Numerical Study of the High-Frequency Wave Loads and Ringing Response of A Bottom-Hinged Vertical Cylinder in Focused Waves

This paper presents a numerical study on the high-frequency wave loads and ringing response of offshore wind turbine foundations exposed to moderately steep transient water waves. Input wave groups are generated by the technique of frequency-focusing, and the numerical simulation of focused waves is based on the NewWave model and a Fourier time-stepping procedure. The proposed model is validated by comparison with the published laboratory data. In respect of both the wave elevations and the underlying water particle kinematics, the numerical results are in excellent agreement with the experimental data. Furthermore, the local evolution of power spectra and the transfer of energy into higher frequencies can be clearly identified. Then the generalized FNV theory and Rainey’s model are applied respectively to calculate the nonlinear wave loads on a bottom-hinged vertical cylinder in focused waves. Resonant ringing response excited by the nonlinear high-frequency wave loads is found in the numerical simulation when frequency ratios (natural frequency of the structure to peak frequency of wave spectra) are equal to 3–5. Dynamic amplification factor of ringing response is also investigated for different dynamic properties (natural frequency and damping ratio) of the structure.


Introduction
In ocean engineering, high-frequency wave loads occurring in steep wave conditions have been identified since the 1990s. The high-frequency wave loads are typically excited on tension-leg platforms (TLPs) or gravity-based platforms (GBSs), and they may induce a rapid build-up of vibrations at the resonant frequency of the lightly damped offshore structure (Grue and Huseby, 2002). The phenomenon of the high-frequency resonant response is so-called 'ringing', due to its sudden appearance and much slower decay. Unlike springing, ringing usually occurs during the passage of steep wave crests, and can generate fairly high levels of stress within a burst of only a few oscillations. The ringing response was firstly recorded in the model tests of the Heidrun TLP and Draugen GBS, as shown in the Ringing Report (Stansberg, 1993). This subject has been widely studied in the oil and gas industry from the 1990s. Ringing may also occur in offshore wind turbine foundations, which has recently been a topic of particular interest since ringing induced loads may be relevant to extreme loads and fatigue damage. However, proper and efficient numerical models of predicting these loads and responses are still lacking.
The ringing phenomenon has motivated a range of experimental and theoretical studies. The experimental studies have been implemented in wave tanks or wave basins with incoming focused wave groups (Grue et al., 1993;Chaplin et al., 1997;Scolan et al., 1997;Zang et al., 2010), steep random waves (Stansberg et al., 1995;Bachynski et al., 2017;Riise et al., 2018) and regular waves (Huseby and Grue, 2000;Kristiansen and Faltinsen, 2017). In focused wave events, a secondary load cycle was found and it might contribute to a build-up of ringing response. In terms of theoretical studies, we have known that the high-frequency wave loads cannot be well modeled by conventional analysis tools such as linear and second-order wave diffraction-radiation models. Thus the main focus of the theoretical studies has been to capture the wave loads up to the third-harmonic component in regular waves, e.g. Faltinsen et al. (1995) (hereafter referred to as the FNV theory), and Malenica and Molin (1995). The original FNV theory was firstly obtained for the case of regular incident waves in deep water, and secondly generalized by Newman (1996) to irregular waves. The first three harmonics of the horizontal wave force on a slender surface-piercing vertical cylinder were derived analytically based on a third-order perturbation analysis in the long-wave regime. Kristiansen and Faltinsen (2017) then generalized the FNV theory to finite water depth, and to allow for arbitrary incident wave kinematics. The solution of Malenica and Molin (1995) was derived based on a complete third-order diffraction model in finite water depth, which could capture the third-harmonic force in regular waves by complicated integrals of the first-, second-and third-order velocity potentials. However, the approach is only applicable to the wave diffraction of vertical cylinders in regular waves with small steepness and is really difficult to extend to actual irregular sea states. Unlike the theoretical studies outlined above, an alternative nonlinear load model was developed by Rainey (1989Rainey ( , 1995 for slender structural members, which was derived from energy considerations instead of direct pressure integration based on a conventional perturbation approach. The overriding advantage of Rainey's model is that it could be applied to important cases involving extreme wave events, provided that the appropriate nonlinear undisturbed wave kinematics is adopted as input. Some fully nonlinear potential flow models (e.g. Ferrant, 1998;Shao and Faltinsen, 2014) have also been developed as well as the CFD models (Paulsen et al., 2014;Chen et al., 2014) in recent years. However, both the fully nonlinear potential flow models and CFD models remain time consuming for practical applications and present some weaknesses in terms of robustness.
For now, theories and analysis models of the high-frequency wave loads in realistic sea states still have some drawbacks. Wave load mechanisms are not fully understood, particularly in steep transient waves. They may contain large wave-exciting inertia forces, the secondary load cycle, wave slamming due to steep and breaking waves, and possible flow separation effects, as pointed out by Riise et al. (2018). Regardless of which load mechanisms exist, the occurrence of ringing necessarily involves the development of significant nonlinearities, which are in both the wave motion and the wave-structure interactions. As mentioned by Atkins et al. (1997), it appears that large transient waves are more likely to induce ringing instead of a regular wave train. Therefore, efficient analysis models that can capture such events in irregular waves are needed for practical design. As a result, the simplicity and efficiency of the FNV theory and Rainey's model make them appropriate to the prediction of higher-harmonic wave loading in realistic sea states.
The purpose of this study is to get access to a deeper insight into efficient modeling of wave impacts on offshore wind turbine foundations in steep transient wave events. The large-diameter monopile support structures are widely used in the offshore wind industry, thanks to the simple but robust design through comparison with other foundation concepts. In the field conditions, the diameter of the monopile is typically smaller than 8 m, the first mode natural peri-od is 3-5 s and the damping ratio is 1%-4% (Kallehave et al., 2015), while the typical spectral peak period of the extreme sea state in the North Sea is T p~1 5 s (Grue and Huseby, 2002), which is about three to five times longer than the resonance period of the structure. In general, the waves are moderately steep or even breaking when the ringing response occurs, and thus the wave height is comparable to the diameter of the monopile. Nevertheless, the diameter of the monopile is relatively small compared with the characteristic wavelengths so that the diffraction effect of the structure in this phenomenon is generally slight. In this paper, the monopile foundation is simplified as a bottom-hinged vertical cylinder, and the focused wave groups are used to provide a good representation for large transient wave events, then the FNV theory and Rainey's model are adopted to predict the higher-frequency wave loads and ringing response on the bottom-hinged vertical cylinder.
The remainder of this paper is organized as follows. Section 2 describes the focused wave model and hydrodynamic models adopted in this paper in detail. In Section 3, the convergence and accuracy of the proposed focused wave model are verified by the comparison with the published laboratory data. Then the high-frequency wave loads and ringing response of a bottom-hinged vertical cylinder in focused wave groups are investigated. Furthermore, the effects of nonlinear high-frequency wave loads, natural frequency and damping ratio on the ringing response of the structure are systematically investigated. Conclusions are drawn in Section 4.

Mathematical formulation
2.1 Focused wave model ϕ (x, z, t) A Cartesian coordinate system Oxyz is defined. The coordinate system is fixed in space with its origin O located at the calm water surface and the positive z-axis pointing vertically upwards. The positive x-axis is along the propagation direction of incident waves. In this section, a two-dimensional focused wave model is introduced to simulate large transient wave events. The fluid is assumed to be incompressible and inviscid, and the flow is irrotational. Therefore, the velocity potential can be introduced, which satisfies the Laplace equation in the fluid domain, (1) and the nonlinear kinematic and dynamic free surface boundary conditions can be written after some re-arrangement as: η where is the free surface elevation and g is the gravitational acceleration. It can be seen that in this form there is no 514 ZHANG Yi, TENG Bin China Ocean Eng., 2020, Vol. 34, No. 4, P. 513-525 η ϕ η ϕ time derivative on the right side of Eq. (2). In consequence, supposing that a spatial description of and can be defined at some initial time, the initial solution can be timestepped to give new solutions for both and at subsequent times.
In this study, a Fourier time-stepping procedure outlined by Johannessen and Swan (1997) is adopted, and some improvements are made in setting the initial conditions, which will be introduced in detail in the following text. By using Fourier modal expansion in space, the free surface elevation is defined by a sum of Fourier series: and the corresponding velocity potential can be expressed by: where d is the water depth, and is the large length scale (i.e. the fundamental wavelength) over which the solution is assumed to be periodic.
The velocity potential expressed by Eq. (4) satisfies both the governing equation ( ) and the bottom boundary condition ( ) exactly. The unknown coefficients and are merely functions of time. Thus the resulting solution only involves the calculation of the unknown coefficients at each time step. To accomplish this, the initial values of and are firstly specified at a total of 2N spatial locations that are equally spaced over the length scale . Then we apply the nonlinear free surface boundary conditions (i.e. Eq. (2)) at each spatial location, which gives two sets of 2N linear simultaneous equations. Afterwards, time derivatives of the unknown coefficients (i.e. , and , ) can be solved. It is noted that the water surface elevation, and thus its time derivative, is expressed in terms of a standard Fourier series, therefore and can be rapidly solved by using the fast Fourier transform (FFT) technique, which corresponds to an Nlog 2 N process. However, the approach cannot be applied in the case of velocity potential due to the depth-variation included within Eq. (4). Accordingly, a standard lower-upper matrix decomposition method (Press et al., 1992) is applied to solve and . Once time derivatives of the coefficients have been determined, the initial coefficients can be time-stepped to get new solutions at subsequent times, then the new solution of both and can be obtained. In this study, the Adams-Bashford-Moulton predictor-corrector scheme (Press et al., 1992) is adopted as the time-stepping scheme, which appears to provide a satisfactory combination of accuracy and efficiency. Since this is a multi-step method, the calculations are initially implemented by the fourth-order RungeKutta scheme to obtain values of the first three time steps.
Unfortunately, the description of an extreme wave event at the initial time is not known a priori, which means that both and cannot be exactly determined at the initial time. However, since the present paper is mainly concerned with the representation of steep wave events generated by the technique of multi-frequency focusing, we can know that if the initial conditions are specified at some early stage well in advance of the focal time, the wave energy will be widely distributed within the spatial domain due to the dispersion property of water waves. Thus, the wave-wave interactions at this initial time will be generally weak so that the underlying wave components can be approximatively modeled based on the linear or second-order theory.
Within the framework of linear theory, the water surface elevation and corresponding velocity potential of a focused wave group are specified as: α n k n ω n where N is the total number of the discrete wave components, is the amplitude of the n-th wave component, is the n-th wavenumber component and is the corresponding wave frequency component defined by the following dispersion relation: To obtain realistic modeling of an extreme wave event, the amplitude spectrum consisting of discrete wave components at the initial time can be determined by the NewWave model outlined by Tromans et al. (1991). On the basis of the statistical predictions of Lindgren (1970) and Boccotti (1983), the NewWave model relates the averaged properties of extreme events to the power spectrum for a linear random Gaussian sea state. In its original form, this model describes the most probable or average shape of an extreme wave event, conditional on the presence of a large crest elevation at some arbitrary time (i.e. and ). The most probable shape of an extreme wave around a large crest is given by a scaled autocorrelation function as: where is the autocorrelation function of the water surface elevation, is the underlying frequency spectrum and is the variance defined by: α n ω n By adopting the linear surface profile given in Eq. (5), the amplitudes of the frequency components can be determined by Fourier analysis such that: where is the linear sum of the component wave amplitudes and is the frequency interval. Provided that the linear focused wave group corresponding to a maximum elevation of occurs at , the spatial description of the surface profile at this time is given by: k n ω n where the wavenumber components , corresponding to the uniformly spaced frequencies , are derived from the dispersion relation given in Eq. (7). However, the spatial distribution specified in Eq. (12) is not periodic in space and thus the Fourier time-stepping procedure outlined above cannot be directly applied. To overcome it, instead of using a least squares method adopted in Johannessen and Swan (1997), an alternative approach is proposed in this study. With the continuity of the spectrum, the power spectrum in the frequency domain can be directly transformed into a power spectrum in the wavenumber domain as: As an example, the schematic diagram of this transformation is shown in Fig. 1.
Afterwards, a modified version of the NewWave model can be applied, and the amplitude of each discrete wavenumber component is simply given by: α ω n By re-applying Eq. (7), the frequency components , based on the evenly distributed wavenumber components , can be determined. For now, the resulting surface profile and the corresponding velocity potential are periodic in space, and hence the initial conditions for the Fourier timestepping procedure can be specified by Eqs. (5) and (6). If the focal event occurs at , some negative time is chosen as the starting time, such that both and at are relatively small everywhere within the spatial domain due to the dispersion property of water waves. The choice of depends on the convergence analysis. Once an appropriate has been chosen, the initial coefficients are given by: .
Accordingly, by adopting this NewWave-type solution to specify the initial conditions, as the input to the Fourier time-stepping procedure, the evolution of a large design wave can be simulated in a fully nonlinear sense. In addition, the nonlinear water particle kinematics beneath a wave crest can be accurately predicted, without the ambiguity in the existing design models. Although several empirically corrected or stretched wave models (e.g. Wheeler, 1970) are widely adopted in design practice to overcome the effects of high-frequency contamination in predicting the irregular wave kinematics, they do not exactly satisfy the governing equation and neglect the nonlinear wave-wave interactions. Further discussion on this point can be found in Sobey (1990).

Ringing response
A build-up of ringing response is illustrated with the model of a bottom-hinged rigid vertical cylinder. As shown in Fig. 2, the incident wave groups propagate along the positive x-axis, and the vertical cylinder is hinged at the bottom and supported at the top with springs in the horizontal plane. In this study, as mentioned above, the incident wave amplitude A and cylinder radius a are of the same order, and both are small compared with the wavelength . The vertical cylinder is free to rotate with an angle in the pitch mode of motion. As a simplification, it can be represented as a single-degree-of-freedom system. According to the law of conservation of angular momentum, the equation for pitch motion of the bottom-hinged cylinder is then given by: θθθ where I is the moment of inertia, K is the rotational stiffness, C is the structural damping and M y is the overturning moment. , and denote the angular displacement, velocity and acceleration of the cylinder, respectively. I a is the added moment of inertia and can be obtained by: It can be seen that the added moment of inertia is fluctuating and varying with the wave elevation.
The specific hydrodynamic models for predicting the higher-harmonic wave loads and corresponding overturning moment of the cylinder will be introduced in Section 2.3. Afterwards, the motion equation of the cylinder is solved by using the Newmark integration scheme. It is noted that the equation of motion contains the fluctuating added moment of inertia, which is extended up to the instantaneous water surface. As a result, the effect of the added mass force and the variation of added mass due to changes in the wetted area of the cylinder are considered here.

Hydrodynamic models
In the original FNV theory, a small parameter was introduced and both the incident wave steepness kA and the non-dimensional wavenumber ka were of . Nevertheless, the incident wave amplitude A and cylinder radius a are in the same order, i.e.
, which is different from the conventional diffraction theory in which . Potential flow is assumed, and the total velocity potential was written as , where denotes the incident wave potential, is the linearized scattered potential that can be found by the approach of slender-body approximations and matched asymptotic expansions, and is a nonlinear potential of in the inner domain and tends to zero in the outer domain, which satisfies the 3-D Laplace equation and approximate free-surface conditions to third order. Following the FNV theory, in the near field of the cylinder, the linearized diffraction potential can be expressed as: ) .
Here, and satisfy the 2-D Laplace equation and the body-boundary condition in each horizontal plane, which can be found as simple harmonic functions. These characteristics are all retained in the generalized FNV theory proposed by Kristiansen and Faltinsen (2017).
In the present study, we combine the generalized FNV theory with incoming focused wave groups. The total horizontal wave force is obtained by integrating the distributed load term due to along the cylinder axis up to the undisturbed free surface, and adding the load term due to , The distributed load term per unit length due to can be written as: where is the mass density of water, is the added mass per unit length in surge, which takes the value of for slender circular cross-sections. u and w are the incident horizontal and vertical particle velocity components ( , ) at the cylinder axis , which can be obtained from the proposed focused wave model in Section 2.1. The horizontal force due to the nonlinear potential is: where the quantities are calculated at , representing a point force. The derivation of can be referred to Faltinsen et al. (1995), and thus is not elaborated here. The overturning moment is obtained by multiplying with the moment arm within the expressions of Eqs. (20) and (21).
Unlike FNV theory, Rainey's model (1989Rainey's model ( , 1995 was not based on a conventional perturbation approach, but was derived from energy considerations. For the limiting case of a slender cylinder, he argued that the potential flow loading could be written by the sum of three terms corresponding to the Morison inertia force, the axial divergence force and the surface intersection force: u ∂u/∂t where is the water particle acceleration in the incident wave and the Lagrangian acceleration (including the convective terms) given by Eq. (23) is used instead of .
In general, the contribution of the convective terms in Eq. (23) is small so that the difference between these two expressions is often slight, but the convective terms in Eq. (23) contribute to higher-harmonic load components and thus may produce disproportionate effects on dynamically sensitive structures, as pointed out by Manners and Rainey (1992). The second term on the right side of Eq. (22) is the axial divergence force, which represents the rate of change of transverse momentum due to the axial divergence of the F SI incident flow. Further, the surface intersection force in Eq. (22), as expressed by Eq. (24), represents a point force acting at the intersection point of the cylinder axis with the instantaneous water surface, which is relevant to the change in kinetic energy of flow when the wetted area of the cylinder changes.
F ′ For circular cross-sections, we can know that the distributed load terms in Rainey's model (i.e. the sum of the Morison inertia term and the axial divergence term) agree with in the FNV theory. Therefore, the only difference between these two load models is the point force term, which contributes to higher-harmonic load components differently. θ Further, since the vertical cylinder is free to rotate with an angle in the pitch mode of motion, some additional load terms come into play due to the effects of feedback from the cylinder's motion to the loading. In addition, in view of the viscous effects, a Morison-type quadratic drag term is also added to the above hydrodynamic models. Thus, the final expression of the overturning moment with respect to the bottom can be written as (e.g. Rainey's model): is the integral variable, is the instantaneous immersion length, and C D is an empirical drag coefficient.

Numerical results and discussion
The proposed numerical scheme of modeling focused wave groups is first validated by comparison with the laboratory data in Baldock et al. (1996). The experiments were conducted in a wave flume at Imperial College. The wave flume has an overall length of 20 m, a width of 0.3 m and a working depth of 0.7 m. In the experiments, a total of 29 wave components were simultaneously generated by the wave paddle, and their relative phases were adjusted so that the wave crest was focused at a given instant and position to produce an extreme wave event. This technique of generating extreme waves in the laboratory has been adopted by many researchers (Rapp and Melville, 1990). In total four specified frequency spectra were investigated in the experiments and two of them were considered here, which were corresponding to the period ranges or frequency bands outlined in Table 1. In each of these cases, the underlying wave components were equally spaced within the given period range, and their amplitudes were equal to each other. Consequently, the corresponding power spectrum of each test ω −4 case decays in terms of . In addition, a series of input amplitudes were considered to investigate the effects of nonlinearity, including A I = 22 mm, 38 mm and 55 mm.
The present paper particularly focuses on the cases with an input amplitude of A I = 55 mm (hereafter referred to as Cases A55 and B55). These wave events are highly nonlinear and near-breaking, which have been found to be within 10% of the wave breaking limit. Afterwards, we investigate the high-frequency wave loads and ringing response on a bottom-hinged vertical cylinder exposed to the extreme wave event of Case B55. The vertical axis of the cylinder is located at the focal position of incident focused wave groups. As outlined above, the cylinder is hinged at the bottom and supported at the top with springs in the horizontal plane. Main structural and mechanical parameters of the cylinder are outlined in Table 2.
In the present cases, the wave amplitude A and cylinder radius a are of the same order, thus the values of the Keulegan-Carpenter number KC and the Reynolds number Re are crucial parameters in the problem. They are defined as: where is the maximum horizontal particle velocity, T is the characteristic wave period, D is the cylinder diameter, and is the kinematic viscosity of water. Here is adopted. In this study, 1.4<KC<5.5 for the above wave conditions, and Re ~ 55000 for the largest waves. Sarpkaya (1986) shows that the viscous drag force on a circular cylinder is relatively small for these values of KC and Re. However, it may be necessary to assess the magnitude of viscous damping arising through the Morison drag term for obtaining accurate predictions of resonant ringing response. According to the experimental work of Sarpkaya (1977) and Chaplin et al. (1997), a drag coefficient of C D =1.0 is estimated and adopted for the cases discussed below.

Convergence study
The convergence analysis of these cases is firstly car-  (1) is sufficiently small so that there are enough wave components within the input range to describe most of the underlying wave spectrum; (2) N is large enough to ensure that no significant wave energy is lost due to the truncation errors of the series solution, but also it should not be excessively large since this may lead to the growth of spurious high-frequency contamination and numerical instabilities.
In the present cases, the focused wave groups with a truncated power spectrum decaying in terms of are considered. Preliminary convergence tests imply that it is sufficient to ensure that approximately equals (8-10)k p , where k p is the corresponding wavenumber of the spectral peak frequency. Figs. 3a and 3b depict the time series of water surface elevation at the focal position for Cases A55 and B55 respectively. Three different wavenumber intervals are chosen, corresponding to a fine, an intermediate and a coarse one, as illustrated in Figs. 3a and 3b. In both cases, the results obtained using the fine and intermediate wavenumber intervals are almost identical, indicating that the numerical results are convergent. On the basis of convergence analysis as indicated above, numerical calculations for each test case are undertaken using the model parameters outlined in Table 3 with an appropriate initial time . In addition, a relatively small time step should be adopted for both the accurate modeling of the evolution of a focused wave group and the solution of the cylinder motion by using the Newmark scheme. Several different time steps have been tested and ≤T p /100, where T p is the spectral peak period, is found to be sufficient for each of the cases discussed below.

Validation tests of focused wave
Figs. 4a and 4b provide the temporal and spatial profile of the wave elevation at focus in Case A55 respectively. In both of the figures, the numerical results are compared with the experimental data, as well as a linear solution and a second-order solution first identified by Longuet-Higgins and Stewart (1960). It is noted that as a large wave evolves, a downstream shifting of both the focal position and the focal time has been found in both the experiments and numerical simulations (see Baldock et al., 1996;Johannessen and Swan, 2003;Ning et al., 2009) due to the third-order or even higher-order nonlinear wave-wave interactions. In this study, the actual focal position and focal time of the numer-   ZHANG Yi, TENG Bin China Ocean Eng., 2020, Vol. 34, No. 4, P. 513-525 ical results have been shifted in order to facilitate the comparison with the linear and second-order solutions as well as the experimental data. The measured central wave crest is higher and narrower than the linear or second-order solution, while the adjacent wave troughs are broader and less deep. This suggests the importance of the high-order nonlinear interactions. Although the second-order solution accounts for the effects to some extent, the largest wave crest that predicted by this solution is still much smaller than the experimental data. In contrast, the numerical results seem to be in good agreement with the experimental data, no matter the temporal or the spatial description of the water surface profile. A similar series of data associated with Case B55 are presented in Figs. 5a and 5b. In this case a similar trend of the relation between the experimental data and theoretical solutions is found. It can be seen visibly that the measured largest wave crest is about 30% larger than the linear solution and 25% larger than the second-order solution. However, the present numerical results provide a much improved prediction of the water surface profile, which give a satisfactory agreement with the experimental data in both the temporal and spatial profiles.
Figs. 6a and 6b show the vertical distribution of the maximum horizontal velocities, u(z), arising beneath the largest wave crest of Cases A55 and B55 respectively. In each case, the numerical results are shown together with the linear solution, the second-order solution and the measured data. The comparisons imply that the linearly predicted velocities tend to overestimate the measured data at locations well beneath the water surface. The deviation is in relation to the frequency-difference terms, which can be reasonably well modeled by the second-order solution. Nevertheless, the measured velocity data close to the water surface show significant differences from both the linear and second-order solutions. This is particularly prominent in Fig. 6b where the measured near-surface velocities are 20% larger than the linearly predicted velocities. However, the present solution shows a quite good description of the horizontal velocity profiles arising throughout the flow field. As a result, the present numerical model demonstrates the ability and accuracy in terms of predicting the water particle kinematics.
To move the analysis forward, the proposed numerical model is used to reproduce the evolution of these experimental wave groups. The spatial evolution of power spectra for these two cases is first investigated to see if a redistribution of wave energy occurs during the spatial-temporal focusing process. In each case, the time series of wave elevation is recorded at several spatial locations, and then the fast Fourier transform (FFT) with a Hanning window is adopted to obtain the corresponding power spectrum. Normalized power spectra of five representative positions (i.e. the four upstream, and the focal position), as well as the power spectrum of linear input, are shown in Fig. 7 for comparison. In each curve, the amplitudes are normalized by dividing the  peak amplitude of the input spectrum. It is noticeable that the power spectrum in the vicinity of the focal position shows a reduction of energy within the input frequency range, and a redistribution of energy to the higher frequencies. The transfer of energy into the higher frequencies as the wave group approaches the focal position is clearly identified.
Furthermore, according to the results of the temporal and spatial profiles of water surface, a two-dimensional discrete Fourier transform (DFT) has been applied to investigate the distribution of wave amplitude in both the wavenumber and frequency domains. Therefore, the dispersive properties of focused wave groups can be identified. As an example, Fig. 8 shows the results of the two-dimensional DFT in terms of contour plots of wave amplitude for Case B55. In the figure, the linear dispersion relation is indicated by the dashed curve, except within the linear input range, where the curve becomes continuous. Figs. 8a and 8b show the results on the basis of the linear and second-order surface profiles, respectively. In Fig. 8a, the results are exactly as expected that the contour plots are nearly distributed around the linear dispersion relation and show no significant distributions beyond the linear input range. In Fig. 8b, in addition to the linear wave components, the results also contain the second-order sum and difference components. Further, Fig. 8c shows the results based on the numerically predicted surface profiles. It is seen that in addition to both the second-order sum and difference components, the third-order sum wave components even appear. Furthermore, Fig. 8c also highlights noticeable changes to the linear free-wave components. In particular, there is a growth of a spread of wave energy that satisfies or nearly satisfies the linear dispersion relation, which is present far beyond the initial input range. This indicates a locally widened free-wave regime, which might be due to the third-order resonant interactions that can cause rapid and significant transfer of energy in a short duration, as mentioned by Gibson and Swan (2007). It may lead to much higher and steeper wave profiles than the expected results from the predictions of linear or second-order theory.

Ringing of a vertical cylinder in focused waves
In this section, ringing of a bottom-hinged vertical cylinder is investigated with incoming focused wave groups. As stated above, the incoming wave conditions are the same as Case B55 and the main parameters of the cylinder are outlined in Table 2. Some typical ringing responses are shown below in the form of the pitch angle of the cylinder and horizontal acceleration at the still water level. Then the effects of nonlinear high-frequency wave loads, natural frequency and damping ratio on the ringing response of the structure are systematically investigated.

Effects of nonlinear wave loads
Three different hydrodynamic models are adopted to Fig. 7. Spatial evolution of normalized power spectra for Cases A55 and B55.  Ocean Eng., 2020, Vol. 34, No. 4, P. 513-525 521 K= 1.49 × 10 3 N · m/rad ω c /ω p = 4.0 ω c ω p predict the wave loads and pitch response. A time-domain linear diffraction model based on the boundary element method is also added for comparison except for the two hydrodynamic models outlined in Section 2.3. Here, through the adjustment of spring stiffness ( ), the frequency ratio is set as , where is the natural frequency in pitch motion of the cylinder and is the peak frequency of the incoming focused waves. ω c /ω p = 4.0 ω 2 Fig. 9 presents the time history of pitch moment on the cylinder. It is seen that the results based on the FNV theory and Rainey's model have higher peak values compared with the linear solution. In addition, the results based on the FNV theory and Rainey's model contain more high-frequency components compared with the linear solution, some of which arise from the added mass force due to the resonant response of the cylinder. The pitch response of the cylinder with is shown in Fig. 10. In both the results from the FNV theory and Rainey's model, the bottomhinged vertical cylinder is excited at its natural frequency yielding large pitch responses, and the high-frequency resonant response appears to be prominent. It appears clearly that no resonance is produced if the nonlinear terms are not taken into account in the hydrodynamic models. Fig. 11 shows the corresponding horizontal acceleration of the cylinder at the still water level. As expected, the resonant response is further amplified in the acceleration due to effect on accelerations. The high level of horizontal acceleration (i.e. 0.15g) of the cylinder is induced in this condition.
In general, the results from these two load models have a good agreement, although the results based on the FNV theory appear to be slightly larger. As stated above, the FNV theory predicts a larger point force term, and it leads to larger higher-harmonic load components. As mentioned by Kristiansen and Faltinsen (2017), the FNV theory tends to overestimate the third-harmonic load component in their regular wave tests with a bottom-mounted vertical cylinder, and the discrepancy between the theory and experiments increases monotonically with increasing wave steepness. The discrepancy was demonstrated to be due to flow separation effects, and clearly related to KC number. In the following calculations, Rainey's model is adopted for further investigations.

Comparisons with a quasi-static model
Assuming small displacement and velocity, the ringing response of the vertical cylinder can be estimated by a simplified analysis tool that is the so called quasi-static model. The quasi-static model is based on the wave loads calculated in the stiffly supported condition. All effects of feedback from the cylinder's motion to the loading other than added mass have been neglected, such as those arising through the Morison drag term and other additional slenderbody terms. The results including the feedback from the cylinder's motion to the loading, as shown in Eq. (25), are hereafter referred to as the coupled dynamic results. Fig. 12 presents the comparison between the quasi-static and coupled dynamic results of pitch response. Similarly, Fig. 13 shows the comparison of the horizontal acceleration at the still water level. In general, the difference between the quasi-static and coupled dynamic results is quite small, implying that it makes possible an approximate response analysis for a cylinder of different dynamic properties based on the same loading record on the fixed cylinder. Nevertheless, as shown in Figs. 12a and 13a, in terms of the time history of pitch response and horizontal acceleration, the coupled dynamic results present a faster decline after the peak compared with the quasi-static results. In addition, with regard to the corresponding amplitude spectra of pitch response and horizontal acceleration, the amplitudes of the coupled dynamic results at the resonant frequency are slightly smaller than those of the quasi-static results, as shown in Figs. 12b and 13b. It may be because the quasi-static model neglects the hydrodynamic damping arising through the cylin-  der's motion.

Effects of natural frequency and damping ratio of the structure
In this section, a numerical sensitivity study about the effects of natural frequency and damping ratio on the ringing response of the structure is implemented. Firstly, numerical calculations are carried out with the adjustment of spring stiffness to achieve natural frequencies of the cylinder (in still water) from 2.55 to 16.99 Hz (or frequency ratios from 3 to 20). Fig. 14 shows pitch responses of the cylinder for different frequency ratios varying from 3 to 20 with no structural damping. Pitch responses are normalized by the maximum static pitch angle defined by: It is seen that dynamic effects of the pitch response are prominent at low frequency ratios, especially in the range of 3-5. The pitch response approaches the static response as the frequency ratio is larger than 10.
ζ ω c /ω p = 4.0 It has been presented by Davies et al. (1994) that ringing responses are insensitive to the damping level. To investigate that, four different damping ratios ( = 0, 0.5%, 2.0%, 5.0%) are applied in this study. The small damping ratio implies a lightly damped vibration system relevant to monopile support structures for offshore wind turbines, as suggested by Kallehave et al. (2015). Normalized pitch responses for different damping ratios are shown in Fig. 15 with . It shows that the magnitude of damping ratio has slight influence on the peak value of pitch responses. However, the large damping ratio leads to a faster decay of the resonant response after the peak. This indicates that the damping ratio has prominent effects on the resonant response of the cylinder in pitch motion studied in this paper.   ZHANG Yi, TENG Bin China Ocean Eng., 2020, Vol. 34, No. 4, P. 513-525 523 β DA β DA A more systematic investigation of the dynamic response behavior is conducted by comparing the dynamic amplification factor of pitch response for various frequency ratios and damping ratios. The dynamic amplification factor is defined as the largest absolute value of the normalized pitch response. Fig. 16 shows the results of the sensitivity study on the relationship between the dynamic properties and the resulting dynamic amplification factor. It is seen that dynamic effects of the pitch response are significant at low frequency ratios, where is much larger than 1.0, and it decreases and approaches 1.0 with the increase of frequency ratios. In addition, the magnitude of damping ratio plays a certain role in the numerical results particularly in the condition of low frequency ratios.

Conclusions
In this paper, we propose an efficient and simple method to compute the higher-order wave loads and induced ringing responses of a monopile exposed to moderately steep transient wave events. The method combines a Fourier time-stepping model to obtain fully nonlinear incident wave kinematics with existing calculation methods of higher-order wave loads (FNV theory and Rainey's model). A series of numerical simulations of ringing response have been implemented on a bottom-hinged vertical cylinder in focused waves, over frequency ratios (natural frequency of the structure to peak frequency of wave spectra) between 3 and 20.
The numerical predictions of incident focused waves appear to be in almost perfect agreement with the experimental data in respect of both the wave elevations and the underlying water particle kinematics. The local evolution of power spectra and the transfer of energy into higher frequencies are clearly identified during the spatial-temporal focusing process of extreme wave groups.
Resonant ringing response excited by the nonlinear high-frequency wave loads has been found when frequency ratios are equal to 3-5, which are shown in the form of the pitch angle of the cylinder and horizontal acceleration at the still water level. Comparisons with a quasi-static model are also given. Dynamic amplification factor of ringing response has also been investigated for different dynamic properties of the structure, including damping ratio and natural frequency. It is noted that dynamic effects of the ringing response are significant at low frequency ratios where the dynamic amplification factor is much larger than 1.0, and it decreases and approaches 1.0 with the increase of frequency ratios. In addition, the magnitude of damping ratio plays a certain role in the numerical results particularly in the condition of low frequency ratios.
The proposed method is straightforward to implement, which can be applied to quickly providing high-frequency wave loads and ringing response on the monopile-supported offshore wind turbines exposed to moderately steep transient water waves. Thus, it offers an efficient and simple analysis model for practical design. For further research, a new nonlinear potential flow model based on the weak-scatterer approximation is under consideration to better deal with the nonlinear wave-structure interaction problems.