Extreme Wave Loading on Offshore Wave Energy Devicesusing CFD: a Hierarchical Team Approach

J. Westphalen1, D. M. Greaves1, C. K. Williams2, P. H. Taylor3, D. M. Causon4,C. G. Mingham4, Z. Z. Hu4, P. K. Stansby5, B. D. Rogers5 and P. Omidvar5

1PRIMaRE: Peninsula Research Institute for Marine Renewable Energy,School of Engineering,University of Plymouth,

Drake Circus, Plymouth, PL4 8AA, UKE-mail: jan.westphalen@plymouth.ac.uk

2Department of Architecture and Civil Engineering,University of Bath,Bath, BA2 7AY, UK

E-mail: c.j.k.williams@bath.ac.uk

3Department of Engineering Science,University of Oxford,

Parks Road, Oxford, OX1 3PJ, UKE-mail: paul.taylor@eng.ox.ac.uk

4Department of Computing and Mathematics,Manchester Metropolitan University,

Chester Street, Manchester, M1 5GD, UKE-mail: Z.Hu@mmu.ac.uk

5School of Mechanical, Aerospace and Civil Engineering,University of Manchester,

Manchester, M60 1QD, UKE-mail: pourya.omidvar@postgrad.manchester.ac.uk

AbstractMany different types of wave energy converters have

been proposed in recent years. The two primary de-sign considerations are the need to generate energy atcompetitive economic rates in average sea states and theneed for the wave energy converters (WEC) to surviveextreme wave conditions. Due to the complexity of mostoffshore wave energy devices and their motion responsein different sea states, model scale tank tests are commonpractice for WEC design. Full scale tests are also neces-sary, but are expensive and only considered once the de-sign has been optimised. Computational Fluid Dynamics(CFD) is now recognised as an important complement totraditional physical testing techniques in offshore engi-neering. Once properly calibrated and validated to theproblem, CFD offers a high density of test data andresults in a reasonable timescale to assist with designchanges and improvements to the device. This paper

c© Proceedings of the 8th European Wave and Tidal EnergyConference, Uppsala, Sweden, 2009

deals with the results of test cases leading towards sim-ulation of the full dynamics of Pelamis and the Manch-ester Bobber. The test cases presented involve the inter-action between waves and fixed horizontal cylinders andresults are compared with experimental data to validatethe CFD codes. Also results for fluid-structure interac-tion of an oscillating cone on the water surface are pre-sented. The surface elevation and diffraction effects arediscussed, as well as the forces on the structures due tothe waves and motion respectively. Four different CFDcodes are applied to simulate the test cases: Smooth Par-ticle Hydrodynamics, a Cartesian Cut Cell method basedon an artificial compressibility method with shock cap-turing for the interface, and two pressure-based Navier-Stokes codes, one using a Finite Volume and the other acontrol volume based Finite Element approach.

Keywords: Cartesian-Cut-Cell, FEM, FVM, SPH, Wave-Structure Interaction

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NomenclatureA = Wave Amplitude or max. excursionA = Surface areaBBB = Body forceCV = Control volumed = DisplacementD = Diameter of cylinderF = ForceFFF = Flux vector functiong = Acceleration due to gravityh = Waterlevelk = Wavenumberl = Length of cylinderm = Integer between 1 and 12n = Frequency componentnnn = Normal vectorN = Number of frequency componentsNKC = Keulegan-Carpenter numberp = PressureQQQ = Vector of conserved variablesr = Radius of cone at still water levelS = Source termS = Surfacet = Timeuuu = Velocity vectoru,v,w = Components of uuuV = Domain of interestV = Volume of CV or particleW = Weighting functionxxx = Position vectorx,y,z = Coordinate directionsβ = Coefficient of artificial compressibilityΓ = Diffusion coefficient∆t = Timestep∆ωn = Frequency stepρ = Densityφ = General flow propertyΦ = Flow property gradientµ = Viscosityη = Surface elevationω = Frequency∇ = Gradient

Subscripts

0 = Initiala = Particleb = at boundaryh = Smoothing kerneli, j = mid pointn = Frequency componentrms = Root mean squaret = Timex,y,z = Coordinate directionsφ = General flow property

Superscripts′ = relative

1 IntroductionIn recent years the necessity for sustainable energy

has become a topic, which is addressed almost every-where in everyday-life. For example, many car manufac-turers are developing low emission or fuel cell poweredvehicles and efforts are being made to incrase energy ef-ficiency in buildings. The European Governments havethe reduction of CO2 emissions on their agenda. For ex-ample, the UK aim to cut the volume of carbon diox-ide blown in the atmosphere by 20% by 2010 relativeto 1990 levels. In offshore engineering wave and tidalenergy is available in almost unlimited resources, whichmakes this area very interesting for a contribution to Eu-ropean renewable energy targets. The major drawbackstill is the high cost of developing such technologies, andespecially operating offshore. High strains act on everystructure at sea, making survivability a key design driver.Maintenance is difficult due to harsh weather conditionsand highly qualified staff and specialist equipment is ex-pensive.

CFD provides a design tool, both cost effective andaccurate, that can help to reduce the overall design costsof such wave energy converters (WEC). The UK basedand EPSRC funded project “Extreme Wave Loading onOffshore Wave Energy Devices using CFD” follows a hi-erarchical approach. In this project, four different CFDtechniques are being applied to simulate two WECs:Pelamis and the Manchester Bobber.

The ultimate aim of the simulations is to investi-gate the very complex dynamics of these devices withmulti-body-wave interaction and 6-degree-of-freedommotion using different CFD techniques. Towards thisaim, a number of test problems of increasing com-plexity are being investigated. Wave-wave interactionhas successfully been modelled [1]. This paper fo-cusses on wave-structure interaction. The four compu-tational techniques, i.e. Smooth Particle Hydrodynamics(SPH), a Cartesian Cut Cell method based on an artifi-cial compressibility method with shock capturing for theinterface (AMAZON), and two pressure-based Navier-Stokes codes, one using a Finite Volume (FV) and theother a control volume Finite Element approach (CV-FE), are applied to a fixed cylinder case and also to onesimulation involving motion.

First, regular waves interacting with a fixed horizon-tal cylinder are modelled. Here the forces generatedby the waves are analysed. Next, the simulation of acone-shaped body near the water surface is discussed,in which the interaction between the cone and water isreduced to act in a single direction. As the cone is os-cillating at the water surface following the motion of aGaussian wave packet, the motion is independent of theforces generated at the surface by the surrounding water.Here the vertical forces on the cone surface are of inter-est and the relative water surface elevation near the coneis discussed.

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2 Mathematical Models2.1 Finite Volume

The Finite Volume solver uses the Navier-Stokesequations discretised on a 3-dimensional mesh to cal-culate the velocities and pressures in the flow field ina segregated iterative way. The flow is described by theequation of mass conservation as

∂u∂x

+∂v∂y

+∂w∂ z

= 0, (1)

and the three momentum equations as

∂uρ∂ t

+div(ρuuuu) =−∂ p∂x

+div(µgrad u)+Sx, (2)

∂vρ∂ t

+div(ρvuuu) =−∂ p∂y

+div(µgrad v)+Sy (3)

and

∂wρ∂ t

+div(ρwuuu) =−∂ p∂ z

+div(µgrad w)+Sz. (4)

u, v and w are the components of the velocity vector uuupointing in x, y and z direction, respectively. ρ is thefluid density, p the pressure, µ the fluid viscosity and tstands for the time. Si, with i being x, y or z, is the sourceterm acting in i-direction, in which e.g. gravity forces areincluded. In integral form equations (1), (2), (3) and (4)can be rewritten as the general transport equation∫

∆t

∂∂ t

(∫

CVρφdV

)dt +

∫

∆t

∫

Annn.(ρφuuu)dA dt =

∫

∆t

∫

Annn.(Γgrad φ)dA dt +

∫

∆t

∫

CVSφ dV dt, (5)

which is the starting point for the solution by the FiniteVolume method. ρ is the fluid density, φ is the trans-ported flow property such as velocity or pressure, Γ is thediffusion coefficient, A is the surface area of the controlvolumes (CV) face, V is the volume of the CV and Sφ isthe source term. nnn is the vector normal to a CV face anduuu is the velocity vector.

The domain, here a numerical wave tank (NWT) in-cluding the structure, i.e. a horizontal cylinder or a coneshaped body at the water surface, is subdivided into dis-crete volumes. The surface and volume integrals (5) per-formed on the control volumes are used to calculate thevariable values at the centre node of the CV. This ap-proach makes the Finite Volume method conservative byconstruction. Calculations are performed for both fluids,i.e. water and air, using the well known Volume of Fluid(VoF) method. For the CVs containing the fluid interfacethe CISCAM scheme, described by [2–4], is applied.

2.2 Control-Volume Finite Element

The control-volume Finite Element approach com-bines the Finite Volume method considering the con-trol volumes and the Finite Element method by using

shape functions and a different discretisation scheme.The shape functions are used to calculate the change ofa variable across the CV [5–7]. As described for the Fi-nite Volume method in 2.1 the CV-FE also solves theNavier-Stokes equations for incompressible fluids. Thegeneral transport equation as stated in (5) is discretisedon a 3-dimensional grid containing hexahedral cells. TheCVs are arranged around the mesh nodes and therebythis technique ensures the conservation of flow quanti-ties such as mass and momentum. In all simulations thefluid fractions of air and water are solved using the Vol-ume of Fluid formulation. The fluid interface is treatedby the method described by [5, 8–10], which is depen-dent on the filling level of the surrounding cells ratherthan the Courant number as in [2]. This solver is usedfor the horizontal and vertical cylinder cases and alsothe oscillating cone. The latter involves motion, which ismodelled by moving the nodes and edges of the mesh atthe beginning of each timestep before the fluid dynamicsare solved.

2.3 Smooth Particle Hydrodynamics

Smoothed Particle Hydrodynamics (SPH) is a flexibleLagrangian technique for computational fluid dynamicssimulations. In this method the fluid system is repre-sented by a set of particles which have individual ma-terial properties and move according to governing con-servation equations [11]. There is no mesh constructionin SPH, therefore in certain problems, for instance sim-ulation of waves, the SPH method may be easier to de-velop and use than Eulerian methods. The value of aflow quantity Φ at a position vector xxx is approximated as

Φ(xxx) = ∑j

Φ jWh (xxx−xxx j)V j (6)

where V j is the volume of the j-th particle located atxxx j with scalar quantity Φ j, and Wh is the weighting func-tion referred to as the smoothing kernel, h. The differen-tiable form of the interpolant of the function φ accordingto 6 leads the kernel function to be also differentiable.Therefore, one can write the gradient of the scalar fieldΦ relative to the particle as

(∇Φ)i = ∑j

(Φ j−Φi)∇iWh (xxxi−xxx j)V j. (7)

To simulate water, the fluid is allowed to be weaklycompressible using an artificial equation of state [11] sothat the time step is not prohibitively small. The sim-ulations are run using the open source code SPHysics[12]. The symplectic algorithm [13], often known askick-drift-kick, is used as the time stepping method. Fol-lowing the work of Vila [14], the governing equations forcompressible Navier-Stokes flow written in SPH forminclude position, conservation of mass and momentum:

∂xxxi

∂ t= uuu(xxxi, t) (8)

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∂∂ t

(Viρi)+Vi ∑j∈p

V j2ρa(uuua,i j−uuu0 (xxxi j, t)

)·∇iWi j = 0

(9)

∂∂ t

(Viρivi)+

Vi ∑j∈p

V j2[ρa +ρauuu0

a,i j⊗(uuu0

a,i j−uuu0 (xi j, t))]

∇iWi j =ViSi

(10)where all the symbols have their usual meaning with

Vi being the volume of the i-th particle, and Si is thebody force. The interaction between each particle pairis solved as a Riemann problem, so that the solution atthe mid-point, xi j, is uuu0

a,i j , pa, ρa for velocity, pressureand density respectively. The scheme is rendered secondorder accurate using TVD reconstruction and there is noexplicit viscosity formulation in the model.

The repulsive boundary condition, developed byMonaghan [15] and modified by [16], is used which pre-vents a water particle crossing a solid boundary. Thistechnique is used to simulate the horizontal cylindercases. By the principle of equal and opposite reaction,the forces on the cylinder can be estimated by summingthe forces exerted on the body particles by water parti-cles. The dimensionless force F ′ is determined as theratio of wave force to the weight of water displaced by atotally submerged cylinder in still water to compare withexperimental data.

2.4 Cartesian-Cut-Cell

The AMAZON-3D numerical wave tank (NWT) fora study of wave loading on a wave energy converter(WEC) device has been developed in MMU, which isbased on the free surface capturing method for two fluidflows with moving bodies developed by [17], whichdemonstrated a rigid 2D wedge-shaped body enteringcalm water and its subsequent total immersion. TheNWT, based on a two fluid free surface capturing andCartesian cut cell method, is being developed for thesimulation of wave loadings on the Manchester Bobberdevice under extreme wave conditions.

The AMAZON-3D code uses a Cartesian cut cellmethod to provide a boundary-fitted grid for both staticand moving boundaries in 3D. The main advantages ofthe Cartesian cut cell approach has been outlined previ-ously [18, 19], including particularly its flexibility fordealing with complex geometries and moving bodies.There is no requirement to re-mesh globally or even lo-cally for a moving boundary problem which only re-quires changes locally at cells in the background Carte-sian mesh that are cut by the moving boundary contour.The AMAZON-3D code has been extended to handle a3D floating bobber moving in the vertical direction inextreme waves. For this paper, however, the code is ap-plied to a fixed horizontal cylinder and the simulation ofa cone shaped body, which oscillates at the water surface.

The AMAZON-3D code is based on the integral formof the Euler equations for 3D incompressible flow withvariable density in integral form and can be written as

∂∂ t

∫∫∫

VQQQdV +

∮

SF.nF.nF.ndS =

∫∫∫BBBdV (11)

where QQQ is the vector of conserved variables whichencloses the time dependent domain of interest V , FFF isthe flux vector function and nnn is the outward unit vectornormal to the boundary S . BBB is a source term for bodyforces. QQQ , FFF and BBB are given by

QQQ = [ρ,ρu,ρv,ρw, p/β ]T , (12)

FFF = f Inx +gIny +hInz (13)

andBBB = [0,0,0,−ρg,0]T , (14)

with

f I =

ρ (u−ub)ρu(u−ub)+ p

ρv(u−ub)ρw(u−ub)

u

, (15)

gI =

ρ (v− vb)ρu(v− vb)

ρv(v− vb)+ pρw(v− vb)

v

, (16)

hI =

ρ (w−wb)ρu(w−wb)ρv(w−wb)

ρw(w−wb)+ pw

, (17)

where u, v and w are the flow velocity components andub , vb and wb are the velocity components of the bound-ary S , which are zero when the boundary is stationary.ρ is the density, p is the pressure, β is the coefficient ofartificial compressibility and g is the gravitational accel-eration.

The free surface is treated as a contact surface in thedensity field that is captured automatically during a time-marched calculation without special provision in a man-ner analogous to shock capturing in compressible flow.A time-accurate artificial compressibility method andhigh resolution Godunov-type scheme replaces the pres-sure correction solver used in many current VoF meth-ods. AMAZON-SC can handle break-up and recombina-tion of the free surface as well as air entrainment into thewater and, in principle, associated local compressibilityeffects. The total force is obtained by integration of thepressure field along the body F =−∫Sb

pnnndS , where Sbis the body surface and is defined by the cut cell surface.

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3 Horizontal CylinderThe first set of tests concern a fixed horizontal cylin-

der described by [20]. The authors carried out physicaltank tests in order to improve Morison’s formula for thecalculation of the forces on a horizontal cylinder. For dif-ferent levels of submergence and wave amplitudes theyrecorded the vertical forces acting on the cylinder due tothe waves. Here, all four numerical techniques were ap-plied to several tests and compared with the experimentalresults [20].

The forces calculated using each of FV, CV-FE, SPHand AMAZON are compared for several different waveamplitudes and cylinder axis depths as it can be seen inTables 1 and 2. The relative amplitude A′ and axis depthor displacement d′ are defined as the ratios between ei-ther the amplitude A or axis depth d and the cylinder di-ameter D, respectively. To compare the numerical resultswith those obtained by Dixon et al. [20] the verticalforces Fz on the cylinder resulting from drag and pres-sure on the surface are exported. The forces F ′z shown inall figures are non-dimensionalised using the followingexpression

F ′ =Fz

gρ(1/4πD2l

) (18)

with Fz being the measured vertical force on the cylinder,g the acceleration due to gravity, ρ the density of water,D the cylinder diameter and l the length of the cylinder.As in the physical experiment 1st order regular wavesare generated in the NWT to hit the structure. For theFV and CV-FE approach the wave velocity componentsu and w and the surface elevation η as described by

u =gAk cosh(k (z+h))cos(kx−ωt)

ω cosh(kh), (19)

w =−gAk sinh(k (z+h))sin(kx−ωt)ω cosh(kh)

(20)

andη = Acos(kx−ωt) . (21)

are applied for the water fraction at the upstream end ofthe 3-dimensional NWT using a velocity inlet boundarycondition. The velocities for the air fraction are set to0.0m/s. The top boundary is a pressure outlet, the sidesare modelled as symmetry planes and the bottom, thecylinder and downstream end of the domain are definedas walls. The total number of cells for the FV meshes are113856, 113606 and 114599. The CV-FE meshes con-tain 79495, 69537 and 695375 cells. The number of cellson the cylinder itself however is 250 for the FV solverand 236 for CV-FE. The timestep to achieve a convergedsolution is set to 0.001 s and 0.05 s for FV and CV-FE re-spectively. Figures 1 and 2 show a section of the meshesused for the Navier-Stokes solvers for the relative axisdepth d′ of 0.0, where the centre of the cylinder sits atstill water level.

In AMAZON, the left-hand boundary condition is avelocity inlet, where (19), (20) and (21) are applied forthe water fraction to generate the wave in the domain.

The air part of the left-hand boundary and the top andright boundary are specified as non-reflecting boundaryconditions allowing air to leave or enter the domain. Thecylinder surface is defined as a reflective wall bound-ary. The remaining sides are slip boundaries. The NWTcontains 438850 cells with a minimum edge length of0.015m. A timestep of 0.00025s is applied.

For the SPH method the calculations are carried outin a 2-dimensional NWT using 13000 particles in the do-main. The boundaries are treated as walls, i.e. the wave-maker, the bottom, the downstream end and the cylinder.The waves are generated by moving the upstream wallsimilar to a piston wavemaker.

In Figures 3 and 5 the time histories of the non-dimensionalised vertical forces over one wave periodare shown for AMAZON, and the two Navier-Stokessolvers. For d’ = 0.0 and A’ = 0.5 the three codes givevery good agreement with the experimental data. For d’= -0.3 and A’ = 0.2 the results look even better. Fig. 4shows the surface elevation around the cylinder (d′ = 0.0)for different times for the FV solver. Especially for t/T= 0.6 and 0.73 air entrainment can be observed. For thefully submerged case shown in Fig. 6 the relative forcescalculated by CFD differ somewhat from the physicalexperiment. Although the computational results agree inquantity, the phases do not match between FV, CV-FEand the physical experiment.

Table 1: Properties of Horizontal Cylinder Simulations for FV,CV-FE and AMAZON (simulation 1 and 2 only)

Simulation1 2 3

d′ [m] 0.0 -0.3 -0.6A′ [m] 0.5 0.2 0.3

kA 0.2 0.01 0.12kh 1.61 1.61 1.61

NKC 3.1 1.3 1.9

Figure 1: CV-FE Mesh Section around Cylinder

Fig. 7 shows the particle distributions at differenttimes for the SPH method. The wave propagation near

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Figure 2: FV Mesh Section around Cylinder

Figure 3: Relative vertical forces on horizontal cylinder(d’ = 0.0, A’ = 0.5)

the cylinder has clearly been altered by the presence ofthe cylinder. Since the simulations are mono-phase (i.e.only water particles) the compressibility of air aroundthe cylinder is not taken into account. To compare theforces obtained by the SPH method with the experimen-tal data, the root mean square vertical force is calculatedas

Frms =

√1n

n

∑n=1

F2i (22)

where Frms is the root mean square force, Fi is the to-tal force component at each time step (evaluated by sum-ming the force contribution from nearby fluid particles)and n is the number of time step.

Table 2 shows the comparisons of dimensionless rmsforce between the SPH results and the experimental datapresented by [20] for different relative axis depths andrelative amplitudes. As shown in the table, one can seethat the experimental and SPH results are in reasonableagreement especially for the half submerged cylinderwhere the interaction between fluid particles and bodyparticles are less than the fully submerged case. Forthe submerged cylinder, the results also show promising

(a) (b) (c)

(d) (e) (f)

Figure 4: Surface elevation around cylinder, d′ = 0.0, A’ =0.5; (a) t/T = 0.0, (b) t/T = 0.12, (c) t/T = 0.36, (d) t/T = 0.6,(e) t/T = 0.73, (f) t/T = 1.0.

Figure 5: Relative vertical forces on horizontal cylinder(d’ = -0.3, A’ = 0.2)

agreement but with a discrepancy that might be due toproblems with the experimental data.

4 Oscillating ConeFor the simulation of floating bodies, the authors’

main interest lies in Pelamis and the Manchester Bobber,so it is important to be able to calculate the forces on amoving body and the surface elevations around it cor-rectly. For the purpose of the validation of the codes asimplified test setup accommodating a cone shaped bodynear the water surface is chosen. The motion of the coneis driven and not influenced by the forces generated onits surface by the surrounding fluids. The physical tanktests are described in [21]. In the experiments, the verti-cal forces on the cone surface due to its motion and therelative water surface elevation at a distance of 0.03mfrom the cone surface were recorded. Here comparisonsbetween AMAZON, the CV-FE solver and the physical

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Figure 6: Relative vertical forces on horizontal cylinder(d’ = -0.6, A’ = 0.3)

(a) t = 5s

(b) t = 7.5s

Figure 7: Particle configuration for half submerged cylinderwith the relative amplitude of 0.5 at

experiments are shown.The motion of the cone is defined by the displacement

d(t) from the initial position at t = 0s following the formof a Gaussian wave packet, which is described by

d (t) = AN

∑n=1

Z (ωn)cos[

ωn (t− t0)−hπ2

]∆ωn, (23)

where

Z (ωn) =1

ωn2π√

2πexp

[− (ωn−ω0)

2

2(ω0

2π)2

](24)

with h = 0 or 1. A denotes the largest excursion from the

Figure 8: Computational domain used for CV-FE with meshand position of initial water surface

Table 2: Comparison of SPH rms force, normalised by sub-merged buoyancy force, with experiment for different relativewave amplitude and axis depth

A′ d′ SPH Frms Exp. Frms

0.10 0 0.075 0.0620.50 0 0.259 0.2640.20 -0.1 0.115 0.1120.20 -0.2 0.120 0.1010.20 -0.3 0.109 0.0870.30 -0.4 0.149 0.1180.30 -0.5 0.179 0.089

still water level. N is the number of frequency compo-nents and ωn is the appropriate circular frequency. Thecentral circular frequency ω0 [rad/s] is defined by

ω0 =mπ3

(25)

with m being an integer between 1 and 12.For the CV-FE approach the simulations are per-

formed in a three-dimensional domain with a length andwidth of 2.5m and a height of 2.0m. The cone is placedin the centre, as it can be seen in Fig. 8. It has a topdiameter of 0.6m and a steepness of the deadarise angleof 45. The slope itself is 0.3m high. The initial draughtof the cone is 0.15m at a waterdepth of 1.0m. The coneis modelled as a cavity in the mesh. The outer bound-aries, the bottom and the cone are modelled as free slipwalls. The top boundary is defined as a pressure outletwith constant atmospheric pressure. The mesh consistsof 820,000 hexahedral cells, where the regions aroundthe water surface and the cone surface are highly refinedto achieve cell edges of approximately 0.01m. The outerregions are relatively coarse to save computational re-sources and encourage numerical damping, thus avoid-ing reflections from the walls. The simulations were car-ried out using high performance computing on 16 CPUs.The timestep is 0.0005s.

In AMAZON a 2m x 1.6m axisymmetric domain isused. The still water level is set to 1.02m and the ini-tial draught of the cone is 0.148m. The calculations areperformed on a hexahedral grid using an axisymmetric(2D) version of the code with cell sizes of 0.02 x 0.02m.The timestep is 0.00005s. The exported vertical forcesFz from the CFD codes are non-dimensionalised usingthe expression

F ′(t) =Fz(t)

ρgπr2A(26)

with ρ being the density of fresh water, g the acceler-ation due to gravity, r the cone radius at still water leveland A the maximum excursion. Also the time is dividedby the corresponding period of the central frequency ω0.The measured relative motion of the water surface is di-vided though the maximum excursion A = 0.05m.

Fig. 9 compares the force data obtained by AMA-ZON and the CV-FE solver with those of the physical ex-periment. Generally the agreement is satisfactory. Both

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codes generate little differences especially in the forceminima and maxima. AMAZON slightly overestimatesthe forces and the Navier-Stokes solver underestimatesthem. Looking at the relative motion of the free sur-face at the cone for this experiment shown in Fig. 10,the differences become larger. Here, both codes gener-ate results, which are smaller in the crests and larger inthe troughs. this difference is currently under investiga-tion and may be due to the difficulty in generating anorthogonal mesh for a body with sloping sides at the freesurface.

Figure 9: Relative vertical force on cone

Figure 10: Relative surface motion at cone

5 ConclusionsResults are presented for four different computational

methods, i.e. SPH, a Cartesian Cut Cell method and twopressure-based Navier- Stokes codes, one using a FiniteVolume and the other a control volume based Finite El-ement approach. These were applied to cases involvingfluid structure interaction with a fixed horizontal cylin-der in different wave conditions and levels of submer-gence. Also results for an experiment including motionof the structure are shown. All the numerical simulations

show good overall agreement with the experimental re-sults they are compared with.

The investigation of the horizontal cylinder showsthat all solvers calculate the forces on the structure wellfor axis displacements d’ = 0.0 and d’ = -0.3. Here thepressures and shear stresses are in good agreement withmeasurements in the physical tank test. For the fullysubmerged cases with d’ smaller than -0.5 the numericalresults in all solvers generate differences. The FV andCV-FE methods can predict the forces quantitatively, butthe quality of the time histories is different. For the SPHmethod the root mean squares were analysed, which givethe same trend with good agreement for the cases withthe cylinder axis closer to the water surface.

The CV-FE and AMAZON solvers performed well onthe oscillating cone case. The main design interest forWEC probably lies in the calculation of the forces actingon the structure. These are in very good agreement. Theprediction of the free surface at the body boundary canstill be improved however. For the CV-FE method thisis believed to occur due to mesh effects coming fromnon-vertical and -horizontal cell edges aggravated by thevery small body displacement in the test case. This willbe investigated further in both codes and also using theFV solver and SPH.

The outcome so far shows, that all the applied com-putational techniques are very powerful tools. TheCartesian-Cut-Cell method (AMAZON), SPH and thetwo Navier-Stokes solvers, i.e. the FV and CV-FE ap-proach, perform well in all experiments they were ap-plied to.

AcknowledgementsThe authors would like to acknowledge the support of

the Engineering and Physical Sciences Research Coun-cil.

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