1. Introduction
Ocean thermal energy conversion (OTEC) can be utilized as a technique for generating electricity by exploiting the temperature difference between warm surface seawater in tropical regions and cold deep seawater. Currently, this technique is mainly limited by its low efficiency due to the small temperature difference [1,2]. This renewable energy source utilizes the temperature difference of seawater as its heat source and can be applied in various applications, including freshwater aquaculture and seawater desalination. OTEC, compared to other forms of ocean energy, offers the advantage of providing a more stable energy yield with fewer cyclic fluctuations.
The riser serves as a crucial component in the ocean temperature difference energy system, facilitating the transportation of cold water from the deep sea to the surface and the extraction of deep seawater as the condensing working fluid for power generation. However, the riser is susceptible to vortex-induced vibrations. When the fluid flows across the circular cylinder, it engenders a phenomenon known as alternating vortex shedding. When the frequency of vortex shedding closely approaches the inherent frequency of the structure, a captivating event known as lock-in occurs, resulting in vortex-induced vibration. The frequency and amplitude of vibrations depend on the flow rate, diameter, and length of the cylinder, as well as fluid properties. Abundant studies have been conducted on cylinder flow and vortex-induced vibration [3,4,5]. Vortex-induced vibration (VIV) is one of the primary challenges in riser design that needs to be addressed.
The vibration problem of a long, flexible riser is an important engineering issue. When the exciting frequency approaches the natural frequency of the risers, lock-in phenomena may occur and also cause multi-mode vibration problems.
Therefore, understanding the VIV response characteristics of risers is critical for providing a basis for the strength and fatigue design of OTEC infrastructures. Currently, the vibration characteristics of risers are typically studied in controlled conditions through flume experiments, wind tunnel experiments, and numerical simulations. The researcher summarized recent studies of cylindrical cross-sectional vortex vibration and concluded that in the determination of the maximum amplitude of vortex vibration in the cylinder, the relationship between peak amplitude, mass, damping, and the existence of multi-modal vibration of flexible risers is yet to be studied [6,7]. Reynolds number has a significant impact on the amplitude, frequency, and phase angle of the low-damping system. Birkhoff [8] proposed a wake oscillator model for cylindrical vibration prediction. Zanganeh et al. [9] established a three-dimensional cylindrical wake oscillator model for uniform flow conditions. They studied an “8-shape” motion trajectory caused by the interaction of multi-order modes and high-order harmonic components. Saravanan et al. [10] used the measured data of field currents as input to perform a flume experiment on the riser of the inverted catenary. They also conducted numerical analysis with Shear7 and Orcaflex, which yielded results in good agreement with the experiment. Adiputra et al. [11] derived an analytical solution for different constraints under the influence of the internal flow of the riser and verified it by numerical simulation with the fluid-solid coupling method. Prawira et al. [12] used numerical simulation to analyze the riser’s diameter of the OTEC system in the Makassar Strait, Indonesia, and concluded that with an increased Reynolds number, the drag coefficient decreased, the lift coefficient increased, and the shedding frequency increased. Wang et al. [13] studied a floating platform and suspended riser in pool tests. Two conditions of VIV caused by the floating platform’s motion and an equivalent uniform current load were studied. The results showed that the VIV responses caused by the floating platform’s motion were similar to those produced by uniform flow when the Keulegan–Carpenter number is large. Numerical simulation results by Duanmu et al. [14], Song et al. [15], Pavlovskaia et al. [16], and Srinil [17] indicated that long flexible risers are subject to multi-mode vibrations, which become stronger with increasing flow velocity. Moreover, Duanmu et al. [18] concluded that riser vibrations undergo a transition from single mode to multi-order mode as the aspect ratio increases. Although these studies have given insight into the vortex-induced vibration of cylindrical flow and flexible risers, research on the dynamic effects of floating platforms and their risers is limited. Hence, further investigations on the vibration of the floating platform and the riser system are necessary.
The present article first presents a study in which the modal parameters of the riser under the fixed-bottom condition were identified. Flume experiments were conducted to study the acceleration response of a single riser under different flow velocities. Further experiments were carried out on a floating platform and the suspended riser system under different flow velocities to analyze and compare the vibration characteristics of the floating platform and the suspended riser system with those of the single riser. Finally, a finite element model was created in Orcaflex software, and the numerical simulation was compared with experimental results. The characteristics of vortex-induced vibrations in the riser were investigated. The work presented in this article contributes to improving the reliability of the riser in ocean thermal energy conversion systems and enhancing our understanding of the development and utilization of OTEC’s riser design. Additionally, the research methods and results are also of reference value for related research in other marine engineering fields.
2. Flume Experiment of Riser
Flume experiments were conducted in the Laboratory of the Hunan University of Science and Engineering, China. The flume spanned 22 m in length, 0.6 m in section width, and 0.8 m in height. Propelled by a variable-frequency pump, the maximal flow velocity exceeded 1400 m3/h. The flume flow velocity was instrumented by an ADV ultrasonic Doppler flow meter with a measurement range of 0.1 m/s to 10 m/s. Moreover, a waterproof acceleration sensor (uTekl-11133) was employed to measure the vibration response, which possessed a resolution of 0.001 g and a measurement range of ±500 g (where “g” denotes the acceleration of gravity). The variable-frequency pump produced 10 distinct uniform current velocities. The sampling rate was 128 Hz (This sampling frequency was used for all subsequent working conditions), and the test duration was 32 s.
2.1. Single Riser Natural Vibration Characteristics Test
Experimental modal analysis was carried out to identify the dynamic parameters of the riser. The tested riser sample was a hollow PVC cylindrical conduit. Its external diameter was 16 mm and its wall thickness was 1 mm. Its length was 0.4 m, resulting in an aspect ratio of 25. The material’s Young’s modulus and density were 220 MPa and 1380 kg/m3, respectively. The height of the measuring point was located 0.49 m from the base of the flume. The boundary conditions have been established using a fixed-end approach, and an acceleration sensor configuration has been implemented in the cross-flow direction. The acceleration response of the riser was measured under two separate conditions: with a water depth of 0.4 m and in a water-free environment. Free vibration tests were conducted, and the vibration acceleration was measured by initiating a specific displacement at the top of the riser and releasing it to observe its free vibration. To identify the damping of the riser, the Random Decrement Technique was employed.
The schematic diagram of the model was depicted in Figure 1. Figure 2 and Figure 3 illustrate the attenuation of the acceleration time data and acceleration power spectrum of the riser sample. In order to facilitate comparison, the power spectrum’s vertical coordinate of acceleration is uniformly represented using a semilogarithmic coordinate system. The results of the power spectrum analysis indicate that the energy of the riser model is predominantly concentrated in the low-frequency range with a distinct peak, and the inherent frequency in water-free conditions is approximately 5.75 Hz (with a damping ratio of 2.3%). With 0.4 m water depth, the fundamental frequency was measured at 5.25 Hz (with a damping ratio of 2.7%. The observed reduction in the natural frequency of underwater conditions is attributable to the influence of an additional mass.
2.2. Vortex-Shedding Parameters of the Riser
The vortex-shedding pattern of the riser is predominantly influenced by the Reynolds number, which is defined by the following:
(1)
where U represents the velocity of the incoming flow, D is the cylindrical diameter, and ν denotes the kinematic viscosity of the liquid. In this experiment, the Reynolds number lies in the range of 1700 to 1.23 × 105.In the subcritical Reynolds number region, it is typical to assume the Strouhal number has a value of 0.2 [3]. The cylindrical vortex-shedding frequency f is related to the Strouhal number:
(2)
The reduced velocity Ur is a dimensionless parameter related to the cylindrical vortex vibration and can be determined by the following:
(3)
where fn represents the cylindrical natural vibration frequency.2.3. Flume Experiment and Analysis of Vibration Response of Single Riser
Initially, by gauging the response of cross-flow vibration emanating from the apex of a riser constrained by a stationary pedestal under varying flow velocities. The experimental parameters dictate a uniform incoming flow at a range of flow velocities spanning from 0.25 m/s to 0.8 m/s, with increments of approximately 0.05 m/s. By evaluating the cross-flow acceleration response of the riser in response to current loading under diverse incoming flow conditions and subsequently calculating the power spectrum and root mean square (RMS) of acceleration response, the vibration pattern is ultimately elucidated by analyzing the relationship between frequency and the natural vibration frequency corresponding to the maximum response value across various flow velocities.
The power spectrum of acceleration response Is illustrated in Figure 4. The results of the spectral analysis reveal a positive correlation between the vibration frequency and flow velocities, and multiple peaks appear at certain flow velocities. The acceleration power spectrum is mainly dominated by only one prominent peak, and it is accompanied by locking-in phenomenon with the increase of flow velocity. Table 1 shows the acceleration response and peak frequency results. As the flow velocities increase, the vibration frequency in the cross-flow direction of the riser rises gradually. The root mean square of acceleration shows a distinct peak value, and at the flow velocity of 0.435 m/s (which corresponds to the reduced velocity Ur = 5.18), the cross-flow acceleration’s RMS value increases steeply. This frequency is nearly identical to the natural vibration frequency of the riser in water conditions, and at the current flow velocity, the riser initiates vortex-induced vibration. As the flow velocity rises, there is a significant decrease in the RMS value of the cross-flow acceleration. This indicates that the riser vibration frequency gradually shifts away from the natural frequency and starts moving out of the lock-in region as the flow velocity rises further. Additionally, the acceleration response decreases as the flow velocity rises.
2.4. Floating Platform with Suspended Riser System Flume Experiment and Analysis
The floating platform adopted the semi-submersible platform scaled-down model; the model was made of ABS plastic with a thickness of 2 mm; please refer to the Appendix A, Table A1 for specific dimensions; and its positioning system utilized a multi-point mooring method. The mooring chains consisted of four 0.7-m-long wires, 2 mm in diameter, and were linked to a spring for stiffness adjustment. The floating platform is entirely constrained by the wire tensioning method to limit vibration. Affix the upper section of the riser (with a length of 0.45 m) to the base of the buoyant platform using fixed-end constraints, similar to operating condition 1. The lower part of the riser remains unattached. At a distance of 10 mm from the bottom of the riser, a riser acceleration sensor was installed, and at the center of the side of the floating platform, a floating platform acceleration sensor was instrumented. The cross-flow acceleration response of the riser and the floating platform was measured at various flow velocities. Please refer to the Appendix A, Table A1 for the dimensions of the floating platform.
The schematic diagram of model and section is shown in Figure 5 and Figure 6. The experimental results of the riser and floating platform are presented in Table 2, and the acceleration power spectrum is illustrated in Figure 7. As the flow rate increased, the vibration characteristics of the floating platform and the suspended riser system were different from those of the bottom-fixed riser system. The introduction of the floating platform resulted in a significant decrease in the vibration frequency of the riser as compared to the single riser case. This is mainly due to the fact that the constraints on the riser imposed by the floating platform were weaker than those imposed by the bottom-fixed single riser configuration. When the suspended riser fixed to the floating platform reaches peak acceleration, the corresponding frequency was 3.94 Hz, which was 25% lower than the frequency of the bottom-fixed single riser, 5.26 Hz, when the flow rate was 0.392 m/s (corresponding to the reduced velocity Ur = 4.67). There was a noticeable peak acceleration at this velocity.
It is evident from the results of the acceleration power spectrum that there were multiple peaks in the cross-flow vibration of the floating platform and the riser, with both experiencing coupled vibration. From Figure 7, it is evident that the power spectrum of acceleration exhibits a prominent peak, with an additional peak around 5 Hz. This could potentially be attributed to the diminished stiffness of the buoyant platform. Therefore, both the floating platform and the riser will have peaks with frequencies in proximity. The maximum peak value of the vibration frequency of the floating platform surpasses that of the riser since the floating platform possesses a greater stiffness.
The floating platform and suspended riser system exhibited a different vibration characteristic from that of a single riser. Specifically, upon reaching the resonance frequency, acceleration response initially decreased and subsequently gradually increased as the flow rate increased. This was due to the excitation of higher-order vibrations in the riser as the flow rate increased. Similar phenomena have been observed in the top-tensioned riser flume experiments conducted by Yunqing et al. [19]. Despite the differences in boundary conditions between their experiments and this study and the fact that the bottom is hinged, high flow rates still induced higher-order mode vibration in the riser.
3. Numerical Simulation
3.1. Single Riser Modal Analysis and Response Analysis
The analysis of two distinct condition modes is depicted in Figure 8 and Figure 9. To verify the model’s reliability, numerical simulations were conducted utilizing the Orcaflex software to establish a single riser model. The boundary condition adopts the bottom fixed end constraint (similar to the flume experiment), with a drag coefficient of 1.2, a lift coefficient of 0.4, and an additional mass coefficient of 1 [13,20]. In the numerical simulation, identical inflow parameters as in the flume experiment were used in the time-domain calculations. The riser’s length and pedestal parameters were as follows: the base plate height is 0.01 m, the sleeve height is 0.10 m, the length of the riser is 0.5 m, and the actual length is 0.4 m, as depicted in Figure 10. The boundary conditions were set as fixed at the bottom, and the riser adopted the same parameters as the flume experiment (outer diameter: 16 mm, wall thickness: 1 mm, elastic model: 220 MPa, density: 1380 kg/m3). Orcaflex calculated the bending stiffness based on the input elastic modulus and moment of inertia.
The modal analysis results indicated that the fundamental natural frequency of the single riser is 5.55 Hz. The Orcaflex numerical simulation results were presented in Table 3, with the acceleration power spectrum depicted in Figure 11. The comparison between numerical simulation results and experimental results in the flume experiment is illustrated in Figure 12.
The velocity exhibits a positive correlation with the frequency. The numerical simulation and modal analysis reveal that the riser exhibits a first natural frequency of 5.75 Hz in water-free conditions, whereas, in wet conditions, it exhibits a first natural frequency of 5.55 Hz. The outcomes evince the presence of a sharp peak in the acceleration power spectrum when flow velocities are varied. The frequency of vortex-induced resonance was 5.58 Hz, which was close to the flume experimental result of 5.26 Hz. At a flow velocity of 0.435 m/s (reduced velocity Ur = 5.18), the acceleration response attained its maximum value. The greatest discrepancy between the numerical acceleration response and the flume experiment is approximately 10%. Hence, the outcomes manifested the capability of numerical simulation to emulate the vortex-induced vibration.
3.2. Numerical Simulation of Floating Platform and Suspension Riser System
The semi-submersible platform and riser system model was established using Orcaflex. The standpipe represented a cylindrical riser crafted from PVC, possessing an external diameter of 16 mm, a thickness of 1 mm, and an extent spanning 0.5 m. It exhibited an elastic modulus of 220 MPa and a density of 1380 kg/m3, aligning harmoniously with the outcomes of the flume experiment. The riser’s boundary conditions included that the upper end of the riser was rigidly fixed at the base of the platform, while the lower end of the riser was unrestrained. The buoyant platform remains attached through a multi-point mooring arrangement. This entails the utilization of four iron wires, measuring 2 mm in diameter and spanning a length of 0.70 m, as confirmed by the flume test. Furthermore, these wires possess an elastic modulus of 200 GPa. The schematic diagram of the finite element model was depicted in Figure 13. Modal analysis revealed that the fundamental natural frequency of the floating platform and riser system was 3.9 Hz. The mode shape is depicted in Figure 14. The results indicated that the fundamental vibration was primarily governed by the oscillation of the riser.
The riser’s solution for addressing VIV utilized the MILAN wake oscillator model integrated within Orcaflex, employing its default parameters. Nevertheless, a distinction arose in the corresponding flow velocity during vortex-induced resonance, while the frequency exhibited a fundamental concurrence. The summary of the numerical simulation results was compiled in Table 4. This observation signified that the incorporation of the floating platform exerted a discernible influence on the resonance flow velocity of the riser, plausibly attributable to the variance between the test mooring and the simulation constraints. In essence, the numerical simulation results obtained from Orcaflex exhibited good fidelity in reproducing the vortex-induced vibration occurring within the floating platform and riser system. The numerical simulation results revealed that the lateral vibration frequency of the floating platform is higher than that of the riser. The power spectrum of acceleration was illustrated in Figure 15, while the comparison between numerical simulation and flume experiments was depicted in Figure 16.
4. Conclusions
The oceanic vortex-induced vibration of risers is a self-limiting and highly nonlinear fluid-structure interaction phenomenon. In this study, the feasibility of the numerical simulations was validated by comparing them to the results of the flume experiment. Since this paper mainly focuses on scaled model experiments and numerical simulations, if there is a need to compute the vortex-induced vibration of the riser within the physical structure, the Strouhal number similarity criterion could be used for conversion. This would provide a theoretical foundation for subsequent calculations pertaining to the full-scale model. However, using a contact acceleration sensor would impact the measurement outcomes to a certain extent. Nonetheless, the results indicated that the frequency of the vortex-induced vibration at the peak of the acceleration response did not significantly deviate from the natural vibration frequency of the structure. Thus, its accuracy aligns with the engineering requirements. In the future, non-contact methods, such as employing DIC to capture photographs and discern structural displacement responses, could be explored. The main conclusions of this study are as follows:
(1) Numerical calculations conducted in Orcaflex showed that when the flow velocity matches the frequency of vortex shedding, the vortex-induced vibration of a single riser can be accurately estimated, and the results are consistent with those observed in the flume experiment. When the analysis was extended to the suspended riser system and floating platform, the numerical results were largely consistent with the response and vibration frequency calculation of the flume experiment, although they showed some discrepancy in the flow velocity at resonance.
(2) The vibration characteristics of the floating platform and suspended riser system differ from those of the single riser at different flow velocities. Taking into account the two resonance segments in the riser system of the floating platform, even if the vibration frequency is remote from the riser’s natural frequency, the acceleration will still escalate with the rise in the flow velocity. This is mainly due to the higher-order modal vibrations of the riser.
(3) Unlike the vibration of a single riser, the natural frequency of the suspended riser system decreased when the floating platform was incorporated. The real structural riser aspect ratio is more substantial, leading to a further reduction in the overall structural frequency, which should be taken into account when analyzing the entire response of the floating platform and the riser.
(4) Two different Orcaflex model calculations were validated via hydrodynamic flume experiments. However, conducting actual OTEC tests on the floating platform and riser response is challenging. Future research can utilize full-scale models for time-domain analysis, providing a robust basis for calculating the floating platform and riser structure of the thermal energy differential system.
Conceptualization, Z.L.; methodology, Z.L. and B.H.; software, Q.C.; validation, Q.C.; formal analysis, Q.C. and B.H.; investigation, Q.C., H.L., G.D. and H.W.; resources, Z.L.; data curation, Q.C.; writing—original draft preparation, Q.C.; writing—review and editing, Z.L. and R.W.K.C.; visualization, Q.C. and B.H.; supervision, Z.L.; project administration, Z.L.; funding acquisition, Z.L. and B.H. All authors have read and agreed to the published version of the manuscript.
Not applicable.
Not applicable.
Not applicable.
Thanks to Hunan University of Science and Technology for providing the flume experiment laboratory site and technical support.
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
Footnotes
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Enlarge this image.Figure 1. Diagram of riser flume experiment. (a) Water-free condition; (b) Water condition; (c) Pedestal.
Enlarge this image.Figure 2. Water-free condition. (a) Acceleration attenuation curve for water-free conditions; (b) Acceleration power spectrum under water-free conditions.
Enlarge this image.Figure 3. Water depth 0.4 m condition. (a) Acceleration attenuation curve for 0.4 m water depth; (b) Acceleration power spectrum at 0.4 m water depth.
Enlarge this image.Figure 4. Power spectrum of cross—flow acceleration of riser. (a) riser’s power spectrum of flow velocity of 0.263 m/s, (b) riser’s power spectrum of flow velocity of 0.316 m/s, (c) riser’s power spectrum of flow velocity of 0.392 m/s, (d) riser’s power spectrum of flow velocity of 0.435 m/s, (e) riser’s power spectrum of flow velocity of 0.494 m/s, (f) riser’s power spectrum of flow velocity of 0.546 m/s, (g) riser’s power spectrum of flow velocity of 0.605 m/s, (h) riser’s power spectrum of flow velocity of 0.676 m/s, (i) riser’s power spectrum of flow velocity of 0.746 m/s, (j) riser’s power spectrum of flow velocity of 0.79 m/s.
Enlarge this image.Figure 4. Power spectrum of cross—flow acceleration of riser. (a) riser’s power spectrum of flow velocity of 0.263 m/s, (b) riser’s power spectrum of flow velocity of 0.316 m/s, (c) riser’s power spectrum of flow velocity of 0.392 m/s, (d) riser’s power spectrum of flow velocity of 0.435 m/s, (e) riser’s power spectrum of flow velocity of 0.494 m/s, (f) riser’s power spectrum of flow velocity of 0.546 m/s, (g) riser’s power spectrum of flow velocity of 0.605 m/s, (h) riser’s power spectrum of flow velocity of 0.676 m/s, (i) riser’s power spectrum of flow velocity of 0.746 m/s, (j) riser’s power spectrum of flow velocity of 0.79 m/s.
Enlarge this image.Figure 5. Diagram of floating platform, riser model, and acceleration sensor layout.
Enlarge this image.Figure 7. Cross−flow acceleration power spectrum of the floating platform and riser system. (a) riser’s power spectrum of flow velocity of 0.263 m/s, (b) platform’s power spectrum of flow velocity of 0.263 m/s, (c) riser’s power spectrum of flow velocity of 0.316 m/s, (d) platform’s power spectrum of flow velocity of 0.316 m/s, (e) riser’s power spectrum of flow velocity of 0.392 m/s, (f) platform’s power spectrum of flow velocity of 0.392 m/s, (g) riser’s power spectrum of flow velocity of 0.435 m/s, (h) platform’s power spectrum of flow velocity of 0.435 m/s, (i) riser’s power spectrum of flow velocity of 0.494 m/s, (j) platform’s power spectrum of flow velocity of 0.494 m/s, (k) riser’s power spectrum of flow velocity of 0.546 m/s, (l) platform’s power spectrum of flow velocity of 0.546 m/s, (m) riser’s power spectrum of flow velocity of 0.605 m/s, (n) platform’s power spectrum of flow velocity of 0.605 m/s, (o) riser’s power spectrum of flow velocity of 0.676 m/s, (p) platform’s power spectrum of flow velocity of 0.676 m/s, (q) riser’s power spectrum of flow velocity of 0.746 m/s, (r) platform’s power spectrum of flow velocity of 0.746 m/s, (s) riser’s power spectrum of flow velocity of 0.79 m/s, (t) platform’s power spectrum of flow velocity of 0.79 m/s.
Enlarge this image.Figure 7. Cross−flow acceleration power spectrum of the floating platform and riser system. (a) riser’s power spectrum of flow velocity of 0.263 m/s, (b) platform’s power spectrum of flow velocity of 0.263 m/s, (c) riser’s power spectrum of flow velocity of 0.316 m/s, (d) platform’s power spectrum of flow velocity of 0.316 m/s, (e) riser’s power spectrum of flow velocity of 0.392 m/s, (f) platform’s power spectrum of flow velocity of 0.392 m/s, (g) riser’s power spectrum of flow velocity of 0.435 m/s, (h) platform’s power spectrum of flow velocity of 0.435 m/s, (i) riser’s power spectrum of flow velocity of 0.494 m/s, (j) platform’s power spectrum of flow velocity of 0.494 m/s, (k) riser’s power spectrum of flow velocity of 0.546 m/s, (l) platform’s power spectrum of flow velocity of 0.546 m/s, (m) riser’s power spectrum of flow velocity of 0.605 m/s, (n) platform’s power spectrum of flow velocity of 0.605 m/s, (o) riser’s power spectrum of flow velocity of 0.676 m/s, (p) platform’s power spectrum of flow velocity of 0.676 m/s, (q) riser’s power spectrum of flow velocity of 0.746 m/s, (r) platform’s power spectrum of flow velocity of 0.746 m/s, (s) riser’s power spectrum of flow velocity of 0.79 m/s, (t) platform’s power spectrum of flow velocity of 0.79 m/s.
Enlarge this image.Figure 7. Cross−flow acceleration power spectrum of the floating platform and riser system. (a) riser’s power spectrum of flow velocity of 0.263 m/s, (b) platform’s power spectrum of flow velocity of 0.263 m/s, (c) riser’s power spectrum of flow velocity of 0.316 m/s, (d) platform’s power spectrum of flow velocity of 0.316 m/s, (e) riser’s power spectrum of flow velocity of 0.392 m/s, (f) platform’s power spectrum of flow velocity of 0.392 m/s, (g) riser’s power spectrum of flow velocity of 0.435 m/s, (h) platform’s power spectrum of flow velocity of 0.435 m/s, (i) riser’s power spectrum of flow velocity of 0.494 m/s, (j) platform’s power spectrum of flow velocity of 0.494 m/s, (k) riser’s power spectrum of flow velocity of 0.546 m/s, (l) platform’s power spectrum of flow velocity of 0.546 m/s, (m) riser’s power spectrum of flow velocity of 0.605 m/s, (n) platform’s power spectrum of flow velocity of 0.605 m/s, (o) riser’s power spectrum of flow velocity of 0.676 m/s, (p) platform’s power spectrum of flow velocity of 0.676 m/s, (q) riser’s power spectrum of flow velocity of 0.746 m/s, (r) platform’s power spectrum of flow velocity of 0.746 m/s, (s) riser’s power spectrum of flow velocity of 0.79 m/s, (t) platform’s power spectrum of flow velocity of 0.79 m/s.
Enlarge this image.Figure 11. Numerical simulation of single riser acceleration power spectrum. (a) riser’s power spectrum of flow velocity of 0.263 m/s, (b) riser’s power spectrum of flow velocity of 0.316 m/s, (c) riser’s power spectrum of flow velocity of 0.392 m/s, (d) riser’s power spectrum of flow velocity of 0.435 m/s, (e) riser’s power spectrum of flow velocity of 0.494 m/s, (f) riser’s power spectrum of flow velocity of 0.546 m/s, (g) riser’s power spectrum of flow velocity of 0.605 m/s, (h) riser’s power spectrum of flow velocity of 0.676 m/s, (i) riser’s power spectrum of flow velocity of 0.746 m/s, (j) riser’s power spectrum of flow velocity of 0.79 m/s.
Enlarge this image.Figure 12. Comparison of response results between single riser flume experiment and numerical simulation. (a) Acceleration RMS of the single riser; (b) Vibration frequency of the single riser.
Enlarge this image.Figure 14. The first-order vibration diagram of the floating platform and the riser.
Enlarge this image.Figure 15. Numerical simulation of the floating platform and riser cross−flow acceleration power spectrum. (a) riser’s power spectrum of flow velocity of 0.263 m/s, (b) platform’s power spectrum of flow velocity of 0.263 m/s, (c) riser’s power spectrum of flow velocity of 0.316 m/s, (d) platform’s power spectrum of flow velocity of 0.316 m/s, (e) riser’s power spectrum of flow velocity of 0.392 m/s, (f) platform’s power spectrum of flow velocity of 0.392 m/s, (g) riser’s power spectrum of flow velocity of 0.435 m/s, (h) platform’s power spectrum of flow velocity of 0.435 m/s, (i) riser’s power spectrum of flow velocity of 0.494 m/s, (j) platform’s power spectrum of flow velocity of 0.494 m/s, (k) riser’s power spectrum of flow velocity of 0.546 m/s, (l) platform’s power spectrum of flow velocity of 0.546 m/s, (m) riser’s power spectrum of flow velocity of 0.605 m/s, (n) platform’s power spectrum of flow velocity of 0.605 m/s, (o) riser’s power spectrum of flow velocity of 0.676 m/s, (p) platform’s power spectrum of flow velocity of 0.676 m/s, (q) riser’s power spectrum of flow velocity of 0.746 m/s, (r) platform’s power spectrum of flow velocity of 0.746 m/s, (s) riser’s power spectrum of flow velocity of 0.79 m/s, (t) platform’s power spectrum of flow velocity of 0.79 m/s.
Enlarge this image.Figure 15. Numerical simulation of the floating platform and riser cross−flow acceleration power spectrum. (a) riser’s power spectrum of flow velocity of 0.263 m/s, (b) platform’s power spectrum of flow velocity of 0.263 m/s, (c) riser’s power spectrum of flow velocity of 0.316 m/s, (d) platform’s power spectrum of flow velocity of 0.316 m/s, (e) riser’s power spectrum of flow velocity of 0.392 m/s, (f) platform’s power spectrum of flow velocity of 0.392 m/s, (g) riser’s power spectrum of flow velocity of 0.435 m/s, (h) platform’s power spectrum of flow velocity of 0.435 m/s, (i) riser’s power spectrum of flow velocity of 0.494 m/s, (j) platform’s power spectrum of flow velocity of 0.494 m/s, (k) riser’s power spectrum of flow velocity of 0.546 m/s, (l) platform’s power spectrum of flow velocity of 0.546 m/s, (m) riser’s power spectrum of flow velocity of 0.605 m/s, (n) platform’s power spectrum of flow velocity of 0.605 m/s, (o) riser’s power spectrum of flow velocity of 0.676 m/s, (p) platform’s power spectrum of flow velocity of 0.676 m/s, (q) riser’s power spectrum of flow velocity of 0.746 m/s, (r) platform’s power spectrum of flow velocity of 0.746 m/s, (s) riser’s power spectrum of flow velocity of 0.79 m/s, (t) platform’s power spectrum of flow velocity of 0.79 m/s.
Enlarge this image.Figure 16. Comparison of Numerical Simulation and flume experiment. (a) RMS acceleration of floating platform and riser; (b) Frequency of floating platform and riser.
Single riser cross-flow vibration response test results.
| Velocity (m/s) | Reynolds Number | Reduced Velocity | Acceleration RMS (m/s2) | Vibration Frequency (Hz) |
|---|---|---|---|---|
| 0.263 | 4208 | 3.13 | 0.59 | 3.56 |
| 0.316 | 5056 | 3.76 | 0.83 | 4.48 |
| 0.392 | 6272 | 4.67 | 1.49 | 4.99 |
| 0.435 | 6960 | 5.18 | 3.66 | 5.26 |
| 0.494 | 7904 | 5.88 | 2.00 | 5.41 |
| 0.546 | 8736 | 6.50 | 1.37 | 5.40 |
| 0.605 | 9680 | 7.20 | 1.18 | 5.31 |
| 0.676 | 10,816 | 8.05 | 0.97 | 5.39 |
| 0.746 | 11,936 | 8.88 | 0.79 | 5.37 |
| 0.79 | 12,640 | 9.40 | 0.77 | 5.35 |
Test results of cross-flow vibration response of floating platform and riser system.
| Velocity |
Reynolds Number | Reduced |
Riser Acceleration RMS (m/s2) | Platform Acceleration RMS (m/s2) | Riser |
Platform |
|---|---|---|---|---|---|---|
| 0.263 | 4208 | 3.13 | 0.46 | 0.29 | 3.63 | 5.38 |
| 0.316 | 5056 | 3.76 | 0.60 | 0.29 | 3.59 | 5.31 |
| 0.392 | 6272 | 4.67 | 3.29 | 1.47 | 3.90 | 3.94 |
| 0.435 | 6960 | 5.18 | 1.70 | 0.87 | 4.13 | 4.16 |
| 0.494 | 7904 | 5.88 | 1.63 | 0.89 | 4.68 | 4.41 |
| 0.546 | 8736 | 6.50 | 1.39 | 0.60 | 5.31 | 4.34 |
| 0.605 | 9680 | 7.20 | 0.90 | 0.61 | 4.68 | 4.75 |
| 0.676 | 10,816 | 8.05 | 0.77 | 0.55 | 4.75 | 5.37 |
| 0.746 | 11,936 | 8.88 | 0.73 | 0.64 | 5.13 | 5.31 |
| 0.790 | 12,640 | 9.40 | 0.80 | 0.65 | 5.31 | 5.31 |
Numerical simulation cross-flow response results of Orcaflex single riser.
| Velocity (m/s) | Reynolds Number | Reduced Velocity | Acceleration RMS (m/s2) | Vibration Frequency (Hz) |
|---|---|---|---|---|
| 0.263 | 4208 | 3.13 | 0.53 | 3.88 |
| 0.316 | 5056 | 3.76 | 0.65 | 4.25 |
| 0.392 | 6272 | 4.67 | 1.35 | 5.11 |
| 0.435 | 6960 | 5.18 | 3.15 | 5.58 |
| 0.494 | 7904 | 5.88 | 1.56 | 5.59 |
| 0.546 | 8736 | 6.50 | 1.30 | 5.63 |
| 0.605 | 9680 | 7.20 | 0.98 | 5.8 |
| 0.676 | 10,816 | 8.05 | 0.75 | 6.56 |
| 0.746 | 11,936 | 8.88 | 0.71 | 7 |
| 0.790 | 12,640 | 9.40 | 0.65 | 7.31 |
Numerical analysis of floating platform and riser system.
| Velocity |
Reynolds Number | Reduced Velocity | Riser Acceleration RMS (m/s2) | Platform |
Riser |
Platform Frequency |
|---|---|---|---|---|---|---|
| 0.263 | 4208 | 3.13 | 0.42 | 0.26 | 2.93 | 3.1 |
| 0.316 | 5056 | 3.76 | 0.55 | 0.26 | 3.1 | 3.28 |
| 0.392 | 6272 | 4.67 | 1.58 | 0.82 | 3.63 | 3.75 |
| 0.435 | 6960 | 5.18 | 2.92 | 1.34 | 3.9 | 3.91 |
| 0.494 | 7904 | 5.88 | 1.51 | 0.8 | 4.32 | 4.53 |
| 0.546 | 8736 | 6.50 | 1.3 | 0.57 | 4.62 | 4.88 |
| 0.605 | 9680 | 7.20 | 0.85 | 0.56 | 5.08 | 5.37 |
| 0.676 | 10,816 | 8.05 | 0.68 | 0.49 | 5.61 | 5.89 |
| 0.746 | 11,936 | 8.88 | 0.69 | 0.57 | 6.08 | 6.37 |
| 0.79 | 12,640 | 9.40 | 0.74 | 0.59 | 6.44 | 6.73 |
Appendix A
Main Parameters of Semi-Submersible Platform Model.
| Parameter | Size (m) | Parameter | Size (m) |
|---|---|---|---|
| Platform pontoon | Platform column | ||
| Length | 0.38 | Height of column | 0.2 |
| Width | 0.067 | Length of column | 0.053 |
| Height | 0.05 | Width of column | 0.058 |
| Deck | Brace | ||
| Deck width | 0.387 | Length of brace | 0.142 |
| Deck length | 0.372 | Section radius | 0.004 |
References
1. Abbas, S.M.; Alhassany, H.D.S.; Vera, D.; Jurado, F. Review of enhancement for ocean thermal energy conversion system. J. Ocean. Eng. Sci.; 2022. [DOI: https://dx.doi.org/10.1016/j.joes.2022.03.008]
2. Zhang, W.; Li, Y.; Wu, X.; Guo, S. Review of the applied mechanical problems in ocean thermal energy conversion. Renew. Sustain. Energy Rev.; 2018; 93, pp. 231-244. [DOI: https://dx.doi.org/10.1016/j.rser.2018.05.048]
3. Blevins, R.D. Flow-Induced Vibration. New York. 1977; Available online: https://ui.adsabs.harvard.edu/abs/1977vnr.book...B/abstract (accessed on 1 May 2023).
4. Van Dyke, M. An Album of Fluid Motion; Parabolic Press Stanford: Stanford, CA, USA, 1982.
5. Coutanceau, M.; Defaye, J.-R. Circular Cylinder Wake Configurations: A Flow Visualization Survey. Appl. Mech. Rev.; 1991; 44, pp. 255-305. [DOI: https://dx.doi.org/10.1115/1.3119504]
6. Liu, G.; Li, H.; Qiu, Z.; Leng, D.; Li, Z.; Li, W. A mini review of recent progress on vortex-induced vibrations of marine risers. Ocean Eng.; 2020; 195, 106704. [DOI: https://dx.doi.org/10.1016/j.oceaneng.2019.106704]
7. Williamson, C.; Govardhan, R. A brief review of recent results in vortex-induced vibrations. J. Wind Eng. Ind. Aerodyn.; 2008; 96, pp. 713-735. [DOI: https://dx.doi.org/10.1016/j.jweia.2007.06.019]
8. Birkhoff, G. Jets, Wakes, and Cavities; Elsevier: Amsterdam, The Netherlands, 2012.
9. Zanganeh, H.; Srinil, N. Three-dimensional VIV prediction model for a long flexible cylinder with axial dynamics and mean drag magnifications. J. Fluids Struct.; 2016; 66, pp. 127-146. [DOI: https://dx.doi.org/10.1016/j.jfluidstructs.2016.07.004]
10. Saravanan, R.; Bhattacharya, S.; Ramana Murthy, M. Dynamic Behaviour of Inverted Catenary Cold Water Pipelines for Seawater Desalination Project. Proceedings of the Fourth International Conference in Ocean Engineering (ICOE2018); Springer: Singapore, 2019; Volume 2, pp. 463-477. [DOI: https://dx.doi.org/10.1007/978-981-13-3134-3_35]
11. Adiputra, R.; Utsunomiya, T. Stability based approach to design cold-water pipe (CWP) for ocean thermal energy conversion (OTEC). Appl. Ocean Res.; 2019; 92, 101921. [DOI: https://dx.doi.org/10.1016/j.apor.2019.101921]
12. Prawira, M.Z.; Koto, J. Vortex Induced Vibration on Cold Water Pipe of OTEC. Proceedings of the 11th International Conference on Marine Technology, MARTEC; Kuala Lumpur, Malaysia, 13–14 August 2018; pp. 33-41.
13. Wang, J.; Fu, S.; Wang, J.; Li, H.; Ong, M.C. Experimental investigation on vortex-induced vibration of a free-hanging riser under vessel motion and uniform current. J. Offshore Mech. Arct. Eng.; 2017; 139, 4. [DOI: https://dx.doi.org/10.1115/1.4036370]
14. Duanmu, Y.; Zou, L.; Wan, D. Numerical analysis of multi-modal vibrations of a vertical riser in step currents. Ocean Eng.; 2018; 152, pp. 428-442. [DOI: https://dx.doi.org/10.1016/j.oceaneng.2017.12.033]
15. Song, L.; Fu, S.; Cao, J.; Ma, L.; Wu, J. An investigation into the hydrodynamics of a flexible riser undergoing vortex-induced vibration. J. Fluids Struct.; 2016; 63, pp. 325-350. [DOI: https://dx.doi.org/10.1016/j.jfluidstructs.2016.03.006]
16. Pavlovskaia, E.; Keber, M.; Postnikov, A.; Reddington, K.; Wiercigroch, M. Multi-modes approach to modelling of vortex-induced vibration. Int. J. Non-Linear Mech.; 2016; 80, pp. 40-51. [DOI: https://dx.doi.org/10.1016/j.ijnonlinmec.2015.11.008]
17. Srinil, N. Multi-mode interactions in vortex-induced vibrations of flexible curved/straight structures with geometric nonlinearities. J. Fluids Struct.; 2010; 26, pp. 1098-1122. [DOI: https://dx.doi.org/10.1016/j.jfluidstructs.2010.08.005]
18. Duanmu, Y.; Zou, L.; Wan, D.-C. Numerical simulations of vortex-induced vibrations of a flexible riser with different aspect ratiosin uniform and shear currents. J. Hydrodyn. Ser. B; 2017; 29, pp. 1010-1022. [DOI: https://dx.doi.org/10.1016/S1001-6058(16)60815-6]
19. Yunqing, S.; Zhiqiang, W.; Guoqi, Z. Bifurcation Analysis of Dual-Mode Dynamics for Marine Risers. Appl. Math. Mech.; 2020; 41, 480. [DOI: https://dx.doi.org/10.21656/1000-0887.400257]
20. Norberg, C. Fluctuating lift on a circular cylinder: Review and new measurements. J. Fluids Struct.; 2003; 17, pp. 57-96. [DOI: https://dx.doi.org/10.1016/S0889-9746(02)00099-3]
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Details
; Li, Zhengnong 1 ; Chan, Ricky W K 2
; Luo, Han 1 ; Duan, Guodi 1 ; Huang, Bin 3
; Wu, Honghua 1 1 Key Laboratory of Building Safety and Energy Efficiency of the Ministry of Education, Hunan University, Changsha 410082, China;
2 School of Engineering, RMIT University Melbourne, Melbourne, VIC 3001, Australia
3 School of Civil Engineering and Architecture, Hainan University, Haikou 570228, China;