1 Motivation and Objectives

Over the years, offshore wind farms have moved further from the shore and into deeper waters. At the end of 2014, the average water depth of grid connected wind farms was 22.4 m and the average distance to shore 32.9 km. Projects under construction, consented and planned confirm that average water depths and distances to shore are likely to increase (Arapogianni et al. 2013). Shallow water regions suitable for seabed-fixed, offshore wind turbines are limited, and for sea depths exceeding 30–60 m, floating structures become more economic. Hence, emphasis is placed on the development of floating offshore wind turbines (FOWTs) with several prototypes already operational across the world (Arapogianni et al. 2013). Unlike onshore machines, the FOWT is a highly dynamic system subjected to the wind and wave loads and only constrained by a mooring system. Further, the rotor frequency is low due to the large size of the blades, and wave frequencies may come close or coincide with the rotational frequency of the rotor. It is, therefore, important to develop a method for the analysis of this air-structure-water system.

The common approach is to combine simplified tools into one hybrid model to predict wind turbine responses under wind and wave loads. The Blade Element Momentum (BEM) method is frequently used to calculate aerodynamic loads on the blades and tower (Jonkman 2007; Skaare et al. 2007; Karimirad and Moan 2013). Sometimes analytical models are used that take the form of algebraic equations for the applied thrust that is proportional to the area of the rotor and the relative velocity between the wind and the hub as in Roddier et al. (2009) and Karimirad and Moan (2012). If aero-elasticity is considered, it is often included in BEM methods, where the structure is described by a multi-body formulation, in which wind turbine structures are subdivided into a number of bodies and each body consists of an assembly of Timoshenko beam elements (Larsen and Hanson 2007). Another approach is to characterise flexible bodies using linear modal representation, which usually assumes small deflections.

The hydrodynamic loads on the support structure are often modelled with a linear potential theory assuming inviscid, incompressible and irrotational flow, also known as Airy wave theory (Jonkman 2007; Rieper 2011; Karimirad and Moan 2013). In this case, frequency dependent hydrodynamic-added-mass and hydrodynamic-damping matrices, along with wave-excitation force vector are precomputed for a given problem, and serve as input to the coupled model. At the beginning of the computation, the wave-radiation-retardation kernel is obtained by integrating user-supplied added-mass or damping coefficients (Jonkman 2007). This way, external computer routines can be linked to the aerodynamic solver as a function that employs convolution integrals and returns hydrodynamic loads at given instances. The non-linear hydrodynamic viscous drag is included from Morison’s equation (Morison et al. 1950) using strip theory. The drag coefficient involved in Morison’s equation is often determined based on experiments. Since the drag coefficient depends on many factors, including the Reynolds number, geometry, and the presence of a free surface and a free end of a body, the experimental data is not always directly applicable. The drag coefficient can be obtained from a CFD computation for given support platform and then applied to Morison’s equation improving the results as was shown by Benitz et al. (2015).

Linearization of the hydrodynamic problem implies that the translational displacements of the support platform are small relative to the size of the body, and that amplitudes of the incident waves are much smaller than their wavelengths i.e. steep or breaking waves cannot be modelled. Some extensions to the second-order potential flow was performed e.g. by Marino et al. (2011) and Roald et al. (2013). Even with second-order hydrodynamic terms included, however, the potential hydrodynamic theory might not completely apply to floating wind turbine platforms due to the large displacements encountered (Matha et al. 2011). Mooring lines constraining the FOWT can be modelled using springs (Savenije et al. 2010), flexible beams (Skaare et al. 2007) or multi-body chains of rigid bodies (Matha et al. 2011). Sometimes, precomputed nonlinear force-displacement relationships are employed, as in Karimirad and Moan (2012). Some of the works in the field of FOWT modelling are summarised in Table 17.1.

Table 17.1 Works relevant for the complete FOWT models
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The purpose of this chapter is to present a coupling algorithm that brings together two Naver-Stokes solvers. For this, the Helicopter Multi-Block (HMB3) solver (Barakos et al. 2005) is used to solve for the aerodynamic forces acting on the wind turbine (WT) blades. Hydrodynamic forces on the support platform are solved using the Smoothed Particle Hydrodynamics (SPH) method (Gomez-Gesteira et al. 2012; Woodgate et al. 2013). Both solvers are coupled by exchanging information while the FOWT is represented by a lumped mass model.

2 Numerical Methods

HMB3 is a 3D multi-block structured solver for the Navier-Stokes equations in 3D. HMB3 solves the Navier-Stokes equations in integral form using the arbitrary Lagrangian-Eulerian formulation for time-dependent domains with moving boundaries (Dehaeze and Barakos 2012a, b; Carrión et al. 2014a). The solver uses a cell-centred finite volume approach combined with an implicit dual-time method (Jameson 1991). Osher’s upwind scheme (Osher and Chakravarthy 1983) is used to resolve the convective fluxes. Central differences (CD) spatial discretisation is used for the viscous terms. The non-linear system of equations that is generated as a result of the linearization is solved by integration in pseudo-time using a first-order backward difference method. A Generalised Conjugate Gradient (GCG) method is then used (Eisenstat et al. 1983) in conjunction with a Block Incomplete Lower-Upper (BILU) factorisation as a pre-conditioner (Axelsson 1994). The HMB3 solver has a library of turbulence closures including several one- and two- equation models. Turbulence simulation is also possible using either the Large-Eddy or the Detached-Eddy simulation approach (Spalart et al. 1997). The solver was designed with parallel execution in mind and the MPI library along with a load-balancing algorithm are used to this end. The flow solver can be used in serial or parallel fashion for large-scale problems. Depending on the purposes of the simulations, steady and unsteady wind turbine CFD simulations can be performed in HMB3 using single or full rotor meshes generated using the ICEM-Hexa tool. Rigid or elastic blades can be simulated using static or dynamic computations. HMB3 allows for sliding meshes to simulate rotor-tower interaction cases as described in Steijl and Barakos (2008). Alternatively, overset grids can be used with the details presented in Jarkowski et al. (2013). To account for low-speed flows, the Low-Mach Roe scheme (LM-Roe) developed by Rieper (2011) is employed for wind turbine cases (Carrión et al. 2013). The chosen methodology allows for easy updating of the solver with new functions. One example presented here, is the coupling with a hydrodynamic solver.

The sea is modelled with the SPH method (Gomez-Gesteira et al. 2012). Each SPH particle has individual material properties and moves according to the Navier-Stokes equations solved in the Lagrangian form. SPH offers a variety of advantages for fluid modelling, particularly those with a free surface and moving bodies. Due to the Lagrangian nature of the SPH method, the free surface requires no special treatment. Further, submerged bodies can be represented with particles. Therefore, it is natural for the method to include floating objects.

The motion of the FOWT components is computed with a multi-body model (MBDM) of rigid bodies and frictionless joints. Mooring cables are modelled as a set of springs and dampers, according to Savenije et al. (2010). The coordinate partitioning method of Nikravesh (1988) is used to solve the resulting system of mixed differential-algebraic equations. The time integration scheme for independent variables is explicit and various schemes are implemented up to the Runge-Kutta method of fourth order. The non-linear position equations for dependent variables are solved using the Newton-Raphson method with exact, an analytical, Jacobian.

The current implementation is schematically presented in Fig. 17.1, where coupling is between both fluids. Another option would be to employ a multi-phase solver (e.g. Volume of Fluid as in Beyer et al. (2013)). This approach does not tackle the problem of coupling, but shifts it to the structure-fluid side.

Fig. 17.1
figure 1

Schematic of the solvers employed in the floating offshore wind turbine model

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2.1 Validation of the Aerodynamic Solver

The HMB3 CFD solver has so far been validated for several wind turbine cases, including the NREL Annex XX experiments (Gómez-Iradi et al. 2009), where the effect of the blades passing in front of the tower was captured, as can be seen by the deficit of the thrust values presented in Fig. 17.2a. The under-prediction of 3 % is due to the fact that computations were performed at the nominal conditions for the experiment, and not for the measured pitch. A small change of pitch accounts for this difference. The pressure and PIV data of the MEXICO project (Schepers and Snel 2007, 2012) have also been used for validation (Carrión et al. 2014b), where the wake was resolved on a fine mesh capable to capture and preserve the vortices downstream the rotor (Fig. 17.2), which enabled the prediction of the onset of wake instabilities (Carrión et al. 2015).

Fig. 17.2
figure 2

Thrust prediction over a full revolution of the NREL Annex XX wind turbine at 7 m/s wind speed (a); and prediction of MEXICO rotor wake, including axial velocity profile (b) (Carrión et al. 2015)

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2.2 Validation of the Hydrodynamic Solver

The hydrodynamic loads are estimated using the SPH method validated against the experiments of Greenhow and Lin (1983) for the high speed entry of a half-buoyant solid cylinder into calm water. As shown in Fig. 17.3a a cylinder of density of 500 kg/m3 was allowed to fall freely from the height of 0.8 m under gravity acceleration; the water depth was 0.3 m. The density of the cylinder was assigned by defining the relative weight between fluid and cylinder particles to be w = 0.5. Simulations were run with a cubic spline kernel, artificial viscosity with viscosity parameter α = 0.1, adiabatic index γ = 7, and Courant-Friedrichs-Lewy number CFL = 0.2. The viscosity between the cylinder SPH particles and the fluid particles was neglected. Five cases were compared with different distances d between the particles. The penetration depth of the cylinder for all cases, along with the experimental results, are shown in Fig. 17.3b, whereas Fig. 17.4 shows the water surface deformation. The results were used for estimating the particle density and viscosity necessary for computations of floating bodies. Note that the best agreement with the experiment was obtained with distances between the particles d = 0.23 cm, what corresponds to 25 particles per radius of the cylinder.

Fig. 17.3
figure 3

Validation case for the SPH solver. (a) Schematic of the SPH validation setup; (b) Depth of penetration of a cylinder of density 500 kg/m3: SPH results for different distances between particles d and experimental results of Greenhow and Lin (1983)

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Fig. 17.4
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Surface deformation during water entry of a cylinder for time t = 0.32s from the beginning of the fall. Comparison between CFD results with distance d = 0.23 m between particles

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2.3 Validation of Multi-body Dynamics Solver

The MBDM was validated using simple mechanical systems of known solution as presented in Leble and Barakos (2016) like 2D and 3D slider-crank mechanisms. The gyroscopic wheel mechanism was used to validate that the gyroscopic effect is properly accounted for in the multi-body formulation. The ground body was placed at the origin at the global coordinate system. A short rod of length 0.1 m was attached to the ground body at height 1.0 m using a universal joint. The other end of the rod was connected to the centre of mass of the steel wheel with a revolute joint. A constant rotational speed of 60 rad/s was applied to the wheel by a revolute driver. The gravitational force acting in negative z direction was applied to all bodies, and at time t = 0 system was assumed to have no precession.

The system is presented in Fig. 17.5a, while the mechanical properties of all bodies are shown in Table 17.2. The analytical solution was obtained from Eq. (17.1) using the gyroscopic approximation, i.e. assuming that precession is much slower than rotation of the wheel \( {\omega}_p\ll {\omega}_w \), so that the magnitude of the angular velocity \( \left|\overrightarrow{\omega}\right|\cong \left|{\omega}_w\right| \) and that precession and rotation rates are nearly constant. Eq. (17.1) is shown below:

Fig. 17.5
figure 5

Validation of the MBDM to account for the gyroscopic effect. Test case setup (a), and rate of precession of the wheel (b)

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Table 17.2 Properties of the bodies employed to model the gyroscopic effect
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$$ {\omega}_p=\tau /L={m}_wgl/{I}_{xx}{\omega}_w $$
(17.1)

In Eq. (17.1), ω p is the angular velocity of precession, τ is the moment due to gravity about the pivot point, and L is the angular momentum of the wheel. The expansion to the right-hand side involves the mass of the wheel m w , the length of the rod l, the gravitational acceleration g, the mass moment of inertia of the wheel about the axis of rotation I xx , and the rotational velocity of the wheel ω w . Substitution of values from Table 17.2 into Eq. (17.1) yields the rate of precession as \( {\omega}_p\approx 0.319\;\mathrm{rad}/\mathrm{s} \). The results are presented in Fig. 17.5b, where the Runge-Kutta integration scheme of fourth order was employed, with a time step Δt = 0.0001s. As can be seen, the rate of precession developed in less than 0.05s, and then maintained almost constant value that agreed with the one obtained using the gyroscopic approximation.

2.4 Coupling Algorithms

Coupling problems arise in many engineering problems, like fluid-structure interaction (FSI), but can also result from domain decomposition, where each sub-domain employs different discretisation or is solved with different method (Zienkiewicz et al. 2005). A multi-physics problem with adjacent domains can be simulated in a monolithic or in partitioned way. The former refers to the flow equations and structural equations being solved simultaneously, while the latter means that they are solved separately. The monolithic approach requires a specific solver for each particular combination of physical problems, whereas the partitioned approach allows for solver modularity. The partitioned approach also allows one to solve the fluid equations with different techniques developed specifically for the air and water. Further, this approach reduces the computational complexity per time-step, simplifies explicit/implicit treatment, facilitates sub-cycling, and eases replacements when better mathematical models and methods emerge in the fluid sub-disciplines. On the other hand, the partitioned simulation requires a special treatment to account for the interaction between the involved domains. Hence, computational efficiency over a monolithic approach is not necessarily guaranteed (Fellipa et al. 1999). The monolithic solution—which is the ultimate form of strong coupling, does not recognise the differences between the mathematical properties of the subsystems. Furthermore, it tends to ignore the issues of software modularity, availability, and integration, even though each of these issues can be in practice a major obstacle (Farhat et al. 2006). Considering that two available and validated solvers (HMB3 and SPH) can be used in this work, the emphasis is placed on partitioned algorithms.

Partitioned coupling can be weak or strong. Explicit algorithms are weak (or loose) as the solvers exchange information once per time step, and the coupled equations are not exactly satisfied due to explicit treatment. Depending on the formulation, one side of the coupling boundary conditions is usually lagging behind another. This can be improved with staggering or extrapolation techniques, but the scheme remains weak, and coupling errors may be introduced. However, loosely coupled algorithms are attractive, since among all solution methods, they are the simplest to implement for realistic applications, and the most computationally inexpensive per time step.

Implicit algorithms are strong (or tight), and enforce exactly the coupling conditions at each time level. This is obtained by conducting iterations until boundary equations are satisfied to certain, prescribed accuracy. The coupling problem can be formulated either as fixed-point or root-finding problem. For the former, fixed-point Jacobi or Gauss-Seidel methods can be employed. Although easy to implement, those methods converge slowly if at all. Under-relaxation techniques can be used to improve convergence of the fixed-point iterations. Methods like fixed under-relaxation, adaptive Aitken’s under-relaxation or steepest descent relaxation are some of the possible choices (Küttler and Wall 2008; Degroote et al. 2010). Newton’s method can also be used. This method requires Jacobians relating the solutions of both solvers that are usually not known. This can be circumvented by employing approximation of Jacobian or Jacobian-vector product. Those types of coupling methods are called Quasi-Newton. Recently, new strongly coupled algorithms have been proposed.

Vierendeels et al. (2007) proposed an Interface Quasi-Newton algorithm with an approximation for the inverse of the Jacobian from a Least-Squares model (IQN-ILS). This approach was further investigated by Degroote et al. (2010), where they compared its performance with the Interface Block Quasi-Newton with an approximation for the Jacobian from a Least-Squares model (IBQN-LS), Aitken relaxation, and the Interface Generalised Minimal Residual method (Interface-GMRES(R)) algorithms. Demonstrated results showed that IQN-ILS and IBQN-LS performed similarly, using three times less evaluations and converging four times faster than the Aitken’s relaxation method. IQN-ILS and IIBQN-LS were also found to use two times less evaluations and be almost three times faster than the Interface-GMRES algorithm.

Fernández and Moubachir (2005) reformulated fluid-structure interaction as a non-linear problem in the state of the structure, with the flow states considered as internal variables of the problem. This system was subsequently solved with the Newton-Raphson method using an exact Jacobian. The performance of this algorithm was compared with the performance of the Aitken relaxation and Quasi-Newton GMRES methods, for the inviscid flow in an elastic tube. Results showed that Aitken’s relaxation was twice as slow as the Quasi-Newton and the exact Jacobian methods, and required almost 40 times more iterations. Further, for time steps of Δt = 10−4 s, both latter algorithms showed similar behaviour in convergence. However, for time steps of Δt = 10−3 s, the fixed-point and Quasi-Newton algorithms failed to converge. This implies sensitivity of the methods to the employed Jacobian.

The strong coupling may be important if the phenomena occurring in both fluids have similar time scales. Due to frequency similarities, resonances may occur and the exact response of a system will deviate from what is predicted by a loosely coupled algorithm. On the other hand, if time scales are largely different, loosely coupled algorithm may be sufficient. The exact bounds when the strong coupling is required for particular FOWT must be carefully assessed. Some indication comes from the waves and rotor frequency analysis. The sea state, wave height, wave frequency, and wind speed are empirically related in terms of range and most probable values e.g. in Lee et al. (1985). On the other hand, every wind turbine is designed to operate at a particular rotational frequency for a given wind speed. This allows one to construct a “Campbell” diagram for the FOWT investigated in this work (Fig. 17.6). It is clear that for sea states between 3 and 4 (or wind speed about 9 m/s) resonances may occur. The rated power production for this 10-MW FOWT corresponds to the wind speed of 11.4 m/s, or sea state 4. This indicates that for rated conditions, the weakly coupled algorithm may be sufficient.

Fig. 17.6
figure 6

Campbell diagram for the investigated FOWT showing frequencies of the rotor and the waves as function of sea state and wind speed

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2.5 Coupling Scheme and Its Implementation

In general, the exchange of information without stopping the computations can be implemented in three ways: through files, shared memory or the Message Passing Interface (MPI). Writing a file is the simplest solution. Both solvers can be launched separately and write files whenever exchange of information is required. This approach calls for very minor changes to both codes.

In the shared memory approach multiple processes have access to the same memory, allowing them to change it and read changes made by other processes. If the random access memory (RAM) is to be used, it requires a shared memory machine, which may not be available on a general High Performance Computer (HPC). The file system can be used instead by mapping the memory on the hard drive. This approach suffers from the same drawback as the case of writing files. That is, writing and reading from hard drive creates a bottleneck, and slows down the computation especially if information is exchanged often, and large amount of data is to be exchanged.

Both employed CFD solvers are parallelised using MPI and the Single Program, Multiple Data (SPMD) paradigm, where each instance of the solver is assigned to perform the same task on different sets of data. Therefore, the easiest way to combine solvers is to employ MPI, but in Multiple Program, Multiple Data (MPMD) approach, where different programs operate on different sets of data.

However, direct MPMD implementation of SPMD solvers requires additional effort to split the global communicator, such that each of the solvers is in a separate communicator (MPI COMM WORLD) with a separate ordering of processes, as detailed in Castain et al. (2015). This can be avoided by dedicating one process to be in charge of executing both solvers with MPI_Comm_spawn routine.

In the present work, the communication between the solvers was established through the Message Passing Interface (MPI), where the MBDM is executed as a single process and is dedicated to start SPH and HMB3 parallel solvers. The data flow diagram of the implementation is presented in Fig. 17.7.

Fig. 17.7
figure 7

Flow chart of the MPI implementation and data exchange for coupled model

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The communication was validated by executing separately SPH or HMB3 and comparing with the results were the body motion was introduced by MBDM. Due to the Lagrangian nature of the SPH method, the submerged bodies can be represented with particles and do not require specific coupling. Therefore, by utilising MPI, the MBDM substituted the body motion routines of the SPH solver and reduced the number of coupled codes to two—SPH and HMB3. This implies that MBDM is advancing in time with the same integration scheme as SPH using a symplectic method in this case (Leimkuhler et al. 1996).

Fig. 17.8
figure 8

The parallel conventional staggered method employed in present work

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In the present work, a weakly coupled approach is employed, namely the parallel, conventional, staggered method shown in Fig. 17.8. Both solvers are advancing with different but constant time steps. SPH employs a time step of ΔtSPH = 2 × 10−4 s with CFL = 0.2, whereas HMB3 employs a time step of ΔtHMB3 = 2 × 10−2 s = 100ΔtSPH with implicit CFL = 5.0. The small time step for the SPH method is required by the explicit integration scheme. The HMB3 solver employs an implicit dual-time method by Jameson (1991) that is superior for larger time steps. Synchronisation of the solvers is performed at the end of each HMB3 step.

At the beginning of each synchronisation time step, the position and velocities of the rotor are transferred to the HMB3 aerodynamic solver, and forces and moments on the rotor are passed to the SPH. The two solvers are then advancing to a new time level with different methods and different number of steps. SPH performs 100 symplectic steps, while HMB3 performs 350 implicit pseudo-time steps. During the symplectic steps of the SPH code, the aerodynamic loads are kept constant (frozen). In return, the position and velocities of the rotor are kept constant during the implicit steps of HMB3. Once the synchronisation point is reached, the new position and velocities of all bodies, and rotor loads are obtained. Then, the algorithm proceeds to the new time level and information between the solvers is exchanged.

3 Test Case Description

A 10-MW wind turbine design by Bak et al. (2013) was used in this work. The blade consists of the FFA-W3 aerofoil family (Björck 1990) with the thickness ranging from 24 to 60 % of the chord. The blade has a non-linear distribution of the chord, the relative thickness of the section and the twist. The rotor diameter is 178.3 m, and the wind turbine operates at a wind speed of 11 m/s with a rotational speed of 8.8 rpm. The blades have a pre-coning of 2.5° and nonlinear pre-bending with 3.3 m displacement at the blade tip. The mass of the rotor is 228 tons, whereas mass of the nacelle and tower is 446 tons and 605 tons, respectively. The tilt of the nacelle in the original design is 5° nose up, but this was not included in the present model.

The wind turbine is attached to the floating support which consists of three cylindrical floats that increase the buoyancy and stability of the structure. A similar concept of the support platform was investigated by Roddier et al. (2009). Unlike that design, the present support is simplified to be symmetric with respect to the location of the tower and the floats are connected to the base of the tower with a solid frame. The size of the tower is taken from Bak et al. (2013), and the dimensions of the support were calculated to provide sufficient buoyancy. A schematic of the studied FOWT is shown in Fig. 17.9.

Fig. 17.9
figure 9

Schematic of the employed model of FOWT (a), and dimensions of the semi-submersible support and tower (b). FOWT model consists of three mooring lines and two rigid bodies: the rotor (blue) and combined body representing nacelle, tower and support (red). Adapted from Leble and Barakos (2016)

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In the present model, the FOWT is represented by three mooring lines and two bodies, as shown in Fig. 17.9a. The first body represents the rotor (three blades with the spinner), and the second body represents the combined nacelle, tower and floating support rigidly linked to each other. The two bodies are connected by a revolute joint and a constraint of constant rotational speed is applied to the rotor. The resulting system has 6 unconstrained degrees of freedom. The mechanical properties of the bodies and mooring lines are presented in Table 17.3.

Table 17.3 Mechanical properties of the employed bodies and mooring lines
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The FOWT is placed in a shallow tank of length 500 m, width 150 m and height 30 m. The tank is filled with water to a depth of 20.6 m. The waves are generated using a paddle on one side, and dissipated using a beach-like slope on the other side of the tank. The tank is presented in Fig. 17.10. Waves are generated to represent the specific sea state corresponding to a given wind speed. Based on the measurements of annual sea state occurrences in the North Atlantic and North Pacific (Lee et al. 1985), the wind speed of 11 m/s corresponds to a sea state 4 with a mean wave height of 1.88 m and a period of 8.8 s.

Fig. 17.10
figure 10

The FOWT model placed in a shallow tank. Mooring lines are shown with dashed lines. Adapted from Leble and Barakos (2016)

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3.1 CFD Mesh

The aerodynamic grid consists of the rotor and nacelle i.e. the tower is not included and the effect of the blade passing on the tower is not investigated. The grid consists of 8M cells, where 24 cells are used in the first layer, and 166 cells are distributed around the aerofoil section as presented in Fig. 17.11a. The surface of the blade is resolved with 90 cells along the span, as shown in Fig. 17.11b. The size of the first cell in the direction normal to the surface was 10−5c, where c = 6.2 m is the maximum chord of the blade. Based on the free-stream condition and the size of the first cell, the y + parameter was estimated to be y + = 1.2. It must be noted that the grid was relatively coarse as compared with the one used by Carrión et al. (2015) to capture the wake of the MEXICO rotor. However, a grid convergence study showed that this density is sufficient to produce meaningful, grid-independent results.

Fig. 17.11
figure 11

8M mesh used to solve for aerodynamic loads. Slice through the volume close to the blade surface (a), surface mesh (b), and computational domain (c)

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The density of the air was assumed to be ρ = 1.225 kg/m3, the dynamic viscosity of the air was assumed to be μ = 1.8 × 10−5 Ns/m2, and the speed of sound was assumed to be 340 m/s. Further, the k-ω SST turbulence model was employed with the free-stream level of turbulence at 2.6 %. The flow was assumed to be fully turbulent, and the atmospheric boundary layer was not modelled. The uniform inflow boundary was set 3R upstream of the rotor, and the outflow boundary was set 6R downstream of the rotor, where R is the radius of the blade. The far-field boundary was assigned 3R from the centre of rotation. In addition, the sliding plane was used to connect rotor to the nacelle and allow relative motion. The computational domain with corresponding boundaries, a slice through the mesh close to the blade surface, and the surface mesh of the blade are presented in Fig. 17.11.

3.2 SPH Setup and Resolution

The hydrodynamic domain is resolved using 5M particles with initial uniform spacing of d = 0.625 m. Note that the best agreement with experimental data was obtained for 25 particles per radius of the cylinder, as shown in Sect. 17.2.2. Here, the employed spacing corresponds to 9 particles per radius of the cylindrical leg, or to spacing d = 0.69 cm in Fig. 17.3b. The coarse particle distribution was chosen for economies in CPU time, where coarse domain is obviously solved faster, but tends to under-predict the slamming loads on the structure. Three test were performed to investigate the influence of the domain width and particle spacing on the force acting on the support structure, as presented in Table 17.4. The average hydrodynamic forces acting on the support during 1 s of simulation were used for comparison. This time interval was chosen such that it leads to direct comparison of the average loads per unit of time. Percentage difference is computed relative to the size and spacing employed for the coupled computation. As can be seen, the size of the hydrodynamic domain has little effect on the average hydrodynamic force. On the other hand, improving the spatial resolution results in about 18 % difference in the hydrodynamic force. This agrees with observations made in Sect. 17.2.2. A spacing of d = 0.3125 m would have been better, but to improve computational performance a spacing of d = 0.625 m was employed.

Table 17.4 Test cases investigating the influence of the domain width and particle spacing on the forces acting on the support structure
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3.3 Initial Conditions

Each of the solvers was executed separately before coupling to obtain a periodic solution of the loads. During this phase of computation the floating support was fixed, and the waves were generated for approximately 30 s. The rotor was set to spin about the axis aligned with the direction of the incoming wind, and was first solved using HMB3 “hover” formulation with 20,000 steps during which the L2 norm of the residual vector dropped below 10−6. Then, the unsteady computation was initiated and the flow was solved for an additional 30° of azimuth. The aerodynamic loads were almost constant during unsteady computation. Once the initial conditions were obtained, the coupled computations were initiated.

3.4 Demonstration Cases

The first demonstration case of the FOWT has the described configuration (see Fig. 17.10) with the difference that rotor was not included in multi-body formulation. Instead, the mass of the rotor was concentrated in the centre of gravity of the support to produce correct mass of the floating structure. In this way, the shift of the centre of mass due to rotor overhung was not considered. Further, the rotor inertia was not included, and the associated gyroscopic effects were not taken into account. The importance of these effects for the system at hand is assessed in the results section. Calm sea was considered, and the time varying thrust with the mean value of 1500 kN was applied at the location of the rotor. The thrust variation is shown in Fig. 17.12, and was estimated from a separate CFD computation of the rotor with the tower included.

Fig. 17.12
figure 12

Thrust as function of azimuth angle of the rotor for decoupled case

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This test case was solved for 150 s. Note that the demonstration case is not a coupled simulation, since the thrust force is prescribed and independent of the platform motion. The last test case was a coupled computation, as described in Sect. 17.2.5. This case was solved for 60 s, and allowed for almost 7 wave passages and about 9 revolutions of the rotor.

4 Results and Discussion

4.1 Decoupled Case

The results of the first case are presented in Fig. 17.13. As can be seen, the FOWT moves in the direction of the thrust by about 0.215 m (displacement in x). The FOWT also sinks in the water for about 0.603 m (displacement in z), and tends to settle at a pitch angle of around 0.09 rad or 5.2° (rotation about y axis). The SPH particles are settling for the first 15 s as is visible in the acceleration plot. This cannot be avoided even if the floating body is fixed and particles are let to settle. This is because releasing the floating structure is equivalent to a drop, and therefore does not represent equilibrium.

Fig. 17.13
figure 13

Lateral and rotational dynamics of the support for decoupled case

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The last 20s of lateral and rotational accelerations are presented in Fig. 17.14. The effect of time varying thrust on the angular acceleration in pitch (about y axis) can be seen in Fig. 17.14c. The variation in the shape and frequency corresponds to the applied time dependent thrust. The effect of time varying thrust on the lateral accelerations can be seen in Fig. 17.14. Again, the frequency of accelerations corresponds to the frequency of the thrust, but some phase shift is present and the shape of the response does not follow the shape of the thrust. This is because the motion in heave is linked to the applied thrust only through the rotational motion of the support i.e. through the second time integral of the angular acceleration that does follow the shape of the thrust as shown in Fig. 17.14c.

Fig. 17.14
figure 14

The last 20 s of lateral and rotational accelerations of the support for decoupled case

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The acceleration in the x direction is directly linked to the applied thrust, and the frequency dependence on thrust without the phase shift is clearly visible. However, the shape of the acceleration is not following the shape of applied thrust. This is a result of high stiffness of the mooring lines in this direction, where high frequency response of the mooring system augments the overall response of the support platform.

There are three sources of momentum for the decoupled computation: hydrodynamics, prescribed aerodynamics and mooring lines. Time histories of forces and moments are presented in Fig. 17.15. Note that for clarity, the time starts at 25 s. Also, note the differences in magnitude of the computed moments, where moments about y axis are three orders of magnitude bigger, as compared to the other moment components.

Fig. 17.15
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Forces and moments acting at CoG of the support for decoupled case

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First, it should be noted that mooring lines are in general opposing the hydrodynamic forces introduced by the SPH solver. This is not true for the pitching moment, where hydrodynamics and mooring lines are acting together to counter the imbalance of the moment due to the thrust. For the mooring lines, moment is created by the displacements of the fairleads, whereas for the hydrodynamics, moment is created by the change of the buoyancy introduced by the rotation of the support. As can be seen, the mooring lines contribute about 30 %, whereas buoyancy about 70 % of the restoring moment in this system. One would expect similar, cooperative behaviour for the forces in surge (in x direction). The obtained results suggest otherwise, as shown in Fig. 17.15a. As can be seen, only the mooring lines are responsible for balancing the thrust force.

Since the water is considered calm for the decoupled case, the only source of hydrodynamic force acting in x direction is the hydrodynamic damping. Therefore, it is acting in the opposite direction of the motion, and as a result in opposite direction to the mooring force, which is a main source of motion in this direction. Lastly, small spurious moments and forces are noted, e.g. force in sway (y direction), which is normal to the plane of symmetry of the support. This is due to the SPH, where motion of the particles is never indeed symmetric. However, these discrepancies diminish with the number of particles, as was seen when test cases from Table 17.4 were computed.

Further, the SPH method is known for its pressure instabilities, where the pressure field of the particles exhibits large pressure oscillations due to acoustic waves present in compressible fluids. This is commonly tackled with solution smoothing techniques, also termed particles smoothing. Schemes up to the second order were proposed in the literature (Belytschko et al. 2000; Bilotta et al. 2011). In the present work, no particles smoothing was applied, including validation test cases. In fact, stability issues were encountered when a zero-order Shepard density filter was applied to the decoupled test case every 50 and 100 SPH steps. However, smoothing was shown to have a small effect on the overall pressure distribution for the artificial viscosity formulation used in this work (Gomez-Gesteira et al. 2012).

4.2 Coupled Case

Coupled computations were also performed, and results are presented in Fig. 17.16. As was mentioned in Sect. 17.2.5, the time step for SPH was set to ΔtSPH = 2 × 10−4 s, whereas HMB3 employed a time step of ΔtHMB3 = 2 × 10−2 s = 100ΔtSPH, or 1.06° of revolution per time step. The aerodynamic forces acting on the rotor as functions of time are shown in Fig. 17.17a. The platform motion shows similar trend as for the previous, decoupled test case. However, the rotor thrust is now dependent on the position and velocity of the rotor. As the wind turbine pitches under the thrust force, the rotor moves in the direction of the wind (velocity in x direction in Fig. 17.17b). In return, the thrust force decreases due to the reduced inflow speed and the orientation of the rotor disk. As the applied force is reduced, the rotor velocity decreases. The inverse relation between the aerodynamic force and velocity of the hub in x direction is clear in Fig. 17.17. Further, due to the pitch angle, a component of the thrust is acting along the z axis. As a result, the FOWT experiences higher displacement in heave: −0.8 m as compared to −0.6 m for the decoupled solutions. The initial motion of the FOWT is dominated by the imbalance of the forces due to the applied thrust, and the effect of the first wave passage is not visible. However, the effect of every consecutive wave is clearly visible in periodic variation of the moment about the y axis, as shown in Fig. 17.16f.

Fig. 17.16
figure 16

Lateral and rotational dynamics of the support platform for coupled test case

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Fig. 17.17
figure 17

Forces acting on the rotor and velocity of centre of gravity of the rotor as function of time for coupled computation

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To facilitate the analysis of forces and moments acting on the system, the aerodynamic moments were transferred to the centre of gravity of the support platform. The resulting time histories of forces and moments for the coupled test case are presented in Fig. 17.18. First, we observe lasting for about 10 s high frequency hydrodynamic forces and moments due to initial particles settling. Similar was observed for decoupled test case. After an initial phase, the hydrodynamic forces show periodic variation related to the frequency of the passing waves. Next, the mooring line forces are opposing the SPH forces in all directions. Finally, periodic variation of the aerodynamic forces with frequency of the waves is noted. A phase shift is present, since the aerodynamic forces are dependent on velocity and position, rather than on forces, as was discussed in previous paragraphs.

Fig. 17.18
figure 18

Forces and moments acting at CoG of the support for the coupled test case

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For the moments, pitching moment (about y) is dominating and after the initial phase the solvers tend to a periodic solution. The aerodynamic moment follows the inverse relation to the hydrodynamic pitching moment. The phase shift for the mooring lines moment is present, as it depends on the orientation of the support. The aerodynamic moment about x axis applied at the rotor is a result of a driving force created by the lift and drag. Clearly, the driving force follows the same trend as the thrust force i.e. inverse relation with the velocity of the hub. The aerodynamic moment is transferred to the structure, and hydrodynamic and mooring lines moments are trying to compensate for this moment. Finally, the mooring lines are opposing the hydrodynamic moments for the moment about z axis (yawing).

Note that no significant gyroscopic effect was observed for this FOWT. The value of gyroscopic moment can be estimated using gyroscopic approximation as \( \tau ={I}_{zz}{\omega}_r{\omega}_p \). In this case the precession rate ω p is caused by the waves, and gyroscopic torque τ should develop about body-fixed yaw axis. The pitching angular velocity is shown in Fig. 17.16d and follows sinusoidal shape with amplitude \( {\omega}_p\approx 0.006\;\mathrm{rad}/\mathrm{s} \). Given that the angular velocity of the rotor \( {\omega}_r=0.92\;\mathrm{rad}/\mathrm{s}\gg {\omega}_p \), some of the gyroscopic approximation assumptions are still valid. Substituting the above values and the mass moment of inertia of the rotor from Table 17.3 into the equation of gyroscopic approximation, it follows that the amplitude of gyroscopic torque is \( \tau =0.86\;\mathrm{M}\mathrm{N}\mathrm{m} \).

Since FOWT is oscillating about a mean pitch angle of about 0.11 rad (6.3°), the gyroscopic torque has two components when projected on the direction of global axes: one about the global z-axis \( {\tau}_z=0.77\;\mathrm{M}\mathrm{N}\mathrm{m} \), and one about the global x-axis \( {\tau}_x=0.09\;\mathrm{M}\mathrm{N}\mathrm{m} \). As can be seen, the estimated magnitude of the rolling gyroscopic torque is about 0.75 % of the mean aerodynamic moment in roll. Therefore, it can be considered negligible. On the other hand, the gyroscopic torque in yaw is comparable to other moments about the z-axis. However, those small moments did not cause significant rotation of the FOWT about this axis due to large inertia of the floater. The estimated magnitude of the gyroscopic torque is about 0.35 % of the mean aerodynamic moment in pitch. This agrees with the observations made by Velazquez and Swartz (2012) that gyroscopic effect and resulting moment is small (less than 5 %) as compared to the pitching moment for horizontal axis wind turbines with low speed rotors.

Figure 17.19 presents different positions of the FOWT during the computation. The wave breaking effect of the support structure is visible, and the recovery of the waves behind the FOWT can be seen. The change of the pressure on the rotor can also be observed, especially at the tip of the nacelle. Note that the tower was not included in the aerodynamic domain; it is, however, shown in the figure, as the presence of the tower was accounted for in the multi-body model.

Fig. 17.19
figure 19

Position and orientation of the FOWT at times t = 0 s and t = 30 s during coupled computation. Contours on the rotor correspond to pressure coefficient Cp, contours on the water surface correspond to surface elevation z in meters

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4.3 Computational Performance

For all cases, the SPH solver with MBDM were executed on a single 8 cores Intel® Xeon® CPU machine with 16 threads. Each of the CPU cores had a clock rate of 2 GHz, and 6.6 GB of dedicated memory. As no interconnect switch was involved, the message passing delay between SPH and MBDM solvers was reduced to minimum. For the coupled case, HMB3 was executed on 29 dual-core AMD Opteron™ processors with 4 threads, giving in total 116 parallel instances of the solver. Each of the CPU cores had a clock rate of 2.4 GHz, and 4 GB of random access memory. It should be noted that the SPH method requires only local (limited by the kernel function) weighted average in the vicinity of the given particle, whereas HMB3 solves the complete set of equations involving all the cells in the domain. Hence, more processing units were assigned to the aerodynamic side of the coupled problem.

The average time required to compute a second of the solution for the coupled case is 27.26 h, where about 27.25 h were spent to solve aerodynamics, 21.3 h to solve hydrodynamics, and 0.24 h to solve multi-body equations. The average time spent to exchange information for a second of the solution is 0.53 s, and was mostly dictated by the communication between the SPH and the MBDM solvers.

It should be noted that time accuracy can be improved, if the coupling step is reduced. In the presented coupled case, the information is exchanged every 100 SPH steps (Δt = 2 × 10−2). When information between the solvers is exchanged every 50 SPH steps (Δt = 1 × 10−2), the average time required to compute a second of the solution becomes 45.0 h. If information is exchanged every single SPH step (Δt = 2 × 10−4), the average time per one second extends to about 438.9 h. In the former case, HMB3 requires on average 237 pseudo-time steps to achieve the level of convergence below 10−2, and 45 pseudo-time steps for the latter case. The convergence is defined as L2-norm of the residual vector. This suggests that computational cost can be further reduced by employing explicit schemes for both solvers and performing less evaluations (four for Runge-Kutta scheme of 4th order). However, the biggest possible explicit step for HMB3 that would satisfy explicit CFL condition of 0.4 for the smallest cell in the domain is about 3.6 × 10−9 s. Therefore, the aerodynamic time-step becomes the limiting factor for this approach and for the problem at hand. More information about the computational performance is presented in Table 17.5. Stability issues were encountered for a time step Δt = 2 × 10−2 and HMB3 implicit CFL number 10.0, where the residual vector does not converge as fast as for CFL number 5.0. This indicates that CFL number of about 8.0 would be an optimal choice for this time step.

Table 17.5 Computational performance of the coupling algorithm for various coupling time steps
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5 Conclusions

The chapter presented a coupling method for the analysis of the dynamics of floating offshore wind turbines. The HMB3 CFD solver was used for the analysis of blade aerodynamics and via a multi-body dynamics method it was coupled to a smoothed particle hydrodynamics tool to model the floating part of the turbine. The results showed that the weak coupling method is adequate for the solution of the problem at hand. Due to the lack of experimental data for a coupled system, validation was only possible for the components of the model. Data from the MEXICO project were used for aerodynamics; good overall agreement has been seen between CFD and test data. For the hydrodynamics solver, experiments related to drops of solid objects in water were used. Again, with a refined set of particles, the SPH method delivered good results. The third component of the method was the multi-body dynamics and this was validated using simple slider-crank problems.

Presented results demonstrated that a FOWT is a highly dynamic system. To obtain a deeper understanding of how rotor thrust and torque vary under dynamic conditions, efforts should be put forward to study the aerodynamic flow and loads as a wind turbine undergoes prescribed motion in pitch and yaw. It should be noted that the spatial resolution of water employed in this work can be improved. In the future, a finer set of SPH particles will be employed and the tower will be included in the aerodynamic domain. Also, in the future, the work will continue with the validation of the method against experimental data, when available, and comparisons with a strong coupling technique. Another aspect that should be addressed is the experimental measurements. Clearly, each of the components can be validated separately, but the set of comprehensive data for the complete FOWT system is crucial for the model validation. The following measurements would be an asset: forces and moments due to the mooring system, water basin tests with small- or full-scale wind turbine including pressure distributions on support and rotor, and the overall FOWT time response including transient and periodic states.