Suppression of three-dimensional flow instabilities in tube bundles

Transcription

1 Suppression of hree-dimensional flow insabiliies in ube bundles N. K.-R. Kevlahan Deparmen of Mahemaics & Saisics, McMaser Universiy, Hamilon, Canada J. Wadsley Deparmen of Physics & Asronomy, McMaser Universiy, Hamilon, Canada Absrac We sudy he generaion of hree-dimensional voriciy in ighly packed ube bundles. In paricular, our goal is o invesigae which condiions (if any) enable he flow o remain wo-dimensional for Re > 18. We calculaed wo- and hree-dimensional flow hrough periodic roaed square ube bundles wih igh packing, P/D = 1.5, using a high resoluion pseudo-specral code wih penalizaion. The ubes are cylinders whose response is modelled as a rigid harmonic oscillaor forced by he flowinduced lif. We find ha a Re = 2 ube moion compleely suppresses he hree-dimensional insabiliy. A Re = 1 ube moion does no suppress he hree-dimensional insabiliy, alhough he flow does have increased spanwise correlaion and he Srouhal number for he wo- and hree-dimensional flows is approximaely he same. The igh packing alone does no suppress he hree-dimensional insabiliy. Three-dimensional voriciy drasically reduces fluid forces acing on he ube compared wih an equivalen wo-dimensional flow. 1 Inroducion The wake of an isolaed cylinder firs becomes hree-dimensional a Re 18 via he formaion of regular sreamwise vorices wih a spacing of abou hree cylinder diameers [he mode A insabiliy idenified by Williamson (1989)]. A Re 23 a second vorex mode appears [he mode B insabiliy idenified by Williamson (1989)], characerized by irregular sreamwise vorices wih Corresponding auhor. address: kevlahan@mcmaser.ca (N. K.-R. Kevlahan). Preprin submied o Journal of Fluids and Srucures 9 November 24

2 a spacing of one cylinder diameer (Williamson, 1989). As Reynolds number increases furher, he wake becomes increasingly complicaed (possibly via period-doubling) unil i is compleely urbulen. In conras, he ransiion o hree-dimensionaliy in ighly coupled ube bundles is sill no well undersood. Indeed, in some experimens i appears ha he flow and cylinder response remain roughly wo-dimensional for Reynolds numbers well beyond 18 (Weaver, 21). For example, Price e al. (1995) find ha Srouhal frequency and rms drag do no change grealy wih Reynolds number for Re > 15. Blevins (1985) demonsraed ha acousic forcing of an isolaed cylinder a is Srouhal frequency is able o produce nearly perfec spanwise correlaion of pressure for 2 Re 4. He conjecured ha similar effecs migh be observed in ube bundles. Blevins invesigaions confirmed earlier work by Toebes (1969) who showed ha a cylinder vibraion ampliude of A/D.125 was required o achieve a high degree of spanwise correlaion. In he case of ube bundles, i has also been suggesed ha i is he igh spacing of he bundle ha suppresses hree-dimensional insabiliy. In his paper we aemp o undersand which condiions (if any) enable he flow o remain wo-dimensional for Re > 18. I is also possible ha some aspecs of he flow remain wo-dimensional (e.g. spanwise correlaion of vorex shedding), while ohers become fully hree-dimensional (e.g. developmen of sreamwise voriciy). To invesigae hese quesions we calculae wo- and hree-dimensional flow hrough periodic roaed square ube bundles wih spacing P/D = 1.5. We consider wo ypes of bundles: fixed cylinder and moving cylinder bundles. The naural frequency of he moving cylinders is uned o mach he Srouhal frequency in order o maximize he moving cylinder effec. All cylinders move in phase, which corresponds o he acousic resonance condiions invesigaed by Blevins (1985). The mahemaical formulaion of he problem is described in Secion 2 and he penalized pseudo-specral mehod used o solve he equaions is oulined in Secion 3. The simulaions are described in Secion 4 and resuls are presened in Secion 5. The main conclusions are summarized in Secion 6. 2 Problem formulaion Le us consider a viscous incompressible fluid governed by he Navier Sokes equaions u + (u + U ) u + P = ν u, (1) u =, (2) 2

3 where u is he velociy, P is he pressure and U is an imposed upsream flow velociy. We focus here on he case where he fluid occupies he complemen in he plane R 3 of a periodic laice of cylinders O i (wih diameer D) oriened wih heir axes in he x 3 direcion. The exernal boundary condiions associaed wih his problem are herefore ha u is Q-periodic on ], L 1 [ ], L 2 [ ], L 3 [, wih no-slip boundary condiions on he surface of he (moving) obsacle, u + U = U o on O i, i (3) where U o is he velociy of he obsacle. Noe ha all quaniies (e.g. Srouhal number, Reynolds number and ime) will be nondimensionalized using he upsream flow velociy U, he cylinder diameer D and he kinemaic viscosiy ν. To model he no-slip boundary condiions wihou explicily imposing Eqs. (3) we follow Ango e al. (1999) by replacing Eqs. (1 3) by he following se of L 2 -penalized equaions: u η + (u η + U ) u η + P η = ν u η 1 η χ(x, )(u η + U U o ), (4) u η =, (5) where U o is he obsacle s velociy. Noe ha Eqs. (4 5) are valid in he enire domain Ω: he las erm on he righ hand side of (4) is a volume penalizaion of he flow inside he obsacle. Here < η 1 is a penalizaion coefficien and χ denoes he characerisic funcion (or mask) 1 if x O i, χ(x, ) = oherwise. (6) This approximaion is called Brinkman penalizaion, and models flow hrough a porous medium wih fixed porosiy and (very small) permeabiliy η. Ango (1999) proved ha he soluion of he penalized Eqs. (4-5) converge o ha of he Navier Sokes Eqs. (1-2) wih he correc boundary condiions (3) as η. More precisely, he upper bound on he global error of he L 2 -penalizaion was shown o be (Ango, 1999) u u η H 1 (Ω) = O(η 1/4 ). (7) In he specific case of impulsively sared flow over a plane Kevlahan and Ghidaglia (21) showed analyically ha he error in approximaing (3) is 3

4 acually lower: O(η 1/2 ). I seems reasonable ha his is he sharp esimae in he general case as well. Noe ha he velociy is coninuous and differeniable, due o he insananeous smoohing acion of he Laplacian operaor erm (i.e. viscous diffusion). This volume penalizaion has been implemened in a finie difference code (Khadra e al., 2) for wo-dimensional flow around an isolaed cylinder, and was found o give good resuls. I is imporan o noe ha η is an arbirary parameer, independen of he spaial or emporal discreizaion, and hus he boundary condiions can be enforced o any desired accuracy by choosing η appropriaely. This propery disinguishes he Brinkman mehod from oher penalizaion schemes and allows he error o be conrolled precisely. Anoher advanage of he Brinkman penalizaion is ha he force F i acing on an obsacle O i can be found by simply inegraing he penalizaion erm over he volume of he obsacle: F i = 1 η O i (u η + U U o ) dx. (8) Thus, he calculaion of lif and drag on an obsacle can be made simply, accuraely and a low cos. This is helpful when calculaing fluid srucure ineracion, where he force mus be updaed a each ime sep. Kevlahan and Ghidaglia (21) showed analyically ha he error in calculaing he force over a fla plae using (8) is only O(η). We have found numerically ha η = 1 4 gives drag curves correc o wihin 1%. The cylinders move as forced simple harmonic oscillaors according o he equaion (m + m A ) d2 x o d 2 + bdx o d + kx o = F w (), (9) where x o () is he cylinder posiion, m is he cylinder mass, m A is he added mass, b is he damping, k is he spring consan. The fluid forcing F w () (due o wake voriciy) is calculaed as a volume inegral using (8). Thus he penalized Navier Sokes Eqs. (4 5) are fully coupled o he cylinder moion equaion (9) via he fluid force Eq. (8). In his paper ake he cylinder diameer D = 1, and consider square cylinder arrays where he raio of pich (cylinder spacing) o diameer P/D = 1.5 (ighly packed). Figure 1 shows he ube bundle geomery. This configuraion models flow in he inerior of a very large periodic array. The periodic domain conains only one cylinder, and hus all (image) cylinders move in phase. This corresponds o he case of acousic resonance, caused by a sanding acousic 4

5 D = 1 P = 1.5 U D = 1 L 3= 6 (Re= 2) L 3= 1.5 (Re= 1 ) Fig. 1. Tube bundle geomery. Noe ha here is one ube per periodic domain (indicaed by dashed lines), and he mean flow is a 45 o he array axis (i.e. his is a saggered or roaed square array). The ubes are free o oscillae in he ransverse direcion only. wave beween he duc walls. Alhough modelling all cylinders as moving in phase is unrealisic for some flows, i should maximize any suppression of hree-dimensional insabiliies via spanwise correlaion of he voriciy. The cylinders move freely in he ransverse direcion due o he fluid-induced lif force. 3 Numerical mehod The penalized Navier Sokes equaions are solved using a Fourier-ransformbased pseudo-specral mehod in space [e.g. Vincen and Meneguzzi (1991)] and a Krylov mehod in ime (Edwards e al., 1994). The pseudo-specral mehod is compuaionally efficien and highly accurae for spaial derivaives, while he Krylov mehod is a siffly sable explici mehod wih an adapive sepsize o mainain error o a specified olerance. In he pseudo-specral mehod derivaives are calculaed in Fourier space wih exponenial accuracy, while nonlinear erms (i.e. he advecion and penalizaion erms) are calculaed in physical space. This approach has zero numerical dissipaion, which is imporan for accurae simulaion of moderae o high Reynolds number calculaions. The incompressibiliy condiion is enforced by projecing he velociy in Fourier space ono he plane normal o he wave vecor k. A consan equivalen upsream flow U is imposed by seing he ime evoluion of he zero wavenumber mode o zero. Noe ha he pich velociy U p (i.e. he average velociy beween he ubes) is greaer han U : U p = U (V Ω /V F ), where V Ω /V F is he raio of oal volume o fluid volume. The difference beween U and U p should be remembered when comparing he presen resuls o hose for isolaed cylinders. All quaniies are nondimen- 5

6 sionalized wih respec o U. Noe ha enforcing a consan upsream flow is equivalen o adding energy o he sysem o compensae he energy los o dissipaion. Alhough he mask χ(x, ) used o define he posiion of he ubes is disconinuous, he velociy remains coninuous and differeniable. This ensures ha Gibbs oscillaions are small, and we have found hey do no perurb he calculaion, even in he voriciy formulaion (Kevlahan and Ghidaglia, 21). One of he main advanages of he Brinkman penalizaion is ha i allows us o use a Caresian (recangular) compuaional grid, which is compuaionally efficien. By performing grid convergence sudies, we have found ha a grid spacing of h = δ/6 in he wall normal direcion (where δ = Re 1/2 is he boundary layer hickness) is sufficien o give fully converged resuls. The hree-dimensional calculaions are large ( grid poins) and ime consuming (especially a Re = 1 ), and so we have developed a parallel version of he code. The Fourier ransforms are performed using ffw (Frigo and Johnson, 1998), which parallelizes he calculaion by decomposing he domain ino spanwise slabs (one per processor). The Krylov ime scheme is also parallelized using mpi. The resuling code scales surprisingly well for such a ighly coupled calculaion: for example, each ime sep is 2.5 imes faser on 48 processors han on 16. All parallel calculaions were carried on McMaser Universiy s sharcne cluser idra, which has 128 processors linked by a Quadrics nework. One drawback of he penalizaion approach is ha he small parameer η makes he equaions siff. Because of his siffness, sepsizes are bounded by he penalizaion parameer, i.e < η, unless we employ a siffly sable mehod in ime. To obain an accurae force calculaion we mus also use a highorder mehod ha auomaically adjuss he ime sep o mainain he desired olerance. We have found ha he Krylov ime-sepping mehod developed by Edwards e al. (1994) works very well for he presen problem. This approach is based on he Krylov mehod for solving a linear sysem, and is accuracy (in he linear case) is O(K), where K is he dimension of he Krylov subspace used o approximae he marix represening he righ hand side of he equaion. We have found ha K = 1 is opimal for compuaional efficiency, and gives sufficien accuracy. The sep size is se o ensure ha he L 2 error in he approximaion of u/ is smaller han he desired olerance. This is done by comparing he exac value for u/ (given by he righ hand side of he equaion) o he approximaion o u/ given by differeniaing he Krylov approximaion o u(). We have found ha he Krylov approach gives good resuls, especially when he cylinder is moving. In our calculaions we use K = 1, and se he L 2 error olerance o 1 3. Wih hese parameers we achieve a cfl value of 2 o 6. 6

7 Pseudo specral Vorex Asympoic C d Fig. 2. Drag for impulsively sared flow pas a cylinder a Re = 55. To eliminae noise in he force calculaion when he cylinder is moving which is caused when grid poins move in or ou of he mask, we slighly smooh he edge of he mask. The mask is smoohed over a disance of only h/6 (i.e. 1/6 of a grid poin). This smoohing compleely eliminaes he noise, and changes he force by less han 1%. One can view his sligh smoohing as rounding he sep-like edge of he cylinder due o he Caresian grid approximaion of is boundary. Figure 2 shows he drag for impulsively sared flow around a wo-dimensional cylinder a Re = 55 calculaed wih he presen mehod (on a large L 1 = L 2 = 2 domain so L 1 U T ) compared wih resuls from a vorex mehod (Koumousakos and Leonard, 1995), and he shor ime asympoic soluion (Bar-Lev and Yang, 1975). The good agreemen demonsraes ha our penalized pseudo-specral mehod gives reliable force resuls. 4 Simulaions In each of he following cases we do four simulaions: wo-dimensional wih fixed and moving cylinders, and hree-dimensional wih fixed and moving cylinders. This allows us o direcly compare he wo- and hree-dimensional flows in order o deermine he degree of suppression of hree-dimensional insabiliy. The Reynolds number is defined as Re = U D/ν (where he upsream velociy U = 1, and he cylinder diameer D = 1), and he flow is a 45 o he axis of he square cylinder array (his is referred o as a roaed square array). The compuaional domain conains one cylinder and has dimensions L 1 L 2 = in he plane perpendicular o he 7

8 Re Dimension Resoluion L 3 m b f 2 2-D D D D Table 1 Parameers for he numerical simulaions. Fixed and moving cylinder runs are done for each of he above cases. cylinder axis (see Figure 1). In each case he spring frequency f is chosen o mach he Srouhal frequency of he fixed wo-dimensional cylinder, he cylinder is undamped (b = ), and he nondimensional ube mass per uni lengh is m = m/(ρd 2 /2) = 5. We se he oal nondimensional mass o be m + π/2 (where π/2 = m A is he nondimensional added mass), and hus he ube is forced by he wake force only, as in Shiels e al. (21). The parameers for each simulaion are summarized in Table 1. Noe ha we consider wo Reynolds numbers: Re = 2 (jus above he mode A hree-dimensional ransiion), and Re = 1 (when he flow should be fully hree-dimensional). This allows us o compare he effec of ube moion on weakly and srongly hree-dimensional flows. The compuaional grid is for he Re = 2 simulaions, and he lengh of he domain in he spanwise direcion is 6 (sufficien o capure he sreamwise voriciy, which has a wavelengh of abou 3 D). Alhough he spanwise direcion is no as finely resolved as he oher direcions (h = 3/4δ, compared wih h = δ/6 for he oher direcions), we have checked a poseriori ha hree-dimensional voriciy is sufficienly well resolved. This is acually slighly more han he spanwise resoluion used in he finie volume calculaions of Persillon and Braza (1998). The normalized spring consan is se o k = k/(ρu 2 /2) = in order o mach he Srouhal number S = 1.32 of he fixed wo-dimensional cylinder. A Re = 1 we kep he same sreamwise resoluion as for Re = 2 (i.e. h = δ/6), bu found ha we needed o increase he spanwise resoluion o h = δ/2. The compuaional grid is hus , and he lengh of he domain in he spanwise direcion is 1.5 (i.e. he domain is cubic). As we will see, he shorer spanwise domain is sufficien due o he smaller size of he sreamwise vorices a his Reynolds number. The normalized spring consan is se o k = 13 in order o mach he peak Srouhal number S = 1. of he fixed wo-dimensional cylinder. 8

9 (a) 1 (b) 5 5 C L C L (c) 5 (d) C L C L Fig. 3. Lif curves for Re = 2 case. (a) Fixed cylinder, shor imes: 2-D and 3-D resuls mach. (b) Fixed cylinder, long imes: 2-D has fixed ampliude, while 3-D has modulaed ampliude. (c) Moving cylinder, shor imes: 2-D and 3-D resuls mach. (d) Moving cylinder, long imes: 2-D and 3-D resuls mach. 5 Resuls 5.1 Reynolds number 2 Figure 3 shows he lif curves a shor and long imes for he fixed and moving cylinders a Re = 2. In boh cases he lif curves agree a shor imes (before he hree-dimensional insabiliy has developed). However, a longer imes he hree-dimensional fixed cylinder s lif ampliude is modulaed in a nonsaionary way, whereas he he moving cylinder s lif sill closely maches ha of he moving wo-dimensional cylinder. Table 2 liss he peak lif frequencies for each of he cases. The frequencies of he wo- and hree-dimensional moving cylinder cases mach, while hose of he fixed cases differ by abou.14. Noe ha he Srouhal number of he hree-dimensional fixed ube is This value is much higher han he experimenal value of.18 for an isolaed ube (Williamson, 1989), bu is consisen wih he value of 1.25 ±.1 found for roaed square ube bundles wih P/D = 1.5 by Price e al. (1995). Figure 4 shows he voriciy field a = 5 (he poin of maximum lif during he vorex shedding cycle). I is clear from Figure 4(c) ha he moving cylinder compleely suppresses he producion of sreamwise voriciy, while Figure 4(a) shows he developmen of sreamwise vorices a a spacing of abou 9

10 Fig. 4. Isosurfaces of hree-dimensional voriciy a = 5 for Re = 2. The spanwise voriciy isosurfaces are a.25 of maximum voriciy magniude, and he ransverse and sreamwise voriciy isosurfaces are a.25/4 of maximum voriciy. (a) Fixed cylinder, hree componens. Four sreamwise vorices are eviden, wih a spacing of abou 1.5D. (b) Fixed cylinder, spanwise voriciy only. (c) Moving cylinder, spanwise voriciy only. 1.5D, wice as close as he spacing of 3D seen in simulaions of flow pas an isolaed cylinder (Persillon and Braza, 1998). In fac, he spacing observed here is much closer o he mode B spacing of abou 1D. Finally, Figure 4(b) shows ha, alhough significan sreamwise voriciy has been generaed, he spanwise voriciy is sill largely wo-dimensional excep for some perurbaion a he railing edge of he vorex shees due o ineracion wih he sreamwise vorices. These resuls show ha a Re = 2 he in-phase acousic resonance movemen of he cylinders compleely suppresses he developmen of any hreedimensionaliy in he flow. Since he ampliude of cylinder oscillaion is A/D =.23 >.125, hese resuls are consisen wih he observaions of Toebes (1969) ha sufficienly large ampliude cylinder vibraion ampliude should produce a high degree of spanwise correlaion. We also find ha he igh packing alone does no suppress hree-dimensional insabiliies. 5.2 Reynolds number 1 The drag and lif curves for he Re = 1 simulaions shown in Figure 5 demonsrae ha he hree-dimensional forces are decorrelaed from he wodimensional forces, and are of much lower ampliude for boh he fixed and 1

11 Case Re = 2 Re = 1 2-D, fixed D, fixed D, moving D, moving.95.8 Table 2 Srouhal numbers for each simulaion. The resul for he Re = 1 moving cylinder case is he average of he peaks a.92 and.67. moving cylinder cases. The small ampliude of he force flucuaions in he hree-dimensional simulaion is somewha surprising. I is probably due o he sreamwise vorices decorrelaing he spanwise voriciy, and hence reducing he flucuaing force caused by vorex shedding (see Figure 6). The igh packing of he cylinders may enhance he decorrelaing effec of sreamwise voriciy. However, Figures 5(a,d) indicae ha cylinder movemen does sill increase he spanwise correlaion of spanwise voriciy compared wih he fixed cylinder case. The very large force ampliudes of he wo-dimensional simulaion are clearly unphysical, as is seen from he fac ha he wo-dimensional drag is ofen negaive (see Figures 5(a,c)). The lack of suppression of he hree-dimensional insabiliy is consisen wih he fac ha, as shown in Figure 7, he cylinder vibraion ampliude is only A/D.5. This is less han he minimum hreshold of.125 idenified by Toebes (1969). Somewha surprisingly, as shown in Table 2 and Figure 8, he peak Srouhal frequency of he moving wo-dimensional cylinder is approximaely equal o he average of he wo peak Srouhal frequencies of he moving hree-dimensional cylinder. In conras, he peak Srouhal frequencies of he fixed woand hree-dimensional cylinders differ by.16. Recall ha a Re = 2 he Srouhal numbers of wo- and hree-dimensional cylinders differ by a similar amoun. These resuls show ha, alhough hree-dimensional voriciy is no suppressed by cylinder moion a Re = 1, he average Srouhal numbers of he wo-dimensional and hree-dimensional flows are approximaely he same. This suggess ha cylinder moion decorrelaes non-spanwise voriciy, reducing is effec on Srouhal number. Noe also ha he larges peak Srouhal frequency is he same for he moving and fixed hree-dimensional cylinders. This shows ha cylinder moion adds an addiional shedding mode, bu does no perurb he fixed cylinder shedding mode. Figure 6(a,d) indicaes ha he spanwise voriciy is more srongly correlaed in he moving cylinder case han in he fixed cylinder case. These resuls indicae ha uning he cylinder response o he Srouhal frequency of he fixed cylinder does no produce sufficienly large cylinder vibraions o suppress he hree-dimensional insabiliy. On he oher hand, cylinder moion does reduce he effec of hree-dimensional 11

12 4 4 3 (a) 3 (b) 2 2 C D 1 C D (c) (d) C L C L Fig. 5. Drag and lif a Re = 1 case. In each figure he hree-dimensional curve is he one wih he lower ampliude. (a) Drag, fixed cylinder. (b) Drag, moving cylinder. (c) Lif, fixed cylinder. (d) Lif, moving cylinder. voriciy on he average Srouhal frequency. 6 Conclusions We have used high resoluion penalized pseudo-specral simulaions o invesigae how and when in-phase cylinder vibraion of ube bundles can suppress hree-dimensionaliy. The in-phase cylinder vibraion models he case of acousic resonance, which was observed by Blevins (1985) o induce nearly perfec spanwise correlaion of pressure for an isolaed cylinder. We performed wo- and hree-dimensional simulaions of fixed and free ransverse vibraions of ube bundles a he relaively low Reynolds number Re = 2 (when only he mode A insabiliy is presen), and a he moderae Reynolds number Re = 1 (when he flow around he fixed cylinder is highly hreedimensional and he hree-dimensional insabiliy is more complicaed and much sronger). We found ha a Re = 2 he cylinder vibraion is large enough (A/D = 12

13 (a) (b) (c) (d) (e) (f) Fig. 6. Isosurfaces of hree-dimensional voriciy magniude a = 15 for Re = 1. All voriciy isosurfaces are a.15 of maximum voriciy magniude. (a) Fixed cylinder, spanwise voriciy. (b) Fixed cylinder, sreamwise voriciy. (c) Moving cylinder, ransverse voriciy. (d) Moving cylinder, spanwise voriciy. (e) Moving cylinder, sreamwise voriciy. (f) Moving cylinder, ransverse voriciy..2 (a) A/D (b) V o Fig. 7. Cylinder moion a Re = 1 : 3-D, ; 2-D, (a) Ampliude. (b) Velociy. 13

14 1.8 (a) Fixed Moving 1.8 (b) Fixed Moving Energy.6.4 Energy f f Fig. 8. Lif specra a Re = 1 : (a) wo-dimensional; (b) hree-dimensional..23 >.125) o compleely suppress he hree-dimensional insabiliy. The Srouhal frequency of he moving hree-dimensional cylinder is herefore exacly he same as ha of he moving wo-dimensional cylinder, and no sreamwise or ransverse voriciy is produced. Noe ha he igh packing P/D = 1.5 of he ube bundle is no sufficien by iself o suppress he hree-dimensional insabiliy. In conras, a Re = 1 he cylinder vibraion is insufficien (A/D.5 <.125) o suppress he hree-dimensional insabiliies, alhough he spanwise voriciy is slighly more correlaed. This may be due o he fac ha significan ransverse and sreamwise voriciy (which is uncorrelaed in he spanwise direcion) is generaed before he cylinder vibraion is large enough o suppress he hree-dimensional insabiliy. This resul does no agree wih Blevins (1985) s observaion ha acousic forcing a he Srouhal frequency should correlae he flow in he spanwise direcions, even a 2 Re 4. In his case he igh packing of he ube bundle may acually reduce he cylinder vibraion ampliude compared o he single cylinder case considered by Blevins. Despie he presence of hree-dimensional voriciy, our resuls show ha he average Srouhal frequencies of he wo- and hree-dimensional moving cylinder flows are similar. This is no rue for he fixed cylinder case, and suggess ha cylinder moion decorrelaes non-spanwise voriciy so much ha is effec on Srouhal frequency is negligible. In his sense, he moving cylinder flow a Re = 1 may be described as parly wo-dimensional. This effec is likely o persis for larger Reynolds numbers. By comparing wo- and hree-dimensional simulaions, we showed ha hreedimensional voriciy drasically reduces he ampliude of fluid forces on he cylinder (by as much as 1 imes a Re = 1 ). This reducion in fluid forces produces a corresponding reducion in cylinder vibraion ampliude. I is clear ha he large forces of he wo-dimensional flow a Re = 1 are unphysical; indeed he drag is ofen negaive in he wo-dimensional case. In summary, we have used wo- and hree-dimensional numerical simulaions 14

15 o show ha resonan cylinder moion a Re = 2 compleely suppresses he generaion of hree-dimensional voriciy. A Re = 1 cylinder moion eliminaes he effec of non-spanwise voriciy on he Srouhal number, alhough he generaion of hree-dimensional voriciy is no suppressed. Tigh packing does no suppress he hree-dimensional insabiliy, and may acually reduce cylinder vibraion ampliude. Acknowledgemens The auhors graefully acknowledge he use of sharcne s parallel compuers and an undergraduae sharcne fellowship. NKRK s research is suppored by nserc. We would like o hank J. Simon and N. Tonne for help developing he parallel version of he code, and L. Tuckerman for suggesing he Krylov ime inegraion mehod. References Ango, P., Analysis of singular perurbaions on he brinkman problem for ficiious domain models of viscous flows. Mahemaical Mehods in he Applied Science 22, Ango, P., Bruneau, C.-H., Fabrie, P., A penalizaion mehod o ake ino accoun obsacles in viscous flows. Numerische Mahemaik 81, Bar-Lev, M., Yang, H., Iniial flow field over an impulsively sared circular cylinder. Journal of Fluid Mechanics 72, Blevins, R., The effec of sound on vorex shedding. Journal of Fluid Mechanics 161, Edwards, W. S., Tuckerman, L. S., Friesner, R. A., Sorensen, D. C., Krylov mehods for he incompressible Navier Sokes equaions. Journal of Compuaional Physics 11, Frigo, M., Johnson, S. G., FFTW: an adapive sofware archiecure for he FFT. In: ICASSP conference proceedings. Vol. 3. pp Kevlahan, N., Ghidaglia, J.-M., 21. Compuaion of urbulen flow pas an array of cylinders using a specral mehod wih Brinkman penalizaion. European Journal of Mechanics/B 2, Khadra, K., Ango, P., Parneix, S., Calagirone, J. P., 2. Ficiious domain approach for numerical modelling of Navier Sokes equaions. Inernaional Journal of Numerical Mehods in Fluids 34, Koumousakos, P., Leonard, A., High-resoluion simulaions of he flow around an impulsively sared cylinder using vorex mehods. Journal of Fluid Mechanics 296,

16 Persillon, H., Braza, M., Physical analysis of he ransiion o urbulence in he wake of a circular cylinder by hree-dimensional Navier Sokes simulaion. Journal of Fluid Mechanics 365, Price, S. J., Païdoussis, M. P., Mark, B., Flow visualizaion of he inersiial cross-flow hrough parallel rangular and roaed square arrays of cylinders. Journal of Sound and Vibraion 181, Shiels, D., Leonard, A., Roshko, A., 21. Flow-induced vibraion of a circular cylinder a limiing srucural parameers. Journal of Fluids and Srucures 15, Toebes, G., The unseady flow and wake near an oscillaing cylinder. Transacions of he ASME D: Journal of Basic Engineering 91, Vincen, A., Meneguzzi, M., The spaial srucure and saisical properies of homogeneous urbulence. Journal of Fluid Mechanics 225, 1 2. Weaver, D. S., 21. Privae communicaion. Williamson, C. H. K., Oblique and parallel modes of vorex shedding in he wake of a circular cylinder a low reynolds numbers. Journal of Fluid Mechanics 26,