Hurricane Wave Loads on Spar-Type Floating Wind Turbines: A Comparison of Simulation Schemes

Abstract

Floating wind turbines are sensitive to hurricane events. Since the turbine rotors are parked and the blades are feathered during hurricanes, the aerodynamic loads due to boundary-layer winds are relatively small compared to the hydrodynamic loads due to sea surface elevations. Hence, accurate modeling of the hurricane wave loads is crucial to ensure the safety of floating wind turbines. During a hurricane, large wave heights with severe flow separation make it inaccurate to use either linear panel method-based models (without nonlinear consideration associated with fluid viscosity) or Morison equation-based models (without unsteady consideration associated with fluid memory). Efforts have been made to advance simulation schemes of hurricane wave loads on spar-type floating wind turbines. This study systematically compares and assesses the efficacy of six hydrodynamic models available in the literature along with a newly proposed model. The ability of these seven hydrodynamic models to capture nonlinear and/or unsteady effects is investigated. As a demonstration example, the wave loads on a spar-type wind turbine are calculated using these seven models to highlight the underlying role of each simulation scheme in accurately acquiring the dynamic responses of this type of offshore floating structure in severe hurricane seas. It is found that the nonlinear viscous term in the Morison equation and hybrid model serves as an important nonlinear damping mechanism. The reduction of the low-frequency wave load and added mass in the modified hybrid model collectively leads to larger displacements compared to those based on the hybrid model. While the displacements based on the stretching method and Rainey’s equation are similarly larger than those based on the Morison equation, their nonlinear wave loads are much smaller than those in FNV theory.

1. Introduction

There are growing interests worldwide in floating wind turbines due to their ability to exploit the abundant offshore wind resource in deep water [1,2]. Extensive deployment of this novel structure in practice requires the accurate dynamic analysis of floating wind turbines to evaluate their survivability under extreme weather conditions. Hurricanes are considered one of the most catastrophic events that threaten the safety of offshore floating wind turbines. Most of the existing research focuses on hurricane-induced effects on fixed-bottom offshore wind turbines [3] while only a few studies address the dynamic response of floating wind turbines in high winds and large waves [4]. The rotors of the wind turbines are parked and the pitch angle of the blades is set to be 90 degrees during hurricanes [3]. Hence, the aerodynamic loads are relatively small compared to hydrodynamic loads due to the small drag coefficients of turbine blades. As a result, the accurate quantification of hydrodynamic loads is the key to evaluating the risk to offshore floating wind turbines during hurricane events. Efforts have been made to advance the hydrodynamic load models of floating wind turbines. For example, Azcona et al. [5] utilized a hybrid simulation in wave tank-scaled experiments to investigate the low-frequency dynamics of a floating wind turbine. Robertson et al. [6] systematically validated a set of simulation tools for low-frequency hydrodynamic loads using data from wave tank tests. In addition to wave tank tests, computational fluid dynamics (CFD) simulations have been used to characterize the nonlinear wave load and validate existing hydrodynamic models [7,8]. Most of the existing studies focus on the normal operation conditions of floating wind turbines, while the extreme hurricane wave conditions have not been systemically investigated yet. It is noted that the risk analysis of floating wind turbines under extreme waves usually involves Monte Carlo simulations, which make the use of time-consuming numerical simulations (e.g., CFD) very challenging and hence require computationally efficient wave load models.

Justified by the argument that the deformation of a small spherical fluid element can be divided into pure straining motion and rigid-body rotation, the hydrodynamic loads on floating structures are usually classified into irrotational flow-induced force and vortex flow-induced force [9]. While in practice the complicated vortex flow-induced force is often simply modeled by the semi-empirical nonlinear viscous drag term in the Morison equation, the irrotational flow-induced force can be predicted more rigorously based on potential theory. To obtain the wave loads on offshore floating structures, most of the existing research uses the linear potential theory-based model, semi-empirical Morison equation, or a hybrid combination of these two models [10]. In large-amplitude hurricane-generated waves, the linear panel method for hydrodynamic loads assuming no flow separation is inaccurate. Although the semi-empirical Morison equation [11] based on long-wave assumption could take the nonlinear viscous force into account, the “unsteadiness” in the wave load is ignored due to the “quasi-steady” nature of the Morison equation (assuming that each strip of the structure is immersed in a uniform flow field due to the large ratio of wavelength to the structural dimension). The so-called hybrid model simply adds the viscous drag term in Morison equation to the linear panel method-based formula, and hence includes both viscous force and unsteady wave loads [1,12]. However, the wave-excited force and radiation force in the hybrid model is still calculated via the linear panel method, assuming no flow separation. Hence, a modified hybrid model is proposed in this study to better capture the effects of flow separation on the extreme wave loads. Specifically, the low-frequency wave-excited force as well as the added mass in the radiation force is modified using the empirical mass coefficient C_M from the Morison equation for improved consideration of flow separation effects, while the high-frequency wave-excited load is still based on the linear panel method. It is noted that all these four models calculate the wave load based on the mean water elevations. Hence, three advanced models are introduced in this comparison study to account for the nonlinearities from an instantaneous wetted surface, namely the Wheeler stretching-based model that simply stretches the linear wave kinematics based on mean water surface up to the instantaneous wetted surface [13], the FNV theory that adds nonlinear load terms based on perturbation analysis [14] and Rainey’s equation that adds nonlinear terms based on energy conservation [15].

To better inform the practice of modeling the hurricane wave loads for offshore floating wind turbines, the abovementioned seven models are systematically compared in a consistent manner, where the differences and connections between these models are highlighted to show their features when modeling the extreme wave loads (are shown Figure 1). The ability of these models to capture nonlinear and/or unsteady effects is summarized in

Table 1. Consideration of nonlinear and unsteady effects in various models.

Model Type	Consideration of Unsteady Wave Load	Consideration of Nonlinear Wave Load
		Nonlinearity from Viscous Drag	Nonlinearity from Instantaneous Wetted Surface
Linear panel method-based model	✓
Morison equation-based model		✓
Hybrid model	✓	✓
Proposed modified hybrid model	✓	✓
Wheeler stretching-based model		✓	✓
FNV model		✓	✓
Rainey’s model		✓	✓

. Although both the hybrid and modified hybrid models can consider unsteady wave load and viscous drag, the computation of certain terms in these two models is different and will be discussed in Section 3. In the following sections, the structural dynamics for floating wind turbines are first briefly introduced, which are followed by the detailed discussion of seven hydrodynamic load models. Finally, the wave loads on a spar-type wind turbine are calculated with these hydrodynamic models as a demonstration example to highlight the underlying role of each simulation scheme in accurately capturing the dynamic response of this type of floating structures in severe hurricane seas.

2. Structural Dynamics of Floating Wind Turbines

The motion of offshore floating wind turbines subjected to aerodynamic loads F_aero and hydrodynamic loads F_hydro is governed by the following equation:

$$M_{\mathit{s}}\ddot{\mathit{x}}\left(t\right)+{\mathit{C}}_{\mathit{s}}\dot{\mathit{x}}\left(t\right)+{\mathit{K}}_{\mathit{s}}\mathit{x}\left(t\right)={\mathit{F}}_{\mathit{a}\mathit{e}\mathit{r}\mathit{o}}\left(t\right)+{\mathit{F}}_{\mathit{h}\mathit{y}\mathit{d}\mathit{r}\mathit{o}}\left(t\right)$$

(1)

where x represents the degrees of freedom of structural motion; M_s, C_s and K_s are the mass matrix, damping matrix and stiffness matrix of the structure, respectively. The aerodynamic loads on a parked wind turbine are relatively small, and hence not discussed here for the sake of simplicity. Also, the sea current that could introduce complicated wave-current–structure interactions is not considered. The coordinate system is defined in Figure 2, where the origin is located at the center of the turbine on a still water surface. In addition, only the wave loads and structural motions in surge and pitch mode subjected to unidirectional linear incident waves propagating along the x axis are considered in this study.

3. Model Description

3.1. Linear Potential Theory-Based Models

In the potential theory that assumes the fluid is inviscid and irrotational, the velocity potential ϕ could be used to describe the fluid velocity field V:

$$\mathit{V}=\nabla \varphi =\mathit{i}\frac{\partial \varphi }{\partial x}+\mathit{j}\frac{\partial \varphi }{\partial y}+\mathit{k}\frac{\partial \varphi }{\partial z}$$

(2)

where i, j and k are unit vectors along the x, y and z axis, respectively. The velocity potential ϕ satisfies the Laplace equation:

$$\frac{{𝝏}^{\mathbf{2}}\mathit{\varphi }}{𝝏{\mathit{x}}^{\mathbf{2}}}+\frac{{𝝏}^{\mathbf{2}}\mathit{\varphi }}{𝝏{\mathit{y}}^{\mathbf{2}}}+\frac{{𝝏}^{\mathbf{2}}\mathit{\varphi }}{𝝏{\mathit{z}}^{\mathbf{2}}}=\mathbf{0}$$

(3)

The pressure field p could be found via the Bernoulli equation:

$$\mathit{p}+\mathit{\rho }\mathit{g}\mathit{z}+\mathit{\rho }\frac{𝝏\mathit{\varphi }}{𝝏\mathit{t}}+\frac{\mathit{\rho }}{\mathbf{2}}\mathit{V}\mathit{·}\mathit{V}={\mathit{p}}_{\mathit{a}}$$

(4)

where ρ is the water velocity; g is the gravitational acceleration and p_a is the atmospheric pressure. Like a typical fluid–structure interaction problem, the differences among each specific case essentially arise from different boundary conditions. The most general case is to consider the wave load on a structure moving at the velocity U in incident waves, where the undisturbed surface elevation of incident waves is ${\xi }_I\left(x,y,t\right)$. Due to the presence of the moving structure, the disturbed surface elevation is $\xi \left(x,y,t\right)$. Boundary conditions need to be satisfied on both the structure surface and the disturbed free surface. For the boundary conditions of structure surface, the fluid velocity on the structure surface should be equal to the velocity of the structure to ensure no fluid enters or leaves the structure surface, which gives the kinematic boundary condition on the instantaneous wetted surface:

$$\frac{\partial \varphi }{\partial n}=\mathit{U}·\mathit{n} \mathrm{on\; instantaneous\; wetted\; surface}$$

(5)

where $\frac{\partial }{\partial n}$ denotes the differential along the normal direction to the body surface and n is the unit vector normal to the structure surface (positive toward the fluid domain). For the boundary conditions of a free surface, fluid particles on the disturbed free surface should always stay on the disturbed free surface ξ (x, y, t), which gives the kinematic boundary condition on the disturbed free surface:

$$\frac{\partial \xi }{\partial t}+\frac{\partial \varphi }{\partial x}\frac{\partial \xi }{\partial x}+\frac{\partial \varphi }{\partial y}\frac{\partial \xi }{\partial y}-\frac{\partial \varphi }{\partial z}=0 \mathrm{on} z=\xi \left(x,y,t\right)$$

(6)

In addition, the pressure on the disturbed free surface should be equal to the atmospheric pressure, which gives the dynamic boundary condition on the disturbed free surface:

$$g\xi +\frac{\partial \varphi }{\partial t}+\frac{1}{2}\left[{\left(\frac{\partial \varphi }{\partial x}\right)}^2+{\left(\frac{\partial \varphi }{\partial y}\right)}^2+{\left(\frac{\partial \varphi }{\partial z}\right)}^2\right]=0 \mathrm{on} z=\xi \left(x,y,t\right)$$

(7)

After obtaining the disturbed potential ϕ and utilizing Bernoulli’s equation, the hydrodynamic load on the structure could then be calculated by integrating the pressure over the instantaneous wetted surface (IWS):

$${\mathit{F}}_{\mathit{h}\mathit{y}\mathit{d}\mathit{r}\mathit{o}}\left(t\right)=-\int p\mathit{n}\mathrm{d}{S}_{IWS} \mathrm{for\; force}$$

(8a)

$${\mathit{F}}_{\mathit{h}\mathit{y}\mathit{d}\mathit{r}\mathit{o}}\left(t\right)=-\int p\left(\mathit{r}\times \mathit{n}\right)\mathrm{d}{S}_{IWS} \mathrm{for\; moment}$$

(8b)

where r is vector pointing from the origin to the point of calculation. One challenge of calculating the disturbed velocity potential is that the nonlinear boundary conditions are satisfied on the disturbed free surface, which is unknown before solving the problem. Hence, simplification of the complicated boundary condition is needed to obtain the disturbed velocity potential.

Assuming a small amplitude of the incident wave and structural motion, the disturbed surface elevation ξ can be considered to be small. Via Taylor expansion and only keeping the linear terms, the kinematic and dynamic boundary condition at z = ξ (x, y, t) could be moved to the still water level at z = 0:

$$\frac{\partial \xi }{\partial t}=\frac{\partial \varphi }{\partial z} \mathrm{on} z=0$$

(9)

$$g\xi +\frac{\partial \varphi }{\partial t}=0 \mathrm{on} z=0$$

(10)

The kinematic boundary condition is satisfied on the mean wetted surface (MWS):

$$\frac{\partial \varphi }{\partial n}=\mathit{U}·\mathit{n} \mathrm{on\; the\; mean\; wetted\; surface}$$

(11)

The pressure could be found based on the linearized Bernoulli equation:

$$p+\rho gz+\rho \frac{\partial \varphi }{\partial t}=p_a$$

(12)

Due to linearity, the disturbed potential $\varphi$ can then be separated into

$$\varphi ={\varphi }_{wa}-{\varphi }_{rad}$$

(13)

where ${\varphi }_{wa}$ is the disturbed velocity potential with the structure restrained from oscillating in the incident wave field; ${\varphi }_{rad}$ is the radiation potential with the structure forced to oscillate in still water. Hence, the hydrodynamic load can be divided into three sub-components:

$${\mathit{F}}_{\mathit{h}\mathit{y}\mathit{d}\mathit{r}\mathit{o}}\left(t\right)={\mathit{F}}_{\mathit{s}\mathit{t}\mathit{a}}\left(t\right)+{\mathit{F}}_{\mathit{w}\mathit{a}}\left(t\right)-{\mathit{F}}_{\mathit{r}\mathit{a}\mathit{d}}\left(t\right)$$

(14)

with

$${\mathit{F}}_{\mathit{s}\mathit{t}\mathit{a}}\left(t\right)=-\int \left(p_a-\rho gz\right)\mathit{n}\mathrm{d}{S}_{MWS} \mathrm{for\; force}$$

(15a)

$${\mathit{F}}_{\mathit{s}\mathit{t}\mathit{a}}\left(t\right)=-\int \left(p_a-\rho gz\right)\left(\mathit{r}\times \mathit{n}\right)\mathrm{d}{S}_{MWS} \mathrm{for\; moment}$$

(15b)

$${\mathit{F}}_{\mathit{w}\mathit{a}}\left(t\right)=\int \rho \frac{\partial {\varphi }_{wa}}{\partial t}\mathit{n}\mathrm{d}{S}_{MWS} \mathrm{for\; force}$$

(16a)

$${\mathit{F}}_{\mathit{w}\mathit{a}}\left(t\right)=\int \rho \frac{\partial {\varphi }_{wa}}{\partial t}\left(\mathit{r}\times \mathit{n}\right)\mathrm{d}{S}_{MWS} \mathrm{for\; moment}$$

(16b)

$${\mathit{F}}_{\mathit{r}\mathit{a}\mathit{d}}\left(t\right)=\int \rho \frac{\partial {\varphi }_{rad}}{\partial t}\mathit{n}\mathrm{d}{S}_{MWS} \mathrm{for\; force}$$

(17a)

$${\mathit{F}}_{\mathit{r}\mathit{a}\mathit{d}}\left(t\right)=\int \rho \frac{\partial {\varphi }_{rad}}{\partial t}\left(\mathit{r}\times \mathit{n}\right)\mathrm{d}{S}_{MWS} \mathrm{for\; moment}$$

(17b)

where the F_sta is the hydrostatic load resulting from the pressure component $p_a-\rho gz$ in the Bernoulli equation; the wave-excited force F_wa and motion-induced force (radiation force) F_rad result from ${\varphi }_{wa}$ and ${\varphi }_{rad}$, respectively. The negative sign in Equations (12) and (13) is due to the convention that the radiation force impedes structural motion. In the linear analysis with the assumption of a small amplitude of the incident wave and structural motion, the hydrodynamic loads are obtained by integrating the pressure over the mean wetted surface (MWS). Since F_sta is very easy to consider, it will not be discussed in the rest of this study. Essentially, F_rad is obtained by solving the boundary value problem for the radiation potential ${\varphi }_{rad}$, where the fluid velocity on the structure surface is equal to the velocity of the structural motion. The disturbed potential ${\varphi }_{wa}$ and associated wave-excited force F_wa can be further divided into

$${\varphi }_{wa}={\varphi }_I+{\varphi }_{dif}$$

(18)

$${\mathit{F}}_{\mathit{w}\mathit{a}}\left(t\right)={\mathit{F}}_{\mathit{F}\mathit{K}}\left(t\right)+{\mathit{F}}_{\mathit{d}\mathit{i}\mathit{f}}\left(t\right)$$

(19)

where F_FK is the Froude–Kriloff force resulting from the unsteady pressure of an undisturbed incident wave on the structure surface as if the structure is not there, which is relative easy to consider given the undisturbed incident wave potential ${\varphi }_I$; the diffraction force F_dif is due to the change of the pressure field by presence of the structure. The F_dif is obtained by solving the boundary value problem for the diffraction potential ${\varphi }_{dif}$, where the velocity on the structure surface has to be in opposite direction and of the same magnitude as the undisturbed fluid velocity to ensure no fluid particle enters or leaves the structure surface.

Based on linear theory, the hydrodynamic loads of irregular waves can be obtained via linear superposition of the load components subjected to regular waves. The surface elevation of unidirectional incident irregular waves ${\xi }_I\left(x,t\right)$ could be calculated as the linear superposition of the different frequency components ${\xi }_i\left(x,t\right)$ of the wave spectrum S (f):

$${\xi }_I\left(x,t\right)={\sum }_{i=1}^n{\xi }_i\left(x,t\right)={\sum }_{i=1}^n{A}_i\sin \left({\omega }_{i}t-k_{i}x+{\phi }_i\right)$$

(20)

where $A_i=\sqrt{2S\left(f_j\right)\Delta f}$ and the random phase ${\phi }_i$ is uniformly distributed from zero to 2π. The wave number satisfies the dispersion relation $k_i=\frac{{\omega }_i{}^2}{g}$. The horizontal and vertical velocity field of each regular wave component $u_i\left(x,z,t\right)$ and $w_i\left(x,z,t\right)$ in deep water are given by

$$u_i\left(x,z,t\right)={\omega }_i{A}_i{e}^{k_{i}z}\sin \left({\omega }_{i}t-k_{i}x+{\phi }_i\right)$$

(21a)

$$w_i\left(x,z,t\right)={\omega }_i{A}_i{e}^{k_{i}z}\cos \left({\omega }_{i}t-k_{i}x+{\phi }_i\right)$$

(21b)

The dynamic pressure field is given by

$$p_i\left(x,z,t\right)=\rho g{A}_i{e}^{k_{i}z}\sin \left({\omega }_{i}t-k_{i}x+{\phi }_i\right)$$

(22)

It should be noted that only the linear incident wave is considered as the input for all the hydrodynamic models discussed in this study.

3.1.1. Linear Panel Method-Based Model

The hydrodynamic loads on structures of general shapes subjected to waves of an arbitrary wavelength could be calculated numerically using the linear panel method [16]. By distributing sources and sinks over the structure surface, the linear panel method numerically solves the diffraction and radiation potential for regular waves with the linearized boundary conditions. The solution of the diffraction problem is usually expressed in the frequency domain in terms of the complex-valued wave excitation transfer function matrix X_wa(jω) (with both Froude–Kriloff force and diffraction force included):

$${{\mathit{F}}_{\mathit{w}\mathit{a}}^{\mathit{l}\mathit{p}}}_{\mathit{F}}\left(\mathrm{j}\omega \right)={\mathit{X}}_{\mathit{w}\mathit{a}}\left(\mathrm{j}\omega \right){\xi }_I{}_F\left(0,\mathrm{j}\omega \right)$$

(23)

where ${{\mathit{F}}_{\mathit{w}\mathit{a}}^{\mathit{l}\mathit{p}}}_{\mathit{F}}\left(\mathrm{j}\omega \right)$ represents the Fourier transform of the wave-excited force ${\mathit{F}}_{\mathit{w}\mathit{a}}^{\mathit{l}\mathit{p}}\left(t\right)$ and ${\xi }_I{}_F\left(0,\mathrm{j}\omega \right)$ denotes the Fourier transform of surface elevation at the center of the structure ${\xi }_I\left(0,t\right)$. The time-domain expression is expressed as [1]

$${\mathit{F}}_{\mathit{w}\mathit{a}}^{\mathit{l}\mathit{p}}\left(t\right)={\int }_{-\infty }^{\infty }\mathit{K}\left(t-\tau \right){\xi }_I\left(0,\tau \right)\mathrm{d}\tau$$

(24)

$$\mathit{K}\left(t\right)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{\mathit{X}}_{\mathit{w}\mathit{a}}\left(\mathrm{j}\omega \right)e^{\mathrm{j}\omega t}\mathrm{d}\omega$$

(25)

where K(t) is the linear wave excitation impulse response function. The solution of the radiation problem is usually expressed in the frequency domain in terms of added mass A(ω) (in phase with structural acceleration) and added damping B(ω) (in phase with structural velocity) [17]:

$${{\mathit{F}}_{\mathit{r}\mathit{a}\mathit{d}}^{\mathit{l}\mathit{p}}}_{\mathit{F}}\left(\mathrm{j}\omega \right)={\mathit{X}}_{\mathit{r}\mathit{a}\mathit{d}}\left(\mathrm{j}\omega \right){\dot{\mathit{x}}}_F\left(\mathrm{j}\omega \right)=\left[\mathit{B}\left(\omega \right)+\mathrm{i}\omega \mathit{A}\left(\omega \right)\right]{\dot{\mathit{x}}}_F\left(\mathrm{j}\omega \right)$$

(26)

where ${{\mathit{F}}_{\mathit{r}\mathit{a}\mathit{d}}^{\mathit{l}\mathit{p}}}_{\mathit{F}}\left(\mathrm{j}\omega \right)$ and ${\dot{\mathit{x}}}_F\left(\mathrm{j}\omega \right)$ represent the Fourier transform of the wave-excited force ${\mathit{F}}_{\mathit{r}\mathit{a}\mathit{d}}^{\mathit{l}\mathit{p}}\left(t\right)$ and the oscillating velocity of the structure $\dot{\mathit{x}}\left(t\right)$, respectively; ${\mathit{X}}_{\mathit{r}\mathit{a}\mathit{d}}\left(\mathrm{j}\omega \right)$ denotes the radiation force transfer function. The corresponding time-domain expression is usually based on the Cummins equation [18]:

$${\mathit{F}}_{\mathit{r}\mathit{a}\mathit{d}}^{\mathit{l}\mathit{p}}\left(t\right)=A_{\mathit{i}\mathit{n}\mathit{f}}\ddot{\mathit{x}}\left(t\right)+{\int }_0^t\mathit{H}\left(t-\tau \right)\dot{\mathit{x}}\left(\tau \right)\mathrm{d}\tau$$

(27)

The matrix of the motion impulse response function H(t) and the infinite-frequency added-mass matrix A_inf are given by the Ogilvie relations [19]:

$$A_{\mathit{i}\mathit{n}\mathit{f}}=\underset{\omega \to \infty }{\lim }\mathit{A}\left(\omega \right)=\mathit{A}(\infty )$$

(28a)

$$\mathit{H}\left(t\right)=\frac{2}{\pi }{\int }_0^{\infty }\mathit{B}\left(\omega \right)\cos \left(\omega t\right)\mathrm{d}\omega \mathrm{or} \mathit{H}\left(t\right)=-\frac{2}{\pi }{\int }_0^{\infty }\omega \left[\mathit{A}\left(\omega \right)-\mathit{A}(\infty )\right]\sin \left(\omega t\right)\mathrm{d}\omega$$

(28b)

The relation between $\mathit{A}\left(\omega \right)$ and $\mathit{B}\left(\omega \right)$ exists implicitly in the Ogilvie relations. It should be noted that viscous effects are not considered in the panel method-based model assuming no flow separation.

3.1.2. Morison Equation-Based Model

Great simplification can be made for the diffraction and radiation problems of slender structures subjected to incident waves of long wavelength. Consider a strip dz of a vertical circular cylinder subjected to a regular incident wave ${\xi }_i\left(x,t\right)$, where the velocity and pressure fields of the incident wave are given in Equations (21) and (22), the Froude–Kriloff force can be obtained by

$$d{\mathit{F}}_{\mathit{F}\mathit{K}}=-p_i\mathit{n}\mathrm{d}S=-\mathit{n}\rho g{A}_i{e}^{k_{i}z}\sin \left({\omega }_{i}t-k_{i}x+{\phi }_i\right)\mathrm{d}S$$

(29)

Since only the incident wave propagating along the x axis is considered here, the Froude–Kriloff force can be found in the cylindrical coordinate:

$$\mathrm{d}{F}_{FK}=-\rho g{A}_i{e}^{k_{i}z}\mathrm{d}z\frac{D}{2}{\int }_0^{2\pi }\sin \left({\omega }_{i}t-k_i\frac{D}{2}\cos \theta +{\phi }_i\right)\cos \theta \mathrm{d}\theta$$

(30)

If the wavelength is much larger than the diameter of the cylinder D (i.e., $k_{i}D\ll 1$), further simplification can be made as follows [20]:

$$\mathrm{d}{F}_{FK}\approx \rho \pi \frac{D^2}{4}{\omega }_i{}^2{A}_i{e}^{k_{i}z}\cos \left({\omega }_{i}t+{\phi }_i\right)\mathrm{d}z=\rho \pi \frac{D^2}{4}\frac{\partial u_i}{\partial t}{|}_{x=0}\mathrm{d}z$$

(31)

where the Froude–Kriloff force is directly related to the undisturbed fluid particle acceleration at the center of the vertical cylinder $\frac{\partial u_i}{\partial t}{|}_{x=0}$. For the diffraction problem, the velocity at the surface of the strip can be approximated as the fluid velocity at the center of the strip $u_i{|}_{x=0}$ due to the long wavelength assumption. Accordingly, the diffraction force can be found in a method similar to the radiation problem by calculating the motion-induced force of the strip oscillating at the velocity $-u_i{|}_{x=0}$ (or at the acceleration $-\frac{\partial u_i}{\partial t}{|}_{x=0}$). Assuming the strip dz is immersed in unbounded two-dimensional fluid, the diffraction force is in-phase with the oscillating acceleration based on the potential theory

$$\mathrm{d}{F}_{dif}=C_A\rho \pi \frac{D^2}{4}\frac{\partial u_i}{\partial t}{|}_{x=0}\mathrm{d}z$$

(32)

where $C_A$ is the added-mass coefficients. The value of $C_A$ is the unity for a circular cross section in non-separated potential flow, indicating that the diffraction force has the same contribution as the Froude–Kriloff force. The total wave-excited force on the strip dz is then expressed as

$$\mathrm{d}{F}_{wa}=\mathrm{d}{F}_{FK}+\mathrm{d}{F}_{dif}=(C_A+1)\rho \pi \frac{D^2}{4}\frac{\partial u_i}{\partial t}{|}_{x=0}\mathrm{d}z=C_M\rho \pi \frac{D^2}{4}\frac{\partial u_i}{\partial t}{|}_{x=0}\mathrm{d}z$$

(33)

where $C_M=(C_A+1)$ is the mass coefficient, with a value of two for the circular cross section. The radiation force is simply given as

$$\mathrm{d}{F}_{rad}=\rho C_A\frac{\pi D^2}{4}\frac{{\partial }^2\eta }{\partial t^2}\mathrm{d}z=\rho \left(C_{\mathrm{M}}-1\right)\frac{\pi D^2}{4}\frac{{\partial }^2\eta }{\partial t^2}\mathrm{d}z$$

(34)

where η is the horizontal motion of the strip dz. Compared with general expressions based on the panel method, there is no force component resulting from the added damping in Equations (33) and (34). Also, the vertical cylinder is divided into infinite strips in the derivation shown above. Each strip is assumed to be immersed in two-dimensional uniform flow, and the wave load of each strip is calculated independently without considering the free surface. For the three-dimensional surface-piercing structure, the waves on the free surface generated by structural motion provide the physical origin of the added damping and associated fluid-memory effect. However, the force due to added damping is small for slender structures with a low ability to generate waves on the free surface. This justification results in the inertia force term (irrotational potential flow-induced force) in the Morison equation [11].

The semi-empirical Morison equation is widely used to determine the in-line hydrodynamic loads on slender structures in long waves when the viscous effect due to flow separation becomes important. The wave-excited force ${\mathit{F}}_{\mathit{w}\mathit{a}}^{\mathit{m}}$ and radiation force ${\mathit{F}}_{\mathit{r}\mathit{a}\mathit{d}}^{\mathit{m}}$ of a moving cylinder based on the Morison equation are given by

$${\mathit{F}}_{\mathit{w}\mathit{a}}^{\mathit{m}}\left(t\right)={\mathit{F}}^{\mathit{i}\mathit{n}\mathit{e}\mathit{r}}\left(t\right)+{\mathit{F}}^{\mathit{v}\mathit{i}\mathit{s}}\left(t\right)=\left[\begin{array}{c}{\int }_{-d}^0\rho \frac{\pi D^2}{4}{C}_{\mathrm{M}}\frac{\partial u}{\partial t}\mathrm{d}z\\ {\int }_{-d}^{0}z\rho \frac{\pi D^2}{4}{C}_{\mathrm{M}}\frac{\partial u}{\partial t}\mathrm{d}z\end{array}\right]+\left[\begin{array}{c}{\int }_{-d}^0\frac{\rho }{2}{C}_{\mathrm{D}}D\left|u-\frac{\partial \eta }{\partial t}\right|(u-\frac{\partial \eta }{\partial t})\mathrm{d}z\\ {\int }_{-d}^{0}z\frac{\rho }{2}{C}_{\mathrm{D}}D\left|u-\frac{\partial \eta }{\partial t}\right|(u-\frac{\partial \eta }{\partial t})\mathrm{d}z\end{array}\right]$$

(35)

$${\mathit{F}}_{\mathit{r}\mathit{a}\mathit{d}}^{\mathit{m}}\left(t\right)=\left[\begin{array}{c}{\int }_{-d}^0\rho \left(C_{\mathrm{M}}-1\right)\frac{\pi D^2}{4}\frac{{\partial }^2\eta }{\partial t^2}\mathrm{d}z\\ {\int }_{-d}^{0}z\rho \left(C_{\mathrm{M}}-1\right)\frac{\pi D^2}{4}\frac{{\partial }^2\eta }{\partial t^2}\mathrm{d}z\end{array}\right]$$

(36)

where d is the draft of the vertical cylinder; ρ is water density; D is the cylinder diameter; and C_M and C_D are the mass and drag coefficients depending on the Reynolds (Re) number and Keulegan–Carpenter (KC) number [21]. All the fluid velocities are evaluated at the center of the cylinder. Although C_M and C_D are usually obtained by the test results of regular wave loads, the Morison equation is often used without full justification for calculating the wave load in irregular waves. The integration limit is from the cylinder bottom z = −d to the mean water level z = 0 to be consistent with the linear theory. The nonlinear viscous drag load F^vis that takes into account the relative motion of the incident waves and structure is considered here as part of the wave-excited force assuming a small structural motion. For the rest of the models in this study, the terms including relative motions are regarded as wave-excited force, while the terms that only include structural motion are regarded as radiation force. The horizontal motion of the strip η can be related to the global structural motion through $\eta =x_1+z{x}_2$ where x₁ and x₂ are the surge and pitch motion, respectively. Accordingly, Equation (36) can be written as

$${\mathit{F}}_{\mathit{r}\mathit{a}\mathit{d}}^{\mathit{m}}\left(t\right)=\left[\begin{array}{cc}{\int }_{-d}^0\rho \left(C_{\mathrm{M}}-1\right)\frac{\pi D^2}{4}\mathrm{d}z& {\int }_{-d}^0\rho \left(C_{\mathrm{M}}-1\right)\frac{\pi D^2}{4}z\mathrm{d}z\\ {\int }_{-d}^0\rho \left(C_{\mathrm{M}}-1\right)\frac{\pi D^2}{4}z\mathrm{d}z& {\int }_{-d}^0\rho \left(C_{\mathrm{M}}-1\right)\frac{\pi D^2}{4}{z}^2\mathrm{d}z\end{array}\right]\left[\begin{array}{c}{\ddot{x}}_1\\ {\ddot{x}}_2\end{array}\right]=A_{\mathit{m}}\ddot{\mathit{x}}\left(t\right)$$

(37)

where $A_{\mathit{m}}$ is the added-mass matrix based on the Morison equation. Since the structure oscillating at an infinite frequency does not generate waves at the free surface, the radiation force ${\mathit{F}}_{\mathit{r}\mathit{a}\mathit{d}}^{\mathit{m}}$ (t) in the Morison equation with C_M = 2 is equal to the term $A_{\mathit{i}\mathit{n}\mathit{f}}\ddot{\mathit{x}}\left(t\right)$ in Equation (27). As a result, these two models predict similar radiation force if the radiation damping (convolution term in Equation (27)) is negligible. For long waves, it is clear that the inertia term F^iner(t) with C_M = 2 in Morison’s equation is the asymptotic solution of the wave-excited force ${\mathit{F}}_{\mathit{w}\mathit{a}}^{\mathit{l}\mathit{p}}\left(t\right)$ based on the linear panel method. For high-frequency waves, the Morison equation loses its accuracy due to the violation of the assumption of long-wavelength waves.

3.1.3. Hybrid Model

The high Re and KC numbers of a typical spar used for floating wind turbines in severe hurricane-generated sea states (as shown in Figure 3) indicate the occurrence of flow separation [22]. A so-called hybrid model has been proposed to take into account both the linear inertia load and nonlinear viscous effects [1,12]. In the hybrid model, the linear panel method-based wave-excited force is modified by adding the viscous term of Morison’s equation:

$${\mathit{F}}_{\mathit{w}\mathit{a}}^{\mathit{h}}\left(t\right)={\int }_{-\infty }^{\infty }\mathit{K}\left(t-\tau \right){\xi }_I\left(0,\tau \right)\mathrm{d}\tau +{\mathit{F}}^{\mathit{v}\mathit{i}\mathit{s}}\left(t\right)$$

(38)

The radiation force is still based on the formula of the panel method assuming no flow separation:

$${\mathit{F}}_{\mathit{r}\mathit{a}\mathit{d}}^{\mathit{h}}\left(t\right)=A_{\mathit{i}\mathit{n}\mathit{f}}\ddot{\mathit{x}}\left(t\right)+{\int }_0^t\mathit{H}\left(t-\tau \right)\dot{\mathit{x}}\left(\tau \right)\mathrm{d}\tau$$

(39)

While the viscous force resulting from the flow separation is included in the hybrid model, the effects of flow separation on other fluid–structure system properties such as K(t), H(t) and A_inf are not considered.

3.1.4. Proposed Modified Hybrid Model

Figure 4 presents the C_M and C_D of a cylindrical structure at various KC and Re numbers, indicating that there is a significant reduction in C_M at high KC and Re numbers [21]. The reduction of C_M (from a value of two) suggests that the term ${\int }_{-\infty }^{\infty }\mathit{K}\left(t-\tau \right){\xi }_I\left(0,\tau \right)d\tau$ in the hybrid model may overestimate the wave-excited force for low-frequency waves. It also indicates that the term $A_{\mathit{i}\mathit{n}\mathit{f}}\ddot{\mathit{x}}\left(t\right)$ in the hybrid model may overestimate the radiation force because it fails to consider the possible reduction in added mass implied by the reduction of C_M. To address these issues, a modified hybrid model is proposed here. For the wave-excited force ${\mathit{F}}_{\mathit{w}\mathit{a}}^{\mathit{m}\mathit{h}}\left(t\right)$, the incident irregular waves are divided into low-frequency and high-frequency components. For the low-frequency waves, the inertia term in the Morison equation is utilized with the reduced mass coefficient $C_M^r$. The high-frequency components are calculated by the linear panel method-based formula. The viscous drag is treated in the same way as the hybrid model. Accordingly, the wave-excited force in the modified hybrid model could be expressed as

$${\mathit{F}}_{\mathit{w}\mathit{a}}^{\mathit{m}\mathit{h}}\left(t\right)={\mathit{F}}^{\mathit{i}\mathit{n}\mathit{e}\mathit{r}\mathit{\_}\mathit{l}\mathit{o}\mathit{w}}\left(t\right)+{\int }_{-\infty }^{\infty }\mathit{K}\left(t-\tau \right){\xi }_I^h\left(0,\tau \right)d\tau +{\mathit{F}}^{\mathit{v}\mathit{i}\mathit{s}}\left(t\right)$$

(40)

$${\mathit{F}}^{\mathit{i}\mathit{n}\mathit{e}\mathit{r}\mathit{\_}\mathit{l}\mathit{o}\mathit{w}}\left(t\right)=\left[\begin{array}{c}{\int }_{-d}^0\rho \frac{\pi D^2}{4}{C}_M^r\frac{\partial u_l}{\partial t}\mathrm{d}z\\ {\int }_{-d}^{0}z\rho \frac{\pi D^2}{4}{C}_M^r\frac{\partial u_l}{\partial t}\mathrm{d}z\end{array}\right]$$

(41)

where ${\xi }_I^h$ is the surface elevation of high-frequency waves; u_l is the undisturbed fluid particle velocity of low-frequency waves. The determination of the cutting frequency will be discussed later in the numerical example. The formula of the radiation force ${\mathit{F}}_{\mathit{w}\mathit{a}}^{\mathit{m}\mathit{h}}$ (t) is similar to the hybrid model except that the reduced added-mass matrix $A_{\mathit{m}}^{\mathit{r}}$ calculated from the reduced mass coefficient $C_M^r$ is utilized:

$${\mathit{F}}_{\mathit{r}\mathit{a}\mathit{d}}^{\mathit{m}\mathit{h}}\left(t\right)=A_{\mathit{m}}^{\mathit{r}}\ddot{\mathit{x}}\left(t\right)+{\int }_0^t\mathit{H}\left(t-\tau \right)\dot{\mathit{x}}\left(\tau \right)\mathrm{d}\tau$$

(42)

It is noted that C_D and C_M depend on both KC and Re numbers, and their determination requires full-scale experiments or simulations. Due to a lack of carefully validated data, estimated values of C_D (arbitrarily selected as 0.7) and C_M (arbitrarily selected as 1.7) will be used in this study.

3.2. Nonlinear Potential Theory-Based Models

The irrotational flow-induced loads in the abovementioned four models are based on linear potential theory assuming a small incident wave and structural motion. When calculating the disturbed potential in linear theory, the free surface boundary conditions are satisfied on the still water level (Equations (9) and (10)) and body surface boundary conditions are satisfied on the mean wetted surface (Equation (11)). The wave load is obtained by integrating the fluid pressure found from the linearized Bernoulli equation (Equation (12)) over the mean wetted body surface (Equations (15)–(17)). Hence, the obtained hydrodynamic load is linearly proportional to the amplitude of the incident wave and structural motion. Based on linear theory, the incident waves only generate the load within the frequency range of the wave spectrum and hence most of the floating structures are designed so that the natural frequencies do not lie in frequency range of the wave spectrum. For large-amplitude waves during hurricanes, the nonlinear wave load whose frequency is beyond the frequency range of the wave spectrum may lead to a large structural response due to resonance.

The nonlinear potential theory can better consider the boundary condition of zero-normal flow through the instantaneous body surface and more accurately capture the pressure and fluid particle velocity at the instantaneous disturbed free surface. For structures of arbitrary shapes, the nonlinear wave loads can be calculated numerically via the higher-order panel method [23,24,25], where the hydrodynamic load includes higher-order terms with respect to wave amplitude. Currently, most of the existing commercial software using the panel method usually calculates nonlinear wave load up to the second order of wave amplitude by solving the second-order boundary value problems derived from perturbation analysis. The nonlinear wave load ${\mathit{F}}_{\mathit{w}\mathit{a}}^{\mathit{n}\mathit{p}}$ is obtained by adding the second-order term to linear solution:

$${\mathit{F}}_{\mathit{w}\mathit{a}}^{\mathit{n}\mathit{p}}\left(t\right)={\int }_{-\infty }^{\infty }\mathit{K}\left(t-\tau \right){\xi }_I\left(0,\tau \right)\mathrm{d}\tau +{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{\mathit{K}}_{\mathbf{2}}\left(t-{\tau }_1,t-{\tau }_2\right){\xi }_I\left(0,{\tau }_1\right){\xi }_I\left(0,{\tau }_2\right)\mathrm{d}{\tau }_1\mathrm{d}{\tau }_2$$

(43)

$${\mathit{K}}_{\mathbf{2}}\left(t_1, t_2\right)=\frac{1}{{\left(2\pi \right)}^2}{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{\mathit{X}}_{\mathbf{2}\mathit{w}\mathit{a}}\left({\omega }_1,{\omega }_2\right)e^{i{\omega }_1{t}_1+i{\omega }_2{t}_2}\mathrm{d}{\omega }_1\mathrm{d}{\omega }_2$$

(44)

where ${\mathit{K}}_{\mathbf{2}}\left(t_1, t_2\right)$ is the quadratic wave excitation impulse response function and ${\mathit{X}}_{\mathbf{2}\mathit{w}\mathit{a}}\left({\omega }_1,{\omega }_2\right)$ is the quadratic wave excitation transfer function given by the second-order panel method [26]. The radiation force in the nonlinear model is expressed in the same way as the linear model, which only includes terms concerning structural motion:

$${\mathit{F}}_{\mathit{r}\mathit{a}\mathit{d}}^{\mathit{n}\mathit{p}}\left(t\right)=A_{\mathit{i}\mathit{n}\mathit{f}}\ddot{\mathit{x}}\left(t\right)+{\int }_0^t\mathit{H}\left(t-\tau \right)\dot{\mathit{x}}\left(\tau \right)\mathrm{d}\tau$$

(45)

With the perturbation expansion of the quantities of the disturbed fluid field and structural motion, the second-order wave load includes terms resulting from first-order structural motion [27]. Hence, the structural dynamic properties are included in the solution of ${\mathit{X}}_{\mathbf{2}\mathit{w}\mathit{a}}\left({\omega }_1,{\omega }_2\right)$. As a result, the accuracy of the second-order wave load depends on the mass and stiffness matrices of the floating structure, which is used as an input for the second-order panel method [12]. However, the dynamics of the fully coupled aero-servo-hydro-elastic floating wind turbines cannot be accurately modeled by rigid-body mass and stiffness matrices alone. To consider the fully coupled fluid–structure interaction of the floating wind turbine, the disturbed velocity potential should be calculated based on the instantaneous position of the structure in the time domain [28]. It is an extremely complicated task along with the identification of the quadratic wave excitation transfer function. For a slender structure such as the spar, in long waves significant simplifications can be made to derive explicit formulas for a nonlinear wave load.

3.2.1. Wheeler Stretching-Based Model

In the Morison equation, the hydrodynamic load is obtained by integrating the distributed loads of the strips over the mean wetted body. The most straightforward way to extend the Morison equation to consider the nonlinear wave load resulting from the unsteady wetted surface is to integrate the distributed load over the instantaneous wetted surface, which requires wave kinematics up to the instantaneous wave surface. It is known that the linear wave theory only predicts the wave kinematics up to the still water level. The direct extension of the linear wave theory to obtain the kinematics beyond the surface level will result in an unrealistically large value due to the exponential growth of the fluid velocity above the mean water level. Several approaches have been used to predict the wave kinematics up to the instantaneous free surface [29], including Wheeler’s stretching method [13]. Accordingly, the vertical coordinates are stretched from the original level z to a modified level z′ in the evaluation of wave kinematics:

$$z^{\prime }=\frac{z-{\xi }_I}{1+{\xi }_I/h}$$

(46)

where h is the water depth. For deep water, $z^{\prime }=z-{\xi }_I$ simply moves the origin of the original coordinate to the surface intersection of the incident wave (as shown in Figure 5).

The Morison equation-based model modified by Wheeler stretching can be expressed as

$${\mathit{F}}_{\mathit{w}\mathit{a}}^{\mathit{m}\mathit{w}}\left(t\right)={\mathit{F}}_{\mathit{w}}^{\mathit{i}\mathit{n}\mathit{e}\mathit{r}}\left(t\right)+{\mathit{F}}_{\mathit{w}}^{\mathit{v}\mathit{i}\mathit{s}}\left(t\right)=\left[\begin{array}{c}{\int }_{-d}^{{\xi }_I}\rho \frac{\pi D^2}{4}{C}_{\mathrm{M}}\frac{\partial u}{\partial t}\mathrm{d}z\\ {\int }_{-d}^{{\xi }_I}z\rho \frac{\pi D^2}{4}{C}_{\mathrm{M}}\frac{\partial u}{\partial t}\mathrm{d}z\end{array}\right]+\left[\begin{array}{c}{\int }_{-d}^{{\xi }_I}\frac{\rho }{2}{C}_{\mathrm{D}}D\left|u-\frac{\partial \eta }{\partial t}\right|(u-\frac{\partial \eta }{\partial t})\mathrm{d}z\\ {\int }_{-d}^{{\xi }_I}z\frac{\rho }{2}{C}_{\mathrm{D}}D\left|u-\frac{\partial \eta }{\partial t}\right|(u-\frac{\partial \eta }{\partial t})\mathrm{d}z\end{array}\right]$$

(47)

$${\mathit{F}}_{\mathit{r}\mathit{a}\mathit{d}}^{\mathit{m}\mathit{w}}\left(t\right)=\left[\begin{array}{c}{\int }_{-d}^{{\xi }_I}\rho \left(C_{\mathrm{M}}-1\right)\frac{\pi D^2}{4}\frac{{\partial }^2\eta }{\partial t^2}\mathrm{d}z\\ {\int }_{-d}^{{\xi }_I}z\rho \left(C_{\mathrm{M}}-1\right)\frac{\pi D^2}{4}\frac{{\partial }^2\eta }{\partial t^2}\mathrm{d}z\end{array}\right]$$

(48)

where the fluid velocity u is evaluated at the stretched coordinate z′. It is clear that the stretching method can only consider the nonlinearity resulting from the instantaneous wetted surface. The straightforward extension of the Morison equation with Wheeler’s stretching method implies the absolute value of the load per unit length is largest at the free surface, which is unrealistic because the pressure at the free surface is constant and the load per unit length has to be zero at the free surface [20]. A more accurate method to consider the nonlinear load due to the unsteady free surface is to directly integrate the pressure over the instantaneous wetted surface.

3.2.2. FNV Model

Faltinsen, Newman and Vinje (FNV) derived a formula for the nonlinear load of regular waves on a fixed infinitely long vertical circular cylinder in a system where the wave amplitude A and the cylinder radius r are of the same order, and both are small compared to the wavelength λ [14]. Hence, it is suitable to be applied to the case of the spar-type floating wind turbine under extreme waves. The leading-order nonlinear potential in FNV theory includes terms proportional to A²r and A³, and the nonlinear wave load includes the terms proportional to A²r² and A³r which are of comparable magnitude. The FNV theory improves the conventional perturbation analysis in which the wave load proportional to A²r² is included and terms proportional to A³r are ignored because A is assumed to be small compared to both λ and r. Later, Newman [30] extended the original FNV theory to an irregular wave load by assuming the wavelength of each wave component is large compared to the cylinder radius. For the convenience of comparison, the nonlinear wave load based on the FNV theory is expressed as

$${\mathit{F}}_{\mathit{w}\mathit{a}}^{\mathit{F}\mathit{N}\mathit{V}}\left(t\right)={\mathit{F}}_{\mathit{l}\mathit{,}\mathit{l}}^{\mathit{d}}\left(t\right)+{\mathit{F}}_{\mathit{l},\mathit{l}}^{\mathit{p}}\left(t\right)+{\mathit{F}}_{\mathit{n},\mathit{l}}^{\mathit{d}}\left(t\right)+{\mathit{F}}_{\mathit{n},\mathit{l}}^{\mathit{p}}\left(t\right)+{\mathit{F}}_{\mathit{n},\mathit{n}}^{\mathit{d}}\left(t\right)+{\mathit{F}}_{\mathit{n},\mathit{n}}^{\mathit{p}}\left(t\right)$$

(49)

with

$$\begin{array}{l}{\mathit{F}}_{\mathit{l},\mathit{l}}^{\mathit{d}}\left(t\right)=\left[\begin{array}{c}{\int }_{-d}^0\rho \frac{\pi D^2}{2}\frac{\partial u}{\partial t}\mathrm{d}z\\ {\int }_{-d}^{0}z\rho \frac{\pi D^2}{2}\frac{\partial u}{\partial t}\mathrm{d}z\end{array}\right] {\mathit{F}}_{\mathit{l},\mathit{l}}^{\mathit{p}}\left(t\right)=\left[\begin{array}{c}{\int }_0^{\xi }\rho \frac{\pi D^2}{2}\frac{\partial u}{\partial t}\mathrm{d}z\\ {\int }_0^{\xi }z\rho \frac{\pi D^2}{2}\frac{\partial u}{\partial t}\mathrm{d}z\end{array}\right]\\ {\mathit{F}}_{\mathit{n},\mathit{l}}^d\left(t\right)=\left[\begin{array}{c}{\int }_{-d}^0\rho \frac{\pi D^2}{4}\left(2w\frac{\partial w}{\partial x}+u\frac{\partial u}{\partial x}\right)\mathrm{d}z\\ {\int }_{-d}^{0}z\rho \frac{\pi D^2}{4}\left(2w\frac{\partial w}{\partial x}+u\frac{\partial u}{\partial x}\right)\mathrm{d}z\end{array}\right] {\mathit{F}}_{\mathit{n}\mathit{,}\mathit{l}}^{\mathit{p}}\left(t\right)=\left[\begin{array}{c}{\int }_0^{\xi }\rho \frac{\pi D^2}{4}\left(2w\frac{\partial w}{\partial x}+u\frac{\partial u}{\partial x}\right)\mathrm{d}z\\ {\int }_0^{\xi }z\rho \frac{\pi D^2}{4}\left(2w\frac{\partial w}{\partial x}+u\frac{\partial u}{\partial x}\right)\mathrm{d}z\end{array}\right]\\ {\mathit{F}}_{\mathit{n}\mathit{,}\mathit{n}}^{\mathit{d}}\left(t\right)=\left[\begin{array}{c}{\int }_{-d}^0\rho \frac{\pi D}{2g}{u}_0{}^2\frac{\partial u_0}{\partial t}\left(3{\psi }_1+4{\psi }_2\right)\mathrm{d}z\\ {\int }_{-d}^{0}z\rho \frac{\pi D}{2g}{u}_0{}^2\frac{\partial u_0}{\partial t}\left(3{\psi }_1+4{\psi }_2\right)\mathrm{d}z\end{array}\right] {\mathit{F}}_{\mathit{n}\mathit{,}\mathit{n}}^{\mathit{p}}\left(t\right)=\left[\begin{array}{c}{\int }_0^{\xi }\rho \frac{\pi D}{2g}{u}_0{}^2\frac{\partial u_0}{\partial t}\left(3{\psi }_1+4{\psi }_2\right)\mathrm{d}z\\ {\int }_0^{\xi }z\rho \frac{\pi D}{2g}{u}_0{}^2\frac{\partial u_0}{\partial t}\left(3{\psi }_1+4{\psi }_2\right)\mathrm{d}z\end{array}\right]\end{array}$$

where ${\mathit{F}}_{\mathit{l}\mathit{,}\mathit{l}}^{\mathit{d}}$ is the distributed linear load resulting from the linear potential acting on the mean wetted surface of the cylinder (which is the same as the inertia term in the Morison equation with C_M = 2); ${\mathit{F}}_{\mathit{l}\mathit{,}\mathit{l}}^{\mathit{p}}$ is the linear load resulting from the linear potential acting on the unsteady wetted surface (which could be considered as a point load acting near the free surface); ${\mathit{F}}_{\mathit{n}\mathit{,}\mathit{l}}^{\mathit{d}}$ and ${\mathit{F}}_{\mathit{n}\mathit{,}\mathit{l}}^{\mathit{p}}$ are the nonlinear distributed and point loads resulting from quadratic term of the linear potential; ${\mathit{F}}_{\mathit{n}\mathit{,}\mathit{n}}^{\mathit{d}}$ and ${\mathit{F}}_{\mathit{n}\mathit{,}\mathit{n}}^{\mathit{p}}$ are the nonlinear distributed and point loads resulting from the nonlinear potential; ${\psi }_1$ and ${\psi }_2$ are dimensionless functions related to the vertical distribution of the nonlinear potential [14]; and u₀ and w₀ denote the undisturbed vertical horizontal and vertical velocity at the mean surface level z = 0. It is noted that ${\mathit{F}}_{\mathit{l}\mathit{,}\mathit{l}}^{\mathit{p}}$ here is not strictly a linear load because the integration limit involves the wave amplitude. The upper limit of integration is the disturbed surface elevation $\xi$ which is implicitly defined by the free surface boundary condition and can be written in terms of the linear incident wave amplitude ${\xi }_I$ through the perturbation expansion. The unknown wave kinematics beyond the mean water level are also obtained via the Taylor expansion from z = 0. The utilization of the Taylor expansion is similar to stretching in the sense that they both deal with wave kinematics near the unsteady free surface. It can be shown that the integrand $\rho \frac{\pi D}{2g}{u}_0{}^2\frac{\partial u_0}{\partial t}\left(3{\psi }_1+4{\psi }_2\right)$ attenuates rapidly as depth increases, hence ${\mathit{F}}_{\mathit{n},\mathit{n}}^{\mathit{d}}$ combined with ${\mathit{F}}_{\mathit{n}\mathit{,}\mathit{n}}^{\mathit{p}}$ can also be considered as a point force acting locally on the free surface. Further calculation of the nonlinear terms up to third-order accuracy with respect of the wave amplitude A can be found in [30]:

$$\begin{array}{l}{\mathit{F}}_{\mathit{l}\mathit{,}\mathit{l}}^{\mathit{p}}\left(t\right)+{\mathit{F}}_{\mathit{n}\mathit{,}\mathit{l}}^{\mathit{p}}\left(t\right)\\ =\left[\begin{array}{c}\rho \frac{\pi D^2}{4}\left[2\frac{\partial u_0}{\partial t}{\xi }_I+\frac{{\partial }^2{u}_0}{\partial t\partial z}{\xi }_I{}^2+2w_0\frac{\partial w_0}{\partial x}{\xi }_I+u_0\frac{\partial u_0}{\partial x}{\xi }_I-\frac{2}{g}\frac{\partial u_0}{\partial t}\frac{\partial w_0}{\partial t}{\xi }_I-\frac{1}{g}\frac{\partial u_0}{\partial t}\left(u_0{}^2+w_0{}^2\right)\right]\\ 0\end{array}\right]\end{array}$$

(50)

$${\mathit{F}}_{\mathit{n}\mathit{,}\mathit{n}}^{\mathit{d}}\left(t\right)+{\mathit{F}}_{\mathit{n}\mathit{,}\mathit{n}}^{\mathit{p}}\left(t\right)=\left[\begin{array}{c}\frac{\rho \pi D^2}{g}{u}_0{}^2\frac{\partial u_0}{\partial t}\\ 0\end{array}\right]$$

(51)

To extend the FNV theory derived for fixed circular cylinders to a moving structure, the radiation force can be calculated in the same the way as the Morison equation:

$${\mathit{F}}_{\mathit{r}\mathit{a}\mathit{d}}^{\mathit{F}\mathit{N}\mathit{V}}\left(t\right)=\left[\begin{array}{c}{{\displaystyle \int }}_{-d}^0\rho \left(C_{\mathrm{M}}-1\right)\frac{\pi D^2}{4}\frac{{\partial }^2\eta }{\partial t^2}\mathrm{d}z\\ {{\displaystyle \int }}_{-d}^{0}z\rho \left(C_{\mathrm{M}}-1\right)\frac{\pi D^2}{4}\frac{{\partial }^2\eta }{\partial t^2}\mathrm{d}z\end{array}\right]$$

(52)

In addition, the viscous drag load as in the Morison equation can also be added to the FNV-based formula, which is included in the numerical example for effective comparison. It is noted that the effect of structural motion on the disturbed velocity potential is not considered in FNV theory.

3.2.3. Rainey’s Model

Rainey [15,31] also proposed a method to predict the nonlinear wave load for slender structures in long waves. Instead of using perturbation analysis and integrating the pressure over the wetted surface, the principle of energy conservation is utilized assuming that the position of the wave surface is unaffected by the presence of a structure. Rainey’s theory can be applied to moving structures with arbitrary cross sections. In addition, the incident wave field can be more general, indicating that even fully nonlinear wave kinematics can be utilized in Rainey’s equation. The load in Rainey’s original expression [15,31] is reformulated to present the load of the surge and pitch mode on a vertical circular cylinder. The total load is divided into the wave-excited load ${\mathit{F}}_{\mathit{w}\mathit{a}}^{\mathit{R}\mathit{a}\mathit{i}\mathit{n}\mathit{e}\mathit{y}}$ and motion-induced load ${\mathit{F}}_{\mathit{r}\mathit{a}\mathit{d}}^{\mathit{R}\mathit{a}\mathit{i}\mathit{n}\mathit{e}\mathit{y}}$ to be consistent with previous discussion:

$${\mathit{F}}_{\mathit{w}\mathit{a}}^{\mathit{R}\mathit{a}\mathit{i}\mathit{n}\mathit{e}\mathit{y}}\left(t\right)={\mathit{F}}_{\mathit{t}}^{\mathit{d}}\left(t\right)+{\mathit{F}}_{\mathit{t}}^{\mathit{p}}\left(t\right)+{\mathit{F}}_{\mathit{c}}^{\mathit{d}}\left(t\right)+{\mathit{F}}_{\mathit{c}}^{\mathit{p}}\left(t\right)+{\mathit{F}}_{\mathit{a}\mathit{d}\mathit{c}}^{\mathit{d}}\left(t\right)+{\mathit{F}}_{\mathit{a}\mathit{d}\mathit{c}}^{\mathit{p}}\left(t\right)+{\mathit{F}}_{\mathit{l}\mathit{e}}^{\mathit{p}}\left(t\right)+{\mathit{F}}_{\mathit{s}\mathit{i}}^{\mathit{p}}\left(t\right)$$

(53)

with

$$\begin{array}{l}{\mathit{F}}_{\mathit{t}}^{\mathit{d}}(t)=\left[\begin{array}{c}{\int }_{-d}^0\rho \frac{\pi D^2}{2}\frac{\partial u}{\partial t}\mathrm{d}z\\ {\int }_{-d}^{0}z\rho \frac{\pi D^2}{2}\frac{\partial u}{\partial t}\mathrm{d}z\end{array}\right] {\mathit{F}}_{\mathit{t}}^{\mathit{p}}(t)=\left[\begin{array}{c}{\int }_0^{{\xi }_I }\rho \frac{\pi D^2}{2}\frac{\partial u}{\partial t}\mathrm{d}z\\ {\int }_0^{{\xi }_I}z\rho \frac{\pi D^2}{2}\frac{\partial u}{\partial t}\mathrm{d}z\end{array}\right]\\ {\mathit{F}}_{\mathit{c}}^{\mathit{d}}(t)=\left[\begin{array}{c}{\int }_{-d}^0\rho \frac{\pi D^2}{2}(u\frac{\partial u}{\partial x}+w\frac{\partial u}{\partial z})\mathrm{d}z\\ {\int }_{-d}^{0}z\rho \frac{\pi D^2}{2}(u\frac{\partial u}{\partial x}+w\frac{\partial u}{\partial z})\mathrm{d}z\end{array}\right] {\mathit{F}}_{\mathit{c}}^{\mathit{p}}(t)=\left[\begin{array}{c}{\int }_0^{{\xi }_I }\rho \frac{\pi D^2}{2}(u\frac{\partial u}{\partial x}+w\frac{\partial u}{\partial z})\mathrm{d}z\\ {\int }_0^{{\xi }_I}z\rho \frac{\pi D^2}{2}(u\frac{\partial u}{\partial x}+w\frac{\partial u}{\partial z})\mathrm{d}z\end{array}\right]\\ {\mathit{F}}_{\mathit{a}\mathit{d}\mathit{c}}^{\mathit{d}}(t)=\left[\begin{array}{c}{\int }_{-d}^0\rho \frac{\pi D^2}{4}(u-\frac{\partial \eta }{\partial t})\frac{\partial w}{\partial z}\mathrm{d}z\\ {\int }_{-d}^{0}z\rho \frac{\pi D^2}{4}(u-\frac{\partial \eta }{\partial t})\frac{\partial w}{\partial z}\mathrm{d}z\end{array}\right] {\mathit{F}}_{\mathit{a}\mathit{d}\mathit{c}}^{\mathit{p}}(t)=\left[\begin{array}{c}{\int }_0^{{\xi }_I}\rho \frac{\pi D^2}{4}(u-\frac{\partial \eta }{\partial t})\frac{\partial w}{\partial z}\mathrm{d}z\\ {\int }_0^{{\xi }_I}z\rho \frac{\pi D^2}{4}(u-\frac{\partial \eta }{\partial t})\frac{\partial w}{\partial z}\mathrm{d}z\end{array}\right]\\ {\mathit{F}}_{\mathit{l}\mathit{e}}^{\mathit{p}}(t)=\left[\begin{array}{c}\rho \frac{\pi D^2}{4}{w}_{le}(u_{le}-\frac{\partial {\eta }_{le}}{\partial t})\\ -d\rho \frac{\pi D^2}{4}{w}_{le}(u_{le}-\frac{\partial {\eta }_{le}}{\partial t})\end{array}\right] {\mathit{F}}_{\mathit{s}\mathit{i}}^{\mathit{p}}(t)=\left[\begin{array}{c}-\rho \frac{\pi D^2}{8}\cot \alpha {(u_{si}-\frac{\partial {\eta }_{si}}{\partial t})}^2\\ -{\xi }_I\rho \frac{\pi D^2}{8}\cot \alpha {(u_{si}-\frac{\partial {\eta }_{si}}{\partial t})}^2\end{array}\right]\end{array}$$

where ${\mathit{F}}_{\mathit{t}}^{\mathit{d}}$ and ${\mathit{F}}_{\mathit{t}}^{\mathit{p}}$ are, respectively, the linear distributed and point loads proportional to the temporal derivative of the undisturbed velocity $\frac{\partial u}{\partial t}$ (similar to ${\mathit{F}}_{\mathit{l}\mathit{,}\mathit{l}}^{\mathit{d}}$ and ${\mathit{F}}_{\mathit{l}\mathit{,}\mathit{l}}^{\mathit{p}}$ in FNV theory except that the integration limit is the undisturbed incident wave surface ${\xi }_I$); ${\mathit{F}}_{\mathit{c}}^{\mathit{d}}$ and ${\mathit{F}}_{\mathit{c}}^{\mathit{p}}$ are the distributed and point loads proportional to the convective derivative of the undisturbed velocity $(u\frac{\partial u}{\partial x}+w\frac{\partial u}{\partial z})$ (zero for the linear incident wave velocity field in deep water as shown in Equation (21)); ${\mathit{F}}_{\mathit{a}\mathit{d}\mathit{c}}^{\mathit{d}}$ and ${\mathit{F}}_{\mathit{a}\mathit{d}\mathit{c}}^{\mathit{p}}$ are the axial divergence correction terms (equal to ${\mathit{F}}_{\mathit{n}\mathit{,}\mathit{l}}^{\mathit{d}}$ and ${\mathit{F}}_{\mathit{n}\mathit{,}\mathit{l}}^{\mathit{p}}$ in FNV theory except for the difference in integration limit considering $u\frac{\partial w}{\partial z}=2w\frac{\partial w}{\partial x}+u\frac{\partial u}{\partial x}$) [32]; ${\mathit{F}}_{\mathit{l}\mathit{e}}^{\mathit{p}}$ is the point force acting at the lower end resulting from the sudden change in added mass; ${\mathit{F}}_{\mathit{s}\mathit{i}}^{\mathit{p}}$ is point load at the instantaneous surface intersection of the structure and incident wave, resulting from the growing fluid kinetic energy due to the increase of the wetted length of the cylinder; subscript le and si in the velocity terms $u_{le}$, $w_{le}$, $\frac{\partial {\eta }_{le}}{\partial t}$, $u_{si}$ and $\frac{\partial {\eta }_{si}}{\partial t}$ indicate that they are evaluated at the lower end and surface intersection of the cylinder, respectively; and α is the angle between the incident wave surface and the axis of the moving cylinder. For a fixed vertical cylinder with infinitely deep draft in a linear incident wave field, Rainey’s equation predicts exactly the same load as the FNV theory up to the second order of A [14]. The difference in third-order loads between these two theories may result from the energy in the distorted wave surface due to the presence of the cylinder [33]. The radiation force predicted by Rainey’s model ${\mathit{F}}_{\mathit{r}\mathit{a}\mathit{d}}^{\mathit{R}\mathit{a}\mathit{i}\mathit{n}\mathit{e}\mathit{y}}$ is similar to the radiation force based on the Morison equation except for the integration limit:

$${\mathit{F}}_{\mathit{r}\mathit{a}\mathit{d}}^{\mathit{R}\mathit{a}\mathit{i}\mathit{n}\mathit{e}\mathit{y}}\left(t\right)=\left[\begin{array}{c}{\int }_{-d}^{{\xi }_I}\rho \frac{\pi D^2}{4}\frac{{\partial }^2\eta }{\partial t^2}\mathrm{d}z\\ {\int }_{-d}^{{\xi }_I}z\rho \frac{\pi D^2}{4}\frac{{\partial }^2\eta }{\partial t^2}\mathrm{d}z\end{array}\right]$$

(54)

The wave kinematics in Rainey’s model should be evaluated at the instantaneous position of the displaced structure in contrast to the mean position in the FNV theory, indicating that Rainey’s theory can account for large structural motion. Considering that Rainey’s model needs the wave kinematics up to the unsteady free surface as input, it also needs some form of stretching to calculate the wave kinematics (using Taylor expansion or the direct extension of the linear solution). Further simplification can be made in ${\mathit{F}}_{\mathit{s}\mathit{i}}^{\mathit{p}}$:

$${\mathit{F}}_{\mathit{s}\mathit{i}}^{\mathit{p}}\left(t\right)=\left[\begin{array}{c}-\rho \frac{\pi D^2}{8}\left(\frac{\partial {\xi }_I}{\partial x}+x_2\right){\left(u_0-\frac{\partial {\eta }_0}{\partial t}\right)}^2\\ -{\xi }_I\rho \frac{\pi D^2}{8}\left(\frac{\partial {\xi }_I}{\partial x}+x_2\right){\left(u_0-\frac{\partial {\eta }_0}{\partial t}\right)}^2\end{array}\right]$$

(55)

In addition, the viscous drag load as in the Morison equation can also be conveniently introduced to Rainey’s model, which is included in the numerical example for effective comparison.

4. Demonstration Example

4.1. Structural Properties of a Typical Spar-Type Floating Wind Turbine

The dynamic response of a typical spar-type floating wind turbine under extreme waves is calculated as a numerical example to highlight the differences and similarities between the abovementioned seven hydrodynamic models, as well as their unsteadiness and nonlinearity considerations [34]. It should be noted that these models cannot necessarily simulate the hydrodynamic loading from waves generated by nonstationary winds [35]. The design parameters of the floating wind turbine are given by [22]. This floating turbine utilizes a slender 120 m drafted ballasted spar with a tapering cross section (diameter from 9.4 m to 6.5 m) near the mean water level, and it is moored by a system of three catenary lines. Only the rigid-body motion in the surge and pitch mode of the floating wind turbine with linear mooring lines are considered in this study. The mass, damping and stiffness matrices M_s, C_s and K_s of the floating wind turbine are listed as follows [22]:

$$M_{\mathit{s}}=\left[\begin{array}{cc}8.07\times {10}^6& -6.30\times {10}^8\\ -6.30\times {10}^8& 6.78\times {10}^{10}\end{array}\right]$$

(56a)

$$C_{\mathit{s}}=\left[\begin{array}{cc}\mathrm{100,000}& 0\\ 0& 0\end{array}\right]$$

(56b)

$$\begin{array}{l}{K}_{\mathit{s}}=K_{\mathit{s}\mathit{t}\mathit{a}}+K_{\mathit{m}\mathit{o}\mathit{o}\mathit{r}}+K_{\mathit{g}\mathit{r}\mathit{a}}=\left[\begin{array}{cc}0& 0\\ 0& -5.00\times {10}^9\end{array}\right]+\left[\begin{array}{cc}4.12\times {10}^4& -2.82\times {10}^6\\ -2.82\times {10}^6& 3.11\times {10}^8\end{array}\right]+\\ \left[\begin{array}{cc}0& 0\\ 0& 6.18\times {10}^9\end{array}\right]=\left[\begin{array}{cc}4.12\times {10}^4& -2.82\times {10}^6\\ -2.82\times {10}^6& 1.48\times {10}^9\end{array}\right]\end{array}$$

(56c)

where all the variables are in SI units. The mass matrix $M_{\mathit{s}}$ includes the mass component of the spar, tower, nacelle, hub and three blades, while the stiffness matrix $K_{\mathit{s}}$ includes the hydrostatic stiffness $K_{\mathit{s}\mathit{t}\mathit{a}}$, mooring line stiffness $K_{\mathit{m}\mathit{o}\mathit{o}\mathit{r}}$ and gravitational stiffness $K_{\mathit{g}\mathit{r}\mathit{a}}$.

4.2. Result Comparison of Linear Potential Theory-Based Models

Considering that all linear potential theory-based hydrodynamic models discussed in this study utilize the nonlinear viscous drag term of the Morison equation for the consideration of vortex-induced force (if any), the differences arise essentially from the irrotational flow-induced force (inertia force). The surge-mode wave-excited force calculated by the linear panel method, the semi-empirical Morison equation with C_M = 2 and the proposed modified hybrid model are shown in the frequency domain in Figure 6. It is clear from Figure 6 that both the amplitude and phase angle of the wave excitation force predicted by the linear panel method and the Morison equation agree well for incident waves of a frequency up to 0.75 rad/s. Hence, the cutting frequency is chosen to be 0.75 rad/s.

To better compare the differences between the hybrid model and modified hybrid model, the low-frequency wave load in the modified hybrid model ${\mathit{F}}^{\mathit{i}\mathit{n}\mathit{e}\mathit{r}\mathit{\_}\mathit{l}\mathit{o}\mathit{w}}$ in this numerical example is calculated in the following form:

$${\mathit{F}}^{\mathit{i}\mathit{n}\mathit{e}\mathit{r}\mathit{\_}\mathit{l}\mathit{o}\mathit{w}}\left(t\right)=\frac{C_M^r}{C_{\mathrm{M}}}{\int }_{-\infty }^{\infty }\mathit{K}\left(t-\tau \right){\xi }_I^l\left(0,\tau \right)d\tau$$

(57)

where ${\xi }_I^l$ is the low-frequency surface elevation. Due to the lack of the experimental and numerical results for the coefficients in the Morison equation for the KC and Re number of this specific case, the reduced mass coefficient $C_M^r$ used in the proposed modified hybrid model is chosen to be 1.7 for all strips of the spar, which is shown by the reduction in the amplitude of the wave-excited force of low-frequency waves in Figure 6a. The added-mass matrix A_m (A_inf) and reduced added-mass matrix $A_{\mathit{m}}^{\mathit{r}}$ are

$$A_{\mathit{m}}=A_{\mathit{i}\mathit{n}\mathit{f}}=\left[\begin{array}{cc}8.26\times {10}^6& -5.13\times {10}^8\\ -5.13\times {10}^8& 4.12\times {10}^{10}\end{array}\right]$$

(58)

$$A_{\mathit{m}}^{\mathit{r}}=(C_M^r-1)/\left(C_M-1\right)A_{\mathit{m}}=\left[\begin{array}{cc}6.61\times {10}^6& -4.10\times {10}^8\\ -4.10\times {10}^8& 3.30\times {10}^{10}\end{array}\right]$$

(59)

The drag coefficient C_D is chosen to be 0.7 for all strips of the spar.

The generalized JONSWAP spectrum S(f) is used along with Equations (20)–(22) to generate the hurricane wave field [36]:

$$S\left(f\right)={\beta }_s{g}^2{\left(2\pi \right)}^{-4}{f}_p{}^{-\left(5+n\right)}{f}^n\exp \left[\frac{n}{4}{\left(\frac{f}{f_p}\right)}^{-4}\right]{\gamma }_e{}^{\exp \left[\frac{-{\left(f-f_p\right)}^2}{2{\sigma }^2{f}_p{}^2}\right]}$$

(60)

where β_s is the scale parameter; f_p is the spectral peak frequency; γ_e is the spectral peak enhancement factor; σ = 0.11 is the spectral width parameter. In this study, n = −4 is used as suggested by [36]. The scale parameter β_s is determined by satisfying the relationship $\int S\left(f\right)\mathrm{d}f={\left(H_s/4\right)}^2$ given the significant H_s. Considering the scattering measurement data, γ_e = 1 is used here, which is similar to Pierson–Moskowitz spectrum. The simulated hurricane wave condition is shown in Figure 7, including the wave spectrum of a severe sea state with H_s = 12 m and f_p = 0.1 Hz and a surface elevation based on the linear superposition of different frequency components of the given spectrum. The low- and high-frequency elevations with a cutting frequency of 0.75 rad/s (0.12 Hz) are shown in Figure 8.

The wave-excited forces based on the linear panel method, Morison equation, hybrid model and modified hybrid model are shown in Figure 9, and a zoomed-in version is presented in Figure 10. It is clear that the excitation force based on the linear panel method is close to that based on the hybrid model, which indicates that the nonlinear viscous force is relatively small compared to the inertia load. The reduction in the low-frequency load in the modified hybrid model is clearly shown in Figure 10.

The structural displacements based on the linear panel method, Morison equation, hybrid model and modified hybrid model (with original added mass $A_{\mathit{m}}$ and reduced added mass $A_{\mathit{m}}^{\mathit{r}}$) are shown in Figure 11, and a zoomed-in version is presented in Figure 12. The mean value of the root mean square (RMS) of displacements based on simulations of ten different wave fields is shown in Figure 13. The discrepancy between the displacements based on the linear panel method and the Morison equation may result from differences in the excitation force and damping force (nonlinear damping in the Morison equation and linear damping in the linear panel method). The smaller displacement of the hybrid model compared to that of the linear panel method indicates that the nonlinear viscous terms serve as nonlinear damping instead of excitation force. It is clear that the reduction of low-frequency wave loads in the modified hybrid model results in smaller displacements while the reduction in added mass leads to even larger displacements compared with the hybrid model.

4.3. Result Comparison of Nonlinear Potential Theory-Based Models

In the following discussion, results based on nonlinear potential theory-based models (i.e., Wheeler’s stretching, the FNV model and Rainey’s equation) are compared with the reference model of the Morison equation to highlight the contribution of the nonlinearities from the instantaneous wetted surface. It is noted that nonlinear viscous terms are included in all these models for a fair comparison with the Morison equation. The wave excitation force and structural displacement of these four hydrodynamic models are shown in Figure 14 and Figure 15, and the corresponding zoomed-in versions are presented in Figure 16 and Figure 17. The mean value of the RMS of displacements based on ten simulations is shown in Figure 18. While strong nonlinear contributions are observed in the FNV theory, the differences between the rest of three models are relatively small. Compared with that, based on the Morison equation, the displacement based on Wheeler stretching is larger for both surge and pitch modes due to the nonlinear load resulting from the unsteady wetted surface. Noting that the stretching method is utilized in Rainey’s equation in the prediction of the wave kinematics of the unsteady free surface, the small difference in the wave load and displacement of the stretching-based model and Rainey’s equation indicates the insignificant contributions of the unique terms in Rainey’s equation. The nonlinearities of wave loads based on the FNV theory are much larger than the rest of the models. The main reasons are: (1) the wave kinematics beyond mean water level are predicted by the Taylor expansion of the linear solution, which is much larger than that based on Wheeler stretching (Figure 5); (2) the contribution of the diffracted surface elevation is considered when integrating the wave load over the unsteady wetted surface; and (3) the FNV theory is derived for fixed structures and the possible reduction in the nonlinear load resulting from structural motion is not considered.

5. Concluding Remarks

To ensure the safety of offshore floating wind turbines under extreme hurricane waves, seven different hydrodynamic models are systematically compared to better inform the practice of accurately modeling the extreme wave loads. These seven models include the linear panel method-based model (with consideration of the unsteady wave load), the Morison equation-based model (with consideration of viscous drag), the hybrid model (with added viscous drag term to linear panel-based model), the modified hybrid model (with a modified low-frequency wave load and added mass in the hybrid model), Wheeler’s stretching-based model (with stretched wave kinematics up to the instantaneous wetted surface), the FNV model (with perturbation analysis-based higher-order load terms) and Rainey’s model (with energy conservation-based higher-order load terms). As a demonstration example, the dynamic response of a simplified spar-type floating wind turbine is calculated using these models and the results are compared to highlight the nonlinear and unsteady effects on the structural displacement of the floating platform. It is found that the nonlinear viscous term in the Morison equation and the hybrid model serves as nonlinear damping instead of excitation force. While the reduction in the low-frequency wave load alone in the modified hybrid model leads to a smaller dynamic response, the reduced added mass results in even larger displacements compared to that based on the hybrid model. The stretching method and Rainey’s equation predict close nonlinear wave loads. While the displacements based on these two models are larger than those based on the Morison equation, their nonlinear wave loads are much smaller than those in the FNV theory. Future validation efforts involving full-scale experiments or simulations need to be made due to the dependence of the mass/drag coefficients on both KC and Re numbers. Further investigations on the effects of different hydrodynamic models on tower and nacelle dynamics will be included in future work, using more a sophisticated structural dynamics model of the floating wind turbine.