Computational Study on Rogue Wave and Its Application to a Floating Body

Abstract

A rogue wave is a huge wave that is generated by wave energy focusing. Rogue waves can cause critical damage to ships and offshore platforms due to their great wave energy and unpredictability. In this paper, to generate a rogue wave, a bull’s-eye wave, which is a focusing of multi-directional waves, was simulated in a numerical wave tank. A multi-directional wave generating boundary was developed using OpenFOAM, which is an open-source computational fluid dynamics (CFD) library. The wave height and profile of the generated rogue wave were compared to those of the regular wave. In addition, the pressure and velocity contours of water particles and velocity vectors at the free surface of the rogue wave were studied, along with the kinematic and dynamic effects of the rogue wave on a floating body.

1. Introduction

A rogue wave, also called a freak wave, refers to a wave more than twice the surrounding significant wave height. The physical mechanism of the rogue wave can be explained by wave-current interaction, wave-wind interaction, geometrical seabed variation, and spatial-temporal focusing. The great wave height and unpredictability of a rogue wave mean that it can cause significant damage to ships and offshore platforms. Previously, a rogue wave was measured at the Draupner platform in the North Sea [1]. Its significant wave height was 11.9 m, the measured crest height was 18.5 m, and the measured trough height was −7.5 m. The wave height of the rogue wave was 26 m, and it was asymmetric about the still water level. The asymmetry rogue wave’s crest height was extreme and unpredictable [1].

Chabchoub et al. [2] generated rogue waves in a water wave tank and compared the measured results with a Peregrine-type breather solution [3] of the nonlinear Schrödinger equation (NLS) [4,5]. The wave heights from crest to trough measured by the experiment and breather solution were almost 2.5 times larger than the average wave height around the perturbation. Chabchoub et al. [2] observed super rogue waves with the high-order breather solution of the nonlinear wave equation and experiments in a water wave tank. Xiao et al. [6] studied rogue waves in deep water using phase-resolved numerical simulations based on a high-order spectral (HOS) method. Rudman and Cheary [7] simulated the fully non-linear dynamics of a large breaking rogue wave on a tension leg platform using smoothed particle hydrodynamics (SPH). Rudman and Cleary [8] studied a rogue wave’s impact on various mooring systems of an offshore platform using SPH. Hu et al. [9] simulated a rogue wave through a Peregrine-type breather solution [3] and developed a fully coupled fluid-structure interaction (FSI) model to study a rogue wave overtopping. Ha et al. [10] simulated a focusing wave by varying the wave maker velocity using computational fluid dynamics (CFD). Alberello et al. [11] simulated a breaking rogue wave using the CFD-HOS coupling method and assessed the three-dimensional velocity field underneath a breaking wave. Ransley et al. [12] studied the behavior of a wave energy converter system under a rogue wave loading using CFD and experiment, and Dorozhko et al. [13] studied a breaking rogue wave impact on a ship using CFD. Chen et al. [14] simulated oblique focused waves and interaction with a fixed body and compared them with experiments.

The objectives of the present study were (1) to generate rogue waves using a developed wave directional focusing library and (2) to apply it to a floating object. Open-source CFD libraries, termed OpenFOAM, were used to simulate the rogue waves. To capture free-surface wave flows, interFoam, which is a basic application solver in OpenFOAM, was selected.

2. Computational Methods

2.1. Governing Equations

The mass and momentum conservation equations in the incompressible flow were considered to obtain the velocity and pressure values, and could be expressed as follows:

$$\nabla ·({\rho }_m{\overrightarrow{v}}_m)=0$$

(1)

$$\frac{\partial {\rho }_m{\overrightarrow{v}}_m}{\partial t}+\nabla ·({\rho }_m{\overrightarrow{v}}_m{\overrightarrow{v}}_m)=-\nabla p+\nabla ·\stackrel{\overline{¯}}{\tau }+S$$

(2)

where $\overrightarrow{v}$, $p$, $\overline{\tau }$, $S$ are the velocity vector, static pressure, viscous stress tensor, and the source term, respectively. The subscript $m$ indicates the mixture phase.

The interface between the air and water phases was captured by the volume fraction transport equation. The volume fraction transport equation was expressed as follows:

$$\frac{\partial }{\partial t}(\alpha {\rho }_m)+\nabla •(\alpha {\rho }_m\overrightarrow{v_m})+C_{ad}\nabla •(\alpha (1-\alpha )\overrightarrow{v_r})=0$$

(3)

where the third term is reducing the solution smearing, and $C_{ad}$ is the anti-diffusion coefficient. A $C_{ad}$ of 1 was selected for this study [15]. $\overrightarrow{v_r}$ is the artificial compression velocity and the normal vector at the interface, could be written as follows:

$${\overrightarrow{v}}_r=|\overrightarrow{v_m}|\frac{\nabla \alpha }{|\alpha |}$$

(4)

2.2. Wave Generation and Absorption

A bull’s-eye wave was simulated based on the snake-type wave maker theory [16]. A series of monochromatic waves from a circular-shaped inlet boundary were propagating toward a focal point. Figure 1 shows a wavemaker and a focal point.

A multi-directional waves generation and absorption library has been developed and added to OpenFOAM [17,18,19]. The generated wave velocity and profile on the inlet boundary were written as follows:

$$U_H=\frac{H}{2}\omega \frac{\cosh \left(k\left(z+h\right)\right)}{\sinh \left(kh\right)}\cos \left(k_{x}x+k_{y}y-\omega t-ϵ\right)\frac{\overrightarrow{n}}{|\overrightarrow{n}|}$$

(5)

$$U_V=\frac{H}{2}\omega \frac{\sinh \left(k\left(z+h\right)\right)}{\sinh \left(kh\right)}\sin \left(k_{x}x+k_{y}y-\omega t-ϵ\right) \frac{\overrightarrow{g}}{|\overrightarrow{g}|}$$

(6)

$$\eta =A\cos \left(k_{x}x+k_{y}y-\omega t-ϵ\right)$$

(7)

where $U_H$ and $U_V$ are the horizontal and vertical velocity components of the wave, respectively. $\eta$ is the wave elevation, $h$ is the water depth of numerical wave tank, and $t$ is time. $H, k, \omega \mathrm{and} ϵ$ are wave height, wave number, angular frequency, and phase difference of the wave, respectively. $\overrightarrow{g}$ is the gravitational acceleration vector. $\overrightarrow{n}$ is the normal vector which is perpendicular to $\overrightarrow{g}$. x and y are x- and y-coordinates of numerical wavemaker on inlet boundary, respectively. The subscripts $x \mathrm{and} y$ indicate the x- and y-directions, respectively. The normal vector ($\overrightarrow{n}$) could be written as follows:

$$\overrightarrow{n}=\left(x_f-x\right)\widehat{i}+\left(y_f-y\right)\widehat{j}+0\widehat{k}$$

(8)

The phase difference ($ϵ$) could be written as follows:

$$ϵ=k\sqrt{{\left(x_f-x\right)}^2+{\left(y_f-y\right)}^2}$$

(9)

where $x_f$ and $y_f$ are the coordinates of the focal point.

The relaxation zone technique [18,19] was applied to absorb reflection waves. The velocity field in relaxation zone could be written as follows:

$$\overrightarrow{v }=\left(1-w_R\right){\overrightarrow{v}}_{\mathrm{target}}+w_R{\overrightarrow{v}}_{\mathrm{computed}}$$

(10)

where $w_R$ is the weighing function could be written as follows:

$$w_R=1-\frac{\exp {\sigma }^{3.5}-1}{\exp 1-1}$$

(11)

where $\sigma$ is the local coordinates system in the relaxation zone. Where the relaxation zone started, $\sigma$was set to 1 and at the outlet boundary where the relaxation zone ended, $\sigma$ was set to 0. In this study, $|{\overrightarrow{v}}_{\mathrm{target}}|$ was set to 0 m/s.

2.3. Numerical Methods

The time derivative term was discretized using a second-order accurate backward implicit scheme, with the constant time step size of 0.001 s used. The solution gradients at the cell centers were evaluated using Euler’s method. The convection terms were discretized using a total variation diminishing (TVD) scheme with the van Leer limiter, and for the diffusion terms, a central differencing scheme was used. The velocity-pressure coupling, and overall solution procedure were based on the PIMPLE algorithm, which combines the semi-implicit method for pressure linked equation (SIMPLE) algorithm and the pressure-implicit with splitting of operators (PISO) algorithm. SST k-ω turbulence model [20] with the wall function [17] was used for the turbulence closure. The discretized algebraic equations were solved using a Gauss–Seidel iterative algorithm, while an algebraic multi-grid method [21] was employed to accelerate solution convergence.

3. Results and Discussion

3.1. Regular Wave Simulation

A regular wave was simulated to validate the numerical. The parameters of flow are listed in

Table 1. Flow parameters.

Parameters	Value
ρwater	998 kg/m3
νwater	1.004 × 10−6 m2/s
ρair	1.204 kg/m3
νair	1.506 × 10−5 m2/s
$U_H{}_{\max }$	0.0445 m/s
Rewave amplitude	445
Refloating body	88,900

. The density and kinematic viscosity of water are 998 kg/m³, 1.004 × 10⁻⁶ m²/s, respectively. The density and kinematic viscosity of air are 1.204 kg/m³, 1.506 × 10⁻⁵ m²/s, respectively.

The Stokes second wave with the wavelength $(\lambda ) \mathrm{of} 3.2375$m and the wave period $\left(T\right) \mathrm{of} 1.44$ s was generated from the inlet boundary. For the wave height $(H)$, 0.02 m was considered. The domain extent and boundary conditions of a numerical wave tank are shown in Figure 2. The Cartesian coordinate was selected and the generated wave from the inlet boundary were propagated in the x-direction. The whole domain extent was 19.4 m in the x-direction and 5 m in the z-direction. The wave absorption zone was located in front of the outlet boundary. The x-directional length of the absorption zone was set to 6.475 m, twice the wavelength [18,19].

For the velocity, the Dirichlet boundary condition was applied on the inlet boundary while the Neumann boundary condition was applied on the outlet boundary. At the side boundary, a no-slip boundary condition was considered. An atmospheric condition was applied on the top boundary and a no-slip condition was applied on the bottom boundary.

Figure 2 shows a typical structured grid of 27,000 meshes. In a wave, 100 meshes were used in the one wavelength and 20 meshes were used in the wave height. Figure 3 shows the nondimensionalized wave elevations and the analytic solution. The simulated amplitude and period of a regular wave were well matched with the analytic solution.

3.2. Bull’s-Eye Waves Simulation

The parameters of water and air are the same as the regular wave simulation. The domain extent and boundary conditions of a numerical wave tank are shown in Figure 4. The Cartesian coordinate was selected and the generated waves on the inlet boundary were propagated in the x-y plane. The half domain was considered because the domain and boundary conditions were symmetric about the x-z plane (y = 0 plane). The circular-shaped inlet boundary was considered. The generated waves from the inlet boundary were overlapped at the center of the circle and dissipated in the downstream. The radius of the inlet boundary was 7.5 m. The whole domain extent was 20.45 m in the x-direction, 7.5 m in the y-direction, and 5 m in the z-direction. The wave absorption zone was located in front of the outlet boundary. The x-directional length of the absorption zone was set to 6.475 m, twice the wavelength [18,19]. The location of the focal point was x = 0 m and y = 0 m.

For the velocity field, the Dirichlet boundary condition was applied on the inlet boundary while the Neumann boundary condition was applied on the outlet boundary. At the side boundary, a no-slip boundary condition was used. An atmospheric condition was applied on the top boundary and a no-slip condition was applied on the bottom boundary.

Figure 5 shows a typical structured grid of 2,800,000 meshes. In a wave, 100 meshes were used in the one wavelength and 20 meshes were used in the wave height. From the inlet boundary to the center of the circular-shaped inlet boundary, the meshes generated are aligned to the wave propagating direction in order to decrease the numerical error caused by false diffusion.

A regular wave with a wavelength $(\lambda ) \mathrm{of} 3.2375$m and the wave period $\left(T\right) \mathrm{of} 1.44$ s was generated on the inlet boundary. For the wave height $(H)$, 0.02, 0.03, and 0.04 m were considered as listed in

Table 2. Simulated wave condition.

	Wave Height (H)	Wavelength (λ)	Wave Period (T)
Wave condition 1	0.02 m	3.2375 m	1.44 s
Wave condition 2	0.03 m
Wave condition 3	0.04 m

. The regular waves propagated towards the focal point, and 3-D bull’s-eye waves were generated.

The grid convergence test was carried out with a wave height of 0.02 m for four types of grids. As listed in

Table 3. Grid sizes for grid convergence test.

Grid System		Number of Mesh in Wavelength/Wave Height	Total Number of Meshes
Index	Description
1	Coarse mesh	63/12	784,672
2	Medium mesh	80/16	1,486,953
3	Fine mesh	100/20	2,800,530
4	Extra fine mesh	126/24	5,324,560

, the four types of coarse, medium fine, and extra fine grids were considered. The grid size was increased or decreased in all directions by 1.26 times. Figure 6 shows the wave height history in time at the focal point for the four grids. Figure 7 shows the relative errors of wave crest height at the focal point for the four grids. The relative error defined as follows:

$${ϵ}_{ij}=\left|\frac{\mathrm{RMS}\left(A_{\mathrm{crest}}{}_i\right)-\mathrm{RMS}\left(A_{\mathrm{crest}}{}_j\right) }{\mathrm{RMS}\left(A_{\mathrm{crest}}{}_i\right) }\right|$$

(12)

where, the subscripts $i \mathrm{and} j$ indicate the index of the grid system. The relative errors between the four meshes were ε₂₁ = 1.88%, ε₃₂ = 0.88%, ε₄₃ = 0.31%, respectively. Mesh 3 was selected from the grid convergence test results.

Figure 8 shows the rogue wave heights for various incident wave amplitudes. The simulated wave heights were compared with the experimental data [22]. The rogue wave height was calculated as the vertical distance between the crest and the trough. The simulated wave heights showed the same trend as the experimental results. Figure 9 shows the wave profile at the focal point along $x=0$. The simulated wave profile was well matched with the experimental results [22].

Figure 10 shows the wave height history in time at the focal point. The wave heights were captured by a volume fraction of 0.5. As the incident waves overlapped at the focal point, a wave with a high wave height was observed. A transient wave height was shown at first, and then it was slightly lowered, and a constant rogue wave was repeated. The crest was higher than the trough. The height difference between the crest and the trough increased as the amplitude at the inlet boundary increased.

Figure 11 shows the rogue wave’s development over time. The contours showed the z-elevation at the iso-surface with a volume fraction of 0.5. The vertical white column indicates the focal point. The incident wave propagated from the inlet boundary to the focal point, and the rogue wave was generated. The rogue wave was observed at the focal point.

Figure 12 and Figure 13 show the wave profile at the $y=0$plane and $x=0$plane, respectively. In the $y=0$plane, the rogue wave was generated at the focal point and dissipated in the absorption zone. In the wave profile of the rogue wave, the heights of the crest and the trough were different, so the asymmetry was large.

Table 4. Wave asymmetry and slope.

	Wave Height (H)	Acrest/Atrough	Wave Height (H)/Wave Length (λ)
Wave condition 1	0.02 m	0.057/0.022 = 2.61	0.078/3.235 = 0.024
Wave condition 2	0.03 m	0.084/0.031 = 2.73	0.115/3.169 = 0.036
Wave condition 3	0.04 m	0.119/0.039 = 3.07	0.158/3.103 = 0.051

lists the asymmetry and slope of the rogue wave. The asymmetry indicates the ratio between the amplitude of the crest ($A_{crest}$) and the amplitude of the trough ($A_{trough}$), and the slope indicates the ratio of the wave height and the wavelength. The asymmetry of the regular wave was close to 1, while the asymmetry of the rogue wave was more than 2. The asymmetry increased as the wave height of the inlet boundary increased. Because the energy of a wave was proportional to the square of its amplitude, it could be seen that the energy of the rogue wave was greater than that of the regular wave.

For the regular wave heights of 0.02, 0.03, and 0.04, the slopes were 0.0062, 0.0093, and 0.0124, respectively. In the rogue wave, the slopes were 0.024, 0.036, and 0.051 for the wave heights of 0.024, 0.036, and 0.051 on the inlet boundary, respectively. The rogue wave had a significantly increased asymmetry and slope compared to the regular wave.

The pressure change due to the rogue wave was investigated. Figure 14 shows the pressure contours of water particles in the $y=0$plane for the regular and rogue waves. The wave height used at the inlet boundary of the regular and rogue waves was 0.02 m. In the regular wave, the pressure was in the range of −95.3~98.6 Pa, whereas in the rogue wave, the pressure was in the range of −216.6~568.9 Pa. When the rogue wave was generated by overlapping the regular waves, the pressure could increase by about six times, which could act as an important design factor for a floating structure.

The effect of the rogue wave on the velocity of water particles was investigated. Figure 15 and Figure 16 show x-velocity and z-velocity contours in the $y=0$plane for the regular and rogue waves, respectively. The wave height used at the inlet boundary of the regular and rogue waves was 0.02 m. In the regular wave, the x-direction velocity ranged from −0.0578 to 0.0648 m/s, whereas in the rogue wave, the x-direction velocity increased to the range from −0.104 to 0.202 m/s. The x-velocity increased about three times in the direction of the wave propagation and about two times in the opposite direction.

In the z-velocity contours, z-velocity was in the range of −0.0438~0.0437 m/s in the regular wave, whereas the z-velocity was in the range of −0.216~0.195 m/s in the rogue wave. The z-velocity of the rogue wave was wider than that of the z-velocity of the regular wave. The maximum z-velocity increased about five times, which increased the wave height.

The velocity vector at the interface of air and water phases for the regular and rogue waves was investigated. Figure 17 represents the velocity vectors at the free surface, which were captured from the volume fraction of 0.5. The wave height used at the inlet boundary of the regular and rogue waves was 0.02 m. In the regular wave, the maximum velocity was 0.072 m/s, whereas in the rogue wave, the maximum velocity was 0.219 m/s, which was about three times higher. In particular, the velocity increase was very large near x = 0, where the rogue wave was generated.

3.3. Response of Floating Body

The effects of regular and rogue waves on a floating body, such as in onshore wave power generation, were investigated. The wave height used at the inlet boundary of the regular and rogue waves was 0.02 m. The Reynolds number based on water particle velocity $\left(U_H{}_{\max }=0.0445 \mathrm{m}/\mathrm{s}\right)$ and floating body length (2 m) was 88,900.

The shape of the floating body was defined as a rectangular shape and was assumed to be a rigid body. The length, width, and height of the structure were set to 2 m, 0.28 m, and 0.15 m, respectively. The draft was set to 0.1 m. The center of the floating body was placed at the focal point of the rogue wave. Two degrees of freedom of heave (translational motion on the z-axis) and pitch (rotational motion around the y-axis) were allowed.

The motion and forces acting on the floating body by the regular and rogue waves were compared.

Figure 18 shows the heave and pitch motions of the floating body. Initially, the body was fixed and then allowed to move after the solutions were stabilized. Because the height of the rogue wave was higher than that of the regular wave, a higher heave motion was also predicted. The high slope and height of the rogue wave caused an increase in the pitch motion compared to the regular wave. From the results, when the floating body was exposed to the rogue wave, the motions became significantly larger than those of the regular wave.

Figure 19 shows the x-force and z-force acting on the floating body. The rogue wave increased the pressure and velocity of the water particles and increased the velocity at the free surface, which in turn greatly increased the force acting on the floating body. For floating structures that could be exposed to rogue waves, it is necessary to consider such an increased force.

Figure 20 shows the free surface evolution around the floating body. The floating body motions were repeated according to the wave phase.

4. Concluding Remarks

To investigate the characteristics of the rogue wave, simulations were carried out based on computational fluid dynamics (CFD). The multi-directional wave generation boundary f, or the bull’s-eye simulation, was developed using OpenFOAM, a collection of open-source CFD libraries. To minimize the numerical error caused by false diffusion, the circular-shaped inlet boundary was selected and the meshes aligned to the wave propagating direction were used. A structured grid of 2,800,000 meshes was generated.

Rogue waves were generated using the wave heights of 0.02, 0.03, and 0.04 m. The generated rogue wave heights showed the same trend as the experimental results [22]. The generated rogue wave showed significantly greater asymmetry, wave slope, and energy than the regular wave. The rogue wave had larger water particle pressure, x-velocity, and z-velocity than the regular waves. The rogue wave moved about three times faster than the regular wave on the free surface.

To investigate the effect of the rogue wave on a floating body such as a floating onshore wave power plant, the rectangular-shaped rigid body was considered. Under the influence of the rogue wave, the motions and acting forces on the floating body become much larger than those of the regular wave. For structures operating in the sea area where rogue waves may occur, consideration of rogue waves is essential. In future work, the elastic behavior of floating structures related to rogue waves will be studied.