1 Introduction

The utilization of wind energy in offshore has become one of the hot issues in marine renewable energy and large-scale wind power development. There is a long coastal line and many islands in China, where abundant offshore wind power resources need to be developed in the coastal and offshore water. Much attention should be given in the development and utilization of the wind energy in the offshore regions with high safety, great efficiency, and less environmental impact.

The continental shelves of the South and East China Sea are quite gentle. Most of the deposits in the coastal water are silt. There is a large scale of coastal and offshore areas in the shallow water, where the water depth is less than 40 m. Such shallow regions have been considered as suitable places for the development of the large offshore wind farm construction of China. The low cost large wind turbine foundation structures are one of the key technologies. Taking the offshore wind farms of the East China Sea Bridge as an example, the lower part of the support structure and foundation of the 3 MW wind turbines consist of a concrete platform and steel piles. The top elevation of the concrete slab is 5.00 m. The slab is divided into two parts. The lower part is a cylinder with a diameter of 14 m, and the upper part is a truncated cone. There are 8 steel piles with the diameter of 1.70 m under the slab. The construction of the slab and piles for the wind farm near the East China Sea Bridge is shown in Fig. 1. Understanding the mechanism of the interaction between the wind farm structures and the wave-current filed and developing the prediction methods of the hydrodynamic loads on the slab and piles should be addressed in the design and construction of such offshore structures.

In the wave-current field, the wave-current force on the pile is usually calculated by the extended Morison equation. The current analytical methods include the characteristic wave method, the probability distribution method, the spectrum analysis method, and the wave simulation method. For a single pile, one of the key factors associated with calculation of the wave-current force with the Morison equation is to determine the drag coefficient $C_\mathrm d$ and the inertia coefficient $C_\mathrm m$ reasonably. Determining the hydrodynamic coefficients is very complicated. The coefficients $C_\mathrm d$ and $C_\mathrm m$ cannot be derived directly for the cases of waves of large amplitude. Much work has been done in determining the hydrodynamic coefficients and lighting the mechanics of the changes of these coefficients by a large number of experimental studies^{[ 1]}. The experiments of the wave-current force on a single pile and the research of hydrodynamic coefficients calculations can be referred to Ren^{[ 2]}, Li et al.^{[ 3]}, Li^{[ 4]}, Wolfram and Naghipour^{[ 5]}, etc. Sundar et al.^{[ 6]}reviewed the previous studies on hydrodynamic characteristics of the inclined pile and analyzed the wave forces on a pile under regular waves by the least square method. The changes of the hydrodynamic coefficients can be related with the Keulegan-Carpenter number $N_\mathrm {KC}$ for a pile with different obliquities.

There are two kinds of methods to analyze the wave-current loads on pile group. One is to calculate the correlation between the hydrodynamic coefficients and the parameters for each pile in the group by the Morison equation and the measured wave forces. The method and process are the same as those of a single pile, ignoring the effects of other piles on the wave forces on an individual pile. Another method is to establish the relationship between the wave forces on each pile of a group of piles and the wave forces on a single pile using the so-called grouping pile effect coefficient. By multiplying the wave force on a single pile by the grouping pile coefficient and then summing up, the total force on the pile group can be obtained. Based on the experiments in the oscillating water tunnel and regular wave flume, Sarpkaya et al.^{[ 7]}and Chakrabati^{[ 8- 9]}studied the changes of hydrodynamic coefficients with the KC number and pile distance of double pile cases. Yu and Shi^{[ 10]}, He et al.^{[ 11]}, and Li and Wang^{[ 12]}studied the effects of the relative pile distance and KC number on waves forces on double piles and triple piles under action of the regular wave, the irregular wave, and the wave-current conditions.

Although the wave-current force on a single vertical pile has been studied extensively and the design criteria can be used directly to calculate the wave-current force on the single pile^{[ 13]}, there is not a grouping pile coefficient and wave-current loads of inclined pile group structure due to the diversity of these kinds of structure. Because the elevation of the platform bottom is between the design low water and high water level, the wave-current force is much complicated on the platform. Moreover, influenced by the pile group foundation, the wave-current force may be increased. The wave diffusion effect appears near the large scale slab. In this case, the wave-current forces on the pile are different with and without a slab.

More attention focused on the studies on predicting the wave-current forces on such a special foundation for construction of the Donghai Bridge of Yangshan deep water port in Shanghai. The bridge is the first cross-sea bridge in China, which is 32.5 km in the length. The foundation structure with piles and slab is adopted. There are 822 bridge piers and 8 721 piles under the bridge. Since the structure may encounter typhoon and extreme waves during the period of construction, it is necessary to evaluate the safety issues of the bridge pier foundation. It is necesary to carry out a series of physical model experiments of wave-current forces on piles and slab foundation structure, including single pile, pile group, and pile-slab structures. Lan et al.^{[ 14]}and Yao and Liu^{[ 15]}carried out the experiments of the wave-current forces on each pile of the pile group and the slab in various combination of wave and current. The time series of the hydrodynamic loads are measured by the five component balance mounted in each pile. Using the statistical analysis, spectral analysis, and cross spectrum analysis, they presented the grouping pile coefficient of the structure. Considering the influence of the slab on the wave-current force characteristics on the pile foundation, it turns out that the effects of the slab on the grouping pile coefficient are not significant and, however, the non-uniformity of the wave-current forces on the all piles appears.

In this paper, the recent research progress in the experimental investigation and numerical simulation of the wave-current loads acting on the pile and slab foundation structure is summarized. Firstly, for the physical model tests, the experimental method and the measured results of the wave-current force on the pile and slab foundation are introduced, and the effect of pile group and its variation law are analyzed. Then, the numerical simulation of the wave-current force on the pile and slab foundation is reported with the computed results of wave force and instantaneous wave surface. Finally, some problems need to be studied in the field of hydrodynamic loads on the foundation of this kind of offshore wind turbines, and the corresponding local complicated flow phenomena in waves of large amplitude are discussed.

2 Wave-current loads and grouping pile effect 2.1 Instantaneous measurement of hydrodynamics on piles and slab

The experiments of wave-current loads on piles and slab are carried out in the State Key Laboratory of Ocean Engineering at Shanghai Jiao Tong University and the State Key Laboratory of Coastal and Offshore Engineering of Dalian University of Technology. The wave basin in Shanghai Jiao Tong University is 50 m in the length, 30 m in the width, and 6 m in the depth. Various ocean environments can be simulated, and the water depth can be adjusted by moving the false bottom of the basin as required. The main facilities of the basin are the wave generator, the current generating system, the wind generating system, and the large span $XY$-direction towing carriage. Besides, the principal dimensions of the wave flume are 60 m×1.0 m×1.2 m. This facility can generate irregular wave, current, and wind. There is an irregular wave maker at one end of the flume, and a grid-type absorbing beach that damps out the oncoming wave energy at the opposite end of the wave maker.

The wave height and velocity are measured by the wave gauge and propeller flow meters, and the pressure is measured by the pressure sensor. The hydrodynamic loads on piles and slab are measured using a specially designed four component balance and six component balance, respectively. To measure the wave-current force on piles and slab simultaneously, we design a built-in four component balance and install it on the root of the piles. The wave-current force of the upper slab is measured by the suspended six component balance. Because there exists a gap between the top of the piles and the bottom of the slab, their hydrodynamics can be measured simultaneously. Figure 2 shows the physical model of the slab and pile group in the wave basin.

The time series of hydrodynamic loads on Piles 1, 2, $\cdots$, 8 under irregular wave and the same direction flow are shown in Figs. 3 (a) - 3 (h) , where the water depth d is $0.474$ m, the flow velocity U is $0.356$ m/s, the significant incident wave height $H_{\rm s}$ is $0.05$ m, and the average wave period $T_{\rm m}$ is $0.9$ s. The results include the hydrodynamic force in the wave direction F_x, the force normal to the wave direction F_y, and the vertical force F_z.

The experimental data show that the mean value of the forces on the piles is not equal to zero under action of the wave-current combination. The mean values of the wave forces on different piles appear with a significant range due to complicated local flows around the piles. Comparing with the horizontal components of wave force, the vertical force is small. It can be found that the force frequency of F_x is close to the wave frequency, but the force frequency of F_y is larger than that of the frequency of F_x for Piles 1, 2, 3, 5, 6, and 8. The force frequency of F_x is larger than the wave frequency, and the force frequency of F_y is smaller than or close to the frequency of F_x but close to the wave frequency for Piles 4 and 7.

These characteristics of the wave forces appear for the cases of the grouping pile experiments without a slab. It is indicated that the influence of slab on the force frequency of component piles is small. However, the wave forces change for individual piles.

2.2 Effects of grouping piles

The pile grouping coefficient is defined as the ratio between the averaged values of the forces on all inclined piles and a single vertical pile,

(1a)

(1b)

(1c)

where $\sum\limits_{j=1}^{n}{{{F}_{\text{SG}xj}}}$, $\sum\limits_{j=1}^{n}{{{F}_{\text{SG}yj}}}$, and $\sum\limits_{j=1}^{n}{{{F}_{\text{SG}rj}}}$ are the 1/10 large values of the forces along and in perpendicular to the wave propagation direction and the total force on the slab and grouping pile. $F_{Dvx}$, $F_{Dvy}$, and $F_{Dvr}$ are the 1/10 large values of the forces and the total force on a single vertical pile. Due to the existence of the slab, the grouping pile coefficient in Eq. (1) is influenced by the pile group as well as the effect of slab on the pile foundation.

Table 1 shows the grouping pile coefficient of the wave force, the transverse force, and the total force of pile foundation under irregular wave and current in the same direction. It is found that the coefficient of the wave force and the total force are close in the range of 0.41-0.64. The transverse force is small between 0.20-0.47. Comparing with the experimental data without slab, we find that the effect of slab on the pile foundation is not significant. The total coefficient of pile indicates the relationship between the total force on the pile foundation and the single pile under the same wave-current condition. The ratio between the total force of the single vertical pile and the multiplied pile number and the measured total force is 1.56-2.44, which means that the total force of pile foundation is 41%-64% of the value of total force of single vertical pile and multiplied pile number. For the sake of safety, if calculating the total force using this method in the practical application, the grouping pile coefficient of the wave force and the total force is suggested to be 0.65, and the grouping pile coefficient of transverse force is suggested to be 0.5.

Actually, the maximum values of the wave-current force on a slab and pile foundation do not appear at the same time. The phase differences of the wave forces on those components must be considered in predicting the total wave forces on the structure. If taking the sum of the maximum values of wave forces on all components as the reference value for design, the overall structure is undoubtedly to be safe. Generally speaking, when the slab has the greatest force, the force on the pile foundation does not appear to be the maximum value. Vice versa, when the force on the pile foundation is the largest, the force on slab does not appear to be the maximum value. We define the ratio between the maximum force on slab and pile foundation and the actual maximum force as a reduced coefficient caused by the phase difference or the safety factor. The expression can be written as follows:

(2)

where ${{\left( \sum\limits_{j=1}^{n}{{{F}_{\text{SG}rj}}} \right)}_{\max }}$ is the maximum value of the total force on pile foundation, i.e., the maximum value of the sum of the instantaneous force on each component pile. $ (F_{{\rm S}r}) _{\max}$ is the maximum force of the slab. ${{\left( \sum\limits_{j=1}^{n}{{{F}_{\text{SG}rj}}} \right)}_{\max }}+{{F}_{Sr}}$ is the sum of the total force of pile foundation and the force of slab at the same time. ${{\left( {{F}_{Sr}} \right)}_{\max }}+\sum\limits_{j=1}^{n}{{{F}_{\text{SG}rj}}}$ is the sum of the total force of slab and the force of pile foundation at the same time.nis the number of the piles. The reduced coefficient caused by the phase difference of the wave-current force on slab and pile foundation is shown in Table 2. The coefficient changes in the region of 1.20-1.60, which means that it is safe to use the maximum force of slab and pile foundation as the design criteria. The safe factor is 1.20-1.60.

3 Impact on slab

For the foundation structure with a slab and piles for offshore wind farm, the slab is located near the water surface. When the water level changes with the tide, the bottom of the slab is usually suffered from the impact of waves. Since the impact of waves on the structure is closely related to the violent changes of the free surface and wave breaking, the mechanism of the vertical wave force is usually more complicated than the fully submerged slab. For the cylinder near the water surface, the maximum impact coefficient $C_{\rm s}$ is defined by

(3)

where F_z is the maximum vertical impact, D is the diameter of the cylinder, L is the length of the cylinder, and U is the instantaneous vertical velocity of the wave impact. The theoretical value of $C_{\rm s}$ is $\pi$. However, the discreteness of the measured results is large, which changes within the range from 1.0 to 7.79^{[ 1, 17]}.

The physical model experiment results show that^{[ 2]}the maximum impact pressure on a horizontal panel usually appears in the relative clearance height of $s/H=0.2$ and the impact does not appear as $s/H>0.4$. In the analysis of vertical hydrodynamic on a slab, the relative clearance heights affect not only the impact pressure of slab but also the range of the pressure distribution. Figure 4 gives the variation of dimensionless positive and negative vertical forces with the relative clearance heights. Different symbols and lines represent different relative diameters of slab and relative wave heights. From Fig. 4 (a) , the maximum positive vertical force appears when the relative clearance heights are between 0-0.3 in the case of different slab diameters and wave heights. The relative clearance heights increase with the slab diameter. It is interesting to note that, for $D/L=0.296$ in Fig. 4 (a) , the vertical force changes with the relative clearance heights as the shape of the "M". When the relative clearance height is 0.2, the minimum value appears. The reason may be that the distribution range of pressure is small. Even if the impact pressure is large, the impact force is small in this case. Figure 4 (b) shows that, in the cases of different slab diameters and wave heights, the negative vertical force reaches the maximum when the relative clearance height is the minimum, namely, $s/H=-0.2$. When $s/H< 0$, the negative vertical force increases rapidly as the relative clearance height decreases. The reason is that in the range of the relative clearance heights of this slab, the main factor influencing the negative vertical force is the degree that the dynamic uplift forces less than the hydrostatic buoyancy force. The greater the value is, the greater the negative vertical force is. If the relative clearance height is small, the inundation depth of slab is small, and the negative vertical force is large. When $s/H\ge0$, the negative vertical force increases in different degrees with the relative clearance heights. In the range of the relative clearance heights of this slab, the negative vertical force is related to the closed vacuum layer, and the peak value appears when $s/H=0.4$.

4 Effect of slab on wave force of pile foundation

Under the regular waves, the force on the component pile of multi-pile system is quite different with that on the single pile, which is the so-called grouping pile effect. The grouping pile effect is mainly caused by the phase difference between each component pile and the flow field disturbance between the piles. Here, only the grouping pile effect generated by the phase difference is discussed,

(4)

where $\left| \sum\limits_{i=1}^{n}{{{F}_{i}} (t) } \right|$ is the sum of the wave force of each pile at the timet, and ${{\left| {{F}_{i}} (t) \right|}_{\max }}$ is the maximum wave force on theith pile.

Based on the linear potential flow theory, the analytical solution of the linear wave affecting on a fixed truncated cylinder can be obtained by referring to the solution given by Garrett^{[ 18]}. According to the design code of coastal engineering^{[ 13]}, the wave force on the pile is calculated by the Morison equation. The velocity and accelerate of water particles at the axial line of the pile are derived from the linear potential flow solution.

We study the two structure types as shown in Fig. 5. One type is that 8 piles are distributed uniformly in a circle under the slab.ris the center distance of Pile 1. $S_\mathrm D$ is the center distance between two adjacent piles, $S_\mathrm D=0.765r$. Another type is that 9 piles are distributed as a 3×3 matrix.ris the center distance of Pile 1. $S_\mathrm D$ is the center distance between two adjacent piles, $S_\mathrm D=r$, $\theta_\mathrm w$ is the wave direction, andais the diameter of the truncated cone.

When the wave height $H=3$ m, the wave period $T=5$ s, the water depth $d=20$ m, the diameter of slab $a=6$ m, the diameter of pile $D=1$ m, the draught depth of slab $b=$ m, Fig. 6 presents the change rule of grouping pile coefficient $k_2$ of two forms in Fig. 5 with ${{S}_{\text{D}}}$ and ${{\theta }_{\text{w}}}$. It can be found that the grouping pile effect generated by the phase difference decreases with the center distance of two adjacent piles when the slab exists. The greater the distance between two adjacent piles is, the stronger the effect of phase difference is. When the wave direction angle ${{\theta }_{\text{w}}}$ is about 22.5° or 67.5°, the grouping pile coefficient is the largest. When ${{\theta }_{\text{w}}}$ is 45°, the grouping pile coefficient is the smallest. In the case without a slab, the grouping pile effect generated by the phase difference does not change with the wave direction angle, but decreases significantly with the increasing distance between the piles.

5 Numerical simulation of wave-current force on foundation structure of piles and slab

Prediction of the wave-current force on the foundation structure of slab and piles based on the numerical simulation is an advanced issue in the field of computational hydrodynamics. It needs to deal with not only the different scales of pile, slab, and wave length, but also the strong nonlinear process of violent deformation of water surface and wave breaking.

Consider the three-dimensional unsteady incompressible flow with the free surface. The governing equations are the continuity equations, the Reynolds averaged Navier-Stokes (RANS) equations, and the transport equation of the volume ratio function for tracking the free surface deformation. Discretizing the continuity equation can be used to establish the coupling algorithm of the pressure and the velocity, and then a pressure equation can be obtained. The coupled equation of pressure and velocity can be solved by the PISO algorithm. The flow field is computed by the FLUENT solver. The RNG $k\textrm-\epsilon$ model is adopted as the turbulence model.

The analytical relaxation method is used to implement the wave generation and dissipation, which needs to add momentum source in the FLUENT solver^{[ 19]}. Based on the work of two-dimensional numerical wave flume, this work extends to the simulation of wave-current force on a foundation of piles and slab in a three-dimensional numerical wave basin. In the numerical simulation, the geometry of the structure is shown in Fig. 7. The water depth h is $14.0$ m, the wave height H is $7.89$ m, and the wave period T is $7.3$ s. The results show that the maximum run-up of waves on the pile is up to 8 m. The total horizontal forces are $F_{\max}=3 204$ kN and $F_{\min}=-1 716$ kN. The numerical results agree well with the experimental data (see Fig. 8) . The sudden change of the force during wave breaking can be predicted by numerical simulation.

Figure 9 shows the snapshots of the wave surface around the structure. Numerical simulation can carefully predict the water surface of breaking wave, as well as the scattered water distributed on the slab.

6 Summary and discussion

The design and construction of fixed offshore wind turbine foundation structure must consider the effect of wave and tidal current. Because the pile foundation and slab structure is applicable to the soft soil foundation in the coastal areas, this kind of structure has become the main structural form of the offshore wind farm construction in China. The simultaneous measurement system of multi-piles and slab is helpful to obtain the time series of hydrodynamic loads and analyze the variation law of wave-current force through physical model experiment. The experimental data show that the grouping pile effect on wave forces is very significant. The grouping pile effect of wave force and total force can be 0.65, and the grouping pile effect of transverse force can be 0.5. The maximum values of wave-current force on slab and pile foundation do not appear at the same time. It is safe of considering the maximum value of the forces and the safe factor to be 1.20 and 1.60. The interaction between nonlinear waves and pile foundation and slab can be simulated numerically by the RANS equations and the volume of fluid (VOF) method. The violent deformation of water surface and run-up can be obtained by numerical simulation.

Though the wave-current forces on single pile and the pile group have been investigated experimentally and documented well, local complicated flow structures around the pile group under wave and current still need to be studied. Carrying out the experiments of hydrodynamic loads on piles and slab under action of wave and current with high Reynolds numbers is still subject to the restriction of the experimental facilities. Development of the numerical simulation method for nonlinear wave and pile interaction particularly at the mechanics of ringing phenomena remains to be addressed.

The impact of wave on the near surface slab involves not only the strong nonlinear free surface flow but also the significant randomness of transient hydrodynamics due to the free surface shape, air compressible, and air cushion effect. It is necessary to develop effective experimental techniques and theoretical analysis method to simulate and describe the impact of waves on the near surface slab and the effect of breaking wave on the foundation of wind turbine. It can provide a new method to deal with the hydrodynamic loads with strong nonlinear and stochastic characteristics.

Aimed at the shallow coastal waters (water depth less than 40 m) and large wind turbine of more than 6 MW, studying the low cost foundation structure has important values of engineering application. The establishment of a numerical simulation and engineering calculation method of hydrodynamic loads on those unconventional structures for offshore wind farm construction under the action of nonlinear wave and nonuniform current should be an important issue for further studies.

Acknowledgements The authors appreciate for the Shanghai Survey Design and Research Institute and East China Sea Bridge Administration, who provide necessary technical information for the project. Dr. Xi ZHAO of Shanghai Jiao Tong University is appreciated for her help in preparing and revising the paper.