ON THE INSTABILITY OF A SPRING-MOUNTEDCIRCULAR CYLINDER IN A VISCOUS FLOW AT LOWREYNOLDS NUMBERSCarlo CossuLaboratoire d'Hydrodynamique (LadHyX), CNRS-UMR 7646�Ecole Polytechnique, F-91128 Palaiseau Cedex, FRANCELuigi MorinoDipartimento di Ingegneria Meccanica e Industriale, Universit�a di Roma TreVia della Vasca Navale 79, I-00146 Roma, ITALYOctober 30, 1998Submitted to: Journal of Fluids and StructuresNumber of pages: 35Number of �gures: 8Number of tables: 2Number of manuscript copies: 2

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Running headline:CYLINDER INSTABILITY IN A VISCOUS FLOWCorresponding author:Dr. Carlo CossuLadHyX, �Ecole PolytechniqueF-91128 Palaiseau Cedex, FRANCEPhone: + 33 - 1 - 69 33 36 79Fax: + 33 - 1 - 69 33 30 30E{mail: carlo@ladhyx.polytechnique.fr

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AbstractThe �rst instability of a spring-mounted, damped, rigid circular cylinder immersed in aviscous ow and free to move in a direction orthogonal to the unperturbed ow, is investi-gated by a global stability analysis. The ow is modeled by the full Navier-Stokes equations.For low ratios of the uid density to the structure density, the von Karman mode is alwaysthe critical one and the critical Reynolds number, of about 47, is nearly the same as for a sta-tionary cylinder. In this case, for low structural damping, two complex modes are active andchaos is possible near the bifurcation. For higher density ratios the critical Reynolds numberis lowered to less than the half the critical Reynolds number for a stationary-structure. Insuch a case, only a complex mode is active and chaotic behavior seems not to be possiblenear the bifurcation threshold.

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1 INTRODUCTIONIn 1985 Sreenivasan observed di�erent chaotic transition scenarios in the circular cylinder wakeat low Reynolds numbers. These observations led to a number of studies to understand if alow-dimensional chaotic attractor could explain the turbulent dynamics in open ows such aswakes, mixing and boundary layers. Van Atta and Gharib (1987) recognized the aeroelasticnature of the phenomena observed by Sreenivasan. Since then the uid-dynamics communityconcentrated on the problem of the wake of a stationary circular cylinder and of its dynamics(Williamson, 1996, gives a review on recent advances in this �eld). For this problem it is now wellestablished that the �rst instability is a Hopf bifurcation (Mathis et al., 1984; Sreenivasan et al.,1987; Jackson, 1987) which occurs at a Reynolds number of about 47 and a Strouhal numberslightly less than 0:12. At the onset of this global instability a �nite region of the near wakeis absolutely unstable (Monkewitz, 1988; Huerre & Monkewitz, 1990). Concerning the study ofvortex-induced vibrations of blu�-bodies, the aeroelastic community has been interested mainlyin the high Reynolds number regime, where one usually has to resort to some empirical modelingof the uid-structure interaction (see for instance Blevins, 1991).In this study the �rst instability of the aeroelastic system composed of a rigid circularcylinder immersed in a viscous ow is numerically investigated by a global (in the sense de�nedby Huerre & Monkewitz, 1990) stability analysis. The cylinder is spring-mounted, damped andfree to move in the direction perpendicular to the undisturbed ow. The uid is modeled bythe full two-dimensional Navier-Stokes equations without resorting to any empirical modeling.The main objective of the present analysis is to study the spectrum and the linear modes ofthe coupled system near the global instability threshold. The linear analysis will provide adescription of the \active" modes that nonlinearly interact near the bifurcation and is thereforea necessary step before going to the non-linear analysis which may explain the early observations4

of Sreenivasan (1985) and Van Atta & Gharib (1987).An integro-di�erential vorticity-only formulation, that uses Wu's integral representation ofthe velocity �eld (Wu & Thomson, 1973) and integral constraints on the vorticity �eld (Wu,1976), is used for the Navier-Stokes equations. This approach allows one to limit the numerical-solution domain to the vortical region of the ow. The formulation, coupled with the structuremotion equations, is then discretized to obtain a �nite-dimensional system of ordinary di�erentialequations with quadratic nonlinearities. The state vector of this discretized system is composedof the vorticity values at the grid nodes in the uid domain, a global circulation variable andthe cylinder position and velocity. The parameters of the system are the Reynolds numberbased on the cylinder diameter Re = DU1=�, the uid/solid density ratio n = �f=�c and thestructural natural circular frequency !c and structural damping . We consider the Reynoldsnumber Re as a \purely uid" parameter because it is the only parameter governing the systemwhen the structure is stationary. The structural natural circular frequency !c and damping are the \purely structural" parameters as they rule the system in the absence of uid. The uid/solid density ratio n is the coupling parameter. For every set of parameters considered,the equilibrium solution is found and then its stability is determined by numerically evaluatingthe spectrum of the linearized operator. The algorithm has been validated for the stationarycircular cylinder (Cossu, 1997; Cossu & Morino, 1997).For low density ratios n two signi�cant modes are identi�ed: the \nearly-structural" andthe von Karman one. The nearly-structural mode corresponds to eigenvalues that, in the limitn ! 0, tend to the characteristic (complex) frequency of the structure in the absence of uid.The von Karman mode corresponds to eigenvalues almost identical to the critical eigenvaluesof the \purely uid" system with a stationary-structure. The two modes are described as thestructural natural frequency is changed; the in uence of the density ratio on their stability isalso analyzed.5

The mathematical formulation of the problem, i.e. the de�nition of the structure and uidmodels, is introduced in Section 2. The discretization of the coupled system is brie y discussedin Section 3. The linear stability problem is posed in Section 4 and the numerical results arediscussed in Section 5. The main conclusions are summarized in Section 6.2 MATHEMATICAL FORMULATIONThe rigid cylinder is spring-mounted, damped and immersed in a uniform incompressible viscous ow of velocity U1~i and is free to move in the direction ~j orthogonal to the undisturbed ow.We assume the ow two-dimensional and the uid to be initially at rest. We use a polar referencesystem (r; �) centered in the cylinder center. The lengths are made dimensionless on the cylinderradius R, the velocities using the undisturbed ow velocity U1 and the time using the convectivescale R=U1.2.1 STRUCTURE MODELThe law of motion of the rigid cylinder, mounted on a spring and subject to viscous dampingand to the uid action is d2ycdt2 + dycdt + !2cyc = 12n cL(t); (1)where yc is the position of the center of the cylinder with respect to its equilibrium solutionin the absence of uid, !c is the dimensionless structural natural circular frequency, is thedimensionless structural damping (equal to twice the structural damping factor multiplied bythe structural natural frequency !c, see for instance Blevins 1991), n is the uid/solid densityratio and cL is the lift coe�cient of the cylinder per unit length. The lift coe�cient may beobtained (Patel, 1978) as a function of the vorticity distribution � on the body surface (r = 16

in dimensionless polar coordinated) and its normal gradient @�=@r at the wallcL = 4Re Z 2�0 �� � @�@r�r=1 cos � d�: (2)2.2 FLUID MODELThe uid motion is governed by the continuity equation and the vorticity-transport equation,which is fully equivalent to the Navier-Stokes equations,@�@t + ~v � r� = 1Rer2�: (3)In cylindrical coordinates the vorticity transport equation reads@�@t + v� 1r @�@� + vr @�@r = 1Rer2�: (4)The boundary conditions, enforced on the velocity at the cylinder boundary (r = 1 in dimen-sionless coordinated) are ~v(1; �) = ~vB ; (5)where ~v and ~vB denote the velocity of the uid and of the body boundary. We write theequations in a frame of reference rigidly connected with the structure. In this frame the bodyvelocity ~vB is always zero but the uid velocity at in�nity is seen as ~v1 =~i� vc~j, with vc = _yc(in dimensionless variables U1 = 1).For two-dimensional ows, the formulation used here is a modi�cation of the vorticity-stream7

function method, in which the -� relationshipr2 = �� (6)is inverted to yield an integral representation of the velocity in terms of the vorticity. In contrastto Wu (Wu, 1976; Wu, 1982), in order to invert Eq. (6) and obtain the desired integral form ofthe -� relationship, we use an in�nite-space formulation and extend Eq. (6) to the solid region,with � obtained from the prescribed motion of the solid body (assumed to be incompressible).Thus, inverting Eq. (6) extended to the whole R2, we have (~x) = 1 � ZZR2 G(~x; ~y )�(~y )dS(~y ); (7)where G(~x; ~y ) = 12� ln k~x � ~y k is the fundamental solution of the two-dimensional Laplaceoperator. Equation (7) states that if we know the vorticity in the whole space R2 (i.e., in the uid as well in the solid region), then we also know (and hence the velocity) in the whole spaceR2. Since the cylinder is in pure translation (without rotation) with respect to the undisturbed ow � = 0 in the solid region. From the de�nition of stream-function and Eq. (7) one obtainsan integral representation of the velocity �eld ~v in the ow as a function of the vorticity �eld~v(~x) = ~v1 + ~k � ZZR2 rG(~x; ~y )�(~y )dS(~y ); (8)r denotes the gradient with respect to the variable ~x and ~k = ~i � ~j is the unit vector in thedirection normal to the (x; y) plane. In polar coordinates, in a cylinder-based reference frame,Eq. (8) yields v� = ~e� � ~k � ZZR2 rG � dS � sin � � vc cos �;8

vr = ~er � ~k � ZZR2 rG � dS + cos � � vc sin �: (9)Within the context of the integral formulation used here, the boundary condition on the cylindersurface, Eq. (5), need be imposed only along one direction because only the � scalar �eld has tobe determined on the cylinder boundary. It should be noted that for two-dimensional ows, onemay use either the normal or the tangential boundary condition (Wu, 1976; Wu, 1982), whereasfor three-dimensional ows the normal boundary condition appears more suitable (for a detaileddiscussion the reader is referred to Morino 1986 ). Let us assume, in particular, the direction ofthe normal to the wall ~er, which yields:~er � ~k � ZZR2 rG � dS = � cos � + vc sin �: (10)Enforcing this condition yields a zero-thickness layer of vorticity which immediately di�uses,thereby ensuring that the tangential boundary conditions are satis�ed (Lighthill, 1963; Batche-lor, 1967). All the results presented here are obtained using a projection technique (related to thework of Wang and Wu, 1986 ) which yields the same algorithm for both the tangential and thenormal approach. Finally, as shown by Wu (Wu, 1976), in the case of external two-dimensional ows on a body of �nite extension, the conservation of total vorticity must be enforced to obtaina unique solution when Eq. (10) is \inverted" in order to determine the vorticity distribution onthe cylinder boundary. This condition is obtained by noting that, according to Kelvin's theorem,d�1=dt = 0, where �1(t) = ZC1 ~v � d~x = ZZR2 �(~y; t)dS(~y ): (11)Recalling that we have assumed the uid to be initially at rest, we have �1(0) = 0 and hence�1(t) = 0 for all t. 9

3 DISCRETIZED EQUATIONS AND BOUNDARY-CONDITIONSThe above integro-di�erential formulation is discretized in space in order to obtain a set ofordinary di�erential equations that will be studied by a classical dynamical-system approach.The computational domain extends from the cylinder surface to rmax typically of order of 50cylinder radii. This domain is discretized in polar coordinates with a uniformly spaced grid inthe azimuthal coordinate � and an exponential stretch in the radial coordinate r. The nodeswhere the vorticity is always negligible are not considered in the computation. The state vectorx of the discrete system is composed of the vorticity values in the nodes in the vortical region(in the uid), of the total vorticity �E outside the computational domain and of the circularcylinder position yc and velocity vc.We can eliminate, from the discretized vorticity-transport equation, the velocity [using thediscretized form of Eq. (8)] and the vorticity at the body-boundary nodes [using the discretizedform of Eq. (10)] so as to obtain a vorticity-only formulation. As pointed out above, enforcingthe boundary condition yields a zero-thickness vortex layer which then di�uses into a �nite-thickness one. In the discretized formulation, we combine the two processes and assume thata �nite-thickness layer of vorticity is generated at once. This is accomplished automatically byexpressing the vorticity �eld as: �(~x; t) = NTXn=1 �n(t)fn(~x); (12)where fn(~x) denotes a suitable �nite-element-like basis and �n(t) denotes the value of the vorticityat the node n (imposing the boundary condition determines the layer of vorticity connected withthe values of �n at the boundary nodes). The issue of the boundary condition discretization istreated in some detail in (Cossu, 1997) and (Cossu & Morino, 1997). Once the vorticity valuesat the nodes on the cylinder surface are expressed as a function of the state variables (i.e., of10

the vorticity in the �eld, the external total vorticity and the structure position and velocity) wecan proceed to discretize the vorticity transport equation.The di�erential terms appearing in Eq. (4) are discretized using classical �nite di�erenceformulae fourth-order accurate except near the boundaries where they are second-order accurate.We thus obtain, for the nodes ~xi inside the uidnr2�(~xi)o = MDx+ eD;f@�=@r(~xi)g = MRx+ eR;f(1=r)@�=@�(~xi)g = MTx+ eT : (13)The velocity �eld can be evaluated at each node by discretizing the integrals in Eq. (9), theintegrals being evaluated using BEM-like techniques, so as to obtainfvr(~xi)g = MURx+ eUR;nv�(~xi)o = MUTx+ eUT : (14)Finally, combining (see Cossu 1997, or Cossu and Morino,1997 for details) the discretized dif-ferential operators (13), the discretized integral representation of the vorticity �eld (14), thediscretized version of the equation of vorticity conservation (11) and the structure law of motionEq. (1), with a discretized evaluation of the lift coe�cient, as given by Eq. (2), one obtains thespace-discretized system of equations for the state vector x_x = c+Ax+ b(x;x): (15)11

4 GLOBAL STABILITY ANALYSISWe denote by xS the steady-state solution of Eq. (15), which satis�es the equationc+AxS + b(xS ;xS) = 0; (16)and is calculated by a sequential Newton-Raphson algorithm marching in the Reynolds number.By symmetry considerations it is easily seen that this solution corresponds to a zero displacementand velocity of the cylinder and to a velocity �eld symmetric with respect to the axis centeredin the cylinder center and parallel to the undisturbed ow. The vorticity �eld associated to thesteady state solution xS is the same as in the case of a stationary cylinder so that xS is a functionof the Reynolds number but not of the other parameters (!c, and n). Setting x = xS+xP andrecalling that xS satis�es Eq. (16), one obtains_xP = APxP + b(xP ;xP ); (17)withAPxP := AxP+b(xS ;xP )+b(xP ;xS). The matrixAP is the linearized discretized Navier-Stokes operator that is recovered when terms of order higher than the �rst in the perturbationvector xP are neglected. The eigenvalues of AP determine the linear (global) stability of thesystem and its eigenvectors are the global modes. AP , and thus its eigenvalues and eigenvectors,depend upon all the parameters of the problem (Re, !c, , n). The full spectrum of the \purely uid" (stationary structure) system has been studied in some detail (Cossu, 1997) and con�rmsthe well known result that two complex conjugate eigenvalues cross the imaginary axis at aReynolds number of about 47 with an imaginary part of about !0 = 0:37 corresponding toa Strouhal number St = !0=� = 0:117. When the cylinder is free to move in the transversedirection, the system has two more degrees of freedom (the cylinder position yc and velocity vc)12

and then displays two additional eigenvalues with respect to the stationary-structure case. Inthe absence of uid, the system admits only the two structural eigenvalues given by�s = � 2 � is!2c � � 2�2: (18)In the presence of uid, but for n � 1, two \nearly-structural" eigenvalues with � � �s areexpected. The corresponding eigenvectors are structure-driven vorticity �elds in the wake. Assoon as one increases the uid/solid density ratio n, the aerodynamic feedback on the struc-ture becomes more important and the whole spectrum is changed by this interaction. Someinteresting questions arise(a). Can the nearly-structural mode become critical as the density ratio n is increased?(b). Does the critical Reynolds number increase or decrease by increasing the density ratio n?(c). How does the shape of the nearly-structural mode change if we change the frequency ratio0 = !c=!0 at a given Reynolds number?(d). Is chaotic behavior possible for the coupled system near the bifurcation threshold?In the following we try to partially answer these questions by numerically computing the eigen-values and eigenvectors of the discrete linearized operatorAP for some suitable set of parameters.The number of parameters sets considered is limited because each test requires signi�cant com-putational resources, however we think that some insight in the problem can be given from theresults discussed below.5 NUMERICAL RESULTSThe computational domain extends from the cylinder surface to 50 cylinder radii in the radialdirection and has been discretized with 96 intervals in the azimuthal coordinate and with 4813

intervals in the radial direction. As stated above, the grid nodes where the vorticity is alwaysnegligible are not considered in the computation and thus the results were obtained with 2763nodes instead of 4512. The grid is shown in Fig. 1. When the structure is stationary, a criticalReynolds number of 47:0244 and a critical frequency of 0:369 (corresponding to a critical Strouhalnumber equal to 0:1174) are found. The eigenvalues and eigenvectors of AP are numericallycomputed, with a QR algorithm, using the LAPACK routines (Anderson et. al., 1992). Thespectrum at the Hopf bifurcation for the stationary-structure case is shown in Fig. 2: the criticalvon Karman eigenvalues are on the imaginary axis; the associated vorticity �elds are shown inFig. 3. The von Karman mode shape has been extensively analyzed (Jackson, 1987; Noack &Eckelmann, 1994; Cossu, 1997) and the reader is referred to these works for further details.5.1 INFLUENCE OF THE FREQUENCY RATIOWe consider how the the frequency ratio 0 a�ects the spectrum of the coupled system. Theparameter 0 is the ratio of the !c natural circular frequency of the undamped ( = 0) structuraloscillator in the absence of uid to the circular frequency !0 of the undamped purely uid\oscillator", i.e., the frequency of the critical mode at the Hopf bifurcation when the structureis not allowed to move. We analyzed the e�ect of 0 on the spectrum for a low density ratio n =1=7000 (approximatively steel in air) and a small structural damping, = 0:01, so as to avoid thecoalescence of the nearly-structural eigenvalues with the von Karman eigenvalues. The Reynoldsnumber was kept �xed at its critical value in the stationary-structure case (Rec = 47:024), wherethe critical eigenvalues were �i!0 = �0:368865 i. In Table 1 we report the numerical resultsfor the two least stable pairs of eigenvalues �1;2 and �3;4. In the same table the \structural" �seigenvalues, given by Eq. (18), that would have been observed in the absence of uid, are alsoreported. Three frequency ratios were considered: the unitary ratio, the ratio = 1:8 (used bySchumm et. al., 1994, to control the von Karman instability by forced transverse oscillation) and14

its inverse = 0:55. Both the nearly-structural and the von Karman eigenvalues are insensitiveto the aeroelastic coupling and, for the set of parameters considered, the von Karman mode isalways the critical one.When 0 = 1 the shape of the nearly-structural mode almost coincides with the von Karmanone (shown in Fig. 3) except for a phase shift. This is con�rmed by the analysis of Fig. 4 wheretheir vorticity normalized values (the shift has been set equal to zero in order to compare thetwo modes) on the downstream symmetry axis are shown. If we were able to \suppress" the vonKarman mode and let the cylinder free to oscillate in the transverse direction at the stationary-cylinder critical Reynolds number and the corresponding characteristic Strouhal frequency, wewould observe, for small oscillations, the same disturbance vorticity �eld in the wake that for astationary cylinder. It would be interesting to study the nonlinear behavior of the system for0 = 1 because two nearly identical complex modes interact non-linearly with nearly identicalcharacteristic frequencies, so that very strong resonances are expected.In Figs. 5 and 6 the vorticity �elds associated with the real and imaginary parts of the\nearly-structural" modes are shown for the values 0 = 0:55 and 0 = 1:8. Since the Reynoldsnumber is constant the mean advection velocity does not change; therefore if the oscillationfrequency is increased it can be argued that the streamwise wavenumbers will also increase.This is con�rmed by the analysis of the modes shown in Figs. 5 and 6. In the numericalsimulation one has to be careful of this wavenumber change. For too large 0, the wavelengthcan become too small for the resolution of the chosen grid. For low 0, the wavelength canbe too large compared to the numerical solution domain. In Figs. 3, 5 and 6 it can also beobserved that, as the structural frequency is increased, the disturbance vorticity has a tendencyto concentrate upstream. This mode deformation is similar to the one observed as the Reynoldsnumber is increased in stationary blu�-body wakes (Goujon-Durand et al., 1994); indeed, inthe post-critical behavior following the �rst Hopf bifurcation, the Strouhal frequency of the15

stationary cylinder wake increases with the Reynolds number (Williamson, 1996). Therefore,one can conjecture that the von Karman mode deformation, observed when the Reynolds numberis increased, in the classical stationary-structure case, is basically due to the increasing intrinsicfrequency of the oscillator: at a given Reynolds number the deformed mode is thought to besimilar to the one induced by forced transverse oscillation at a frequency corresponding tocharacteristic Strouhal number at that Reynolds number.5.2 INFLUENCE OF THE DENSITY RATIOThe results discussed in Section 5.1 were obtained with a very small density ratio (n = 1=7000)and in that case the structure and the uid were almost uncoupled, i.e. the spectrum of thecoupled system was almost the \sum" of the spectra of the \purely uid" (stationary-structure)system and the \purely structural" (no uid) one. We now concentrate on the e�ects of achange in the density ratio n. When n is increased, the aerodynamic forces become of the sameorder of magnitude of the structural ones and some changes are expected in the spectrum. Weconsidered a frequency ratio 0 = 1:8 and a structural damping = 0:01. For these parametersthe \purely structural" eigenvalues, given by Eq. (18), are �s = �0:5 10�3 � 0:663 i. Four setsof parameters were considered and the results are reported in Table 2. In the �rst two tests(�rst two rows in the table) the Reynolds number was kept at the stationary-structure criticalvalue (Re = 47:0244). In this case, an increase of the density ratio n from 1=7000 to 1=700 didnot substantially a�ect the spectrum as can be seen from Fig. 7, where the spectra of these twocases are reported. From the �rst two rows of Table 2 it is also seen that the von Karmaneigenvalues remain the leading ones (�1;2). The nearly-structural eigenvalues �3;4 slightly shiftto the right in the complex plane as n is increased from 1=7000 to 1=700. Even if they are notstrictly unstable they are almost critical and well separated from the remaining stable part ofthe spectrum so that the corresponding complex mode can also can be considered \active". In16

that case the system admits four real degrees of freedom and chaotic behavior is possible nearthe global bifurcation.In the second two tests (third and fourth rows in Table 2) the Reynolds number was kept�xed at half the critical stationary-structure value, i.e., Re = 23:512. Two density ratios wereconsidered: n = 1=70 and n = 1=7, which approximatively corresponds to steel in water. Asit can be seen from Fig. 8, where the spectra of these two cases are reported, for these sets ofparameters the von Karman mode is stable while the former nearly-structural one is unstable.The stable part of the spectrum does not seem to be a�ected by the change in n. The criticalReynolds number, for n > 1=70, is less than half the one of the stationary-structure case, howeverin that range only a (complex) mode is unstable and chaotic behavior seems not possible nearthe bifurcation threshold. For n = 1=70 this mode seems to be nearly-structural one because ithas almost the same imaginary part. A further increase in n, from 1=70 to 1=7, leads to a strongincrease of the growth rate of the unstable mode and to a decrease of its oscillation frequency.6 CONCLUSIONSThe �rst instability of a spring-mounted, damped rigid circular cylinder in a viscous ow hasbeen numerically investigated without resorting to any semi-empirical modeling. An integro-di�erential vorticity-only formulation has been adopted for the full Navier-Stokes equations,used to model the ow around the moving structure.Two signi�cant modes are identi�ed: the \nearly-structural" one and the von Karmanone. The nearly-structural mode corresponds to eigenvalues which, in the limit of very small uid/solid density ratios n, tend to the characteristic (complex) frequency of the structure in theabsence of uid. The von Karman mode corresponds to a pair of eigenvalues whose frequenciesare almost identical to the leading eigenvalues of the \purely uid" system with a stationary-17

structure near bifurcation. These two modes are well de�ned only for low ratios of the uiddensity to the structure density.We �rst analyzed the e�ect of a change in the frequency ratio 0, for a low density ratio anda Reynolds number equal to the stationary-structure critical value. Both the nearly-structuraland the von Karman eigenvalues have been seen to be quite insensitive to the aeroelastic couplingat the considered very small n = 1=7000, i.e. for steel in air. When 0 = 1 the nearly-structuralcomplex mode almost coincide with the von Karman one except for a phase shift. An increase of0, with a constant Re, produces a deformation of the nearly-structural mode which is similarto the one observed when the Reynolds number is increased in the stationary-structure case.The e�ect of a change in density ratio n was also considered for a �xed 0. An increase ofthe density ratio n from 1=7000 to 1=700 did not substantially a�ect the spectrum. The vonKarman mode remained the critical one while the nearly-structural mode is almost critical butnot strictly unstable. For higher n the situation greatly changes. The critical Reynolds numberfor n = 1=70 is less than the half the one of the stationary-structure case. A further increasein n leads to a strong increase of the growth rate of the unstable mode and to a decrease of itsfrequency. For n > 1=70, for the considered sets of parameters, just a complex mode is unstableand no chaotic behavior seems possible near the bifurcation, while for n < 1=700 and lowstructural damping two complex modes are critical; in that case the system admits four activereal degrees of freedom and chaotic behavior is possible near the bifurcation (Guckenheimer& Holmes, 1986). In future work the nonlinear interaction of the von Karman mode and thenearly-structural one will be studied in a weakly nonlinear framework.

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ACKNOWLEDGMENTSThe authors gratefully acknowledge the computational facilities provided by CINECA Super-computing Center (Casalecchio di Reno, Italy) under a Vector and Parallel Computation Grant.The �rst author acknowledges fruitful discussions with E. de Langre and P. Mannevile, theItalian Ministry of University, Scienti�c and Technologic Research for �nancial support and theAerospace Department of the University of Rome \La Sapienza", where part of the researchwas developed during his Ph.D. Program. A preliminary version this paper was presented atthe CEAS International Forum on Aeroelasticity and Structural Dynamics (Rome, Italy, 17-20June 1997).

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ReferencesAnderson, E., Bai, Z., Bishof, C., Demmel, J., Dongarra, J., Du Croz, J., Green-baum, A., Hammarling, S., Mc Kenney, A., Ostuchov, S., & Sorensen, D. 1992LAPACK Users' Guide. Philadelphia: SIAM.Van Atta, C. W. & Gharib, M. 1987 Ordered and chaotic vortex streets behind circularcylinders at low Reynolds number. Journal of Fluid Mechanics 174, 113{133.Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge, UK: CambridgeUniversity Press.Blevins, R. D. 1991 Flow-Induced Vibrations. New-York: Van Nostrand Reinhold.Cossu, C. 1997 Linear and nonlinear stability analysis of a viscous ow around a circularcylinder (in Italian). Ph.D. Thesis in Aerospace Engineering, Universit�a La Sapienza, Roma,Italy.Cossu, C. & Morino, L. 1997 A vorticity-only formulation and a low-order asymptotic ex-pansion near Hopf bifurcation. Computational Mechanics 20, 229{241.Goujon-Durand, S., Jenffer, P. & Wesfreid, J. E. 1994 Downstream evolution of theB�enard-von K�arm�an instability. Physical Review E 30, 308{313.Guckenheimer, J. & Holmes, P. 1986 Nonlinear Oscillations, Dynamical Systems and Bi-furcations of Vector Fields. New York: Springer.Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing ows. Annual Review of Fluid Mechanics 22, 473{537.Jackson, C. P. 1987 A �nite-element study of the onset of vortex shedding in ow past variouslyshaped bodies. Journal of Fluid Mechanics 182, 23{45.20

Lighthill, M. J. 1963 Introduction to boundary-layer theory. In Laminar Boundary Layers(Ed. L. Rosenhead), pp. 46{113. Oxford, UK: Oxford University Press.Mathis, C., Provansal, M. & Boyer, L. 1984 The B�enard-von K�arm�an instability: anexperimental study near the threshold. Journal de Physique, Lettres 45, 483{491.Monkewitz, P. A. 1988 The absolute and convective nature of instability in two-dimensionalwakes at low Reynolds numbers. Physics of Fluids 31, 999{1006.Morino, L. 1986 Helmholtz decomposition revisited: Vorticity generation and trailing edgecondition, part 1, incompressible ows. Computational Mechanics 65{90.Noack, B. R. & Eckelmann, H. 1994 A global stability analysis of the steady and periodiccylinder wake. Journal of Fluid Mechanics 270, 297{330.Patel, V. A. 1978 K�arm�an vortex street behind a circular cylinder by the series truncationmethod. Journal of Computational Physics 28, 14{42.Schumm, M., Berger, E. & Monkewitz, P. A. 1994 Self-excited oscillations in the wake oftwo-dimensional blu� bodies and their control. Journal of Fluid Mechanics 271, 17{53.Sreenivasan, K. R. 1985 Transition to turbulence in uid ows and low-dimensional chaos.In Frontiers in Fluid Mechanics (Eds. S. H. Davis & J. L. Lumley). New York: Springer.Sreenivasan, K. R., Strykowski, P. J. & Olinger, D. J. 1987 Hopf bifurcation, Landauequation and vortex shedding behind circular cylinders. In Proceedings of the Forum onUnsteady Flow Separation (Ed. K. N. Ghia), volume 52, pp. 1{13, New York. AmericanSociety of Mechanical Engineers.Wang, C. M. & Wu, J. C. 1986 Numerical solutions of Navier-Stokes problems using integralrepresentation with series expansion. AIAA Journal 24, 1305{1312.21

Williamson, C. H. K. 1996 Vortex dynamics in the cylinder wake. Annual Review of FluidMechanics 28, 477{539.Wu, J. C., 1976 Numerical boundary conditions for viscous ow problems. AIAA Journal 14,1042{1049.Wu, J. C. 1982 Problems of general viscous ows. In Developments in Boundary ElementMethods (Eds. R. P. Shaw & P. K. Banerjee). London: Elsevier Applied Sc. Publishers.Wu, J. C. & Thomson, J. F. 1973 Numerical solution of time dependent incompressibleNavier-Stokes equations using an integro-di�erential formulation. Computer and Fluids 1,197{215.

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APPENDIX: NOMENCLATUREA linear part of the discretized Navier-Stokes operatorAP linearized discretized Navier-Stokes operatorb bilinear part of the discretized Navier-Stokes operatorc constant part of the discretized Navier-Stokes operatorcL lift coe�cient, L=(�fU21=2)D cylinder diameter, 2R~er unit vector in the radial direction~e� unit vector in the azimuthal directionG Green's function for the laplacian operatori imaginary unit, p�1~i unit vector parallel to the undisturbed ow~j unit vector orthogonal to the undisturbed ow in the plane of the ow~k unit vector orthogonal to the plane of the own uid/solid density ratio, �f=�cr radial coordinateR cylinder radiusRe Reynolds number, U1D=�St Strouhal number, !=�vc vertical velocity of the cylinder, _ycvr radial velocity component, ~er � ~vv� azimuthal velocity component, ~e� � ~vU1 magnitude of the freestream velocity in the inertial reference frame

23

~v velocity �eld in the uid~v1 velocity �eld of the undisturbed ow~vB velocity of the structurex state vector of the discretized systemxP perturbation from the steady state of the discretized systemxS steady state solution of the discretized systemyc position of the cylinder center in the inertial reference frame structural damping�1 global circulation�E circulation over the domain external to the computational domain� eigenvalue of the linearized aeroelastic operator�1;2 leading eigenpair�3;4 second leading eigenpair (excepted �1;2)�s structural eigenvalue in the absence of uid� kinematic viscosity of the uid!0 circular frequency of the von Karman mode at the bifurcation with a stationary-structure!c structural natural circular frequency0 frequency ratio, !c=!0 stream function�c density of the cylindrical structure�f density of the uid� azimuthal coordinate� vorticity �eld24

TABLES0 �1;2 �3;4 �s0:55 �5:398 10�5 � 0:368 i �4:077 10�3 � 0:204 i �5:0 10�3 � 0:204 i1:00 3:640 10�4 � 0:367 i �4:597 10�3 � 0:369 i �5:0 10�3 � 0:368 i1:80 1:732 10�5 � 0:368 i �4:555 10�3 � 0:663 i �5:0 10�3 � 0:663 iTable 1: E�ect of the frequency ratio 0 on the four leading eigenvalues �1;2, �3;4 of the coupledsystem for Re = 47:024, = 0:01 and n = 1=7000. The purely structural eigenvalues, given byEq. (18), are also reported for comparison. The stationary-structure von Karman eigenvalues,at the same Reynolds number, are �0:368865 i.

n Re �1;2 �3;4 �s1/7000 47.024 1:732 10�5 � 0:368 i �4:555 10�3 � 0:663 i �5:0 10�3 � 0:663 i1/700 47.024 1:705 10�4 � 0:368 i �5:277 10�4 � 0:663 i �5:0 10�3 � 0:663 i1/70 23.512 3:803 10�2 � 0:660 i �4:385 10�2 � 0:302 i �5:0 10�3 � 0:663 i1/7 23.512 6:855 10�1 � 0:097 i �4:360 10�2 � 0:304 i �5:0 10�3 � 0:663 iTable 2: E�ect of the density ratio n on the four leading eigenvalues �1;2, �3;4 of the coupledsystem for 0 = 1:8 and = 0:01. In the two upper rows the Reynolds number is choosed at itscritical value in the stationary-cylinder case while in the third and fourth row it is taken equalto half that value. The purely structural eigenvalues �s, given by Eq. (18), are also reported forcomparison

25

FIGURESFigure 1: Grid used to discretize the Navier-Stokes equations. The grid nodes where the vorticityis negligible are not included in the computational domain.Figure 2: Spectrum of the linearized Navier-Stokes operator at the Hopf bifurcation (Re =47:024), for the stationary-structure.Figure 3: Real and imaginary part of the von Karman mode at the Hopf bifurcation (Re =47:024) for the stationary-structure.Figure 4: Shape of the vorticity modes on the symmetry axis with density ratio n = 1=7000 andfrequency ratio 0 = 1. a) real part of von Karman mode, b) imaginary part of von Karmanmode, c) real part of the structural mode, d) imaginary part of the structural mode.Figure 5: Real and imaginary part of the disturbance vorticity �eld associated to the nearly-structural mode for 0 = 0:55, = 0:01, n = 1=7000 and Re = 47:024.Figure 6: Real and imaginary part of the disturbance vorticity �eld associated to the nearly-structural mode for 0 = 1:8, = 0:01, n = 1=7000 and Re = 47:024.

26

Figure 7: Spectra of the coupled aeroelastic system corresponding to the �rst two rows of Table 2:two density ratios n = 1=7000 and n = 1=700 are considered for Re = 47:024, 0 = 1:8 and = 0:01.Figure 8: Spectra of the coupled aeroelastic system corresponding to the third and fourth rowin Table 2: two density ratios n = 1=70 and n = 1=7 are considered for Re = 23:512, 0 = 1:8and = 0:01.

27

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35