4.1. Proposed CF VIV Analysis Model
In this study, CF VIV analysis of the slender body with the current is carried out using the synchronization I model and lumped-mass line model, which are validated in the previous section. To simulate the VIV of the slender body due to the current, the calculation procedure is proposed as shown in
Figure 13.
The calculation procedure is summarized in three stages. First, the static configuration of the slender body is calculated due to internal forces and external forces. Internal forces consist of tension and shear forces due to bending moments, and external forces consist of self-weight, buoyancy, ground-contact force, and drag force due to the current. In the second step, based on the static configuration of the slender body caused by the current, the effective incoming flow velocity
at each cross-section is calculated at each node as shown in
Figure 14. The unit normal vector
perpendicular to the effective incoming flow direction is also calculated using tangential vector
and
as follows Equation (12).
Finally, dynamic analysis of the slender body is carried out using the VIV load, considered by
and
. In this study, it is assumed that the static configuration of the slender body with current is the mean position for defining
. Therefore, the VIV load always acts perpendicular to
of the static configuration. When the amplitude of the vortex-induced vibration is greater than about 1.2 times the diameter, the lift force decreases dramatically and the damping force becomes dominant. Therefore, it is expected that the maximum dynamic amplitude is very small compared with the length of the slender body. The assumption, that the static configuration is the mean position for defining
, is reasonable. Equations of motion (13) are derived by adding the VIV term to Equation (7). Unlike the study of Thorsen et al. [
26], applying the simplified synchronization II model and Morison drag in the CF direction to which the VIV load is applied was eliminated to apply the more sophisticated synchronization I model. Since the drag force term of VIV load and Morison equation overlap, the drag force term of Morison equation perpendicular to the effective inflow flow is eliminated.
In Equation (13),
means the VIV load using the synchronization I model and is defined considering the mean length
of each node as Equation (14). The added mass term is already defined in Equation (13) and is therefore eliminated in Equation (14).
In Equation (14), means the relative velocity of the slender body. and mean the amplitude and velocity perpendicular to the effective incoming flow and are defined using . In this study, since it is limited only to the CF VIV analysis caused by the current, due to the acceleration of fluid particles is also eliminated in Equation (13).
4.2. Validation of CF VIV Analysis Model
In this study, the CF VIV analysis code for the slender body applying the proposed procedure and synchronization I model is developed. To validate the developed code, tension riser under uniform current and shear current are simulated, where relatively numerous results have been published [
5,
7].
First, the simulations of tensioned riser under uniform current [
7] are carried out. The riser model is divided into 50 elements in these simulations. These tests implement a uniform current through towing equipment moving at a constant speed, as shown in
Figure 15. Simulation conditions for the tensioned riser under a uniform current are shown in
Table 5. Morison drag and added mass coefficients are assumed as shown in
Table 4.
Figure 16a,b show the results of the VIV simulation under the uniform current for the model test case by Song et al. [
7]. In the upper panel of
Figure 16a, the calculated static configuration in the inline direction is also in exact agreement with that of the commercial code. In the lower panel of
Figure 16a, the magnitude of the CF RMS amplitude and the main excitation mode of the RMS of CF displacement in this study are similar to the measurements by Song et al. [
7]. The space-time plot of the CF displacement also shows the main excitation mode over time as shown in
Figure 16b. The 1st mode in CF VIV is observed in Song et al.’s case [
7] with the fairly long model.
In order to validate the VIV simulation of the tension riser under the shear current, simulations for the high-mode VIV model test [
5] performed in the Norwegian deep-water program (NDP) are carried out as shown in
Figure 17.
Table 6 shows the parameters of the tensioned riser model of the high-mode VIV test performed in NDP. The riser model is also divided into 50 elements in these simulations. In the riser simulations under shear current conditions, the Morison drag and the added mass coefficient are also assumed as shown in
Table 4. The VIV simulations of the riser are carried out under shear current conditions of 0.6, 1.3, and 2.0 m/s, respectively, with maximum velocity
. The simulation results are compared with those of Kristiansen and Lie [
5].
For the shear current condition with 0.6m/s of maximum velocity
, the static configuration of the inline displacement and the RMS of CF displacement are shown in
Figure 18a.
In
Figure 18a, the inline static configuration of the present simulation agrees exactly with the result of OrcaFlex 10.1. The magnitude of the CF RMS amplitude and the main excitation mode of the RMS of CF displacement of the present simulation also agree excellently with the results of the model tested by Kristiansen and Lie [
5] and the calculation by Thorsen et al. [
25], respectively, in
Figure 18a.
Figure 18b shows the space–time plot of the CF displacement. Unlike the uniform current condition, the wave of CF displacement under the shear current condition propagates from the section where the velocity of the incoming flow is fast to the section where the velocity of the incoming flow is slow.
Figure 19a and
Figure 20a show the inline static configuration and the RMS of CF displacement in the conditions of the maximum velocity
of shear current 1.3 m/s and 2.0 m/s, respectively. The magnitude of the CF RMS amplitude and the main excitation mode of the RMS of CF displacement in this study also agree well with the results of the model test by Kristiansen and Lie [
5] and the calculation by Thorsen et al. [
25].
Figure 19b and
Figure 20b also show the space-time plot of the CF displacement. The wave of CF displacement also propagates from the section of fast incoming flow to the section of slow incoming flow, such as the conditions of the maximum velocity
of shear current 0.6 m/s. However, in the case of
, it is observed that standing waves are generated due to the reflection of the fixed boundary condition at the section of the slow incoming flow. These standing waves have also been reported by the calculation of Thorsen et al. [
25]. These propagated waves due to VIV mean the propagation of energy from fast velocity region to a slow velocity region and this phenomenon is not observed from the uniform current condition.
In order to analyze the VIV characteristics of a tensioned riser under shear current, VIV responses at three points (near the top, middle, and bottom) of the riser under the shear current of
are presented in the time and frequency domains in
Figure 21.
Figure 21a shows the VIV responses at the three points in the time history. Unlike regular wave response in the CF on the uniform current condition, an irregular response with multi-frequencies is observed at all three points. Multi-frequencies components of VIV responses from other points due to the shear current are presumed to propagate and be superposed along the riser. The phenomenon is further analyzed with the frequency results in
Figure 21b. The peak frequency of 7.5 (Hz), resonance frequency with the 9th natural frequency (7.335 Hz) of the riser, is observed at different three points. Considering the natural frequencies of the riser model, VIV components with 7.5 (Hz) of frequency induced by 1.04 m/s of flow speed at −7.6 m of the riser excites the riser, resulting in resonant with the 9th natural frequency of the riser. It is found that a tensioned riser under the shear current vibrates in crossflow with multi-frequencies riding one dominant frequency.
Figure 22 shows the CF VIV snapshots of the riser in the time history. It is confirmed that the tensioned riser is dominantly excited with the 9th natural frequency by vortex shedding.
In this study, CF VIV analysis of the tensioned riser for the uniform current and shear current condition is carried out and compared with the model test. Through the comparative study, it is observed that the magnitude of the CF RMS amplitude and the main excitation mode of the RMS of CF displacement, which is calculated by the developed code in this study, are similar to the results of the model tests. Therefore, it is confirmed that the developed code provides a reasonable prediction of VIV of a slender structure.