Analytical solution of mean top tension of long flexible riser in modeling vortex-induced vibrations

Applied Ocean Research 41 (2013) 1–8

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Analytical solution of mean top tension of long flexible riser in modeling vortex-induced vibrations Jijun Gua,b,* , Yi Wangb , Yu Zhangb , Menglan Duanb , Carlos Levia a b

Ocean Engineering Program, COPPE, Federal University of Rio de Janeiro, CP 68508, Rio de Janeiro, RJ 21941-972, Brazil Offshore Oil/Gas Research Center, China University of Petroleum, Beijing 102249, China

a r t i c l e

i n f o

Article history: Received 10 July 2012 Received in revised form 25 November 2012 Accepted 17 January 2013 Keywords: Wake oscillator model Riser VIV prediction Top tension Multi-mode

a b s t r a c t The vortex-induced vibration (VIV) of flexible long riser with combined in-line and cross-flow motion has been studied using a wake oscillator in this paper. The analytical solution of mean top tension of long flexible riser is evaluated and compared with experimental results, and good agreement is observed to verify its validity. Then the nonlinear coupled dynamics of the in-line and cross-flow VIV of a long tensiondominated riser were analyzed through wake oscillator model with the consideration of variation of the mean top tension. The in-line and cross-flow resonant frequencies, lift and drag coefficients, dominant mode numbers, amplitudes and instantaneous deflections are reported and compared with experimental results, and excellent agreements are observed. The comparison of mode numbers between the calculation with and without consideration of variation of mean top tension shows that the proposed analytical solution of the mean top tension can produce a better prediction of multi-mode VIV.  c 2013 Elsevier Ltd. All rights reserved.

1. Introduction Deep water, string-like, marine risers subject to strong ocean currents, suffer from vortex-induced vibration (VIV), where vortex shedding interacts with the structural properties of the riser, resulting in large amplitude vibrations in both in-line (IL) and cross-flow (CF) directions. When the vortex shedding frequency approaches the natural frequency of a marine riser, the cylinder takes control of the shedding process causing the vortices to be shed at a frequency close to its natural frequency. This phenomenon is called vortex shedding lock-in or synchronization. Under lock-in conditions, large resonant oscillations will reduce the fatigue life significantly. Despite continued research work on the mathematical model of description of the VIV, the lock-in phenomenon has remained elusive. Computational fluid dynamic codes have been developed and are capable of capturing some insight of the VIV, but the large requirement of the computing capacity gives rise to the limitation of modeling the deep water riser, whereas in some cases the length has exceeded more than 1500 m. Alternatively the semi-empirical model for VIV response analysis has been used prevalently in the engineering. These models normally use the hydrodynamic force coefficients as a database, such as drag coefficient, lift coefficient, added mass coefficient and hydrodynamic

* Corresponding author at: Ocean Engineering Program, COPPE, Federal University of Rio de Janeiro, CP 68508, Rio de Janeiro, RJ 21941-972, Brazil. Tel.: +55 21 3867 6768; fax: +55 21 3867 6768. E-mail addresses: [email protected], [email protected] (J. Gu).

c 2013 Elsevier Ltd. All rights reserved. 0141-1187/$ - see front matter  http://dx.doi.org/10.1016/j.apor.2013.01.004

damping coefficient. These coefficients are obtained from rigid cylinder model tests with forced motions. All these models are based on the assumption that in one mode there is a dominant resonant frequency. However, for the deep water risers, shear flow and travelling waves will cause multiple modes and dense spectrum of the response frequencies. Consequently, a safety factor has been used widely in the design of the risers. A second type of semi-empirical model is the wake oscillator model which was created by Birkoff and Zarantanello [1]. Instead of direct application of the measured fluid forces to the equation of structural motion, wake oscillator models couple the equation of structural motion with a nonlinear oscillator equation that describes the CF fluid force. Three different coupling terms, including acceleration, velocity, and displacement coupling, were evaluated by Facchinetti et al. [2]. Violette et al. [3] and Xu et al. [4] performed the VIV prediction along the spanwise extent of a slender structure under uniform flow, non-uniform flow and linearly sheared flow, and good agreement was achieved. The wake oscillator has been extended to predict IL dynamic response beside the CF response in recent literatures [5,6]. Guo et al. [7] and Li et al. [8] performed dynamic response analysis of VIV of a long flexible riser with the consideration of influence of top tension and internal flow, and the comparison with experimental results presented a successful application of such wake oscillator. Meng and Chen [9] implemented a wake oscillator model to analyze the three-dimensional vibration behaviors of an inclined extensible steel catenary riser (SCR). Srinil [10] successfully carried out numerical prediction of vortex-induced vibration of variable-tension vertical risers in linearly sheared currents through a wake oscillator model. A

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J. Gu et al. / Applied Ocean Research 41 (2013) 1–8

comparison between laboratory measurements and blind predictions of eleven different numerical models was carried out by Chaplin et al. [11] to analyze the capability of different numerical tools to predict VIV. The response prediction of Orcina wake oscillator code was between 85% and 105% of the corresponding measurements, which has the same capability of predicting CF displacements and curvatures as empirical models (such as VIVA, VIVANA, VICoMo, SHEAR7 and ABAVIV), but better than the codes based on CFD (such as Norsk Hydro, USP, DeepFlow and VIVIC). Wake oscillators also have been shown to be able to model VIV of vortex-induced waves along cables [3,12]. Much work has been done for investigation of VIV prediction of long slender structures using wake oscillator model. However some of the important factors have not been evaluated quantitatively, such as the variation of top tension. The purpose of the present study is to present an analytical solution of the mean top tension of long flexible riser under vortex-induced vibration. The nonlinear coupled wake oscillator model has been used to simulate the dynamic response of long flexible risers. A comparison between model and experimental results from literature shows the validity of the present analytical solution. 2. Model description 2.1. Wake oscillator model The marine riser model can be idealized as a beam with low flexural stiffness. The deflection of a generic beam is described by means of the Euler–Bernoulli beam equation. A Cartesian reference with its origin at the bottom of the riser has been used, in which the x axis is parallel to the flow velocity, z coincides with the vertical axis of the riser in its undeflected configuration and y is perpendicular to both (see Fig. 1). The dimensionless coupled fluid–structure dynamical system is represented as

∂2 y + ∂t2



rs γ + mΩ f μ



∂y ∂2 y ∂4 y − c2 2 + b2 4 = Mq, ∂t ∂z ∂z

  ∂q ∂ 2q ∂2 y + q = As 2 , + ε q2 − 1 2 ∂t ∂t ∂t ∂2x + ∂t2



rs γ +2 mΩ f μ



(1)

(2)

  ∂2x ∂4x ∂x C Do − c2 2 + b2 4 = N C Dm + g ∂t 2 ∂z ∂z

(3)

  ∂g ∂2g ∂2x + 4g = Bs 2 + 2ε g2 − 1 ∂t ∂t2 ∂t

(4)

  C Dm 2Yrms 0.65 = 1 + 1.043 C Do D

(5)

where y = Y/D, x = X/D, and z = Z/L. D and L denote diameter and length of riser model, respectively. The time-averaged drag coefficient on a cylinder vibrating at or near the vortex shedding frequency is also a function of CF vibration amplitude [13], and some of them have been evaluated by Huang and Sworn [14]. The expression widely used for the increase in the drag coefficient with vibration is shown in Eq. (5) which is based on the results from field experiment with long flexible cylinders carried out by Vandiver [15]. This relation could couple the IL drag force with the CF response. Yrms is the standard deviation of the anti-node displacement in diameters. The dimensionless tension c, bending stiffness b and mass number M, N are given by: c2 = M=

Ttop

, b2 = mΩ 2f L 2 1 C Lo , 2 8π 2 St2 μ

EI mΩ 2f L 4 N=

, 1

8π 2 St2 μ

(6) .

q, g are reduced fluctuating lift and drag coefficients, q = 2CL /  /C ; C denotes lift coefficient. We separate the drag CLo , g = 2C D Do L

Fig. 1. Model of coupled structure and wake oscillators combining in-line and crossflow motion.

coefficient CD into two terms: one is the mean drag coefficient CDm  . The C and the other is the fluctuating drag coefficient C D Dm is the  is obtained time averaged mean of the drag coefficient CD and C D  . The by removing the time averaged mean CDm , i.e. C D = C Dm + C D coefficients CLo and CDo denote amplitude of fluctuating lift and drag coefficients for a fixed rigid cylinder subjected to vortex shedding. The values of CDo = 1.5 and CLo = 0.8 were selected in all cases [13,16]. Ω f and Ω i are CF and IL reference angular frequencies, and Ω i = 2Ω f . The values of the van der Pol parameter ε and scaling parameter As and Bs can be derived from experimental results from Facchinetti et al. [2]. Under the acceleration coupling model the value of ε is set as 0.3 according to a best-fitting on the lock-in bands for synchronization of vortex shedding with transverse cylinder vibration. The value of the combined parameter As /ε = 40 is proposed from a least-squares interpolation between lift magnification and the imposed structure motion amplitude, thus setting As = 12. Bs = 3 is an empirical coefficient in the present simulation. The riser model was pin-ended, hence displacements and curvatures were zero at each end with the following set of boundary conditions: y (0, t) = 0, y (1, t) = 0, x (0, t) = 0, x (1, t) = 0 ∀t ∂ 2 y (1, t) ∂ 2 x (0, t) ∂ 2 x (1, t) ∂ 2 y (0, t) = 0, = 0, = 0, = 0 ∀t ∂z2 ∂z2 ∂z2 ∂z2

(7)

2.2. The variation of mean top tension As for a pin-ended cylinder towed in a water tank, the applied axial tension Ttop is in fluctuation due to IL and CF vibrations. Using Hooke’s Law, the mean top tension Tmean can be defined as Tmean = Tini + E Ac

L . L

(8)

where Tini is initial tension force, L = S − L, L and S denote initial length and instantaneous length of the cylinder model, respectively;

J. Gu et al. / Applied Ocean Research 41 (2013) 1–8

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Ac is the wall cross section area. As for long flexible cylinder, the deflection induced by drag force is much greater than the lift force; therefore the elongation is mainly induced by drag force, and the approximate relation reads  L S= 1 + Y 2 dZ . (9) 0

Assuming small deflections, Y is sufficiently small to allow Eq. (9) to be approximated as   L 1 S= 1 + Y 2 dZ . (10) 2 0 Simplifying the equation, yielding  1 L 2 Y dZ . L = 2 0

(11)

For a pin-ended beam, an analytical solution can be found if deflection shape is considered to be represented by a sine series [17]: Y = 

F πZ  sin , L EIπ 4 /L 4 + Tmean π 2 /L 2

(12)

where F is external force exerted perpendicularly on the model. Here only deflection induced by drag force is considered; therefore, F can be approximately expressed as F =

1 ρU 2 DC D . 2

(13)

The mean tension, related to flow velocity U, is finally obtained by combining Eqs. (8), (11), (12) and (13): 2  E Ac ρU 2 DC D L  . Tmean = Tini + (14) 16 EIπ 2 + Tmean L 2 Eq. (14) can be calculated to get the relationship of Tmean versus reduced velocity Ur which is defined by Ur = U/fn D, where fn is the fundamental natural frequency of the model. 3. Comparison with experimental data 3.1. Verification of analytical solution A series of numerical calculations were performed as direct simulation of the experiments in uniform flow conducted by Huarte [18]. These simulations could verify the availability of the proposed analytical solution of the variation of mean top tesion and the capability of the wake oscillator model described above. The experiments were carried out at Delft Hydraulics in the Delta flume, which was used as a towing tank. The overall layout of the experiments is shown in Fig. 2. The flume is 230 m in length, 5 m in width and, in these tests, the water depth was 6.5 m. It is equipped with a heavy carriage whose maximum speed in both directions is 1.0 m/s. The flow profile was stepped with water in the lower part of the riser model. The lower 5.94 m of its length was subjected to a uniform current, the rest remained in still water. Since the riser model was not fully surrounded by water, the mean drag force only applied on the submerged part in the numerical simulation. The summary of main parameters used in the experiment is given in Table 1. The variation of mean top tension has been compared between the experimental data from Huarte [18] and the numerical prediction from Eq. (14) as shown in Fig. 3. The dashed lines are the corresponding curve fit simulated by Eq. (14). It is noted that the mean top tension increases dramatically when reduced velocity is increasing. For instance, the mean top tension is about 1300 N when reduced velocity approaches 45. The increased top tension is about 500 N higher than the initial top tension of 810 N (approximately 60% higher). Obviously, this great change of the mean top tension has to be considered

Fig. 2. Layout of experiment. The lower 5.94 m of its length was subjected to a uniform current, the rest was in still water. The speed of the carriage was stepped from the range of 0.1–1 m/s. From Chaplin et al. [21]. Table 1 Summary of main parameters of the experiment (from Huarte [18]). Experiment parameters Total length Outer diameter Submerged Length Bending stiffness Mass ratio Aspect ratio Tension at top Flow speed Reynolds number Damping ratio

Symbol

Unit

Value

L D Ls

m m m

13.12 0.028 5.94

EI

Nm2

29.88

μ Λ Ttop U Re

– – N m/s –

3 470 350–2000 <1 2800–28,000

ξ

–

0.0026–0.014

in numerical simulation. From Fig. 3, the good agreement of the proposed analytical solution with the experimental results verifies the validity of Eq. (14). The influence of the top tension will be further discussed in Section 3.3. 3.2. Dynamic response The coupled Eqs. (1)–(5) are numerically integrated in time and space using a standard centered finite difference method of the second order in both domains. As initial conditions, a random noise with amplitude of order O(10−3 ) is applied to the fluid variable q and g. Zero displacement and zero velocity initial conditions for both IL and CF are applied to the structure. The first time derivative of the fluid variables is also set zero as initial condition [3]. For the spatial discretization, 100 points are used for the simulation and a dimensionless time step of 0.01 is used. The integration is carried for dimensionless time t of 100,000 with 10 cycles. The initial values of CDm = 0.001 have been selected in all simulations. The Strouhal number St = 0.17 is chosen based on the research done by Larsen [19]. 8 cases have been conducted in their experiment, ranging from low mode response to high mode response in order to show the main features of the computed loads. Table 2 shows the results for all 8

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J. Gu et al. / Applied Ocean Research 41 (2013) 1–8

Fig. 3. Top tension variation. ♦ Ttop = 810 N,  Ttop = 1175 N,  Ttop = 1538 N,  Ttop = 1922 N, the dashed lines are the corresponding fit curves simulated by Eq. (14).

Fig. 4. Time trace of x and y motions at midspan, (a) for CF motion, (c) for IL motion, (b) and (d) are the frequency spectrum of CF and IL vibration, case 5.

cases: resonant frequencies, dominant mode number in IL and CF. Fig. 4 depicts a typical result at midspan of IL and CF motions in each simulation. The power spectrum of IL motion was calculated by removing the mean displacement. It is clear that the vibration of CF

has larger amplitude but lower frequency than IL vibration, which agrees with the experimental observation. Since the riser model has high aspect ratio, it is expected that the vibration of the riser model should be in multi-mode. Hence, modal analysis has been conducted to identify different modes of vibration. The methodology can be found in the literature [20]. By applying modal analysis here, the mode number up to 10 has been chosen to separate the original response into each mode weight contribution as shown in Figs. 5 and 6 for the case 3. The left column (a) of Figs. 5 and 6 shows non-dimensional displacement modal amplitudes without mean, the right column (b) shows the spectra of each modal amplitude. It can be seen in this case that a strong multi-modal behavior occurs with several modes running at the same frequency. The each modal amplitude indicates that in the case 3 the IL and CF vibrations have been dominated by the 7th mode and 4th mode, respectively. The other modes have less contribution to the overall response. Meanwhile, all the IL dominant frequencies are the same, which are roughly twice that of all the CF modes. The numerical observation is in good agreement with that found in the experiment from Chaplin et al. [11] and Huarte [18]. The comparison of resonant frequencies and the mode numbers are given in Figs. 7 and 8, respectively. Most of the resonant frequencies from numerical simulation are perfectly in agreement with measurement, except in the cases 2 and 4, where the IL resonant frequencies are 85% and 110% of the measured frequencies, respectively. The prediction of the mode numbers is much better. Only 4 points have 1 mode number discrepancy. This result is better than that published by Xu et al. [4] and Ge et al. [5,6]. Fig. 9 shows the CF and IL maxima amplitudes without mean.

J. Gu et al. / Applied Ocean Research 41 (2013) 1–8

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Table 2 Summary of experimental results (from Huarte [18]). Case 1 2 3 4 5 6 7 8

U (m/s) 0.31 0.55 0.60 0.85 0.49 0.63 0.75 0.85

Re 7622 13,649 14,851 20,928 12,191 15,559 18,422 20,900

T (N) 1175 1922 1538 1922 810 810 810 810

Dominant modes ny

fy

nx

fx

2 3 4 4 4 5 6 7

1.66 3.33 3.52 4.75 3.00 3.66 4.71 5.32

4 6 7 7 8 9 11 12

3.40 6.63 7.04 9.45 6.00 7.33 9.43 10.67

Fig. 5. Non-dimensional IL displacement modal amplitudes without mean at spanwise position z = 4.92 m, case 3. Fig. 8. Dominant mode numbers of IL and CF motions. × +: experiment CF and IL mode numbers;  : CF and IL mode numbers by wake oscillator model.

Fig. 6. Non-dimensional CF displacement modal amplitudes without mean at spanwise position z = 4.92 m, case 3.

Fig. 9. CF maxima amplitudes and IL maxima amplitudes without mean. × +: experiment CF and IL maxima amplitudes;  : CF and IL maxima amplitudes by wake oscillator model.

Fig. 7. Resonant frequencies of IL and CF motions for all 8 cases. × +: experiment CF and IL dominant frequencies;  : CF and IL dominant frequencies by wake oscillator model.

While most of the simulations are in good agreement with experiment, exceptions are cases 2 and 4 in CF direction and case 8 in CF and IL directions. As can be seen from Fig. 9, normally, CF amplitude lays between 100% and 117%, and IL amplitude between 100% and 125% of the corresponding measurement. Fig. 10 shows IL maxima amplitudes with mean. Most of the IL amplitude lays between 76% and 100%, exception case 1. The comparisons of deflection and mode shape from experiments and numerical calculations are given in Fig. 11, for case 3, and Fig. 12, for case 5. The typical instantaneous IL deflections from measurement are shown in first column with the mean deflected shape and without it in second column; RMS of the IL in third column; instantaneous CF deflections in forth column; RMS of the CF in fifth column.

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J. Gu et al. / Applied Ocean Research 41 (2013) 1–8

Fig. 10. IL maxima amplitudes with mean. +: experiment IL maxima amplitudes, : IL maxima amplitudes by wake oscillator model.

Fig. 12. The comparison between experiment (upper) and wake oscillator model (lower). The instantaneous IL deflections are shown in first column with the mean deflected shape and without it in second column; RMS of the IL third column; instantaneous CF deflections forth column; RMS of the CF fifth column, case 5.

Fig. 11. The comparison between experiment (upper) and wake oscillator model (lower). The instantaneous IL deflections are shown in first column with the mean deflected shape and without it in second column; RMS of the IL third column; instantaneous CF deflections forth column; RMS of the CF fifth column, case 3.

The lower row shows the corresponding results from the numerical simulation. The CF deflected shape as shown in cases 3 and 5 depicts a standard standing wave which is identical to the measurement. The instantaneous IL deflections without mean presents a travelling wave more intense than the measurement. The travelling wave was captured by Chaplin et al. [11] with the same experiment set-up but with a different case. As mentioned, in the absence of any knowledge of distribution of added mass, and neglecting the effect of variations in tension, the mode shapes have been assumed to be sinusoids. The fact that all contributing modes defined in this way are neither in phase nor in anti-phase with each other, and the fact that there are no pure nodes in the profiles (as shown in Fig. 12 second column, second row), indicated that the motion in both directions is a travelling wave. It was noticed that in all cases the maxima IL mean displacements were placed in the lower part of cylinder model. This is confirmed both by the experiment and numerical simulation. The probable reasons were: (1) the drag force exerted on the lower part of the riser model; (2) the reducing axial tension force descending along the flexible cylinder. That also might be one of the factors that induced the mode numbers overestimation from Xu et al. [4]. Fig. 13 shows the orbital plot (IL versus CF motion) at each depth along the axis of the riser model for case 3. The left graph shows the trajectories of present numerical results, and the right graph shows the experimental results from Huarte [18]. It is clearly observed that

Fig. 13. IL and CF trajectories. Left graph: present numerical results. Right graph: experimental result from Huarte [18], case 3.

the figure-8 pattern depicts different forms in these horizontal planes in the left graph, which is in agreement with the experimental observation by Huarte [18] in the right graph. 3.3. Influence of mean top tension The mean top tension would increase when the flow velocity increased as we derived the relation in Eq. (14). Fig. 3 verified the good representation. Fig. 14 shows the CF and IL model numbers versus reduced velocity in terms of different initial top tension 810 N, 1175 N,1538 N, and 1922 N. The solid square and cycle symbols represented the experimental results. Tmean = Ttop means the mean top tension is equal to the initial top tension when the towing speed is changed, i.e. the variation of mean top tension is not considered in all simulations; Tmean is varying means all the simulations will consider the variation of mean top tension based on Eq. (14). From Fig. 14 it is noticed that when the reduced velocity Ur is less than 25, the CF and IL mode numbers are almost the same in terms of different consideration of mean top tension; when the reduced velocity Ur is larger than 25, the CF and IL mode numbers of Tmean = Ttop are more or less higher than that of Tmean is varying. The discrepancy of mode numbers increases with the reduced velocity increased, and the experimental results are in better agreement with the simulation of Tmean is varying than that of Tmean = Ttop , such as cases 5–9 in the

J. Gu et al. / Applied Ocean Research 41 (2013) 1–8

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Fig. 14. Mode number versus reduced velocity in terms of Tmean = Ttop and Tmean is varying. Tmean is varying based on Eq. (14). , : CF and IL mode number when Tmean is varying; × , +: CF and IL mode number when Tmean = Ttop ;  , •: experimental result from Huarte [18].

plot of Ttop = 810 N, cases 2 and 3 in the plot of Ttop = 1922 N. It makes sense as we know increasing axial tension will increase the natural frequency of cylinder model, then the frequency of excitation (vortex shedding frequency) has to be increased consequently in order to obtain the same mode number that means a higher towing speed is needed. In another words, when the towing speed is the same, the frequency of excitation only can be “controlled” in lower natural frequency of cylinder model which should be applied with a lower top tension. Meanwhile, as shown in all the plots, the experimental results agreed better when Tmean is varying than Tmean = Ttop , so the variation of mean top tension should not be neglected in the case of long flexible cylinder simulation, especially at the range of high reduced velocity. That also might be one of the factors that induced the mode numbers overestimation from Xu et al. [4].

4. Conclusions An analytical solution of mean top tension of long flexible risers has been proposed to predict the nature of VIV of long flexible risers in this paper. The analytical solution of mean top tension is verified with experimental results to show its validity. Then the nonlinear coupled wake oscillator model has been performed to simulate the dynamic response of long flexible risers. The comparisons of the dominant resonant frequencies, mode numbers and displacements between experimental data and the numerical simulation present the excellent agreement between the numerical simulation and the experimental results. Hence, the variation of the mean top tension is an important factor to affect the prediction of multi-mode VIV; and it should not be negligible in numerical simulation of dynamic response of such long structures. Due to the analytical solution of the mean top tension and the

simplicity of the wake oscillator model, all the results presented in this paper required a short period of computational time, which is more practicable than the computational fluid dynamics (CFD) method. Acknowledgments The authors would like to thank the Brazilian National Research Council (CNPq) and the National Basic Research Program of China (973 Program) Grant No. 2011CB013702 for the financial support of this research. References [1] Birkoff G, Zarantanello E. Jets, wakes and cavities. New York: Academic Press; 1957. [2] Facchinetti ML, de Langre E, Biolley F. Coupling of structure and wake oscillators in vortex-induced vibrations. Journal of Fluids and Structures. 2004;19(2):123– 40. [3] Violette R, de Langre E, Szydlowski J. Computation of vortex-induced vibrations of long structures using a wake oscillator model: comparison with DNS and experiments. Computers & Structures. 2007;85(11–14):1134–41. [4] Xu W-H, Zeng X-H, Wu Y-X. High aspect ratio (L/D) riser VIV prediction using wake oscillator model. Ocean Engineering. 2008;35(17/18):1769–74. [5] Ge F, Long X, Wang L, Hong Y. Flow-induced vibrations of long circular cylinders modeled by coupled nonlinear oscillators. Science in China Series G-Physics Mechanics & Astronomy. 2009;52(7):1086–93. [6] Ge F, Lu W, Wang L, Hong Y-S. Shear flow induced vibrations of long slender cylinders with a wake oscillator model. Acta Mechanica Sinica. 2011;27(3):330– 8. [7] Guo H-y, Li X-m, Liu X-c. Numerical prediction of vortex-induced vibrations on top tensioned riser in consideration of internal flow. China Ocean Engineering. 2008;22(4):675–82. [8] Li X-m, Guo H-y, Meng F-s. Nonlinear coupled in-line and cross-flow vortexinduced vibration analysis of top tensioned riser. China Ocean Engineering. 2010;4:749–58. [9] Meng D, Chen L. Nonlinear free vibrations and vortex-induced vibrations of fluid-conveying steel catenary riser. Applied Ocean Research. 2012;34:52–67, ISSN 0141-1187.

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