Study on Nonlinear Characteristics of Freak-Wave Forces with Different Wave Steepness

The nonlinear wave forces on vertical cylinders induced by freak wave trains were experimentally investigated. A series of freak wave trains with different wave steepness were modeled in a wave flume. The corresponding wave forces on vertical cylinders of different diameters were measured. The experimental wave forces were also compared with the predicted results based on Morison formula. Particular attentions were paid to the effects of wave steepness on the dimensionless peak forces, asymmetry characteristics of the impact forces and high-frequency force components. Wavelet-based analysis methods were employed in revealing the local energy structures and quadratic phase coupling in the freak wave forces.


Introduction
Extreme waves have long been a major threat to the safety of marine structures and offshore works. Extreme waves can result in significant wave run-ups, tremendous wave loads and large motion responses. A freak wave is one special phenomenon of an extreme wave with abnormal wave height and asymmetrical wave profile. In recent decades, reports of the damage and shipwrecks caused by freak waves have accumulated. Based on the statistics, more than 22 super carriers were missing because of the attacks of freak waves between 1969 and 1994 (Kharif and Pelinovsky, 2003). Freak waves have been observed both in deep sea and at nearshore, at both stormy and calm seastates (Chien et al., 2002;Mori et al., 2002). Regarding this background, interactions between freak waves and marine structures have been receiving increasing attention.
Because the formation of a freak wave is often accompanied with a huge volume of water and rapid concentration of wave energy (Rudman and Cleary, 2013), it likely results in strongly nonlinear wave forces when rushing at marine structures. As cylindrical structures are commonly used in offshore constructions, the wave forces on these structures have been widely studied. Among these studies, the Morison formula (Morison et al., 1950) is the most fam-ous and popular approach for predicting the wave forces on slender bodies. MacCamy and Fuchs (1954) presented the theoretical solution of linear diffraction problem for large diameter vertical cylinder. Kriebel (1998) studied the second-order wave forces on large diameter cylinders based on semi-analytical diffraction theory. To investigate the ringing phenomenon of marine structures in steep waves, multiple theories have been proposed to calculate the thirdorder wave forces (Faltinsen et al., 1995;Malenica and Molin, 1995;Rainey, 1989). As being restricted by a variety of assumptions, theoretical approaches are insufficient to predict the extreme wave loads due to freak waves. Chen et al. (2018) experimentally investigated the loading driving a ringing response of a model offshore wind turbine foundation and paid particular attentions to the harmonic structure of the extreme wave loads.
Some experimental explorations have been performed to reveal the nonlinear characteristics of wave forces caused by extreme waves. For example, Stansberg et al. (1995) investigated the wave forces on a vertical cylinder in extreme random wave and found that the high-frequency components could contribute 10% to 20% to the peak forces. Chaplin et al. (1997) studied the wave forces under a series of single giant waves. It was concluded that the measured wave forces are higher than the Morison predictions and the discrepancies tend to increase with increasing wave steepness. Kim et al. (1997) proposed a new stretch model of wave kinematics for extreme waves that was employed in studying the impact force of the Draupner freak wave (Kim and Kim, 2003). Paulsen et al. (2013) investigated wave forces of extreme waves with experiments and CFD simulations. Artificial high-frequency components of wave forces are observed in the measurements that result from the impulsive unset of structural vibrations. Li et al. (2014) presented wave forces on a vertical cylinder in multi-directional focused waves and concluded that the spatial profile of the surface of multi-directional focused wave affects the wave forces. Deng et al. (2016b) investigated the inline forces and pitch moments on vertical cylinders induced by the New Year Wave, in which special attentions were paid to the effects of wave stretching models on Morison predictions and the transfer functions as well as the correlations between the incident wave and the wave forces. Besides, some analytical and numerical studies have been performed on the extreme wave forces. For example, Xue et al. (2017) analytically studied the focusing-wave induced forces acting on a vertical cylinder within the scope of the potential theory and proposed the formulae of the wave-induced horizontal force and bending moment. Gao et al. (2017) numerically investigated the wave forces due to focused waves acting on a semi-submerged horizontal cylinder and the secondary load cycle was observed and carefully examined. Note that wave forces under freak waves are strongly nonlinear and influenced by various factors. Available literature provides us with a preliminary understanding on the extreme wave forces. However, knowledge on the nonlinear wave-structure interactions and strongly nonlinear characteristics of the freak wave forces remains limited.
In this paper, we present a comprehensive study on nonlinear characteristics of wave forces induced by a series of freak wave trains with different wave steepness in order to demonstrate the deficiencies of Morison predictions and to reveal the source of these deficiencies. The asymmetric characteristics and the dimensionless peak forces are ex-amined in depth. The measured wave forces are compared with the predictions based on Morison formula and the differences between them are revealed in the time-frequency domain. The local energy structures of wave forces are presented, and studies on the cut-off frequencies are performed to demonstrate the importance of high-frequency components. Moreover, wavelet bicoherence analysis is conducted to explore the sources of these high-frequencies.

Experimental set-up
The experiments were conducted in the wave flume of State Key Laboratory of Ocean Engineering (SKLOE), Shanghai Jiao Tong University, China. The wave flume is 20.0 m long, 1.0 m wide and the water depth d is 0.9 m. The wave trains were generated using a flap-type wavemaker. Downstream of the wave flume, an absorbing beach was used to eliminate wave reflections. Limited by the dimensions of the wave flume, the model scale is selected as 1:100 in this study. For simplicity, only model-scale values, hereafter, are presented unless otherwise specified. Fig. 1 depicts the schematic of the experimental set-up. In the experiments, the focal point as well as the cylinder location is set at 7.0 m away from the wavemaker. With the typical dimensions of practical columnar structures in marine structures such as semi-submersibles and spars, 3 truncated hollow cylinders with diameters D = 10 cm, 15 cm and 20 cm in model scale were successively mounted on the rigid support frame. The above cylinders of different diameters are hereinafter denoted as C10 cm, C15 cm and C20 cm, respectively. A six-component force transducer with a measurement range of 20 kg from KYOWA was carefully calibrated and installed between the cylinders and rigid support structures to measure wave forces. The drafts of these cylinders are h = 0.3 m, and the distance between the transducer and the still water level (SWL) is l = 0.29 m. A sampling frequency of 100 Hz was selected in the experiments. Prior to formal wave force tests, knocking tests were conducted to examine the natural frequencies of the experimental assembly. The natural periods corresponding to C10 cm, C15 cm, C20 cm are 0.0906 s, 0.0912 s, 0.0917 s, DENG Yan-fei et al. China Ocean Eng., 2019, Vol. 33, No. 5, P. 608-617 609 respectively. The corresponding circular frequencies are all above 68.5 rad/s, ten times larger than the primary wave frequencies. It indicates that the experimental assemblies could be regarded as rigid structures.
3 Wave environments γ A JONSWAP (Joint North Sea Wave Project) spectrum of the significant wave height H s = 11 m, spectral peak period T p = 17 s and peakness factor = 2.0 in prototype (H s = 0.11 m, T p = 1.7 s in model scale) was used to design the target wave train using the embedding model (Kriebel and Alsina, 2000), in which the total wave energy spectrum is split into two parts: one for the background random sea and the other for the transient or freak wave. The target wave train hence could be expressed as follows.
θ n where x c and t c are respectively the target focal position and focal moment, the phases of the random wave are selected randomly, the amplitudes of random waves A Rn and the amplitudes of transient waves A Tn are defined as: where P R and P T respectively denote the percentage of energy in the random sea and in transient wave.
represents the JONSWAP spectrum, and k respectively denote the wave frequency and wave number. In this study, the energy percentage of transient wave P T is 10.6%, and the remaining energy is contained in the random wave. In model scale, the maximum wave height H max is 0.257 m, and the crest height H c is 0.180 m, leading to the following ratios: H max /H s = 2.34 and H c /H s = 1.64. Thus, it meets the criteria of freak wave given by Kharif and Pelinovsky (2003). The designed freak wave is shown in Fig. 2.
Based on the designed freak wave train in Fig. 2, four wave trains denoted as M04, M06, M08 and M10 were modeled. Specifically, wave train M10 is basically identical to the original designed wave train. Wave elevations of M04, M06 and M08 are artificially adjusted as 40%, 60% and 80% of M10. Therefore, within each wave period, the corresponding wave periods of these four wave trains are close to each other, while the corresponding wave steepness increases gradually.
According to the phase-amplitude iteration scheme in available literature (Deng et al., 2016a;Schmittner et al., 2009), these four trains are generated and optimized in the wave flume prior to the wave force tests. The length of each wave train is 60.0 s and the target focal moment for each freak wave is set as 30.0 s. Only the results in the time range of 15.0-45.0 s were used for analysis and discussion to reduce the influences of unstable waves in the beginning and the wave reflections in the end. Fig. 3 displays the angular displacements of the wavemaker employed in the experiments. Fig. 4 presents the comparisons of target and optimized freak wave trains, which were used to investigate the wave forces on vertical cylinders in this study.  The horizontal line denotes the zero level and is used to mark the zero crossing points. The oblique line connecting the peak point and the zero crossing point ahead is used to distinguish the front shape, i.e. 'cvf' or 'ccf'. The 'cvf' in the figures denotes a 'convex front or straight-line front' while the 'ccf' represents a 'concave front'. According to Kim et al. (1997), the crest of a strong asymmetric wave has a distinct concave front and convex rear. The same is true for time histories of wave forces. From enlarged wave trains, we observe that these four freak waves exhibit a crest that has a convex front and a convex rear, or a nearly straight front and a convex rear. Thus, all the incident waves are weakly asymmetric. However, the cases are completely different for the wave forces. It shows that the horizontal wave forces are with crests that have a concave front and a convex rear, indicating that the wave forces in-  DENG Yan-fei et al. China Ocean Eng., 2019, Vol. 33, No. 5, P. 608-617 duced by weak-asymmetric freak waves could be strongly asymmetric. This phenomenon stands somewhat in contrast with the views of Kim et al. (1997), which regards the strong asymmetric force as a consequence of strong asymmetric wave only.
In Fig. 5, it appears that the wave forces profiles on different cylinders are similar for the same incident freak wave. With increasing wave steepness, the force crest becomes sharper and the succeeding trough shallower, leading to more obvious asymmetry. Fig. 6 shows the F c /F t values of the horizontal wave forces, in which F c and F t respectively denote the magnitudes of the crest value and trough value. Most values of F c /F t exceed 1.0 to varying degrees, except for the wave forces due to M04. The vertical asymmetry is more significant for wave forces on small diameter cylinders. The exception of Case M04 might result from the randomness factor. Therefore, the viscous forces, which are proportional to the square of the flow velocity, may contribute to the peak forces. The maximum F c /F t value even reaches 2.96, indicating that these wave forces are strongly nonlinear, and presents a challenge for force predictions.

Comparisons with Morison predictions
The key dimensionless parameters for wave force predictions are D/L and H/D, in which D/L is the scattering parameter determining the scattering effects around the structure and H/D determines the relative importance of drag forces over inertial forces (Deng et al., 2016b). In Fig. 7, the wave conditions are displayed on a (D/L, H/D)-plane based on the measured freak wave trains in Fig. 4. The red dashed line denotes the breaking limits (Miche, 1951), and the pink dashed line reflects the nonlinearity of waves and possible wave breaking (Molin, 2002). The vertical dashed line (D/L=0.2) and the horizontal dashed line (H/D) respectively represent the thresholds of diffraction effects and viscous effects. In this study, D/L values are smaller than 0.2 and most values of H/D are above 1.0, indicating that the prerequisite of Morison formula is satisfied and the drag force components should be included in wave force predictions.
In view of the above, Morison formula (Morison et al., 1950) is employed to predict the hydrodynamic loads. The total horizontal forces on vertical cylinders are regarded as the sum of the inertial and drag forces as: ∂u/∂t where f(t, z) is the wave force acting on unit body length as a function of time t and height z, C M and C D are inertial and drag coefficients, u and are respectively the horizontal water particle velocity and acceleration, as shown below.
where A i , , k i and are respectively the wave amplitude, angular frequency, wave number and phase of each wave component.

η(t)
In this study, C M and C D are respectively selected as 2.0 and 1.0 according to relevant classification rules (CCS, 2016). Since the freak wave train is a kind of irregular waves, the water particle velocity and acceleration could be obtained by superimposing the corresponding values of each wave component. In view of the transient surface elevation, the stretching method of Wheeler (1969) was employed to obtain the particle kinematics below the transient water surface . Thus, an equivalent level z', as shown below, was used to replace variable z in Eq. (3). Fig. 8 presents comparisons between measurement and Morison prediction with respect to Case M10 C10cm. A much sharper crest and a shallower trough are observed in the measured wave force compared with the predicted result. For the comparisons of all cases, Fig. 9 presents ratios of peak forces between the predicted and the measured results. It shows that the Morison predictions are likely to un-  derestimate the peak forces. With the increasing wave steepness, the predicted peak forces become far smaller than the measured values. In extreme cases, the predicted peak force reaches only 64% of the measured value. Therefore, the wave forces induced by freak waves are highly nonlinear and the Morison approach is not qualified in predicting such nonlinear wave force caused by freak waves, particularly for the peak forces.
To clarify the influences of wave steepness on the peak forces, the relationship between the dimensionless wave forces and the wave steepness H/L, the wave front steepness are presented in Fig. 10. The wave front ε steepness is determined by Myrhaug and Kjeldsen (1987) η c ε where is the wave crest height, T rise is the rise period between wave trough and wave crest, and T tt is the troughto-trough period. Compared with the wave steepness H/L, the wave front steepness focuses more on the asymmetry of local wave pattern. It is observed that the dimensionless forces increase with the wave steepness and wave front steepness. This result indicates that the nonlinearity of wave forces becomes increasingly significant. In addition, it appears that the dimensionless forces have a closer relationship with the wave front steepness rather than the wave steepness. Therefore, the wave front steepness might be a better alternative to describe the wave steepness of local extreme waves. In Fig. 10, the horizontal lines represent linear predictions of wave forces induced by regular waves with identical wave periods to the measured freak waves. The linear predictions herein are given by the analytical solution of wave force acting on a vertical cylinder (MacCamy and Fuchs, 1954), as shown below.
where k is the wave number, and are Bessel functions of the first and second kinds, respectively. It appears that the measured peak forces approach the linear predictions for the small wave steepness cases (M04). However, for the large wave steepness cases (M10), the measured peak forces could reach two-to-three times the corresponding regular wave forces. Therefore, the freak wave forces could not be determined simply with the wave forces due to regular waves.

Local energy structures
For inspecting the local energy structures in freak wave  and wave forces, the Wavelet Transform (WT) approaches are used to reveal the energy structures. Wavelet transform is defined by inner product of signal x(t) and a family of continuously translated and dilated wavelets τ where the asterisk denotes the complex conjugate. is the translation parameter, corresponding to the location of the wavelet as it is shifted through the signal and a is the scale dilation parameter determining the width of the wavelet. Based on the wavelet transform, the time-averaged wavelet spectrum E 3 could be obtained by In this study, a complex Morlet wavelet with central frequency f c =1.0 Hz was chosen as a mother wavelet function. The scale factors in wavelet transform were transformed into the corresponding circular frequency values according to where f s is the sampling frequency and a is the scale value. To achieve deeper knowledge regarding local energy structures of both freak waves and wave forces and to identify the differences of local energy structures between the measured and predicted wave forces, the wavelet transform spectra of freak wave trains and wave forces on C10 cm are presented in Fig. 11. It is observed that the frequency bandwidths of the freak waves are close to each other with the maximum frequency below 30 rad/s. For the measured wave forces, the high-frequency components appear to become increasingly noticeable from M04 to M10, and the maximum frequency even reaches 60 rad/s. As for the Mor-ison predictions, the maximum frequencies are approximately twice the corresponding values of incident freak waves. The reason might be that the wave force integrations from the cylinder bottom to the transient surface elevations contribute to a certain amount of second-order components. However, the frequency bandwidths of the predicted forces are significantly smaller than those of the measured freak wave forces. Note that the predicted result is calculated based on the measured freak wave trains, and the nonlinear wave components due to wave-wave interactions are included. Therefore, the significant high-frequency components in the measured wave force result from the nonlinear wave-structure interactions. Such high-frequency components are major differences of local energy structures between the measured and predicted wave forces.
To present a more intuitive and quantitative analysis of differences of energy structures on the occurrences of freak waves, Fig. 12 presents the normalized time-averaged wavelet spectra with respect to freak wave trains and wave forces acting on C10 cm. The integral interval for the time-averaged wavelet spectra is selected from 1.5 s preceding to 1.5 s following the freak wave occurrence, i.e. 28.5−31.5 s. In Fig. 12, values of time-averaged wavelet spectra have been normalized by corresponding maximum values for the purpose of comparisons. It shows that the primary frequencies of both the incident waves and wave forces are approximately 4.0 rad/s. Small secondary peaks are observed in the time-averaged wavelet spectra of freak waves accompanied by noticeable secondary peaks in that of wave forces for Case M04, M06 and M10. It is worth noting that the secondary peaks in Morison predictions might largely result from both the small secondary peaks of incident waves and the large values of wave-force transfer functions at specified frequency interval (near 8.0 rad/s).
Comparing the results between the measured and pre- Fig. 11. Wavelet transform spectra of freak wave trains and wave forces (D = 10 cm).
dicted wave forces, the time-averaged wavelet spectra are close to each other within the frequency interval of major wave energy (2.0−6.0 rad/s). Significant gaps occur in the frequency interval above 6.0 rad/s. For M04 and M06, major differences exist in the secondary peaks. However, the major differences are observed in high-frequency area (above 12 rad/s) for M08 and M10. These gaps of time-averaged wavelet spectra result from the nonlinear wave-structure interactions. Therefore, the deficiencies of Morison predictions as shown in Fig. 9 are largely attributed to some unaccounted second-order components for small wave steepness cases (M04 and M06) and some high-frequency components for large wave steepness cases (M08 and M10). This indicates that the nonlinear force components along with the nonlinear wave−structure interactions become increasingly significant with increasing wave steepness of freak waves.

Cut-off frequencies
To evaluate the contributions of such limited high-fre-ω cut quency components to the maximum wave elevations or the peak forces, analyses on the cut-off frequency were conducted. Fig. 13 presents the relationships between the fitted peak values of the freak waves as well as the wave forces and the cut-off frequencies. In the figures, denotes the cut-off frequency, H c and F c respectively represent the crest values of wave elevations and wave forces while H ccut and F ccut are respectively the fitted values of wave crests and force crests under different cut-off frequencies. It shows that a very high cut-off frequency is demanded to obtain a satisfactory fitted peak value with fitting error smaller than 5%. Specifically, the cut-off frequency needs to increase from M04 to M10 for both freak wave trains and wave forces. As for Case M10, the requirements of cut-off frequencies even reach 26 rad/s and 62 rad/s for freak wave and wave force, respectively. As can be observed in Fig. 12, the energy densities of wave components above 12 rad/s are negligible to a certain degree. However, when adopting 12 rad/s as the cut-off frequency, the fitted peak elevation of M10 only achieves approximately 88% of the corresponding meas-  DENG Yan-fei et al. China Ocean Eng., 2019, Vol. 33, No. 5, P. 608-617 ured crest height, and the fitted peak force of M10 is smaller than 60% of the corresponding measured peak force. Therefore, the high-frequency components make essential contributions to the peak values of freak wave and the wave forces, especially for freak wave forces, although the corresponding energy densities of these high-frequency components are limited.

Wavelet bicoherence
It can be observed from the analyses presented above that the high-frequency components in the freak wave forces are indispensable for the peak forces. To explore the sources of these high-frequency components, wavelet-based bicoherence (Van Milligen et al., 1995) was employed in this study to reveal the quadratic phase coupling phenomenon in the freak wave forces. The wavelet auto-bispectrum B x and the square auto-bicoherence b x are defined as: where, a 1 , a 2 and a meet the following relation  Fig. 14b. In other words, the quadratic phase coupling phenomenon exists widely for such large wave steepness case, not only among the primary frequency components but also between the primary and higher-order harmonic frequency components (above 8.0 rad/s). As the wave energies of force components above 12 rad/s are very limited, the corresponding bispectrum values are not noticeable in Fig. 14a. However, these extremely high-frequency components (30−60 rad/s) of wave force (likely caused by the quadratic phase coupling among high-frequency components) are essential parts of such extreme peak forces.

Conclusions
To investigate the nonlinear horizontal forces induced by freak waves, we performed a series of wave forces tests with freak wave trains of different wave heights and vertical cylinders with different diameters. The effects of wave steepness on the asymmetric characteristics of both freakwaves and wave forces, dimensionless peak forces and local energy structures were discussed. The measured wave forces were compared with the Morison predictions. Wavelet-based analyses including wavelet spectrum, time-averaged wavelet spectrum and bicoherence were performed to inspect the local characteristics of freak waves and wave forces. The major conclusions are as follows.
(1) Weak asymmetric freak waves are likely to cause strongly asymmetric wave forces. The vertical asymmetric parameter F c /F t increases rapidly with increasing wave height and is more significant for the wave forces on the cylinder of smaller diameter.
(2) Conventional predictions based on Morison formula could significantly underestimate the peak forces of freak waves, especially for large wave steepness cases.
(3) The wave front steepness might be a better alternative in describing the wave steepness of local extreme waves.
(4) The primary differences between measured and predicted wave forces of large wave steepness exist in high-frequency components.
(5) The high-frequency components in freak wave trains and wave forces are of small energy proportion, but they do make essential contributions to the peak values.
(6) These high-frequency components come from the quadratic phase coupling not only between primary frequency components but also among high-order harmonic components.