around the wind turbine [73–75]. The results produced by the simple wave model, however, deviate significantly from the real flow field. Recently, the Free-Vortex Wake (FVW) method was developed by Xu et al. [76] based on the Computational Fluid Dynamics techniques to simulate the wind loads acting on the floating wind turbine. In terms of temporal discretization, the FVW method uses a three-order time-marching algorithm. Under the unsteady condition, the FVW model composed of a dynamic stall model and a three-dimensional rotational effect model is designed to improve the realistic level of the aerodynamic loads simulation. Through comparing the numerically simulated dynamic responses of the wind turbine to the experimental data, it has been found that the flapping moment and the aerodynamic response of the turbine produced by the numerical simulation agree well with the experimental data [76]. In the experiments carried out to validate the numerical simulation, the TLP type and the Tjæreborg test turbine [77] were used. The agreements have substantiated that the FVW method is reliable and accurate to calculate both the dynamic response of the floating wind turbine and the flow-field around the floating wind turbine.
5.2. Mooring line dynamics
Aiming to restraining the semi-submersible foundation from drifting under the excitations of environmental loads, mooring lines connect the floating foundation to the fixed points on the seabed. Three mooring line models, with different complexities, are currently available to calculate Fmoor. More specifically, they are:
1. Simple linear or nonlinear six DOF force–displacement relationships.
2. Quasi-static numerical approaches.
3. Fully dynamic Finite Element Method.
(1) Force–displacement relationship method
In a simple force–displacement relationship model [78] which regards the mooring system as a spring, the displacements of the foundation resulted from the six DOF motions of the semisubmersible foundation are used to calculate the restoring forces putting the foundation back. Alternatively, each mooring line can be modeled individually. One end of the mooring line attached to the foundation moves in three independent directions, and the other end is fixed at the seabed. In other words, the single mooring line is modeled as a spring, and the restoring forces of a single mooring line are calculated according to the displacement of the end of the mooring line attached to the foundation. Evidently, the force–displacement method calculates the influence of the mooring system or a single mooring line as a whole, which provides the restoring forces of the entire mooring system or a single mooring line. The dynamic effects occurred in the line itself are, however, ignored. In 2007, Rainey and Camp [79] used this method to calculate the restoring forces retaining a semi-submersible foundation under the survival condition.
(2) Quasi-static catenary equations
Quasi-static catenary equations [80] are popularly used to capture some of the nonlinear behaviors of the mooring lines, including influence of seabed frictions and axial stiffness. Assuming that the weight of the line is balanced by tensions, each mooring line is, by the quasi-static catenary method, modeled as the catenary line sagging in the vertical plane. For a two-dimensional catenary line, a set of two nonlinear equations with two unknowns (the horizontal and vertical forces at the fairlead) can be established to describe the internal forces in the catenary line [81]. Through solving the equations analytically, the tensions and positions of any given point in the catenary line can be solved. Although the motions of the mooring lines and the influence of surrounding waters are neglected in the quasi-static approach, it is still widely accepted due to the relatively low computational cost. For example, Roddier et al. [29] calculated the mooring line dynamics of WindFloat, which essentially is a semisubmersible foundation, using this method. It should be noted that