# Militzer J., Bell T., Ham F. Simulations of Vortex-Induced Vibrations on Long Cylinders 2003

код для вставкиСкачатьSimulations of vortex-induced vibrations on long cylinders with one and two degrees of freedom Julio Militzer 1 , Theo Bell 2 and Frank Ham 3 1,2 Department of Mechanical Engineering, Dalhousie University P.O.Box 1000, Halifax, Nova Scotia, B3J 2X4 Tel. (902) 494 3947, e-mail: Julio.Militzer@dal.ca 3 Center for Turbulence Research, Building 500, Stanford University, Stanford, CA 94305-3035 Abstract Numerical simulations of the vortex-induced vibrations of long cylinders are presented. The simulations use a novel Cartesian grid method with local flow-based anisotropic adaptation. The immersed boundary method (IMBM) is used to model the stationary or moving cylinder boundaries. IMBM does not require the grid lines to correspond with the boundary, thus significantly simplifying the mesh generation process, particularly when flow-based adaptation is combined with moving boundaries, as in the present contribution. The flow was assumed to be two dimensional and Reynolds number was set to 8640. Three configurations were considered: the flow around a stationary cylinder (0DoF), the flow around a cylinder free to move in the cross stream direction (1DoF) and the flow around a cylinder free to move in any direction in the plane of flow (2DoF). Four different mass damping factors were also considered for each of the moving cylinder cases. The results show a strong influence of the number of degrees of freedom on the vortex shedding pattern, the lift and drag coefficients and the oscillation amplitudes. 1. Introduction Long cylindrical risers are required for deep water exploration or production of petroleum or natural gas. Depths can exceed 2,000 m. The flow of seawater around these long cylinders is subject to vortex shedding. This is an unsteady oscillatory phenomenon, which causes the pressure distribution around the cylinders to fluctuate. The vortex shedding frequency sometimes is equal to one of the natural frequency modes of the cylinder, which may cause the cylinder to vibrate, with what is known as Vortex Induced Vibrations (VIV). This phenomenon is also known as lock-in, which means that the vortex shedding frequency is "locked" into the riser natural frequency. Metal fatigue is among the undesirable consequence of VIV. Thus, it is very important to have tools to predict these vortex shedding induced vibrations and the dynamic behaviour of the risers under such forcing functions. The prediction and simulation of this flow has attracted the interest of both experimental and numerical researchers. Both approaches have their limitations. In experimental investigations, for example, it is not possible to model large aspect ratio risers, namely long cylinders with relatively small diameters. Another difficulty is to accurately simulate the attachment of the cylinders ends and their elastic behaviour. Numerical simulations, on the other hand, suffer from limitations such as relatively low maximum Reynolds number and difficulties in simulating flows with moving boundaries. Most researchers report results for fixed cylinders or cylinders with a single degree of freedom. Norberg (2001) studied the effect of the Reynolds number variation (47 to 2.2x10 5 ) for the flow around a fixed circular cylinder. He concluded that for the range 5x10 3 < Re < 8x10 3 there is a change-over from a “high quality” von Karman vortex shedding mode to “low quality” shedding mode displaying significant span-wise undulations and occasional vortex dislocations. Breuer (1998) studied the flow past a circular cylinder with Re =3900. He used a LES turbulence model and several discretization schemes. He concluded the best predictions were produced by three dimensional simulations with a Central Differencing Scheme. He used a body fitted curvilinear system of coordinates, but the cylinder was kept fixed. Many experimental investigations considered cylinders with a single degree of freedom in the cross stream direction, for example Khalak and Williamson (1999). They observed that the maximum amplitude of oscillation increased for decreasing damping. They have reached a maximum value of y/D =1.2 for a mass damping factor (m * ) of 0.006, furthermore they predicted that higher amplitudes should be observed in numerical simulations where the mass damping can be set to zero. Evangelinos et al. (2000) carried out a one degree of freedom DNS of a circular cylinder with Re=1000. They considered free and fixed long and short beam configurations. The cylinder was assumed to vibrate at its natural frequency. The root-mean square (RMS) lift coefficient was found to be about 0.8; the average drag coefficient was about 1.8; while the cross-stream non- dimensional displacement varied between 0.9 and 1.1. Blevins (1990) presents a summary of the findings of several researchers on the effect of transverse cylinder vibrations, with frequencies at or near the vortex shedding frequency. Moving cylinders, when compared with stationary cylinders, show an increase in the strength of the vortices and in the span wise correlation of the wake. Also more importantly, there is an alteration in the phase, sequence and pattern of the vortices in the wake. Moe et al. (1994) carried out an experiment investigation for a cylinder with two degrees of freedom. They observed that the ratio between the stream-wise and cross-stream vibration frequencies is around two. Furthermore, they observed that the cross-stream vibration amplitude is a function of the frequency ratio, reaching a dimensionless amplitude of 1.2 for a frequency ratio of 1.2, while for a frequency ratio of 2, the dimensionless amplitude is 0.9. They also noted that the stream-wise versus cross-stream displacement plot generated an “eight” or “crescent” shaped plot. Jeon and Gharib (2001) also performed an experimental investigation of the motion of a cylinder with two degrees of freedom. They noted that the addition of a stream-wise component changed the vortex shedding pattern from two pairs (2P) per cycle (Williamson & Roshko, 1988) to a single vortex (S) per cycle. Furthermore, they observed that the addition of the stream-wise component affected the relative phase between the force and the body motion, thus affecting even the sign of the energy transfer between fluid and cylinder. Meling et al. (1998) carried a numerical simulation of the flow around a circular cylinder with two degrees of freedom. The method they employed is a two-dimensional arbitrary Lagrangian-Eulerian method with a Spalart-Almaras one equation turbulence model. Their flow Reynolds number varied between 7,000 and 40,000. For a stream-wise cross- stream frequency ratio value of 2, they obtained a dimensionless cross-stream displacement of 0.97, which is within 10% of the result obtained by Moe et al. (1994), however the dimensionless stream-wise displacement they found was 0.12, which does not compare well with Moe et al.’s (1994) result of 0.3. Zhou et al. (1999) used a discrete vortex method with vortex in cell technique to simulate the flow around a vibrating circular cylinder. Even though they considered a relatively low Reynolds number of 200, they observed that there is only a qualitative agreement between the one and two degrees of freedom simulations, namely that the stream-wise oscillations have a substantial effect on the cross-stream vibrations. They observed that in an actual fluid-structure interaction, both drag and lift act on the cylinder; thus it is incorrect to assume that only the transverse forces control the y displacement, in particular at large oscillations where the non-linear effects become significant. Furthermore, they observed that fluid damping is responsible for a limit-cycle oscillation behaviour, even when the natural frequency is close to the shedding frequency. So et al. (2000) confirmed that structural vibrations affect the vortex shedding process, which in turn influences the unsteady flow-induced forces. The modified forces affect the motion of the structure and so on. This clearly indicates that both the vortex shedding and the corresponding forces should be quite different depending whether the cylinder is allowed to move with zero, one or two degrees of freedom. They also observed that many experiments were carried out by introducing vibrations generated by externally imposed forces (Williamson & Roshko, 1988). As a result the only important frequencies were the natural frequency of the structure and the frequency of the imposed vibrations. In actual structures they claim that synchronization occurs when the vortex shedding frequency is equal to the combined fluid-structure natural frequency. They also observed that provided the cylinder is not near resonance the vortex shedding is not affected by the natural frequency of the structure. Wilde et al. (2001) studied experimentally the vibrations of rigid and freely vibrating cylinders. They covered the Reynolds number range from 2x10 4 to 3.7x10 5 . The drag coefficient for the stationary cylinder was found to be 1.04, which is lower than the usual value of 1.16. This was attributed to the cylinder surface roughness. For the case of the moving cylinder they found a significant difference between the amplitudes of cross- stream oscillations depending on whether the cylinder was towed forward or backward, namely 1.03 and 0.68 respectively. The difference was attributed to the stabilizing/de- stabilizing nature of the stream-wise drag. Furthermore, for a reduced velocity of 5.5, they obtained a maximum drag coefficient of 1.3, which is smaller than the typical values (larger than 3) reported by most authors. Mittal et al. (1999) carried out a finite element simulation of a cylinder with two degrees of freedom for Reynolds number equal to 325. They concluded that the trajectory of the cylinders corresponds to a Lissajou figure eight. They also noted that for a certain range of structural frequencies the vortex shedding frequency of the oscillating cylinder does not match the natural frequencies, this phenomenon is called “soft-lock-in”. This phenomenon disappears when the mass of the cylinder is significantly greater than the fluid it displaces. In the present investigation, we chose to determine the influence of the number of the degrees of freedom, that is, zero (0DoF), one (1DoF) and two (2DoF), on the cross- stream and stream-wise amplitudes, the lift and drag coefficients, and the vortex shedding patterns. For a Reynolds number of 8,640, a series of nine different runs were performed: a fixed or rigid cylinder (0DoF) and moving cylinder configurations with 1DoF and 2DoF, each with four different mass damping ( factors. The mass damping factors considered were: = 0.0; 0.010; 0.037 and 0.065. One of the major difficulties encountered in the simulation of this flow is the fact that the cylinder moves. This requires a code capable of simulating moving boundaries. Algorithms capable of simulating such flows tend to be rather complicated, since for most of them the numerical mesh must be recalculated every time the cylinder moves, which can be quite onerous in terms of computer time. In this contribution we present a simulation tool called the “Numerical Wind Tunnel”. The algorithm incorporates a series of innovative Computational Fluid Dynamics techniques, such as an adaptive unstructured Cartesian grid and the Immersed Boundary Method (IMBM) for boundary conditions specification. These features allow the NWT scheme to simulate flows with moving boundaries with relative ease. 2. Solution Method 2.1 Introduction The flow is assumed to be incompressible and two dimensional with moving boundaries. The cylinder is held in place until the vortex shedding regime is established, which takes approximately 1200 time steps. After which, the cylinder is released, in either the cross- stream direction (1DoF) or in both stream-wise and cross-stream directions (2DoF). At every subsequent time step, knowing the pressure distribution around the cylinder, we calculate the lift and drag coefficients. The lift and drag coefficient values are introduced into a simple harmonic model, applied to a spring mounted, damped, rigid cylinder (Blevins, 1990). The solution of the ODE(s) gives a displacement and velocity in either the cross-stream (1DoF) or the cross-stream and stream-wise directions (2DoF). The velocities, used for the IMBM, are applied to the cylinder directly. The displacements are used to move the cylinder in the flow calculation domain before carrying out the next time step calculation. This process is repeated until the calculation is stopped after a sufficient number of cycles have been considered. 2.2 The harmonic model The harmonic motion of the cylinder is governed by the following ODE(s): M tC X Ud dX Ud Xd LorD i r i r i 2 24 2 2 2 (1) where X i = x i /D is the dimensionless distance, = tV 0 /D is the dimensionless time, m is the cylinder mass per unit length (including the added mass, see definition below), M = m/ D 2 is the dimensionless mass per unit length, U r = V 0 / f n D is the reduced velocity, f n is the natural frequency and is the damping factor. The reduced velocity is the inverse of the Strouhal number. The total mass, m, is given by: 2222 444 22 oOHiOHiosteel DDDDm (2) where steel is the density of steel, H 2 O is the density of sea water and D 0 and D i are respectively the external and internal diameters of the cylinder. The last term in equation 2 represents the added mass. For our simulations, we considered a steel cylinder with an external diameter of 0.250 m and an internal diameter of 0.235 m. We included the added mass, which increases the mass per unit length of the cylinder from 88.3 kg/m to 138.5 kg/m. Assuming that the sea water density, H 2 O, is 1025 kg/m 3 , we obtain a value of 2.16 for the dimensionless mass, M. Since the cylinder is subject to a large tensile force, it behaves as a vibrating string and hence a string model can be used to calculate the natural frequencies (Blevins, 1990). Assuming a string model, the i th natural frequency of the cylinder can be calculated using the following formula: m T L i f i 2 (3) where L is the length of the cylinder and T is the tension applied. In our study, the cylinder is 2,000 m long and subject to a tension of 10 6 N. Although the cylinder has an infinite number of natural frequencies, we have chosen to use the second natural mode frequency which is equal to 0.0473 Hz. The lift force on the cylinder is given by: 2 0 sin dAptF L (4) where p is the pressure on the cylinder surface and dA is an area element. The lift coefficient is then given by: AV tF tC L L 2 0 2 (5) The drag force and respective drag coefficient are calculated analogously. 2.3 Flow Simulation with the Numerical Wind Tunnel The Numerical Wind Tunnel is the name given to the code used here to solve the unsteady Navier-Stokes equations for a two-dimensional, incompressible flow with moving boundaries. Figure 1 presents the flow geometry, the calculation domain and the boundary conditions considered. The following are the main characteristics of the code: boundary conditions enforced with the Immersed Boundary Method (IMBM); anisotropic Cartesian adaptive mesh; time advancement by the fractional step method; and large eddy simulation (LES) of turbulence. The general form of the non-dimensional Navier-Stokes equations is: U X P UU U ti i i 2 Re 1 (6) where U i = u i / V 0 are the non-dimensional Cartesian velocity components, P= p/ V 0 2 is the non-dimensional pressure , X i = x i / D are the non-dimensional Cartesian coordinates and Re t = V 0 D/ t is the turbulent Reynolds number. The continuity equation is given by: 0 i i X U (7) The turbulent viscosity is obtained by using a simple eddy viscosity model. Although, an eddy viscosity model with Reynolds Averaged Navier-Stokes (RANS) equations provides good predictions for certain types of flows such as: pipe or channel flows and flat plate boundary layers, it fails to represent accurately flows where large scale unsteadiness is significant – such as the flow over bluff bodies which involves unsteady separation and vortex shedding. For these flows, large eddy simulation (LES), which resolves the large-scale unsteady motion explicitly, is a better option. Conceptually, LES is situated between Direct Numerical Simulation (DNS) and the RANS approach. Thus, in LES the dynamics of large scale (energy containing) motions (which are affected by the flow geometry and are not universal) are computed explicitly whereas the influence of smaller scales (energy dissipating) are represented by simple models such as eddy viscosity. The governing equations employed for LES are obtained by filtering the time-dependent Navier-Stokes equations in either Fourier (wave-number) space or physical space. The filtering process effectively filters out the eddies whose scales are smaller than the filter width, , used in the computations. The resulting equations thus govern the dynamics of large eddies. The small eddies (<), on the other hand, require a turbulence model, called sub-grid model. The most basic sub-grid scale model was proposed by Smagorinsky (1963). The main characteristics of this model are summarized below. The eddy viscosity is given by: ijijst SSC 22 (8) i j j i ij x u x u S 2 1 (9) zyx 3 (10) where S ij is the strain rate of the large or resolved field, C s is a model parameter that usually varies between 0.1 and 0.24 (in our case we adopted C s = 0.15) and is the filter length scale. Most traditional methods for solving complex geometries involve the generation of body- fitted grids. The governing equations must then be transformed to use these coordinates and extra metrics terms must be stored for each cell. For flows with moving boundaries, this grid must be completely regenerated each time step. The Immersed Boundary Method (IMBM) presents an alternative. The original idea for the IMBM was presented by Peskin (1972). Boundary conditions are imposed by including additional force terms in the momentum equations. This method has found wide application in Bio-Fluid Mechanics for solving flows such as a swimming fish, a pulsing heart and flow in a flexible blood vessel. Fadlun et al. (2000) proposed an alternative method for the types of flows considered here, namely those where the coupling between the moving boundaries and flow is one way and the boundary is known as a function of space and time. This method lends itself very well to the simulation of any flow with complex and or unsteady boundaries and uses Cartesian meshes. Figure 2 presents a close up view of the grid around a cylinder. It is notable that the round surface is approached by stair-like successions of rectangular cells and that the approximation can be improved by increasing the number of cells. Fadlun et al. (2000) used the alternate expression proposed by Mohd-Yusof (1997) for the forcing term, and thus were able to use a much larger computational time-step than had previously been possible. Ham et al. (2002) successfully used anisotropic local grid refinement on Cartesian grids as an alternative to embedded-grid techniques proposed by Fadlun et al. (2000). Localized grid refinement allows for fewer cells to be used far from the flow feature and a higher concentration of cells near the boundary. This means that the linearized velocity profile will be more accurate because the grid spacing near the boundary is small. The process of refining and coarsening (combining) cells is called grid adaptation. The grid refinement and coarsening is done by looping through the cells of the grid twice. First, cells are marked for refinement or coarsening based on the grid adaptation criterion and then the actual refining or coarsening is performed. The grid adaptation criterion is based on the finite volume discretization error in a given cell. Assuming that the variable varies smoothly throughout the cell, the error can be estimated using a two-variable Taylor series expansion about the cell centroid. An expression for the error can be derived by integrating the Taylor series expansion and retaining only the lowest-order terms. As stated by Ham et al. (2002), "the optimal mesh is the smallest mesh (i.e. mesh with the fewest cells) for which the error associated with each cell is less than a specified tolerance”. The NWT uses the IMBM as modified by Fadlun et al. (2000) with the anisotropic Cartesian grid of Ham et al. (2002). To implement the IMBM a forcing function is added to the momentum equation as a body force. The momentum equation in integral conservation form including the forcing function is: t uV pressureviscousconvectivef ii i i 1 2/1 2/1 (11) where the forcing function f e nsures that the fluid velocity, u, be equal to boundary velocity, V . For time advancement the NWT uses the Fractional Step Method. A pseudo-pressure is used to correct the velocities and enforce continuity at each time step. The transient anisotropic Cartesian grid used for the NWT is fully unstructured and non-staggered. Non-staggered grids may produce non-physical oscillations in the pressure field. This is caused by variables being defined at cell centres and continuity not necessarily being enforced over the cell (Patankar. 1980). Rhie and Chow (1983) developed a so-called "momentum interpolation" method for the non--staggered grid. Using an appropriate interpolation scheme, they enforce continuity by calculating the velocities normal to cell faces. Following the method proposed by Kim and Choi (2000), the integration of the momentum and continuity equation is carried out in steps. Initially, the momentum equations are solved for a velocity using the pressure values from the previous time step and then the estimated velocity field is used to find the new pressure. This new pressure is then used to calculate the velocities at cell centre and face normal. In their derivations and the method used here, a fully implicit Crank-Nicholson method is used for time advancement. Unlike SIMPLE type methods, which are better suited for steady flows, the fractional step scheme proposed here enforces mass conservation at each time step, which is essential for unsteady flows. 3. Calculation parameters and cases considered Figure 1 presents the calculation domain along with the boundary conditions. The calculation domain was set to be 40 D in the stream-wise direction and 100 D in the cross stream direction. With an anisotropic grid 15,000 cells were found to be adequate, since a test run with 30,000 cells produced results quite similar to those obtained with 15,000 cells. In each case we carried out 30,000 time steps, which gave approximately 60 oscillation periods. The flow Reynolds number was set to be 8,640, which corresponds to a flow where the vortex street is fully turbulent and the boundary layer over the cylinder is still laminar. The corresponding flow velocity was V 0 = 0.0590 m/s. 4. Results Table 1 summarizes the results obtained for the different cases simulated. Figure 3 presents a comparison between the cross-stream 1DoF and 2DoF amplitudes of the cylinder motion. The 1DoF case, after an initial period of irregular oscillations that last approximately 5 periods, settles into a regular oscillation mode, with a periodic variation in the amplitude that is repeated about every 8 to 12 periods. The maximum amplitude observed is 0.68D. In the 2DoF case it takes as many 25 periods to reach a relatively steady amplitude. It is noticeable that the amplitude variations after the initial phase are much smaller in the 2DoF case. In this case the maximum amplitude of the oscillations is 0.7D, which occurs after about 20 periods and eventually settles into a regime where the maximum amplitude is 0.5D . Figure 4 presents a study of the effects of the mass-damping ratio, M , on the stream- wise amplitude. The mass-damping ratio varies from 0.0 to 0.065. For a mass-damping ratio of 0.0, it can be seen that the amplitude is irregular yet somewhat periodic. After the initial settling of approximately 23 periods, a pattern of increasing and then decreasing amplitude is established with the maximums becoming larger as time progresses. This is typical behaviour of a system with no damping. The maximum amplitude observed is 0.66D. For a mass-damping of 0.010, the maximum amplitude is 0.61D, observed during the initial settling phase of 30 periods. After this initial phase, the maximum amplitude is reduced to 0.26D. The plot for mass-damping of 0.065 shows the same type of behaviour as that for 0.010, but the increase in mass-damping decreases the maximum amplitude to 0.53D, which is also observed during the initial settling period. The maximum amplitude begins to decrease after the first 16 periods and the motion settles to an amplitude of 0.09D. The effect of the mass-damping ratio is clearly shown in this figure: as the mass- damping ratio increases, both the maximum amplitude and the average amplitude decrease. This is consistent with the findings of Khalak & Williamson (1999). Figure 5 further illustrates the effect of damping on the amplitude of the oscillations. It shows the root mean square of the amplitude of the oscillations in the stream-wise and cross-stream directions as a function of the mass-damping factor (M ). As expected the amplitude of the oscillations decreases with the damping factor. Even the addition of a very small amount of damping (M = 0.010) reduces the amplitude of the oscillation by 56% in the stream wise and 13% in the cross stream directions, respectively. A further increase in the damping factor to M = 0.037 does not bring further reduction of the RMS of the amplitude of the oscillations in either direction. Only when the damping factor M is increased to 0.065 is that we see further reduction in the amplitude of the oscillations. The amplitude of the cross-stream oscillations, y/D, is also presented in Figure 5. The shape of the curve is remarkably similar to the one for the stream-wise oscillation amplitude, x/D. However, it is worth noticing that the cross stream amplitudes are typically larger, increasing from a ratio of 3.25 for M =0 to a ratio of 8.3 for M =0.065. Figure 6 presents a comparison of the lift coefficients for 1 DoF and 2 DoF. Since the cross-stream amplitude is a direct function of the lift coefficient, there is a noticeable correlation between their corresponding plots. After about 20 periods, the lift coefficient for 2 DoF settles into a quasi-steady state with an amplitude of 0.70, with a RMS of 0.54. The plot for 1 DoF does not show any settling and has a maximum amplitude of 2.43 and a RMS of 0.84. These observations are confirmed by the FFT plots. For 1 DoF, a wide band of resonance can be observed for Strouhal numbers between 0.15 and 0.18, while the 2 DoF plot shows a clear resonant peak at a Strouhal number of 0.21. Figure 7 presents a comparison of the drag coefficients for 1 DoF and 2 DoF. From the 2 DoF plot, it can be seen that with the introduction of the stream-wise motion, a quasi- steady state is achieved with an average drag coefficient of 1.61. For the 1 DoF the drag coefficient behaviour is similar to the 1 DoF lift coefficient, with an average value of 1.53. The FFT plot for 2 DoF shows a single resonant peak at a Strouhal number of 0.42, which is, as expected, twice the resonant lift coefficient Strouhal number. However, the plot for 1 DoF does not show a single resonant Strouhal number, but a wide band between 0.31 and 0.45. This band includes the double of the corresponding lift coefficient band, but is wider. Figure 8 shows four typical vorticity plots for Re=8640 and M Figure 8a shows the solution for a stationary cylinder (0 DoF). The vortex shedding pattern is such that in one cycle two vortices are released, one from the bottom and another from the top. This is known in the literature as 2S , namely two single vortices. Figure 8b shows the results for the cylinder free to move in the cross-stream direction (1 DoF). In this case there is a pair of counter-rotating vortices released from the top of the cylinder and a single vortex from the bottom of the cylinder. This is known as P+S in the literature. Figure 8c and 8d present the results for the cylinder free to move in both the cross-stream and stream-wise directions, (2DoF). In Figure 8c we observe a vortex shedding pattern similar to that in the stationary cylinder case ( 2S ), while in Figure 8d the vortex shedding pattern shows two pairs of vortices being shed in every cycle, this is known as 2P in the literature. This behaviour was observed by Bishop and Hassan (1964) and confirmed by Williamson et al. (1988). Near the lock-in region there is an abrupt change in the character of the body forces causing the vortex formation to go from a 2S formation to a 2P formation. However, for 2 DoF the 2S differs from the 2S type observed for a rigid cylinder, where a vortex is shed every half cycle. For the moving cylinder, below certain amplitude of oscillation, two like-signed vortices combine in each half cycle to create a 2S vortex- street-type wake. For higher amplitudes of oscillation the vortices do not combine but arrange themselves in the form of two vortex pairs ( 2P ). Figure 9 presents the stream-wise oscillation amplitudes for a cylinder with mass damping factor M =0.065. The vortex shedding pattern is 2P (see Fig. 8) up until ≈ 200, after which the amplitude of the oscillations is reduced and a 2S vortex shedding pattern appears. Figure 10 shows six different x/D vs. y/D plots for a cylinder with mass damping factor M =0.010. The first noticeable fact is that the trajectories are far from a symmetrical “eight” figure. The figure “eight” has either a smaller (a), or larger negative loop (b). The figure eight can be leaning backward (c), or forward (d). It can also be a closed “elliptical” loop as shown in (e) and (f). It can be concluded from these observations that since the oscillations in the cross-stream and the stream-wise directions are not perfectly synchronized and furthermore they have variable amplitudes, a perfect figure eight can only be obtained if the amplitude and frequency of the oscillations are externally imposed. Jeon and Gharib (2001) report experimental results for cylinders with 2 DoF. However, the oscillation amplitudes and frequency were externally imposed. The cross- stream vibration amplitudes were set at y/D = 1, while the stream-wise oscillations were limited to x/D = 0.1. The stream-wise oscillation frequency was adjusted such that it was twice the cross-stream oscillation frequency. As a consequence, a plot of y/D vs x/D automatically produces a figure eight. In our simulations, the cylinder was free to move in the plane of motion, and thus some seemingly unusual results were produced. Conclusions Our simulations clearly demonstrate that for VIV studies the 2 DoF case presents results that differ significantly from those obtained for either rigid cylinders (0DoF) or cylinders free to move in the cross-stream directions only (1DoF). For the 2 DoF case we were able to show that as the amplitude of the cross-stream oscillations decreased the vortex shedding pattern changed from 2P to 2S . The RMS of the amplitude of the cross-stream oscillations for the 1 DoF case were about 10 to 20% higher than the 2 DoF, depending on the mass damping factor. As expected, the RMS of amplitude of the oscillations in the cross-stream direction is almost one order of magnitude larger than in the stream-wise direction. The RMS of the lift coefficient is about 35% smaller for the 2 DoF cases than for the 1 DoF cases, with the exception of the case with no damping where an actual increase of about 10% is observed in the RMS of the amplitude of the 2 DoF oscillations. The drag coefficient ratio (stationary/1DoF or stationary/2DoF) between the rigid cylinder case and the different 1 DoF and 2 DoF cases remained almost constant at around 1.41, with the differences less than 5% in all cases. References Blevins, Robert D.(1990):Flow Induced Vibration, Van Nostrand Reinhold, 2 nd Edition. Bishop, R. E. D. and A.Y. Hassan (1964). The lift and drag forces on a circular cylinder oscillating in a flowing fluid. Proceedings of the Royal Society (London), Series A 277, 51-75. Breuer, M. 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Combined Immersed Boundaries / B-Splines Methods for Simulations of Flows in Complex Geometries, CTR Annual Research Briefs, NASA Ames / Stanford University. Norberg, C. (2001): Flow Around a Circular Cylinder: Aspects of Fluctuating Lift. Journal of Fluids and Structures 15, pp. 459-469. Patankar, S.V. (1980). Numerical Heat Transfer and Fluid Flow. Washington, D.C.: Hemisphere. Peskin, C.S. Flow Patterns around Heart Valves: A Numerical Method, J. Comp. Phys. 10, 252 (1972). Raw, M. Robustness of the Coupled Algebraic Multigrid for the Navier-Stokes Equations. 34th Aerospace Sciences Meeting and Exhibit, January 15-18, 1996, Reno NV, AIAA 96-0297. Rhie, C.M. and W.L. Chow. (1996).Numerical Study of the Turbulent Flow Past an Airfoil with Trailing Edge Separation. AIAA Journal 21, 1525-1532 (1983). Smagorinsky, J. (1963): General circulation experiments with the primitive equations, part I: the basic experiment. Monthly Weather Rev., 91, 99-164. So, R.M.C., Y. Zhou and M.H. Liu (2000). Experiments in Fluids 29, pp.130-144. Wilde, J. J. and R.H. M. Huijmans (2001): Experiments fo High Reynolds Number VIV on Risers. Proceedings of the Eleventh International Offshore and Polar Engineering Conference, Stavanger, Norway, June 17-22. Williamson, C. H. K., and A. Roshko (1988): Vortex Formation in the Wake of an Oscillationg Cylinder, Journal of Fluids and Structures 2, pp. 355-381. Zhou, C. Y , R. M. C. So and K. Lam (1999): Vortex-Induced Vibrations of an Elastic Circular Cylinder. Journal of Fluid and Structures13, pp. 165-189. Fig. 1: Flow geometry, calculation domain and boundary conditions. Fig. 2: Sample non-isotropic Cartesian adaptive grid. Fig. 3: Cross-stream amplitudes Fig. 4: Effect of mass-damping on stream-wise amplitude Fig. 5: RMS of oscillation amplitudes in the cross-stream and stream-wise directions. Fig. 6: Lift Coefficients Fig. 7: Drag Coefficients Fig. 8: Vorticity plots Fig. 9: Amplitude of stream-wise oscillations as a function of time. Fig. 10: Sample trajectories of the cylinder centre with 2DoF : a) Vertical figure “eight” with smaller negative loop; b) Vertical figure “eight” with larger negative loop; c) backward leaning figure “eight”; d) forward leaning figure “eight”; e) closed loop trajectory; f) hearth shaped trajectory. Figure captions Fig. 1: Flow geometry, calculation domain and boundary conditions. Fig. 2: Sample non-isotropic Cartesian adaptive grid. Fig. 3: Cross-stream amplitudes Fig. 4: Effect of mass-damping on stream-wise amplitude Fig. 5: RMS of oscillation amplitudes in the cross-stream and stream-wise directions. Fig. 6: Lift Coefficients Fig. 7: Drag Coefficients Fig. 8: Vorticity plots Fig. 9: Amplitude of stream-wise oscillations as a function of time. Fig. 10: Sample trajectories of the cylinder centre with 2DoF : a) Vertical figure “eight” with smaller negative loop; b) Vertical figure “eight” with larger negative loop; c) backward leaning figure “eight”; d) forward leaning figure “eight”; e) closed loop trajectory; f) hearth shaped trajectory. Table 1. Summary of main results obtained. DoF M y/D max y/D RMS x/D max x/D aver C D C D /C D o C L RMS C L max 1 0.000 0.80 0.44 n/a n/a 1.68 1.46 0.87 2.54 2 0.000 0.81 0.39 0.66 0.24 1.59 1.38 0.97 3.45 1 0.010 0.82 0.43 n/a n/a 1.69 1.47 0.80 2.51 2 0.010 0.82 0.34 0.61 0.24 1.61 1.40 0.54 3.16 1 0.037 0.70 0.39 n/a n/a 1.63 1.42 0.79 2.56 2 0.037 0.77 0.35 0.63 0.23 1.59 1.38 0.81 3.44 1 0.065 0.68 0.35 n/a n/a 1.53 1.33 0.84 2.43 2 0.065 0.70 0.32 0.53 0.24 1.61 1.40 0.54 2.97 2 a 0.065 0.68 0.35 0.58 0.24 1.64 1.43 0.98 3.09 a The last row represents the results with 30000 calculations cells.

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