An experiment study of vortex induced vibration of a steel catenary riser under steady current ＊

Tie Ren , Yu-wang Xu Jungao Wang, Hao-jie Ren Meng-meng Zhang Shi-xiao Fu Yao-song Chen

1. State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

2. Collaborative Innovation Centre for Advanced Ship and Deep-Sea Exploration, Shanghai 200240, China

3. Marine Design and Research Institute of China, Shanghai 200011, China

4. Norweigian Public Road Administration, Stavanger, Norway

Abstract: The vortex induced vibration (VIV) of marine risers has been investigated by many researchers in experimental studies of a straight flexible riser model as well as a rigid cylinder to reveal the dynamic response characteristic and the mechanics behind it.However, due to the limitation of experimental apparatus, very few studies are about the VIV of a steel catenary riser (SCR) which is with a complex geometry. To investigate the VIV features and to further develop the corresponding numerical predictions of a SCR under steady current, a large-scale model test of a SCR was towed in an ocean basin at various speeds. Fiber Bragg grating strain sensors are instrumented on the riser model to measure both in-plane and out-of-plane responses. The characteristics of oscillating amplitude and dominating frequency response, the phenomenon of mode competition and travelling wave and the fatigue damage of the steel catenary riser in inline and cross-flow direction under steady current are analyzed.

Key words: Vortex induced vibration (VIV), steel catenary riser (SCR), experimental study

Introduction

As the offshore oil exploitation moves to deep sea gradually, the widely used steel catenary riser(SCR) that connects the floating platform and the oil under the seabed becomes increasingly longer and slender. Vortex shedding behind the riser periodically can induce a quite large response as the shedding frequency is locked with one of the natural frequencies of the marine riser. Therefore, the prediction of vortex induced vibration (VIV) and the induced fatigue damage have attracted tremendous attention from both academia and engineering.

In the last decades, experimental investigations on the VIV of a rigid riser section model and a straight flexible riser model are conducted quite frequently.Feng conducted a famous free oscillation experiment of an elastically supported rigid cylinder with high mass ratio in a wind tunnel laboratory[1]. Lock-in phenomenon and amplitude branches of VIV were firstly revealed. Williamson then performed a similar experiment, but in a water tunnel[2-5]. The mass ratio of the model is close to the real marine riser. A new response brand was found and the vortex shedding modes were investigated deeply. Dahl and Hover conducted an experimental study of a rigid riser model freely oscillating in both cross-flow (CF) and inline(IL) direction[6-7]. A vortex mode which can excite higher response amplitude was discovered. The free oscillation test is mainly used to discover the response characteristics of VIV. To have a better understanding of the vortex induced forces, Gopalkrishnan conducted forced oscillation experiment of rigid cylinders and the obtained hydrodynamic coefficients are widely used in the numerical prediction of the VIV of slender marine risers[8], such as the commercial software Shear7 and VIVANA[9-10]and several state-of-the-art time domain prediction theory[11-12].Carberry employed Particle Image Velocity technology in the forced oscillation experiment and compared the vortex shedding mode with that in free oscillation experiment[13-14].

VIV can lead to responses of marine risers with many modes involved and competing with each other.The dominating frequency then always shows “time sharing” feature. Such complicated phenomenon cannot be observed from the experimental study of rigid cylinders. Therefore, VIV of a straight flexible riser model under uniform and shear currents have been analyzed by many researchers[15-16]. Vandiver conducted a field test of a flexible riser with a length of 152.4 m and studied the characteristics of wave travelling in flexible long riser model[17]. A parameter related to mode number and modal damping was introduced to determine the predominance role of travelling and standing wave. Svithenbank found the time shearing of dominating mode in a slender riser model in shear current[18-19]. The hydrodynamic coefficients along the riser model have also been studied by employing some innovative approaches.Song introduced an inverse beam finite element method in the experimental study of a flexible riser in uniform, shear and oscillatory current[20-22]. The obtained hydrodynamic force coefficients showed significant difference from that achieved in rigid body experiment. These studies have provided a good reference for a better understanding of the vortex shedding mechanism and the VIV responses of a slender cylinder structure.

However, the SCR has a complex geometry as a catenary. How a SCR would behave as vortex sheds around its cross-sections has not been studied sufficiently yet, which is a very challenging topic due to the limitation of laboratory facilities[23-24]. In this paper, a large-scale model test of a steel catenary riser was towed in an ocean basin to simulate uniform current. The strain along the riser model measured by fiber bragg grating (FBG) strain sensors was analyzed to reveal the characteristics of the VIV of SCR.

1. Experiment setup

The SCR model test was performed in the ocean basin in Shanghai Jiao Tong University[25]. The riser model has a length of 23.71 m and a diameter of 0.024 m, as shown in Fig. 1. More details can be referred to [26]. To satisfy both Froud and Cauchy similarity law, the model was made by PPR pipe with outside coating and inside copper cable, as shown in Fig. 2. The main parameters of the SCR model are listed in Table 1.

Table 1 Main parameters of the SCR model in the experiment[26]

The lower part of the model is lying on a steel plate used to simulate the seabed, and the end side is fixed through a universal joint. In the ocean basin,current can only be generated near the free surface. To guarantee a uniform current profile along the SCR model, the steel plate and the top side of the riser are installed on horizontal tracks with a length of 4 m driven by serve motors and the servo motors in the water and air are controlled in synchronization to tow the SCR model to move in still water with a constant velocity. Four different current velocities from 0.1 m/s to 0.4 m/s with incident angle of 0oare analyzed in this paper. FBG strain sensors, uniformly distributed at 25 points along the axis of the riser model, are employed to collect the VIV response in CF and IL directions. There are four groups of FBG sensors at each measurement point, i.e., IL_a, CF_b, IL_c and CF_d, as shown in Fig. 2. The sampling frequency is 250 Hz.

Fig. 1 (Color online) Experimental setup in air

Fig. 2 (Color online) Sketch for SCR model test under uniform current (front view)

2. Data analysis

2.1 Pre-processing

The strain signal measured at each point along the riser model contains four components, i.e., initial strain, high frequency noise, the varied axial and bending strain. The initial strain is removed since the strain for a static model is available. A band-pass filter with a low-pass threshold of 50Hz is used to eliminate the signal noise. After these two steps, the pure VIV induced out-of-plane and in-plane strain can be written as:

2.2 Time-frequency analysis

Although a uniform current is simulated in the experiment, the normal velocity along the riser model is in fact like a shear current due to its catenary geometry, as shown in Fig. 3.

It is then necessary to check whether there exists“mode competition” phenomenon and so on. A wavelet transformation is applied to analyze the dominating frequency of the VIV induced strain in the time domain

Fig. 3 (Color online) Normal velocity of the uniform currents along the SCR model

Hereε(t) is the time history of the strain,WTf(a,τ) is its wavelet transformation coefficient,ais the scale factor andψ(t) represents the mother wavelet. Morlet wavelet equation is used in this paper.

2.3 Fatigue damage calculation

The repeated oscillation with high frequency due to vortex shedding will accelerate the fatigue damage and hence, bring a great threat to the safety of marine risers. In this paper, fatigue damages induced by the VIV in CF and IL direction in steady incident current are calculated by employing rain-flow counting method. Combining the linear Palmgren-Miner damage accumulation law and S-N curve, the fatigue damage can be expressed as

Fig. 4 (Color online) Strain in time-space domain under uniform current of 0.1 m/s

Heremdefines the slope of the S-N curve,lgis the intersection between the lgN- axis and the S-N curve, Δiσand Num are the stress range and total cycles obtained in the rain-flow counting process.

3. Results and discussion

3.1 Response along the riser model

Fig. 5 (Color online) Contour of the FFT analysis of the strain along the SCR

Fig. 6 (Color online) The IL strain in time-space domain corresponding to P_1 and P_2 frequency components shown in Fig. 5(b)

Fig. 7 (Color online) Time series and corresponding dominating frequency in the time domain of the CF and IL bending strain at 16th measurement point ( =0.1m/s)V

In this paper, the out-of-plane and in-plane strain distribution along the riser are plotted in time-space domain by using the original strain data, in frequencyspace domain by performing Fourier transform and in frequency-time domain based on wavelet transform algorithm. The properties of the traveling wave and multi-mode coupling and the phenomenon of ‘mode competition’ are investigated deeply.

3.1.1 Current velocity of 0.1 m/s

(1)Response in time space domain

Fig. 8 (Color online) Strain in time-space domain under uniform current of 0.2 m/s

The time series of the strain in CF and IL direction along the riser model (25 measurement points) under current velocity of 0.1 m/s are plotted in Figs. 4(a) and 4(b). Figure 4(c) shows the RMS value of the strain along the riser. There is a peak in the strain in CF direction at the 2ndmeasurement point which is near the touch-down point (TDP). The strain is too large than that in the other measurement point.Hence, the time series of the strain in Figs. 4(a) and 4(b) is plotted from the 3rdmeasurement point. In this case, the response in CF direction is dominated by standing wave while it is not very clear in IL direction.

Fig. 9 (Color online) Contour of the FFT analysis of the strain along the SCR (V =0.2 m/s)

Fig. 10 (Color online ) The IL stra in in time-space domain corresponding to P_1 and P_2 frequency components shown in Fig. 9(b) (V =0.2 m/s)

Fig. 11 (Color online) Time series and corresponding dominating frequency in the time domain of the CF and IL bending strain at 7th measurement point ( =0.2 m/s)V

Normally, the oscillation frequency in IL direction is twice of that in CF direction. However, for the strain in IL direction, an unexpectedly high oscillation frequency component is observed. It should be emphasized that instead of the displacement, it is the strain along the riser that is analyzed in this paper.Once a high-order natural mode is excited, a quite high value of strain can be induced even though the amplitude of the displacement is small since the strain is proportional to the second derivative of the displacement with respect to the position. Due to the high frequency component, the amplitude of the strain in IL direction is even larger than that in CF direction at several measurement points of the riser model. The strain component with high frequency should be paid more attention in the numerical prediction of the response in IL direction as well as in the evaluation of fatigue damage.

(2)Response in frequency space domain

Fig. 12 (Color online) Time series and corresponding dominating frequency in the time domain of the CF and IL bending strain at 20th measurement point ( =0.2 m/s)V

To have a deep understanding of the frequency components in the strain response, the data is further analyzed by performing fast Fourier transform (FFT).The contour plot of the FFT results in CF and IL direction at the 25 measurement points are plotted in Fig. 5.Thex- andy-axis refer to the frequency and sensor positions, respectively. The contour can not only tell the frequency component involved at each position, but also indicate whether standing wave or traveling wave is dominant. If there are obvious spectrum peaks along they-axis, it means standing wave is the main component. Meanwhile, a continuous response spectrum along the riser is a sign referring to the dominance of travelling wave. Here,the dominance of standing wave in CF direction is re-confirmed by Fig. 5(a).

Fig. 13 (Color online) Strain in time-space domain under uniform current of 0.3 m/s

For the strain in IL direction, there are mainly two frequency components, which are referred as P_1 and P_2 in Fig. 5(b). One is twice of the dominant frequency in CF direction represented byfCF, as expected, and the other one reaches almost 7.5fCF.Similar to the strain in CF direction, it is standing wave that dominates the IL response. The dominance of standing wave is more obvious in Fig. 6 which shows the strain in time-space domain corresponding to P_1 and P_2 frequency components. In addition,the amplitudes for the two components are almost the same. It means the high frequency component will make more contributions to the fatigue damage in this case.

As can be seen from Fig. 6, there are about four half-sinewaves in the natural mode shape corresponding to P_1 while thirteen half-sinewaves exist in P_2. Therefore, although the strain corresponding to P_2 component is very large, the displacement will only be2

Fig. 14 (Color online) Strain in time-space domain under uniform current of 0.4 m/s

(4/13) of P_1 component since(Dis the diameter of the riser,wandxare the displacement and position along the riser, respectively, andεrefers to the strain).Hence, it should be made clear that the displacement in IL direction is still dominated by the frequency of 2fCF.

(3)Response in frequency-time domain

Fig. 15 (Color online) Contour of the FFT analysis of the strain along the SCR (V =0.2 m/s)

“Time-shearing” or “mode competition” is one of the main characteristics for the VIV of marine risers in shear current. The strain response at the 16thmeasurement point in time-frequency domain is obtained by using wavelet transform algorithm, as shown in Fig. 7.The blackline in the contour plots represents the varying dominating mode or frequency. It keeps constant in CF direction, while in IL direction, it changes from 0.82 Hz to 3.20 Hz, back and forth.

3.1.2 Current velocity of 0.2 m/s

(1)Response in time space domain

The same analysis procedure was applied in the case with current velocity of 0.2 m/s. In this case, the strain response is a combination of traveling and standing wave, as can be seen from Fig. 8. In the region near the bottom end, standing wave is dominant. This is why the variation of RMS value along the riser has a lot of fluctuations there. Similar to the case with current velocity of 0.1 m/s, the maximum strain occurs near TDP of the riser.

(2)Response in frequency space domain

The contour of the FFT results of the strain along the riser in the case is shown in Fig. 9. There are also two dominating frequency components in IL direction.The high-frequency component strain, P_2, is clearly dominated by traveling wave, which is different from that in the case of 0.1m/s current. The traveling wave can also be seen from Fig. 10(b). It travels from top to bottom since the normal velocity of the uniform currents at the top of the SCR model is the largest and there is likely more energy imported to the top side of the riser. The traveling wave has a period of 0.25 s and a wave length of 2.7 m. The traveling velocity reaches 10.8 m/s. It disappears near the bottom end due to reflection. The amplitude of the strain component in high frequency range is only half of that in low frequency range, as shown in Figs. 10(a) and 10(b).

Fig. 16 (Color online) Contour of the FFT analysis of the strain along the SCR (V =0.4 m/s)

(3)Response in time frequency domain

The frequency response of the strain in the time domain at the 7th(near the bottom end) and 20th(near the top end) measurement points are plotted in Figs.11 and 12, respectively. Similar to the case of 0.1 m/s current, the strain in CF direction is dominated by a single mode. For the strain in IL direction, the power of P_1 component is very strong and dominates the response in the bottom. In the top, it is relatively weak.Therefore, “time sharing” of the dominant frequency is obvious at the 20thmeasurement point where the natural modes in P_1 range and P_2 range are competing with each other, as shown in Fig. 12. In the region near the bottom part, the strain in IL direction is mainly dominated by the frequency component of 2fCF.

3.1.3 Current velocity of 0.3 m/s-0.4 m/s

Fig. 17 (Color online) The fatigue damage in CF and IL direction under uniform current with different velocities

The strain in CF and IL direction in time-space domain under uniform current ranging from 0.3 m/s to 0.4 m/s is shown in Figs. 13 and 14. Compared with the case of 0.2 m/s current, traveling wave becomes more dominant in CF direction, as also can be seen in Figs. 15 and 16, while standing wave still exists which are marked by a “black square”. The maximum value of the strain is more likely to occur at the position near the touch down point. In these two cases, the strain in CF direction is much larger than that in IL direction.

In addition, in the case with current velocity of 0.4m/s, there are more frequency-components involved in the response in CF direction as can be seen from the FFT results of the strain in Fig. 16(a).Near the top-end of the riser where is subject to a larger normal current velocity, a frequency component of 2.00 Hz is dominating. As it goes bottom, the dominating frequency decreases to 1.76 Hz and 1.49 Hz. Unlike the other cases, the highest frequency component of the strain in IL direction in this case is dominated by standing wave.

3.2 Fatigue analysis results

The fatigue damage at the 25 measurement points due to vortex induced vibration in CF and IL direction is calculated based on Palmgren-Miner Damage accumulation law and rain-flow counting method, as shown in Fig. 17. The inverse of they-axis represents the life of the scaled model. In the uniform current with a velocity of 0.1m/s, the fatigue damage in IL direction is even larger than CF direction except at several specific points. It can be attributed to the fact that a much higher natural mode is excited. As the current velocity increases, the fatigue damage in CF direction increases more rapidly and becomes much larger than that in IL direction. The maximum fatigue damage in CF direction is prone to occur near the touch down point where standing wave is usually dominant.

4. Conclusions

In this paper, a large-scale model of a SCR with a total length of 23.71 m was towed in an ocean basin at various speeds from 0.1 m/s to 0.4 m/s. Instead of the displacement, the experimental results are investigated deeply from the perspective of the strain data which is more related to the fatigue damage evaluation and safety design. The unique experimental results provide guidance as well as validations for numerical simulation of VIV in SCR.

Some unexpected phenomena are observed in the strain in IL direction along the SCR model. In addition to a frequency component which is two times of that in CF direction, there is also a quite high frequency component. It is dominated by standing and traveling wave under 0.1 m/s and 0.2 m/s current,respectively. Due to the high frequency component,the strain as well as the fatigue damage induced by the VIV in IL direction is even larger than that in CF direction.

The response in CF direction is characterized by a single-frequency oscillation. Meanwhile, for the strain in IL direction, there are always multi-modes involved.Besides, the dominant mode or frequencies varies with time, which is known as ‘time-sharing’ property,especially in the region near the topside of the riser model.

The fatigue damages at different positions along the SCR model under different current velocities are also evaluated. The maximum fatigue damage in CF direction is more prone to occur near the TDP where standing wave is dominant. The trend is not obvious in IL direction. As the current velocity increase, the fatigue damage in CF direction increase more rapidly than that in IL direction.