XIE Fang-fang, DENG Jian, ZHENG Yao

School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, China, E-mail: xiefangfang921@yahoo.com.cn

MULTI-MODE OF VORTEX-INDUCED VIBRATION OF A FLEXIBLE CIRCULAR CYLINDER*

XIE Fang-fang, DENG Jian, ZHENG Yao

School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, China, E-mail: xiefangfang921@yahoo.com.cn

The vortex-induced vibration of a flexible circular cylinder is investigated at a constant Reynolds number of 1 000. The finite-volume method on moving meshes is applied for the fluid flow, and the Euler-Bernoulli beam theory is used to model the dynamic response of a flexible cylinder. The relationship between the reduced velocity and the amplitude response agrees well with the experimental results. Moreover, five different vibrating modes appear in the simulation. From the comparisons of their vortex structures, the strength of the wake flow is related to the exciting vibrating mode and different vortex patterns arise for different vibrating modes. Only 2P pattern appears in the first vibrating mode while 2S-2P patterns occur in the other vibrating modes if monitoring at different sections along the length of the cylinder. The vibration of the flexible cylinder can also greatly alter the three-dimensionality in the wake, which needs further studies in our future work, especially in the transition region for the Reynolds number from 170 to 300.

multi-mode, Vortex-Induced Vibration (VIV), three-dimensional instability, flexible cylinders, Euler-Bernoulli beam

Introduction

As were reviewed by Refs.[1,2], much progress has been made both numerically and experimentally, toward many fundamental insights into the vortex dynamics in the wake of a vibrating cylinder. Most of the previous studies were focused on the hysteresis phenomena, lock-in/lock-out region determination, hydrodynamic forces on the cylinder, vortex-shedding patterns, and the effect of the mass-damping parameters on the dynamic response of the cylinder. Aiming at the complicated nonlinear wake-body coupling characteristics in vortex-induced vibrations, semiempirical models, such as wake-oscillator model based on the van der Pol equation, have been developped in recent studies[3-6].

The studies on Vortex-Induced Vibration (VIV) have been expanded into 3-D and flexible cylinderswith the increasing demands for more comprehensive understanding of VIV in realistic engineering problems during recent years. Ren et al.[7]employed an Arbitrary Lagrangian-Eulerian (ALE) method in dealing with the flow-induced vibration of an elastic circular cylinder at Re=200, in which the transition of the wake patterns or vortex shedding modes was captured. Deng et al.[8,9]investigated the wake characteristics of two circular cylinders arranged perpendicular to each other in a uniform flow and found that the peak amplitude of vibrations for the cruciform arrangement is lower than that of an isolated cylinder, and the resonance region is wider than that of an isolated cylinder, and the distinct vortex shedding patterns in the wake were studied as well. Flow-structure coupling problems in VIV have attracted more attentions, where the cylinders were flexible, differing from the rigid cylinders in previous studies. Except for the detailed hydrodynamic explanations that have been given by the laboratory experiments with springmounted rigid cylinders and for first mode standing wave vibrations of flexible cylinders, the authors[10]have recently conducted experiments on very long flexible cylinders in the ocean, as well as by laboratory experiments. These experiments have revealedthe unexpected dominance of traveling waves with strong higher harmonic components and stable figure eight trajectories. For numerical simulations, the studies have been classified into two categories: cable-like[10,11]and beam-like[12-15], according to the type of support for the cylinder. If the tension dominates, it is termed as a cable, while if the bending rigidity dominates, it is termed as a beam. Lee et al.[16]suggested that the vibration frequency of a cylinder would rise with the flow speeds for tension-dominated structures but do not rise significantly for bending rigidity controlled structures. In addition, the lock-in bandwidth is still broad in the bending rigidity controlled cases, indicating a weak association between the change of the vibration frequency and the lock-in bandwidth.

As suggested in previous studies that a flexible cylinder can be excited into multiple-modal VIV by varying the exciting frequency or enduring different current conditions along the span-wise direction. Evangelinos et al.[17]used a three-dimensional DNS code to capture the second vibrating mode of a cablelike flexible cylinder in both standing wave and travelling wave responses. Four different vibrating modes of a marine riser were excited in Refs.[14,15], and were confirmed to be directly linked to the reduced velocity. Bearman and Huera-Huarte[18], Huera-Huarte et al.[13]used an indirect finite element method to study the fluid force distributions on a vertical tension riser under 4thand 7thVIV modes in a stepped current. Willden Richard et al.[19]investigated the multi-modally vortex-induced vibrations of a long flexible pipe and found that all of the excited modes vibrated at the excitation (Strouhal) frequency.

In this article, multi-modal VIV of a flexible cylinder is simulated at the Reynolds number Re=1000. The definition of the Reynolds number is Re=UD/ ν, where U=1m/s is the velocity of the incoming flow, D=1m is the diameter of the cylinder, and ν=0.001m2/s is the kinetic viscosity of the fluid. The flexible cylinder is modeled as an Euler-Bernoulli beam, which is supposed to only deal with cross-flow motion. The mass ratio of the cylinder is assumed to be 10. The main contribution of this article is to study the dynamic response of a flexible cylinder with small spanwise length, thus distinct streamwise vortex strcuture will be observed, which vary with different vibrating modes. The following of this article will proceed as follows. In the next section, the physical models and numerical method of fluid-structure interaction are described. In Section 2, the numerical results are presented and discussed, which focuses on the effect of reduced velocity on the amplitude response, the vortex structure in the wake, and the excited multimodal response. The conclusions are drawn in the last section.

1. Physical models and numerical methods

1.1 Computational domain and boundary conditions The geometry and the corresponding dimensions considered for the analysis are schematically shown in Fig.1. The dimensions related to the geometry in this figure are D=1, YD=16, XI=12, XO=25 and ZD=12. A body-fitted structured grid system is generated in the flow domain. The grid is clustered near the cylinders and the grid spacing is increased in a proper ratio away from the surface of the cylinder.

Fig.1 Geometrical model and computational domain

The boundary conditions employed in the present investigation are: (1) The free-slip conditions are imposed at both the transverse and span-wise confining surfaces, in order to not confine the artificial simulated flow, (2) At the inlet, a constant streamwise velocity is imposed with other components set to zero, and the nonreflecting boundary condition is imposed at the outlet. (3) The pressure Neumann condition is applied to the inlet, surface of the cylinder, and the outlet boundaries.

1.2 Numerical methods for fluid-structure system

1.2.1 The finite-volume method on moving meshes

The finite volume method on moving meshes is applied for simulating the flow field. The moving finite-volume cells must be adjusted to the timevarying shape of the interface. A Laplace equation based deforming mesh approach is used in this study where the internal Control-Volume (CV) vertices are moved based on a prescribed motion of the boundary vertices, while the topology of the mesh stays unchanged. The displacement equation for mesh moving is discretized on vertices of the finite-volume mesh and solved by the finite element method.

The fluid flow is supposed to be incompressible and Newtonian, for which the integral form of the governing equations on moving meshes is given by

where U denotes the fluid velocity, Ubthe cell velocity, the subscript p the cell values, and Vpthe cell volume. As the volume Vpis no longer fixed in space, its motion is captured by the motion of its bounding surface S by the velocity Ub.

The general form of the Gauss theorem and the second-order finite-volume discretization are used. The second-order upwind spatial discretization scheme is applied to the convective term and the secondorder central differencing corrected scheme is used for the viscous term. The temporal discretization of all equations is performed by using a second-order implicit three-time-level scheme. During the numerical simulation, where the PISO is used for dealing with the inter-equation coupling in the pressure-velocity system, the mesh motion flux returns to be critical and is calculated as the volume swept by the face f in motion during the current time-step rather than from the grid velocity Ub, making it consistent with the cell volume calculation. The geometrical conservation rule can be well kept in the simulations.

1.2.2 Dynamic equation for the three-dimensional flexible cylinder

In this section, the Euler-Bernoulli beam theory based governing equations for the cylinder is introduced. For simplicity, the damping characteristic and the tension along the beam are not considered here.

Here, it is assumed that the cylinder will bend only at the cross flow direction and the motion of the cylinder is determined by

Note that the physical boundary conditions at the two fixed ends are both described as zero displacement and zero gradient

wherelρ is the mass per unit length of the cylinder, EI the stiffness, l the length of the cylinder, and f(z,t) the distributed lift force acting on the cylinder. The finite-difference method is utilized to discretize the beam equation, which is represented as

where yiis the displacement at the ith node, and Δz the finite-difference grid size (Δz=l/n ), where n is the number of elements or segments along the spanwise direction.

The boundary condition is also presented by central difference method. Two virtual nodes, i=?1, i=n+1 should be established to discretize the Eq.(4)

The influence of the number of segments for the cylinder and the suitability of the numerical scheme in the structural analysis is investigated. The free vibration of fixed-fixed beam is first studied here. From the frequency equation it follows that

The first three roots of the above equation are the 4.73, 7.854 and 10.996. And when i≥3, Eq.(7) can be simplified as

Then the root of the equation is

The corresponding n th eigenfrequency is

In the validation cases (for free vibration), wechoose the first eigenfrequency to be approximately equal to 2πfo(wherefois the natural vortex shedding frequency of the flow around a rigid cylinder atRe=1000). With the first eigen-mode employed as the initial deformation of cylinder, three cases are compared at different numbers of segments (n=6, 12, 24). Compared with the assumed first eigen-frequency, the errors of the simulated frequencies for these three cases are 8.65%, 4.09% and 0% respectively. However, the segment of the cylinder is related to the time intervals in the simulation of the forced vibration. To save computational time and at the same time meet the demand for convergence and accuracy, it is recommended thatn=12 is the best choice in our following simulations. The discretized fixed-fixed beam system can be solved by using the Runge-Kutta algorithm for the internal nodes (i=1-n?1) with the boundary condition satisfied. The node displacements and velocities are updated immediately for the next time step.

1.3Fluid-structure coupling method

The loosely coupling method is utilized between the fluid and structural solvers. It does not require major modification of the pre-existing flow and structure solvers. The coupling process can be briefly described as: the flow field is firstly simulated, and then the fluid forces on the cylinder are calculated, which are passed to the structural solver to determine the motion of the cylinder. According to the updated position of the cylinder, the fluid mesh is modified. The Fluid-Structure Interaction (FSI) iteration loop is repeated with evolution of the flow time. Various test computations have shown that the coupling scheme is rather sensitive to the time intervals.

2. Results and discussions

In this section, firstly, the validation is carried out to confirm the finite-volume code on moving meshes. Then, we will focus on the three-dimensional VIV on a flexible cylinder, with plenty of results presented and discussed, including the contribution of reduced velocity to the dynamic response, as well as the different vortex patterns in the wake for multimodal responses. It should be mentioned to notice the definition of the reduced velocity as follows

whereUis the velocity of the incoming flow,fnis the first natural frequency of the cylinder, andDthe diameter of the cylinder.

2.1Two-dimensional validation

In our previous studies, a finite different method based solver as well as an immersed-boundary method based solver have been implemented in the simulations of VIV for both two-dimensional[8]and threedimensional cylinders[9]. However, it is the first time for us to involve a finite-volume solver in VIV simulations. Its validation must be extensively performed.

By using a single degree-of-freedom model of the structure (spring-damper-mass model), the VIV for a rigid cylinder is simulated. The maximum responses of the vibrating circular cylinder for the current simulations are compared with other computational results which are also obtained atRe=1000. With the same mass-damping coefficient, the peak amplitude value is 0.55 by using 2-D DNS code and 0.74 by using 3-D DNS code in Evangelinos’s simulations[20]. In the present study, the peak amplitude response is 0.57 by 2-D solution. It is shown that the response amplitude is lower than that of the 3-D DNS result in Ref.[20], and close to its 2-D DNS result. It is due to the unsolved three-dimensionality effect in the 2-D studies. This divergence highlights the necessity for 3 implementing -D studies.

Fig.2 Vibration amplitude as a function of the reduced velocity

2.2Three-dimensional simulation of VIV

2.2.1The reduced velocity effect

People generally make research on VIV by changing the reduced velocity (thus the Reynolds number), or the frequency ratio. For experimental studies, the inflow velocity in a wind tunnel or a water tunnel can be conveniently changed. While for numerical studies, people prefer to change the natural frequency of the structures, which is also followed by this work. The dynamic response at different frequency ratios, the beating behavior, and the locking phenomenon for VIV on a flexible cylinder have been well discussed in our previous studies. We suggested that the 2S pattern appear at smaller amplitude responses while the 2P pattern appears at the relatively larger amplitude responses. The wake pattern would be different at the different sections along the length of the cylinder for the bending of flexible cylinder with two fixed ends. Moreover, we have found that in higher amplitude response cases the vortex structures exhibit much more three-dimensionality, with plentyof stream-wise vortex formed in the wake.

Fig.3 Three-dimensional vortex structure for two different reduced velocities

Fig.4 Instantaneous vorticity contours at different z-positions for two different reduced velocities

To study the relationship between the reduced velocities and the dynamic responses, we first exhibit the amplitudes of vibration as a function to the reduced velocities in Fig.2. The results in the current simulations are compared to the experimental results for free vibration of a rigid cylinder at low massdamping[21]. Despite the different conditions between the simulations and experiments, the general trend of the amplitude versus reduced velocity curve obtained by the simulations agrees well with the experimental results. The maximum amplitude in our simulations for U?=5.4 is 1.03 d, a little higher than the value of 0.97 d for U?=5.7 in experiments for m?=10.3. Nevertheless, there are still many differences: the Reynolds number in the experiment is an order higher than that of the simulations, the structure is regarded as a rigid cylinder in experiments and a bending Euler-Bernoulli beam in our simulations. Therefore, it is not surprising that the amplitude responses in our study are well below the experiment results with the reduced velocity values greater than 6, however the peak values still match well. It is noted that the main purpose for Fig.2 is not to make a quantitative comparison with experiments, since the limits for the model used in this paper have been discussed above. different reduced velocities U?=4.61 and U?= 5.43, which represent the smaller and larger amplitude cases respectively. It can be exactly figured out that under the larger amplitude response, the wake flow is much more unstable and the stream-wise vortex is more intensive than the smaller amplitude case.

Fig.5 Dynamic response for a flexible cylinder

Furthermore, we present the three-dimensional vortex structure in Fig.3 for the cases under twoFigure 4 exhibits the 2S and 2P vortex patterns at the planar vortical contours in separate cases. The wake patterns are found to vary in different sections along the length due to the vibrating movement of the flexible cylinder and then different amplitude responses along the length.

Table 1 Different frequencies for VIV response

2.2.2Theexcited multi-modal response

In further study of the larger reduced velocities, second to five vibration modes are presented in the cross-flow response besides the first mode. Compared to Willden et al’s studies[17], in which they suggested that a long pipe can be simultaneously excited into multi-modal VIV if the excitation frequency (the vortex shedding frequency) varies along the pipe length and a large length ratio of the paper is selected, in the current simulations, the simulations are conducted at a uniform excitation frequency in one case, and multimodal vortex vibration modes can be observed for different cases, i.e., with different natural frequencies of the structure. Nevertheless, owing to the restriction of cylinder length, the upper bound of the vibrating mode for the present research is confined to five.

Figure 5 shows the dynamic response of the flexible cylinder under five different reduced velocities. They agree well with the results obtained in Huera’s study[13]except that in our study the foremost five vibrating modes are presented but just 4thand 7thmodes in the cross direction were provided in Ref.[13]. ForU?=5.13, one can clearly see that the structure vibrates in the first mode. If the reduced velocity is increased toU?=15.38, there is a change of oscillating mode, and the flexible cylinder starts vibrating predominantly in the second mode, as shown in Fig.5(b). When the reduced velocity reaches the value ofU?=30.76, the flexible cylinder oscillates mainly in the third mode, as presented in Fig.5(c). For the reduced velocity varied toU?=46.15, the vibration occurs mostly in the fourth mode, as shown in Fig.5(d). And when the reduced velocity reachesU?=61.53, we can get the fifth vibrating mode exactly. The foremost five natural frequencies of the Euler-Bernoulli beam and the vortex shedding frequenciesfvof the flexible cylinder in motion correspondingtospecificreducedvelocitiesare presented in Table 1. One can find that the vortex shedding frequency in VIV is closer to the first natural frequency forU?=5.13, to the second natural frequency forU?=15.38, and to the third natural frequency whenU?=30.76. While at the reduced velocity ofU?=46.15, there are three peak frequencies, which are 0.18, 0.189 and 0.195, respectively. Even though, they are close to the fourth natural frequency. However, forU?=61.53, the vortex shedding frequency lies between the fourth and fifth natural frequencies, and more approaching to the fifth. On the whole, the results coincide well with the dynamic response of flexible cylinder around the respective eigenfrequencies under different reduced velocities.

Fig.6 Three-dimensional vortex structure for different reduced velocities

In order to examine the relationship between the three-dimensionality in the wake and the dynamic response in different vibrating modes, the wake structures are shown in Fig.6, expressed by the iso-surfaces of both the spanwise and streamwisevorticities. The qualitative comparisons among these different vibrating modes can be made. The wakes for these cases are all disordered and show much 3-D instability. However, the intensity of the vortices in the wake has been pretty weakened in the other vibrating modes including the span-wise vortices and the stream-wise vortices, compared to the first mode. It suggested that the intensity of the vortices in the wake is related to the maximum dynamic response of the flexible cylinder. The more seriously the cylinder vibrates (it can be determined by the amplitude of the response), the stronger vortices shed or form in the wake.

Fig.7 Planar vortical contours at different positions along spanwise direction

Furthermore, the planar vortical contours at different positions along the span-wise direction are presented in Fig.7. It is found that the 2P pattern with a pair of vortices shedding in each half cycle of oscillation exists in the first vibrating mode. And in the second and third vibrating modes, the hybrid pattern of 2P-2S exists, in which the 2S pattern behaves like a single vortex shed in each half cycle of the oscillation. We can also find that in the fourth and fifth vibrating modes, the 2S pattern dominates in the planar contours of the wake. It is indicated that in the planar contours of the different positions, if the response displacement is higher than a certain value, the 2P pattern would appear, otherwise, the 2S pattern dominates.

3. Conclusions

This article contributes to the knowledge of the vortex-induced vibration of a flexible cylinder. From the discussion above, the following conclusions can be drawn.

(1) Five different vibrating modes of the cylinder are excited for their respective reduced velocities in the current studies. If the cylinder is made longer and more slender, more higher-order modes can be observed based on the numerical methods involved in this article.

(2) The comparison of the vortex patterns for these five vibrating modes has shown that the three-dimensionality in the wake can be greatly altered, and the intensity of the vortices in the wake is associated to the vibrating modes.

In the future work, we will focus on the studies of the effect of the Reynolds number on the VIV for a flexible cylinder, and try to excite higher-order modal vibrations of the beam-like cylinder.

Acknowledgements

The authors wish to thank the all these numerical computations have been performed at the Center for Engineering and Scientific Computation (CESC), Zhejiang University. The authors are grateful for this assistance.

[1] SARPKAYA T. A critical review of the intrinsic nature of vortex-induced vibrations[J].Journal of Fluids and Structures,2004, 19(4): 389-447.

[2] WILLIAMSON C. H. K., GOVARDHAN R. A brief review of recent results in vortex-induced vibrations[J].Journal of Wind Engineering and Industrial Aerodynamics,2008, 96(6-7): 713-735.

[3] LIN Li-ming, LING Guo-can and WU Ying-xiang et al. Nonlinear fluid damping in structure-wake oscillators in modeling vortex-induced vibrations[J].Journal of Hydrodynamics,2009, 21(1): 1-11.

[4] SRINIL N., WIERCIGROCH M. and O’BRIEN P. Reduced-order modelling of vortex-induced vibration of catenary riser[J].Ocean Engineering,2009, 36(17-18): 1404-1414.

[5] XU Wai-hai, WU Ying-xiang and ZENG Xiao-hui et al. A new wake oscillator model for predicting vortex induced vibration of a circular cylinder[J].Journal of Hydrodynamics,2010, 22(3): 381-386.

[6] SRINIL N. Multi-mode interactions in vortex-induced vibrations of flexible curved/straight structures withgeometric nonlinearities[J].Journal of Fluids and Structures,2010, 26(7-8): 1098-1122.

[7] REN An-Lu, CHEN Wen-qu and LI Guang-wang. An ALE method and DDM with high accurate compact schemes for vortex-induced vibrations of an elastic circular cylinder[J].Journal of Hydrodynamics, Ser. B,2004, 16(6): 708-715.

[8] DENG Jian, REN An-Lu and CHEN Wen-qu. Numerical simulation of flow-induced vibration on two circular cylinders in tandem arrangment[J].Journal of Hydrodynamics, Ser. B,2005, 17(6): 660-666.

[9] DENG Jian, REN An-Lu and SHAO Xue-ming. The flow between a stationary cylinder and a downstream elastic cylinder in cruciform arrangement[J].Journal of Fluids and Structures,2007, 23(5): 715-731.

[10] DAVIS M., DEMETRIOU M. A. and OLINGER D. J. Low order modelling of freely vibrating flexible cables[J].Flow Turbulence and Combustion,2003, 71(1-4): 75-91.

[11] VIOLETTE R., De LANGRE E. and SZYDLOWSKI J. Computation of vortex-induced vibrations of long structures using a wake oscillator model: Comparison with DNS and experiments[J].Computers and Structures,2007, 85(11-14): 1134-1141.

[12] HUANG K., CHEN H. C. and CHEN C. R. Vertical riser VIV simulation in uniform current[J].Journal of Offshore Mechanics and Arctic Engineering-Transactions of the Asme,2010, 132(3): 031101.

[13] HUERA-HUARTE F. J., BEARMAN P. W. and CHAPLIN J. R. On the force distribution along the axis of a flexible circular cylinder undergoing multi-mode vortex-induced vibrations[J].Journal of Fluids and Structures,2006, 22(6-7): 897-903.

[14] MENEGHINI J. R., SALTARA F. and FREGONESI R. D. et al. Numerical simulations of VIV on long flexible cylinders immersed in complex flow fields[J].European Journal of Mechanics B-Fluids,2004, 23(1): 51-63.

[15] YAMAMOTO C. T., MENEGHINI J. R. and SALTARA F. et al. Numerical simulations of vortex-induced vibration on flexible cylinders[J].Journal of Fluids and Structures,2004, 19(4): 467-489.

[16] LEE L., ALLEN D. Vibration frequency and lock-in bandwidth of tensioned, flexible cylinders experiencing vortex shedding[J].Journal of Fluids and Structures,2010, 26(4): 602-610.

[17] EVANGELINOS C., LUCOR D. and KARNIADAKIS G. E. DNS-derived force distribution on flexible cylinders subject to vortex-induced vibration[J].Journal of Fluids and Structures,2000, 14(3): 429-440.

[18] BEARMAN P. W., HUERA-HUARTE F. J. the hydrodynamic forces acting on a long flexible circular cylinder responding to VIV[C].The Proceedings of 6th FSI, AE and FIV and N Symposium.Vancouver, Canada, 2006, 9: 611-619.

[19] WILLDEN RICHARD H. J., GRAHAM J. and MICHAEL R. Multi modal vortex induced vibrations of a vertical riser pipe subject to a uniform current profile[J].European Journal of Mechanics B/Fluids,2004, 23(1): 209-218.

[20] EVANGELINOS C., KARNIADAKIS G. E. M. Dynamics and flow structures in the turbulent wake of rigid and flexible cylinders subject to vortex-induced vibrations[J].Journal of Fluid Mechanics,1999, 400: 91-124.

[21] KHALAK A., WILLIAMSON C. H. K. Dynamics of a hydroelastic cylinder with very low mass and damping[J].Journal of Fluids and Structures,1996, 10(5): 455-472.

February 26, 2011, Revised April 22, 2011)

* Project supported by the National Natural Science Foundation of China (Grant No. 10802075).

Biography: XIE Fang-fang (1987-), Female, Ph. D. Candidate

DENG Jian,

E-mail: zjudengjian@zju.edu.cn

2011,23(4):483-490

10.1016/S1001-6058(10)60139-4