Validation and application of a fully nonlinear numerical wave tank for simulating floating offshore wind turbines

Ocean Engineering 147 (2018) 647–658

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Validation and application of a fully nonlinear numerical wave tank for simulating floating offshore wind turbines N. Bruinsma *, B.T. Paulsen, N.G. Jacobsen Department of Hydraulic Engineering, Deltares, Boussinesqweg 1, 2629 HV Delft, The Netherlands

A R T I C L E I N F O

A B S T R A C T

Keywords: Floating offshore wind turbines OpenFOAM Coupled CFD/6-DOF Waves2Foam Mooring lines

Numerical models are becoming a valid supplement, and even a substitute, to physical model testing for the investigation of fluid-structure interaction due to improved methods and a continuous increase in computational power. This research presents an extensive validation of a fully nonlinear numerical wave tank for the simulation of complex fluid-structure interaction of moored floating structures. The validation is carried out against laboratory measurements including recent and unpublished laboratory measurements of the OC5 floating offshore wind turbine subjected to waves. The numerical wave tank is based on the Navier-Stokes/6-DOF solver, interDyMFoam, provided by the open-source CFD-toolbox OpenFOAM, extended with the wave generation and absorption toolbox, waves2Foam, and an implementation for restraints of floating structures. Two methods are evaluated to address the instability issues of the partitioned Navier-Stokes/6-DOF solver, which are associated with artificial added mass. The model has been shown capable of computing detailed fluid-structure interactions, including dynamic motion response of a rigid structure in waves. However, instabilities due to numerical added mass is observed and discussed and it is concluded that further research is needed in order to establish a efficient yet stable scheme. Overall, good agreement is achieved between the numerical model and the physical model results.

1. Introduction In recent years a transformation in energy supply from burning of fossil fuels towards renewable energy sources such as solar, wind and hydropower is seen. In northern Europe this change is mainly driven by wind energy and lately also offshore wind energy. The increase in wind energy capacity is expected to continue for several decades partly by replacing existing capacity with newer, larger and more efficient turbines, but for the greater part by the installation of new wind farms (Pineda et al., 2014). Offshore wind energy installations are expected to produce approximately 2.9% of the total European Union energy demand by 2020. This foreseen growth will move offshore wind energy from an emerging and immature technology to a key component of the energy mix (Arapogianni et al., 2011). Current developments of offshore wind are primarily based on bottom mounted foundations and the vast majority of those on monopile foundations. Gravity based substructures are the second most common foundation type, followed by jacket structures. However, many countries have limited suitable sites in sufficiently shallow water to economically install offshore wind turbines on bottom mounted substructures. Floating

offshore wind turbine (FOWT) concepts are seen as a promising solution to unlock the offshore potential in deeper waters. This technology is still at an early stage of development and well validated modelling tools capable of simulating dynamic behaviour are necessary to improve the design of these floating structures (Arapogianni et al., 2013). One of the challenges for FOWT at intermediate water depth (50 m < h < 120 m) is optimizing the dynamic behaviour of the mooring system while retaining a limited footprint. Optimization of mooring systems for semi-submersibles and FOWT foundations are discussed in Robertson et al. (2011); Brommundt et al. (2012); Huijs (2015). The common approach for these studies is to use numerical models to analyse different load cases including the effect of wind and wave. However not discussed in this paper, the presented and validated numerical model is suitable for such analysis and one of the reasons why this research was initialized. The International Energy Agency has facilitated a series of international collaboration projects to improve offshore wind modelling tools. These projects aim to verify and validate the accuracy of complex engineering tools through code-to-code and code-to-data comparisons (Jonkman and Musial, 2010; Popko et al., 2012; Robertson et al., 2014b). The most recent of these projects is the Offshore Code Comparison

* Corresponding author. E-mail address: [email protected] (N. Bruinsma). https://doi.org/10.1016/j.oceaneng.2017.09.054 Received 18 January 2017; Received in revised form 17 August 2017; Accepted 24 September 2017 Available online 23 October 2017 0029-8018/© 2017 Elsevier Ltd. All rights reserved.

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2.1. Governing equations Navier-Stokes/VOF

Collaboration, Continued, with Correlation (OC5). This project continues with the improvement of modelling tools based on the dynamic analysis of the DeepCwind design (Koo et al., 2014; Robertson et al., 2014a). This semi-submersible design features a 5 MW horizontal-axis wind turbine with pitch controlled blades developed by the National Renewable Energy Laboratory (Jonkman et al., 2009; Popko et al., 2012). The simulation tools applied in these research projects are frequencydomain potential flow models such as; AQWA (ANSYS, 2010) and WAMIT (Lee and Newman, 2006). These lower order numerical models are used to determine the frequency-dependent added mass and damping coefficients. The output from these models is used in a time-domain model, e.g FAST (Jonkman et al., 2014), to perform the transient analysis. Especially for the analysis of a FOWT in extreme conditions, where these lower order models may be less reliable, it could be beneficial to use a higher order numerical model for describing the fluid-structure interaction. Higher order numerical models, like Computational Fluid Dynamics (CFD) combined with 6-DOF simulations, have been successfully used as a tool in many areas of engineering. With CFD models it is possible to use higher order wave models and even accurately simulate wave breaking. The major benefit of using a higher order CFD/6-DOF model is the capability of solving the fluid-structure interaction problem based on the geometry and mass properties of the floating structure, without predetermining coefficients. However, the accuracy of predictions and numerical convergence needs to be validated in order to use CFD/6-DOF models as a reliable design tool for floating offshore structures. The objective of this research is to establish a well validated fully nonlinear numerical wave tank for the accurate simulation of complex fluid-structure interaction of moored floating offshore structures using a higher order CFD/6-DOF model based on the opensource CFD-toolbox OpenFOAM (Weller et al., 1998). Section 2 provides an overview of the numerical models used to set up the numerical wave tank. In Section 3 the numerical model is validated against theoretical and experimental data. This validation consists of three approaches. First, a wave load and decay case with a two-dimensional cylinder are presented, where the model is validated against benchmark experimental and theoretical data from Dixon et al. (1979), Ito (1977) and Maskell and Ursell (1970). Secondly, the model is used to simulate three-dimensional moored decay of a FOWT from the OC5 projects. Lastly, the moored FOWT is subjected to regular incoming waves. Both the decay and wave loading cases with the OC5 FOWT are validated against physical experiments. These tests were carried out in the concept basin at MARIN and besides focusing on hydrodynamic loading also the effect of the wind turbine controller on the dynamics of the floater was investigated.

The waveDyMFoam solver utilizes the two-phase incompressible Navier-Stokes equations in combination with a VOF-surface capturing scheme (Hirt and Nichols, 1981) to compute fluid structure interaction. The governing equations, used in the Navier-Stokes/VOF solver, for conservation of mass and momentum of an incompressible flow of air and water are given by

∇⋅u ¼ 0;

(1)

∂ρu þ ∇⋅ðρuÞuT ¼ ∇p þ ðg⋅xÞ∇ρ þ ∇⋅ðμ∇uÞ; ∂t

(2)

where ∇ ¼ ð∂x ; ∂y ; ∂z Þ is the three-dimensional gradient operator, u ¼ ðu; v; wÞ is the velocity field in Cartesian coordinates, g is the gravitational acceleration and p is pressure in excess of the hydrostatic pressure, which relates to the total pressure, p, by

p ¼ p  ρðg⋅xÞ:

(3)

Furthermore, the local density, ρ, and viscosity, μ, are given by the water volume fraction, α, consistent with

ρ ¼ αρwater þ ρair ð1  αÞ;

(4)

μ ¼ αμwater þ μair ð1  αÞ;

(5)

where α is zero for air, one for water and a mixture of the two for all intermediate values. OpenFOAM uses a VOF method for tracking the air-water interface. After obtaining the velocity field by solving equations (1) and (2) for the two-phase flow of air and water, the VOF method (Hirt and Nichols, 1981) can be used to advance the α field in time with the following scalar advection equation

∂α þ ∇⋅uα þ ∇⋅ur αð1  αÞ ¼ 0: ∂t

(6)

Using a standard finite-volume approximation for solving the hyperbolic advection equation (6) would lead to significant smearing of the interface. This is significantly reduced by the introduction of an interface compression term as discussed in Berberovi
c et al. (2009). In equation (6) this is the last term of the left-hand side. The interface compression term is only active in the vicinity of the interface, 0 < α < 1, where its strength is governed by the relative velocity, ur . It is stressed that even though ur has the dimension of m/s, it lacks any physical meaning. To ensure stability, a multi-dimensional flux limited scheme (MULES) is used for solving the scalar advection equation (6).

2. Material and methods The multiphase interDyMFoam solver of OpenFOAM is a segregated fluid-structure interaction solver, where the flow-dependent motions of a rigid body are obtained by solving the Navier-Stokes and 6-DOF equations in a coupled manner. The coupled Navier-Stokes/6-DOF solver has an unstable nature, as discussed in Seng (2012) and Dunbar et al. (2015). In the present work, two different methods for stabilizing such a solver are evaluated. Here the standard interDyMFoam solver, provided by the opensource CFD-toolbox OpenFOAM (Weller et al., 1998), version 2.3.1., is extended with the wave generation and absorption toolbox waves2Foam, developed by Jacobsen et al. (2012). This combination is referred to as the waveDyMFoam solver. Furthermore, an implementation for the restraints of floating structures was developed. The structure of this section is as follows: first the governing equations for the Navier-Stokes/Volume-Of-Fluid (VOF) solver are presented in Section 2.1, the differences between the two stabilizing methods are described in Section 2.2, in Section 2.3 the restraints implementation is presented, the applied boundary conditions are presented in Section 2.4, wave generation and absorption with relaxation zones is described in Section 2.5 and finally the spatial and temporal discretization is described in Section 2.6.

2.2. Stability of a coupled Navier-Stokes/6-DOF solver In order to simulate the flow-dependent motion response of a floating structure, a partitioned fluid-structure interaction model is used. Here, the rigid body motion is obtained by separately solving the Navier-Stokes and 6-DOF motion equations for the fluid and the solid. For numerical models with a loos coupling between the fluid solver and the motion of the body, the discrete time integration may become unstable leading to non-physical motion of the object. Effectively these instabilities appear as a numerical added-mass and hence change the behaviour of the structure. If the density difference between the fluid and the body becomes smaller, the fluid has a larger influence on the motions of the body and numerical simulations are more prone to instabilities. The temporal discretization of the numerical simulation was also found to have a significant effect on the stability of the solution. Thorough discussions on instabilities in partitioned fluid-structure interaction models as a result of artificial added mass can be found in the work by 648

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make an initial ”predictor” step and one or more ”corrector” steps. In the predictor step, first the displacement of the body is estimated from the forces acting on the body as is also done in the standard solver. The second step is to update the pressure field and recalculate the corrected displacement of the body. To smooth the convergence an underrelaxation factor, fp , is applied to the pressure field according to

Causin et al. (2005), F€ orster et al. (2007) and Seng (2012). A common solution to counter the instabilities due to artificial added mass is to implement a tighter coupling. In tighter coupled solvers the solutions for both the fluid and the rigid body motion are more converged before the simulation advances to the next time step. Both Piro and Maki (2013) and Dunbar et al. (2015) were able to increase the numerical stability in complex fluid-structure interaction problems by introducing a subiteration loop around the Navier-Stokes and 6-DOF motion solvers and applying dynamic relaxation to the body force. The open-source CFD-toolbox OpenFOAM, version 2.3.1, provides two methods for in increasing numerical stability in fluid-structure interaction problems: an under-relaxation and a predictor-corrector method. Both methods will be evaluated in the present work. The difference between the two methods is in the PIMPLE loop, which is used to perform time stepping. For a comprehensive review of the PIMPLE method, the reader is referred to Ferziger and Peric (2012).


 pi ¼ fp pi þ 1  fp pi1 ;

(8)

where, pi , is the under-relaxed pressure field. In contrast to the work of Dunbar et al. (2015), no dynamic relaxation was applied, however a constant under-relaxation factor, fp of 0.5 was applied. Note that the under-relaxation factor for the pressure field is reset to 1 for the last subiteration loop to ensure time consistence. Throughout the present work the number of subiterations was fixed to 3.

2.2.1. Under-relaxation method The under-relaxation method is embedded in the 6-DOF loop, as can be observed from the algorithm presented at the end of this section. In this method an under-relaxation factor is applied to the acceleration of the rigid body. In the 6-DOF loop, first, the forces and moments on the body are calculated. From these forces and moments, the displacement and rotation of the body is determined for a certain time step. With the under-relaxation method, the acceleration of the center of gravity (COG) of the structure, is ’relaxed’ by an under-relaxation factor, fa , according to

ai ¼ fa ai þ ð1  fa Þai1 ;

(7)

where, ai , is the under-relaxed acceleration of the COG of the structure at the instantaneous time step. Please note that a can be directional and rotational acceleration. Furthermore, the relaxation of a, only starts from the second time step. The applied under-relaxation method effectively acts as a low-pass filter removing high frequency oscillations from the time integration. By doing so, the stability of the solver is improved, however it also introduces a diffusive term to the numerical model which can have a negative effect on the convergence rate of the solution. Throughout the present research the default under-relaxation factor of 0.5 is used, this value was found to provide acceptable stability while maintaining sufficient level of accuracy.

2.3. Restraints for moving boundaries Two types of restraints for moving boundaries are implemented: a linear spring and a simple catenary type mooring line. It should be noted that this simple mooring line implementation does not take the interaction with the fluid into account, thus it is not influenced by waves or current. 2.3.1. Linear spring The linear spring is implemented without any damping and the line is given a simple constant stiffness and a rest length. Force formulation of the spring follows Hooke's law. The spring is defined such that the force in the spring is zero when the length of the spring is equal to the rest length. 2.3.2. Mooring line The simple catenary type mooring line implementation is used for anchoring a floating structure to the seabed, where the shape and force formulations are adopted from Krenk (2001). It should be noted that a flat horizontal seabed is assumed in the presented mooring line implementation. Three different mooring line states are defined: simple state, resting state and hanging state, see Fig. 1. 2.3.2.1. Simple state. In the simple state, see Fig. 1a, the mooring line is completely in suspension, i.e. the anchor point, p0 , is equal to the touchdown point, pt . The formulation follows the restoring force in a catenary line between two attachment points, where the distance between the attachment points is less than the length of the line (Krenk, 2001). The line cannot break, while the distance between the connected

2.2.2. Predictor-corrector method The predictor-corrector method implements a subiteration loop around the Navier-Stokes and 6-DOF solvers as well as under-relaxation of the pressure field, as can be seen from the algorithm presented at the end of this section. The subiteration loop allows the numerical model to

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the free surface, is on-going research. The viscous boundary layer at the seabed is neglected, since typically the propagation of waves with a CFD-solver is limited to only a couple of wave lengths and in such simulations the influence of the seabed on the wave dissipation is small. A slip condition is applied to the side-walls of the three-dimensional numerical wave-tank, as presented in Section 3.3, while for the two-dimensional computations, as presented in Section 3.1 and 3.2, an empty condition was applied to the front and back walls. At the atmosphere boundary layer of the numerical domain, an atmospheric boundary condition is applied for the velocity, u, and the α field. This means that air and water are allowed to leave the numerical domain, while only air is allowed to flow back in. Furthermore, the total pressure at the atmosphere boundary is equal to zero.

Fig. 1. Schematic representation of the three different states of the catenary mooring line, where (a) is the simple state, (b) the resting state and (c) the hanging state.

objects can exceed the defined length of the line. Beyond this defined length the line is treated as a linear spring, where the stiffness of the line is based on the magnitude of the restraining force in the catenary line at the limit of its length.

2.5. Relaxation zones In present work the wave generation toolbox, waves2Foam, developed by Jacobsen et al. (2012), was used to implement relaxation zones in the numerical domain. For the validation cases without incoming waves, as presented in Section 3.2 and 3.3.1, relaxation zones are implemented at the in- and outlet of the domain to absorb waves generated by the structure's motion. The relaxation zones for the two validation cases where a structure is subjected to incoming waves, as presented in Section 3.1 and 3.3.2, are set up to generate a continuous propagating wave. The relaxation zone at the inlet is used to generate the incoming waves, while the relaxation zone at the outlet is set up to produce the undisturbed incoming wave profile, thus minimizing reflections from the inlet and outlet boundary layers. See Bruinsma (2016) for a more detailed description of this particular relaxation zone set-up. As the physical wave maker signal is unknown, an exact reproduction of hydrodynamic conditions used in the various benchmark cases is not possible. Instead, fully nonlinear stream function waves (Fenton, 1990) of the required height and period are used.

2.3.2.2. Resting state. The resting state, see Fig. 1b, is the mooring line state, where the floating structure is so close to the anchor point that a part of the catenary line is resting on the sea bed. The remainder of the cable will behave as a catenary line, as defined in the simple state. The touchdown point, pt , is evaluated by finding the resting length of the line that ensures that the slope of the line at the touchdown point is horizontal (Krenk, 2001). 2.3.2.3. Hanging state. The hanging state is defined as the state where the floating object is so close to the anchor point that the piecewise linear configuration of the mooring line is shorter than the length of the mooring line, see Fig. 1c. Only a vertical force acts in the attachment point, which is equal to the weight of the suspended part of the line. 2.4. Boundary conditions

2.6. Spatial and temporal discretization

To solve the governing equations of the Navier-Stokes/VOF solver, boundary conditions are specified for all surfaces in the numerical domain. Before going into the allocation of particular boundary conditions, the general denomination of the boundary surfaces is given in a schematic representation of the numerical domain, see Fig. 2. It should be noted that a structure will be present inside the numerical domain. At the inlet and outlet boundary surfaces the velocities, u, and the α field are given by a closed form analytic wave theory. This technique works well and allows minimisation of the start-up time by initialising the free surface with the incident wave profile. At the surface of the body, a slip condition is applied, which means that the viscous boundary layer near the body is not resolved. However, the governing equations still contains the effect of viscosity and turbulence generation due to internal stresses and the free surface motion near the structure i.e. viscous effects due to wave breaking or splashing € near the structure are included. The viscous boundary layer is not resolved as this requires the usage of a turbulence model, which often influences the evolution of the free surface (Brown et al., 2014). Development of turbulence models, which can model a structural boundary layer without influencing the propagation of

The numerical domains used in this research are created using the blockMesh and snappyHexMesh utilities provided by OpenFOAM. The numerical wave tank is formed as a box domain which is generated with blockMesh and snappyHexMesh is used to sculpt the surface boundary of the structure inside the mesh. Generation of the structure surface boundary is done by local refinement and reconstruction of cells, where 2-2 and 4-4 levels of refinement were used for the two- and three-dimensional domains. This local refinement allows for a high resolution interface while keeping the total number of computational cells relatively low. The spatial discretization of the numerical domains is based on the cell size next to the structure surface boundary, which in turn is defined by the number of points per cylinder diameter (p.p.c.d.) as,

Δ¼

D : p:p:c:d:

(9)

This means that for the two-dimensional cases, the cells closest to the structure surface boundary have a dimension equal to the number of p.p.c.d., whereas the cells outside the refinement zone are 4 times as large. For the temporal discretization, i.e., determining the time step of the numerical simulations, the Courant-Friedrichs-Lewy (CFL) condition is used. Throughout the present research a Courant number equal to 0.25 is applied. 3. Results This section presents the results of the benchmark cases performed to validate the numerical model. The first case, presented in Section 3.1,

Fig. 2. Schematic representation of the computational domain, where the shaded areas, I and II, are relaxation zones implemented with the waves2Foam toolbox. 650

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regards the simulation of wave loading on a fixed, partially submerged horizontal cylinder in a two-dimensional fixed mesh. Here, the numerical results are compared against experimental data from Dixon et al. (1979). In Section 3.2, the two stabilizing methods for the Navier-Stokes/6-DOF coupling are evaluated on the basis of the heave decay of a free floating horizontal cylinder in a two-dimensional numerical domain with a dynamic mesh, where the numerical result is compared to theoretical work by Maskell and Ursell (1970) and experimental work by Ito (1977). Please note that for these two validation cases the application of a 2D domain was part of the validation and since there is good agreement with the experimental results the 2D approach is justified. In Section 3.3 the numerical model is validated with a number of three-dimensional physical model tests with the moored OC5 FOWT. First, the moored heave, pitch and roll decay tests are evaluated in Section 3.3.1 and finally the floating structure is subjected to unidirectional regular waves, the results from this numerical simulation are presented in Section 3.3.2.

Fig. 3. Schematic representation of the numerical domain for the two-dimensional validation cases.

Table 1 Spatial discretization of the background mesh, Δ0 , the size of the cells closest to the surface boundary, Δ1 , and total number of computational cells, nCells.

3.1. Flow-induced pressures on a fixed body In this first validation case the numerical model is used to reproduce results from the wave loading experiments from the work of Dixon et al. (1979). These simulations are performed to determine whether the Navier-Stokes solver is capable of accurately simulating fluid-structure interaction, so without the additional complexities of moving meshes and the coupling of the Navier-Stokes and 6-DOF solvers. The work by Dixon et al. (1979) presents a series of physical experiments where vertical and horizontal forces on a fixed horizontal cylinder, induced by regular waves, were determined for various wave amplitudes and levels of cylinder submergence. It was discovered that the cylinder's varying buoyancy can play an equally large role as the inertial force in the resultant vertical force. In particular cases, this led to negative vertical force profiles which act at twice the wave frequency. In these cases the inertial coefficient, CM , in Morison's equation becomes negative. By introducing a varying volume and buoyancy term to Morison's equation, Dixon et al. were able to reformulate the analytical description, thus achieving better correspondence with the experimental results. Table 2 shows a selection of the experimental and theoretical results of Dixon et al. for various relative wave amplitudes and a constant submergence depth of the cylinder, d' equal to zero. Here d0 is measured from the cylinders centroid to the still water level, i.e., the cylinder is half submerged. The relative wave amplitude, A0, is the ratio of the wave amplitude, A, to the cylinder diameter, D. For the comparison of the various results, the dimensionless Root Mean Square (RMS) vertical force is calculated using

Frms

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 1X ¼ F2 ; n k¼1 k

p.p.c.d

Δ0 , [m]

Δ1 , [m]

nCells

20 30 40

0.020 0.013 0.010

0.0050 0.0033 0.0025

16,752 31,493 52,291

Fig. 4. Vertical forcing, normalized by the buoyancy force of the fully submerged cylinder, presented as a function of time normalized by the wave period. Grid sensitivity analysis, where spatial resolutions of 20, 30, and 40 p.p.c.d are applied. Comparison between the numerical solution and the theoretical and experimental data from Dixon et al. (1979). The relative wave amplitude and axis depth are 0.5 and 0.0.

(10)

highest resolution was selected for all further simulations in this section, in order to ensure a high level of convergence. Five simulations are performed with a constant relative axis depth, d0 ¼ 0:0, while the relative amplitude of the wave is varying from 0.1 to 0.5. For all simulations the RMS force, Frms , is computed and these are presented in Table 2, where the error with respect to the experimental data from Dixon et al. (1979) is also presented. It can be observed that the numerically computed RMS forces are slightly underestimating the

where Frms is the RMS force and Fk the relative vertical force component at each time step, which is the total vertical force on the cylinder divided by the buoyancy force of the totally submerged cylinder. Fig. 3 shows a schematic representation of the numerical domain used for the two-dimensional cases presented in this section and Section 3.2. For this section the domain dimensions are: l ¼ 4λ, d ¼ 0:5 m, a ¼ 0:25 m, D ¼ 0:1 m and d0 ¼ 0:0 m. The boundary conditions, relaxation zones and numerical discretization are defined according to the twodimensional set-up described in Section 2.4, 2.5 and 2.6. To verify grid independence, a grid sensitivity analysis is carried out for one of the wave loading cases. The applied spatial discretizations are presented in Table 1. In Fig. 4, the numerical results for the vertical force on the cylinder are presented along with the experimental solutions from Dixon et al. (1979). It can be observed that the numerical solution is converging with increasing grid resolution. Please note that since the CFL condition is used to determine the time step, an increase in the spatial resolution also results in an increase in the temporal resolution. The

Table 2 Comparison of numerically computed RMS forces with the experimental and theoretical data from Dixon et al. (1979). The numerical error relative to the experimental Frms is expressed as a percentage.

651

Amp. A0

Theo. Frms

Exp. Frms

Num. Frms

Error ½%

0.10 0.20 0.30 0.40 0.50

0.063 0.126 0.185 0.238 0.284

0.062 0.121 0.172 0.221 0.264

0.058 0.113 0.165 0.198 0.225

7 7 4 11 15

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to accurately calculate flow induced pressures on a fixed body. In the present section the ability of the numerical wave tank to simulate flowinduced motions of a free floating two-dimensional horizontal cylinder is evaluated. As described in Section 2.2, artificial added mass may cause instabilities in loosely coupled partitioned solvers. This section explores whether the under-relaxation and predictor-corrector method can improve the stability of the Navier-Stokes/6-DOF solver while maintaining a high level of accuracy. The validation is performed on a benchmark case of the heave decay of a free floating horizontal cylinder with theoretical results from Maskell and Ursell (1970) and experimental results from Ito (1977). In their work, Maskell and Ursell (1970), present a theoretical formulation for the decay of heave motion of a free floating horizontal cylinder. This theory was later validated by experimental work from Ito (1977). The numerical domain set-up for the simulation of the heaving cylinder is similar to the one used for the validation cases presented in Section 3.1, a schematic representation is presented in Fig. 3, where l ¼ 6 m, d ¼ 1:24 m, a ¼ 0:2 m and D ¼ 0:1524 m. The initial vertical displacement of the cylinders COG with respect to the still water level is equal to one sixth of the cylinder diameter, or d0 ¼ 0:0254 m. Relaxation zones, with a length of 1 m, are implemented at both ends of the domain to absorb waves generated by the heaving cylinder. The boundary layer conditions, relaxation zones and numerical discretization are defined according to the two-dimensional set-up described in Section 2.4, 2.5 and 2.6. The vertical displacement of a the cylinder's COG relative to the still water level is presented in Fig. 6. Please note that the cylinder density of the cylinder is half that of the fluid and that the cylinder is half submerged in the equilibrium. The cylinder was dropped from an initial displacement of d0 ¼ 0:0254 m. Note that the initial velocity of the cylinder is equal to zero. The spatial discretization of the computational domain is presented in Table 3, where the grid resolution of 40 p.p.c.d. is adopted from the grid sensitivity study of the wave loading simulation discussed in Section 3.1. Overall it can be concluded from the transient heave responses, presented in Fig. 6, that, in spite of the small differences introduced by different stabilizing methods, the numerical solver is able to provide an accurate solution to the physical problem. From the error in the period and amplitude of the first peak response in the cylinder's transient motion, presented in Table 4, it can be observed that both transient motions of the simulations without a stabilizer and with the under-relaxation method show slight overestimations of amplitude, while result obtained with the predictor-corrector method is found to be in better

experimental data and the offset with the experimental RMS force increases for the higher amplitude waves. This underestimation could be a result of not resolving the viscous boundary layer near the body and this is an interesting topic for future research. The RMS forces show that the numerical model is able to determine the general forcing over a single wave period, however for a more detailed evaluation of the accuracy of the numerical model, the force profiles are presented in Fig. 5. It may be observed that for the smaller amplitude waves, both the theoretical and numerical forcing show good correspondence with the experimental data. However, for relative amplitudes larger than 0.2 the theoretical formulation lacks accuracy in the loading caused by the wave crest passing the cylinder. Much better consensus is achieved between the experimental data and the numerical results. The force profiles are in general better aligned and especially the prediction of the peak forces caused by the passing wave crest is more in line with the experimental results. Overall it may be concluded that the Navier-Stokes solver was able to accurately simulate the wave loading on a fixed horizontal cylinder.

3.2. Flow-induced motions of a free floating body From Section 3.1 it follows that the numerical wave tank can be used

Fig. 5. Vertical forcing, normalized by the buoyancy force of the fully submerged cylinder, as a function of time normalized by the wave period. Comparison between theoretical and experimental data from Dixon et al. (1979) and numerical simulation. The relative axis depth, d0, is kept constant at 0.0, while the relative amplitude, A0, is varying from 0.1 to 0.5.

Fig. 6. Vertical displacement of the center of mass normalized by the initial displacement, pffiffiffiffiffiffiffiffi d', as a function of time normalized by g=D. Comparison of different numerical methods to theoretical results from Maskell and Ursell (1970) and experimental results from Ito (1977). 652

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Table 3 Spatial discretization of the background mesh, Δ0 , the size of the cells closest to the surface boundary, Δ1 , and total number of computational cells, nCells. p.p.c.d

Δ0 , [m]

Δ1 , [m]

nCells

40

0.0152

0.0038

61,372

Table 4 Comparisonp offfiffiffiffiffiffiffiffi the period and amplitude of the first peak in the cylinder's transient motion, at around t⋅ g=D ¼ 10, between the numerical solutions and experimental data from Ito (1977). Numerical method

Error period [%]

Error amplitude [%]

no stabilizer under-relaxation predictor-corrector

1.6 1.6 0.8

11.2 16.2 0.9

correspondence with the experimental result from Ito (1977). To examine the reliability of the solver in terms of stability, the velocity and force profiles are evaluated. Fig. 7 presents the normalized vertical velocity of the cylinder's COG and in Fig. 8 the vertical force on the cylinder is presented. Oscillation in the velocity and force profiles are a sign of instabilities in the numerical model. No significant disturbances could be observed in the velocity profiles of the three numerical simulations, while in the force profiles clear disturbances can be observed. For the un-stabilized numerical simulation severe oscillations start to appear as the maximum negative displacement is reached, at around pffiffiffiffiffiffiffiffi t⋅ g=D ¼ 4. When the under-relaxation method is applied it can be seen that the force profile stabilizes to some degree, however, some oscillations and a number of large force spikes can still be observed. It may be observed that the predictor-corrector method provides the most stability as only minor oscillations and no force spikes are visible in the resulting force profile. It should be noted that the added stability comes at a price. For the predictor-corrector method, the execution time increases due to the additional iteration loops and for the under-relaxation method the accuracy of the solution decreases, as a result of under-relaxation of the acceleration while time-stepping, furthermore the execution time is slightly increased with respect to the unstabilized simulation. The predictor-corrector method was selected for all further computations, because of its high level of accuracy and stability. For engineering purposes it might, in some cases, be more useful to apply the underrelaxation method, as decent result are obtained at lower computational cost.

Fig. 7. Vertical velocity, normalized by

Fig. 8. Vertical force, normalized by the buoyancy p force ffiffiffiffiffiffiffiffiof the fully submerged cylinder and presented as a function of time normalized by g=D:

3.3. Flow-induced motions of a moored floating wind turbine From Section 3.2 it follows that the numerical wave tank can be used to accurately calculate flow-induced motions of a free floating twodimensional horizontal cylinder and that the numerical stability can be improved significantly by using the predictor-corrector method described in Section 2.2.2. The present section evaluates the capability of the numerical wave tank to simulate the motions of a moored threedimensional floating body. The aim of this section is to validate the numerical model against physical experiments with the generic OC5 FOWT that were performed in the concept basin at MARIN. These scale model tests were part of the ’TO2 Floating Wind’ research project which was carried out by Dutch research institutes Deltares, ECN, MARIN, and NLR, see Bruinsma (2016). First, a series of three simulations of moored decay tests is presented in Section 3.3.1, after which the simulation of a wave loading test is evaluated in Section 3.3.2. The wave tank used for the physical experiments is depicted in Fig. 9. Note that a 1:50 Froude scaling is applied and that unless otherwise

Fig. 9. Schematic representation of the physical wave tank with the floating wind turbine model. The dimensions of the wave tank are: l ¼ 200 m, w ¼ 4:0 m and d ¼ 4:0 m. The top and bottom figures represent the side and top view of the physical model set-up respectively.

pffiffiffiffiffiffiffi g⋅D and presented as a function of time

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stated all dimensions and properties are presented in model scale. Due to the limited width of the physical wave tank, the OC5's original three line spread mooring system could not be installed. Instead, the mooring system consisted of a front and back mooring line in combination with two horizontal linear springs, which were designed so that restoring properties of the system would be close to those of the original three line system. A schematic representation of the numerical domain, used for the validation cases in this section, is presented in Fig. 10. For the simulation of heave, pitch, and roll decay, presented in Section 3.3.1, the length, l, is equal to 4.0 m, as is the width of the physical domain and the height is equal to 5.0 m, of which 4.0 m is the water depth. Relaxation zones, with a length of 1 m, are implemented at both ends of the domain to absorb waves generated by the motions of the floating structure. For the simulation of the wave loading on the OC5 FOWT, presented in Section 3.3.2, the length of the domain is slightly larger to allow for more lateral movement of the floating structure. The total length of the domain is equal to one wave length, l ¼ 6:4 m, and the relaxation zones, used for the generation and absorption of waves, are equal to a quarter of a wave length. As the physical wave-maker signal is unknown, an exact reproduction of the experiments is not possible. Instead, fully nonlinear stream function waves of the same height, H ¼ 0:14 m, and period, T ¼ 1:71 s, is used Fenton (1990). The boundary layer conditions, relaxation zones and numerical discretization are defined according to the descriptions in Section 2.4, 2.5 and 2.6. It may be noted that the length of the numerical domain is much smaller than that of the physical wave tank. This is possible due to the simple mooring line implementation, see Section 2.3.2, which is simulated as a force in the attachment points on the FOWT and thereby they do not limit the domain size. This also means that, although the adjusted mooring system is used, the numerical simulation could be performed equally well with the original OC5 three line spread mooring system. Furthermore, the sub-structure, turbine tower, nacelle and blades are considered as one rigid body with a single moment of inertia and COG. Since mainly the fluid-structure interaction is considered, only the substructure is modelled in the numerical domain.

Table 5 Physical properties of the scale model of the floating wind turbine and the mooring system. These are the physical properties including the floater, turbine and tower. The values for the inertia are given about the COG of the floating structure. Description

Unit

Magnitude

Mass Roll inertia Pitch inertia Yaw inertia Mooring line Linear spring

[kg] [kg/m2] [kg/m2] [kg/m2] [N/m] [N/m]

111.66 49.77 47.56 43.81 0.043 1.52

Table 6 Spatial discretization of the background mesh, Δ0 , the size of the cells closest to the surface boundary, Δ1 , and total number of computational cells, nCells:

3.3.1. Decay tests with a moored floating wind turbine For all the heave, pitch, and roll simulations the initial position of the sub-structure is modelled in the base mesh. The physical properties of the floating wind turbine and the mooring system are presented in Table 5. The pre-tension in the horizontal springs in the physical mooring system is not measured, therefore MARIN provided a best estimate pre-tension of 0:256N. For a more detailed discussion on the numerical set-up and the physical properties of the floating structure and mooring system the reader is referred to Bruinsma (2016). To verify grid independence, a grid sensitivity analysis is carried out on the heave decay test with the moored FOWT. The applied spatial discretizations are presented in Table 6. In Fig. 11, the numerical results for the vertical displacement of the COG of the FOWT are presented along with the results from the physical model tests. It can be observed that the transient response of the FOWT is converging with increasing spatial and temporal resolution. It may be argued that the lowest resolution, of 5

p.p.c.d

Δ0 , [m]

Δ1 , [m]

nCells

5 10 15

0.102 0.051 0.034

0.0064 0.0032 0.0021

249,222 971,427 2,493,552

Fig. 11. Heave displacement of the COG of the moored FOWT, normalized by the initial displacement, d0 ¼ 0:027 m, presented as a function of time, normalized by the experimentally determined heave period, T ¼ 2:47 s. Grid sensitivity analysis, where spatial resolutions of 5, 10, and 15 p.p.c.d are applied.

p.p.c.d., is not high enough to resolve the challenging physics of this heave decay test. However, for the higher resolution, the numerical solution matches the experimental result quite well. This is especially true for the first two heave periods, after which the numerical and experimental results start to disagree more significantly. It may be noted that in the experimental results, a slight increase in the downward amplitude of the heave displacement can be observed at t=T ¼ 2:5. The source is not explained, however it is speculated that it is related to the mooring system, in particular to the horizontal springs. To evaluate the simulation in terms of the numerical stability, as was done for the two-dimensional decaying cylinder, the vertical velocity of the COG of the floating structure and vertical force on the floating structure are presented in Figs. 12 and 13 respectively. From these figures it may be concluded that, as only minor disturbances can be observed in the force profiles, the predictor-corrector method is able to provide a stable solution. To keep the computational demand relatively low, the spatial resolution of 10 p.p.c.d is selected for all further computations, since the obtained accuracy with respect to the experimental solution was seen to be in the order of one percent, both in term of amplitude and period, see Table 7, where the error after one heave period is presented. The numerical and experimental results of the moored pitch decay

Fig. 10. Schematic representation of the numerical set-up for the three-dimensional validation cases. 654

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Ocean Engineering 147 (2018) 647–658 Table 7 Comparison between the numerical solutions and physical model data for the moored decay tests. Here the absolute period and amplitude error were determined after one wave period. Description

Initial displacement

Error period [%]

Error amplitude [%]

Heave Pitch Roll

0.0272 m 3.34 deg 3.54 deg

0.52 2.01 2.45

0.55 18.48 2.94

Fig. 12. Vertical velocity presented as a function of time normalized by the experimenpffiffiffiffiffiffiffi tally obtained heave period, T ¼ 2:47 s. Here, the velocity is normalized by. g⋅D:

Fig. 14. Pitch angle of the moored FOWT, normalized by the initial angle of inclination, θ0 ¼ 3:34 degrees, presented as a function of time, normalized by the experimentally determined pitch period, T ¼ 4:68 s.

Fig. 13. Vertical force on the floating structure, presented as a function of time normalized by the experimentally obtained heave period, T ¼ 2:47 s. Here, the force is normalized by the buoyancy force of the moored FOWT, Fb :

test are presented in Fig. 14. To isolate the pitch rotation, the numerical model was restricted to the pitch rotation, while it was free to translate in the x, y, and z direction. It may be observed that the damping of the pitch motion is slightly lower in the numerical simulation, resulting in a larger response amplitude with respect to the experimental solution. An overestimation of approximately 18 per cent was observed for the first response amplitude, presented in Table 7. On the other hand, the period of the pitch decay motion is slightly lower than the experimental result. The numerical and experimental results of the moored roll decay test are presented in Fig. 15. It can be observed that an excellent correspondence with the experimental result was achieved, as only small differences are observed in terms of amplitude and period of the pitch response, see Table 7. Note that the experimental roll angle shows a irregularity at negative roll angles, so when the floater is leaning to port side. Although this abnormality does not seem to have a significant influence on the general progression of the roll signal it should be noted that the interpretation of these results needs to be done with great care. It was observed that the horizontal springs used in the mooring system were at times hitting the water surface when they became slack. This unintended interaction with the water is not modelled in the numerical simulation. It is argued that these and other assumptions with respect to

Fig. 15. Roll angle of the moored FOWT, normalized by the initial angle of inclination, ϕ0 ¼ 3:54 degrees, presented as a function of time, normalized by the experimentally obtained roll period, T ¼ 4:84 s.

the numerical representation of the model scale experiment may have resulted in the differences observed in the accuracy between the description of the pitch and roll motion. 3.3.2. Wave loading on a moored floating wind turbine Finally, the numerical model is used to simulate the interaction of unidirectional regular waves with the moored OC5 FOWT. A snapshot of the numerical simulation is presented in Fig. 16. The numerical model is validated against measurements from the physical experiments. The numerical domain set-up used in this section is similar to the one 655

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the generation and absorption of waves. The total length of the domain is equal to one wave length, l ¼ 6:4 m, and the relaxation zones are equal to a quarter of a wave length. The wave generation and absorption technique is similar to the one discussed in Section 3.1, as is also presented in Section 2.5. As the physical wave-maker signal is unknown, a fully nonlinear stream function waves (Fenton, 1990) of the same height, H ¼ 0:14 m, and period, T ¼ 1:71 s, is used as in the experiments. The wave amplitude is ramped up over 4 wave periods. It should be noted that it can be quite difficult to generate regular linear waves in a physical wave tank. A slight overshoot of the wave amplitude is observed in the free surface elevation signal of the experimental simulation, at t=T ¼ 4:4, and afterwords a small decrease in the wave amplitude is noticeable, see the free surface elevation presented in Fig. 17a. The motions of the floating structure are mainly in heave and pitch direction as the waves come from a head on position and the floating structure and mooring system are symmetrical in the vertical plane parallel to the wave direction. The heave response of the structures COG

Fig. 16. Illustration of the simulation of the moored OC5 FOWT subjected to regular waves in OpenFOAM.

described in Section 3.3.1, where the main differences are that the length of the domain and the relaxation zones were slightly larger to improve

Fig. 17. Free surface elevation, (a), and the heave, (b), and pitch, (c), displacement of the moored FOWT, where the free surface elevation and the heave displacement are normalized by the wave height, H ¼ 0:14 m, and all three are presented as a function of time, normalized by the wave period, T ¼ 1:71 s. Please note that the free surface elevations are obtained from wave gauges in front of the floating structure and extrapolated in time to represent the undisturbed free surface elevation at the position of the FOWT. 656

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detailed fluid-structure interactions, including dynamic motion response of a ridgid structure, further improvement and validation is still required. Multi-directional phase-focused waves show the potential to introduce larger peak forces than unidirectional waves. To extend the spatial and temporal reach of the model to investigate these types of wave loading phenomena a domain decomposition strategy can be applied. The fully nonlinear Navier-Stokes/6-DOF solver can be coupled with a fully nonlinear potential flow solver, OceanWave3D, developed by Engsig-Karup et al. (2009). This fully parallel one-way domain coupling is established and validated by Paulsen (2013) and Paulsen et al. (2014) for fixed mesh problems, however, the use of this method remains to be validated for the simulation of complex three-dimensional fluid-structure interactions with moored floating structures. For future research it is also important to extend and further validate the mooring line implementation, as for design purposes it is important to know the forces in mooring line and anchor points during extreme wave events and the current mooring line implementation only allows for the evaluation of forces in the attachment points on the structure.

is presented as a function of time in Fig. 17b, and the vertical displacement relative to the initial position of the floater is normalized by the wave height of the undisturbed incoming waves. In terms of the heave motion of the floating structure it can be observed that in general there is a build-up in the motion response over the first four wave periods. During this build-up the numerically determined motion response is higher than the experimental result. Directly after this build-up period, the numerically computed heave response slightly reduced after which it stabilizes over the next couple of wave periods. In the experimental simulation it takes more time for the heave response to stabilize. This could be a result of the irregularities found in the experimental wave signal. The pitch motion of the floating structure is presented in Fig. 17c. Similar to the heave motion response the pitch motion is seen to build-up over the first four wave period. Again a higher response can be observed in the numerical result. However, this may be expected as the pitch decay test, presented in Section 3.3.1, shows that the damping of the pitch motion is lower in the numerical simulation. The numerically computed pitch response shows a small decrease after the initial build-up, after which also the pitch motion stabilizes. In the experimental result a higher order motion response can be observed. This could indicate that natural pitch frequency of the FOWT is exited by the wave. Overall the numerical simulation has shown to provide stable solution with quite good correspondence with the experimental result.

Acknowledgements We thank the anonymous reviewers for their insightful comments and suggestions on the written material. Furthermore, we thank the Dutch research institutes Deltares, ECN, MARIN, and NLR who participated in the ’TO2 Floating Wind’ project for their contribution to the research. This work was carried out partly under the ’TO2 Floating Wind’ program and partly under the ”Coastal and Offshore Engineering” program, both of which are funded by the Dutch Ministry of Economic Affairs.

4. Conclusions The presented work describes the extensive validation of a state-ofthe-art numerical model with respect to complex fluid-structure interaction. The fully nonlinear Navier-Stokes/6-DOF solver is used to successfully reproduce results from benchmark physical model experiments with circular cylinders and a moored FOWT. The accuracy of the fluid-structure interaction in a fixed mesh is evaluated with a comparison against experimental data for the wave loading on a two-dimensional fixed horizontal cylinder. Excellent comparison is achieved in terms of the force profiles of the vertical loading on the cylinder. The numerical model is used for the simulations of the free heave decay of a two-dimensional horizontal cylinder and good agreement is obtained with both experimental and theoretical results. However, the standard coupling between the Navier-Stokes and 6-DOF solvers is shown to cause stability issues, which were observed in the form of force oscillations. The use of two stabilizing methods, the under-relaxation and the predictor-corrector method, is evaluated. Both of these stabilizing methods significantly improve the stability of the numerical model, however the predictor-corrector method is seen to be more effective at eliminating the oscillations in the force profile. Further validation of the numerical model including the predictorcorrector method and the mooring line implementation is done with data from a number of three-dimensional physical model tests with the moored OC5 FOWT. It is concluded from the performed moored heave, pitch and roll decay tests that the numerical model is capable of providing stable numerical results that are in good agreement with the physical results, especially considering the difficulties involved in evaluating such experiment. The numerical model is used to simulate the heave and pitch motions of the moored OC5 FOWT subjected to regular incoming waves. The results from this numerical simulation compares well with the physical model experiments. It is therefore concluded that the model can be used to provide accurate and stable numerical solutions for complex fluid-structure interaction using the predictor-corrector method for stabilizing the coupling between the Navier-Stokes and 6DOF solvers. Furthermore, the simple mooring line implementation presented in the present work is used to provide results with an high level of correspondence with physical model experiments. Even though the model has been shown capable of computing highly

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