# CFD Investigation of a Complete Floating Offshore Wind Turbine

## Abstract

This chapter presents numerical computations for floating offshore wind turbines for a machine of 10-MW rated power. The rotors were computed using the Helicopter Multi-Block flow solver of the University of Glasgow that solves the Navier-Stokes equations in integral form using the arbitrary Lagrangian-Eulerian formulation for time-dependent domains with moving boundaries. Hydrodynamic loads on the support platform were computed using the Smoothed Particle Hydrodynamics method. This method is mesh-free, and represents the fluid by a set of discrete particles. The motion of the floating offshore wind turbine is computed using a Multi-Body Dynamic Model of rigid bodies and frictionless joints. Mooring cables are modelled as a set of springs and dampers. All solvers were validated separately before coupling, and the loosely coupled algorithm used is described in detail alongside the obtained results.

## Keywords

Wind Turbine Smooth Particle Hydrodynamic Thrust Force Message Passing Interface Smooth Particle Hydrodynamic## Nomenclature

## Latin

- d
distance between particles (m)

- I
inertia tensor (kg m

^{2})- m
mass (kg)

- w
relative weight between the fluid and body particles (–)

## Greek

*α*artificial viscosity parameter (–)

*γ*adiabatic index (–)

*ω*rotational velocity (rad/s)

*τ*gyroscopic torque (Nm)

## Acronyms

- BEM
Blade Element Momentum method

- BILU
Block-Incomplete Upper Lower factorisation

- FOWT
Floating Off-shore Wind Turbine

- FSI
Fluid Structure Interaction

- GCG
Generalised Conjugate Gradient

- GMRES
Generalised Minimal Residual method

- HMB3
Helicopter Multi-Block CFD Solver

- HPC
High Performance Computer

- IBQN-LS
Interface Block Quasi-Newton with an approximation for the Jacobian from a Least-Squares mode

- IQN-ILS
Interface Quasi-Newton algorithm with an approximation for the inverse of the Jacobian from a Least-Squares model

- MBDM
Multi-Body Dynamic Model

- MPI
Message Passing Interface library

- SPH
Smoothed Particle Hydrodynamics Method

## 17.1 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).

Works relevant for the complete FOWT models

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.

## 17.2 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.

### 17.2.1 Validation of the Aerodynamic Solver

### 17.2.2 Validation of the Hydrodynamic Solver

^{3}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.

### 17.2.3 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.

Properties of the bodies employed to model the gyroscopic effect

Name | Mass (kg) | Inertia tensor (kg m |
---|---|---|

Wheel | 28.3 | \( \left[\begin{array}{ccc}\hfill 1.45\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0.73\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0.73\hfill \end{array}\right] \) |

Rod | 0.1 | \( \left[\begin{array}{ccc}\hfill {10}^{-6}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 8.3\times {10}^{-5}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 8.3\times {10}^{-5}\hfill \end{array}\right] \) |

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.

### 17.2.4 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.

### 17.2.5 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, 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 Δt_{SPH} = 2 × 10^{−4} s with CFL = 0.2, whereas HMB3 employs a time step of Δt_{HMB3} = 2 × 10^{−2} s = 100Δt_{SPH} 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.

## 17.3 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.

Mechanical properties of the employed bodies and mooring lines

Rotor | |

m (kg) | 227,962 |

I (kg m | \( \left[\begin{array}{ccc}\hfill 1.56\times {10}^8\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 7.84\times {10}^7\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 7.84\times {10}^7\hfill \end{array}\right] \) |

Nacelle, support and tower | |

m (kg) | 4,223,938 |

I (kg m | \( \left[\begin{array}{ccc}\hfill 2.03\times {10}^{10}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 2.03\times {10}^{10}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 2.81\times {10}^9\hfill \end{array}\right] \) |

Mooring lines | |

120.0 | Angle between adjacent lines () |

20.6 | Depth of anchors below SWL (m) |

7.0 | Depth of fairleads below SWL (m) |

116.73 | Length of the relaxed line (m) |

400×10 | Mooring line extensional stiffness (N/m) |

40,000 | Mooring line damping coefficient (Ns/m) |

### 17.3.1 CFD Mesh

^{−5}c, 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.

The density of the air was assumed to be ρ = 1.225 kg/m^{3}, the dynamic viscosity of the air was assumed to be μ = 1.8 × 10^{−5} Ns/m^{2}, 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.

### 17.3.2 SPH Setup and Resolution

Test cases investigating the influence of the domain width and particle spacing on the forces acting on the support structure

Domain size x × y (m) | Spacing d (m) | 1s averaged hydrodynamic force (N) | Difference (%) |
---|---|---|---|

500 × 150 | 0.6250 | 1.070 × 10 | – |

500 × 300 | 0.6250 | 1.068 × 10 | 0.20 % |

500 × 150 | 0.3125 | 1.267 × 10 | 18.40 % |

### 17.3.3 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.

### 17.3.4 Demonstration Cases

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.

## 17.4 Results and Discussion

### 17.4.1 Decoupled Case

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.

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).

### 17.4.2 Coupled Case

_{SPH}= 2 × 10

^{−4}s, whereas HMB3 employed a time step of Δt

_{HMB3}= 2 × 10

^{−2}s = 100Δt

_{SPH}, 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.

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.

### 17.4.3 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.

^{−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.

Computational performance of the coupling algorithm for various coupling time steps

Coupling | HMB3 | HMB3 | SPH | Time per | Time per 1s |
---|---|---|---|---|---|

Δt (s) | CFL no. | Newton steps | steps | coupling step (s) | of solution (s) |

2 × 10 | 5.0 | 315 | 100 | 1.95 × 10 | 9.81 × 10 |

2 × 10 | 10.0 | 350 | 100 | 2.29 × 10 | 1.15 × 10 |

1 × 10 | 5.0 | 237 | 50 | 1.61 × 10 | 1.62 × 10 |

1 × 10 | 10.0 | 105 | 50 | 1.04 × 10 | 1.06 × 10 |

2 × 10 | 5.0 | 45 | 1 | 3.13 × 10 | 1.58 × 10 |

2 × 10 | 10.0 | 23 | 1 | 1.59 × 10 | 7.97 × 10 |

## 17.5 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.

## Notes

### Acknowledgments

Results were obtained using the EPSRC funded ARCHIE-WeSt High Performance Computer (www.archie-west.ac.uk). EPSRC grant no. EP/K000586/1.

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