Numerical Investigation on the Interaction between a Tsunami-like Solitary Wave and a Monopile on a Sloping Sandy Seabed

Abstract

An integrated numerical model was developed to investigate the interaction between a tsunami-like solitary wave and a monopile on a sloping sandy seabed in this study. The solitary wave motion is governed by the RANS equations with the k-ε turbulence model. The porous sloping sandy seabed is governed by Biot’s equation (u-p approximation). The solitary wave is validated with previous experimental data. Meanwhile, a further comparison of solitary wave scattering by the monopile is carried out to verify the numerical model. Then, the effects of different monopile locations were examined in investigating the solitary wave–monopile interaction problem. The velocity magnitudes and the free-surface elevation changes in the solitary wave around the monopile are investigated at various monopile locations. In addition, the response of the sloping sandy seabed and monopile under the solitary wave are examined. The numerical results demonstrate the accuracy of the current method in simulating solitary waves and wave height variation around monopiles. Wave run-up is observed in front of the monopile, with a high-velocity forward-moving water jet forming behind it. The maximum fluid velocity, wave run-up height in front of the monopile, excess pore water pressure (EPWP), and bending moment of the monopile increase as the monopile approaches the shoreline. However, at the closest location to the shoreline, due to the strong dynamic interaction between the solitary wave and the monopile, significant wave shoaling and breaking are observed, resulting in a slight decrease in the wave force acting on the monopile.

1. Introduction

The destructive power of tsunamis against coastal areas is enormous, as evidenced by the 2004 Indian Ocean tsunami and the 2011 Tohoku tsunami, which caused significant damage to coastal structures [1,2,3]. Therefore, the stability of coastal structures under the influence of tsunamis is of great importance to coastal engineers and researchers. Offshore wind energy has rapidly developed into an attractive renewable energy source in the world, and a large number of offshore wind turbines have been built [4,5]. Therefore, understanding the behavior of monopiles, the commonly used foundation type for offshore wind turbines, and the responses of the seabed under tsunami loading is essential.

Due to the unpredictability of tsunamis, most previous research was carried out through experimental and numerical methods, with solitary waves commonly used to simulate tsunami waves [6,7,8,9,10,11]. Many researchers have studied the mechanism of solitary wave run-up and breaking on a sloping seabed through potential flow theory and turbulent flow theory [12,13,14,15,16,17,18], but note that they did not consider the existence of offshore structures. Recently, the interactions between solitary waves and offshore structures, such as breakwaters and monopiles on sandy seabeds have attracted much attention [19,20]. Given that breakwaters along coastlines are extremely long, the interactions between solitary waves and breakwaters are commonly considered as two-dimensional (2D) problems in numerical investigations [7].

This is completely different from the interaction between solitary waves and monopiles, where strong three-dimensional (3D) features can be observed. When a solitary wave passes through a monopile, the characteristics of the solitary wave’s run-up in front of the monopile, the acceleration of the solitary wave on the monopile’s sides, and the reconnection wave behind the monopile are obvious [21,22,23,24]. Zhao et al. numerically studied the interaction between a solitary wave and an array of surface-piercing vertical circular cylinders [25]. Mo et al. investigated the interaction between a solitary wave and a vertical cylinder on a constant slope through both experimental and numerical methods [26]. Their results indicated that laboratory waves appear to be a little faster than numerical simulation waves. Recently, some researchers studied the interaction between solitary waves and cylinders by using different CFD models, such as OpenFOAM and REEF3D [27,28,29].

Numerous investigations on wave-induced seabed response have been conducted over the past decades [30,31,32,33,34,35,36]. However, only few researchers have focused on seabed response under solitary waves. Ye et al. numerically studied the stability of a breakwater—that was built on a sloped coast—against a solitary wave [7]. Zhao developed a 2D integrated model to investigate the effects of wave and soil characteristics on pore pressure and liquefaction [13]. Additionally, differences in wave motions on a 1:15 slope and a 1:6 slope were studied in detail. Leng et al. developed a 2D coupled approach to investigate tsunami-like solitary wave-induced soil response, where the effect of various slopes (1:5, 1:10, 1:15, and 1:20) on slope response was studied [37].

In this study, numerical simulations were conducted to study the dynamic interaction between a tsunami-like solitary wave and a monopile on a sloping sandy seabed. Solitary wave motion is governed by the RANS equations and the k-ε model. Under solitary wave loading, the response of the sloping sandy seabed is governed by Biot’s equation (u-p approximation). The numerical results are verified by previous experimental and numerical results. Subsequently, the solitary wave motions around the monopile, as well as the dynamic responses of the monopile and sloping seabed, are examined at various monopile locations.

2. Integrated Numerical Model

2.1. Governing Equations for Waves

Fluid motion is described by the RANS equations, mass conservation, and momentum conservation equations.

$${\displaystyle \frac{\partial 〈u_{fi}〉}{\partial x_i}}=0$$

(1)

$${\displaystyle \frac{\partial {\rho }_f〈u_{fi}〉}{\partial t}}+{\displaystyle \frac{\partial {\rho }_f〈u_{fi}〉〈u_{fj}〉}{\partial x_j}}=-{\displaystyle \frac{\partial 〈p_f〉}{\partial x_i}}+{\displaystyle \frac{\partial }{\partial x_j}}\left[\mu \left({\displaystyle \frac{\partial 〈u_{fi}〉}{\partial x_j}}+{\displaystyle \frac{\partial 〈u_{fj}〉}{\partial x_i}}\right)\right]+{\displaystyle \frac{\partial }{\partial x_j}}\left(-{\rho }_f〈u_i^{\prime }{u}_j^{\prime }〉\right)+{\rho }_f{g}_i$$

(2)

in which $〈u_{fi}〉$ is the ensemble mean fluid velocity, x_i are the coordinates x, y, and z, ρ_f is the fluid density, μ is dynamic viscosity, $〈p_f〉$ is fluid pressure, t is time, and g_i is the gravitational acceleration. The term $-{\rho }_f〈u_i^{\prime }{u}_j^{\prime }〉$ in Equation (2) is the Reynolds stress which can be written as

$$-{\rho }_f〈u_i^{\prime }{u}_j^{\prime }〉={\mu }_t\left[{\displaystyle \frac{\partial 〈u_{fi}〉}{\partial x_j}}+{\displaystyle \frac{\partial 〈u_{fj}〉}{\partial x_i}}\right]-{\displaystyle \frac{2}{3}}{\rho }_f{\delta }_{ij}k$$

(3)

where μ_t is the turbulent viscosity, k is the turbulence kinetic energy, and δ_ij is the Kronecker delta. Substituting Equation (3) into Equation (2), the Equation (2) can be rewritten as

$${\displaystyle \frac{\partial {\rho }_f〈u_{fi}〉}{\partial t}}+{\displaystyle \frac{\partial {\rho }_f〈u_{fi}〉〈u_{fj}〉}{\partial x_j}}=-{\displaystyle \frac{\partial }{\partial x_i}}\left[〈p_f〉+{\displaystyle \frac{2}{3}}{\rho }_{f}k\right]+{\displaystyle \frac{\partial }{\partial x_j}}\left[{\mu }_{eff}\left({\displaystyle \frac{\partial 〈u_{fi}〉}{\partial x_j}}+{\displaystyle \frac{\partial 〈u_{fj}〉}{\partial x_i}}\right)\right]+{\rho }_f{g}_i$$

(4)

where ${\mu }_{eff}=\mu +{\mu }_t$.

The k-ε turbulence model is employed to enclose turbulence [38]:

$${\displaystyle \frac{\partial \rho k}{\partial t}}+{\displaystyle \frac{\partial \rho 〈u_j〉k}{\partial x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_k}}\right){\displaystyle \frac{\partial k}{\partial x_j}}\right]+\rho P_k-\rho \epsilon$$

(5)

$${\displaystyle \frac{\partial \rho \epsilon }{\partial t}}+{\displaystyle \frac{\partial \rho 〈u_j〉\epsilon }{\partial x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_{\epsilon }}}\right){\displaystyle \frac{\partial \epsilon }{\partial x_j}}\right]+{\displaystyle \frac{\epsilon }{k}}\left(C_{\epsilon 1}\rho P_k-C_{\epsilon 2}\rho \epsilon \right)$$

(6)

$${\mu }_t=\rho C_{\mu }{\displaystyle \frac{k^2}{\epsilon }}$$

(7)

in which k is TKE, ε is the rate of turbulent kinetic energy dissipation, and μ_t is the turbulent kinematic viscosity. The values of constants in Equations (5)–(7) are as follows: σ_k = 1.00, σ_ε = 1.00, C_ε1 = 1.00, C_ε2 = 1.00, and C_μ = 0.99 [39].

2.2. Governing Equations of Porous Sloping Sandy Seabeds

In this study, Biot’s partly dynamic theory (u-p approximation) [31] is used to control the dynamic response of the sloping sandy seabed. The governing equations can be written as

$$k_z{\nabla }^{2}p-{\gamma }_{w}n\beta {\displaystyle \frac{\partial p}{\partial t}}+k_s{\rho }_f{\displaystyle \frac{{\partial }^2{\epsilon }_s}{\partial t^2}}={\gamma }_w{\displaystyle \frac{\partial {\epsilon }_s}{\partial t}}$$

(8)

$$\begin{array}{l}{\displaystyle \frac{\partial {\sigma }_x^{\prime }}{\partial x}}+{\displaystyle \frac{\partial {\tau }_{xy}}{\partial y}}+{\displaystyle \frac{\partial {\tau }_{xz}}{\partial z}}={\displaystyle \frac{\partial p}{\partial x}}+\rho {\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}\\ {\displaystyle \frac{\partial {\tau }_{xy}}{\partial x}}+{\displaystyle \frac{\partial {\sigma }_y^{\prime }}{\partial y}}+{\displaystyle \frac{\partial {\tau }_{yz}}{\partial z}}={\displaystyle \frac{\partial p}{\partial y}}+\rho {\displaystyle \frac{{\partial }^{2}v}{\partial t^2}}\\ {\displaystyle \frac{\partial {\tau }_{xz}}{\partial x}}+{\displaystyle \frac{\partial {\tau }_{yz}}{\partial y}}+{\displaystyle \frac{\partial {\sigma }_z^{\prime }}{\partial z}}={\displaystyle \frac{\partial p}{\partial z}}+\rho {\displaystyle \frac{{\partial }^{2}w}{\partial t^2}}\end{array}$$

(9)

in which p is the pore pressure; γ_w is the unit weight water, n is the soil porosity, ρ_f is the fluid density, ρ is the average density of porous sloping sandy seabed, u, v, and w are the soil displacements, ${\sigma }_x^{\prime }$, ${\sigma }_y^{\prime }$ and ${\sigma }_z^{\prime }$ are the effective normal stresses, and τ_xy, τ_yz and τ_xz are shear stresses. ε_s can be written as

$${\epsilon }_s={\displaystyle \frac{\partial u}{\partial x}}+{\displaystyle \frac{\partial v}{\partial y}}+{\displaystyle \frac{\partial w}{\partial z}}$$

(10)

where β is the compressibility of the pore fluid that can be written as

$$\beta ={\displaystyle \frac{1}{K^{\prime }}}={\displaystyle \frac{1}{K_w}}+{\displaystyle \frac{1-S_r}{P_{w0}}}$$

(11)

where K_w is the bulk modulus of pore water, and P_w0 is the absolute water pressure. The effective normal stress and shear stress can be written as

$$\begin{array}{l}{\sigma }_x^{\prime }=2G\left[{\displaystyle \frac{\partial u}{\partial x}}+{\displaystyle \frac{\mu }{1-2\mu }}{\epsilon }_s\right] , {\tau }_{xz}=G\left[{\displaystyle \frac{\partial u}{\partial z}}+{\displaystyle \frac{\partial w}{\partial x}}\right]={\tau }_{zx}\\ {\sigma }_y^{\prime }=2G\left[{\displaystyle \frac{\partial v}{\partial y}}+{\displaystyle \frac{\mu }{1-2\mu }}{\epsilon }_s\right] , {\tau }_{yz}=G\left[{\displaystyle \frac{\partial v}{\partial z}}+{\displaystyle \frac{\partial w}{\partial y}}\right]={\tau }_{zy}\\ {\sigma }_z^{\prime }=2G\left[{\displaystyle \frac{\partial w}{\partial z}}+{\displaystyle \frac{\mu }{1-2\mu }}{\epsilon }_s\right] , {\tau }_{xy}=G\left[{\displaystyle \frac{\partial u}{\partial y}}+{\displaystyle \frac{\partial v}{\partial x}}\right]={\tau }_{yx}\end{array}$$

(12)

where G is the shear modulus.

3. Boundary Conditions and Numerical Method

The integrated numerical model consists of a solitary wave sub-model and sloping bed sub-model, as shown in Figure 1. In the solitary wave sub-model, the symmetry boundary condition is applied to the two lateral boundaries (y-direction) to weaken the wave reflection. Since the sloping seabed is sufficiently long to simulate the solitary wave run-up and run-down, the wave does not reach the outlet area. Therefore, the outflow boundary is applied at the outlet boundary even if it has no effects on the calculation. The solitary wave solution is based on McCowan’s theory, which offers higher order accuracy than Boussinesq’s theory [40]. The governing equations for the water elevation μ, x-velocity u_f1, z-velocity u_f3, and wave speed c can be expressed as

$${\displaystyle \frac{\eta }{h}}={\displaystyle \frac{N}{M}}{\displaystyle \frac{\sin \left[M\left(1+\frac{\eta }{h}\right)\right]}{\cos \left[M\left(1+\frac{\eta }{h}\right)\right]+\cosh \left(M\frac{x-ct}{h}\right)}}$$

(13)

$${\displaystyle \frac{u_{f1}\left(x, z, t\right)-\overline{u}}{\sqrt{g\left(h+H\right)}}}=N{\displaystyle \frac{1+\cos \left(\frac{Mz}{h}\right)\cosh \left(\frac{MX}{h}\right)}{{\left[\cos \left(\frac{Mz}{h}\right)+\cosh \left(\frac{MX}{h}\right)\right]}^2}}$$

(14)

$${\displaystyle \frac{u_{f3}\left(x, z, t\right)}{\sqrt{g\left(h+H\right)}}}=N{\displaystyle \frac{\sin \left(\frac{Mz}{h}\right)\sinh \left(\frac{MX}{h}\right)}{{\left[\cos \left(\frac{Mz}{h}\right)+\cosh \left(\frac{MX}{h}\right)\right]}^2}}$$

(15)

$$c=\overline{u}+\sqrt{g\left(h+H\right)}$$

(16)

where H is the wave height, h is the undisturbed water depth, $X=x-ct$, g is the gravitational acceleration, and M and N satisfy

$${\displaystyle \frac{H}{d}}={\displaystyle \frac{N}{M}}\tan \left[{\displaystyle \frac{1}{2}}M\left(1+{\displaystyle \frac{H}{h}}\right)\right]$$

(17)

$$N={\displaystyle \frac{2}{3}}{\sin }^2\left[M\left(1+{\displaystyle \frac{2H}{3h}}\right)\right]$$

(18)

The initial estimates of M and N are $M=\sqrt{3H/h}$ and $N=2H/h$. The initial estimate of water elevation η is from Boussinesq’s solution.

$${\displaystyle \frac{\eta }{h}}={\displaystyle \frac{H}{h}}{\mathrm{sech}}^2\left(\sqrt{{\displaystyle \frac{3H}{4h}}}{\displaystyle \frac{x-ct}{h}}\right)$$

(19)

The VOF method is used to capture the free water surface in this research [41]. The volume fraction F is defined as

$$F=\left\{\begin{array}{cc}F=0\hfill & \hfill \mathrm{air}\hfill \\ 0<F<1\hfill & \hfill \mathrm{interface}\hfill \\ F=1\hfill & \hfill \mathrm{water}\hfill \end{array}\right.$$

(20)

in which F satisfies the following equation:

$${\displaystyle \frac{\partial F}{\partial t}}+{\displaystyle \frac{\partial \left(u_{i}F\right)}{\partial x_i}}=0$$

(21)

In the porous sloping sandy seabed sub-model, the four vertical sides of the sloping sandy seabed are all fixed and impermeable.

$$u=v=w={\displaystyle \frac{\partial p}{\partial n}}=0$$

(22)

where n donates the normal vector of the boundaries. The sloping sandy seabed bottom is treated as impermeable and rigid. The wave-induced dynamic pressure is applied on the sloping sandy seabed surface through linear interpolation.

$$p=p_b=p_f-{\rho }_{f}gd$$

(23)

where p_b is the dynamic wave pressure on the sloping sandy seabed surface.

In the solitary wave sub-model, the commercial software Flow3D v11.2 was adopted to simulate the wave propagation and determine the wave pressure acting on the seabed and monopile. The scale of the model is 250 m long, 30 m wide, and 20 m high. The bed slope is 1/10. The computation domain is subdivided into the mesh of fixed hexahedron cells. The fluid motion is solved through the finite-difference method (FDM). The computational mesh of the solitary wave sub-model is shown in Figure 2. The total element number is 12,000,000. The minimum mesh size around the monopile is 0.2 m (0.1D) in x-, y-, and z-directions to achieve a satisfactory accuracy in simulating the wave motion around the monopile. The computation time for each numerical model is approximately 1 week. The presented numerical model as well as the commercial software Flow3D v11.2 were carried out with up to 16 cores on a desktop computer (CPU 3.8 GHz and 16.0 GB RAM).

In the sloping sandy seabed sub-model, FEM codes are constructed within COMSOL Multiphysics to solve Biot’s u-p equations. The sloping sandy seabed sub-model is discretized into structured tetrahedral elements, as shown in Figure 3. The maximum element size of the monopile is 0.4 m. The total element number is 103,223. In the integrated model, the solitary wave sub-model is adopted to simulate the wave motion and determines the dynamic pressures acting on the seabed and monopile. The porous sloping bed sub-model simulates the responses of the sloping sandy seabed and monopile under solitary wave loading. More details about the numerical method can be found in Zhang’s study [32].

4. Results and Discussion

4.1. Comparison of the Solitary Wave Surface Elevation

To examine the accuracy of the numerical model, a comparison of the free-surface elevation in a solitary wave was conducted firstly [26]. In the experiment, the normalized solitary wave height is H/h = 0.33, and the water depth is h = 0.205 m. As shown in Figure 4, the numerical simulation of the solitary wave agrees well with Mo’s experimental result, which confirms that the numerical approach is accurate enough to generate solitary waves.

The numerical model is further validated against a classic example of solitary wave reflection from a vertical wall. Figure 5 illustrates the comparison of the maximum solitary wave run-up height on the vertical wall with the previous experimental results [42,43], analytical results [44,45], and numerical results [46,47], in which Byatt-Smith’s analytical result is based on the Bousinessq equation that can be written as

$${\displaystyle \frac{R_{\max }}{h}}=2{\displaystyle \frac{H}{h}}+{\left({\displaystyle \frac{H}{h}}\right)}^2$$

(24)

where R_max is the maximum solitary wave run-up height on the vertical wall. Su and Mirie gave the following expression [45]:

$${\displaystyle \frac{R_{\max }}{h}}=2{\displaystyle \frac{H}{h}}+{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{H}{h}}\right)}^2+{\displaystyle \frac{3}{4}}{\left({\displaystyle \frac{H}{h}}\right)}^3$$

(25)

Figure 5 shows that the differences among these results are relatively small. The present numerical result is closer to Su and Mirie’s analytical result than to Byatt-Smith’s analytical result, especially for larger H/h, which is probably due to the fact that a larger wave height has strong nonlinearity so that the third-order component may have a greater impact. Additionally, the present result is close to Chen’s experimental result and Chambarel’s numerical result.

4.2. Comparison of the Maximum Run-Up Height on a Vertical Cylinder

A further comparison of solitary wave scattering by a vertical cylinder is conducted to verify the present model. Previous experimental and numerical results [27] are used for the validation. The experiment was conducted in a wave tank with dimensions of 7.6 m in length, 0.76 m in width, and 0.6 m in depth. The water depth is h = 0.04 m, the radius of the cylinder is R = 0.0635 m, and the normalized wave height is H/h = 0.4. Eight wave probes were arranged around the cylinder, as shown in Figure 6. The comparisons of the free-surface elevation along the upstream center line ($\theta =0^{°}$, where $\theta$ is the polar angle) are illustrated in Figure 7, in which r/R is the radial distance from the cylinder center to the wave probe. The numerical results by Chen et al. are also included in the comparison [28].

It can be observed that the calculated free-surface elevations at different wave probes upstream of the cylinder generally agree well with previous results. As the wave probes move closer to the cylinder, the maximum wave surface elevation increases. Additionally, a secondary peak is separately observed for r/R = 4.5 and 2.61, attributed to reflected waves as illustrated in Figure 7a,b. A similar result was also observed in Zhao’s numerical results [25].

Comparing with two numerical results, the present result is closer to the Yates’s experimental data for r/R = 4.5, 2.61, and 1.66. However, both the present and previous numerical results slightly overestimate the maximum wave surface elevation at r/R = 1.03. Figure 8 shows the time histories of free-surface elevations at four wave probes along the downstream centerline of the cylinder ($\theta =180^{°}$). The present results and Chen’s numerical results show good agreement with previous findings. Both models offer improved accuracy in calculating the free-surface elevation compared to Yates and Wang’s numerical results. In general, these comparisons confirm that the numerical model used in this research is accurate for simulating the interaction between solitary waves and cylinders.

4.3. Solitary Wave Interaction with the Monopile on the Sloping Sandy Seabed

The sloping seabed is modeled with dimensions of 300 m in length, 20 m in width, and a slope ratio of 1:10. As shown in Figure 1, five different monopile locations (C1, C2, C3, C4, and C5) are simulated to investigate the solitary wave–monopile interaction problem. The other solitary wave, monopile, and sloping sandy seabed characteristics are listed in

Table 1. Calculation parameters of the numerical model.

Water Characteristics
Water Depth (h)	10 (m)	Wave Height (H)	3 (m)
Slopping seabed characteristics
Density (ρs)	1850 (kg/m3)	Young’s modulus (Es)	4.05 × 107 (N/m2)
Poisson ratio (vs)	3.448	Permeability (kz)	1 × 10−3 (m/s)
Porosity (n)	0.4	Degree of saturation (Sr)	0.98
Monopile characteristics
Young’s modulus (Em)	0.5	Density (ρm)	2300 (kg/m3)
Poisson ratio (vm)	0.25	Monopile diameter (D)	2 (m)

.

Figure 9 illustrates the velocity magnitudes and the variations in free-surface elevation in the solitary wave interacting with the monopile positioned at C1 at different times. Given that C1 is situated at the tip of a sloping sandy seabed, distant from the shoreline, the solitary wave’s interaction with the monopile mirrors that of a flat bed. As the solitary wave approaches the monopile, the wave run-up is observed in front of the monopile. The solitary wave separates into two symmetric waves with respect to the center line in y-direction, which then accelerate downstream, as shown in Figure 9c. The free-surface elevation decreases along the monopile surface from upstream to downstream. The two symmetric waves merge again behind the monopile. Meanwhile, a forward-moving water jet with high velocity is formed behind the monopile. As the wave progresses downstream, the wave run-down in front of the monopile is observed, with the height and velocity of the water jet decreasing gradually (Figure 9d).

Figure 10, Figure 11 and Figure 12 illustrate the velocity magnitudes and changes in free-surface elevation when the monopile is located at C2, C3, and C4. It is evident that the wave processes through the monopiles at C2 and C3 are similar to those at C1. For these three conditions, the solitary wave is distant from the shoreline as it passes through the monopile, with no significant wave shoaling or breaking observed on the sloping seabed (Figure 9, Figure 10 and Figure 11). However, the maximum fluid velocity and wave run-up height in front of the monopile increase as the monopile moves closer to the shoreline. As shown in Figure 12, the wave approaches the shoreline at the downstream of the monopile, where wave shoaling, run-up, and breaking are prominently observed. The wave front sharpens gradually as it reaches to the shoreline.

Figure 13 shows the entire process of wave shoaling, breaking, run-up, and drawdown on the sloping sandy seabed when the monopile is located at C5. The solitary wave breaks before its peak reaches the monopile. Simultaneously, the velocity magnitude of the water increases gradually as the wave climbs the sloping sandy seabed leading to a strong dynamic interaction between the solitary wave and the monopile, as shown in Figure 13c. The water rises up higher in front of the monopile, and the fluid velocity intensifies, with some sea water even splashing into the air. At this moment, the water jet is formed gradually behind the monopile. After the solitary wave passes the monopile, the water jet and the sea water, along with the splashed waters falling from the air, rapidly climb the sloping sandy seabed (Figure 13d,e). The height of the water jet and the maximum fluid velocity gradually decrease. At t = 30 s, the solitary wave reaches its farthest distance on the sloping sandy seabed and begins to draw down under the effect of gravity (Figure 13f). As the fluid flows down from the sloping sandy seabed, the water rises behind the monopile, forming a water jet in front of the monopile (Figure 13g,h).

Figure 14 presents the time histories of free-surface elevation at three wave probes for the monopile positioned at C1, C2, C3, C4, and C5, respectively. The maximum wave elevation at wave probe A is significantly higher than at wave probes B and C when the monopile is located at C1 (Figure 14a). Additionally, a secondary peak is observed in front of the monopile at t = 14.5 s, due to the effects of the reflection wave. As the monopile shifts from C1 to C4, the maximum free-surface elevation in wave probe A increases, and the secondary peak gradually disappears (Figure 14a–c). When the monopile is located at C5, the maximum free-surface elevation in front of the monopile rapidly increases to approximately 15.2 m, then decreases to a constant value of about 14.8 m. The maximum free-surface elevation at wave probe B surpasses that at wave probe A, due to the strong dynamic interaction between the solitary wave and the monopile. These results are also reflected in Figure 15a. It is evident that as the monopile location moves from C1 to C5, the maximum free-surface elevations at wave probes A and B increase significantly. However, the maximum free-surface elevations at wave probe C remain almost unchanged across different monopile locations. Figure 15b displays the maximum fluid velocity when the solitary passes the monopile at different locations. Comparing these results, it is evident that the fluid velocity gradually increases as the solitary climbs the sloping sandy seabed. The maximum fluid velocity increases from approximately 4.1 m/s to 15.5 m/s as the monopile moves from C1 to C5.

Figure 16 illustrates the cloud figures of the solitary wave-induced excess pore water pressure (EPWP) in the sloping sandy seabed, capturing the moment when the solitary wave passes the monopile. At this point, the EPWP around the monopile achieves its peak value. It is observed that the EPWP decreases in vertically from the surface of the sloping sandy seabed. However, the effect of the solitary wave on the seabed is confined to a narrow region.

Figure 17 illustrates the distributions of EPWP along the vertical direction at three gauge locations around the monopile, with the monopile bottom located at z = −25 m. As depicted in Figure 17a, the EPWP in front of the monopile (G1) along the vertical direction decreases rapidly. The maximum EPWP at the seabed surface increases as the monopile moves from C1 to C4. However, when the monopile is located at C5, the maximum EPWP at G1 is significantly lower than that at C3 and C4. This is mainly because C5 is at the shoreline, where the solitary wave is already breaking upon arrival. Additionally, the effects of the monopile bottom on the EPWP distributions around z = −25 m are significant. Figure 17b illustrates the distributions of the EPWP at the lateral side of the monopile (G2). Similar trends in EPWP distributions are observed as observed in Figure 17a. However, the maximum EPWP at G2 for both monopile locations is smaller than at G1. Figure 17c displays the vertical distributions of EPWP at G3 for different monopile locations. The maximum EPWP at the seabed surface increases as the monopile moves from C1 to C2. However, as the wave breaks while climbing the sloping seabed, the maximum EPWP at the seabed surface becomes irregular as the monopile moves from C3 to C5. Overall, the effects of the solitary wave on the sloping sandy seabed are significant.

The response of the monopile under the solitary wave is also examined. Figure 18 shows the distribution of the bending moment M of the monopile along the vertical direction for various monopile locations. It is evident that the maximum M occurs near the sloping sandy seabed surface for each monopile locations with different height. In addition, the maximum M slightly decreases from about 450 kN·m to 400 kN·m as the monopile moves from C1 to C5, indicating that the solitary wave forces acting on the monopile decrease as the monopile moves from C1 to C5. The bending moments at both the monopile’s top and bottom are zero, which is reasonable since the monopile head is unconstrained in the air and the insertion depth of the monopile below the sloping sandy seabed is sufficiently deep.

5. Conclusions

In this study, the dynamic interaction between the tsunami-like solitary wave and monopile on a sloping sandy seabed was investigated numerically. An integrated numerical model was developed for the investigation, in which the solitary wave sub-model is governed by the RANS equations and the porous sloping sandy seabed sub-model is governed by Biot’s partly dynamic theory (u-p approximation). The main conclusions are as follows:

(1) The present numerical model is accurate for simulating solitary waves. Due to the larger wave height’s strong nonlinearity, the numerical result of the maximum run-up height on a vertical wall is closer to Su and Mirie’s analytical result.

(2) The comparison of the free-surface elevations at different wave probes around the cylinder is set up between the present model and previous results. Good agreement is observed. The maximum wave surface elevation increases as the wave probes move towards the cylinder upstream. Due to the reflected waves, a secondary peak is separately observed for r/R = 4.5 and 2.61 upstream the cylinder.

(3) When the monopile is located at C1, C2, and C3, the wave run-up is observed in front of the monopile, and a forward-moving water jet with a high velocity is formed behind it. When the monopile is located at C4 and C5, wave shoaling and breaking on the sloping sandy seabed are observed. Strong dynamic interaction between the solitary wave and the monopile is observed when the monopile is located at C5, with some sea water even splashing into the air.

(4) The EPWP decreases quickly in a vertical direction from the seabed surface. The maximum EPWP at the seabed surface at G1 and G2 increases as the monopile moves from C1 to C4. Due to the strong dynamic interaction between the solitary wave and monopile at C4 and C5, the maximum EPWP at G3 becomes irregular as the monopile moves from C3 to C5. The maximum M occurs near the sloping sandy seabed surface. The maximum M slightly decreases as the monopile moves from C1 to C5.

(5) This study investigates the dynamic response of a monopile and sloping sandy seabed under the excitation of a solitary wave, focusing on variations in the interaction between the monopile and solitary wave at different slope locations, differences in wave flow around the monopile, and variations in the bending moment response of the monopile. This study provides a valuable reference for understanding the response of offshore monopile foundations subjected to solitary wave excitation on sloping sandy seabeds.