Hydrodynamics of A Flexible Riser Undergoing the Vortex-Induced Vibration at High Reynolds Number

This study proposed a method to obtain hydrodynamic forces and coefficients for a flexible riser undergoing the vortex-induced vibration (VIV), based on the measured strains collected from the scale-model testing with the Reynolds numbers ranging from 1.34E5 to 2.35E5. The riser is approximated as a tensioned spatial beam, and an inverse method based on the FEM of spatial beam is adopted for the calculation of hydrodynamic forces in the cross flow (CF) and inline (IL) directions. The drag coefficients and vortex-induced force coefficients are obtained through the Fourier Series Theory. Finally, the hydrodynamic characteristics of a flexible riser model undergoing the VIV in a uniform flow are carefully investigated. The results indicate that the VIV amplifies the drag coefficient, and the drag coefficient does not change with time when the CF VIV is stable. Only when the VIVs in the CF and IL directions are all steady vibrations, the vortex-induced force coefficients keep as a constant with time, and under “lock-in” condition, whether the added-mass coefficient changes with time or not, the oscillation frequency of the VIV keeps unchanged. It further shows that the CF excitation coefficients at high frequency are much smaller than those at the dominant frequency, while, the IL excitation coefficients are in the same range. The axial distributions of the excitation and damping region at the dominant frequency and high frequency are approximately consistent in the CF direction, while, in the IL direction, there exists a great difference.


Introduction
As the exploitation of offshore oil resources moves into deeper waters, the risers that transfer oil from the seabed to facilities on the surface are becoming increasingly slender (Mekha, 2001). Under the actions of ocean current, vortices are generated and alternately shed from the sides of these flexible, slender risers. This vortex shedding leads to the periodic pressure variation around the riser, producing a vortex-induced force on the riser; then a vibration of the riser is induced, which is termed as the vortex-induced vibration (VIV). When the frequency of the vortex-induced force is near one of the natural frequencies of the risers, the VIV amplitude will be significant. When the VIV occurs in the riser, the vibrating riser would in turn affect the fluid flow distribution, which affects the distribution of the hydrodynamic forces along the riser. For instance, the VIV can disrupt the hydrodynamic force distribution along the riser in the inline (IL) direction, and make the drag coefficient increase from 1.2 (the drag coefficient of a stationary, a rigid cylinder at the sub-critical Reynolds numbers) to 4.0 (Zhao et al., 2010). The increase of the drag coefficient would make a larger drag force on the riser, and then cause a great threat to the structural strength of the riser.
Moreover, the VIV can result in severe fatigue damage in the riser and decrease its lifetime. Thus, it is important to accurately predict the VIV of the riser. For the prediction of the VIV, the hydrodynamic force will directly determine the accuracy of the results. In the current VIV prediction, the hydrodynamic forces are calculated based on the coefficient database including the excitation coefficient and the addedmass coefficient, which are obtained by the forced-oscillation tests of a rigid cylinder in a two-dimensional flow. In the forced-oscillation tests, a rigid cylinder was forced to oscillate in only the cross flow (CF) (Gopalkrishnan, 1993) or IL direction (Aronsen, 2007) under the assumption that the VIVs of the riser in the CF and IL directions do not interfere with each other. However, under real conditions, the VIV of a flexible riser exists in both directions, and the vibrations in the two directions are strongly coupled. This kinematic coupling leads to an interaction of the hydrodynamic forces in the two directions, and then makes the vortexinduced force coefficients different from the results obtained from the forced oscillation test of a rigid cylinder in single direction (Marcollo and Hinwood, 2006;Sumer and Fredsoe, 2006;Song et al., 2016a;Wu et al., 2016).
Since the hydrodynamics of risers subjected to the VIV are not well understood so far, a factor of safety larger than 10 is normally used in the riser design to prevent failure (API, 1998). However, with the development of oil and gas resources into deeper waters, the structural integrity of risers cannot be insured simply by increasing the safety factor. Hence, the investigations on the hydrodynamic characteristics of flexible risers undergoing the VIV, including the drag force in the IL direction and the vortex-induced forces in both CF and IL directions, are becoming more important.
Owing to the development of scientific computation, the computational fluid dynamics (CFD) has become a very powerful and efficient tool. CFD is an ideal method with which to study complex fluid-structure interaction (FSI) problems such as the VIV (Sarpkaya, 2004;Kaiktsis et al., 2007). In most CFD studies of a flexible cylinder's VIV, the structure is simplified as a tensioned-beam model and divided into a large number of rigid segments; while the flow along each segment is assumed to be two-dimensional or three-dimensional. The related FSI studies of a flexible cylinder have been extensively conducted to study the hydrodynamic characteristics and structural response of flexible risers (Evangelinos et al., 2000;Willden and Graham, 2004;Yamamoto et al., 2004;Zhao et al., 2010;Liu et al., 2012). However, CFD requires detailed meshes for both the fluid and the structure, which leads to large effort of computation. For example, one study involving a 6-meter-long riser required several months to complete (Chen et al., 2007). Hence, the use of CFD methods to systematic analyze the hydrodynamic forces on a flexible slender riser is still limited so far.
Scale-model test in the water basin is another effective method to study the hydrodynamics of flexible risers undergoing the VIV. However, in model testing, it is challenging to directly measure the hydrodynamic force at the cross-sections of the riser model under a current for the restriction of the experimental technology. To systematically investigate the hydrodynamics of a flexible riser under the coupled VIV responses in the IL and CF directions. Dahl (2008) conducted two dimensional forced oscillation tests of a rigid cylinder. In the tests, the rigid cylinder was forced to oscillate sinusoidally with a series of flow speed, IL and CF motion amplitude, CF oscillation frequency (IL frequency is assumed to be twice of the CF one) and the motion phase between IL and CF displacements. A force coefficient database to be used for the prediction of the forces on a riser, particularly the third harmonic forces were generated. However, the coefficients in the database are still considered to be sparse because of the large number of parameters required and the accuracy of the prediction using the database compared to the field measurement needs to be examined in the future. To obtain the vortex-induced force of a flexible riser with realistic cross sectional orbits, Soni (2008) and Yin and Larsen (2010) studied forced oscillations of a smooth, rigid cylinder, where the oscillations replicated the motion at various cross-sections on a flexible riser as measured from the flexible riser model tests. Based on these experiments, the hydrodynamics of flexible risers undergoing VIV in uniform and shear flows were studied, and the results revealed that the hydrodynamic force coefficients obtained from the observed motion may differ significantly from using the harmonic motions. However, a large number of experiments is needed in order to produce enough data for the force coefficient database. To predict the VIV in the CF direction more accurately, Mukundan (2008) parameterized the database of the excitation coefficients obtained from the forced oscillation experiments of Gopalkrishnan (1993). Using an optimization method to minimize the error between the response from an empirical model of the VIV and the test results, Mukundan (2008) established a new database of the excitation coefficients in the CF direction. The new database had a larger excitation range, and the main excitation region overlapped the secondary excitation region. However, this method could not capture higher-order harmonic force components in the CF direction or the vortex-induced force in the IL direction. Huera-Huarte et al. (2006) analyzed the fluid forces by inputting the displacement measurements at various sites into a finite element analysis of a vertically tensioned riser subjected to a stepped current. However, the coefficients of the vortex-induced force, including both the excitation and added mass coefficients, were not analyzed further. Wu et al. (2016) obtained the vortex-induced forces and its coefficients on a flexible riser in the CF direction using an inverse method. However, the hydrodynamic forces of a flexible riser in the IL direction, including the mean drag force and vortex-induced force were not analyzed. Combining the modal analysis method and the Euler-Bernoulli beam vibration equations, Song et al. (2016a) analyzed the hydrodynamic force of a flexible riser using the strain information measured in the scaled model test.
The above analysis indicates: The existing database describing the vortex-induced forces is based on the forced oscillations of a rigid cylinder in only the CF or IL direction in a two-dimensional flow. An actual riser undergoing the VIV, however, experiences the kinematic coupling in the CF and IL directions and a three-dimensional flow field. The differences in the hydrodynamics of a rigid cylinder undergoing forced oscillations in a single direction and those of a flexible riser require further investigation. How to obtain the hydrodynamics more accurately requires more investigation. In addition, in the aforementioned studies, the largest Re number investigated was only on the order of 10 4 , whereas in practical situations, the risers are subjected to the Re number on the order of 10 5 or more. It is well known that Re significantly affects the flow separation modes and hydrodynamic forces. For Re below 300, the wake flow is laminar, and the flow pattern gradually becomes turbulent as Re increases. When Re is in the critical and super-critical regimes, the wake pattern becomes even more disordered. The boundary layer around the cylinder becomes asymmetric with laminar and turbulent sections, resulting in different hydrodynamic forces than those in the subcritical regions. The performances of the drag force (Sarpkaya, 1977;Vandiver, 1983;Humphries and Walker, 1988;Chaplin et al., 2005) and the VIV of the risers (Sarpkaya, 1978) under high Re are different from those under low Re. Thus, further studies are required to determine whether the test results for the Re number below 10 4 are representative of the VIV amplification on the drag coefficient at higher Re.
In this study, a method to identify the hydrodynamic forces of flexible risers undergoing the VIV based on the strain information measured from the model test is presented. The riser model with large slenderness ratio is idealized as a tensioned spatial beam, and the structural responses in the CF and IL directions are described by the governing equation of the spatial beam based on the Finite Element Method (FEM); The CF and IL structural responses are obtained by the modal superposition method based on the measured strain signals. Consequently, the hydrodynamic forces in two directions, including the mean drag force in the IL direction and the vortex-induced forces in both CF and IL directions are obtained by the inverse analysis procedures. Based on the structural responses and the hydrodynamic forces, the hydrodynamic force coefficients in two directions, namely the drag coefficients, excitation coefficients and added-mass coefficients are calculated using the Fourier series theory. The hydrodynamic coefficients in both CF and IL directions of a flexible riser undergoing the VIV in uniform flows are finally investig-ated using this proposed method.

Basic theory
2.1 Hydrodynamic forces on a flexible riser undergoing the VIV A submerged flexible riser with a tensional force T o in the uniform current is illustrated in Fig. 1a. The central axis of the riser lies on the z-axis. The direction of the flow is parallelled to the xoz plane and orthogonal to the riser. The mean drag force acts on the riser in the IL plane, which causes the riser a mean deflection in the flow direction. This deflection will be referred to as the mean bending, and the corresponding position of the riser will be referred to as the equilibrium position, as illustrated by Fig. 1b. As the fluid flows over the riser, the vortices form and are shed periodically. This vortex shedding generates the periodic vortex-induced forces in the CF and IL planes with the mean values of zero, and causes the riser to vibrate in the two planes, which is called the VIV, as illustrated by Figs. 1b and 1c. Thus, the hydrodynamic forces on a flexible riser include three parts: the vortex-induced forces in the CF and IL directions, and mean drag force in the IL direction. Correspondingly, the structural responses shall also include three parts: the VIV in the CF and IL directions and mean bending in the IL direction.
According to the FEM, the governing equation of spatial beam can be expressed as: where M, C and K are the global mass matrix, damping matrix and stiffness matrix of the riser, respectively; δ, δ′ and δ′′ are the displacement matrix, velocity matrix and acceleration matrix, respectively; F is the hydrodynamic force matrix. For a riser with N nodes and six degrees of freedom at each node, the dimensions of M, C, and K are and the dimensions of δ and F are .
It should be noted that, for a tensioned beam, the local element stiffness matrix would be expressed as: where is the pre-stress stiffness matrix caused by the axial tension; is the small displacement linear stiffness matrix caused by the bending stiffness.

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REN Tie et al. China Ocean Eng., 2018, Vol. 32, No. 5, P. 570-581 C The damping matrix can be obtained based on the Rayleigh damping model, which can be expressed as: where α and β are the Rayleigh damping coefficients, which can be obtained by the natural frequency and structural damping ratio of the riser. The detailed derivations of the bending strains and bending displacement in the CF and IL directions using the strain information measured by the strain sensors in the model test are illustrated by Song et al. (2016b). With displacement matrix δ, the velocity matrix δ′ and acceleration matrix δ′′ can be obtained by using the central-difference method to calculate the first-and the second-order partial derivatives of δ with respect to time. After obtaining the structural response matrixes δ, δ′ and δ′′, the hydrodynamic force F on the right side of Eq. (1) can be obtained by the inverse analysis.

Hydrodynamic force coefficients
ω ω, 3ω, 5ω · · · 2ω, 4ω, 6ω · · · y(z, t) x (z, t) Assume that the VIV responses in the CF and IL directions are composed of the periodic vibrations with different frequencies. If we adopt the dominant vibration frequency of the VIV in the CF direction as the fundamental frequency, , the odd times harmonic components are generally associated with the CF vibration, i.e. , while the even times components are characteristics of the IL vibration, i.e. (Vandiver et al., 2009). Besides, as stated before, the response in the IL direction includes the mean bending apart from the VIV. Thus, the displacements in the CF and IL directions at Node z, i.e. and , can be expressed as: is the mean bending displacement of the riser in the IL direction. and are the VIV displacements at the frequency in the CF and IL directions, respectively, as shown in Eq. (5), where and are the displacement amplitude and phase angle at the frequency in the CF direction; and are the amplitude and phase angle in the IL direction.
From Eq.(5), the velocities, i.e. and , and accelerations, i.e. and in the CF and IL directions at Node z can be expressed as: where, where and are the amplitude of the velocity at the frequency in the CF and IL directions, respectively; and are the amplitude of the acceleration in each directions, respectively.
For the VIV response with single frequency, Song et al. (2016b) proposed that the vortex-induced force can be expressed as: and are the excitation coefficient and add-mass coefficient, respectively; is the hydrodynamic diameter; is the fluid density; is the flow velocity; and are the velocity and acceleration of the VIV, respectively; is the velocity amplitude.
Thus, for the VIV response with multi-frequencies in the CF direction, the vortex-induced force at Node z can be expressed as the superposition of the vortex-induced force at each frequency component, namely: where and are the excitation coefficient and added-mass coefficient at the frequency component in the CF direction at Node z.
The hydrodynamic force in the IL direction consists of the mean drag force and vortex-induced force, where the mean drag can be expressed by the Morison equation, and the vortex-induced force in the IL direction can be obtained according to Eq. (8). Therefore, the hydrodynamic force at Node z in the IL direction can be expressed as:
where and are the excitation coefficient and added-mass coefficient at the frequency component in the IL direction at Node z; is the drag coefficient. According to the Fourier Series theory, the vortex-induced force coefficient at each frequency component in both CF and IL directions and the drag coefficient in the IL direction can be obtained from Eqs. (11) to (15): where nT means the integral number of times of the vibration period in the CF or IL direction. Since the hydrodynamic force coefficients depend on the response, the hydrodynamic force coefficients may vary with time, especially when the response features are not constant. In a non-stationary response, the hydrodynamic force coefficients may vary from the cycle to cycle. Herein, in order to investigate the time-varying characteristics of the hydrodynamic coefficients, the hydrodynamic force coefficients are calculated on a per-cycle basis, namely setting n=1.

t, iω)
In Eqs. (11)-(15), the hydrodynamic forces, i.e. and , can be obtained by Eq. (1); the responses of the velocity and acceleration at each frequency component in the CF and IL directions, i.e. , , and , can be achieved using the bandpass filtering method from the total responses, which can be obtained by the modal superposition method from the bending strains. The vibration period T in Eqs. (11)-(15) is reciprocal of the vibration frequency which is also the frequency of the hydrodynamic forces exerted on the pipe. The vibration frequency can be obtained from the FFT results of the measured strain.

Scale-model tests
The experiments were performed in the towing tank of the Shanghai Shipping Science Institute with the dimensions of 192 m×10 m×4.2 m (length×width×depth). A flexible riser model was horizontally towed with a constant speed to simulate the uniform flow. The two ends of the riser were connected to the towing carriage with universal joints, and maintained a constant pre-tension. A photograph of the experimental setup is shown in Fig. 2.
The hydrodynamic diameter of the model riser was 0.1683 m with an effective length of 7.9 m. The slenderness ratio was 46.94. The detailed parameters and the first two orders of natural frequencies in still water (with addedmass coefficient of 1.0) of the riser model are listed in Table 1. The structural damping ratio shown in Table 1 was obtained from a free-decay test in air.
Fiber Bragg Grating (FBG) strain sensors were installed on the surface of the riser model in the CF and IL planes to measure the strain response information. The strain sensors were placed on the opposite sides of a cross section at the points denoted as CF1, CF2, IL1 and IL2, as shown in Fig. 3; the corresponding strains were denoted as ε CF1 , ε CF2 , ε IL1 and ε IL2 , respectively. In total, 36 sensors are installed on the surface of the riser model (each nine at CF1, CF2, IL1 and IL2), and uniformly distributed in the CF and IL planes. Hence, the distance between two adjacent sensors was 0.9 m in both CF and IL plane. All of the strain signals were synchronously acquired at a rate of 125 Hz, which assured none phase lag between the signals. (1) the dominant frequency in the IL direction was two times of the fundamental frequency, i.e. . Such a relation between the frequencies in the two directions has also been revealed by several studies (for example, Fu et al. (2011) andFang et al. (2014)). (2) At U=0.8 m/s, the VIVs in the CF and IL directions were multi-frequency vibrations, and contained high frequency component, i.e. three times of the fundamental frequency in the CF direction, , and four times of the fundamental frequency in the IL direction, .
The restructured displacement responses at the stable stage at z=3.95 m under the three test cases were shown in Fig. 7. As shown in this figure, the mean value of the IL displacement response was not zero due to the mean bending; the "amplitude modulation" phenomenon existed in the IL displacement response at U=1.4 m/s.
Since the RMS value of the noise errors' weights in each mode order decays by a factor of 1/n 2 as the number of the mode order increases (Lie and Kaasen, 2006;Li, 2012), which means that the noise error has a relatively large weight in the lower order modes, using the modal analysis    REN Tie et al. China Ocean Eng., 2018, Vol. 32, No. 5, P. 570-581 575 to reconstruct the VIV displacements with low-order dominant modes may introduce some errors. In order to evaluate this error, we present the RMS distribution of the VIV displacements in the CF and IL directions under the three flow rates, as shown in Fig. 8. It can be seen that the dominant modes of the VIV displacements are the low-order modes, and then the displacement weights in the higher-order modes can be considered as being generated by the noise errors. In this paper, the RMS value of the 5-th order modal weight is regarded as the noise error weight. Then, based on the relationship between the RMS value of the noise error weights at each mode order (Lie and Kaasen, 2006;Li, 2012), the weights of the noise errors on the lower order modes (the 1-st to 4-th order) can be obtained, as shown in Fig. 8. Results illustrated that the error weight on the lower order modes (the 1-st to 4-th order) is very small compared to the total displacement weight, which is basically smaller than 5% of the total displacement weight. Therefore, it can be considered that the displacement results reconstructed by the modal superposition method are accurate. The reason for the small error is the choice of a fiber-optic strain sensor instead of the traditional resistive strain gauge to measure the strains of the model, which is immune to electromagnetic interference and thus has a small noise error.
The VIVs in the CF and IL directions were multi-frequency responses at U=0.8 m/s. The VIV displacement response at each frequency component can be obtained using the band-pass filtering method. Fig. 9 shows the CF displacement responses at the frequency component of and , and the IL displacement responses at the frequency component of and at z=3.95 m. As shown in Fig. 9, the VIV displacement responses in the CF and IL directions were stable at the dominant frequency component, but had certain "amplitude modulation" characteristics at high frequency component, especially for the CF direction. The VIV displacement amplitude at the dominant frequency component was much larger than that at the high frequency component.

Drag coefficient
The drag coefficient at each node of the riser model in each IL vibration period can be obtained by Eq. (15) in Section 2.2 using the hydrodynamic forces in the IL direction. The contour plot for the drag coefficient at each node and the axial distribution of the mean drag coefficient at U=0.8, 1.2, and 1.4 m/s are plotted in Fig. 10. Fig. 10 shows that the drag coefficient at each node remained constant with time under the three test cases. It should be noted that at U=1.4 m/s, the vibration in the IL direction had obvious "amplitude modulation" characteristics. However, such VIV response characteristics did not lead to time varying characteristics of the drag coefficient.  This maybe because that compared with the VIV in the CF direction, the VIV in IL direction had little influence on the drag coefficient (Vandiver, 1983;Chaplin et al., 2005). The stable vibration in the CF direction kept the drag coefficient in the IL direction as a constant.
6 Vortex-induced force coefficient 6.1 Vortex-induced force coefficient at the dominant frequency The vortex-induced force coefficients of the flexible riser model at the dominant frequency in the CF and IL directions, i.e., the excitation coefficient and added-mass coefficient, in each vibration period can be obtained by Eqs. (11) to (15) in Section 2.2 with the hydrodynamic forces and the corresponding velocity and acceleration responses at the dominant frequency. Figs. 11, 12 and 13 are the contour plot of the vortex-induced force coefficients in the CF and IL directions at each node at U=0.8, 1.2, and 1.4 m/s. Meanwhile, the time history of the vortex-induced coefficients in the CF and IL directions at z=1.58, 3.95, and 6.32 m and the axial distribution of the vortex-induced coefficients at different time instants were also given in those figures. The reduced velocity in Figs. 11, 12 and 13 is defined as: where U is the flow velocity; D is the hydrodynamic diameter of the pipe; are the dominant frequencies in the CF and IL directions.
As shown in Figs. 11, 12 and 13, the excitation coefficient and added-mass coefficient in the CF and IL directions kept constant basically with time at U=0.8 and 1.2 m/s, for which, the VIV responses at the dominant frequency were stable response and the corresponding reduced velocities are V rCF =7.81 and 7.65 for the CF VIV and V rIL =3.90 and 3.86 for the IL VIV. The excitation coefficient and added-mass coefficient in the IL direction changed with time at U=1.4 m/s (V rCF =8.35, V rIL =4.17), for which, the "amplitude modulation" phenomenon of the VIV response in the IL direction occurred. The VIV response in the CF direction was stable at U=1.4 m/s, but the corresponding vortexinduced force coefficients showed a strong time varying characteristics. It is because the VIVs in the CF and IL directions are always strongly coupled. This kinematic coupling leads to an interaction of the vortex-induced forces in the two directions so that the time varying VIV response in the IL direction resulted in time varying vortex-induce forces and then the time varying vortex-induce force coefficients in the CF direction.
Based on the calculated added-mass coefficients in each vibration period in the CF and IL directions at U=1.4 m/s, the natural frequencies that correspond to the dominant mode in the CF and IL directions were computed using the finite element analysis conducted by ABAQUS. The results shown that although the added-mass coefficients in the CF  and IL directions changed with time, the calculated natural frequencies kept constant with time and were equal to the dominant frequency of the VIV responses, which agreed with the conclusion obtained by the Wavelet Analysis above. This indicates that under the "lock-in" condition, whether the added-mass coefficients change with time or

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REN Tie et al. China Ocean Eng., 2018, Vol. 32, No. 5, P. 570-581 not, the natural frequencies of the riser would keep constant in both directions, leading to the vibration frequency of the VIV unchanged. As shown in Figs. 11-13, the excitation coefficients in the CF direction were much larger than the excitation coefficients in the IL direction. At U=0.8 m/s, the excitation coefficients varied from -1.1 to 1.1 in the CF direction, and -0.2 to 0.2 in the IL direction; At U=1.2 m/s, the excitation coefficients varied from -0.8 to 1.6 in the CF direction, and -0.2 to 0.3 in the IL direction. The region with positive excitation coefficient means energy is transferred from the flow field to the riser, namely the excitation region, while the region with negative excitation coefficient means energy is transferred from the riser to the flow field, namely the damping region. Figs. 11-13 indicate that the distribution of the excitation region along the riser in the CF and IL directions was different from each other, and it was also true for the damping region.
6.2 Vortex-induced force coefficient at high frequency The above analysis shows that VIVs in both CF and IL directions are multi-frequency responses at U=0.8 m/s. VIV in the CF direction contained response at three-times fundamental frequency, i.e. the dominant frequency, and VIV in the IL direction contained response at four-times fundamental frequency. Using the same processing method of vortex-induced force coefficients at dominant frequency, Fig. 14 is the contour plot of vortex-induced force coeffi-cients at the high frequency component in CF and IL directions at each node at U=0.8 m/s (Re=1.35E5). Meanwhile, the time history of the vortex-induced coefficients at z=1.58, 3.95 and 6.32 m and the axial distribution of the vortex-induced coefficients at different time instants in CF and IL directions were also given in the figure. Fig. 14 shows that due to the "amplitude modulation" characteristics of VIV response at high frequency in the CF direction, time varying characteristics excited for the vortex-induced coefficients, including the excitation coefficients and the addmass coefficients in both directions, especially for the CF direction.
By comparing Fig. 11 with Fig. 14, we can find that: (1) Differing from the characteristics of the excitation coefficients at the dominant frequency component, the excitation coefficients in the CF and IL directions at high frequency component were in the same range, all varied from -0.2 to 0.2; (2) The excitation coefficients at high frequency component were much smaller than those at the dominant frequency component in the CF direction, while in the IL direction, the excitation coefficients at the two frequency components all ranged from -0.2 to 0.2; (3) In the CF direction, the axial distribution of the excitation and damping region at the dominant frequency and high frequency was approximately consistent. The space of the riser ranging from 2.5 m to 5 m was approximately the damping region; the rest of the riser was at the excitation region. However, in the IL direction, there is a big difference in the distribution of the excitation and damping region between the dominant frequency and high frequency. At the dominant frequency, the damping region ranged from 3.3 m to 6 m and the rest of the riser was at the excitation region, while, at high frequency, the excitation region ranged from 0 m to 4.25 m, and the damping region ranged from 4.25 m to 7.9 m.
Figs. 11 and 14 show that the axial distributions of the add-mass coefficients at the dominant frequency and high frequency were different: the add-mass coefficients in the CF and IL directions were all positive at the dominant frequency, while at high frequency, the add-mass coefficients in the CF direction were all negative, and the coefficients in the IL direction had negative values at the region from 1 m to 6 m.

Conclusions
In this study, a method to obtain the hydrodynamic forces and coefficients for a flexible riser undergoing the VIV based on the measured strains collected from the scalemodel testing was proposed. The riser is approximated as a tensioned spatial beam, and an inverse method based on the FEM of spatial beam is adopted for the calculation of the hydrodynamic forces in the CF and IL directions using the structural responses in both directions. Based on these hydrodynamic forces, and combined with the velocities and accelerations responses of the riser, the drag coefficients and vortex-induced force coefficients are obtained through the Fourier Series Theory. As an illustration example, the hydrodynamic characteristics of a flexible riser model un-dergoing the VIV in the uniform flow were carefully investigated. The following conclusions can be drawn.
(1) The drag coefficient does not change with time when the CF VIV is a steady response.
(2) Only when the VIVs in the CF and IL directions are all steady vibrations, the vortex-induced force coefficients keep constant with time, otherwise, the vortex-induced coefficients change with time. Under the "lock-in" condition, whether the added-mass coefficient changes with time or not, the natural frequency of the riser keeps as a constant, leading to the vibration frequency of the VIV unchanged.
(3) The CF excitation coefficients at a high frequency are much smaller than those at the dominant frequency, while, the IL excitation coefficients are in the same range at two frequency components. The axial distributions and values of the add-mass coefficients at two frequency components are different in both directions.
(4) The axial distributions of the excitation and damping region at the dominant frequency and high frequency are approximately consistent in the CF direction, while, in the IL direction, there exists a large difference.