- Email: [email protected]
Experimental study on motion responses of a moored rectangular cylinder under freak waves (I: Time-domain study)
Ocean Engineering 153 (2018) 268–281
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Experimental study on motion responses of a moored rectangular cylinder under freak waves (I: Time-domain study) Wenbo Pan, Ningchuan Zhang, Guoxing Huang *, Xiangyu Ma State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, People's Republic of China
A R T I C L E I N F O
A B S T R A C T
Keywords: Motion response Rectangular cylinder Freak wave Random wave Time-domain
Extensive experiments on the motion responses of a rectangular cylinder under random and freak waves have been conducted in the present study. The effects of the relative wave height, relative period and freak wave parameters on the motion responses were investigated. Analysis in time-domain revealed that the freak wave parameter α1 has significant effect on the motion responses of the cylinder, especially for surge and heave. However, α2, α3, α4 have not such significant effect on the motion components. With α1 ¼ 2–2.8, the maximum surge and heave under freak waves were 2.5 and 1.5 times, respectively, larger than those under irregular waves. In addition, the maximum pitch under freak waves was approximately 1.3 times of that under irregular waves. The difference in motion response under freak waves and irregular waves decreased with the increasing relative wave heights. With Hs/d ¼ 0.03–0.1 and α1 ¼ 2–2.2, the maximum surge and heave under freak waves were approximately 30–60% and 20–40%, respectively, larger than those under irregular waves. For the effect of the relative period, the critical surge, heave and pitch occur at period around Tp/T0heavee1.0, 1.5, 2.0 and Tp/T0pitche1, respectively. With the natural periods and α1 ¼ 2–2.2, surge, heave and pitch under freak waves were approximately 44%, 40% and 30%, respectively, larger than those under irregular waves.
1. Introduction
Few researches concerned the interaction between freak waves and marine structures. For the generation mechanism of freak waves, Tromans et al. (1991) proposed NewWave methodology as an efficient approach to simulate extreme wave. Osborne (2001) and Slunyaev et al. (2002) used nonlinear Schrodinger (NLS) equation to simulate freak waves numerically. Pei et al. (2007) superposed a random wave train with two transient wave trains to generate freak waves in a short wave train. Cui et al. (2012) applied the constrained NewWave theory in the simulation of freak waves and studied the bottom influence on the propagation of the freak waves. As to the probability of occurrence, numerical and experimental studies have demonstrated that waves with freak wave features can be generated frequently in a two-dimensional wave flume without current, refraction or diffraction (Stansberg, 1990; Mori et al., 1992). Chien et al., (2002) analyzed the probability of nearshore freak waves and pointed out that the probability of freak wave occurrences increases significantly after a typhoon. On the aspect of interaction between freak waves and structures, Clauss et al. (2003) analyzed the motions behavior and resulting splitting forces of a semisubmersible under freak wave with both the
During offshore structures design, the extreme wave conditions are commonly adopted to determine the design wave loads according to most of design codes, which ignore freak waves. Generally, the distribution of the freak wave height does not follow the classic distribution for sea waves, i.e. Rayleigh distribution in deepwater and Γлуховский distribution in shallow water (Yu, 2000). Although it is hard to forecast, the existence of freak waves still imposes serious potential risks on marine structures and vessels. Random waves can be described as irregular wave sequence in time-domain and wave spectrum in frequency-domain. With the same wave spectrum, there are various wave sequences, in which freak waves could occur. It is therefore important to clarify if the occurence of freak waves leads to significant difference on the dynamic response of the offshore structures. In this study, the wave sequence with and without freak wave are referred to “freak wave sequence” and “conventional irregular wave sequence”, or “freak wave” and “irregular wave”, respectively. The existing study of freak waves foucses on revealing the formation mechanism, the probability of occurrence and numerical simulation, etc.
* Corresponding author. E-mail address: [email protected] (G. Huang). https://doi.org/10.1016/j.oceaneng.2018.01.084 Received 7 June 2017; Received in revised form 21 December 2017; Accepted 21 January 2018 0029-8018/© 2018 Elsevier Ltd. All rights reserved.
W. Pan et al.
Ocean Engineering 153 (2018) 268–281
Cartesian grid system. The comparison of the numerical results and measured data reveals that the proposed model is applicable in predicting the nonlinear dynamics of the floating body. Deng et al. (2014, 2015) investigated the effects of wave group characteristics on a semi-submersible under freak wave condition and found that the surge response increased significantly. Gao et al. (2015) studied numerical simulation of deterministic freak wave series and wave-structure interaction. Freak wave interactions with fixed cylinders submerged in different depths are investigated. The results suggest that the most critical vertical loads appear at the process of the freak wave approaching which may cause severe vertical responses on offshore structures. The present investigation compares the motion response of a floating rectangular cylinder under freak and irregular waves and quantifies the difference through experimental measurements. This provides a basis for understanding the mechanism of interaction between freak waves and structures. 2. Experiments 2.1. Experimental setup
Fig. 1. Relationship of tension (Tm) versus deformation (Δs).
The tests were carried out in a wave flume at the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology. The flume is 60 m long, 4.0 m wide and 2.5 m deep. The wave generation system is Hydro-servo irregular wave maker system, and it generates waves with a wave period ranging from 0.5s to 5.0s. At the other end of the flume, an absorbing beach is installed for wave dissipation. The motion responses were measured by a non-touched 6DOF (degree of freedom) measurement system, consisting of dual-CCD cameras and a data acquisition system. Three light markers were arranged in a plane and fixed on the top of the cylinder to track the motion behaviors by the dual-CCD cameras. The images of the markers were continuously acquired by the two cameras at 30 frames per second, and the signals were processed to recover the instantaneous position of each marker in a calibrated coordinate system. The mooring tension was measured by a tension sensor with an accuracy of 0.1 N. The wave heights were measured by DS30 waves measuring system, which controlled 64 wave gauges synchronously. The wave gauges were calibrated before tests with an accuracy is 0.1 mm.
time-domain simulation and model tests. Schmittner. (2005) investigated the motions and bending moments of an FPSO and a heavy lift vessel as well as the motions and splitting forces of a semisubmersible due to Rogue Waves. The results show that the vertical bending moments, heave and pitch motions of an FPSO and a heavy lift vessel as well as the airgap and splitting forces of a semisubmersible under Rogue Waves are larger than the maximum values predicted by the classification rules and frequency-domain analysis. Bunnik et al. (2008) used a VOF-based model to predict the extreme wave loads on fixed offshore structures due to focused wave groups. Shen and Yang, (2013) investigated the stress, wave climbing and wave slamming force on the column and semi-submersible platform by freak wave through numerical simulations. The results showed that the crest of a freak wave is the main parameter affecting the platform motion. Rudman and Cleary (2013) adopted the Smoothed Particle Hydrodynamics (SPH) method to simulate the fully non-linear interactions between a TLP and freak waves. They considered the effect of wave impact angle and mooring line pre-tension on the subsequent motions of the platform. Gu et al. (2013) studied the hydrodynamic of a TLP under freak waves. Zhao et al. (2014) developed an advanced tool to model freak waves impact on a floating body under large amplitude motions. The model is solved by a Constrained Interpolation Profile(CIP)-based high order finite-difference method on a fixed
2.2. Model design and experimental parameters The floater is an airtight rectangular cylinder made of acrylic (Fig. 1). The cylinder is a rectangular solid with a height of h ¼ 62 cm, and a square base with a side length of D ¼ 50 cm. The sharp corners of the cylinder were grinded to smooth arcs with a radius of R ¼ 6.6 cm. The
Table 1 Summary of hydrodynamic parameters of the floating cylinder model. Parameters Moulded height Draft Weight Center of gravity measured from bottom Center of buoyancy measured from bottom Transverse metacentric radius Longitudinal metacentric radius Transverse metacentric height Longitudinal metacentric height Transverse moment of inertia Longitudinal moment of inertia Natural surge period Natural heave period Natural pitch period
269
unit
magnitude
cm cm kg cm
62 34 73 20
cm cm cm cm cm cm4 cm4 s s s
17 6.13 6.13 5.13 5.13 5.2 105 5.2 105 10.0 1.4 2.0
W. Pan et al.
Ocean Engineering 153 (2018) 268–281
Fig. 2. Mooring pattern and layout of the model in the wave flume.
clump weight was set on the bottom of the cylinder to achieve the design center of gravity (COG), center of buoyancy (COB) and draft. The prototype of the rectangular cylinder (with the smooth arc corners) is a floating moored structure for 30,000 ton vessel operating in a wind farm in China. The hydrodynamic parameters of the model are presented in Table 1. The floating rectangular cylinder was anchored by four mooring lines as illustrated in Fig. 2. Each mooring line is made of nylon ropes to simulate the weight, length and tension-deformation curve. The length of each mooring line is 4 m, and the relation between the tension (Tm) and deformation (Δs) is:
Tm ¼
Cp dp ðΔS=SÞn λ3
(1)
Tm mooring tension of the model cable (N);Cp Elasticity coefficient of prototype cable, with nylon rope of Cp ¼ 1:540 104 Mpa;dp Diameter of prototype cable (m);λModel scale, λ ¼ 35;nIndex, with nylon rope adopting n ¼ 3. The simulation of mooring lines needs to match the elastic similarity as well as gravity similarity. An example of the tension ~ deformation curve for the prototype and model is presented in Fig. 1. It can be observed that excellent agreement was achieved between the model and prototype.
270
W. Pan et al.
Ocean Engineering 153 (2018) 268–281
Table 2 Summary of experimental wave parameters. significant wave height Hs(cm)
spectrum peak period Tp(s)
parameter of freak wave α1
parameter of freak wave α2~α4
relative wave height Hs/d
4.54 6.51 8.52 10.54 11.80 13.62 7.86–8.15
1.40–1.45
2.10<α1 < 2.2
1.72<α2 < 2.1 1.55<α3 < 2.1 0.53<α4 < 0.62
1.14 1.26 1.56 1.83 2.01 2.16 2.87 3.10 1.56–1.6
2.01<α1 < 2.2
1.40<α2 < 2.14 1.40<α3 < 2.37 0.51<α4 < 0.66
0.032 0.047 0.061 0.075 0.084 0.097 0.056–0.058
1.55 1.70 1.91 2.07 2.32 2.60 2.83
1.30<α2 < 1.52 1.50<α3 < 1.94 0.51<α4 < 0.56
7.77–8.11
0.056–0.058
remarks
Tp / T0heave
Tp/ T0pitch
0.14–0.145
1.0–1.04
0.7–0.73
To investigate the effect of wave height
0.114 0.126 0.156 0.183 0.201 0.216 0.287 0.310 0.156–0.160
0.81 0.90 1.11 1.31 1.44 1.54 2.05 2.21 1.11–1.14
0.57 0.63 0.78 0.92 1.01 1.08 1.44 1.55 0.78–0.8
To investigate the effect of wave period
To investigate the effect of parameter of freak wave
αn
m ¼ 1, Бm ¼ Surge/D; T0m ¼ T0surge; m ¼ 2, Бm ¼ Heave/D; T0m ¼ T0heave; m ¼ 3, Бm ¼ pitch; T0m ¼ T0pitch;
Fig. 2 shows the layout of the model in the wave flume mooring pattern and mooring configuration. The lower ends of the mooring lines were anchored on the bottom of the wave flume. The top ends together with tension sensors were connected to the four corners of the rectangular cylinder, and each connecting point was 25 cm measured from the bottom of the cylinder. For three-dimensional floater, the motion responses are sway, surge, heave, roll, pitch and yaw. For two-dimensional case, three motion components are investigated: surge, heave and pitch. To ensure the scope of investigation in a practical level, dimentional analysis is performed firstly. It is assumed that the motion response depends on the following variables: the dimensions of the rectangular cylinder (the side length of D, the moulded height of h), hydrodynamic parameters of the floating body (refer to Table 1) and wave parameters (the significant wave height Hs, peak period Tp, freak wave parameters αn and spectral pattern, etc.). Generally, the characteristics of the freak waves can be defined by four parameters (Klinting and Sand, 1987): α1 ¼ Hmax/Hs, α2 ¼ Hmax/Hmax-1, α3 ¼ Hmax/Hmaxþ1, α4 ¼ ηmax/Hmax, where Hmax is the maximum wave height in the measured sequence, Hmax-1 the wave height before the maximum wave height, Hmaxþ1 the wave height after the maximum wave height, ηmax is surface elevation of the maximum wave height in the irregular wave sequence. In the experiments, JONSWAP spectrum was applied with a water depth of d ¼ 1.4 m and a draft of floating body of h0 ¼ 34 cm. To compare the experimental results collected under freak and irregular wave conditions, the same spectrum with identical spectral parameters were adopted in the wave simulation. In the current study, the freak wave was generated using a random wave train combined with two transient wave trains (Pei et al., 2007). Based on the above parameters, the dimensional analysis can be performed as, Бm ¼ fm(H/d, Tp/T0m, αn)
relative period Tp/T0m Tp/T0surge
The detailed wave parameters in the experiments are summarized in Table 2. 2.3. Analysis methodology Firstly, the design waves were calibrated at the center of the model in the wave flume. The model was then installed and moored with initial tension of 1–1.2 N. The measurements for each test typically lasted for approximately 100 waves, and were performed 2 or 3 times to ensure repeatability. During each test, the motions and mooring tensions were recorded synchronously with a frequency of 30 Hz. The measured data can be analyzed statistically in time-domain and spectrually in frequency-domain. Statistical analysis in time-domain provides the maximum value, 1/10 value, 1/3 value (effective value), average value and other statistical characteristics.
3. Analysis of experiments results 3.1. Comparison on the time history of the wave series and motion responses With identical significant wave height Hs and peak period Tp (Hs ¼ 13.62 cm, Tp ¼ 1.41s) using JONSWAP spectrum, two wave series were generated during the experiment. One was a freak wave series with freak parameters of α1 ¼ 2.11, α2 ¼ 1.72, α3 ¼ 2.03, α4 ¼ 0.62, as shown in Fig. 3(a1). The other was a random wave series with freak parameters of the most critical wave α1 ¼ 1.51, α2 ¼ 1.40, α3 ¼ 1.45, α4 ¼ 0.61 as shown in Fig. 3(a2). The corresponding motion responses (surge, heave and pitch) are illustrated in Fig. 3(b-d) under freak and irregular waves, respectively. It is observed that the lowfrequency components of surge occur under both freak and irregular
(2)
Where. m ¼ 1–3;
271
W. Pan et al.
Ocean Engineering 153 (2018) 268–281
Fig. 3. Time history of motion response of the floater under freak and irregular waves (Freak wave α1 ¼ 2.11, α2 ¼ 1.72, α3 ¼ 2.03, α4 ¼ 0.62; Maximum wave of the irregular wave α1 ¼ 1.51, α2 ¼ 1.4, α3 ¼ 1.45, α4 ¼ 0.51).
3.2. Effect of relative wave height
wave. In particular, the observed critical surge in freak wave series is significant larger than that under random waves. To understand the difference, the critical wave heights and motion components in Fig. 3 were zoomed in and shown in Fig. 4. It shows that the maximum surge and heave of the floater were observed at the phase with the occurrence of the freak wave. However, similar phenomenon was not observed for pitch. It indicates that the freak wave has direct effect on the surge and heave of the floater, while not for pitch. The maximum surge and heave under the freak wave condition are significantly larger than those under irregular wave condition.
To investigate the effect of the relative wave heights, two wave series were generated with the identical peak period of Tp ¼ 1.4 s and relative wave heights of Hs/d ¼ 0.03–0.1 using JONSWAP spectrum. The freak wave series are generated with the characteristics of 2.01<α1 < 2.20, 1.71<α2 < 2.15, 1.35<α3 < 2.13, 0.53<α4 < 0.62, and the random waves are generated with 1.51<α1 < 1.59, 1.33<α2 < 1.69, 1.92<α3 < 1.99, 0.51<α4 < 0.63. In Fig. 5, the extreme surge, heave and pitch are plotted and compared between freak and irregular wave conditions. It is shown that the statistical characteristic of surge increases with the relative
272
W. Pan et al.
Ocean Engineering 153 (2018) 268–281
Fig. 3. (continued).
maximum pitch transfer from each other alternatively with relative wave heights. More information can be obtained by plotting the motions under freak waves normalized by the motions under random waves against the relative wave heights. The parameters of surge(f)/surge(i), heave(f)/ heave(i) and pitch(f)/pitch(i) are the pair extreme value for the surge, heave and pitch under freak and irregular waves, respectively. The graph is shown in Fig. 6 with Hs/d ¼ 0.03–0.1. In the range of Hs/d ¼ 0.03–0.1, the maximum surge, heave and pitch under freak waves are 30–60%, 20–40% and 20–33%, respectively, larger than those under irregular waves. It should also be noted that the extreme pitch in freak waves is almost identical to that under random waves in the relative wave heights range of Hs/d ¼ 0.06–0.1.
wave heights for both freak and irregular waves, but the extreme surge under freak waves is significantly larger. While, the statistical average value of surge in freak wave series is identical to that under random wave series. It indicates that the extreme surge depends on the type of wave series. Comparing the surge and heave under freak and irregular waves in Fig. 5(a) and (b), the tendency of statistical characteristics varying with the relative wave height is indentical, although the range of variation is not exactly the same. Similarly, the dimensionless effective value and average value of heave under freak and irregular waves are almost indentical. Fig. 5(c) shows that there is no significant increment in maximum pitch under freak waves than that under irregular waves with Hs/d > 0.05. It is different from the surge and heave, the
273
W. Pan et al.
Ocean Engineering 153 (2018) 268–281
Fig. 4. Zoom-in details of extreme waves in Fig. 3 (Freak wave α1 ¼ 2.11, α2 ¼ 1.72, α3 ¼ 2.03, α4 ¼ 0.62; Maximum wave of the irregular wave α1 ¼ 1.51, α2 ¼ 1.4, α3 ¼ 1.45, α4 ¼ 0.51).
In addition, there are some cases with pitch(f)/pitch (i) < 1 in Fig. 6. The probably reasons could be that there is no one-to-one relationships between the maximum pitch and the maximum wave height under irregular waves (Fig. 5).The pitch under the maximum wave height is related to the posture of the floater at the moment. It depends on the wave sequence before the maximum wave height. Even if the maximum wave heights (and the corresponding period) are identical, the maximum pitches could be different. In other words, the maximum pitches may have insignificant difference under irregular waves with various maximum wave heights (Fig. 7(c)). This has been pointed out in previous studies, i.e. Bennett et al. (2013).
The surge usually has wave frequency component and lowfrequency component, which probably caused by the non-linear towing force. However, the heave is dominated by the wave frequency component only. Fig. 6 shows the same trend of the specific value of the maximum surge and heave under freak wave and irregular waves, both decrease with the relative wave height. This can eliminate the possibility that the phenomenon is caused by the low-frequency motions of the floating body; On the other hand, the motion of the floater is related to the mooring pattern and material of mooring ropes. Using nylon material, when the displacement of the floating body increases, the rapid growth of the cable tension will limit the movement significantly.
274
W. Pan et al.
Ocean Engineering 153 (2018) 268–281
Fig. 4. (continued).
dimensionless surge with the relative period shows the same trend. The surge increases with the relative periods and reaches the local peak at approximately Tp/T0surge ¼ 0.14 (Tp/T0heave ¼ 1). This may be due to the peak spectral period of the wave is consistent with the natural period of heave. The maximum surge is affected by the resonant phenomenon of heave. It then decreases, and then increases slightly again towards larger Tp/T0surge. However, the maximum surge observed under freak wave condition is approximately 1.5 times to that under irregular waves. In Fig. 7(b), the variation of the statistical characteristic of heave with the relative period is oscillating with increasing relative period Tp/ T0heave. However, the critical heave under freak waves is significantly larger than that under irregular waves. The extreme heave occurs at the ‘resonance’ period of Tp/T0heave ¼ 1–1.2, 1.4–1.6 and 2.2–2.3,
3.3. Effect of relative period To investigate the effect of relative wave period, two wave series were generated with the identical wave height of Hs ¼ 8 cm and relative periods of Tp/T0surge ¼ 0.114～0.310, Tp/T0heave ¼ 0.8–2.2, Tp/ T0pitch ¼ 0.5–1.6 using JONSWAP spectrum. The freak wave series are generated with the characteristics of 2.01<α1 < 2.20, 1.40<α2 < 2.14, 1.40<α3 < 2.37, 0.51<α4 < 0.66, and the random waves are generated with 1.45<α1 < 1.53, 1.06<α2 < 2.16, 1.29<α3 < 1.99, 0.51<α4 < 0.64. Under freak and irregular waves, Fig. 7(a), (b) and (c) illustrate the dimensionless statistical motions versus the relative period for surge, heave and pitch, respectively. It is observed from Fig. 7(a) that, for both freak and irregular wave conditions, the variation of the
275
W. Pan et al.
Ocean Engineering 153 (2018) 268–281
Fig. 5. Comparison of the dimensionless statistical characteristics of surge, heave and pitch under freak and irregular wave (Freak wave: 2.0<α1 < 2.20, 1.71<α2 < 2.15, 1.35<α3 < 2.13, 0.53<α4 < 0.62; Maximum wave of the irregular wave: 1.51<α1 < 1.59, 1.33<α2 < 1.69, 1.92<α3 < 1.99, 0.51<α4 < 0.63).
respectively. However, the ‘resonance’ period under random waves are Tp/T0heave ¼ 1.0, 1.5 and 2.0, respectively. Fig. 7(a) and (b) shows that the variation of statistical characteristic of surge and heave with the relative periods are different, which could be attributed to the various natural periods of surge and heave. For the maximum heave, the impact of freak wave action is larger than that under irregular waves. It is observed from Fig. 7(c), for both freak and irregular wave conditions, the variation of the statistical characteristic of pitch are generally the same, and with approximately the same ‘resonance’ period of Tp/ T0pitch ¼ 1. The minor difference is the pitch under freak waves oscillating with the relative period Tp/T0pitch, this could be attributed to the freak wave parameters which have an impact on pitch. More information can be obtained by plotting the motions under freak waves normalized by the motions under random waves against the relative periods. The graph is shown in Fig. 8 with surge, heave and pitch periods of Tp/T0surge ¼ 0.114–0.310, Tp/T0heave ¼ 0.8–2.2 and Tp/ T0pitch ¼ 0.5–1.6, respectively. It is observed that the maximum variation occurs at the ‘resonance’ period of the floater under both freak and irregular waves. The variation of surge, heave and pitch are 44%, 40%
and 30%, respectively. 3.4. Effect of freak wave parameters In this section, the effect of the four freak wave parameters (α1~α4) on the motion responses are discussed, respectively. 3.4.1. The effect of α1 To investigate the effect of the freak wave parameter α1, wave series were generated with various α1 (α1 ¼ 1.7–2.83). All wave series were generated under constant wave height of Hs ¼ 8 cm, peak period of Tp ¼ 1.6s and other parameters of α2 ¼ 1.30–1.52, α3 ¼ 1.42–1.60, α4 ¼ 0.51–0.56. In Fig. 9, the extreme surge, heave and pitch are plotted against the freak wave parameter α1 and compared between freak and irregular wave conditions. The irregular waves are generated with the characteristics of 1.57<α1 < 1.60, 2.15<α2 < 2.31, 1.92<α3 < 1.99, 0.54<α4 < 0.55. In Fig. 9(a), the surge of the rectangular cylinder is increases with α1 almost linearly, which indicates that α1 has significant effect on surge.
276
W. Pan et al.
Ocean Engineering 153 (2018) 268–281
Fig. 5. (continued).
With α1 ¼ 2–2.83, the maximum surge under freak waves is 1.4–2.5 times larger than that under irregular waves. Fig. 9(b) shows that the heave of interceptive rectangular cylinder increases with the freak wave parameter α1, and reaches the local peak at approximately α1 ¼ 2.3. It then decreases mildly with father increment. With α1 ¼ 2–2.83, the maximum heave under freak waves is 1.4–1.6 times larger than that under irregular waves. Fig. 9(c) shows that the pitch of interceptive rectangular cylinder has minor dependence on the freak wave parameter α1. It depends on the wave sequence before the maximum wave height. In the range of α1 ¼ 2–2.83, the maximum pitch under freak waves increases 30% than that under irregular waves.
wave height Hs/d ¼ 0.03–0.1, and the other three freak wave parameters
α1 ¼ 2.0–2.2; α3 ¼ 1.92–1.99; α4 ¼ 0.53–0.62. The test results demonstrating the effect of α2 and α3 on motion responses are shown in
Fig. 10(a) and b. Test sequence 2: with α2 ¼ 1.14–2.14, significant wave height of Hs ¼ 8 cm, relative period Tp/T0surge ¼ 0.114～0.310; Tp/ T0heave ¼ 0.8–2.2; Tp/T0pitch ¼ 0.5–1.6, and the other three freak wave parameters are α1 ¼ 2.0–2.2; α3 ¼ 1.35–2.37; α4 ¼ 0.5–0.66. The test results demonstrating the effect of α2 and α3 on motion responses are shown in Fig. 11(a) and b. It is observed from Figs. 10 and 11 that the motion responses of the floater fluctuate with the variation of the freak wave parameters α2 and α3, which suggests α2 and α3 have no significant effects on the motion response.
3.4.2. The effect of α2 and α3 Freak wave parameters α2 and α3 reflect the characteristics of the waves before and after the maximum wave, respectively. The effect of α2 and α3 on motion responses are expected to be similar, and thus discussed together. Test sequence 1: with α2 ¼ 1.7–2.15, peak period Tp ¼ 1.6s, relative
3.4.3. The effect of α4 To investigate the effect of α4 on the motion response, the same analysis method is chosen as α2 and α3. The results are shown in Fig. 12(a) and b, respectively.
277
W. Pan et al.
Ocean Engineering 153 (2018) 268–281
Fig. 6. The variation of maximum motion component versus the relative wave height under the freak and irregular wave (Freak wave: 2.0<α1 < 2.20, 1.71<α2 < 2.15, 1.35<α3 < 2.13, 0.53<α4 < 0.62; Maximum wave of the irregular wave: 1.51<α1 < 1.59, 1.33<α2 < 1.69, 1.92<α3 < 1.99, 0.51<α4 < 0.63).
Fig. 8. The variation of maximum motion component versus the relative period under the freak and irregular wave (Freak wave: 2.0<α1 < 2.20, 1.40<α2 < 2.14, 1.40<α3 < 2.37, 0.51<α4 < 0.66; Maximum wave of the irregular wave: 1.45<α1 < 1.53, 1.06<α2 < 1.94, 1.29<α3 < 2.16, 0.51<α4 < 0.64).
It is observed from Fig. 12 that the surge and heave fluctuate with the variation of the freak wave parameter α4, which suggests α4 has no significant impact on the motion response with of α4 0.66. However, the pitch motions for the two wave series are different: the results of sequence 1 show that the pitch increases with α4, which could be attributed to the relative wave height of Hs/d ¼ 0.03–0.1. The increment of pitch is caused by the increment of the relative wave height. The results of sequence 2 show that the pitch fluctuates with α4, which shall be investigated further in future studies.
4. Conclusion In this study, the motion response of a rectangular cylinder under the freak and random wave conditions was investigated experimentally. Extensive tests were conducted to quantify the effect of relative wave height, period and freak wave parameters, and the main conclusions are summarized as follows. 1) Given identical wave spectrum, the maximum motion response of the mooring rectangular cylinder differs substantially under the action of
Fig. 7. Comparison of the dimensionless motion response varying with the relative period under freak and irregular wave (Freak wave: 2.0<α1 < 2.20, 1.40<α2 < 2.14, 1.40<α3 < 2.37, 0.51<α4 < 0.66; Maximum wave of the irregular wave: 1.45<α1 < 1.53, 1.06<α2 < 1.94, 1.29<α3 < 2.16, 0.51<α4 < 0.64).
278
W. Pan et al.
Ocean Engineering 153 (2018) 268–281
Fig. 9. Motion response varying with α1 under freak and irregular wave (Freak wave: 1.70<α1 < 2.83, 1.30<α2 < 1.52, 1.42<α3 < 1.6,0.51<α4 < 0.56; Maximum wave of the irregular wave: 1.57<α1 < 1.6, 2.15<α2 < 2.30, 1.92<α3 < 1.99, 0.54<α4 < 0.55).
Fig. 10. Motion response varying with α2 under freak wave (Test sequence 1)( Freak wave: 2.0<α1 < 2.20, 1.7<α2 < 2.15, 1.92<α3 < 1.99, 0.53<α4 < 0.62). (b)Motion responses varying with α3 under freak wave (Test sequence 1). (Freak wave: 2.0<α1 < 2.20, 1.7<α2 < 2.15, 1.92<α3 < 1.99, 0.53<α4 < 0.62). 279
W. Pan et al.
Ocean Engineering 153 (2018) 268–281
Fig. 11. Motion responses varying with α2 under freak wave (Test sequence 2)( Freak wave: 2.0<α1 < 2.20, 1.14<α2 < 2.14, 1.35<α3 < 2.37, 0.5<α4 < 0.66). (b)Motion responses varying with α3 under freak wave(Test sequence 2).
30–60% larger than that under irregular wave; with Hs/ d ¼ 0.03–0.05, the heave and pitch are approximately 20–40% and 20–33% larger than those under irregular wave, respectively. However, the influence becomes insignificant for Hs/d ¼ 0.06–0.1. 5) For the effect of relative period on the motion response, significant difference was observed at Tp/T0surge ¼ 0.14, Tp/T0heave ¼ 1.0, 1.5,2.0 and Tp/T0pitch ¼ 1. It is observed that the maximum variation occurs at the ‘resonance’ period of the floater under both freak and irregular waves. Given identical relative period and α1 ¼ 2–2.2, the variation of surge, heave and pitch under freak and irregular waves were 44%, 40%, 30%, respectively. 6) The freak wave parameters α1 has direct impact on surge and heave, but not on pitch. Within the range of α1 ¼ 2–2.8, the maximum surge and heave under freak waves are 1.4–2.5 and 1.4–1.6 times larger than those under irregular waves and the maximum pitch increases approximately 30%. Based on the
the freak versus random waves. The difference is closely correlated with the relative wave height, relative period and freak wave parameter α1. 2) The freak wave has direct effect on the surge and heave of the floater, but not for pitch. The maximum surge and heave under the freak wave are significantly larger than those under irregular wave. It indicates that the maximum surge and heave depends on the type of wave series. 3) There is no one-to-one relationships between the extreme pitch and wave height under irregular waves. The pitch under maximum wave height is related to the posture of the floater. In other words, it depends on the wave sequence prior to the maximum wave height. 4) The variation of motion response under freak and irregular waves decreases with the increasing relative wave heights. With Hs/ d ¼ 0.03–0.1, α1 ¼ 2–2.2, the maximum surge under freak wave is
280
W. Pan et al.
Ocean Engineering 153 (2018) 268–281
Fig. 12. Motion responses varying with α4 under freak wave (Test sequence 1)( Freak wave: 2.0<α1 < 2.20, 1.7<α2 < 2.15, 1.92<α3 < 1.99, 0.53<α4 < 0.62). (b)Motion responses varying with α4 under freak wave (Test sequence 2)( Freak wave: 2.0<α1 < 2.20, 1.14<α2<2.14, 1.35<α3 < 2.37, 0.5<α4 < 0.66).
observation in the current study, α2, α3 and α4 have no direct impact on the motion response of the floating cylinder.
Gu, J.Y., Lv, H.N., Yang, J.M., 2013. Dynamic response study of four column TLP in freak waves. Ocean Eng. 31 (5), 25–35 (in chinese). Klinting, P., Sand, S., 1987. Analysis of Prototype Freak Waves, Conference on Nearshore Hydrodynamics. ASCE, pp. 618–632. Mori, N., Yasuda, T., Ito, K., 1992. Freak waves in a unidirectional wave train and their kinematics. In: Proceedings of the 23rd International Conference on Coastal Engineering. Osborne, A.R., 2001. The random and deterministic dynamics of ‘rogue waves’ in unidirectional deep water wave trains. Mar. Struct. 14 (3), 275–293. Pei, Y.G., Zhang, N.C., Zhang, Y.Q., 2007. Efficient generation of freak waves in laboratory. China Ocean Eng. 21 (3), 515–523. Rudman, M., Cleary, P.W., 2013. Rogue wave impact on a tension leg platform: the effect of wave incidence angle and mooring line tension. Ocean Eng. 61 (6), 123–138. Schmittner, C.E., 2005. Rogue Wave Impact on Marine Structures. Universit€atsbibliothek. Shen, Y.G., Yang, J.M., 2013. Numerical investigation on the motion response of semisubmersible platform under extreme waves. Ocean Eng. 31 (3), 9–17. Slunyaev, A., Kharif, C., Pelinovsky, E., Talipova, T., 2002. Nonlinear wave focusing on water of finite depth. Phys. Nonlinear Phenom. 173 (1–2), 77–96. Stansberg, C.T., 1990. Extreme Waves in Laboratory Generated Irregular Wave Trains, vol. 178. Springer Netherlands, pp. 573–589. Tromans, P., Anaturk, A.R., Hagemeijer, P., 1991. A new model for the kinematics of large ocean waves-application as a design wave. In: Proceedings of 1st International Offshore and Polar Engineering Conference, Edinburgh, UK. Yu, Y.X., 2000. Random Wave and its Applications for Engineering. Dalian University of Technology Press, pp. 222–267. Zhao, X.Z., Ye, Z.T., Fu, Y.N., Cao, F.F., 2014. A CIP-based numerical simulation of freak wave impact on a floating body. Ocean Eng. 87, 50–63.
References Bunnik, T., Veldman, A., Wellens, P., 2008. Prediction of extreme wave loads in focused wave groups. In: Proceedings of the 18th International Offshore and Polar Engineering Conference. Vancouver, BC, Canada, pp. 32–38. Bennett, S.S., Hudson, D.A., Temarel, P., 2013. The influence of forward speed on ship motions in abnormal waves: experimental measurements and numerical predictions. J. Fluid Struct. 39 (5), 154–172. Chien, H., Kao, C., Chuang, L.H., 2002. On the characteristics of observed coastal freak waves. Coast Eng. J. 44 (04), 301–319. Clauss, G.F., Schmittner, C.E., Stutz, K., 2003. Freak wave impact on semisubmersibles time-domain analysis of motions and forces. In: Proc. 13th International Offshore and Polar Engineering Conf., Honolulu, USA. JSC-371. Cui, C., Zhang, N.C., Yu, Y.X., 2012. Numerical study on the effects of uneven bottom topography on freak waves. Ocean Eng. 54 (9), 132–141. Deng, Y.F., Yang, J.M., Xiao, L.F., 2014. Influence of wave group characteristics on the motion of a semisubmersible in freak waves. In: Proceedings of 33rd International Conference on Ocean, Offshore and Arctic Engineering. American Society of Mechanical Engineers. Deng, Y.F., Yang, J.M., Xiao, L.F., 2015. An experimental investigation on the motion and dynamic Responses of A Semisubmersible in freak waves. In: Proceedings of the 17th National Conference on Ocean Engineering, Nanning, China. Gao, N.B., Yang, J.M., Zhao, W.H., 2015. Numerical simulation of deterministic freak wave sequences and wave-structure interaction. Ships Offshore Struct. 1–16.
281
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Experimental study on motion responses of a moored rectangular cylinder under freak waves (I: Time-domain study) Wenbo Pan, Ningchuan Zhang, Guoxing Huang *, Xiangyu Ma State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, People's Republic of China
A R T I C L E I N F O
A B S T R A C T
Keywords: Motion response Rectangular cylinder Freak wave Random wave Time-domain
Extensive experiments on the motion responses of a rectangular cylinder under random and freak waves have been conducted in the present study. The effects of the relative wave height, relative period and freak wave parameters on the motion responses were investigated. Analysis in time-domain revealed that the freak wave parameter α1 has significant effect on the motion responses of the cylinder, especially for surge and heave. However, α2, α3, α4 have not such significant effect on the motion components. With α1 ¼ 2–2.8, the maximum surge and heave under freak waves were 2.5 and 1.5 times, respectively, larger than those under irregular waves. In addition, the maximum pitch under freak waves was approximately 1.3 times of that under irregular waves. The difference in motion response under freak waves and irregular waves decreased with the increasing relative wave heights. With Hs/d ¼ 0.03–0.1 and α1 ¼ 2–2.2, the maximum surge and heave under freak waves were approximately 30–60% and 20–40%, respectively, larger than those under irregular waves. For the effect of the relative period, the critical surge, heave and pitch occur at period around Tp/T0heavee1.0, 1.5, 2.0 and Tp/T0pitche1, respectively. With the natural periods and α1 ¼ 2–2.2, surge, heave and pitch under freak waves were approximately 44%, 40% and 30%, respectively, larger than those under irregular waves.
1. Introduction
Few researches concerned the interaction between freak waves and marine structures. For the generation mechanism of freak waves, Tromans et al. (1991) proposed NewWave methodology as an efficient approach to simulate extreme wave. Osborne (2001) and Slunyaev et al. (2002) used nonlinear Schrodinger (NLS) equation to simulate freak waves numerically. Pei et al. (2007) superposed a random wave train with two transient wave trains to generate freak waves in a short wave train. Cui et al. (2012) applied the constrained NewWave theory in the simulation of freak waves and studied the bottom influence on the propagation of the freak waves. As to the probability of occurrence, numerical and experimental studies have demonstrated that waves with freak wave features can be generated frequently in a two-dimensional wave flume without current, refraction or diffraction (Stansberg, 1990; Mori et al., 1992). Chien et al., (2002) analyzed the probability of nearshore freak waves and pointed out that the probability of freak wave occurrences increases significantly after a typhoon. On the aspect of interaction between freak waves and structures, Clauss et al. (2003) analyzed the motions behavior and resulting splitting forces of a semisubmersible under freak wave with both the
During offshore structures design, the extreme wave conditions are commonly adopted to determine the design wave loads according to most of design codes, which ignore freak waves. Generally, the distribution of the freak wave height does not follow the classic distribution for sea waves, i.e. Rayleigh distribution in deepwater and Γлуховский distribution in shallow water (Yu, 2000). Although it is hard to forecast, the existence of freak waves still imposes serious potential risks on marine structures and vessels. Random waves can be described as irregular wave sequence in time-domain and wave spectrum in frequency-domain. With the same wave spectrum, there are various wave sequences, in which freak waves could occur. It is therefore important to clarify if the occurence of freak waves leads to significant difference on the dynamic response of the offshore structures. In this study, the wave sequence with and without freak wave are referred to “freak wave sequence” and “conventional irregular wave sequence”, or “freak wave” and “irregular wave”, respectively. The existing study of freak waves foucses on revealing the formation mechanism, the probability of occurrence and numerical simulation, etc.
* Corresponding author. E-mail address: [email protected] (G. Huang). https://doi.org/10.1016/j.oceaneng.2018.01.084 Received 7 June 2017; Received in revised form 21 December 2017; Accepted 21 January 2018 0029-8018/© 2018 Elsevier Ltd. All rights reserved.
W. Pan et al.
Ocean Engineering 153 (2018) 268–281
Cartesian grid system. The comparison of the numerical results and measured data reveals that the proposed model is applicable in predicting the nonlinear dynamics of the floating body. Deng et al. (2014, 2015) investigated the effects of wave group characteristics on a semi-submersible under freak wave condition and found that the surge response increased significantly. Gao et al. (2015) studied numerical simulation of deterministic freak wave series and wave-structure interaction. Freak wave interactions with fixed cylinders submerged in different depths are investigated. The results suggest that the most critical vertical loads appear at the process of the freak wave approaching which may cause severe vertical responses on offshore structures. The present investigation compares the motion response of a floating rectangular cylinder under freak and irregular waves and quantifies the difference through experimental measurements. This provides a basis for understanding the mechanism of interaction between freak waves and structures. 2. Experiments 2.1. Experimental setup
Fig. 1. Relationship of tension (Tm) versus deformation (Δs).
The tests were carried out in a wave flume at the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology. The flume is 60 m long, 4.0 m wide and 2.5 m deep. The wave generation system is Hydro-servo irregular wave maker system, and it generates waves with a wave period ranging from 0.5s to 5.0s. At the other end of the flume, an absorbing beach is installed for wave dissipation. The motion responses were measured by a non-touched 6DOF (degree of freedom) measurement system, consisting of dual-CCD cameras and a data acquisition system. Three light markers were arranged in a plane and fixed on the top of the cylinder to track the motion behaviors by the dual-CCD cameras. The images of the markers were continuously acquired by the two cameras at 30 frames per second, and the signals were processed to recover the instantaneous position of each marker in a calibrated coordinate system. The mooring tension was measured by a tension sensor with an accuracy of 0.1 N. The wave heights were measured by DS30 waves measuring system, which controlled 64 wave gauges synchronously. The wave gauges were calibrated before tests with an accuracy is 0.1 mm.
time-domain simulation and model tests. Schmittner. (2005) investigated the motions and bending moments of an FPSO and a heavy lift vessel as well as the motions and splitting forces of a semisubmersible due to Rogue Waves. The results show that the vertical bending moments, heave and pitch motions of an FPSO and a heavy lift vessel as well as the airgap and splitting forces of a semisubmersible under Rogue Waves are larger than the maximum values predicted by the classification rules and frequency-domain analysis. Bunnik et al. (2008) used a VOF-based model to predict the extreme wave loads on fixed offshore structures due to focused wave groups. Shen and Yang, (2013) investigated the stress, wave climbing and wave slamming force on the column and semi-submersible platform by freak wave through numerical simulations. The results showed that the crest of a freak wave is the main parameter affecting the platform motion. Rudman and Cleary (2013) adopted the Smoothed Particle Hydrodynamics (SPH) method to simulate the fully non-linear interactions between a TLP and freak waves. They considered the effect of wave impact angle and mooring line pre-tension on the subsequent motions of the platform. Gu et al. (2013) studied the hydrodynamic of a TLP under freak waves. Zhao et al. (2014) developed an advanced tool to model freak waves impact on a floating body under large amplitude motions. The model is solved by a Constrained Interpolation Profile(CIP)-based high order finite-difference method on a fixed
2.2. Model design and experimental parameters The floater is an airtight rectangular cylinder made of acrylic (Fig. 1). The cylinder is a rectangular solid with a height of h ¼ 62 cm, and a square base with a side length of D ¼ 50 cm. The sharp corners of the cylinder were grinded to smooth arcs with a radius of R ¼ 6.6 cm. The
Table 1 Summary of hydrodynamic parameters of the floating cylinder model. Parameters Moulded height Draft Weight Center of gravity measured from bottom Center of buoyancy measured from bottom Transverse metacentric radius Longitudinal metacentric radius Transverse metacentric height Longitudinal metacentric height Transverse moment of inertia Longitudinal moment of inertia Natural surge period Natural heave period Natural pitch period
269
unit
magnitude
cm cm kg cm
62 34 73 20
cm cm cm cm cm cm4 cm4 s s s
17 6.13 6.13 5.13 5.13 5.2 105 5.2 105 10.0 1.4 2.0
W. Pan et al.
Ocean Engineering 153 (2018) 268–281
Fig. 2. Mooring pattern and layout of the model in the wave flume.
clump weight was set on the bottom of the cylinder to achieve the design center of gravity (COG), center of buoyancy (COB) and draft. The prototype of the rectangular cylinder (with the smooth arc corners) is a floating moored structure for 30,000 ton vessel operating in a wind farm in China. The hydrodynamic parameters of the model are presented in Table 1. The floating rectangular cylinder was anchored by four mooring lines as illustrated in Fig. 2. Each mooring line is made of nylon ropes to simulate the weight, length and tension-deformation curve. The length of each mooring line is 4 m, and the relation between the tension (Tm) and deformation (Δs) is:
Tm ¼
Cp dp ðΔS=SÞn λ3
(1)
Tm mooring tension of the model cable (N);Cp Elasticity coefficient of prototype cable, with nylon rope of Cp ¼ 1:540 104 Mpa;dp Diameter of prototype cable (m);λModel scale, λ ¼ 35;nIndex, with nylon rope adopting n ¼ 3. The simulation of mooring lines needs to match the elastic similarity as well as gravity similarity. An example of the tension ~ deformation curve for the prototype and model is presented in Fig. 1. It can be observed that excellent agreement was achieved between the model and prototype.
270
W. Pan et al.
Ocean Engineering 153 (2018) 268–281
Table 2 Summary of experimental wave parameters. significant wave height Hs(cm)
spectrum peak period Tp(s)
parameter of freak wave α1
parameter of freak wave α2~α4
relative wave height Hs/d
4.54 6.51 8.52 10.54 11.80 13.62 7.86–8.15
1.40–1.45
2.10<α1 < 2.2
1.72<α2 < 2.1 1.55<α3 < 2.1 0.53<α4 < 0.62
1.14 1.26 1.56 1.83 2.01 2.16 2.87 3.10 1.56–1.6
2.01<α1 < 2.2
1.40<α2 < 2.14 1.40<α3 < 2.37 0.51<α4 < 0.66
0.032 0.047 0.061 0.075 0.084 0.097 0.056–0.058
1.55 1.70 1.91 2.07 2.32 2.60 2.83
1.30<α2 < 1.52 1.50<α3 < 1.94 0.51<α4 < 0.56
7.77–8.11
0.056–0.058
remarks
Tp / T0heave
Tp/ T0pitch
0.14–0.145
1.0–1.04
0.7–0.73
To investigate the effect of wave height
0.114 0.126 0.156 0.183 0.201 0.216 0.287 0.310 0.156–0.160
0.81 0.90 1.11 1.31 1.44 1.54 2.05 2.21 1.11–1.14
0.57 0.63 0.78 0.92 1.01 1.08 1.44 1.55 0.78–0.8
To investigate the effect of wave period
To investigate the effect of parameter of freak wave
αn
m ¼ 1, Бm ¼ Surge/D; T0m ¼ T0surge; m ¼ 2, Бm ¼ Heave/D; T0m ¼ T0heave; m ¼ 3, Бm ¼ pitch; T0m ¼ T0pitch;
Fig. 2 shows the layout of the model in the wave flume mooring pattern and mooring configuration. The lower ends of the mooring lines were anchored on the bottom of the wave flume. The top ends together with tension sensors were connected to the four corners of the rectangular cylinder, and each connecting point was 25 cm measured from the bottom of the cylinder. For three-dimensional floater, the motion responses are sway, surge, heave, roll, pitch and yaw. For two-dimensional case, three motion components are investigated: surge, heave and pitch. To ensure the scope of investigation in a practical level, dimentional analysis is performed firstly. It is assumed that the motion response depends on the following variables: the dimensions of the rectangular cylinder (the side length of D, the moulded height of h), hydrodynamic parameters of the floating body (refer to Table 1) and wave parameters (the significant wave height Hs, peak period Tp, freak wave parameters αn and spectral pattern, etc.). Generally, the characteristics of the freak waves can be defined by four parameters (Klinting and Sand, 1987): α1 ¼ Hmax/Hs, α2 ¼ Hmax/Hmax-1, α3 ¼ Hmax/Hmaxþ1, α4 ¼ ηmax/Hmax, where Hmax is the maximum wave height in the measured sequence, Hmax-1 the wave height before the maximum wave height, Hmaxþ1 the wave height after the maximum wave height, ηmax is surface elevation of the maximum wave height in the irregular wave sequence. In the experiments, JONSWAP spectrum was applied with a water depth of d ¼ 1.4 m and a draft of floating body of h0 ¼ 34 cm. To compare the experimental results collected under freak and irregular wave conditions, the same spectrum with identical spectral parameters were adopted in the wave simulation. In the current study, the freak wave was generated using a random wave train combined with two transient wave trains (Pei et al., 2007). Based on the above parameters, the dimensional analysis can be performed as, Бm ¼ fm(H/d, Tp/T0m, αn)
relative period Tp/T0m Tp/T0surge
The detailed wave parameters in the experiments are summarized in Table 2. 2.3. Analysis methodology Firstly, the design waves were calibrated at the center of the model in the wave flume. The model was then installed and moored with initial tension of 1–1.2 N. The measurements for each test typically lasted for approximately 100 waves, and were performed 2 or 3 times to ensure repeatability. During each test, the motions and mooring tensions were recorded synchronously with a frequency of 30 Hz. The measured data can be analyzed statistically in time-domain and spectrually in frequency-domain. Statistical analysis in time-domain provides the maximum value, 1/10 value, 1/3 value (effective value), average value and other statistical characteristics.
3. Analysis of experiments results 3.1. Comparison on the time history of the wave series and motion responses With identical significant wave height Hs and peak period Tp (Hs ¼ 13.62 cm, Tp ¼ 1.41s) using JONSWAP spectrum, two wave series were generated during the experiment. One was a freak wave series with freak parameters of α1 ¼ 2.11, α2 ¼ 1.72, α3 ¼ 2.03, α4 ¼ 0.62, as shown in Fig. 3(a1). The other was a random wave series with freak parameters of the most critical wave α1 ¼ 1.51, α2 ¼ 1.40, α3 ¼ 1.45, α4 ¼ 0.61 as shown in Fig. 3(a2). The corresponding motion responses (surge, heave and pitch) are illustrated in Fig. 3(b-d) under freak and irregular waves, respectively. It is observed that the lowfrequency components of surge occur under both freak and irregular
(2)
Where. m ¼ 1–3;
271
W. Pan et al.
Ocean Engineering 153 (2018) 268–281
Fig. 3. Time history of motion response of the floater under freak and irregular waves (Freak wave α1 ¼ 2.11, α2 ¼ 1.72, α3 ¼ 2.03, α4 ¼ 0.62; Maximum wave of the irregular wave α1 ¼ 1.51, α2 ¼ 1.4, α3 ¼ 1.45, α4 ¼ 0.51).
3.2. Effect of relative wave height
wave. In particular, the observed critical surge in freak wave series is significant larger than that under random waves. To understand the difference, the critical wave heights and motion components in Fig. 3 were zoomed in and shown in Fig. 4. It shows that the maximum surge and heave of the floater were observed at the phase with the occurrence of the freak wave. However, similar phenomenon was not observed for pitch. It indicates that the freak wave has direct effect on the surge and heave of the floater, while not for pitch. The maximum surge and heave under the freak wave condition are significantly larger than those under irregular wave condition.
To investigate the effect of the relative wave heights, two wave series were generated with the identical peak period of Tp ¼ 1.4 s and relative wave heights of Hs/d ¼ 0.03–0.1 using JONSWAP spectrum. The freak wave series are generated with the characteristics of 2.01<α1 < 2.20, 1.71<α2 < 2.15, 1.35<α3 < 2.13, 0.53<α4 < 0.62, and the random waves are generated with 1.51<α1 < 1.59, 1.33<α2 < 1.69, 1.92<α3 < 1.99, 0.51<α4 < 0.63. In Fig. 5, the extreme surge, heave and pitch are plotted and compared between freak and irregular wave conditions. It is shown that the statistical characteristic of surge increases with the relative
272
W. Pan et al.
Ocean Engineering 153 (2018) 268–281
Fig. 3. (continued).
maximum pitch transfer from each other alternatively with relative wave heights. More information can be obtained by plotting the motions under freak waves normalized by the motions under random waves against the relative wave heights. The parameters of surge(f)/surge(i), heave(f)/ heave(i) and pitch(f)/pitch(i) are the pair extreme value for the surge, heave and pitch under freak and irregular waves, respectively. The graph is shown in Fig. 6 with Hs/d ¼ 0.03–0.1. In the range of Hs/d ¼ 0.03–0.1, the maximum surge, heave and pitch under freak waves are 30–60%, 20–40% and 20–33%, respectively, larger than those under irregular waves. It should also be noted that the extreme pitch in freak waves is almost identical to that under random waves in the relative wave heights range of Hs/d ¼ 0.06–0.1.
wave heights for both freak and irregular waves, but the extreme surge under freak waves is significantly larger. While, the statistical average value of surge in freak wave series is identical to that under random wave series. It indicates that the extreme surge depends on the type of wave series. Comparing the surge and heave under freak and irregular waves in Fig. 5(a) and (b), the tendency of statistical characteristics varying with the relative wave height is indentical, although the range of variation is not exactly the same. Similarly, the dimensionless effective value and average value of heave under freak and irregular waves are almost indentical. Fig. 5(c) shows that there is no significant increment in maximum pitch under freak waves than that under irregular waves with Hs/d > 0.05. It is different from the surge and heave, the
273
W. Pan et al.
Ocean Engineering 153 (2018) 268–281
Fig. 4. Zoom-in details of extreme waves in Fig. 3 (Freak wave α1 ¼ 2.11, α2 ¼ 1.72, α3 ¼ 2.03, α4 ¼ 0.62; Maximum wave of the irregular wave α1 ¼ 1.51, α2 ¼ 1.4, α3 ¼ 1.45, α4 ¼ 0.51).
In addition, there are some cases with pitch(f)/pitch (i) < 1 in Fig. 6. The probably reasons could be that there is no one-to-one relationships between the maximum pitch and the maximum wave height under irregular waves (Fig. 5).The pitch under the maximum wave height is related to the posture of the floater at the moment. It depends on the wave sequence before the maximum wave height. Even if the maximum wave heights (and the corresponding period) are identical, the maximum pitches could be different. In other words, the maximum pitches may have insignificant difference under irregular waves with various maximum wave heights (Fig. 7(c)). This has been pointed out in previous studies, i.e. Bennett et al. (2013).
The surge usually has wave frequency component and lowfrequency component, which probably caused by the non-linear towing force. However, the heave is dominated by the wave frequency component only. Fig. 6 shows the same trend of the specific value of the maximum surge and heave under freak wave and irregular waves, both decrease with the relative wave height. This can eliminate the possibility that the phenomenon is caused by the low-frequency motions of the floating body; On the other hand, the motion of the floater is related to the mooring pattern and material of mooring ropes. Using nylon material, when the displacement of the floating body increases, the rapid growth of the cable tension will limit the movement significantly.
274
W. Pan et al.
Ocean Engineering 153 (2018) 268–281
Fig. 4. (continued).
dimensionless surge with the relative period shows the same trend. The surge increases with the relative periods and reaches the local peak at approximately Tp/T0surge ¼ 0.14 (Tp/T0heave ¼ 1). This may be due to the peak spectral period of the wave is consistent with the natural period of heave. The maximum surge is affected by the resonant phenomenon of heave. It then decreases, and then increases slightly again towards larger Tp/T0surge. However, the maximum surge observed under freak wave condition is approximately 1.5 times to that under irregular waves. In Fig. 7(b), the variation of the statistical characteristic of heave with the relative period is oscillating with increasing relative period Tp/ T0heave. However, the critical heave under freak waves is significantly larger than that under irregular waves. The extreme heave occurs at the ‘resonance’ period of Tp/T0heave ¼ 1–1.2, 1.4–1.6 and 2.2–2.3,
3.3. Effect of relative period To investigate the effect of relative wave period, two wave series were generated with the identical wave height of Hs ¼ 8 cm and relative periods of Tp/T0surge ¼ 0.114～0.310, Tp/T0heave ¼ 0.8–2.2, Tp/ T0pitch ¼ 0.5–1.6 using JONSWAP spectrum. The freak wave series are generated with the characteristics of 2.01<α1 < 2.20, 1.40<α2 < 2.14, 1.40<α3 < 2.37, 0.51<α4 < 0.66, and the random waves are generated with 1.45<α1 < 1.53, 1.06<α2 < 2.16, 1.29<α3 < 1.99, 0.51<α4 < 0.64. Under freak and irregular waves, Fig. 7(a), (b) and (c) illustrate the dimensionless statistical motions versus the relative period for surge, heave and pitch, respectively. It is observed from Fig. 7(a) that, for both freak and irregular wave conditions, the variation of the
275
W. Pan et al.
Ocean Engineering 153 (2018) 268–281
Fig. 5. Comparison of the dimensionless statistical characteristics of surge, heave and pitch under freak and irregular wave (Freak wave: 2.0<α1 < 2.20, 1.71<α2 < 2.15, 1.35<α3 < 2.13, 0.53<α4 < 0.62; Maximum wave of the irregular wave: 1.51<α1 < 1.59, 1.33<α2 < 1.69, 1.92<α3 < 1.99, 0.51<α4 < 0.63).
respectively. However, the ‘resonance’ period under random waves are Tp/T0heave ¼ 1.0, 1.5 and 2.0, respectively. Fig. 7(a) and (b) shows that the variation of statistical characteristic of surge and heave with the relative periods are different, which could be attributed to the various natural periods of surge and heave. For the maximum heave, the impact of freak wave action is larger than that under irregular waves. It is observed from Fig. 7(c), for both freak and irregular wave conditions, the variation of the statistical characteristic of pitch are generally the same, and with approximately the same ‘resonance’ period of Tp/ T0pitch ¼ 1. The minor difference is the pitch under freak waves oscillating with the relative period Tp/T0pitch, this could be attributed to the freak wave parameters which have an impact on pitch. More information can be obtained by plotting the motions under freak waves normalized by the motions under random waves against the relative periods. The graph is shown in Fig. 8 with surge, heave and pitch periods of Tp/T0surge ¼ 0.114–0.310, Tp/T0heave ¼ 0.8–2.2 and Tp/ T0pitch ¼ 0.5–1.6, respectively. It is observed that the maximum variation occurs at the ‘resonance’ period of the floater under both freak and irregular waves. The variation of surge, heave and pitch are 44%, 40%
and 30%, respectively. 3.4. Effect of freak wave parameters In this section, the effect of the four freak wave parameters (α1~α4) on the motion responses are discussed, respectively. 3.4.1. The effect of α1 To investigate the effect of the freak wave parameter α1, wave series were generated with various α1 (α1 ¼ 1.7–2.83). All wave series were generated under constant wave height of Hs ¼ 8 cm, peak period of Tp ¼ 1.6s and other parameters of α2 ¼ 1.30–1.52, α3 ¼ 1.42–1.60, α4 ¼ 0.51–0.56. In Fig. 9, the extreme surge, heave and pitch are plotted against the freak wave parameter α1 and compared between freak and irregular wave conditions. The irregular waves are generated with the characteristics of 1.57<α1 < 1.60, 2.15<α2 < 2.31, 1.92<α3 < 1.99, 0.54<α4 < 0.55. In Fig. 9(a), the surge of the rectangular cylinder is increases with α1 almost linearly, which indicates that α1 has significant effect on surge.
276
W. Pan et al.
Ocean Engineering 153 (2018) 268–281
Fig. 5. (continued).
With α1 ¼ 2–2.83, the maximum surge under freak waves is 1.4–2.5 times larger than that under irregular waves. Fig. 9(b) shows that the heave of interceptive rectangular cylinder increases with the freak wave parameter α1, and reaches the local peak at approximately α1 ¼ 2.3. It then decreases mildly with father increment. With α1 ¼ 2–2.83, the maximum heave under freak waves is 1.4–1.6 times larger than that under irregular waves. Fig. 9(c) shows that the pitch of interceptive rectangular cylinder has minor dependence on the freak wave parameter α1. It depends on the wave sequence before the maximum wave height. In the range of α1 ¼ 2–2.83, the maximum pitch under freak waves increases 30% than that under irregular waves.
wave height Hs/d ¼ 0.03–0.1, and the other three freak wave parameters
α1 ¼ 2.0–2.2; α3 ¼ 1.92–1.99; α4 ¼ 0.53–0.62. The test results demonstrating the effect of α2 and α3 on motion responses are shown in
Fig. 10(a) and b. Test sequence 2: with α2 ¼ 1.14–2.14, significant wave height of Hs ¼ 8 cm, relative period Tp/T0surge ¼ 0.114～0.310; Tp/ T0heave ¼ 0.8–2.2; Tp/T0pitch ¼ 0.5–1.6, and the other three freak wave parameters are α1 ¼ 2.0–2.2; α3 ¼ 1.35–2.37; α4 ¼ 0.5–0.66. The test results demonstrating the effect of α2 and α3 on motion responses are shown in Fig. 11(a) and b. It is observed from Figs. 10 and 11 that the motion responses of the floater fluctuate with the variation of the freak wave parameters α2 and α3, which suggests α2 and α3 have no significant effects on the motion response.
3.4.2. The effect of α2 and α3 Freak wave parameters α2 and α3 reflect the characteristics of the waves before and after the maximum wave, respectively. The effect of α2 and α3 on motion responses are expected to be similar, and thus discussed together. Test sequence 1: with α2 ¼ 1.7–2.15, peak period Tp ¼ 1.6s, relative
3.4.3. The effect of α4 To investigate the effect of α4 on the motion response, the same analysis method is chosen as α2 and α3. The results are shown in Fig. 12(a) and b, respectively.
277
W. Pan et al.
Ocean Engineering 153 (2018) 268–281
Fig. 6. The variation of maximum motion component versus the relative wave height under the freak and irregular wave (Freak wave: 2.0<α1 < 2.20, 1.71<α2 < 2.15, 1.35<α3 < 2.13, 0.53<α4 < 0.62; Maximum wave of the irregular wave: 1.51<α1 < 1.59, 1.33<α2 < 1.69, 1.92<α3 < 1.99, 0.51<α4 < 0.63).
Fig. 8. The variation of maximum motion component versus the relative period under the freak and irregular wave (Freak wave: 2.0<α1 < 2.20, 1.40<α2 < 2.14, 1.40<α3 < 2.37, 0.51<α4 < 0.66; Maximum wave of the irregular wave: 1.45<α1 < 1.53, 1.06<α2 < 1.94, 1.29<α3 < 2.16, 0.51<α4 < 0.64).
It is observed from Fig. 12 that the surge and heave fluctuate with the variation of the freak wave parameter α4, which suggests α4 has no significant impact on the motion response with of α4 0.66. However, the pitch motions for the two wave series are different: the results of sequence 1 show that the pitch increases with α4, which could be attributed to the relative wave height of Hs/d ¼ 0.03–0.1. The increment of pitch is caused by the increment of the relative wave height. The results of sequence 2 show that the pitch fluctuates with α4, which shall be investigated further in future studies.
4. Conclusion In this study, the motion response of a rectangular cylinder under the freak and random wave conditions was investigated experimentally. Extensive tests were conducted to quantify the effect of relative wave height, period and freak wave parameters, and the main conclusions are summarized as follows. 1) Given identical wave spectrum, the maximum motion response of the mooring rectangular cylinder differs substantially under the action of
Fig. 7. Comparison of the dimensionless motion response varying with the relative period under freak and irregular wave (Freak wave: 2.0<α1 < 2.20, 1.40<α2 < 2.14, 1.40<α3 < 2.37, 0.51<α4 < 0.66; Maximum wave of the irregular wave: 1.45<α1 < 1.53, 1.06<α2 < 1.94, 1.29<α3 < 2.16, 0.51<α4 < 0.64).
278
W. Pan et al.
Ocean Engineering 153 (2018) 268–281
Fig. 9. Motion response varying with α1 under freak and irregular wave (Freak wave: 1.70<α1 < 2.83, 1.30<α2 < 1.52, 1.42<α3 < 1.6,0.51<α4 < 0.56; Maximum wave of the irregular wave: 1.57<α1 < 1.6, 2.15<α2 < 2.30, 1.92<α3 < 1.99, 0.54<α4 < 0.55).
Fig. 10. Motion response varying with α2 under freak wave (Test sequence 1)( Freak wave: 2.0<α1 < 2.20, 1.7<α2 < 2.15, 1.92<α3 < 1.99, 0.53<α4 < 0.62). (b)Motion responses varying with α3 under freak wave (Test sequence 1). (Freak wave: 2.0<α1 < 2.20, 1.7<α2 < 2.15, 1.92<α3 < 1.99, 0.53<α4 < 0.62). 279
W. Pan et al.
Ocean Engineering 153 (2018) 268–281
Fig. 11. Motion responses varying with α2 under freak wave (Test sequence 2)( Freak wave: 2.0<α1 < 2.20, 1.14<α2 < 2.14, 1.35<α3 < 2.37, 0.5<α4 < 0.66). (b)Motion responses varying with α3 under freak wave(Test sequence 2).
30–60% larger than that under irregular wave; with Hs/ d ¼ 0.03–0.05, the heave and pitch are approximately 20–40% and 20–33% larger than those under irregular wave, respectively. However, the influence becomes insignificant for Hs/d ¼ 0.06–0.1. 5) For the effect of relative period on the motion response, significant difference was observed at Tp/T0surge ¼ 0.14, Tp/T0heave ¼ 1.0, 1.5,2.0 and Tp/T0pitch ¼ 1. It is observed that the maximum variation occurs at the ‘resonance’ period of the floater under both freak and irregular waves. Given identical relative period and α1 ¼ 2–2.2, the variation of surge, heave and pitch under freak and irregular waves were 44%, 40%, 30%, respectively. 6) The freak wave parameters α1 has direct impact on surge and heave, but not on pitch. Within the range of α1 ¼ 2–2.8, the maximum surge and heave under freak waves are 1.4–2.5 and 1.4–1.6 times larger than those under irregular waves and the maximum pitch increases approximately 30%. Based on the
the freak versus random waves. The difference is closely correlated with the relative wave height, relative period and freak wave parameter α1. 2) The freak wave has direct effect on the surge and heave of the floater, but not for pitch. The maximum surge and heave under the freak wave are significantly larger than those under irregular wave. It indicates that the maximum surge and heave depends on the type of wave series. 3) There is no one-to-one relationships between the extreme pitch and wave height under irregular waves. The pitch under maximum wave height is related to the posture of the floater. In other words, it depends on the wave sequence prior to the maximum wave height. 4) The variation of motion response under freak and irregular waves decreases with the increasing relative wave heights. With Hs/ d ¼ 0.03–0.1, α1 ¼ 2–2.2, the maximum surge under freak wave is
280
W. Pan et al.
Ocean Engineering 153 (2018) 268–281
Fig. 12. Motion responses varying with α4 under freak wave (Test sequence 1)( Freak wave: 2.0<α1 < 2.20, 1.7<α2 < 2.15, 1.92<α3 < 1.99, 0.53<α4 < 0.62). (b)Motion responses varying with α4 under freak wave (Test sequence 2)( Freak wave: 2.0<α1 < 2.20, 1.14<α2<2.14, 1.35<α3 < 2.37, 0.5<α4 < 0.66).
observation in the current study, α2, α3 and α4 have no direct impact on the motion response of the floating cylinder.
Gu, J.Y., Lv, H.N., Yang, J.M., 2013. Dynamic response study of four column TLP in freak waves. Ocean Eng. 31 (5), 25–35 (in chinese). Klinting, P., Sand, S., 1987. Analysis of Prototype Freak Waves, Conference on Nearshore Hydrodynamics. ASCE, pp. 618–632. Mori, N., Yasuda, T., Ito, K., 1992. Freak waves in a unidirectional wave train and their kinematics. In: Proceedings of the 23rd International Conference on Coastal Engineering. Osborne, A.R., 2001. The random and deterministic dynamics of ‘rogue waves’ in unidirectional deep water wave trains. Mar. Struct. 14 (3), 275–293. Pei, Y.G., Zhang, N.C., Zhang, Y.Q., 2007. Efficient generation of freak waves in laboratory. China Ocean Eng. 21 (3), 515–523. Rudman, M., Cleary, P.W., 2013. Rogue wave impact on a tension leg platform: the effect of wave incidence angle and mooring line tension. Ocean Eng. 61 (6), 123–138. Schmittner, C.E., 2005. Rogue Wave Impact on Marine Structures. Universit€atsbibliothek. Shen, Y.G., Yang, J.M., 2013. Numerical investigation on the motion response of semisubmersible platform under extreme waves. Ocean Eng. 31 (3), 9–17. Slunyaev, A., Kharif, C., Pelinovsky, E., Talipova, T., 2002. Nonlinear wave focusing on water of finite depth. Phys. Nonlinear Phenom. 173 (1–2), 77–96. Stansberg, C.T., 1990. Extreme Waves in Laboratory Generated Irregular Wave Trains, vol. 178. Springer Netherlands, pp. 573–589. Tromans, P., Anaturk, A.R., Hagemeijer, P., 1991. A new model for the kinematics of large ocean waves-application as a design wave. In: Proceedings of 1st International Offshore and Polar Engineering Conference, Edinburgh, UK. Yu, Y.X., 2000. Random Wave and its Applications for Engineering. Dalian University of Technology Press, pp. 222–267. Zhao, X.Z., Ye, Z.T., Fu, Y.N., Cao, F.F., 2014. A CIP-based numerical simulation of freak wave impact on a floating body. Ocean Eng. 87, 50–63.
References Bunnik, T., Veldman, A., Wellens, P., 2008. Prediction of extreme wave loads in focused wave groups. In: Proceedings of the 18th International Offshore and Polar Engineering Conference. Vancouver, BC, Canada, pp. 32–38. Bennett, S.S., Hudson, D.A., Temarel, P., 2013. The influence of forward speed on ship motions in abnormal waves: experimental measurements and numerical predictions. J. Fluid Struct. 39 (5), 154–172. Chien, H., Kao, C., Chuang, L.H., 2002. On the characteristics of observed coastal freak waves. Coast Eng. J. 44 (04), 301–319. Clauss, G.F., Schmittner, C.E., Stutz, K., 2003. Freak wave impact on semisubmersibles time-domain analysis of motions and forces. In: Proc. 13th International Offshore and Polar Engineering Conf., Honolulu, USA. JSC-371. Cui, C., Zhang, N.C., Yu, Y.X., 2012. Numerical study on the effects of uneven bottom topography on freak waves. Ocean Eng. 54 (9), 132–141. Deng, Y.F., Yang, J.M., Xiao, L.F., 2014. Influence of wave group characteristics on the motion of a semisubmersible in freak waves. In: Proceedings of 33rd International Conference on Ocean, Offshore and Arctic Engineering. American Society of Mechanical Engineers. Deng, Y.F., Yang, J.M., Xiao, L.F., 2015. An experimental investigation on the motion and dynamic Responses of A Semisubmersible in freak waves. In: Proceedings of the 17th National Conference on Ocean Engineering, Nanning, China. Gao, N.B., Yang, J.M., Zhao, W.H., 2015. Numerical simulation of deterministic freak wave sequences and wave-structure interaction. Ships Offshore Struct. 1–16.
281