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Experimental study on hydrodynamic characteristics of a moored square cylinder under freak wave (II: Frequency-domain study)
Ocean Engineering 219 (2021) 108452
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Experimental study on hydrodynamic characteristics of a moored square cylinder under freak wave (II: Frequency-domain study) Wenbo Pan , Chen Liang , Ningchuan Zhang , Guoxing Huang * State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, 116024, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Hydrodynamic characteristics Square cylinder Freak wave Random wave Frequency-domain
Previous studies indicated that the occurrence of freak wave has a significant impact on the time-domain characteristics of a moored floater dynamic response. In this study, extensive investigations have been per formed to figure out further dynamic response characteristics of a moored square cylinder under freak wave in frequency-domain. In the experiments, the wave sequences with and without the freak wave are defined as freak and random waves, respectively. The results show that, the freak wave parameter α1 and spectral peak period have significant impacts on the dynamic response characteristics of the moored floater in frequency-domain. The effect of freak wave on surge and mooring tension of the floater is figured out through the low frequency components (0–0.5fp) which leads to significant increase with α1. Followed the occurrence of freak wave, surge increases sharply and oscillates for longer than 20 wave periods in low frequency. Without the occurrence of freak wave, however, both amplitude and oscillating of surge is significantly weaker. The mooring tension response is subjected to surge and hence the trend of variation is similar to surge.
1. Introduction In the recent decades, more and more occurrences of freak wave have been recorded and reported in the oceanographic observations globally. In the meantime, many maritime accidents demonstrate that freak wave is a serious hazard to offshore vessels and structures. Therefore, it is worth to investigate the security of marine structures under freak wave. Progresses have been made on freak wave generation mechanism, nu merical and physical simulation as well as the interaction with struc tures. Thus far, the techniques of generating freak wave in both numerical and physical ways have been well developed. Osborne (2001) and Slunyaev et al. (2002) used NLS (Nonlinear Schrodinger) equation to simulate freak wave numerically. Pei et al. (2007) generated freak wave at a target location in wave tank by superposing a random wave train with two transient wave trains. Cui et al. (2012) applied the VOF (Volume of Fluid) method to establish fully nonlinear wave numerical model with the governing equation of Reynolds-averaged Navier-Stokes equation and k-ε equation as well as adopted the model to simulate the generation of freak wave, evolution process and the effect of water bottom on the freak wave propagation. There are extensive numerical investigations on the interaction be tween freak wave and structures. Moctar et al. (2007) investigated the
freak wave impacts on a mobile jack-up drilling platform with CFD (Computational Fluid Dynamic) and FEM (Finite Element Method) methods and established a numerical model to predict freak wave load on self-elevating drilling platform. The model solved the Reynolds-averaged Navier-Stokes equations applying the VOF interface-capturing technique. Rudman and Cleary (2013) adopted the SPH (Smoothed Particle Hydrodynamic) method to simulate fully non-linear interactions between a TLP (Tension Leg Platform) and freak wave. They considered the effect of incident angle and mooring line pre-tension on the subsequent motions of the platform. Zhao et al. (2014) developed a CIP (Constraint Interpolation Profile)-based nu merical model to simulate large amplitude motions of a floater under freak wave. Gao et al. (2016) numerically investigated the interaction between deterministic freak wave and fixed cylinders submerged in different depths. The results indicated that the most critical vertical loads were observed during the freak wave approaching which may cause severe vertical responses on the fixed horizontal structures. In terms of model experiments, Clauss et al. (2003) investigated the motions behavior and splitting forces of a semi-submersible under freak wave through the experimental measurements and time-domain simu lation. It was found that the maximum response is subjected to the freak wave height. Schmittner (2005) investigated the motions and bending
* Corresponding author. E-mail address: [email protected] (G. Huang). https://doi.org/10.1016/j.oceaneng.2020.108452 Received 17 July 2020; Received in revised form 26 October 2020; Accepted 29 November 2020 Available online 8 December 2020 0029-8018/© 2020 Elsevier Ltd. All rights reserved.
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contact 6DOF (degree of freedom) measurement system, consisting of dual-CCD (charge-coupled device) cameras and a data acquisition sys tem. Three light markers were arranged in a plane and fixed on top of the cylinder to track the motion behaviors by the dual-CCD cameras. The images of the markers were acquired continuously by the dual-CCD at 30 Hz, 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, with an accuracy of 0.1 mm, were calibrated before each test.
moments of a FPSO (Floating Production Storage and Offloading) and a heavy lift vessel as well as the motions and splitting forces of a semi-submersible under freak wave. The results show that the vertical bending moments, heave and pitch of FPSO and heavy lift vessel as well as the airgap and splitting forces of semi-submersible under freak wave are larger than the maximum values predicted by codes and frequency-domain analysis. Shen and Yang (2013) compared the motion response of semi-submersible under two predetermined extreme wave sequences and analyzed the impacts of freak wave and “Three Sisters” wave (three huge waves in succession) on platform motions. The results showed that the peak value of wave height is the key parameter in determining the dynamic response of the platform, which should be a concern in Front End Engineering Design study. The adjacent effect in the “Three Sisters” wave has impacts on the platform motion, the surge and heave of the platform are both increased with the adjacent wave heights. Deng et al. (2015) investigated the effect of mooring stiffness on the motion behaviors and mooring tensions of a semi-submersible platform. The measurements showed that the freak wave can lead to critical horizontal motions on the flexible mooring system and extremely large mooring tensions on the tight mooring system. With the identical wave spectra, Pan et al. (2018) compared the motion response of a moored square cylinder under freak and random waves and quantified the effects of relative wave height, relative period and freak wave pa rameters on motion response of the floater based on time-domain measurements. The above researches show that, the effect of freak wave on various structures is more critical than the random wave. However, the existing researches mainly focus on the time-domain statistical characteristics investigation of the interaction between freak wave and structures, few researches investigated the frequency-domain characteristics on the dynamic response of the floaters. Through frequency-domain analysis, it is convenient to obtain energy distribution, peak frequency, total energy and each order spectral moment etc. But the frequency spectrum cannot provide the variation of energy in time series. For a stationary process, the local frequency characteristics in time series probably is not so important. However, as a spike in a random wave series, the local characteristics of freak wave in time-domain are exactly our concerns. Compared to Fourier transfer, wavelet analysis method is effective in obtaining the energy spectral density and the energy distribution of each frequency in time-domain, especially the instantaneous physical change under freak wave. Hence, the wavelet analysis method is adopted in this study. Based on the work of Pan et al. (2018), the present study compared the frequency-domain dynamic response of a moored square cylinder under freak and random waves with identical wave spectra through experimental measurements, which are designed to determine the variation of frequency-domain characteristics of dynamic response and the quantitative relationship with the freak wave parameter. To further investigate the significant impact on the frequency-domain character istics of the dynamic response of the floater under freak wave. The time-frequency spectra of the dynamic response under freak and random waves were calculated by wavelet analysis to investigate the time-frequency structure characteristics and variation.
2.2. Model parameters and layout The prototype of moored square cylinder (with smooth arc corners) is the columns of a 30,000 ton floating structure operating at a wind farm in East China Sea. The cylinder is a rectangular solid with a height of h = 21.7 m and a square base with a side length of D = 17.5 m. The corners were arcs with a radius of 2.31 m. The cylinder was scale by λ = 35. To achieve the design Center of Gravity (CoG), Center of Buoyancy (CoB) and draft, certain clump weights were placed on the bottom of the cylinder. The mechanical parameters of the prototype and model are presented in Table 1. As illustrated in Fig. 2, the square cylinder was anchored by four mooring lines. Each of the prototype mooring lines is nylon rope with 75 mm diameter and 140 m length. Based on the Wave Model Test Regu lation (JTJ/T234-2001), to simulate the model mooring line, not only the length and weight are scaled, but also the curve of tension (Tm) deformation (Δs) should be matched. Tm =
Cp dp (ΔS/S)n λ3
(1)
Tm − mooring tension of the model cable (N); Cp − elasticity coefficient of prototype cable, with nylon rope of Cp = 1.540 × 104MPa; dp − diameter of prototype cable (m); ΔS/S− relative elongation of prototype cable; λ − model scale, λ = 35; n − index, with nylon rope adopting n = 3. The simulation of mooring lines matches both elastic and gravity ˜ deformation curve and similarity. An example of theoretical tension measured scatters are presented in Fig. 1. It shows that excellent agreement was achieved. The model was placed 21 m away from the wave maker, where the freak wave occurred. The lower ends of the mooring lines were anchored Table 1 Mechanical parameters of the square cylinder.
2. Experiments 2.1. Experimental setup The tests were carried out in a wave flume at the State Key Labora tory of Coastal and Offshore Engineering, Dalian University of Tech nology. The flume is 60.0 m long, 4.0 m wide and 2.5 m deep. The wave generation system is Hydro-servo random wave maker system and can generate waves with periods ranging from 0.5s to 5.0s. At the other end of the flume, an absorbing beach is installed for wave dissipation effectively. The motion of moored square cylinder were measured by a non2
Parameters
prototype size
unit
model size
unit
Length Height Arc radius Draft Weight Center of gravity (from bottom) Center of buoyancy (from bottom) Transverse metacentric radius Longitudinal metacentric radius Transverse moment of inertia Longitudinal moment of inertia Natural surge period T0S Natural heave period T0H Natural pitch period T0P
17.5 22 2.31 12.0 3130 7.0 6.0
m m m m t m m
50 62 6.6 34 73 20 17
cm cm cm cm kg cm cm
2.15 2.15 1.04 × 108 1.04 × 108 59.0 8.3 11.8
m m kg∙m2 kg∙m2 s s s
6.13 6.13 1.98 1.98 10.0 1.4 2.0
cm cm kg∙m2 kg∙m2 s s s
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variation (within 3%) to the target values, which is allowable (Table 2). 2.3.2. Experiment methods In the present study, the freak wave was generated at the target location using a random wave train combined with two transient wave trains, which was proposed and used by Pei et al. (2007). With the linear superposition method, the wave surface can be expressed as:
η(x, t) = η1 (x, t) + η2 (x, t) + η3 (x, t) = M ∑
+
M ∑
A1i cos(ki x + wi t + εi )
i=1
A2i cos(ki (x − xc ) + wi (t − tc ) + εi )
(5)
i=1 M ∑
+
A2i cos(ki (x − xc ) + wi (t − tc ) + εi )
i=1
Fig. 1. Theoretical mooring line.
and
measured
tension-deformation
curve
of
where η denotes the elevation of water surface above the mean water level; A1i is amplitude of the random wave; A2i and A3i are amplitudes of the transient waves. The model was installed and moored with pre-tension of 1–1.2 N under hydrostatic status and collected 0 point at equilibrium position. The decay test was then performed by applying an artificial displace ment/rotation on the floater and releasing. The decay curve of motion process had been completely collected until the floater resumed to static state. RAO test: the time history of the motion responses were recorded with a frequency of 30 Hz under the regular wave (Sequence III in Table 2). The measurements for each test typically lasted for more than 10 waves. Freak wave test: the freak wave was generated using a random wave train combined with two transient wave trains with P-M spectrum. The measurement for each test typically lasted for more than 100 waves, and were performed 2 or 3 times to ensure the repeatability. During each test, the motions and mooring tensions were recorded synchronously with a frequency of 30 Hz. Random wave test: to compare the experi mental results measured under freak and random waves, the same spectrum with identical spectral parameters to the freak wave were applied and the measurement was similar to the freak wave test. Based on the measured time histories, spectral analyses were per formed to obtain the frequency-domain characteristics of the motions and mooring tensions of the floating square cylinder, i.e. the spectral pattern, spectral peak, 0th/1st/2nd order moments and other statistics.
the
on the bottom of wave flume. The top ends together with tension sensors were connected to the four corners of the square cylinder, and each connecting point was 25 cm measured from the bottom of the cylinder. Fig. 2 shows the layout and mooring pattern of the model in wave flume. The origin of the coordinate system is defined at the center of the model. Positive x is along the wave propagation, positive z is vertical up along the water depth, and positive y is determined by right-hand rule. For two-dimensional case, three motion components are investigated: surge, heave and pitch. Surge is the longitudinal motion along x-axis (wave propagation direction is +); heave is the vertical motion along z-axis (vertical up is +); and pitch is rotation around y-axis (clockwise direc tion is +). 2.3. Experiments 2.3.1. Experimental parameters The experiments include the decay tests getting natural frequency of the mooring system, RAO of the motion and mooring tension (regular wave), motion and mooring tension under freak and random waves. Detailed wave parameters are summarized in Table 2. Noted that water depth is d = 1.4 m and floater draft is h0 = 34 cm. It is assumed that the dynamic response subjects to the dimensions of the square cylinder (the side length of D, the module height of h), me chanical parameters of the floater (refer to Table 1), wave parameters (the significant wave height Hs, the maximum wave height Hmax, spec tral peak period Tp, freak wave parameters αn, αn includes: α1 = Hmax/Hs, α2 = Hmax/Hmax-1, α3 = Hmax/Hmax+1, α4 = ηmax/Hmax, where Hmax-1 is the wave height before the maximum wave height, Hmax+1 is the wave height after the maximum wave height, ηmax is surface elevation of the maximum wave height in the random wave sequence), mooring configuration, water depth, etc. In the experiments, P-M spectrum (Yu, 2000) was applied as: ] [ (2) S(w) = Aw− 5 exp − Bw− 4
2.3.3. Wavelet analysis method Wavelet analysis method is adopted to compute the time-frequency spectra of the dynamic responses under freak and random waves. The variation of the time-frequency energy structure has been investigated. The Morlet wavelet is selected as the mother wavelet. That can be expressed as: Ψ (t) = π−
1/4 − t2 /2 iw0 t
e
(3)
− 4 B = 691T0.1
(4)
(6)
where ω0 is the non-dimensional frequency, here taken to be 6 to satisfy the admissibility condition (Torrencr and Compo, 1998). The contin uous wavelet transform of a discrete sequence xn is defined as the convolution of xn with a scaled and translated version ofΨ (t): N− 1 ∑
Wn (s) = n
− 4 A = 173Hs2 T0.1
e
xn′ ψ *
′
[(n′ − n)δt] s
(7)
where the superscript “*” is the complex conjugate, s is the wavelet scale and n is the time index. It is more efficient to do the calculations in Fourier space. By the convolution theorem, the wavelet transform is the inverse Fourier transform of the product:
where Hs and T0.1 are the significant wave height and the average period calculated from the spectral moment, respectively. To figure out the effect of various parameters, two types of random wave sequences have been designed and generated for the experiments. In the tests, the constant wave height and period generated has minor
N− 1 ∑
Wn (s) = k=0
3
*
̂ (swk )eiwk nδt ̂ xk ψ
(8)
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Fig. 2. Mooring pattern and layout of the model in the wave flume (Pan et al., 2018).
where k = 0 … N-1 is the frequency index. The spectral density S(f,t) of the wavelet energy spectrum can be defined as: S(f , t) = |W(s, t)|2
The time-frequency spectrum (wavelet spectrum) of a physical quantity integrated in the frequency at any time, and the result can be regarded as a time history of “generalized energy spectrum”, which is denoted as E(t): ∑ Si,j ∗Δfj (10) E(t) =
(9)
The spectral density S(f,t) describes the variation of a physical parameter from time domain and frequency domain. For surge, heave and pitch of moored square cylinder, we can determine the variation of time-frequency domain characteristics followed the occurrence of the freak wave and the maximum wave of random wave. In order to investigate the variation quantitatively, the following parameters are defined:
j
The “generalized” total energy of each frequency component of a physical quantity when the freak wave (or the maximum wave of random wave) occurred is denoted as Ec = E (t)|t=c ∑ Ec = Sc,j ∗Δfj (11) j
(1) “Generalized energy spectrum” E(t)
(2) “Time distribution parameter of energy concentration”ΔTE 4
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Table 2 Summary of experimental conditions and objectives. Test Sequences
Freak and random waves
Freak wave
Significant wave height Hs(cm)
Peak period Tp(s)
Relative wave height Hs/d
I: Test sequences with different spectrum peak periods and identical significant wave height
8.0
0.057
II: Test sequences with different freak wave parameter α1 and identical significant wave height and spectrum peak period
8.0
1.14 1.26 1.56 1.83 2.01 2.16 2.87 3.10 1.6
Wave period T (s) 0.07
Relative wave height H/d 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34
Regular waves Wave height H (cm) III: Additional test sequences of regular waves
10.00
1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40
0.057
Relative period Tp / T0s
Tp / T0h
Tp / T0p
0.11 0.13 0.16 0.18 0.20 0.22 0.29 0.31 0.16
0.81 0.90 1.11 1.31 1.44 1.54 2.05 2.21 1.14
0.57 0.63 0.78 0.92 1.01 1.08 1.44 1.55 0.80
Relative period Tp/T0s 0.71 0.86 1.00 1.14 1.29 1.43 1.57 1.71 1.86 2.00 2.14 2.29 2.43
The problem we focus on is the variation of dynamic response of a moored square cylinder under freak and random waves. The maximum value of the “generalized energy spectrum” E(t) of a physical quantity under random wave is denoted as EImax; For the freak wave, TEmin is the starting time of E(t) ≥ EImax after the occurrence of freak wave. TEmax is the ending time of E(t) ≥ EImax, The response time of the freak wave is defined as Δ TE: ΔTE = TEmax-TEmin
Objectives
Parameter of freak wave α1
Parameter of freak wave α2~α4
2.00 < α1 < 2.2
1.40 < α2 < 2.14 1.40 < α3 < 2.37 0.51 < α4 < 0.66
To investigate the effect of wave period, measured Hs is in the range of 7.86–8.15 cm
1.71 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
To investigate the effect of parameter of freak wave αn measured Hs and Tp is in the range of 7.77–8.11 cm and 1.56–1.60s.
Tp/T0h
Tp/T0p
0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70
To obtain the RAO and related phase response of the floater
Objectives
through regular wave test Sequence III in Table 2. Fig. 3 illustrates that heave and pitch increase with the relative period and reach the local peak at the natural periods. The maximum heave can be up to 0.9H (H = 10 cm) at the natural period T = T0H. With further increasing of the wave period, heave oscillates in the range of 0.4–0.6H (close to the wave amplitude), which is consistent with the variation range of conventional moored floaters. The maximum pitch occurs at the natural period T = T0P. With further increasing of the wave period, pitch varies in the range of θ ≈ 1/3–1/2θmax. In addition, the amplitudes of heave and pitch are approximately symmetrical in the positive and negative components. Due to the limitation of the wave maker, the test period cannot cover the natural period of surge. Therefore, the test curve may not show the completed characteristics of surge RAO. In the test ranges of wave period, the positive amplitude of surge is larger than the negative one in the wave propagating direction. The maximum surge occurs at T/T0p = 0.7 (T/T0h = 1.0), the natural period of heave. It implies that the maximum surge is affected by the resonance of heave. When the wave period is smaller than the natural period of pitch (T/T0p < 1), the surge shows a considerable value, which has been addressed by Song et al. (2018) in the experimental study on motions of tunnel element. This is related to the fact that the larger relative motion of the floater under short waves due to stronger diffraction effects and larger mean 2nd order horizontal forces (i.e. Pinkster, 1979).
(12)
3. Analysis of experiments results It is known that the dynamic response of the floater is affected by the natural period and RAO characteristics significantly. Therefore, the experimental results of natural period and RAO response are presented firstly. Based on the motion responses and mooring tensions of the floating cylinder measured synchronously, the spectra of motion re sponses and mooring tensions under freak and random waves are analyzed and the characteristics in frequency-domain of each motion component and mooring tension are discussed. Also, the effect of spec tral peak period and freak wave parameter on the dynamic response are discussed, and the difference in dynamic response under freak and random waves is analyzed in frequency domain. 3.1. Decay tests and response amplitude operators (RAOs)
3.2. Comparison of frequency domain characteristics of moored floater under freak and random waves
After performing decay tests of the moored floater in still water, the natural periods of surge, heave and pitch were obtained T0S = 10.0s, T0H = 1.4s, and T0P = 2.0s, respectively. The response amplitude operators (RAO) of the floater were obtained
The dynamic response of moored square cylinder includes the mo tion response, mooring tension and wave load. The frequency domain characteristics of motion response and mooring tension are discussed in this study. The wave loads on square cylinder, including the total wave 5
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Fig. 3. Response Amplitude Operators (RAO) of the model floater.
force and wave pressures, will be discussed in a separate study. 3.2.1. Effect of spectral peak period The same wave spectrum can generate various random wave series by applying different random seeds. To compare the experimental re sults measured under freak and random waves, the identical spectrum were applied and the measurement was similar for the freak and random wave tests. In Fig. 4, the comparison shows that the measured freak and random wave spectra match well with the target spectrum. Under freak and random waves generated by the same spectra, the frequency domain characteristics of motion response with various spectral peak periods were shown in Fig. 5. Two wave series were generated with the identical wave height of Hs = 8 cm and relative periods of TP/T0H = 0.81–2.21, TP/T0P = 0.57–1.55. The freak wave series are generated with the characteristics of 2.01 < α1 < 2.20. Fig. 5 shows: 1) The surge spectra of the floater are consisted of low frequency and wave frequency components. The surge spectra show the similar trends under freak and random waves with various relative wave period. 2) The variation of surge in frequency-domain is subjected to the relative wave period. With respect to the relative wave period, there are three typical scenarios on surge spectra. Firstly, with small relative wave period (0.57≤Tp/T0p ≤ 0.92), the surge spectra show significant lowfrequency characteristics and the wave frequency component can be neglected. Secondly, when the relative wave period is close to the nat ural pitch period T0p (0.92
Fig. 5. Comparison of motion responses spectra varying with the relative period under freak and random waves.
under freak and random waves and both are subjected to the relative wave period. 4) With relative wave period small (0.81≤Tp/T0H ≤ 1.11), the heave spectra are unimodal. The peak frequency is observed at the natural frequency of heave. However, with the relative wave period larger than the natural period of heave (Tp/T0H > 1.11), the heave spectra show dual peaks at spectral peak frequency and natural fre quency of heave. With the wave period increasing, the spectral peak frequency deviates from the natural frequency of heave gradually. As a consequence, the peak of heave spectrum at the wave frequency
Fig. 4. Comparison between the measured and target spectra (Hs = 8.0 cm, Tp = 1.14s). 6
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increases gradually until reaches the peak. In this case, the heave is dominated by wave frequency, and the effect of natural frequency of heave is weaken with the wave period gradually. 5) The pitch spectra are unimodal and the peak occurs at the natural frequency of pitch in the test range. When the wave spectral peak frequency is close to the natural frequency of pitch, the critical pitch spectrum was observed. It implies that the pitch of moored square cylinder is dominated by the natural period, which is different to the pitch of a moored ship. The characteristics of mooring tension in frequency-domain are addressed in Fig. 6. The same wave conditions are adopted in motion response analysis. The results indicate: 1) With respect to the relative wave period, there are two typical scenarios on frequency response of mooring tension. Firstly, with small relative wave period (0.57≤Tp/T0p ≤ 0.78), the peak of the mooring tension spectra under freak wave is larger than that under random wave. The frequency response of mooring tension is dominated by the 2ndorder drift force under both freak and random waves. In other words, the mooring tension is dominated by the low frequency component and the wave frequency component is minor. With relative wave period increasing (0.92≤Tp/T0p ≤ 1.55), the varia tion of mooring tension between the freak and random waves becomes minor since the 2nd order drift force decreases with the relative period and the wave frequency component of mooring tension has the same order of magnitude as the low frequency component. In summary, the dynamic response of surge and mooring tension in frequency-domain is dominated by the first-order wave force (wave frequency) and second-order drift force (far lower than wave fre quency). The response of heave in frequency-domain is dominated by the resonant effect of the natural frequency and wave spectral peak period. However, the response of pitch in frequency-domain is subjected to the natural frequency only. In general, the frequency domain char acteristics of the dynamic responses show the similar trends under freak and random waves with the relative wave period.
parameter α1 is large (α1 ≥ 2.32), the low frequency component of surge under freak wave is significantly larger than that under random wave and the variation increases with α1. However, the wave frequency component of surge are basically identical under freak and random waves. 2) With respect to the heave and pitch, the spectral character istics of heave and pitch are basically identical under freak and random waves and the variation is insignificant with freak wave parameter α1. The characteristics of mooring tension in frequency domain are addressed in Fig. 8. The variation of the frequency domain character istics of the mooring tension with α1 is similar to the surge of moored floater under freak and random waves. In general, the freak wave parameter α1 has significant impact on the frequency-domain characteristics of the dynamic response which leads to significant increase of surge and mooring tension in low frequency component. Such variation increases rapidly with the freak wave parameter α1, but has seldom impact on heave and pitch. 3.3. The effect of freak wave parameter on dynamic response in frequency domain The above discussion shows that the freak wave parameter α1 has direct impact on the surge and mooring tension of the floating body in frequency domain. In this section, the effect of the freak wave parameter α1 on the dynamic response are analyzed quantitatively in frequency domain under freak wave. 3.3.1. Quantitative analysis of the effect of freak wave parameter Fig. 9 shows the test results of the dynamic responses spectral pa rameters varying with the freak wave parameter α1. All wave series were generated under constant relative wave height is Hs/d = 0.057, relative period is Tp/T0H = 1.14 and Tp/T0P = 0.8, the freak wave parameter varies at α1 = 1.71–2.83. The results indicate: 1) Compared with the random wave of same spectrum, the 0th/1st order moments of surge increase with α1 signifi cantly (m0f/m0i = 0.89–2.63, m1f/m1i = 1.0–1.42), but the 2nd order moments vary not significantly (m2f/m2i = 1.0–1.07). 2) The 0th/1st/2nd order moments of heave and pitch are basically unchanged with α1. It indicates that the effect of the freak wave parameter α1 on the heave and pitch are weaker than that on surge. 3) The variation of the 0th/1st/2nd order moments of mooring tension with α1 is similar to the surge of floating body. The 0th/1st order moments increase with α1 significantly (m0f/m0i = 1.05–2.81, m1f/m1i = 1.04–1.53), the variation of 2nd order
3.2.2. Effect of freak wave parameter To investigate the effect of the freak wave parameter α1, the freak wave were generated with various α1 (α1 = 1.71–2.83). All wave series were generated under constant relative wave height is Hs/d = 0.057, relative period is Tp/T0H = 1.14 and Tp/T0P = 0.8. Fig. 7 shows the motion responses spectra of the floater under various freak wave parameter α1. The results indicate: 1) When the freak wave parameter α1 is small (α1 ≤ 2.07), the surge spectrum under freak wave is basically identical with the random wave. When the freak wave
Fig. 6. Comparison of mooring tension spectra varying with the relative period under freak and random waves. 7
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Fig. 7. Comparison of motion responses spectra varying with varying with α1 under freak and random waves.
moments is insignificant with freak wave parameter α1 (m2f/m2i = 0.99–1.17).
capturing instantaneous changes of each spectrum. In Fig. 10, the amplitude of surge increases shapely after the occur rence of the freak wave and oscillates for longer than 20 wave periods in low frequency. During the oscillating, the wave frequency component of surge is minor and can be neglected. Under the random wave, the surge component in low frequency is also significant, but the amplitude is significantly smaller than that under freak wave which is a substantial difference to freak wave. The low-frequency surge responses is signifi cantly enlarged with the freak wave action. This is because the freak wave provide an extra impulse driving the floater further away before pulling it back by the restoring forces. The mooring tension response is similar to the surge. There is a significant variation in the time-frequency spectra of the surge and mooring tension under freak and random waves. With the
3.3.2. Analysis on the key factor for the frequency response of the floater It is found in the previous discussion that with α1 = 1.71–2.83, the 0th order moment of surge and mooring tension under freak wave are 0.89–2.63 and 1.05–2.81 times of those under random waves. As a solo extreme wave in a random wave series, the freak wave makes substantial change to the spectral characteristics of surge and mooring tension. To figure out the reason, the synchronous time history and time-frequency spectra of surge and mooring tension calculated by wavelet analysis under freak and random waves are compared in Fig. 12 under various α1, respectively. The wavelet analysis provides the energy spectral density and energy distribution of each frequency in time-domain, especially in 8
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Ocean Engineering 219 (2021) 108452
Fig. 8. Comparison of mooring tension spectra varying with varying with α1 under freak and random waves.
Fig. 9. Comparison on moments of dynamic responses versus freak wave parameter α1 under freak and random waves.
occurrence of freak wave, the time-frequency spectra of the surge and mooring tension increase significantly and show a concentrated distri bution of the energy. It is also shown that the peak value of timefrequency spectra increases significantly with the freak wave param eter α1.
To further investigate the significant impact on the frequencydomain characteristics of the dynamic response of the moored floater under freak wave. The “generalized energy spectrum” E(t) of the dy namic response (surge and mooring tension) are illustrated in Fig. 11 under freak and random waves, respectively. It is observed that the 9
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α1 under freak and random waves. It is defined the frequency of 0–0.5fp
and 0.5fp~∞ as low frequency and wave frequency, respectively (fp is the wave spectral peak frequency). With α1 increases from 1.71 to 2.83, the low frequency component of surge increases rapidly under freak wave. The ratio of low frequency component of surge under freak wave to that under random wave increase from 0.83 to 3.61, but the ratio of wave frequency components of surge has minor variation within 0.94–1.0. It implies that the key effect of freak wave on the motion response of the floater is subjected to the low frequency component of surge. Also, the variation of mooring tension with α1 is similar to the surge of floating body. The low frequency component of the mooring tension under freak wave is larger than that under random wave. With α1 = 1.71–2.83, the ratio of low frequency component of mooring tension under freak wave to that under random wave increase from 0.92 to 5.34, the ratio of wave frequency components has minor variation within 1.04–1.44, respectively. The results show that the effect of freak wave on the mooring tension of the floater is also realized through the low fre quency component. 4. Conclusion Based on the experimental measurements in time-domain, the dy namic response of a moored square cylinder under freak wave has been investigated in frequency-domain. Given identical wave spectrum, the dynamic response of moored square cylinder in frequency domain and time-frequency energy structure are substantially different under freak and random waves. Extensive analyses have been conducted to quantify the effect of relative wave period and freak wave parameter, and main conclusions can be drawn: 1. The freak wave has direct effect on the frequency-domain charac teristics of surge and mooring tension of the floater, but seldom impact on heave and pitch. The freak wave parameter α1 leads to significant variation on surge and mooring tension. 2. The effect of freak wave on the surge and mooring tension of the floater is figured out through the low frequency components which leads to significant increase with α1. This is because the freak wave provides an extra impulse for driving the floater further away before pulling it back by the restoring forces. The mooring tension response is similar to the surge. 3. The amplitude of surge increases shapely after the occurrence of freak wave and oscillates for longer than 20 wave periods in low frequency. In random waves, however, both amplitude and oscilla tion of surge are significantly weaker. The response of mooring tension is subjected to the surge response. 4. The wavelet analysis is effective in investigating the dynamic response of moored floater under freak wave. Compared with sta tistical or spectral analysis, wavelet analysis can outstandingly demonstrate the typical characteristics of dynamic response on the floater under freak wave. It can obtain the energy spectral density and the energy distribution of each frequency in time domain, especially the instantaneous physical change with the occurrence of freak wave. It is found that the freak wave have significant effect on the time-frequency domain characteristics of the dynamic response than the random wave with the same wave spectrum. With the occurrence of freak wave, the time-frequency spectra of the dynamic responses increase significantly and show a concentrated distribu tion of the energy. This implies that the occurrence of freak wave imposes serious potential risks on marine structures and vessels. 5. The freak wave parameter α1 has direct impact on the time-frequency spectra characteristics of surge and mooring tension. The peak value of time-frequency spectra of surge and mooring tension increases significantly with the freak wave parameter α1. The critical “gener alized energy spectra” E(t) and the “time distribution parameter of
Fig. 10. Time history and time-frequency spectra of dynamic responses under freak and random waves.
critical “generalized energy spectrum” E(t) and the time distribution parameter of energy concentration ΔTE of surge and mooring tension under freak wave is significant larger than that under random wave. It is also observed from Fig. 11 that the freak wave parameter α1 has direct impact on the E(t) andΔTE which both increase with the freak wave parameter α1. For further investigating the variation of the low frequency and wave frequency components of the dynamic responses under freak wave, the normalized spectral moment m0f/m0i (subscript 0 represents the 0-order spectral moment, f and i represent freak and random waves) are plotted in Fig. 12, which shows the comparison of the low frequency and wave frequency components of the surge and mooring tension spectra versus 10
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Fig. 11. “Generalized energy spectrum” of dynamic responses under freak and random waves.
Fig. 12. Comparison of low frequency and wave frequency components with various α1 under freak and random waves.
energy concentration”ΔTE of surge and mooring tension under freak wave also increase with the freak wave parameter α1.
the work reported in this paper. References
Funding
Clauss, G.F., Schmittner, C.E., Stutz, K., 2003. Freak wave impact on semisubmersibles -time-domain analysis of motions and forces. In: Proceedings of 13th International Offshore and Polar Engineering Conference, 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., 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., 2016. Numerical simulation of deterministic freak wave sequences and wave-structure interaction. Ships Offshore Struct. 11 (8), 1–16. Moctar, O.E., Schellin, T.E., Jahnke, T., Peric, M., 2007. Wave load and structural analysis for a jack-up platform in freak waves. J. Offshore Mech. Arctic Eng. 131 (2), 623–632. Osborne, A.R., 2001. The random and deterministic dynamics of ‘rogue waves’ in unidirectional deep-water wave trains. Mar. Struct. 14 (3), 275–293. Pan, W.B., Zhang, N.C., Huang, G.X., Ma, X.Y., 2018. Experimental study on motion responses of a moored rectangular cylinder under freak waves (I: time-domain study). Ocean Eng. 153 (APR.1), 268–281. Pinkster, J.A., 1979. Mean and low frequency wave drifting forces on floating structures. Ocean Eng. 6 (6), 593–615. Pei, Y.G., Zhang, N.C., Zhang, Y.Q., 2007. Efficient generation of freak waves in laboratory. China Ocean Eng. 21 (3), 515–523.
The work is supported by China Natural and Scientific Fund through No. 51509120 and the Fundamental Research Funds for the Central Universities (DUT19LAB13). CRediT authorship contribution statement Wenbo Pan: Investigation, Writing - original draft. Chen Liang: Investigation, Data curation. Ningchuan Zhang: Supervision, Method ology, Project administration. Guoxing Huang: Conceptualization, Methodology, Writing - review & editing, Funding acquisition. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence 11
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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. Ph.D thesis. Technischen Universit¨ at Berlin, Germany. 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. Song, Y., Zhang, N.C., Huang, G.X., Sun, Z.X., 2018. Experimental study on motions of tunnel element during immersion standby stage in long wave regime. Ocean Eng. 161, 29–46.
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. Torrencr, C., Compo, G.P., 1998. A practical guide to wavelet analysis. Bull. Am. Meteorol. Soc. 79 (1), 61–78. Yu, Y.X., 2000. Random Wave and its Applications for Engineering. Dalian University of Technology Press (in chinese). 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.
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Experimental study on hydrodynamic characteristics of a moored square cylinder under freak wave (II: Frequency-domain study) Wenbo Pan , Chen Liang , Ningchuan Zhang , Guoxing Huang * State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, 116024, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Hydrodynamic characteristics Square cylinder Freak wave Random wave Frequency-domain
Previous studies indicated that the occurrence of freak wave has a significant impact on the time-domain characteristics of a moored floater dynamic response. In this study, extensive investigations have been per formed to figure out further dynamic response characteristics of a moored square cylinder under freak wave in frequency-domain. In the experiments, the wave sequences with and without the freak wave are defined as freak and random waves, respectively. The results show that, the freak wave parameter α1 and spectral peak period have significant impacts on the dynamic response characteristics of the moored floater in frequency-domain. The effect of freak wave on surge and mooring tension of the floater is figured out through the low frequency components (0–0.5fp) which leads to significant increase with α1. Followed the occurrence of freak wave, surge increases sharply and oscillates for longer than 20 wave periods in low frequency. Without the occurrence of freak wave, however, both amplitude and oscillating of surge is significantly weaker. The mooring tension response is subjected to surge and hence the trend of variation is similar to surge.
1. Introduction In the recent decades, more and more occurrences of freak wave have been recorded and reported in the oceanographic observations globally. In the meantime, many maritime accidents demonstrate that freak wave is a serious hazard to offshore vessels and structures. Therefore, it is worth to investigate the security of marine structures under freak wave. Progresses have been made on freak wave generation mechanism, nu merical and physical simulation as well as the interaction with struc tures. Thus far, the techniques of generating freak wave in both numerical and physical ways have been well developed. Osborne (2001) and Slunyaev et al. (2002) used NLS (Nonlinear Schrodinger) equation to simulate freak wave numerically. Pei et al. (2007) generated freak wave at a target location in wave tank by superposing a random wave train with two transient wave trains. Cui et al. (2012) applied the VOF (Volume of Fluid) method to establish fully nonlinear wave numerical model with the governing equation of Reynolds-averaged Navier-Stokes equation and k-ε equation as well as adopted the model to simulate the generation of freak wave, evolution process and the effect of water bottom on the freak wave propagation. There are extensive numerical investigations on the interaction be tween freak wave and structures. Moctar et al. (2007) investigated the
freak wave impacts on a mobile jack-up drilling platform with CFD (Computational Fluid Dynamic) and FEM (Finite Element Method) methods and established a numerical model to predict freak wave load on self-elevating drilling platform. The model solved the Reynolds-averaged Navier-Stokes equations applying the VOF interface-capturing technique. Rudman and Cleary (2013) adopted the SPH (Smoothed Particle Hydrodynamic) method to simulate fully non-linear interactions between a TLP (Tension Leg Platform) and freak wave. They considered the effect of incident angle and mooring line pre-tension on the subsequent motions of the platform. Zhao et al. (2014) developed a CIP (Constraint Interpolation Profile)-based nu merical model to simulate large amplitude motions of a floater under freak wave. Gao et al. (2016) numerically investigated the interaction between deterministic freak wave and fixed cylinders submerged in different depths. The results indicated that the most critical vertical loads were observed during the freak wave approaching which may cause severe vertical responses on the fixed horizontal structures. In terms of model experiments, Clauss et al. (2003) investigated the motions behavior and splitting forces of a semi-submersible under freak wave through the experimental measurements and time-domain simu lation. It was found that the maximum response is subjected to the freak wave height. Schmittner (2005) investigated the motions and bending
* Corresponding author. E-mail address: [email protected] (G. Huang). https://doi.org/10.1016/j.oceaneng.2020.108452 Received 17 July 2020; Received in revised form 26 October 2020; Accepted 29 November 2020 Available online 8 December 2020 0029-8018/© 2020 Elsevier Ltd. All rights reserved.
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contact 6DOF (degree of freedom) measurement system, consisting of dual-CCD (charge-coupled device) cameras and a data acquisition sys tem. Three light markers were arranged in a plane and fixed on top of the cylinder to track the motion behaviors by the dual-CCD cameras. The images of the markers were acquired continuously by the dual-CCD at 30 Hz, 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, with an accuracy of 0.1 mm, were calibrated before each test.
moments of a FPSO (Floating Production Storage and Offloading) and a heavy lift vessel as well as the motions and splitting forces of a semi-submersible under freak wave. The results show that the vertical bending moments, heave and pitch of FPSO and heavy lift vessel as well as the airgap and splitting forces of semi-submersible under freak wave are larger than the maximum values predicted by codes and frequency-domain analysis. Shen and Yang (2013) compared the motion response of semi-submersible under two predetermined extreme wave sequences and analyzed the impacts of freak wave and “Three Sisters” wave (three huge waves in succession) on platform motions. The results showed that the peak value of wave height is the key parameter in determining the dynamic response of the platform, which should be a concern in Front End Engineering Design study. The adjacent effect in the “Three Sisters” wave has impacts on the platform motion, the surge and heave of the platform are both increased with the adjacent wave heights. Deng et al. (2015) investigated the effect of mooring stiffness on the motion behaviors and mooring tensions of a semi-submersible platform. The measurements showed that the freak wave can lead to critical horizontal motions on the flexible mooring system and extremely large mooring tensions on the tight mooring system. With the identical wave spectra, Pan et al. (2018) compared the motion response of a moored square cylinder under freak and random waves and quantified the effects of relative wave height, relative period and freak wave pa rameters on motion response of the floater based on time-domain measurements. The above researches show that, the effect of freak wave on various structures is more critical than the random wave. However, the existing researches mainly focus on the time-domain statistical characteristics investigation of the interaction between freak wave and structures, few researches investigated the frequency-domain characteristics on the dynamic response of the floaters. Through frequency-domain analysis, it is convenient to obtain energy distribution, peak frequency, total energy and each order spectral moment etc. But the frequency spectrum cannot provide the variation of energy in time series. For a stationary process, the local frequency characteristics in time series probably is not so important. However, as a spike in a random wave series, the local characteristics of freak wave in time-domain are exactly our concerns. Compared to Fourier transfer, wavelet analysis method is effective in obtaining the energy spectral density and the energy distribution of each frequency in time-domain, especially the instantaneous physical change under freak wave. Hence, the wavelet analysis method is adopted in this study. Based on the work of Pan et al. (2018), the present study compared the frequency-domain dynamic response of a moored square cylinder under freak and random waves with identical wave spectra through experimental measurements, which are designed to determine the variation of frequency-domain characteristics of dynamic response and the quantitative relationship with the freak wave parameter. To further investigate the significant impact on the frequency-domain character istics of the dynamic response of the floater under freak wave. The time-frequency spectra of the dynamic response under freak and random waves were calculated by wavelet analysis to investigate the time-frequency structure characteristics and variation.
2.2. Model parameters and layout The prototype of moored square cylinder (with smooth arc corners) is the columns of a 30,000 ton floating structure operating at a wind farm in East China Sea. The cylinder is a rectangular solid with a height of h = 21.7 m and a square base with a side length of D = 17.5 m. The corners were arcs with a radius of 2.31 m. The cylinder was scale by λ = 35. To achieve the design Center of Gravity (CoG), Center of Buoyancy (CoB) and draft, certain clump weights were placed on the bottom of the cylinder. The mechanical parameters of the prototype and model are presented in Table 1. As illustrated in Fig. 2, the square cylinder was anchored by four mooring lines. Each of the prototype mooring lines is nylon rope with 75 mm diameter and 140 m length. Based on the Wave Model Test Regu lation (JTJ/T234-2001), to simulate the model mooring line, not only the length and weight are scaled, but also the curve of tension (Tm) deformation (Δs) should be matched. Tm =
Cp dp (ΔS/S)n λ3
(1)
Tm − mooring tension of the model cable (N); Cp − elasticity coefficient of prototype cable, with nylon rope of Cp = 1.540 × 104MPa; dp − diameter of prototype cable (m); ΔS/S− relative elongation of prototype cable; λ − model scale, λ = 35; n − index, with nylon rope adopting n = 3. The simulation of mooring lines matches both elastic and gravity ˜ deformation curve and similarity. An example of theoretical tension measured scatters are presented in Fig. 1. It shows that excellent agreement was achieved. The model was placed 21 m away from the wave maker, where the freak wave occurred. The lower ends of the mooring lines were anchored Table 1 Mechanical parameters of the square cylinder.
2. Experiments 2.1. Experimental setup The tests were carried out in a wave flume at the State Key Labora tory of Coastal and Offshore Engineering, Dalian University of Tech nology. The flume is 60.0 m long, 4.0 m wide and 2.5 m deep. The wave generation system is Hydro-servo random wave maker system and can generate waves with periods ranging from 0.5s to 5.0s. At the other end of the flume, an absorbing beach is installed for wave dissipation effectively. The motion of moored square cylinder were measured by a non2
Parameters
prototype size
unit
model size
unit
Length Height Arc radius Draft Weight Center of gravity (from bottom) Center of buoyancy (from bottom) Transverse metacentric radius Longitudinal metacentric radius Transverse moment of inertia Longitudinal moment of inertia Natural surge period T0S Natural heave period T0H Natural pitch period T0P
17.5 22 2.31 12.0 3130 7.0 6.0
m m m m t m m
50 62 6.6 34 73 20 17
cm cm cm cm kg cm cm
2.15 2.15 1.04 × 108 1.04 × 108 59.0 8.3 11.8
m m kg∙m2 kg∙m2 s s s
6.13 6.13 1.98 1.98 10.0 1.4 2.0
cm cm kg∙m2 kg∙m2 s s s
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variation (within 3%) to the target values, which is allowable (Table 2). 2.3.2. Experiment methods In the present study, the freak wave was generated at the target location using a random wave train combined with two transient wave trains, which was proposed and used by Pei et al. (2007). With the linear superposition method, the wave surface can be expressed as:
η(x, t) = η1 (x, t) + η2 (x, t) + η3 (x, t) = M ∑
+
M ∑
A1i cos(ki x + wi t + εi )
i=1
A2i cos(ki (x − xc ) + wi (t − tc ) + εi )
(5)
i=1 M ∑
+
A2i cos(ki (x − xc ) + wi (t − tc ) + εi )
i=1
Fig. 1. Theoretical mooring line.
and
measured
tension-deformation
curve
of
where η denotes the elevation of water surface above the mean water level; A1i is amplitude of the random wave; A2i and A3i are amplitudes of the transient waves. The model was installed and moored with pre-tension of 1–1.2 N under hydrostatic status and collected 0 point at equilibrium position. The decay test was then performed by applying an artificial displace ment/rotation on the floater and releasing. The decay curve of motion process had been completely collected until the floater resumed to static state. RAO test: the time history of the motion responses were recorded with a frequency of 30 Hz under the regular wave (Sequence III in Table 2). The measurements for each test typically lasted for more than 10 waves. Freak wave test: the freak wave was generated using a random wave train combined with two transient wave trains with P-M spectrum. The measurement for each test typically lasted for more than 100 waves, and were performed 2 or 3 times to ensure the repeatability. During each test, the motions and mooring tensions were recorded synchronously with a frequency of 30 Hz. Random wave test: to compare the experi mental results measured under freak and random waves, the same spectrum with identical spectral parameters to the freak wave were applied and the measurement was similar to the freak wave test. Based on the measured time histories, spectral analyses were per formed to obtain the frequency-domain characteristics of the motions and mooring tensions of the floating square cylinder, i.e. the spectral pattern, spectral peak, 0th/1st/2nd order moments and other statistics.
the
on the bottom of wave flume. The top ends together with tension sensors were connected to the four corners of the square cylinder, and each connecting point was 25 cm measured from the bottom of the cylinder. Fig. 2 shows the layout and mooring pattern of the model in wave flume. The origin of the coordinate system is defined at the center of the model. Positive x is along the wave propagation, positive z is vertical up along the water depth, and positive y is determined by right-hand rule. For two-dimensional case, three motion components are investigated: surge, heave and pitch. Surge is the longitudinal motion along x-axis (wave propagation direction is +); heave is the vertical motion along z-axis (vertical up is +); and pitch is rotation around y-axis (clockwise direc tion is +). 2.3. Experiments 2.3.1. Experimental parameters The experiments include the decay tests getting natural frequency of the mooring system, RAO of the motion and mooring tension (regular wave), motion and mooring tension under freak and random waves. Detailed wave parameters are summarized in Table 2. Noted that water depth is d = 1.4 m and floater draft is h0 = 34 cm. It is assumed that the dynamic response subjects to the dimensions of the square cylinder (the side length of D, the module height of h), me chanical parameters of the floater (refer to Table 1), wave parameters (the significant wave height Hs, the maximum wave height Hmax, spec tral peak period Tp, freak wave parameters αn, αn includes: α1 = Hmax/Hs, α2 = Hmax/Hmax-1, α3 = Hmax/Hmax+1, α4 = ηmax/Hmax, where Hmax-1 is the wave height before the maximum wave height, Hmax+1 is the wave height after the maximum wave height, ηmax is surface elevation of the maximum wave height in the random wave sequence), mooring configuration, water depth, etc. In the experiments, P-M spectrum (Yu, 2000) was applied as: ] [ (2) S(w) = Aw− 5 exp − Bw− 4
2.3.3. Wavelet analysis method Wavelet analysis method is adopted to compute the time-frequency spectra of the dynamic responses under freak and random waves. The variation of the time-frequency energy structure has been investigated. The Morlet wavelet is selected as the mother wavelet. That can be expressed as: Ψ (t) = π−
1/4 − t2 /2 iw0 t
e
(3)
− 4 B = 691T0.1
(4)
(6)
where ω0 is the non-dimensional frequency, here taken to be 6 to satisfy the admissibility condition (Torrencr and Compo, 1998). The contin uous wavelet transform of a discrete sequence xn is defined as the convolution of xn with a scaled and translated version ofΨ (t): N− 1 ∑
Wn (s) = n
− 4 A = 173Hs2 T0.1
e
xn′ ψ *
′
[(n′ − n)δt] s
(7)
where the superscript “*” is the complex conjugate, s is the wavelet scale and n is the time index. It is more efficient to do the calculations in Fourier space. By the convolution theorem, the wavelet transform is the inverse Fourier transform of the product:
where Hs and T0.1 are the significant wave height and the average period calculated from the spectral moment, respectively. To figure out the effect of various parameters, two types of random wave sequences have been designed and generated for the experiments. In the tests, the constant wave height and period generated has minor
N− 1 ∑
Wn (s) = k=0
3
*
̂ (swk )eiwk nδt ̂ xk ψ
(8)
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Fig. 2. Mooring pattern and layout of the model in the wave flume (Pan et al., 2018).
where k = 0 … N-1 is the frequency index. The spectral density S(f,t) of the wavelet energy spectrum can be defined as: S(f , t) = |W(s, t)|2
The time-frequency spectrum (wavelet spectrum) of a physical quantity integrated in the frequency at any time, and the result can be regarded as a time history of “generalized energy spectrum”, which is denoted as E(t): ∑ Si,j ∗Δfj (10) E(t) =
(9)
The spectral density S(f,t) describes the variation of a physical parameter from time domain and frequency domain. For surge, heave and pitch of moored square cylinder, we can determine the variation of time-frequency domain characteristics followed the occurrence of the freak wave and the maximum wave of random wave. In order to investigate the variation quantitatively, the following parameters are defined:
j
The “generalized” total energy of each frequency component of a physical quantity when the freak wave (or the maximum wave of random wave) occurred is denoted as Ec = E (t)|t=c ∑ Ec = Sc,j ∗Δfj (11) j
(1) “Generalized energy spectrum” E(t)
(2) “Time distribution parameter of energy concentration”ΔTE 4
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Table 2 Summary of experimental conditions and objectives. Test Sequences
Freak and random waves
Freak wave
Significant wave height Hs(cm)
Peak period Tp(s)
Relative wave height Hs/d
I: Test sequences with different spectrum peak periods and identical significant wave height
8.0
0.057
II: Test sequences with different freak wave parameter α1 and identical significant wave height and spectrum peak period
8.0
1.14 1.26 1.56 1.83 2.01 2.16 2.87 3.10 1.6
Wave period T (s) 0.07
Relative wave height H/d 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34
Regular waves Wave height H (cm) III: Additional test sequences of regular waves
10.00
1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40
0.057
Relative period Tp / T0s
Tp / T0h
Tp / T0p
0.11 0.13 0.16 0.18 0.20 0.22 0.29 0.31 0.16
0.81 0.90 1.11 1.31 1.44 1.54 2.05 2.21 1.14
0.57 0.63 0.78 0.92 1.01 1.08 1.44 1.55 0.80
Relative period Tp/T0s 0.71 0.86 1.00 1.14 1.29 1.43 1.57 1.71 1.86 2.00 2.14 2.29 2.43
The problem we focus on is the variation of dynamic response of a moored square cylinder under freak and random waves. The maximum value of the “generalized energy spectrum” E(t) of a physical quantity under random wave is denoted as EImax; For the freak wave, TEmin is the starting time of E(t) ≥ EImax after the occurrence of freak wave. TEmax is the ending time of E(t) ≥ EImax, The response time of the freak wave is defined as Δ TE: ΔTE = TEmax-TEmin
Objectives
Parameter of freak wave α1
Parameter of freak wave α2~α4
2.00 < α1 < 2.2
1.40 < α2 < 2.14 1.40 < α3 < 2.37 0.51 < α4 < 0.66
To investigate the effect of wave period, measured Hs is in the range of 7.86–8.15 cm
1.71 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
To investigate the effect of parameter of freak wave αn measured Hs and Tp is in the range of 7.77–8.11 cm and 1.56–1.60s.
Tp/T0h
Tp/T0p
0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70
To obtain the RAO and related phase response of the floater
Objectives
through regular wave test Sequence III in Table 2. Fig. 3 illustrates that heave and pitch increase with the relative period and reach the local peak at the natural periods. The maximum heave can be up to 0.9H (H = 10 cm) at the natural period T = T0H. With further increasing of the wave period, heave oscillates in the range of 0.4–0.6H (close to the wave amplitude), which is consistent with the variation range of conventional moored floaters. The maximum pitch occurs at the natural period T = T0P. With further increasing of the wave period, pitch varies in the range of θ ≈ 1/3–1/2θmax. In addition, the amplitudes of heave and pitch are approximately symmetrical in the positive and negative components. Due to the limitation of the wave maker, the test period cannot cover the natural period of surge. Therefore, the test curve may not show the completed characteristics of surge RAO. In the test ranges of wave period, the positive amplitude of surge is larger than the negative one in the wave propagating direction. The maximum surge occurs at T/T0p = 0.7 (T/T0h = 1.0), the natural period of heave. It implies that the maximum surge is affected by the resonance of heave. When the wave period is smaller than the natural period of pitch (T/T0p < 1), the surge shows a considerable value, which has been addressed by Song et al. (2018) in the experimental study on motions of tunnel element. This is related to the fact that the larger relative motion of the floater under short waves due to stronger diffraction effects and larger mean 2nd order horizontal forces (i.e. Pinkster, 1979).
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3. Analysis of experiments results It is known that the dynamic response of the floater is affected by the natural period and RAO characteristics significantly. Therefore, the experimental results of natural period and RAO response are presented firstly. Based on the motion responses and mooring tensions of the floating cylinder measured synchronously, the spectra of motion re sponses and mooring tensions under freak and random waves are analyzed and the characteristics in frequency-domain of each motion component and mooring tension are discussed. Also, the effect of spec tral peak period and freak wave parameter on the dynamic response are discussed, and the difference in dynamic response under freak and random waves is analyzed in frequency domain. 3.1. Decay tests and response amplitude operators (RAOs)
3.2. Comparison of frequency domain characteristics of moored floater under freak and random waves
After performing decay tests of the moored floater in still water, the natural periods of surge, heave and pitch were obtained T0S = 10.0s, T0H = 1.4s, and T0P = 2.0s, respectively. The response amplitude operators (RAO) of the floater were obtained
The dynamic response of moored square cylinder includes the mo tion response, mooring tension and wave load. The frequency domain characteristics of motion response and mooring tension are discussed in this study. The wave loads on square cylinder, including the total wave 5
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Fig. 3. Response Amplitude Operators (RAO) of the model floater.
force and wave pressures, will be discussed in a separate study. 3.2.1. Effect of spectral peak period The same wave spectrum can generate various random wave series by applying different random seeds. To compare the experimental re sults measured under freak and random waves, the identical spectrum were applied and the measurement was similar for the freak and random wave tests. In Fig. 4, the comparison shows that the measured freak and random wave spectra match well with the target spectrum. Under freak and random waves generated by the same spectra, the frequency domain characteristics of motion response with various spectral peak periods were shown in Fig. 5. Two wave series were generated with the identical wave height of Hs = 8 cm and relative periods of TP/T0H = 0.81–2.21, TP/T0P = 0.57–1.55. The freak wave series are generated with the characteristics of 2.01 < α1 < 2.20. Fig. 5 shows: 1) The surge spectra of the floater are consisted of low frequency and wave frequency components. The surge spectra show the similar trends under freak and random waves with various relative wave period. 2) The variation of surge in frequency-domain is subjected to the relative wave period. With respect to the relative wave period, there are three typical scenarios on surge spectra. Firstly, with small relative wave period (0.57≤Tp/T0p ≤ 0.92), the surge spectra show significant lowfrequency characteristics and the wave frequency component can be neglected. Secondly, when the relative wave period is close to the nat ural pitch period T0p (0.92
Fig. 5. Comparison of motion responses spectra varying with the relative period under freak and random waves.
under freak and random waves and both are subjected to the relative wave period. 4) With relative wave period small (0.81≤Tp/T0H ≤ 1.11), the heave spectra are unimodal. The peak frequency is observed at the natural frequency of heave. However, with the relative wave period larger than the natural period of heave (Tp/T0H > 1.11), the heave spectra show dual peaks at spectral peak frequency and natural fre quency of heave. With the wave period increasing, the spectral peak frequency deviates from the natural frequency of heave gradually. As a consequence, the peak of heave spectrum at the wave frequency
Fig. 4. Comparison between the measured and target spectra (Hs = 8.0 cm, Tp = 1.14s). 6
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increases gradually until reaches the peak. In this case, the heave is dominated by wave frequency, and the effect of natural frequency of heave is weaken with the wave period gradually. 5) The pitch spectra are unimodal and the peak occurs at the natural frequency of pitch in the test range. When the wave spectral peak frequency is close to the natural frequency of pitch, the critical pitch spectrum was observed. It implies that the pitch of moored square cylinder is dominated by the natural period, which is different to the pitch of a moored ship. The characteristics of mooring tension in frequency-domain are addressed in Fig. 6. The same wave conditions are adopted in motion response analysis. The results indicate: 1) With respect to the relative wave period, there are two typical scenarios on frequency response of mooring tension. Firstly, with small relative wave period (0.57≤Tp/T0p ≤ 0.78), the peak of the mooring tension spectra under freak wave is larger than that under random wave. The frequency response of mooring tension is dominated by the 2ndorder drift force under both freak and random waves. In other words, the mooring tension is dominated by the low frequency component and the wave frequency component is minor. With relative wave period increasing (0.92≤Tp/T0p ≤ 1.55), the varia tion of mooring tension between the freak and random waves becomes minor since the 2nd order drift force decreases with the relative period and the wave frequency component of mooring tension has the same order of magnitude as the low frequency component. In summary, the dynamic response of surge and mooring tension in frequency-domain is dominated by the first-order wave force (wave frequency) and second-order drift force (far lower than wave fre quency). The response of heave in frequency-domain is dominated by the resonant effect of the natural frequency and wave spectral peak period. However, the response of pitch in frequency-domain is subjected to the natural frequency only. In general, the frequency domain char acteristics of the dynamic responses show the similar trends under freak and random waves with the relative wave period.
parameter α1 is large (α1 ≥ 2.32), the low frequency component of surge under freak wave is significantly larger than that under random wave and the variation increases with α1. However, the wave frequency component of surge are basically identical under freak and random waves. 2) With respect to the heave and pitch, the spectral character istics of heave and pitch are basically identical under freak and random waves and the variation is insignificant with freak wave parameter α1. The characteristics of mooring tension in frequency domain are addressed in Fig. 8. The variation of the frequency domain character istics of the mooring tension with α1 is similar to the surge of moored floater under freak and random waves. In general, the freak wave parameter α1 has significant impact on the frequency-domain characteristics of the dynamic response which leads to significant increase of surge and mooring tension in low frequency component. Such variation increases rapidly with the freak wave parameter α1, but has seldom impact on heave and pitch. 3.3. The effect of freak wave parameter on dynamic response in frequency domain The above discussion shows that the freak wave parameter α1 has direct impact on the surge and mooring tension of the floating body in frequency domain. In this section, the effect of the freak wave parameter α1 on the dynamic response are analyzed quantitatively in frequency domain under freak wave. 3.3.1. Quantitative analysis of the effect of freak wave parameter Fig. 9 shows the test results of the dynamic responses spectral pa rameters varying with the freak wave parameter α1. All wave series were generated under constant relative wave height is Hs/d = 0.057, relative period is Tp/T0H = 1.14 and Tp/T0P = 0.8, the freak wave parameter varies at α1 = 1.71–2.83. The results indicate: 1) Compared with the random wave of same spectrum, the 0th/1st order moments of surge increase with α1 signifi cantly (m0f/m0i = 0.89–2.63, m1f/m1i = 1.0–1.42), but the 2nd order moments vary not significantly (m2f/m2i = 1.0–1.07). 2) The 0th/1st/2nd order moments of heave and pitch are basically unchanged with α1. It indicates that the effect of the freak wave parameter α1 on the heave and pitch are weaker than that on surge. 3) The variation of the 0th/1st/2nd order moments of mooring tension with α1 is similar to the surge of floating body. The 0th/1st order moments increase with α1 significantly (m0f/m0i = 1.05–2.81, m1f/m1i = 1.04–1.53), the variation of 2nd order
3.2.2. Effect of freak wave parameter To investigate the effect of the freak wave parameter α1, the freak wave were generated with various α1 (α1 = 1.71–2.83). All wave series were generated under constant relative wave height is Hs/d = 0.057, relative period is Tp/T0H = 1.14 and Tp/T0P = 0.8. Fig. 7 shows the motion responses spectra of the floater under various freak wave parameter α1. The results indicate: 1) When the freak wave parameter α1 is small (α1 ≤ 2.07), the surge spectrum under freak wave is basically identical with the random wave. When the freak wave
Fig. 6. Comparison of mooring tension spectra varying with the relative period under freak and random waves. 7
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Fig. 7. Comparison of motion responses spectra varying with varying with α1 under freak and random waves.
moments is insignificant with freak wave parameter α1 (m2f/m2i = 0.99–1.17).
capturing instantaneous changes of each spectrum. In Fig. 10, the amplitude of surge increases shapely after the occur rence of the freak wave and oscillates for longer than 20 wave periods in low frequency. During the oscillating, the wave frequency component of surge is minor and can be neglected. Under the random wave, the surge component in low frequency is also significant, but the amplitude is significantly smaller than that under freak wave which is a substantial difference to freak wave. The low-frequency surge responses is signifi cantly enlarged with the freak wave action. This is because the freak wave provide an extra impulse driving the floater further away before pulling it back by the restoring forces. The mooring tension response is similar to the surge. There is a significant variation in the time-frequency spectra of the surge and mooring tension under freak and random waves. With the
3.3.2. Analysis on the key factor for the frequency response of the floater It is found in the previous discussion that with α1 = 1.71–2.83, the 0th order moment of surge and mooring tension under freak wave are 0.89–2.63 and 1.05–2.81 times of those under random waves. As a solo extreme wave in a random wave series, the freak wave makes substantial change to the spectral characteristics of surge and mooring tension. To figure out the reason, the synchronous time history and time-frequency spectra of surge and mooring tension calculated by wavelet analysis under freak and random waves are compared in Fig. 12 under various α1, respectively. The wavelet analysis provides the energy spectral density and energy distribution of each frequency in time-domain, especially in 8
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Fig. 8. Comparison of mooring tension spectra varying with varying with α1 under freak and random waves.
Fig. 9. Comparison on moments of dynamic responses versus freak wave parameter α1 under freak and random waves.
occurrence of freak wave, the time-frequency spectra of the surge and mooring tension increase significantly and show a concentrated distri bution of the energy. It is also shown that the peak value of timefrequency spectra increases significantly with the freak wave param eter α1.
To further investigate the significant impact on the frequencydomain characteristics of the dynamic response of the moored floater under freak wave. The “generalized energy spectrum” E(t) of the dy namic response (surge and mooring tension) are illustrated in Fig. 11 under freak and random waves, respectively. It is observed that the 9
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α1 under freak and random waves. It is defined the frequency of 0–0.5fp
and 0.5fp~∞ as low frequency and wave frequency, respectively (fp is the wave spectral peak frequency). With α1 increases from 1.71 to 2.83, the low frequency component of surge increases rapidly under freak wave. The ratio of low frequency component of surge under freak wave to that under random wave increase from 0.83 to 3.61, but the ratio of wave frequency components of surge has minor variation within 0.94–1.0. It implies that the key effect of freak wave on the motion response of the floater is subjected to the low frequency component of surge. Also, the variation of mooring tension with α1 is similar to the surge of floating body. The low frequency component of the mooring tension under freak wave is larger than that under random wave. With α1 = 1.71–2.83, the ratio of low frequency component of mooring tension under freak wave to that under random wave increase from 0.92 to 5.34, the ratio of wave frequency components has minor variation within 1.04–1.44, respectively. The results show that the effect of freak wave on the mooring tension of the floater is also realized through the low fre quency component. 4. Conclusion Based on the experimental measurements in time-domain, the dy namic response of a moored square cylinder under freak wave has been investigated in frequency-domain. Given identical wave spectrum, the dynamic response of moored square cylinder in frequency domain and time-frequency energy structure are substantially different under freak and random waves. Extensive analyses have been conducted to quantify the effect of relative wave period and freak wave parameter, and main conclusions can be drawn: 1. The freak wave has direct effect on the frequency-domain charac teristics of surge and mooring tension of the floater, but seldom impact on heave and pitch. The freak wave parameter α1 leads to significant variation on surge and mooring tension. 2. The effect of freak wave on the surge and mooring tension of the floater is figured out through the low frequency components which leads to significant increase with α1. This is because the freak wave provides an extra impulse for driving the floater further away before pulling it back by the restoring forces. The mooring tension response is similar to the surge. 3. The amplitude of surge increases shapely after the occurrence of freak wave and oscillates for longer than 20 wave periods in low frequency. In random waves, however, both amplitude and oscilla tion of surge are significantly weaker. The response of mooring tension is subjected to the surge response. 4. The wavelet analysis is effective in investigating the dynamic response of moored floater under freak wave. Compared with sta tistical or spectral analysis, wavelet analysis can outstandingly demonstrate the typical characteristics of dynamic response on the floater under freak wave. It can obtain the energy spectral density and the energy distribution of each frequency in time domain, especially the instantaneous physical change with the occurrence of freak wave. It is found that the freak wave have significant effect on the time-frequency domain characteristics of the dynamic response than the random wave with the same wave spectrum. With the occurrence of freak wave, the time-frequency spectra of the dynamic responses increase significantly and show a concentrated distribu tion of the energy. This implies that the occurrence of freak wave imposes serious potential risks on marine structures and vessels. 5. The freak wave parameter α1 has direct impact on the time-frequency spectra characteristics of surge and mooring tension. The peak value of time-frequency spectra of surge and mooring tension increases significantly with the freak wave parameter α1. The critical “gener alized energy spectra” E(t) and the “time distribution parameter of
Fig. 10. Time history and time-frequency spectra of dynamic responses under freak and random waves.
critical “generalized energy spectrum” E(t) and the time distribution parameter of energy concentration ΔTE of surge and mooring tension under freak wave is significant larger than that under random wave. It is also observed from Fig. 11 that the freak wave parameter α1 has direct impact on the E(t) andΔTE which both increase with the freak wave parameter α1. For further investigating the variation of the low frequency and wave frequency components of the dynamic responses under freak wave, the normalized spectral moment m0f/m0i (subscript 0 represents the 0-order spectral moment, f and i represent freak and random waves) are plotted in Fig. 12, which shows the comparison of the low frequency and wave frequency components of the surge and mooring tension spectra versus 10
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Fig. 11. “Generalized energy spectrum” of dynamic responses under freak and random waves.
Fig. 12. Comparison of low frequency and wave frequency components with various α1 under freak and random waves.
energy concentration”ΔTE of surge and mooring tension under freak wave also increase with the freak wave parameter α1.
the work reported in this paper. References
Funding
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The work is supported by China Natural and Scientific Fund through No. 51509120 and the Fundamental Research Funds for the Central Universities (DUT19LAB13). CRediT authorship contribution statement Wenbo Pan: Investigation, Writing - original draft. Chen Liang: Investigation, Data curation. Ningchuan Zhang: Supervision, Method ology, Project administration. Guoxing Huang: Conceptualization, Methodology, Writing - review & editing, Funding acquisition. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence 11
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