Offshore Hydrogen Infrastructure: Insights from CFD Simulations of Wave–Cylinder Interactions at Various Cross-Sections

Abstract

CFD-based numerical wave tank models are valuable tools for analyzing the nonlinear interaction between waves and structures. This paper aims to examine the deformation of high-order free surfaces near a vertical, surface-piercing fixed cylinder with various cross-sections under regular head waves, assuming no wave breaking. Additionally, the study investigates the effects of wavelength on wave evolution, nonlinear wave amplification, and the harmonics around the cylinder. The numerical analysis is performed using the CFD toolbox OpenFOAM. The comparison of numerical results for different cross-sections reveals the influence of corner ratio on lateral edge waves and highlights its significant impact on the nonlinear wave field around the cylinder, particularly for short incident waves. The numerical results indicate the important contribution of the cross-section shape together with the corner effect on the lateral edge waves and accordingly the nonlinear wave field surrounding the given column, which involves high harmonics wave amplification up to fourth. The reduction in corner ratio results in a reduction in maximum run-up height from 2.57 to 2.2 in short waves, while for the long waves, it is from 1.61 to 1.45. This research not only enhances our understanding of fluid–structure interactions but also has implications for the design and safety of hydrogen storage and transportation systems. Understanding dynamic pressures and structural responses is crucial for these applications. CFD simulations of wave–cylinder interactions are essential for designing and optimizing offshore hydrogen infrastructure. These simulations model how waves interact with cylindrical structures, such as wind turbine foundations, hydrogen production platforms, and storage tanks. Understanding these interactions is vital for ensuring the structural integrity, efficiency, and sustainability of offshore hydrogen facilities.

1. Introduction

Offshore hydrogen infrastructure is an emerging field that integrates renewable energy generation, primarily from offshore wind, with hydrogen production, storage, and transportation. This infrastructure aims to leverage the abundant and consistent wind resources available offshore to produce green hydrogen, a clean energy carrier that can decarbonize various sectors of the economy.

The application of the CFD simulations of wave–cylinder interactions is a cornerstone in the development of offshore hydrogen infrastructure. By providing critical insights into the forces and dynamics at play in marine environments, these simulations ensure the structural integrity, efficiency, and sustainability of offshore hydrogen projects. As the demand for clean energy continues to rise, the role of CFD in optimizing and safeguarding offshore hydrogen infrastructure will become increasingly vital. Offshore hydrogen infrastructure must endure extreme weather conditions, including high waves and strong currents. CFD simulations provide detailed insights into how these conditions affect structural stability and longevity. As offshore hydrogen infrastructure scales up, the importance of robust and optimized design becomes even more critical. CFD simulations allow for scalable solutions that can be adapted to larger and more complex projects. Regulatory bodies require detailed assessments of structural safety and environmental impact. CFD simulations provide the necessary data to meet these regulatory requirements, facilitating project approval and implementation. Offshore hydrogen production is a key component of global decarbonization efforts. CFD simulations contribute to the sustainability of these projects by ensuring that they are designed for maximum efficiency and minimal environmental impact.

Surface-piercing cylindrical columns are pivotal in supporting various offshore platforms, including semi-submersibles, Gravity-Based Structures (GBSs), Tension Leg Platforms (TLPs), Spars, and wind turbine foundations, which are integral to the renewable energy sector, including hydrogen production. These offshore structures face harsh marine environments and extreme wave conditions that can induce complex nonlinear wave–structure interactions (WSIs). Such interactions are critical in the context of steep storm waves, leading to significant wave amplification and localized free-surface effects like upwelling and water jets. These phenomena are crucial for designing air gaps and mitigating the impact of green water and wave slamming, which can cause severe damage to structures and jeopardize their integrity. A deep understanding of the nonlinear dynamics of wave evolution and amplification around these columns is, thus, essential for ensuring the safety, reliability, and cost-effectiveness of hydrogen infrastructure in marine settings.

Recent studies emphasize hydrogen vehicles’ crucial role in the UK’s net-zero goals by 2050, highlighting fuel cell electric vehicles (FCEVs) for reducing fossil fuel reliance and supporting a circular economy. They enhance resource use, enable zero-emission mobility, and integrate renewable energy. The research also stresses the need for strong safety measures, using quantitative risk assessment (QRA) and the DEMATEL framework for safe and efficient operation [1].

Another study highlights the need for resilient hydrogen energy infrastructure within sustainable development. It proposes a decision-making framework with resilience indicators and sustainability factors to assess hydrogen systems, offering insights into the vulnerabilities of hydrogen production, use, and storage for sustainable progress [2,3,4].

The exploration of natural hydrogen in the Earth’s crust is promising, but a more effective approach may be generating hydrogen artificially in underground reservoirs. This involves using biotic and abiotic processes to convert materials into hydrogen. Some reactions may allow for a slow “seed and leave” approach, while others could produce hydrogen more quickly [5].

Numerous experimental and numerical investigations on the interactions between waves and offshore structures have been reported over the last several decades. The works related to the potential flow theory, primarily, are based on classical diffraction, first- and second-order, and further fully nonlinear solutions. The analytic solutions for a bottom-mounted cylinder in regular waves [6,7] have been improved and developed with time. For steep waves, the linear assumption significantly under-predicts crest heights around columns [8]. Refs. [9,10] investigated the second- and higher-order effects.

For experimental research, the works are mainly focusing on wave run-up height, wave loading, and nonlinear wave field around a column in addition to presenting empirical and semi-empirical formulas or wave run-up height prediction in regular and irregular wave conditions. Kriebel [8] reported important benchmark experiments for wave run-up assuming a circular cylinder located in finite water depth. Morris-Thomas et al. [11] experimentally studied the effect of cross-section on the wave run-up harmonics. Swan et al. [12,13] carried out experiments and investigated the high-frequency wave scattering around the cylinder and also investigated the higher harmonic wave force. Grue et al. [14] experimentally quantified wave run-up level on a cylindrical column that is slender and subject to long and steep waves with the assumption of both breaking and non-breaking wave events at finite depth in the wave channel.

Paulsen et al. [15] studied the wave loads on a surface-piercing cylinder with a circular cross-section using OpenFOAM based on no-turbulence mode. Yoon et al. [16] studied the wave run-up employing CFDShip-Iowa with and without turbulence model. Mohseni et al. [17] simulated the high-frequency wave scattering around a single circular cylinder subject to the different wave conditions by OpenFOAM. Mohseni and Guedes Soares [18] studied the influence of the change in underwater volume on the nonlinear wave field surrounding a fixed cylinder using the same numerical model. Afterward, Mohseni and Guedes Soares [19] performed a detailed study of the nonlinear wave loading together with the wave amplification and corresponding harmonics for two tandem columns parallel to the wave direction.

Presently, to the authors’ knowledge, the experimental and numerical simulations by Nielsen [20], Morris-Thomas et al. [11], and Mohseni and Guedes Soares [21] are the only instances where the wave amplification and wave run-up and associated harmonics on the vertical cylinders with circular and non-circular cross-section have been investigated. Thus, a more detailed study is necessary to have a deeper insight into the wave amplification phenomena due to the change in cross-section.

Based on the findings by Swan et al. [13] and Chaplin et al. [22], the present work intends to study the deformation of the high-order free surface and the violent fluid motion around a surface-piercing cylinder which is stationary with different cross-sections and corner ratios. The objective is to obtain better insight into the physics behind the nonlinear wave evolution and wave amplification and corresponding harmonics and patterns around the given cylinder subject to non-breaking, regular short, and long waves propagating in deep water. Mohseni and Guedes Soares [21] and Mohseni, M. and Soares, C.G. [23] presented the definition and physical evolution of the wave scattering Type-1 and Type-2 together with edge waves around a circular cylinder.

The novelty and contribution of this paper lie in its focused investigation of wave–cylinder interactions specifically within the context of offshore hydrogen infrastructure, which is critical for renewable energy applications. Unlike previous works, this study explores the impact of varying cross-sectional geometries (based on the change in corner ratio) on nonlinear wave amplification and run-up, and presents comprehensive harmonic analysis. By examining the effects of different corner ratios and wave conditions, this research provides new insights into how these geometric variations influence hydrodynamic forces, offering practical design implications for the safety and efficiency of hydrogen production, storage, and transportation systems in marine environments.

The remainder of the paper is organized as follows: Section 2 shows the details of the CFD-based NWT including the methodology, flow regime, wave conditions, computational domain, and mesh. Afterward, the obtained results including the physical mechanism of wave evolution and then the wave interaction with a cylinder with different cross-sections subject to short and long waves and the maximum run-up height are analyzed in Section 3. The concluding remarks are presented in Section 4.

2. Methodology

2.1. Numerical Model

This study uses the open-source CFD toolbox OpenFOAM/InterFoam^® v5.0 (2017) to set up a CFD-based numerical wave tank and perform the necessary simulations. InterFoam solves the three-dimensional Unsteady Navier–Stokes equations using the finite-volume method (FVM) for two incompressible phases, assuming a static mesh and utilizing a VOF-based (Volume of Fluid) interface-capturing approach.

2.1.1. Governing Equations

The basic governing equations of mass and momentum conservation with the viscous fluid assumption are expressed as follows:

$$\nabla .\mathbf{U}=0$$

(1)

$${\displaystyle \frac{\partial \rho \mathbf{U}}{\partial t}}+\nabla .\left(\rho \mathbf{U}\mathbf{U}\right)=-\nabla p^{*}-\mathbf{g}·\mathbf{X}\nabla \rho +\left(\nabla .\left(\mu \nabla \mathbf{U}\right)+\nabla \mathbf{U}·\nabla \mu \right)+\sigma \kappa \nabla \mathsf{\alpha }$$

(2)

where $\left(\rho \right)$ represents the fluid density, $\left(\mathbf{U}\right)$ is the velocity vector in Cartesian coordinates, $\left(\nabla \right)$ is the vector differential operator, $\left(\mu \right)$ is the fluid molecular dynamic viscosity, and time is denoted by $\left(t\right)$. Furthermore, $\left(\mathbf{g}\right)$ is the acceleration due to gravity, and $\left(\mathbf{X}\right)$ is the position vector. $\left(p^{\mathrm{*}}=p-\rho \mathbf{g}·\mathbf{X}\right)$ is the pseudo-dynamic pressure where $\left(p\right)$ stands for the total pressure. The last term in Equation (2) represents the surface tension force.

In OpenFOAM, the VOF function is tracked using the transport (advection) equation, where $\left(\alpha \right)$ denotes the volume fraction:

$${\displaystyle \frac{\partial \alpha }{\partial t}}\left(\mathbf{U}\alpha \right)+\nabla ·\left({\mathbf{U}}_{\mathbf{r}}\alpha \left(1-\alpha \right)\right)=0$$

(3)

The function of the last term in Equation (3) is to limit the smearing of the interface, and it is active only at the interface. The properties of the fluid; local density, $\left(\rho \right)$; and dynamic viscosity, $\left(\mu \right)$ in each cell are weighted by the volume fraction, $\left(\alpha \right)$, as follows:

$$\left\{\begin{array}{c}\rho ={\alpha \rho }_w+{\alpha \rho }_a\\ \mu ={\alpha \mu }_w+{\alpha \mu }_a\end{array}\right\}$$

(4)

where subscripts$\left(w\right)$ and$\left(a\right)$ denote water and air, respectively. In the following simulations, the physical parameters of the case are specified: $\left({\sigma }_{air/water}=0.07 \mathrm{N}/\mathrm{m}\right)$, $\left({\rho }_{water}={10}^3 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3\right)$, $\left({\nu }_{water}={10}^{-6} {\mathrm{m}}^2/\mathrm{s}\right)$, $\left({\rho }_{air}=1 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3\right)$, and $\left({\nu }_{water}={1.48}^{-5} {\mathrm{m}}^2/\mathrm{s}\right)$.

In this work, the decision to exclude additional “turbulent” viscosity was made due to the specific flow regime under investigation. The flow conditions, characterized by a low Keulegan–Carpenter (KC) number, are primarily dominated by inertial forces, making the contribution of turbulence negligible. According to studies like Sumer and Fredsøe [24], turbulence modeling becomes more critical when the KC number exceeds 4, where viscous effects begin to play a more significant role. In our case, including turbulent viscosity would add unnecessary complexity without substantially improving the accuracy of the results, as validated by previous work. If incorporated, turbulent viscosity could slightly modify boundary layer behavior and wave amplification patterns, but the overall impact on large-scale hydrodynamic forces would be minimal, given the inertia-dominated nature of the flow. Therefore, its exclusion allows for a more streamlined analysis without compromising the study’s objectives or accuracy.

2.1.2. Numerical Inputs

The numerical wave generation in this study is based on the static boundary approach of the olaFlow toolbox. Additionally, the cell stretching method is combined with active wave absorption at the outlet boundary using olaFlow. An overview of the performance of an empty numerical wave tank (two-dimensional) and an evaluation of the effectiveness of the damping zone have been previously conducted [21]. The initial and boundary conditions for all the domain boundaries, as well as for each of the flow variables, are consistent with the work by Mohseni and Guedes Soares [21].

During post-processing, the vertical coordinate of the free-surface interface at each time step is determined using the iso-contour of the phase fraction α = 0.5. For harmonic analysis, the mean of the measured wave elevation time series is first subtracted. Then, by applying the Fourier series over a time window of the last 15 wave periods from the steady-state solution, the harmonics at each wave probe are calculated. The wave elevation time history is sampled at a frequency of 100 Hz.

Following the approach of Mohseni and Guedes Soares [21], in this work, the Courant number for the solver (Co_max) and for the free surface (Co_α) is maintained at less than 0.25 while keeping the time step fixed.

2.2. Selected Wave Conditions and Flow Regime

In this study, the surface-piercing cylinder is fixed and truncated with various cross-sections. The incident waves are regular, non-breaking steep waves propagating over a flat seabed in deep water.

Table 1. Selected incident wave conditions.

Case Name	T [s]	H [m]	L [m]	H/L	D/L	KC	Fr	Re	We
T7 s116	7	4.777	76.44	1/16	0.210	0.9381	0.031	Full: 0.6186 × 107	Full: 184.8623
								Model: 0.1733 × 105	Model: 3.6742
T15 s116	15	21.938	351.00	1/16	0.046	4.3074	0.065	Full: 1.3022 × 107	Full: 389.1055
								Model: 0.3648 × 105	Model: 7.7335

presents the selected incident wave conditions. All the simulations are conducted at the model scale based on the Froude similarity law, following the benchmark experiments provided by the ITTC, as shown in

Table 2. Main particulars for single circular cylinders following the benchmark by ITTC (OEC), (2013).

Description	Symbol [Unit]	Full Scale	Model Scale
Diameter	[m]	16	0.318
Draft	hs [m]	24	0.477
Draft ratio	hs/D	1.5	1.5
Scale	$\Gamma$	-	1/50.314

. The wave periods T = 7 s and T = 15 s, based on the Fifth-Order Stokes wave theory, represent short and long waves in deep water, respectively.

Following Table 1, the flow regime around the cylinder is turbulent, with the Keulegan–Carpenter number KC varying between 0.9 and 4.5. For the short incident waves, where KC < 3, the flow is dominated by inertia, as described by Sumer and Fredsøe [24]. For long incident waves, where 3 < KC < 7, the flow lies within a combination of drag-inertia and unsteady inertia regimes. Additionally, previous experiments by Sarpkaya [25] suggest that the effect of fluid viscosity can be neglected in the region where KC < 4. According to Mohseni et al. [17], the interaction between waves and the cylinder involves a weakly turbulent flow, primarily due to the localized free-surface breaking around the circular cylinder. They also concluded that the current Navier–Stokes model can reasonably and efficiently account for viscous and turbulent effects. Therefore, in this study, the turbulent viscosity is set to $\left({\mathsf{\mu }}_{\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{b}}=0\right)$ for all the simulations.

2.3. Computational Model

The computational domain consists of the main zone and the damping zone. Based on the physical observations by Swan et al. [13], wave scattering around the cylinder is symmetric. Therefore, a symmetry plane is utilized in this study, as shown in Figure 1. Table 1 provides the details of the computational mesh for both the short- and long-wave cases. The spatial mesh is characterized by two parameters: the number of cells per wavelength $\left(\mathrm{L}/\left(∆\mathrm{X}=∆\mathrm{Y}\right)\right)$, and the number of cells per wave height, $\left(\mathrm{H}/∆\mathrm{Z}\right)$. More information on the computational domain and mesh can be found in the work by Mohseni and Soares (2022) [21]. Figure 2 shows the top view of 10 wave probe locations at angles $\left(\theta =0^{°},{45}^{°}, {90}^{°}, {135}^{°}, {\mathrm{a}\mathrm{n}\mathrm{d} 180}^{°}\right)$. The distance of $\left(r/D=0.513\right)$ represents the on-body probes and $\left(r/D=1\right)$ represents the off-body probes.

2.4. Verification and Validation of the Numerical Model

This part summarizes the findings while still conveying the critical outcome of the study regarding the mesh’s efficiency and effectiveness. Mohseni and Guedes Soares [21] and Mohseni, M., Soares, C.G. [23] conducted a comprehensive verification and validation of the numerical model, focusing on the harmonics of wave elevation around and the inline force on the circular cylinder under both wave conditions described in Table 2. Their study concluded that the optimum spatial and temporal resolution (Mesh 2), provides a converged, sufficiently accurate, and cost-effective solution.

3. Results and Discussion

This section investigates the influence of changes in cross-section and corner ratio on wave evolution and the resulting nonlinear wave amplification around the given cylinder, subjected to both short and long waves. First, the physical mechanism of wave amplification around the cylinder is described based on the spatial contours of wave evolution. Following this, the associated harmonic patterns around each cross-section are analyzed about these findings.

Four key incident wave phase angles for the front stagnation point of the cylinder are employed to describe the physical mechanism of wave scattering. These wave phases with the assumption of the incident wave propagation in a positive direction from left to right, are $\left(\varphi =0\right)$, the arrival of a crest; $\left(\varphi =\pi /2\right)$, the arrival of a zero down-crossing; $\left(\varphi =\pi \right)$, the arrival of a trough; and$\left(\varphi =3\pi /2\right)$, the arrival of a wave zero up-crossing. The non-dimensional number of corner ratio, $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}\right)$, is defined as the ratio of the corner radius,$\left({\mathrm{r}}_{\mathrm{c}}\right)$, to the half-cylinder width,$\left(R=D/2\right)$ identifies the five cross-sections within the range of $\left({0<\mathrm{r}}_{\mathrm{c}}/\mathrm{R}<1\right)$. These cross-sections vary from circular,$\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=1\right)$, to the round,$\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0.75\right)$, and eventually sharp,$\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0\right)$; see Figure 3.

3.1. Wave Interaction with a Cylinder with Different Cross-Sections Subject to Short Waves

Figure 4, Figure 5, Figure 6 and Figure 7 illustrate the normalized spatial contour of wave evolution around each cylinder subjected to a short incident wave, viewed from both the top and front perspectives. As shown in Figure 4(I(a)), Type-1 wave scattering for $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=1\right)$ is associated with a concentric and symmetric wave field. During the interaction of a single cycle of the incident wave with the cylinder, there are two instances of Type-1 wave scattering, occurring at the front and back stagnation points, as seen in Figure 4(I(a)) and Figure 6(I(f)).

The incident wave traveling downstream bends around the cylinder and is divided into two lateral edge waves on either side, as depicted in Figure 4(I(b)). At the back-corner point, these lateral edge waves move downstream and reach the back stagnation point of the cylinder, resulting in high-frequency and fully developed Type-2 wave scattering, as shown in Figure 5(I(c,d)). When the wave trough reaches the front stagnation point, the lateral edge waves circulate upstream, as illustrated in Figure 6(I(e,f)). For short incident waves, these lateral edge waves barely reach the front-corner point of the cylinder, as seen in Figure 7(I(g)). After the passage of the wave zero up-crossing at the front point, the waves move downstream, as shown in Figure 7(I(h)).

For other cross-sections subjected to short incident waves, there are two instances of Type-1 wave scattering at the front and back stagnation points, as shown in Figure 4(a(II–V)) and Figure 6(f(II–V)). Additionally, there is one instance of Type-2 wave scattering at the back of the cylinder, as seen in Figure 5(d(II–V)). The transition from a circular to a square corner ratio is highlighted by an increase in the maximum run-up height at the front of the cylinder, as shown in Figure 4(a(II–V)). This change in corner ratio from circular to sharp square results in increased wave–structure interactions, leading to a systematic decrease in the energy and momentum of the lateral edge waves. As the corner ratio decreases gradually,$\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0.75\right)$ to $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0.25\right)$, the edge wave energy decreases, and the corner becomes more pronounced. Consequently, there is a stronger interaction with the back-corner point, which is particularly evident at $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0.25\right)$, as shown in Figure 6(f(IV)).

Following the change in the corner ratio from $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0.75\right)$ to $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0\right)$, the energy of the lateral edge waves reduces and they slow down as they circulate the cylinder. This results in a wave–wave interaction at the shoulder and consequently weaker interaction; see Figure 7(g(V)). There is an occurrence of slightly developed wave scattering Type-2 at the front of the given cylinder for $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0.25 \mathrm{a}\mathrm{n}\mathrm{d} 0\right)$; see Figure 6(IV(e,f),V(e,f)). For the corner ratio of $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0.25 \mathrm{a}\mathrm{n}\mathrm{d} 0\right)$, where the corner is visible, the interaction of the negative horizontal velocity with the front corner forms a new set of lateral edge waves, see Figure 6(e(IV,V),(f(IV,V)) and Figure 7(g(IV,V)). Similar to the $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=1\right)$ for the other cross-sections, the lateral edge waves traveling upstream can barely reach the front-corner point; see Figure 7g.

The overall view of the harmonic values of second, third, and fourth of the total wave elevation at front-corner and also front stagnation points, see Figure 8c–e, interestingly, indicates the occurrence of slightly developed wave scattering Type-2 at the front of the given cylinder for $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0.25 \mathrm{a}\mathrm{n}\mathrm{d} 0\right)$; see Figure 6(IV(e,f),V(e,f)). Comparing the occurrence of wave scattering Type-2 for $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0.25 \mathrm{a}\mathrm{n}\mathrm{d} 0\right)$ at the front and back of the given cylinder, there is less dynamic pressure and rise of water level and accordingly less steep waves; see Figure 5d. This is related to the weak wave–wave interaction during the wave phase, $\left(\pi /2<\varphi <\pi \right)$.

Concerning the information from the analysis of time history and spatial contour of wave evolution, harmonic analysis can provide a better and more quantitative understanding of the nonlinear nature of wave amplification and high-frequency wave field around each given cylinder. Here, the mean value and harmonics of the total wave elevation at given wave probes up to the fifth are analyzed. In continuation, in this study, it is shown to what extent the corner ratio contributes to the nonlinear wave field around the given cylinders.

The comparison of the mean value and harmonics of the total wave elevation around the given cross-section in terms of wave probe location, $\left(\mathbf{x}/\mathbf{D}\right)$, normalized by incident-minded $\left({\mathrm{A}}_1\right)$, is presented in Figure 8. In addition, the harmonic contours of the diffracted wave for all the given five cross-sections in the short wave are presented in Figure 9, Figure 10 and Figure 11. In a general view of Figure 8, the systematic variation in corner ratio, there is a systematic change in harmonic pattern and values for cross-sections in the range,$\left({0.25\le \mathrm{r}}_{\mathrm{c}}/\mathrm{R}\le 1\right)$, for harmonics rather than second. In addition, a different pattern is observed for $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0\right)$, or harmonics rather than first, particularly for second harmonic which is related to the sharp corner effect.

The first harmonic, $\left({\mathit{\eta }}_{\mathbf{1}}/{\mathit{A}}_{\mathbf{1}}\right)$, pattern of $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=1\right)$ includes two peaks at front and back stagnation points with the values of $\left({\mathit{\eta }}_{\mathbf{1}}/{\mathit{A}}_{\mathbf{1}}\approx \mathbf{1.75} \mathrm{a}\mathrm{n}\mathrm{d} \mathbf{1}\right)$, respectively, and one depression at the back-corner point with the values of $\left({\mathit{\eta }}_{\mathbf{1}}/{\mathit{A}}_{\mathbf{1}}\approx \mathbf{0.5}\right)$; see Figure 8b, Figure 9a, and Figure 10a. The front peak value is much larger than the back peak. These two peaks are more related to the run-up/-down with the arrival of incident wave crest/trough during the occurrence of wave scattering Type-1 and Type-2 at the front and back stagnation points, respectively, see Figure 4(I(a)), Figure 5(I(c)), and Figure 6(I(f)). As mentioned before, the lateral edge waves traveling around the cylinder experience large energy and momentum reduction caused by both wave–wave and wave–structure interaction. Therefore, there is a significant reduction in first harmonic value from the front to the back-corner point which results in a depression value. Changing the cross-section in the range of,$\left({0\le \mathrm{r}}_{\mathrm{c}}/\mathrm{R}\le 1\right)$, a similar harmonic pattern with a smooth change in the value is also observed. As expected, with the reduction in corner ratio, there is a systematic increase in first harmonics at the front stagnation point from $\left({\mathit{\eta }}_{\mathbf{1}}/{\mathit{A}}_{\mathbf{1}}\approx \mathbf{1.75}\right)$ for $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=1\right)$ to the $\left({\mathit{\eta }}_{\mathbf{1}}/{\mathit{A}}_{\mathbf{1}}\approx \mathbf{1.9}\right)$ for $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0\right)$; see Figure 8b. It was mentioned before that a reduction in corner ratio increases the cylinder circumference and number of stagnation points. It finally, results in larger wave amplification and accordingly systematic rise of 1st harmonics in front of the given cylinder.

Increasing the cylinder circumference from, $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=1\right)$, to the, $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0\right)$, enhances the wave–cylinder interaction, and causes more lateral edge wave damping. Consequently, this affects the wave scattering Type-2 and suppresses the rise of the water level at the back of the cylinder, see Figure 5c,d and Figure 6e. Furthermore, there is more reduction in 1st harmonics at the back-corner and also the back stagnation point which is more apparent for,$\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0\right)$, see Figure 8b. As seen in Figure 8c, the second harmonic, $\left({\mathit{\eta }}_{\mathbf{2}}/{\mathit{A}}_{\mathbf{1}}\right)$, pattern of $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=1\right)$ consists of three peaks at the front, shoulder, and back with the values of $\left({\mathit{\eta }}_{\mathbf{2}}/{\mathit{A}}_{\mathbf{1}}\approx \mathbf{0.18}, \mathbf{0.27}, \mathrm{a}\mathrm{n}\mathrm{d} \mathbf{0.28}\right)$, respectively. In addition, there are two depression points at the front corner and back corner, with values of, $\left({\mathit{\eta }}_{\mathbf{2}}/{\mathit{A}}_{\mathbf{1}}\approx \mathbf{0.12} \mathrm{a}\mathrm{n}\mathrm{d} \mathbf{0.2}\right)$. In Figure 9b and Figure 10b, the second harmonics show a spreading pattern like a bird’s footprint behind the given cylinder.

The large wave amplification and consequently, steep wave run-up during the occurrence of wave scattering Type-1 mainly contribute to the front peak of the second harmonic at the front of the cylinder, see Figure 4(I(a)), Figure 9b, and Figure 10b. The back peak value firstly and mainly is related to the steep and high-frequency wave field caused by wave scattering Type-2; see Figure 5(I(d)). Secondly, it is due to the reduced wave run-up caused by wave scattering Type-1 at the back of the given cylinder, secondary crest; see Figure 6(f(I)). Considering the short incident wave interacting with the cylinder,$\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=1\right)$, there is a strong interaction between edge waves and negative horizontal velocity related to the arrival of a wave trough. This results in the large peak value at the shoulder, see Figure 6(f(I)) and Figure 8c.

Following the passage of zero-down crossing and trough at the back-corner and front-corner, respectively, see Figure 4(I(a)) and Figure 6(I(e)), there are two depression points, see Figure 8c. They are related to the weak interaction of edge waves traveling upstream with the opposite incident wave. The energy of edge waves at the back corner initiated from the back is larger. On the other hand, there is a considerable suppression of edge wave energy at the shoulder which leads to weak edge wave interaction with the front corner. Thus, the value of the 2nd harmonic at the back-corner is larger compared to the front-corner value.

For the variation in the cross-section in the range of, $\left({0\le \mathrm{r}}_{\mathrm{c}}/\mathrm{R}\le 1\right)$, slight and moderate systematic reduction for the second harmonic is observed at the front and back stagnation points, respectively; see Figure 8c. The reduction in corner ratio and accordingly increase in the cross-section circumference leads to more wave–structure interaction during wave scattering Type-1 and more lateral edge wave energy suppression during wave scattering Type-2. Thus, there is a weaker edge wave collision which results in a less steep wave run-up at the front and back stagnation points, respectively. Changing the corner ratio, the harmonic pattern changes, and a considerable variation in second harmonic at back-corner, shoulder, and front-corner points are observed as well; see Figure 9b and Figure 10b. As seen in Figure 6e, the interaction of edge waves with opposite incident waves occurs at the arrival of zero-down crossing at the back-corner point. For the cross-section in the range of,$\left({0.25\le \mathrm{r}}_{\mathrm{c}}/\mathrm{R}\le 1\right)$, there is a systematic increase of 2nd harmonics at the back-corner point.

It was explained before that the energy of lateral edge waves circulating the cylinder in the upstream direction decreases with the reduction in corner ratio. In contrast, as the geometrical presence of the corner becomes visible, there is strong wave–structure interaction at the back-corner point which is highlighted for $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0.25\right)$; see Figure 10(b(IV)). For,$\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0\right)$, although the corner is sharp, regarding the back stagnation point second harmonic value, there is even more edge wave energy suppression, see Figure 10(b(V)).

This leads to weaker wave–structure interaction at the back-corner point compared to $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0.25\right)$. Wave–structure interaction at the back-corner point wave–wave interaction with the opposite incident wave and then wave–structure interaction with the given cylinder circumference slows down the progressive lateral edge waves. So, as seen in Figure 6f, the lateral edge waves interact with the opposite incident wave at different wave phases between zero-down crossing to the trough. As a result, there is weaker wave–wave interaction with the corner ratio reduction, which is apparent as a depression point for both $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0.25 \mathrm{a}\mathrm{n}\mathrm{d} 0\right)$ at the shoulder; see Figure 8c.

Energy reduction due to continuous wave–wave and wave–structure interaction, consequently, slows down lateral edge waves. It is observed for all the given cross-sections that the lateral edge waves cannot reach the front-corner point subject to short incident waves, see Figure 6f, Figure 7g, Figure 9b, and Figure 10b. However, interestingly, there is another set of lateral edge waves for $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0.25 \mathrm{a}\mathrm{n}\mathrm{d} 0\right)$ where the geometrical presence of the corner is visible. It is caused by the interaction of negative horizontal velocity with the passage of incident wave zero-down crossing and then trough at the front-corner point, see Figure 6(e(IV,V),f(IV,V)). Hence, a notable and systematic rise of the second harmonic is observed at the front-corner point for these cross-sections, see Figure 10b.

The mean value,$\left({\mathit{\eta }}_{\mathbf{0}}/{\mathit{A}}_{\mathbf{1}}\right)$, pattern of, $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=1\right)$, includes two peaks at front and back stagnation points with the positive values of $\left({\mathit{\eta }}_{\mathbf{0}}/{\mathit{A}}_{\mathbf{1}}\approx \mathbf{0.2} \mathrm{a}\mathrm{n}\mathrm{d} \mathbf{0.1}\right)$, respectively. There are also two depressions at the shoulder and back-corner point with the negative values of,$\left({\mathit{\eta }}_{\mathbf{0}}/{\mathit{A}}_{\mathbf{1}}\approx \mathbf{0.1} \mathrm{a}\mathrm{n}\mathrm{d} \mathbf{0.08}\right)$, see Figure 8a. Steep and high wave run-up during wave scattering Type-1 shifts up the mean value significantly and results in a peak at the front stagnation point, see Figure 4(I(a)) and Figure 11(I). Similarly, steep wave run-up and rise of water level due to wave scattering Type-2 and then wave scattering Type-1 shifts up the mean value which is observed as a peak at the back stagnation point, see Figure 5d and Figure 6f. The interaction of edge waves traveling upstream with the opposite incident wave at about the arrival of the zero-down crossing phase shifts down the mean value, see Figure 8a. This leads to the negative value of depression at the back-corner point, see Figure 6(I(e)). Similarly, as the edge waves interact with the opposite incident wave at about the arrival of the trough phase at the shoulder point, it also shifts down the mean value, see Figure 6(I(f)). As a result, there is a larger depression with a negative value at the shoulder compared to the back-corner point. As the corner ratio decreases, a slight and systematic increment of peak mean value at the front stagnation point is observed due to a smooth increase in wave run-up height, see Figure 11(II–V). On the opposite, a drop in wave run-up height at the back stagnation point leads to a moderate systematic reduction in peak mean value. For,$\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0.25\right)$, the corner effect becomes strong. Hence, the wave–structure interaction in addition to the interaction of edge waves with the opposite incident wave at different phases between zero-down crossing to the trough shifts down the mean value. Therefore, a considerable negative value of depression is observed at the back-corner point, see Figure 8a. The interaction of edge waves with the opposite incident wave at the shoulder point occurs at phases between zero-down crossing to the trough for the corner ratio in the range of,$\left({0.25\le \mathrm{r}}_{\mathrm{c}}/\mathrm{R}\le 1\right)$; see Figure 6f. Consequently, the mean value shifts up, systematically, from negative for $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=1\right)$ to the positive value for $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0.25\right)$ at the shoulder point. For $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0\right)$, the change in mean value follows the systematic variation at all the wave probes, except, shoulder and back-corner points. As seen in Figure 6e,f, although the corner is sharp, the weak wave–structure interaction at both back-corner and shoulder points shifts up the mean value significantly compared to the other cross-sections.

For higher harmonics rather than second, excluding,$\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0\right)$, there is a systematic change for all cross-sections at all the given wave probe locations; see Figure 8d–f, Figure 9c,d, and Figure 10c,d. In Figure 9c,d and Figure 10c,d, the third harmonics indicate diffraction waves with a stronger pattern in a much wider region behind the given cylinder and the pattern is similar to that of the second harmonics. Following the second to the fifth harmonics, the harmonic values at the back stagnation point for,$\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0\right)$, compared to the other cross-section suggests the occurrence of considerably weak wave scattering Type-2. It is related to the significant suppression of lateral edge wave energy interacting with the cross-section circumference at the back of the given cylinder, see Figure 5d.

It was mentioned before that, as the corner effects become strong,$\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0.25 \mathrm{a}\mathrm{n}\mathrm{d} 0\right)$, another set of edge waves is induced at the front-corner point. This finally results in a slightly developed wave scattering Type-2 front stagnation point, see Figure 6e,f and Figure 7g. As these new sets of edge waves move to the front stagnation point, interact with the given cross-section circumference. This interaction suppresses their energy and leads to a small rise of water and accordingly, a small steep wave run-up at the front stagnation point, see Figure 6(f(IV,V)) and Figure 7(g(IV,V)). It is worth noting that, this second wave run-up at front of the cylinder for,$\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0.25 \mathrm{a}\mathrm{n}\mathrm{d} 0\right)$, does not contribute to the 2nd harmonics but a notable contribution to the 3rd and 4th harmonics, see Figure 9c,d and Figure 10c,d. The overall analysis of higher harmonics rather than first harmonic for all cross-sections reveals that the wave amplification and free-surface nonlinearly are more observed at the backward part of the given cylinder; see Figure 9c,d and Figure 10c,d. Besides, as the geometrical presence of corners becomes visible, corners also contribute significantly to the nonlinear wave field around the given cylinder which is evident in mean value and third and particularly in second harmonics; see Figure 9b–d, Figure 10b–d and Figure 11(II–V).

3.2. Wave Interaction with a Cylinder with Different Cross-Sections Subject to Long Waves

Here, for the long-wave cases, Figure 12, Figure 13, Figure 14 and Figure 15 show the normalized spatial contour of wave evolution around each given cylinder with the top and front view. With the change in the incident wavelength from T = 7 s to T = 15 s, a smaller wave–structure interaction and accordingly reduced dynamic pressure and wave amplification at both front and back stagnation points are observed. Besides, the wave run-up at the back point is less steep and the height is smaller compared to the short-wave case. The wavelength variation is highlighted with the occurrence of fully developed wave scattering Type-2 at the front of the given cylinder, where the lateral edge wave can reach the front of the cylinder before the arrival of zero-up crossing, see Figure 14e. In general, the formation of two-wave scattering Type-2 is observed during the interaction of one wave cycle for, $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=1\right)$, at the back and then the front; see Figure 12(I(a)) and Figure 14(I(e)). Unlike the short-wave case, there is a shift back for the phase of edge wave interaction with the back-corner, shoulder, and front-corner points.

Change in the cross-section from circular to square, $\left({0\le \mathrm{r}}_{\mathrm{c}}/\mathrm{R}\le 1\right)$, increases the wave–structure interaction. The corner effect is observed as the main contribution to the further lateral edge wave energy and momentum damping. For the cross-sections in the range of, $\left({0.25\le \mathrm{r}}_{\mathrm{c}}/\mathrm{R}\le 1\right)$, the lateral edge waves experience energy reduction mostly due to the interaction with the circumference. The lateral edge waves passing the front-corner point collide and form wave scattering Type-2 at the front of the given cylinders. For square cross-sections which are identified with sharp corners,$\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0\right)$, the evolution of the free surface is somewhat different.

At the front of the cylinder with sharp corners, there are lateral edge waves that are from the remaining of the previous incident wave cycle. They get damped more than the other cross-sections and therefore, cannot propagate downstream; see Figure 13(c(II–V)). However, interestingly, a set of lateral edge waves is observed due to the interaction of positive horizontal velocity with the sharp back-corners after the passage of incident wave zero-up crossing. They collide at the back stagnation point, and initiate the formation of wave scattering Type-2 slightly sooner than the other cross-sections; see Figure 13(V(c,d)). In the case of the square cross-section, the lateral edge waves (damped) at the back cannot prorogate upstream and contribute to the formation of wave scattering Type-2 at the front.

Here, the harmonic analysis of total wave evolution at given wave probes can provide useful information on the wavelength effect on the nonlinear wave field around each cylinder. The comparison of normalized mean value and harmonics of the total wave elevation around the given cross-section subject to the long incident wave is presented in Figure 16. In addition, the harmonic contours of the diffracted wave for all the given five cross-sections in the long wave are presented. Overall, following the systematic variation in corner ratio in the range of $\left({0.25\le \mathrm{r}}_{\mathrm{c}}/\mathrm{R}\le 1\right)$, a systematic but with a slight change in harmonic pattern and value is observed compared to short wave cases. The harmonics are somewhat different for square cross-sections,$\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0\right)$, which is related to the sharp corner effect. In general, the first harmonics are small at given wave probes with an average of $\left({\eta }_1/A_1=0.9\right)$. This is related to the smaller wave diffraction and wave amplification by increasing wavelength from T = 7 s to T = 15 s. The wavelength effect on the harmonic contours of the diffracted wave for a circular cylinder is presented in Appendix A.

The first harmonic, $\left({\eta }_1/A_1\right)$, pattern of, $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=1\right)$, similar to the short-wave case, includes two peaks at front and back stagnation points with the values of, $\left({\eta }_1/A_1\approx 1.1 \mathrm{a}\mathrm{n}\mathrm{d} 0.9\right)$, respectively. In addition, there is one depression at the back-corner point with the values of, $\left({\eta }_1/A_1\approx 0.8\right)$, see Figure 16b. As mentioned before, the front and back peaks are caused by the run-up/-down during wave scattering Type-1 and Type-2 at the front and back stagnation points, respectively. Moreover, energy and momentum reduction from the front- to the back-corner point leads to the mentioned depression in first harmonic. Systematic reduction in corner ratio increases the dynamic pressure and accordingly, wave amplification which is observed as a slight and systematic rise in first harmonic at the front stagnation point; see Figure 12a. On the contrary, with a change in cross-section from circular to square, the lateral edge waves experience more energy damping due to more wave–structure interaction. Therefore, there is a weaker lateral edge wave collision that causes less steep and smaller height wave run-up and leads to a slight and systematic drop in first harmonic at shoulder, back-corner, and back stagnation points, see Figure 16b.

Figure 16c, indicates the second harmonic, $\left({\mathit{\eta }}_{\mathbf{2}}/{\mathit{A}}_{\mathbf{1}}\right)$, the pattern, which is completely different from the short wave case, see Figure 9b and Figure 10b. The harmonic pattern of $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=1\right)$ consists of two peaks at front and back stagnation points with the values of $\left({\mathit{\eta }}_{\mathbf{2}}/{\mathit{A}}_{\mathbf{1}}\approx \mathbf{0.27} \mathrm{a}\mathrm{n}\mathrm{d} \mathbf{0.09}\right)$, respectively, and one depression point at the back-corner with the value of $\left({\mathit{\eta }}_{\mathbf{2}}/{\mathit{A}}_{\mathbf{1}}=\mathbf{0.08}\right)$. The back peak value is mainly related to the steep and high-frequency wave field caused by wave scattering Type-2. This value is considerably smaller than the short-wave case because of weak lateral edge wave collision due to the smooth rate of change in velocity components. Furthermore, the contribution of wave scattering Type-1 is small due to the weak interaction of the negative horizontal velocity of the wave trough with the back stagnation point. There is a considerable suppression of edge wave energy at the back of the cylinder which results in the depression value of the second harmonic. Firstly, it is the result of the weak wave–wave interaction of edge waves traveling upstream with the opposite incident wave. Secondly, it is related to the weak wave interaction with the back-corner point before the arrival of the zero-down crossing. It was mentioned before, unlike the short-wave case, the phase in which the edge waves interact with the back-corner, shoulder, and front-corner points shift back. After the passage of the zero-down crossing at the shoulder point, a weak interaction of lateral edge waves with negative velocity components of the incident wave is observed. This weak wave–wave interaction somewhat enhances lateral edge wave energy and causes to some extent stronger wave–structure interaction compared to the back-corner point. Hence, there is an increase in the second harmonic value from the back corner to the shoulder point, see Figure 16c, Figure A2b and Figure A3b. Similarly, the interaction of lateral edge waves passing the front-corner point with the negative horizontal velocity of the incident wave before the arrival of the wave trough considerably increases their energy.

It is observed that a continuous rise in the second harmonic value, initiated from the shoulder point, indicates that the peak value at the front is significantly higher than the other points around the cylinder, see Figure A2b and Figure A3b. This reveals the important contribution of the occurrence of wave scattering Type-2 at the front, see Figure 14e, besides the small contribution by less steep wave run-up during the occurrence of wave scattering Type-1, compared to the short-wave case; see Figure 8c and Figure 16c. Unlike the short-wave case, there is no considerable variation in second harmonic at corners, and the corner effect is observed as more further damping of edge wave energy. Therefore, change in cross-section from circular to square, $\left({0\le \mathrm{r}}_{\mathrm{c}}/\mathrm{R}\le 1\right)$. As seen in Figure 16c, there is a systematic and small reduction in second harmonics from the back to shoulder points. Afterward, the slight augmentation of lateral edge wave energy leads to a slight rise in second harmonics from the shoulder to the front point. For,$\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0\right)$, the harmonic pattern is similar and the change in value is systematic (except the front point) but moderately evident.

It was shown that there is more edge wave energy reduction for $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0\right)$ compared to the other cross-sections, and the edge waves get damped before being able to travel around the cylinder; see Figure A3b(V). Therefore, there is no record of lateral edge waves from the back to the shoulder point, and consequently, a notable reduction in second harmonics is observed. The interaction of the induced lateral edge waves by the front-corner point is weaker than one for the other cross-sections at the front to form wave scattering Type-2 and it is apparent as a considerable reduction of second harmonics at the front for, $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0\right)$.

The mean value, $\left({\mathit{\eta }}_{\mathbf{0}}/{\mathit{A}}_{\mathbf{1}}\right)$, pattern of $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0\right)$ includes two peaks at front and back stagnation points with the positive values of $\left({\mathit{\eta }}_{\mathbf{2}}/{\mathit{A}}_{\mathbf{1}}\approx \mathbf{0.1} \mathrm{a}\mathrm{n}\mathrm{d} \mathbf{0.06}\right)$, respectively; see Figure A3(IV). There is also one depression at the shoulder point with the negative value of $\left({\mathit{\eta }}_{\mathbf{0}}/{\mathit{A}}_{\mathbf{1}}\approx \mathbf{0.13}\right)$; see Figure 16a. The occurrence of two wave run-ups during wave scattering Type-1 and Type-2 shifts up the mean value and results in a notable peak value at front stagnation. The back peak which is smaller than the front peak is mainly related to the shift-up in the mean value caused by wave run-up and the rise of water level during wave scattering Type-2; see Figure A4. The negative value of depression is the outcome of lateral edge waves’ weak interaction with the shoulder point following the passage of incident wave zero-down crossing that shifts down the mean value.

The interaction of lateral edge waves with the back corner at the wave phase before the arrival of zero-down crossing results in a shift-down in the mean value (negative). In contrast, the interaction of lateral edge waves with the negative horizontal velocity at the front corner causes a considerable rise in mean value (positive) which is also larger than the value at the back corner. As the corner ratio decreases in the range of, $\left({0.25\le \mathrm{r}}_{\mathrm{c}}/\mathrm{R}\le 1\right)$, a slight rise in the mean value is observed due to the smooth increase in wave run-up at the front, see Figure A4(II–V), whereas a smooth decrease in wave run-up at the back results in a slight drop of the mean value; see Figure A4(IV). In continuation, a weaker interaction of lateral edge waves at the back-corner and then shoulder leads to a moderate rise in mean value at these points which is evident fore, $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0.25\right)$; see Figure A4(V).

The mean value at the front corner indicates a moderate drop, which is related to the contribution of the corner effect in lateral edge wave enhancement at this point. For, $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0\right)$, the harmonic pattern is similar but unlike the other cross-sections, the value does not follow a systematic change. The second wave run-up height and accordingly the mean value is smaller at the front stagnation point, comporting to the other cross-sections. As seen in Figure 13(V(c)), there is no record of the interaction of lateral edge waves at the shoulder. Furthermore, there is a considerably weak interaction of lateral edge waves at the back corner and also a weak interaction of induced lateral edge waves at the front corner. In general, these interactions shift up significantly the mean value at these points compared to the other cross-sections.

For higher harmonics rather than second, the harmonic patterns are similar and there is mostly systematic slight change for all the cross-sections at all the given wave probe locations. The overall analysis of higher harmonics rather than first harmonic, reveals that in all the cross-sections subject to the long wave, the wave amplification and free-surface nonlinearly are more observed at the forward part of the given cylinder, see Figure A2c,d and Figure A3c,d. This is mainly related to the interaction of negative velocity components of the incident wave with the forward part of the given cylinder and the occurrence of wave scattering Type-2 at the front. Besides, as the geometrical presence of the corner becomes visible, unlike short-wave cases, corners weakly contribute to the nonlinear wave field around the given cylinder.

3.3. Maximum Run-Up Height

In the case of offshore structures, assessing the wave run-up height and measuring the maximum value is the most concerning, particularly in extreme wave conditions. Figure A5 and Figure A6 show the contour of the maximum run-up height ratio, (${\mathsf{\eta }}_{\mathrm{M}\mathrm{a}\mathrm{x}}/{\mathrm{A}}_1$), around each given cylinder and the relationship between the $\left({\mathsf{\eta }}_{\mathrm{M}\mathrm{a}\mathrm{x}}/{\mathrm{A}}_1\right)$ and the $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}\right)$ for both short and long incident waves is presented in Figure A6, as well. Here, the maximum run-up height, $\left({\mathsf{\eta }}_{\mathrm{M}\mathrm{a}\mathrm{x}}\right)$, is calculated by averaging the crest values of the wave elevations from the last 15 wave periods in steady state solution.

Based on the assumption of non-breaking regular incoming waves, propagating from left to right, Figure A5 and Figure A6 reveal that the maximum run-up height occurs at the front of each cylinder and it is related to the wave scattering Type-1 whilst there is a second but very reduced run-up height (compared to the front) at the back stagnation point and it is related to the wave scattering Type-2. It is also found that the minimum value of the maximum run-up which is $\left({\mathsf{\eta }}_{\mathrm{M}\mathrm{a}\mathrm{x}}/{\mathrm{A}}_1\approx 1\right)$ is observed at the part between the shoulder to the back stagnation point, for all the given cylinders in both short and long incident waves.

It was mentioned before that regardless of the wavelength effect, changing the cross-section from the circular to the sharp square and accordingly the reduction in the corner ratio from $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=1\right)$ to $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0\right)$ increases the column circumference and the number of stagnation points at the front and back of the cylinder. This eventually increases the wave–structure interaction and accordingly, the dynamic pressure which leads to a larger diffraction and wave amplification. It is apparent as the maximum run-up height at the front part of the given cylinder. It is also known that increasing the wavelength and accordingly the wave diffraction reduction leads to a decrease in the wave amplification and the resultant wave run-up height around each cylinder, particularly at the front, see Figure A5 and Figure A6.

Figure A6 indicates that when increasing the $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}\right)$, there is a reduction in $\left({\mathsf{\eta }}_{\mathrm{M}\mathrm{a}\mathrm{x}}/{\mathrm{A}}_1\right)$, from 2.57 (circle) to 2.21 (square) for short-wave cases and in a similar way a reduction from 1.61 (circle) to 1.47 (square) for long-wave cases. The relationship between the $\left({\mathsf{\eta }}_{\mathrm{M}\mathrm{a}\mathrm{x}}/{\mathrm{A}}_1\right)$ and the $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}\right)$ is based on the quadratic curve fitting where the reduction rate for the short waves is larger than the long-wave cases.

4. Conclusions

The numerical results from the Navier–Stokes/VOF model were validated against the experimental data for wave amplification around a circular cylinder, showing good agreement. This confirms the model’s accuracy in simulating nonlinear wave–column interactions. Key findings on the impact of corner effects and wavelength variation are as follows:

• Corner Ratio Reduction: Decreasing the corner ratio increases diffraction, wave amplification, and run-up height, especially at the front, regardless of wavelength effects.
• Short Incident Waves: For cross-sections with $\left({0<r}_{\mathrm{c}}/\mathrm{R}<0.75\right)$, the wave behavior around the cylinder resembles that of the circular cylinder. Notably, for $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0.25 \mathrm{a}\mathrm{n}\mathrm{d} 0\right)$, Type-2 wave scattering develops at the front due to corner interaction with the incident wave’s horizontal velocity. For $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0\right)$, the second harmonic shows a distinct pattern due to the sharp corner effect, whereas the first harmonic remains unaffected. Significant nonlinearity is observed at the back of each cross-section, highlighting the corner effect’s impact, especially on the mean value, second harmonics, and fifth harmonics.
• For long incident waves, increasing the wavelength shifts the phase of lateral edge wave interactions with the cylinder’s corners, leading to fully developed Type-2 wave scattering at the front. For $\left({0.25\le \mathrm{r}}_{\mathrm{c}}/\mathrm{R}\le 1\right)$, this scattering is caused by upstream-traveling edge waves, while for $\left({\mathrm{r}}_{\mathrm{c}}/\mathrm{R}=0\right)$, it results from waves induced by the sharp front corner. Compared to short waves, the corner effect reduces edge wave energy more than affects the nonlinear wave field. Type-2 scattering enhances the mean value and harmonics up to the fifth, shifting nonlinearity forward. Overall, for $\left({0.25\le \mathrm{r}}_{\mathrm{c}}/\mathrm{R}\le 1\right)$, harmonic patterns show slight changes, with notable differences for the square cross-section due to the sharp corner effect.

Based on the findings of our study, we recommend several areas for future research: Inclusion of Dynamic Effects: Investigate the impact of small vibrations and dynamic responses of cylindrical structures on wave interactions, including nonlinear wave amplification and harmonic structures. Wave Breaking Phenomena: Extend the model to incorporate wave-breaking effects and explore their influence on wave–cylinder interactions, particularly in practical scenarios where wave-breaking is common.