### 1. Introduction

*D*) and draft (

*d*) of the cylinder at various wavelengths on the wave run-up was studied by analyzing the incident waves of various periods. The effect of wave nonlinearity on the wave run-up at various wave steepness values was studied by comparing changes in the wave run-up under various wave steepness conditions.

### 2. Problem Formulation

*ϕ*) is introduced, and the governing equation becomes the Laplace equation as in Eq. (1).

*ϕ*) is introduced, the governing equation becomes the Laplace equation, which is shown below:

*α*denotes a three-dimensional angle, and

*G*denotes the kernel function. The three-dimensional basic Rankine source expression in 3D is 1/4

*πr*, where

*r*denotes the distance between the source and field points (Kim and Koo, 2019).

*v⃗*= (0,0,

*δη/δt*) is applied to the nonlinear free surface boundary conditions, which can be expressed by Eqs. (5)–(6), to consider the effects of nonlinear waves.

*η*denotes the displacement of the free surface. As the boundary condition of the incident wave, a progressive wave is generated by substituting the incident wave component in the left end of the computational domain. The linear wave is applied as the incident wave for linear analysis, and the second-order Stokes wave is applied in the nonlinear analysis. Eq. (7) shows the boundary condition of the incident wave according to the second-order Stokes wave that is applied in the analysis of the nonlinear wave. In the case where the incident wave is linear, only the first term of Eq. (7) was applied for the boundary condition. where

*g*denotes the gravitational acceleration,

*A*denotes the amplitude of the incident wave,

*k*denotes the wave number, and

*w*denotes the wave frequency.

*∂ϕ*/

*∂n*= 0), and the bottom surface is expressed using the image method. Fig. 1 shows the overall computational domain used to determine wave run-up around the cylinder. To create an open sea condition while eliminating the unnecessary reflected waves that may be generated on the free surface, an artificial damping zone is applied in the frontal, side, and end damping zones, and the length of each damping zone is set to one wavelength (1

*λ*). Moreover, the analysis domain is represented by x-axis symmetry to shorten the analysis time by reducing the number of computational elements. The least square technique is employed for reconstructing the gradient for spatial differentiation, and the inverse distance weighting (IDW) method is employed for interpolation of the nodes. Further details can be found in Kim and Koo (2019).

### 3. Numerical Analysis Model and Results

### 3.1 Numerical Analysis Model and Analysis Conditions

*d*) from 1.5 m to 9 m while maintaining the diameter (

*D*) at 3 m. Fig. 3 shows the appearance of the panel (mesh) in the numerical analysis model. The number of elements in the numerical analysis model is 250 at

*d*= 1.5 m; 350 at

*d*= 3 m; 450 at

*d*= 4.5 m; 550 at

*d*= 6 m; and 750 at

*d*= 9 m. Table 2 shows the 13 incident wave conditions that were calculated for linear and nonlinear analyses. First, the effect of the diffraction parameter was examined by changing

*D*/

*λ*from a minimum of 0.1203 to a maximum of 0.4360, and analysis was performed with varying wave steepness from 0.006 to 0.03 under the condition of a specific diffraction parameter. The water depth was fixed at 15 m.

### 3.2 Wave Run-up Analysis Using a Three-dimensional Linear NWT

*D*= 16 m,

*d*= 24 m, and

*h*= 60 m. The incident wave height was fixed at 0.3 m, and the wave run-up height (

*R*) was calculated by averaging time series results in which the steady state lasted for five cycles after the incident wave reached the cylinder. Fig. 4 shows that the results of the numerical analysis of this study agree well the results of Lee et al. (2013) for all circumferential angles.

*D*/

*λ*) and cylinder draft (

*d*) on the wave run-up were determined. Fig. 5 shows a comparison of the wave run-ups for various diffraction parameters and the representative circumferential angles, which were set to 0°, where wave run-up was at its maximum as shown in Fig. 4; 135°, where wave run-up was at its minimum; and 90°, which was the central point of the cylinder. The draft and the diameter of the cylinder were set to 3 m each. As the diffraction parameter increased, wave run-up decreased when the circumferential angle was 135°, but it gradually increased at 0° and 90°. In particular, at 0°, where the wave run-up reached its maximum value, it converged to approximately 1.7 times the incident wave height as the diffraction parameter increased. An increased diffraction parameter indicates that the incident wavelength is relatively small compared to the cylinder diameter, and a considerable portion of the incident wave is reflected from the front of the cylinder. If total reflection occurs, such as under completely blocked conditions, which could be achieved with a sea wall, a standing wave is generated, and the maximum wave run-up that can be measured is twice the incident wave height.

*R*/

*H*gradually approaches 1. This is considered to be due to the decreased diffraction effect of the incident wave caused by the cylinder as the diffraction parameter falls below 0.2. Conversely, as the diffraction parameter increases, the difference in wave run-up according to the circumferential angle increases. In particular, when the diffraction parameter is the highest at 0.3634, wave run-up of approximately 1.6 times the wave height occurs around the circumferential angle of 0° to 50°. At a circumferential angle of 90° or greater, the wave run-up is generally reduced, and only approximately 0.6 times the incident wave height at 135°.

*D*/

*λ*< 0.2), the maximum wave run-up difference according to the draft is approximately 26%, and when the diffraction parameter is large (

*D*/

*λ*> 0.25), the maximum difference is approximately 7.4%. When the draft is small compared to the cylinder diameter (0.5 times), the wave run-up is considerably smaller than under other cylinder conditions with a small diffraction parameter. It appears that the incident wave does not affect the cylinder owing to the low draft, and most of it passes through. As shown in Fig. 5, the wave run-up increases and converges to 1.7 as the diffraction parameter increases. Based on this, it can be confirmed that the free-board of the cylindrical structure should be at least 1.7 times the incident wave height regardless of the cylinder draft.

*d*/

*D*≥ 1), a similar wave run-up is generated irrespective of the value of the draft.

*F*/

*ρgA*(

*D*/2)

*d*, where

*A*denotes the amplitude of the incident wave. As the cylinder draft increases and the diffraction parameter decreases, the horizontal force generally increases, which is due to the relatively long wavelength of the incident wave increasing the area where the wave energy acts on the cylinder. This relationship between the wavelength and cylinder draft can be observed more prominently by comparing the horizontal force when the diffraction parameter is large. When the diffraction parameter increases, the incident wavelength becomes relatively small, and its effect on the lower surface of the water plane of the circular cylinder is reduced. Because of this, the horizontal force remains almost similar for a deep draft cylinder. In Fig. 8(b), it can be seen that the vertical forces decrease unlike the horizontal forces. The vertical force decreases as the diffraction parameter increases, that is, as the wavelength is relatively reduced. This is because the effect of the wave on the bottom surface of the cylinder, based on which the vertical force is calculated, is reduced.

### 3.3 Wave Run-up Analysis Using a Full Nonlinear NWT

*H*/

*λ*) values were selected as 1/150 and 1/35 for the linear and relatively large nonlinear effects, respectively. All results used were from the period when

*t*/

*T*was between 6 and 11, during which the time series data reached a steady state. When the wave steepness of a typical nonlinear wave (Stokes wave) condition (1/35) is applied, the crest height of the wave run-up increases by 8% and the trough height decreases by 6% compared to the linear analysis results. When the wave steepness is 1/150, it is almost identical to the time series obtained by the linear NWT, which confirms that the nonlinear effect occurs as the wave steepness increases.

*H*/

*λ*) of each cylinder at a cylinder circumferential angle of 0°, where the wave run-up is at its maximum in the linear analysis, and 135°, where the wave run-up is at its minimum. The diffraction parameter (

*D*/

*λ*) was fixed at 0.3634. The wave run-up increases as the wave steepness increases in all cylinders irrespective of the cylinder draft. For 0°, the relative increase rate of the wave run-up increases when the wave steepness is 0.02 or more. As the linear and nonlinear waves are usually demarcated at the wave steepness of 0.02, it can be understood that a larger crest height occurs in the nonlinear wave section, resulting in a greater wave run-up. At 135°, the measuring point with the minimum wave run-up, a relatively high wave run-up occurs when the ratio of the draft and diameter is the smallest (

*d*/

*D*= 0.5). As the light draft allows the incident wave to easily pass through the lower part of the cylinder, a relatively small wave run-up occurs at 0° (Fig. 7 (a)), and a relatively large run-up occurs at 135°.

*d*/

*D*of 3.0 measured at 0° for the maximum wave steepness of 1/35 and minimum wave steepness of 1/150 were separated for each frequency component (Fast Fourier Transform applied), as shown in Table 3. A comparison of the frequency components for various wave steepness values reveals that the primary wave frequency components of the wave run-up remain the same as the wave run-up values irrespective of the wave steepness. As the wave steepness increases, the mean value and double frequency components of the wave run-up that are proportional to the square of the wave amplitude and triple frequency components of the wave run-up that are proportional to the cube of the wave amplitude increase. The wave run-up increases by 4% when the wave steepness is approximately 1/35. In particular, when the wave steepness is 1/35, the mean value of the wave run-up (zero order frequency components) is 5% of the primary frequency components and 3% for the secondary frequency components.

### 4. Conclusion

*D*/

*λ*), which is the ratio of the diameter of the cylinder to the incident wavelength, the circumferential position (angle) of the cylinder, and the change in cylinder draft. In addition, the effect of wave nonlinearity on the wave run-up under various wave steepness conditions was investigated using a nonlinear NWT.

*D*/

*λ*was 0.25 or greater. Based on this, the free-board of a fixed structure composed of a cylindrical lower body should be at least 1.7 times the incident wave height.