Proceedings of the ASME nd International Conference on Ocean, Offshore and Arctic Engineering OMAE2013 June 9-14, 2013, Nantes, France

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1 Proceedings of the ASME nd International Conference on Ocean, Offshore and Arctic Engineering OMAE2013 June 9-14, 2013, Nantes, France VIV OF FLEXIBLE CYLINDER IN OSCILLATORY FLOW OMAE Shixiao Fu State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University Shanghai, China Rolf Baarholm Statoil Trondheim, Norway Jungao Wang State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University Shanghai, China Jie Wu Marintek Trondheim, Norway C. M. Larsen Dep. of Marine Technology Centre for Ships and Ocean Structures, NTNU Trondheim, Norway ABSTRACT VIV in oscillatory flow is experimentally investigated in the ocean basin. The flexible test cylinder was forced to harmonically oscillate in various combinations of amplitude and period. VIV responses at cross flow direction are investigated using modal decomposition and wavelet transformation. The results show that VIV in oscillatory flow is quite different from that in steady flow; novel features such as intermittent VIV, amplitude modulation, mode transition are observed. Moreover, a VIV developing process including Building-Up, Lock-In and Dying-Out in oscillatory flow, is further proposed and analyzed. Keywords:, Vortex-induced Vibration, Oscillatory Flow, number,flexible Cylinder, Intermittent INTRODUCTION As natural gas and oil production extend to deep-water and even ultra-deep-water areas, the VIV of flexible risers under such severe marine environments shows new characteristics such as multi-mode participation, strong randomness, etc. Prediction of hydrodynamic loads and dynamic response of flexible risers is hence becoming one of the most urgent and essential issues in ocean engineering. In general, two methods are now utilized by the industry: numerical empirical model [1-2] and model test [3-12]. In these tests, particular phenomena are observed, including high-order and multi-mode response, traveling wave and coupling between CF and IL direction, which surely provide clearer views for us to understand and further predict VIV. However, only steady flows such as uniform current, sheared current and stepped current are considered during these tests. Recently, an intermittent VIV was found occasionally in the sag bend close to TDP of a SCR from the JIP called STRIDE carried out in the US [13]. Later, it is proved that this sort of VIV is excited not by the steady flow but by the relative oscillatory flow. Under real marine conditions, complicated motions and responses with large amplitude may take place in offshore structures, which would force the underwater pipelines such as risers and umbilical cables to move periodically. Such kind of motion between pipelines and water would generate a relatively oscillatory flow. And the magnitude of this oscillatory flow can be described by the number expressed as Eq. (1), where A m is the amplitude and D is the diameter of the structure. In terms of steel catenary risers (SCR), steady shedding vortices can hardly be formed on the locations where the number is relatively small. Yet as the number increases, there is sufficient distance to generate steady vortices which would further lead to VIV. Therefore, the VIV characteristics in oscillatory flow are essential to the accurate prediction of the dynamic response of flexible risers like SCR. 2 Am (1) D So far, there have been some researches on VIV in oscillatory flow. Chang [14] predicted the riser s VIV in unsteady 1 Copyright 2013 by ASME

2 flow generated by the heave motion of the platform based on the wake oscillator and discrete vortex model. But the accuracy of this prediction has not been examined by model test. Liao [15] adopted the reduced damping parameter and the wave propagation parameter n, to predict such VIV. And he also proposed a numerical method to address the moving boundary problem and analyzed the relationship between excited frequency, natural frequency, shedding frequency and response frequency of the numerical model. Nevertheless, no relevant tests have been conducted to verify these methods. Enrique [16] calculated the distributions of maximum velocity, maximum shedding frequency and number distribution along axial direction of a SCR with forced motion on the top by FEM and estimated whether VIV would occur. Meanwhile, an indoor SCR model test with top-excited motion was carried out in still water by Enrique, but only the top reaction force was measured. Together with observation, FFT result of the reaction force also showed that VIV occurred on the SCR under top excited motion. But obviously, the information only from the top reaction force is not sufficient in revealing the overall situation of the whole riser, especially the sag bend region where VIV is expected to occur. In summary, most of the studies concerning a riser s VIV only take the steady current into consideration. As for VIV in unsteady flow like oscillatory flow, academics have launched certain theoretical and experimental researches, yet the mechanism of VIV in oscillatory flow remains undiscovered, as does the intrinsic nature between such VIV and dominant parameters such as number and oscillatory period. In this paper, a model test of a flexible cylinder in oscillatory flow was conducted aimed at ascertaining the VIV response characteristics under different numbers and oscillatory periods. Modal decomposition and wavelet analysis are applied to study VIV along CF direction. VIV in different test groups is discussed and a special VIV developing process in oscillatory flow is proposed. MODEL TEST DETAILS Experimental setup The model test was carried out in the ocean basin at Shanghai Jiao Tong University, and the whole experimental setup was installed under the bottom of the carriage (Figure 1), which mainly contains two horizontal tracks, two vertical tracks and the test model as shown in Figure 2. Units at the end of the model were assembled as Figure 3 shows, where one end of the universal joint was connected to the model by a clamp and the other end was connected to the force transducer, which was in turn fixed to the slider of the vertical track. Meanwhile, two circular endplates were installed to avoid the perturbation caused by the motion of the supporting frame. Main physical properties of the test model were listed in Table 1 below. Sg Table: 1 Physical Properties of the Test Cylinder Item Value Model Length( m ) 4 Outer Diameter( m ) Mass in air( Kg/ m ) Bending Stiffness( N m ) 10.5 Tensile Stiffness( N ) Pretension( N ) st natural frequency in water( Hz ) nd natural frequency in water( Hz ) 5.46 Force transducers at both ends of the model were used to measure the real-time axial tension. Besides, four groups of FBG strain sensors were instrumented along both CF and IL directions to capture the VIV responses. From the schematic diagram in Figure 4, line CF_a and line CF_c along CF direction are symmetric with respect to the neutral layer of the model while line IL_b and IL_d belong to IL direction. Detailed sensor information is given in Table 2. The sampling rate in the model test is 250Hz. Table: 2 Arrangements of the FBG Strain Sensors Test content Location Sensor No. Spacing( m ) CF _ a IL _ b CF _ c IL _ d In the model test, number and oscillatory period are two key parameters where number determines the in line distance while oscillatory period controls the maximum shedding frequency. So we vary number from 26 to 178 and then calculate a corresponding oscillatory period to maintain a constant maximum shedding frequency even though the number changes. In addition, three groups of cases are designed in which the shedding frequencies are all near the first natural frequency to deal with the problem that the natural frequencies may change as the axial tension and added mass coefficient vary due to the oscillation. All the test cases are listed in Table 3. 2 Copyright 2013 by ASME

3 Table: 3 Test Cases Group A B C Cases No Ur f st _max (Hz) A m (m) [0.1,0.02,0.68] [0.1,0.02,0.68] [0.1,0.02,0.68] Re_max So, the amplitude and velocity of the model can be expressed as Eq. (2) and Eq. (3). 2 A( t) Am sin ( t) (2) T 2 2 V ( t) Am cos( t) (3) T T DATA ANALYSIS Tension Influence Elimination Before the test, a constant pretension was acted on the ends of model, but it varied over time due to the oscillation and deformation of the model. Accordingly, the measured strain contains three parts: initial axial strain caused by pretension, axial strain due to tension variation and bending strain caused by VIV. Therefore, the pure VIV strain _ () in CF direction is calculated by CF VIV t CF _ VIV CF _ a CF _ a_ initial CF _ c CF _ c_ initial ( t)=[( ( t)- )-( ( t)- )]/2 (4) where () and () CF a t CF c t denote the original strain time histories sampled at location CF _ a and CF _ c, is the initial axial strain caused by pretension. Modal Decomposition Modal decomposition is based on the assumption that the displacement of the cylinder could be expressed as a sum of mode-shapes with different modal weights at any time step: n i i (5) i1,, 0, y z t p t z z l where z denotes the position along the cylinder, pi () t is the i th modal weight of displacement, and i () z is the i th mode-shape of the displacement. Based on the small deformation assumption, the curvature can be calculated by: 2 n d y ( z, t) 2 pit iz (6) dz i1 where i () z is the i th mode-shape of the curvature. And according to the geometric relationship between bending strain and curvature, we have zt, zt, (7) R where R is the outer radius of cylinder. Combing Eq. (6) and Eq. (7), CF displacements can be obtained from Eq. (5). In our model, modal shapes of displacement are sinusoidal, which can be expressed as i z φi z sin, i 1,2, (8) l The mode-shapes of curvature are also sinusoidal and Eq. (6) can be rewritten as n n i i1 l i1 2 z, t pi t i z ui t i z (9) 2 i ui t pi t (10) l where ui () t is the i th modal weight of curvature with respect to () z. Therefore, the strain becomes i n z, t, 2 z t R n i R p t z e t z i i i i i1 l i1 (11) i ei t R pi t (12) l where ei () t is the i th modal weight of strain with respect to i () z. Wavelet Analysis According to Eq. (3), the model velocity would change periodically, which leads to a changing shedding frequency according to Eq. (13). Since the shedding frequency is not a constant, it is supposed that the VIV characteristics in oscillatory flow would also be time-varying. This should be the major difference between VIV in steady flow and that in oscillatory flow. Thus, it is necessary to introduce wavelet analysis here to obtain the time-frequency distribution of VIV in oscillatory flow. St V () t fst ( t)= (13) D The continuous wavelet transform equation is expressed as Eq. (13). + -1/2 t- WTf ( a, )= f ( t), a, ( t) = a f ( t) *( ) dt (14) - a where WT ( a, ) is the coefficient of the time domain signal f after wavelet transformation which represents the change of 2 3 Copyright 2013 by ASME

4 frequency on the time scale. While a is the scale factor, t is the shift factor and () t is the mother wavelet. In this paper, we choose Morlet wavelet equation as the mother wavelet which can be defined as 2 - /2 ( t)=ce i cos(5 x) (15) Seven messages can be obtained from a typical result of case (T=16.5s, =178) after data processing shown in Figure 5: 1) Location of the measuring point is CF4, midpoint of the model, as shown in Figure 4. The oscillatory amplitude A =0.68 m m, oscillatory periodt 16.5sand 178 are also exhibited. 2) Row a is shedding frequency varying with time which is calculated by Eq. (13). 3) Row b is displacement time history at CF4 calculated from VIV strain by modal analysis. It clearly illustrates the fluctuation of the VIV response. In this paper, we assume that VIV occurs when response amplitude is greater than 0.05D. Moreover, the maximum response amplitude is also marked in the graph. 4) Row c is wavelet contour plot of strain time history, where horizontal axis represents time and vertical axis represents frequency, while the color shows the concentration level of the vibrational energy. Both instant response frequency and strength of the vibrational energy are quite evident through the contour plot. 5) Real-time natural frequencies are also plotted in the same coordinate in row c which is calculated combing with real-time tension by Eq. (16), assuming the added mass coefficient C A =1. In the graph, the solid lines denote the first and the second natural frequencies separately. It set up a reference for the analysis of VIV response frequency in oscillatory flow. 2 2 n F_ Axial () t n EI fnatu _ n ( t)= +, = + ( 1) 2 m ms mh CA (13) 2l m l m 6) Row d is the modal weight time history of the first and second modes. It shows the mode participation with time which would help us to understand the mode transition more directly. 7) Time interval between the two black vertical dashed lines is one complete flow oscillatory period marked as T. The moments located by red dashed lines are the times that VIV starts, while the green ones indicate when VIV ends. Locations marked by the symbols S and E are the instant frequencies when VIV respectively starts or ends. RESULTS AND DISCUSSION Throughout the data processing, it is discovered that VIV occurs in all the cases in Table 3, but such VIV in oscillatory flow is quite distinct from that in steady flow as we supposed. A novel feature, intermittent VIV, is observed. And a VIV developing process in oscillatory flow is divided into three regions: Building-up, Lock-in and Dying-out. Furthermore, maximum response amplitude, dominant frequency and dominant mode are depend on both number and maximum shedding frequency as determined by oscillatory period. The main features of VIV in different oscillatory flow, as given in Table 3, are introduced and explained. Results in Group A In group A, the maximum shedding frequency is designed to be less than the first natural frequency. Result of case (T=16.5s, =178) shown in Figure 5 is taken as a typical example to demonstrate VIV features. Firstly from the displacement time history in row b, it is obvious that VIV is intermittent, as it only occurs in part of the half oscillatory period, about 0.2T. During the VIV time interval, the response amplitude becomes greater with the increase of the shedding frequency and reaches a peak at around 0.36D, at the moment when the shedding frequency increases to maxima. It then becomes smaller as the shedding frequency decreases. And from the envelop line of displacement, it is clearly indicated that the response amplitudes are modulated. Moreover, to have a clearer view of the displacement, Figure 6 shows the displacement time history for half the oscillatory period. The envelop line is shaped like a triangle which indicates that the VIV is not quite stable. But we assume that the time period, when amplitude fluctuation is greater than 2 2 (A/D) max, is considered to have steady VIV as marked in Figure 6. Therefore, the half oscillatory period is separated into three parts in terms of the VIV developing process: Building-up region, Lock-in and Dying-out as illustrated in Figure 6. In case (T=16.5s, =178), the Lock-in region is about 17% of the half oscillatory period. Meanwhile, from the wavelet contour plot in row c of Figure 5, VIV time interval is periodically distributed which also indicates that such VIV is intermittent. And the concentration levels of the vibration energy show quite a stable response frequency at around 2.1Hz, which happens to be equal to the maximum shedding frequency. But it is worth noting that we choose C A =1 to calculate the natural frequency. In fact, the added mass coefficient should be greater than 1 to adjust the real natural frequency so that it is closer to the maximum shedding frequency. In row d, mode 1 is the absolute dominant mode all the time. Results in Group B In group B, the maximum shedding frequency is designed to be close to the first natural frequency. Result of case (T=10.2s, =178) in Figure 7 is taken as a typical example demonstrating the VIV features. Firstly, from the displacement time history in row b, VIV occurs in the whole oscillatory period. During the VIV time interval, the response amplitude becomes greater with the increase of the shedding frequency and reaches a peak at around 0.58D, before it then becomes smaller as the shedding frequency decreases. From the envelop line, it is also clearly indicated that the response amplitudes are modulated. 4 Copyright 2013 by ASME

5 And in Figure 8, the envelop line is shaped like a trapezium which means there is more time for steady VIV compared to case (T=16.5s, =178). And in case (T=10.2s, =178), the Lock-in region is about 34% of the half oscillatory period. Moreover, oscillatory flow is in fact a combination of a continuous acceleration and deceleration process; the trend of the envelop line ( ( A/ D) acc in acceleration process is a lot smaller than ( A/ D) dec in deceleration when the instant shedding frequency is the same as shown in Figure 7). This shows apparent hysteresis as a distinguish feature which agrees with the hysteresis loop discovered in the free vibration test of an elastically mounted cylinder [17]. Meanwhile, from the wavelet contour plot in row c of Figure 7, the VIV time interval is still periodically distributed. And the concentration level of the vibration energy shows a stable response frequency at around 3Hz, which accords well with the first natural frequency as the red solid line illustrates. In row d, mode 1 is the dominant mode all the time. Mode 2 only takes part occasionally and to a small extent. Results in Group C In group C, the maximum shedding frequency is designed to be larger than the first natural frequency, but it is also close to the second natural frequency. The measuring point CF4 is at the midpoint of the model, which happens to be the nodal position of the second mode shape. So a relatively larger second modal weight may lead to a weaker response which cannot show the overall VIV information. Therefore, the result at CF2 is chosen to explain the experimental phenomena in group C. Result of case (T=8.45s, =178), as shown in Figure 9, is taken to be a typical example that demonstrates the VIV features. Firstly, from the displacement time history in row b, VIV occurs in the whole oscillatory period, but the displacement is a little bit chaotic because mode 2 also participates in the whole response. During the VIV time interval, the response amplitude goes up slightly at first and reaches a peak at around 0.47D, before it goes down slightly as the shedding frequency decreases. From the envelop line, the response amplitudes are modulated, but it is less obvious compared to those in group A and group B. Moreover, the envelop line is still shaped like a trapezium in Figure 10. And in case (T=8.45s, =178), the Lock-in region is about 50% of the half oscillatory period, which is three times that in case(t=16.5s, =178). Meanwhile, from the wavelet contour plot in row c of Figure 9, the concentration level of the vibration energy is quite different from the levels in group A and group B. There are two response frequencies at different moments. It is obvious that the mode 2 contributes a lot in the middle of half the oscillatory period. So there is such a mode transition during every half oscillatory period in group C. In row d, when time = 60s and time = 64s, we can clearly find out that participation of mode 2 is also considerable and further proves the mode transition. General Discussion It is observed that VIV in oscillatory flow occurs in all cases, whereas intermittent VIV is observed only in group A. Based on these results, VIV characteristics vary when the or oscillatory period changes. But generally speaking, there are some obvious trends in terms of maximum response amplitude and Lock-in region. f f, the maximum response amplitude When st _ max natu _1 always occurs at CF4 at around 0.4D for most of the cases as shown in Figure 11. When the number is small, the amplitude becomes relatively large; this is irregular for the reason that the in-line distance is short in such number cases. Since the response amplitude is on the contrary larger, it is assumed that the oscillatory period is too short to calm the wake and die out the existing VIV. Accordingly, the cylinder is still under the influence of its own wake when it reverses to a next oscillatory period, and it is such wake that magnifies the response of the cylinder. Figure 12 presents the portion variation of three VIV developing regions, where the horizontal axis denotes number, the vertical axis denotes the half normalized oscillatory period and each color denotes one region. The symbols a, b, c and d denote the moments that one region starts or ends, and these also accord with the same symbols in Figure 6. As number increases, the Lock-in region becomes smaller and when >100, it reaches a relatively constant value, about 17% of half the oscillatory period. It is also assumed that the previous wake magnifies the response instead when in small. f f, the maximum response amplitude When st _ max natu _1 occurs at CF4 or CF3 and CF5 in different cases, and keeps a relatively stable value at around 0.5D (Figure 13). This is because of the participation of mode 2. In Figure 14, as number increases, the Lock-in region becomes smaller. It is also assumed that the previous wake magnifies the response instead when is small. And the time for the Building-up region is always longer than that for Dyingout region. This once again shows the hysteresis. f > f, the maximum response amplitude When st _ max natu _1 distribution becomes a little bit chaotic, and the maximum amplitude is about 0.6D for all the cases as shown in Figure 15. This also accounts for the participation of mode 2. In Figure 16, it all belongs to the Lock-in region when <100, and then when increases, the Lock-in region becomes barely the same, 50% of half the oscillatory period. Similarly, the time for Building-up region is always longer than that for Dying-out region. By comparing Figure 11, Figure 13 and Figure 15, we can see that the maximum displacement amplitudes for group B and group C are greater than those for group A. The reason is that the reduced velocities in group B and group C are 6.5 and Copyright 2013 by ASME

6 with respect to the first natural frequency, while the reduced velocity for group A is only 4, since the maximum response amplitude appears at the moment when the reduced velocity is around 5-7 typically [18]. By comparing Figure 12, Figure 14 and Figure 16, two obvious trends are seen: 1) There is a shift from 100% Lock-in to a split between three different regions at a number. The reason for this is that number of VIV response cycles per motion cycles will increase for increasing numbers. Hence, number of cycles in a damping condition will increase, and damping will kill the VIV response for each motion cycle. 2) The shift from pure Lock-in to the process including three different regions takes place for increasing number from group A to group B to group C. The reason for this is that number of VIV response cycles per motion cycle for the same numbers is higher for group A than group B and group C, due to the different maximum shedding frequency. Hence, the number of cycles in a damping condition will be higher for A than B than C. Hence, in terms of real steel catenary risers, the shedding frequency by relative motion is normally larger than its natural frequency [16]. Combining with the observations we obtained from the model test, there exists such a Lock-in region and it becomes wilder in half the oscillatory period when maximum shedding frequency increases. Consequently, VIV in such oscillatory flow, generated by the motion of the top floating structures, should not be neglected. CONCLUSIONS In summary, several conclusions are drawn as follow: 1) A VIV developing process in oscillatory flow is defined as three regions: Building-up, Lock-in and Dying-out. As maximum shedding frequency increases, the Lock-in region becomes wilder and more cases with relatively large numbers tend to have a pure Lock-in region. As the number increases, the Lock in region tends to reach a relatively constant value, and it becomes wider as the maximum shedding frequency increases. 2) VIV occurs in all cases in oscillatory flow, and intermittent VIV is discovered when fst _ max fnatu _1, with relatively smaller maximum response amplitude at about 0.4D. It goes up to 0.6D when maximum shedding frequency increases, and it is barely independent of number. 3) A phenomenon such as amplitude modulation exists for VIV in oscillatory flow, but such a trend becomes inconspicuous when maximum shedding frequency increases. 4) Hysteresis, as a special phenomenon of VIV in steady flow, is also observed from VIV in oscillatory flow. 5) Mode transition exists when the shedding frequency is sufficiently large. ACKNOWLEDGEMENT This study was supported by Statoil, Norway. We would like to express thanks for their support. NOMENCLATURE l Model Length [m] D m s m H Outer Diameter [m] Mass in Air [Kg/m] Added Mass[Kg/m] EI Bending Stiffness[N m 2 ] EA F Axial f natu_n A m V T St f st C A Re Tensile Stiffness[N] Strain Axial Tension[N] Natural Frequency[Hz] number Oscillatory Amplitude[m] Velocity[m/s] Oscillatory Period[s] Strouhal number [s] Shedding Frequency[Hz] Added Mass Coefficient in Still Water Reynolds number REFERENCES [1]. J.Kim.Vandiver and Li Li. SHEAR7 V4.4 PROGRAM THEORETICAL MANUAL [M]. Department of Ocean Engineering, MIT. 2005: 32 [2] C.M.Larsen, K.Vikestad, R.Rttervik, E.Passano and G.S.Baarholm. VIVANA, Theory Manual[M]. MARINTEK, Trondheim, 2001 [3].Griffin,O.M., Vandiver,J.K., Vortex-induced strumming vibrations of marine cables with attached masses[j],asme Journal of Energy Resources Technology, 1984, 106, pp [4] Lie, H., Kaasen, K.E., Modal analysis of measurements from a large-scale VIV model test of a riser in linearly sheared flow [J]. Journal of Fluids and Structures, 2006, 22(4): [5] Karl H. Halse, Kunt Mo, Vortex Induced Vibrations of A Catenary Riser [C]. 3rd International Symposium on Cable Dynamics, 1999 Trondheim [6]. D.W.(Don)Allen and D.L.Henning, Prototype Vortex- Induced Vibration Tests for Production Risers [C], Pro.of the OTC,Houston,USA,2001,Paper OTC [7] M.A. Tognarelli, S.T. Slocum, W.R. Frank, R.B. Campbell, VIV Response of a long Flexible Cylinder in Uniform and Linearly Sheared Currents[C], Pro.of the OTC, Houston,USA,2004,Paper OTC [8] DE Wilde, Jaap J. and Huijsmans, Rene H.M., Laboratory Investigation of Long Riser VIV Response [C] ISOPE Conference, Toulon, 2004 [9]. J.R. Chaplina, P.W. Bearmanb, F.J. Huera Huarteb, R.J. Pattendena, Laboratory measurements of vortex-induced 6 Copyright 2013 by ASME

7 vibrations of a vertical tension riser in a stepped current[j], Journal of Fluids and Structures, 2005,21,pp.3-24 [10] A.D Trim, H. Braaten, H. Lie, M.A. Tognarelli, Experimental investigation of vortex-induced vibration of long marine risers [J]. Journal of Fluids and Structures, 2005, 21: [11] J.K. Vandiver, H. Marcollo, S. Swithenbank and V. Jhingran, High Mode Number Vortex-Induced Vibration Field Experiments [C], Pro.of the OTC,Houston,USA,2005,Paper OTC [12] Shixiao Fu, Tie Ren, Runpei Li and Xuefeng Wang, Experimental Investigation on VIV of the Flexible Model Under Full Scale Re Number [C].30th OMAE, Rotterdam. Paper No. OMAE [13] Robert G. Grant, Richard W. Litton and Partha Mamidipuli. Highly Compliant Rigid (HCR) Riser Model Tests and Analysis [C]. Pro.of the OTC, 1999, Huston. Paper No. OTC [14] S.-H. mark Chang, Mike Isherwood. Vortex-Induced Vibrations of Steel Catenary Risers and Steel Offloading Lines due to Platform Heave Motions [C]. Pro.of the OTC, 2003,Huston. Paper No. OTC [15] Jung-Chi Liao. Vortex-induced Vibration of Slender Structures in Unsteady Flow [D]. Massachusetts Institute of Technology, 2002 [16] Enrique C. Gonzalez. High Frequency dynamic response of marine risers with application to flow-induced vibration [D]. Massachusetts Institute of Technology, 2001 [17] Williamson, C.H.K., Govardhan, R., Vortex-induced vibrations [J]. Annual Review of Fluid Mechanics. 2004, 36, [18] Vikestad, Kyrre, Larsen, Carl M., Vandiver, J.Kim, Experimental study of excited circular cylinder in current [C]. 16th OMAE, Yokohama. Paper No. OMAE Copyright 2013 by ASME

8 Figure: 1 Overview of the whole experimental setup End Plate Serve Motor Flexible Cylinder Vertical Track Horizontal Track Side Support Figure: 2 Simplified sketch of the setup 8 Copyright 2013 by ASME

9 Figure: 3 Detailed view of the end connection Figure: 4 Instrumentation of the model 9 Copyright 2013 by ASME

10 Figure: 5 Result of case (T = 16.5s, = 178) at CF4 Figure: 6 VIV developing process of case (T = 16.5s, = 178) at CF4 10 Copyright 2013 by ASME

11 Figure: 7 Result of case (T = 10.2s, = 178) at CF4 Figure: 8 VIV developing process of case (T = 10.2s, = 178) at CF4 11 Copyright 2013 by ASME

12 Figure: 9 Result of case (T = 8.45s, = 178) at CF2 Figure: 10 VIV developing process of case (T = 10.2s, = 178) at CF2 12 Copyright 2013 by ASME

13 MAX_DISP(A/D) Half Normalized time Interval(t/T) MAX_DISP(A/D) Group A (f st_max < f natu_1 ) CF1 CF2 CF3 CF4 CF5 CF6 CF Figure: 11 Distribution of maximum response amplitude in group A Building up Region Lock-in Region Dying out Region d c b a Figure: 12 Time interval distribution of VIV developing process in group A Group B (f st_max close to f natu_1 ) CF1 CF2 CF3 CF4 CF5 CF6 CF Figure: 13 Distribution of maximum response amplitude in group B 13 Copyright 2013 by ASME

14 Half Normalized time Interval(t/T) MAX_DISP(A/D) Half Normalized time Interval(t/T) Building up Region Lock-in Region Dying out Region d c b a Figure: 14 Time interval distribution of VIV developing process in group B Group C (f st_max >f natu_1 ) CF1 CF2 CF3 CF4 CF5 CF6 CF Figure: 15 Distribution of maximum response amplitude in group C Building up Region Lock-in Region Dying out Region d c b a Figure: 16 Time interval distribution of VIV developing process in group C 14 Copyright 2013 by ASME