. 3. Compute the new locaton and velocty of the body usng equaton (14). j 4. Check for convergence. If e s greater than the prescrbed tolerance, ε, repeat steps 2 to 4. If convergence s acheved goto step Fnd fnal poston and velocty of the body usng equaton (15), and the fnal flud pressure and velocty felds from equatons (4)-(7). 2136

7 J. Yang*, S. Predkman, and E. Balaras* 8 In all computatons reported n ths study a tolerance of ε = 10 was used. The number of teratons requred for convergence at each tme step vared from 2 to 6 dependng from the stffness of the problem. 3 RESULTS Vortex-nduced vbratons of crcular cylnders s a problem that has been extensvely studed both expermentally and numercally (see [20] for a recent revew), and wll be used n the present study to evaluate the accuracy and effcency of the proposed methodology. Intally two dfferent confguratons of a sngle cylnder are consdered: 1. free oscllatons n the cross-stream drecton (one degree-of-freedom); 2. free oscllatons n both the streamwse and cross-stream drectons (two degrees-of-freedom). For both cases the selected parametrc space s as close as possble to well documented lamnar two-dmensonal flow problems n the lterature. Fnally to demonstrate the effcency and robustness of the proposed approach, test problems nvolvng and array of freely oscllatng cylnders are presented. 3.1 Free vertcal oscllatons For a cylnder freely oscllatng n the cross-stream drecton, Y, whch s modeled as mass-damper-sprng system, the non dmensonal form of equaton (3) becomes : 2 2π 2π 1 y + 2 ζ y y CL, U + = red Ured 2n (16) where y = Y0 D s the non-dmensonalzed vertcal dsplacement, ζ = c 2 km s the 1 vscous dampng rato, C 2 L fy ( ρdlu ) = s the lft force coeffcent, 2 n m ρd 2 = s the mass rato wth ρ the flud densty, and U red = U f n D s the reduced velocty wth fn = km the natural vbratng frequency of the structure. In the above m s system 1 2π mass, c the dampng coeffcent, k the sprng constant, Y 0 the transverse dsplacement of the system centrod, and f y the nstantaneous lft force on cylnder. Note that the same reference scales as n the Naver-Stokes equatons (the cylnder dameter, D, and the freestream velocty, U ) were used to obtan the dmensonless form of the equatons. The parametrc space was selected to match the experment by Anagnostopoulos and Bearman 1, whch has been used for valdaton n several other numercal smulatons (see for example [12,19,15,8]). The mass rato was set to n = and dampng rato to ζ = A sketch of the computatonal doman s gven n Fg. 2a. The nflow boundary s located 10D upstream of the cylnder where a unform velocty s specfed. At the outflow boundary, located 30D downstream of the cylnder, a convectve boundary condtons s used 13. Radatve boundary condtons are used n both the freestream boundares. The 2137

8 MECOM 2005 VIII Congreso Argentno de Mecánca Computaconal number of grd ponts n all computatons was n streamwse and transverse drectons, respectvely. The grd was stretched n both drectons and the resultng resoluton near the cylnder was approxmately 0.02D 0.02D. A detal of the grd s shown n Fg. 2b. A constant tme step of 0.005D U was used for all cases. For all cases the flow over a statonary cylnder for the same Reynolds number was ntally computed. Then, the cylnder was allowed to move freely. Integraton n tme was performed untl a perodc state of constant maxmum ampltude was reached. The large mass rato of ths system makes the problem farly stff and very long ntegraton tmes were necessary n all cases. In the lock-n regme and for the low Reynolds number, one can observe a monotoncally growng oscllaton ampltude (see Fg. 3a). Approxmately 2500 nondmensonal tme unts were necessary for a perodc state to be acheved. As the Reynolds number ncreases n the lock-n regme the ampltude stops to ncrease monotoncally (see Fg. 3b). Over tme, however, t also approaches a perodc state of lower maxmum ampltude. Outsde the lock-n regme there s a drastc change n the vbraton ampltude whch s modulated and has a very small magntude as shown n Fg. 3c for Re = 90. In ths case the cylnder oscllaton frequency s f c = 0.165, whch s very close to vortex sheddng frequency of the fxed cylnder at ths Reynolds number. The maxmum oscllaton ampltude and the correspondng frequency s shown n Fg. 4 as a functon of the Reynolds number. The expermental results n [1] and prevous computatons [12,19,15,8] have been added for comparson. All reference computatons have been conducted usng boundary conformng formulatons. In partcular, Nomura 12 and We et al. 19 used and arbtrary Lagrangan Euleran (ALE) fnte element formulaton, where the computatonal grd s deformed/regenerated every tme the locaton of the cylnder changes to conform to the body at all tmes. A smlar strategy was used by Schulz and Kallders 15 n the framework of a fnte-volume formulaton. L et al. 8 used a hghly accurate spectral element method wth a translatng frame of reference to avod the tedous process of grd regeneraton. As shown n Fg. 4a all smulatons agree on the crtcal Reynolds number for lock-n, whch s however, lower compared to the one reported n the experment. The correspondng values are approxmately Re 90 n the smulatons and Re 103 n the experment. Schulz and Kallnders 15 speculated that ths dfference could be due three-dmensonal effects nduced by the absence of end-plates n the expermental apparatus. In all dfferent data sets the maxmum vbraton ampltude n the lock-n regme appears at the lowest Reynolds number as shown n Fg. 4b. Dscrepances, however, can be observed regardng actual values of the ampltude, A D. In general, all smulatons tend to underpredct the values of AD reported n the experment. Our results are n good agreement wth the hghly accurate spectral element computatons reported n [8]. An extensve grd refnement study, however, remans to be performed to establsh the grd ndependency of our solutons. The vortcty and pressure contours for the case of maxmum ampltude n the lock- 2138

9 J. Yang*, S. Predkman, and E. Balaras* n regme ( Re = 95) s shown n Fg. 5. The smoothness of the pressure and vortcty felds near the cylnder are ndcatve of the accuracy of the proposed formulaton. Also, the expected 2S mode pattern (two sngle vortces per cycle of moton) can be seen n the wake, whch s n agreement wth the observatons n [21]. 3.2 Free oscllatons n the X-Y plane Although n many practcal problems cylndrcal structures have the same mass rato and natural frequency n both the X and Y drectons, far fewer studes have been conducted wth the cylnder oscllatng n the X-Y plane. Ths s partally due to the fact that early experments dd not show sgnfcantly dfferent response characterstcs compared to cylnders undergong oscllatons n the vertcal drecton only. Recently, however, Jauvts and Wllamson 6 n ther water channel experments at Reynolds numbers rangng from 1,000 to 15,000, reported a new response branch that nvolves substantal oscllaton ampltudes n the streamwse drecton when the mass rato s very low ( n 25. ). In the lamnar flow regme, whch s the focus of the present work, such a dramatc change n the response of the system would be probably absent as ndcated by a the few two-dmensonal numercal smulatons at low Reynolds numbers (see for example [10], [23]). To further nvestgate ths ssue and also establsh the effcency and accuracy of the proposed formulaton we conducted a number of numercal experments of freely oscllatng cylnders n the X-Y plane at a farly wde parametrc space. We kept the Reynolds number, mass rato, and dumpng rato - Re = 200, n = 204. and ζ = respectvely- constant, and vared the reduced velocty, U red, from 1 to 11. Ths way we covered approxmately the same range as n the experments n [6], at a much lower Reynolds number. The computatonal box and boundary condton set-up s smlar to the one used for the vertcally oscllatng cylnder. The grd has been modfed, however, to ensure approprate resoluton n a wder range. The number of grd ponts n all computatons was n streamwse and transverse drectons respectvely, resultng n near cylnder resoluton of approxmately 002. D 002. D (same as the one-degree-of-freedom case above). As for the prevous case, n all computatons the correspondng statonary cylnder problem was frst solved and then free vbratons n both drectons were allowed. Due to the low mass rato n ths case, a stable perodc state was acheved much faster compared to the prevous problem. Fgure 6 shows the oscllaton frequency, transverse oscllaton ampltude, and the streamwse oscllaton ampltude as functon of the reduced velocty. A lock-n regme rangng from approxmately 4 < U red < 6 s evdent. In ths regme the transverse oscllaton ampltude s farly hgh, but always below the 0.6 lmt for lamnar flows dscussed n [20]. The streamwse ampltude s lower by a factor of 10 n agreement wth prevous computatons. Also, the oscllaton frequences are approxmately equal to the natural oscllaton frequency of the structure. On the same fgure we added the correspondng results from the one-degreeof-freedom case reported n the prevous secton. Qualtatvely the response of the two 2139

10 MECOM 2005 VIII Congreso Argentno de Mecánca Computaconal systems s smlar. The one-degree-of-freedom case, however, has narrower lock-n range wth a lower maxmum ampltude, whch s probably due to the hgher mass rato and dfferences n the Reynolds numbers. The X, Y trajectores of the cylnder center for the case wth the largest ampltude ( U red = 4. 1) are shown n Fg. 7a. Both trajectores are approxmately snusodal leadng to the well known fgure-eght type of moton shown n Fg. 7b. The correspondng force coeffcents on the other hand, devate from the snusodal varaton resultng n the complex pattern shown n Fg. 8. Ths s a result of the streamwse moton and s smlar to correspondng plots reported n [6]. Fnally, n Fg. 9 solnes of the vortcty and pressure are shown at one nstant durng the cycle for the case of U = The 2S vortex pattern s evdent. 4 SUMMARY red In the present paper an embedded-boundary formulaton that s applcable to flud structure nteracton problems s presented. The Naver-Stokes equatons for ncompressble flow are solved on a fxed grd and the body s tracked n a Lagrangan reference frame. A strong couplng scheme s presented, where the flud and the structure are treated as elements of a sngle dynamcal system, and all of the governng equatons are ntegrated smultaneously, and nteractvely n the tme-doman. Results presented for cases of vortex nduced vbratons n a crcular cylnder were n good agreement wth reference data n the lterature. 5 REFERENCES [1] Anagnostopoulos, P., and Bearman, P. W., Response characterstcs of a vortex-excted cylnder at low Reynolds numbers, J. Fluds Struct., 6, 39-50, (1992) [2] Balaras, E., Modelng complex boundares usng an external force feld on fxed Cartesan grds n large-eddy smulatons, Computers & Fluds, 33, , (2004). [3] Carnahan, B., Luther, and H. A., Wlkes, J.O., Appled Numercal Methods, John Wley & Sons, Inc., New York, (1969). [4] Fadlun, E. A., Verzcco, R., Orland, P., and Mohd-Yusof, J., Combned mmersedboundary fnte-dfference methods for three-dmensonal complex flow smulatons, J. Comput. Phys., 161, 35-60, (2000). [5] Hu, H. H., Patankar, N. A., and Zhu, M. Y., Drect numercal smulatons of flud-sold systems usng the arbtrary Lagrangan-Euleran Technque, J. Comput. Phys., 169, , (2001). [6] Jauvts, N., and Wllamson, C. H. K., The effect of two degrees of freedom on vortexnduced vbraton at low mass and dampng, J. Flud Mech, 509, 23-62, (2004). [7] Km, J., Km, D., and Cho, H., An mmersed-boundary fnte-volume method for smulatons of flow n complex geometres, J. Comput. Phys., 171, , (2001). [8] L, L., Sherwn, S. J., and Bearman, P. W., A movng frame of reference algorthm for flud/structure nteracton of rotatng and translatng bodes, Int. J. Numer. Meth. Fluds, 38, , (2002). 2140

11 J. Yang*, S. Predkman, and E. Balaras* [9] Mttal, R., and Iaccarno, G., Immersed boundary methods, Annu. Rev. Flud Mech., 37, , (2005). [10] Newman, D. J., and Karnadaks, G. E., Drect numercal smulatons of flow over a flexble cable, In Proc. 6th Int. Conf. Flow-Induced Vbratons, ed. P. W. Bearman, Rotterdam, Netherlands, , (1995). [11] Newman, D. J., and Karnadaks, G. E., A drect numercal smulaton study of flow past a freely vbratng cable, J. Flud Mech., 344, , (1997). [12] Nomura, T., Fnte element analyss of vortex-nduced vbratons of bluff cylnders, J. Wnd Eng. Ind. Aerodyn., 46-47, , (1993). [13] Orlansk, I., Smple boundary-condton for unbounded hyperbolc flows, J. Comput. Phys., 21, , (1976). [14] Peskn, C. S., Flow patterns around heart valves: a numercal method, J. Comput. Phys., 10, , (1972). [15] Schulz, K. W., and Kallnders, Y., Unsteady flow structure nteracton for ncompressble flows usng deformable hybrd grds, J. Comput. Phys., 143, , (1998). [16] Shen, L., Zhang, X., Yue, D. K. P., and Trantafyllou, M. S., Turbulent flow over a flexble wall undergong a streamwse travelng wave moton, J. Flud Mech., 484, , (2003). [17] Tezduyar, T. E., Fnte Element methods for flow problems wth movng boundares and nterfaces, Archves of Computatonal Methods n Engneerng, 8, , (2001). [18] Tseng, Y. H., and Ferzger, J. H., A ghost-cell mmersed boundary method for flow n complex geometry, J. Comput. Phys., 192, , (2003). [19] We, R., Sekne, and A., Shmura, M., Numercal analyss of 2D vortex-nduced oscllatons of a crcular cylnder, Int. J. Numer. Meth. Fluds, 21, , (1995). [20] Wllamson, C. H. K., and Govardhan, R., Vortex-nduced vbratons, Annu. Rev. Flud Mech., 36, , (2004). [21] Wllamson, C. H. K., and Roshko, A., Vortex formaton n the wake of an oscllatng cylnder, J. Fluds Struct., 2, , (1988). [22] Yang, J., and Balaras, E., An embedded-boundary formulaton for large-eddy smulaton of turbulent flows nteractng wth movng boundares, J. Comput. Phys., Submtted for Publcaton, (2005). [23] Zhou, C. Y., So, R. M. C., and Lam, K., Vortex-nduced vbratons of an elastc crcular cylnder, J. Fluds Struct., 13, , (1999). 2141

12 MECOM 2005 VIII Congreso Argentno de Mecánca Computaconal (a) (b) Fgure 1: (a) Taggng of grd nodes on the Euleran grd accordng to ther locaton relatve to the flud/sold nterface ndcated by the dashed lne; sold ponts; flud ponts; forcng ponts. (b) Example of a lnear local reconstructon stencl nvolvng the projecton of the forcng pont to the nterface (pont 1) and two flud ponts (ponts 2 and 3). Fgure 2: Problem set-up for the free vbratng cylnder n the cross-stream drecton. (a) Sketch of the computatonal doman; (b) close-up of grd near the cylnder (every fve ponts are shown). 2142

13 Re = 100 Re =110 Re = 90 Fgure 3: Tme hstory of the dsplacements for the freely vbratng cylnder n the cross-stream drecton. (a) and (b) are Reynolds number n the lock-n regme; (c) s outsde the lock-n regme. 2143

14 Fgure 4: Comparson of: (a) frequency of the oscllaton f f n ; (b) maxmum oscllaton ampltude A D, as a functon of the Re number. present computatons; --- experment n [1]; computaton n [12]; * computaton n [19]; computaton n [15]; computaton n [8]. 2144

15 (a) Vortcty (b) Pressure Fgure 5: Freely vbratng cylnder n the cross-stream drecton. (a) Spanwse vortcty solnes; (b) pressure solnes Re = 95, U = red 2145

16 Fgure 6: The oscllaton frequency, transverse oscllaton ampltude and streamwse oscllaton ampltude as functons of reduced velocty. two-degrees-of-freedom; one-degree-of-freedom. 2146

17 Fgure 7: Freely vbratng cylnder n the ( X Y ) plane durng lock-n. (a) Tme hstory of the dsplacements, x-drecton and --- y-drecton; (b) fnal, perodc dsplacement pattern. Re = 200, U = 4.1. red Fgure 8: Phase plots for the lft and drag coeffcents. (a) One-degree-of-freedom, Re = 95, U = 51. ; Twodegrees-of-freedom, Re = 200, U = 41.. red red 2147

18 (a) (b) Fgure 9: Freely vbratng cylnder n the X Y plane. (a) Spanwse vortcty solnes; (b) pressure solnes. Re = 200, U = red 2148