CFD Modelling coupled with FloatingStructures and Mooring Dynamics for

Offshore Renewable Energy Devices usingthe Proteus Simulation Toolkit

Tristan de Lataillade∗†, Aggelos Dimakopoulos†, Christopher Kees‡,Lars Johanning§, David Ingram¶, Tahsin Tezdogan‖

∗Industrial Doctoral Centre for Offshore Renewable Energy, Edinburgh, UK(author for correspondence: t.delataillade@ed.ac.uk)

†HR Wallingford, Howberry Park, Wallingford, UK‡Coastal and Hydraulics Laboratory, US Army ERDC, Vicksburg, USA§University of Exeter, College of Engineering, Penryn Campus, UK¶The University of Edinburgh, School of Engineering, Edinburgh, UK

‖University of Strathclyde, Department of Naval Architecture, Glasgow, UK

Abstract—In this work, the coupling of novel open-source tools for simulating two-phase incompressibleflow problems with fluid-structure interaction andmooring dynamics is presented. The open-sourceComputational Fluid Dynamics (CFD) toolkit Proteusis used for the simulations. Proteus solves the two-phase Navier-Stokes equations using the Finite Ele-ment Method (FEM) and is fully coupled with anArbitrary Lagrangian-Eulerian (ALE) formulationfor mesh motion allowing solid body motion withinthe fluid domain. The multi-body dynamics solver,Chrono, is used for calculating rigid body motionand modelling dynamics of complex mooring systems.At each time step, Proteus computes the forces fromthe fluid acting on the rigid body necessary to findits displacement with Chrono which will be usedas boundary conditions for mesh motion. Severalverification and validation cases are presented herein order to prove the successful coupling between thetwo toolkits aforementioned. These test cases includewave sloshing in a tank, floating body dynamics underfree and wave-induced motion for different degrees offreedom (DOFs), and mooring dynamics using beamelement theory coupled with rigid body dynamicsand collision detection. The successful validation ofeach component shows the potential of the coupledmethodology to be used for assisting the design ofoffshore renewable energy devices.

Index Terms—Computational Fluid Dynamics, Fi-nite Element, Fluid-Structure Interaction, OffshoreRenewable Energy, Mooring Dynamics

I. INTRODUCTION

Being able to understand and predict the be-haviour of offshore floating structures under typicalor extreme environmental loads is an importantfactor for assessing their viability. This becomes

particularly critical for offshore renewable energywhere devices are purposely placed in highly ener-getic sites in order to capture a relatively abundantresource. Some of these floating devices aim tolimit the response to wave loads such as floatingwind turbines, while others are tuned to have a highresponse with the most energetic waves such as waveenergy converters. The numerical tools used forsimulating such events must be thoroughly verifiedand validated through established benchmarks, con-ceptual problems and comparison to experimentaldata from well monitored physical experiments.

Numerical modelling is rapidly and increasinglybecoming cost and time efficient, and recent ad-vancements in this domain have made it more reli-able. These scientific efforts have led the world ofComputational Fluid Dynamics (CFD) that tradition-ally preferred the Finite Volume Method (FVM) tolook at the more flexible but more computationallyexpensive Finite Element Method (FEM).

Proteus, an open-source computational methodsand simulation toolkit developed by the EngineerResearch and Development Center (ERDC) of theU.S. Army Corps of Engineers (USACE) and HRWallingford, is used in the work presented here forsolving Navier-Stokes equations, tracking the freesurface of two-phase flows, and moving the mesh.The source code of Proteus is available to all at:https://github.com/erdc-cm/proteus.

Using input forces and moments from the CFDsolver, floating body and mooring dynamics aresolved with the open source library Project Chrono.

This library allows a fully coupled simulation ofrigid and flexible bodies with FEM cable dynamics,where collision detection of the cables with struc-tures is enabled by creating node clouds along thecable. The source code of Chrono is available at:https://github.com/projectchrono/chrono

The focus of the research presented here is thesimulation of wave-structure interaction coupledwith floating rigid body and mooring dynamics usingnovel open-source tools. First, the capabilities of theCFD solver alone for simulating two-phase flows arepresented with a conceptual model of wave sloshingmotion in a fixed tank. Then, numerical results forfloating bodies are validated against experimentaldata for different degrees of freedom (DOF). Thisincludes free oscillation and decay of surface-piercing bodies released from a position away fromequilibrium, and wave-induced oscillation underwave loads of various frequencies. Finally, mooringquasi-statics and mooring dynamics coupled withrigid body motion will be presented.

II. GOVERNING EQUATIONS

A. Fluid Domain

In the context of the simulations presented infollowing sections, the fluid domain Ωf is composedof two phases: the water phase (Ωw) and the airphase (Ωa), so that Ωf = Ωw ∪Ωa. Solution withinΩf is computed with Proteus in the following steps.

1) Navier-Stokes: The fluid velocity field isobtained through the Navier-Stokes equation forincompressible fluid (see eq. (1)).

ρu + ρu · ∇u−∇ · ¯σ = ρg

∇ · u = 0(1)

with the density of the fluid ρ, the fluid velocityvector u, the gravity acceleration vector g, andthe Cauchy-Schwarz tensor ¯σ = −pI + µ∆u withpressure p and dynamic viscosity µ.

Dividing by the density, the first term of eq. (1)is often written as:

u + u · ∇u−∇ ·(ν(∇u +∇ut

))= g − 1

ρ∇p

(2)

With kinematic viscosity ν = µρ

The time integration used for the cases presentedhere is backward Euler due to its robustness andlower computational cost when compared to higherorder methods. The Courant-Friedrichs-Lewy (CFL)condition is used to limit the size of the time step(eq. (3)).

CFL =u∆t

∆x(3)

where ∆t is the time step and ∆x is the character-istic element length. The CFL value is calculated

for each cell, and is limited against a predefinedvalue throughout the numerical domain.

2) Free Surface Tracking: The free surface is de-scribed implicitly using the Volume-of-Fluid (VOF)[5] method coupled with the Level Set (LS) method[12]. The LS method uses a signed distance functionas shown in eq. (4), giving the distance of any pointin space φ from the free surface.

‖∇φ‖ = 1 ∀x ∈ Ωf

φ = 0 ∀x ∈ Γaw(4)

The VOF method returns 0 for all points in thewater phase, 1 for all points in the air phase, andvalues between 0 and 1 in a transition zone aroundthe free surface using a smoothed heaviside function(see eq. (5)).

θε(φ) =

0 φ < −ε12

(1 + φ

ε + 1π sinφπε

)|φ| ≤ ε

1 φ > ε

(5)

Where ε is a user defined smoothing distance (equalto 3 characteristic element length throughout thispaper), and θε the value of the heaviside function.

The LS model is used to correct the VOF model,more details about this implementation are availablein [11].

3) Turbulence: In Proteus, the Navier-Stokesequation are discretised in space using Residual-Based Variational Multiscale (RBVMS) method, asoriginally proposed in [2] and there are 2 turbulencemodels available: k-ε and k-ω.

B. Body Dynamics

As mentioned previously, Project Chrono – anopen-source multi-body dynamics simulation engine– is used for the simulation of floating bodiespresented in this work. It allows the simulation ofrigid and flexible bodies, imposition of constraints(joints, springs, etc.), and collision detection, as wellas finite element modelling of beams and cables(more details about the latter in section IV-C).

To define a rigid body, the necessary variablesare its mass m, inertia tensor ¯It (2×2 in 2D, 3×3in 3D), and the original position of its barycentre r0.In addition, the geometrical description of the outershell of the body must be provided using vertices,segments, and facets, as the mesh on the exteriorsurface of the body conforms to the fluid mesh,while the interior of the body (Ωs) is not mesh asthe structures investigated in this work are assumedfully rigid.

The hydrodynamic forces over the fluid/solidinterface, ∂Ωf∩s = ∂Ωf ∩ ∂Ωs, is computedby retrieving the tensor of the fluid stresses ¯σ

from the solution of the Navier-Stokes equations.Subsequently, stresses are integrated over the bodyboundaries to provide forces Ff and moments Mf

as described in eqs. (6–7).

Ff =

∫∂Ωf∩s

¯σndΓ (6)

Mf =

∫∂Ωf∩s

(x− r)× ( ¯σn)dΓ (7)

with n normal vector to the boundary, x a point onthe boundary, r the position of the barycentre, andΓ the boundary (segment if 2D, surface if 3D).

The total forces and moments are then used tocalculate the new acceleration, velocity, and positionof the body for a given time step (see eqs. (8–9)).

Ft = mr (8)

Mt = ¯Itω (9)

with m the mass, ¯It the inertia tensor and ω theangular acceleration of the body.

The time integration chosen for solving bodydynamics in the simulations presented here isimplicit Euler for systems of second order usingthe Anitescu/Stewart/Trinkle single-iteration method.Other integration methods are available in Chrono,and this one was chosen for its robustness and speed.Because a time step within Chrono takes much lesscomputational time than a time step of Proteus, therigid body solver uses several substeps with the CFDsolver time step, allowing a more refined solutionfor body dynamics.

C. Mesh Motion

In Proteus, the mesh is moved using the ArbitraryLagrangian-Eulerian (ALE) method also known asthe Mixed Interface-Tracking/Interface-CapturingTechnique (MICTICT) [16].

The mesh motion is solved through the equationsof linear elastostatics, which take the form ofeq. (10) for the equilibrium equation.

f + µ∇2u + (λ+ µ)∇ (∇ · u) = 0 (10)

where λ and µ are the Lame parameters, as shownin eq. (11).

µ =E

2(1 + ν)(11)

λ =µE

(1 + ν)(1− 2ν)(12)

For the simulations presented in this work, thenominal mesh Young’s modulus is E = 1 andPoisson ratio is ν = 0.3.

Fig. 1: Snapshots of mesh deformation aroundmoving body for different positions (top left showsthe mesh as it was initially generated).

The mesh is constrained, in our case, at theborders of the domain (walls and solid boundaries)and is updated at each time step by using thedisplacement of the nodes at the boundary of therigid bodies. The mesh then deforms in the fluiddomain following eq. (10).

The mesh motion is taken into account directlyin the Navier-Stokes equations that are solved inthe Eulerian frame. This is achieved by solving theequation in a transformed space, where the meshmotion effect is introduced as an additional transportterm in the conservation equations [1].

The displacement of the nodes placed on a mov-ing body are used as dirichlet boundary conditionsfor the moving mesh model, and are obtainedthrough eq. (13).

∆xn =(xn − xnp

)·Rn+1−(xn−xnp )+hn+1 (13)

with x the coordinates of a point in space, xp thecoordinates of the pivotal point (which is the centreof mass in the case of a single body not experiencingany collisions),h the translational displacement, andRn+1 the rotation matrix of the body from itsposition at time tn to tn+1.

At the tank/domain boundaries (∂Ω), the meshnodes are either fixed ((∆xn) = 0) or are allowedonly a translational motion along the boundary(∆xn · n = 0).

Figure 1 illustrates a close-up the moving mesharound a moving body.

Additional details about the mesh motion areprovided in [1].

III. VERIFICATION AND VALIDATION

The verification and validation cases presented inthis section aim to show the capability of Proteus tosimulate two-phase flow problems, the convergenceof the solution with spatial and temporal refinement,as well as its coupling with Chrono and the use ofthe ALE mesh module.

A. Sine Wave Sloshing

This verification test case looks at spatial andtemporal convergence of the solution of a conceptualtwo-phase flow problem in the absence of anymoving obstacle, meaning that the mesh motionmodule is not activated.

The sloshing motion of a body of water is studiedin a fixed tank given initial conditions for the freesurface. This type of test case is common practicefor verifying that the coupling of the equations oftwo-phase flow CFD solvers work adequately, andto quantify their accuracy and convergence order.The conditions used are the same as in [14] (basedon [17]), with a 2D tank of dimensions 0.1× 0.1m,water depth d = 0.05m, and amplitude of thesinusoidal profile of the free surface of a = 0.05m.The initial conditions are derived using the thirdorder solution as provided by [15] for a sloshingwave of the first mode in a fixed tank. This analyticalsolution is also used as a comparison to the time-varying numerical results. Boundary conditions arefree-slip on the tank bottom and side walls andatmospheric conditions at the top boundary. Inthis conceptual model, the fluid viscosity is turnedoff (ν = 0). For all test cases, the mesh used isunstructured with triangular elements that are of thesame size all across the domain.

The error ε between the numerical results andtheoretical curve is estimated using Pearson product-moment correlation coefficient (PPMCC or Pear-son’s r) over the first 5 periods of the sloshingoscillation.

On fig. 2, the error of the pressure obtainednumerically at the left boundary placed halfwayin the water column is plotted for different timestepping. The order of convergence can be foundby fitting a linear curve to the error data ona logarithmic scale. It clearly appears that thesolution converges with, in this case, a convergenceorder of 0.8 for time convergence with a fixedmesh refinement with characteristic element sizeof he = a

10 . Varying the spatial refinement also hasan effect, but tests shows that it becomes minimalwhen compared to reducing the time step after asufficiently refined mesh value (here he = a

10 ). Thevariation of the CFL value instead of using fixedtime step also gives an order of convergence of 0.8,but the solution gets away from the asymptotic rangefor a CFL value superior to 0.5. For this reason, thesimulations presented in the next sections of thispaper use CFL < 0.5.

B. Floating Body - Free Oscillation

This section presents validation results for floatingbodies with different constraints on their degrees offreedom.

Fig. 2: Error (ε) between numerical results and an-alytical solution for pressure sloshing wave motionusing PPMCC on the 5 first periods for differentfixed time steps (∆t).

1) Roll: This experiment is based on the ex-perimental setup of [10] with a floating caissonof dimensions 0.3m × 0.1m (length×height) thatis constrained for all DOFs apart from roll, in atank of dimensions 35m× 1.2m. The width of thecaisson and the tank are the same: 0.9m, making itpossible to approximate the experiment numericallyby setting a 2D test case, where a slice of theexperimental setup can be used numerically. Thewidth of the caisson is only taken into accountwhen calculating the forces and resulting momentson the floating body using its moment of inertiaIzz = 0.236kg m2.

The initial conditions are as follows: initialangle of 15 for the floating caisson, tank of totaldimensions 5m × 1.8m with absorption zones of2m at each end of the tank, and fluid at rest witha water column of 0.9m. For this experiment, no-slip boundary conditions (dirichlet value of zero forall velocity components) were imposed on bottomand side walls (Γwall) of the tank and on all facetsof the caisson. The top boundary (Γatm) of thetank used atmospheric conditions, with dirichletboundary conditions preventing fluid to travel in amotion parallel to Γatm and a VOF value imposedas the value of air (1 in this case). The simulatedtime was as long as the experimental recordings: 5seconds.

Figure 3 shows the unfiltered results for theoscillation of the floating body and its dampingcompared to the experimental results. While thereis good agreement regarding the natural period,the damping is underestimated numerically. It isbelieved that this significant difference in dampingis not an error of the model, but could be explainedby several factors, such as and mainly by frictionfrom the structure used to fix the caisson to the tankin the experimental setup or, to a minimal extent, thefact that even though the width of the caisson and

Fig. 3: Time series of roll motion of rectangular(2D) floating body for he = Hc

20 (with Hc the heightof the floating body) and fixed ∆t = 5e− 4 withinitial angle φ0 = 15. Experimental results weredigitalised from [10].

the tank are the same, the 2D numerical simulationis just a slice of a 3D experimental setup (the latterinevitably letting some fluid travel across the widthof the tank). Furthermore, the turbulence model wasswitched off during the simulation, similarly to theparticle-in-cell simulation of the same case from[3] that gives similar results to Proteus in terms ofdamping of the oscillation.

In order to investigate the stability of the meshmotion integrated in Proteus, the natural periodwas also calculated for different starting angles (5,10, 15) that lead to different mesh deformations.This resulted in a maximum difference of 0.2%between the results for the natural period of therolling caisson with different initial angle and similardamping profiles, proving the robustness of theALE mesh technique. The small difference couldbe explained by nonlinearities developing in thefluid domain and affecting the motion slightly dueto the difference of volume of water displaced andvelocity/vorticity of the fluid around the floatingbody when using different starting angle.

2) Heave: This test is based on the experimentof [6], where the heave oscillation of a cylinder wasstudied with: cylinder length of L = 1.83m and adiameter of D = 15.24cm (with 1.27cm betweenthe tank walls and end-plates of the cylinder). Thecylinder is halfway submerged when at rest, and allDOFs apart from heave are constrained.

For this simulation, the initial conditions aresimilar to the previously described test case withthe fluid at rest and no-slip boundary condition andcylinder is pushed down from its resting position toa given initial position (−2.54cm from still waterlevel). The mesh refinement is as follows: constantmesh refinement up to a distance equal to theinitial push of the cylinder around the mean waterlevel, and up to a distance of D from the cylinderboundaries. Gradual coarsening of the mesh is then

Fig. 4: Time series of heave motion of a circular(2D) floating body for he = d

50 and fixed ∆t = 5e−4 with initial push of y0 = −2.54cm. Experimentalresults were digitalised from [6].

applied from the refined parts with 10% increase inarea of mesh elements.

A time series of the experimental and numer-ical results is presented in fig. 4, with the timenormalised as t

√gr , and the heave displacement

normalised as yy0

. The numerical curve is in goodagreement with the experimental curve, leading toaccurate prediction of the natural period and ofthe damping of the oscillation. Even though theDOF investigated here is different, one could arguethat the good agreement obtained in damping couldmean that there was indeed frictional effects in theexperiment described in the previous section, wherethe experimental damping was overestimated.

C. Floating Body - Wave-Induced Oscillation

In this experiment, still based on the experimentalsetup [10] with a configuration similar to the onedescribed in section III-B1, waves are generated inorder to compute the Response Amplitude Operator(RAO) of the floating body. With λ the wavelengthof the wave generated, the tank dimensions arechanged depending on the wave characteristics inorder to match a generation zone of 1λ to limitwave reflection upstream and an absorption zone of2λ to absorb waves downstream. The two relaxationzones are separated by a distance of 2λ, and thefloating body is placed 1λ away from the generationzone (see fig. 5). Initially, the floating body is atrest position (no initial rotation).

As the waves generated for this example areweakly nonlinear, their profile is computed using theFenton approach with Fourier Transform in orderto get a wave profile close to fully developed. Seetable I for the characteristics of the waves generated.For each test case, the characteristic element sizewas no less than he = λ

200 . The generation andabsorption zones work with a blending function ina similar manner to [7].

Lgen LabsLtank

Hta

nk

Γatm

Γwall

Γw

all

Γw

all

Wc

Hc

h

Fig. 5: Numerical tank setup for a typical floatingbody test case under wave loads.

Figure 6 gives results for the response of the bodyunder different wave frequencies and amplitudesplotted against the experimental results and lineartheory. Overall, these results are in good agreementwith the experimental results, as explained in moredetails below.

For higher frequency waves, where ωωn

> 1,the results are in good agreement with the linearpotential theory as well as the experimental results.At ω

ωn= 1.15, where the experiment shows a

smaller response than the theory, the divergencein response between the experiment and the nu-merical/theoretical results looks particularly large,possibly due to the steepness of the responsecurve at this particular point, meaning that evena small difference of wave frequency betweenthe experimental setup and the numerical setupcould lead to a very different response result. Atωωn

= 1.32, 3 tests with same frequency wavesbut different wave heights were run, and similarresults were obtained, confirming the experimentalobservations that the wave height has a negligibleeffect on the caisson response at high frequency.

For waves with frequency identical to the naturalfrequency of the caisson ( ωωn

= 1), the magnifi-cation factor varies greatly with wave height. Thenumerical results for different wave heights shownin fig. 6 at ω

ωn= 1 indicate that the magnification

value increases when the wave height decreases,which is the expected behaviour according to [10].

For lower frequency waves ( ωωn< 1), the numer-

ical results are generally closer to the experimentalresults than the linear potential theory. This canbe explained by the fact that nonlinearities have asignificant effect at these frequencies, affecting thecaisson response. The numerical results, however,seem to slightly underestimate this effect whencompared to the experiment. Possible reasons forthis behaviour include the rather large characteristicelement size (with respect to the floating bodydimensions) for longer wave periods (as he is scaledwith λ) and/or the larger time stepping. It is worthnoting that results from [3] using PICIN (ParticleIn Cell - INcompressible) for this case are differentin this frequency range with an overestimation

Fig. 6: RAO of rolling caisson under regular waveloads, with φ the roll response, k the wavenumber,and A the wave amplitude

of the nonlinear effects when compared to theexperiment. Further investigation is necessary, witha finer mesh around the caisson as these low wavefrequencies lead to relatively large wavelengths andthe characteristic element size was scale with thewavelength throughout the whole domain for thesesimulations. Figure 7 shows snapshots of the floatingbody oscillation and fluid velocity and direction fora full wave period in the case of ω

ωn= 0.77.

IV. MOORINGS

Moorings are an important component of floatingoffshore renewables that should not be overlookedas it can significantly influence the response of thedevice. This section presents a quasi-statics modulethat was developed independently and validation ofmooring cable dynamics using the Project Chronolibrary. The quasi-statics model is used for settinginitial conditions to lay out the mooring cables forthe dynamics model.

A. Quasi-Statics

The quasi-statics module developed here supportsmulti-segmented sections (line made out of differentmaterials) and stretching of the cable is taken into

TABLE I: Wave characteristics for computing RAOof floating body (see fig. 6)

Case T H lambda

1 0.8 0.029 1.0062 0.85 0.033 1.1353 0.93 0.016 1.3514 0.93 0.027 1.3545 0.93 0.032 1.3556 0.93 0.04 1.3597 1 0.044 1.5688 1.1 0.057 1.8919 1.2 0.032 2.224

10 1.2 0.06 2.2311 1.2 0.067 2.23312 1.3 0.06 2.58113 1.4 0.061 2.939

(a) (b)

(c) (d)

(e) (f)

(g) (h)

(i) (j)

(k) (l)

Fig. 7: Snapshots of Proteus results for roll motion of floating body under wave-induced oscillation withwave of period T = 1.2s and H = 6cm. Top panel shows the shows roll angle and wave elevation(Fenton wave theory in the absence of the floating body), with map of subfigures (a) to (l). Colour scaleand arrow length correspond to fluid velocity ranging from 0 to 0.5 m/s (see digital version for colours).

Fig. 8: Catenary layout, with x0 the horizontal span,Ls lifted cable length, d horizontal distance betweenanchor and fairlead, h vertical span

account. It is also used for initial conditions inorder to lay out the cable given fairlead and anchorcoordinates, and the length of the cable.

Once the shape of the catenary described withthe parameter a is found (see fig. 8 for a typicalcatenary layout of a mooring line), the catenaryequation eq. (14) can be used to find coordinatesof points along the cable.

y = a cosh(xa

)(14)

s = a sinh(xa

)(15)

with y the vertical position (direction aligned withthe gravitational acceleration), x the horizontalposition, s the distance along the cable, and a thecharacteristic value of the catenary shape.

In the current implementation of the quasi-staticsmodule, an iterative process is used. In case of apartly-lifted line, the right value of the horizontalspan x0 of the mooring line must be found so thatthe lifted line length and the line laying on theseabed are together equal to the total line length. Ifthe line is fully lifted (Ls = L), a transcendentalequation in a is solved (see eq. (16)).√

Ls2 − h2 = 2a sinhd

2a(16)

If elasticity of the line is taken into account, moreiterations are needed to adjust the change in liftedline length as it stretches.

Quasi-statics analysis can be used to estimate theorder of magnitude of tensions that are experiencedby the floating device. It is a simple and fastmethod, but it neglects the dynamic effects of thecable that can become non-negligible under certainconditions (e.g. wave frequencies applying largeand varying loads on the device), often leading toan underestimation of the maximum tensions thatcan occur along the cable.

This module can be used for simple and fastmoorings analysis, but also for initial conditions

Fig. 9: validation of quasi-statics module for dif-ferent surge positions of a WEC moored with 3catenary lines, with x0 the horizontal span, d thevertical and h the horizontal distance from theanchor to the fairlead, and T the tension at thefairlead.

to lay out cables at equilibrium position for amore complex and demanding dynamic analysisas described in the following section. Figure 9shows results of a quasi-statics analysis comparedto experimental results from [4], where a WaveEnergy Converter (WEC) is moored with 3 catenarylines with 4 different segments each (2 chains and2 ropes) and is placed at different surge positions.

B. External Forces

The total external force fe due to the fluidsurrounding the cable is:

fe = fd + fam + fb (17)

With fd the drag force, fam the added mass force,and fb the buoyancy force.

The buoyancy force is applied as a volumetricload and calculated as follows, with ρc the densityof the cable, and ρf the density of the fluid:

fb = (ρc − ρf )g (18)

The drag force and added mass force use dif-ferent coefficients for their tangential and normalcomponents to the cable. The tangential and normalcomponents of a vector are denoted with t and n inthe following equations, and can be found for anyvector following eqs. (19–20).

xt =(x · t

)t (19)

xn = x−(x · t

)t (20)

The drag force is calculated using Morison’s equa-tion (eqs. (21–23)), where the cable is considered aslender cylindrical structure.

fd = fd,n + fd,t (21)

fd,n =1

2ρfCd,ndn ‖ur,n‖ur,n (22)

fd,t =1

2ρfCd,tda ‖ur,t‖ur,t (23)

with Cdn and Cdt the normal and tangential dragcoefficient, dn and da the normal and axial dragdiameters, and ur the relative velocity of the fluidto the cable element as in eq. (24).

ur = uf − r (24)

The added mass force is calculated with theacceleration of the fluid with the cable ar, as shownin eqs. (25–27).

fam = fam,n + fam,t (25)

fam,n = ρfCam,nπd2

4ar,n (26)

fam,t = ρfCam,tπd2

4ar,t (27)

For each element, the drag and added mass forcesare calculated at the nodes and averaged to beapplied as a distributed load over the length ofthe element.

As the mooring cable mesh and the fluid domainmesh do not conform, the loads can be calculatedfrom the velocity and acceleration of the fluid atthe nodes of the cable that can be retrieved fromthe fluid domain using the solution of the Navier-Stokes equations calculated from Proteus in eachcell containing a cable node.

C. Dynamic Analysis

For mooring dynamics, the FEM method wasused within Chrono. The cable is discretised inelements that are described with the Absolute NodalCoordinate Formulation (ANCF). The ANCF beamelements are composed of 2 nodes, each definedwith their position and direction vectors.

The mooring and floating body dynamics are fullycoupled and solved together with Chrono betweeneach CFD time step by applying the external forcescomputed from Proteus. Substeps within a mainCFD step are generally used for stability and betteraccuracy of the multi-body dynamics as these areusually significantly cheaper computationally thanwhen solving the equations for the fluid domain.In the current implementation, the external forcesfrom Proteus computed for each CFD step are keptconstant within the multi-body dynamics substeps.

In the simulations presented here, the material ofthe mooring cables is assumed to be transverselyisotropic. Bending stiffness can be changed bymodifying the moment area of inertia, and collision

Fig. 10: Indicator diagram with axial pretensionTA0 = 11.78N, ω = 0.79rad s−1 and a = 0.04z.Experimental and Orcaflex results from [9].

detection of the cable with obstacles such as theseabed is activated by producing a nodes cloud.

Due to its relatively slender profile and to allowunrestricted displacement of the mooring line, themooring cable has its own separate mesh. Theslender profile of the mooring cable would requirevery small mesh elements relative to the rest ofthe domain, increasing the computational cost ofthe simulation drastically. Furthermore, the displace-ment of the cable within the fluid domain might betoo large for the moving mesh to handle accurately.

In the current implementation, the fluid can acton the mooring (hydrodynamic loads as describedin the previous section), but there is no feedbackof the mooring displacement onto the fluid. Theeffects of the mooring cable on the fluid is thoughtto be negligible compared to the other forces in thesimulation.

The test case chosen for validating the mooringsdynamics is based on [8, 9], where a scaled cylindri-cal floating Wave Energy Converter (WEC) and itscatenary mooring line are modelled experimentally.The WEC is placed in a tank with uniform waterdepth of z = 2.8m and has a diameter of 0.5mand height of 0.2m. Several weights of the WECwere tested, and the one used here to compareresults is m = 30kg. The mooring line is a chainof length L = 6.98m with submerged weight perunit length of w = 1.036N m−1. The fairlead isplaced h = 2.65m above the seabed. The chosenexperiment consists in driving the WEC in thesurge direction at a given angular frequency ωand amplitude a with, in this case, a = 0.04z andw = 0.79rad s−1 with pretension TA0 = 11.78N.

The quasi-statics model is used to initialise theposition of the nodes along the cable for the Chronosimulation. A nodes cloud is generated in Chronoalong the mooring line for collision detection onthe seabed. External forces are taken into account ateach time step when solving the mooring dynamics

by retrieving the relative velocity of the cable withthe fluid velocity. Figure 10 is an indicator diagram,showing the horizontal tension at the fairlead againstthe position in surge of the WEC. The tensionsare well predicted with Chrono, and the dynamiceffect (hysteresis of the curve) due to the drag forceacting on the mooring line can be observed. Thedifference in hysteresis observed between Chronoand the experiment can be influenced by the dragand added-mass coefficients, which are derived fromassumptions and are one of the main source oferrors in mooring dynamics simulations. In thiscase, Cd,n = 1.4, Cd,t = 1.15, Cam,n = 1 andCam,t = 0.5 were used in Chrono, as they aretypical values used for studless chain. This is awork in progress and further testing with differentcoefficients is required.

Preliminary simulations using all the tools pre-sented in this work – CFD with wave generation andALE mesh, floating body and mooring dynamics,and collision detection – have been successfullyproduced and show great potential for simulatingrealistic cases for offshore renewable energy ap-plications. An illustrative snapshot of a 3D proof-of-concept simulation is shown in fig. 11 basedon the floating cylinder and mooring configurationof [13], with 3 catenary mooring lines discretisedwith 100 elements per mooring cable, and regularwave generation with period T = 1s and heighth = 0.15m in a tank with a water level of 0.9m. Formooring cable nodes that extend outside of the CFDdomain, fluid velocity is assumed to be the one givenby the waves that are used as boundary conditionsin the CFD model (assuming that the waves are thenare undisturbed). As mentioned earlier, this is anearly proof-of-concept simulation combining all thetools that have been validated in the work presentedhere, implying that further testing and validationof the models coupled together is necessary tomore confidently use it as a whole for complexCFD simulations making use of floating body andmooring dynamics.

V. CONCLUSION

The numerical results presented in this paperfor two-phase flows CFD using the Proteus toolkitcoupled with body and mooring dynamics usingthe Chrono library show good agreement with avariety of analytical, numerical and experimentalresults. Temporal and spatial convergence of thetwo-phase flow solver alone was shown with thesloshing motion of a body of water, with errorfrom the results laying in the asymptotic range.The simulation of floating bodies with Proteus andChrono was undertaken for i) free oscillation forcomparing their natural period and damping of the

Fig. 11: Snapshot of proof-of-concept simulationcombining wave generation, ALE mesh, and 3catenary mooring lines attached to a 6 DOF floatingcylinder at t = 12.8s. Color map shows velocitymagnitude.

simulation with experiments (roll and heave motion)and ii) the RAO of a floating body under loads ofregular waves of different frequencies. Finally, thesimulation of mooring lines with quasi-static anddynamic models was validated against experimentaldata.

The aforementioned test cases successfullydemonstrate the capabilities of Proteus – an FEMbased solver differing from the majority of the mainopen-source CFD tools that are FVM based – asa viable solution for modelling the key aspects offloating offshore renewable energy devices whencoupled with Chrono. The combination of thesenumerical tools show great potential to reliablysimulate the response of possibly complex offshorefloating structures enduring environmental loads.

ACKNOWLEDGEMENT

Support for this work was given by the EngineerResearch and Development Center (ERDC) and HRWallingford through the collaboration agreement(Contract No. W911NF-15-2-0110). The authorsalso acknowledge support for the IDCORE programfrom the Energy Technologies Institute and the Re-search Councils Energy Programme (grant numberEP/J500847/).

REFERENCES

[1] I. Akkerman, Y. Bazilevs, D. J. Benson, M. W. Farthing, andC. E. Kees. Free-Surface Flow and Fluid-Object InteractionModeling With Emphasis on Ship Hydrodynamics. Journalof Applied Mechanics, 79(1), 2012.

[2] Y. Bazilevs, V. M. Calo, J. A. Cottrell, T. J R Hughes,A. Reali, and G. Scovazzi. Variational multiscale residual-based turbulence modeling for large eddy simulation ofincompressible flows. Computer Methods in AppliedMechanics and Engineering, 197(1-4):173–201, 2007.

[3] Qiang Chen, Jun Zang, Aggelos S. Dimakopoulos, David M.Kelly, and Chris J K Williams. A Cartesian cut cellbased two-way strong fluid-solid coupling algorithm for2D floating bodies. Journal of Fluids and Structures, 62:252–271, 2016.

[4] V. Harnois, S. D. Weller, L. Johanning, P. R. Thies,M. Le Boulluec, D. Le Roux, V. Soule, and J. Ohana.Numerical model validation for mooring systems: Methodand application for wave energy converters. RenewableEnergy, 75:869–887, 2015.

[5] C.W Hirt and B.D Nichols. Volume of fluid (VOF)method for the dynamics of free boundaries. Journal ofComputational Physics, 39(1):201–225, 1981.

[6] Soichi Ito. Study of the Transient Heave Oscillation of aFloating Cylinder. PhD thesis, 1977.

[7] Niels G. Jacobsen, David R. Fuhrman, and Jørgen Fredsøe.A wave generation toolbox for the open-source CFD library:OpenFoam. International Journal for Numerical Methodsin Fluids, 70:1073–1088, 2012.

[8] Lars Johanning and George H Smith. Interaction BetweenMooring Line Damping and Response. In InternationalConference on Offshore Mechanics and Arctic Engineering(OMAE), 2006.

[9] Lars Johanning, George H Smith, and Julian Wolfram.Measurements of static and dynamic mooring line dampingand their importance for floating WEC devices. OceanEngineering, 34:1918–1934, 2007.

[10] Kwang Hyo Jung, Kuang-an Chang, and Hyo Jae Jo.Viscous Effect on the Roll Motion of a RectangularStructure. Journal of engineering mechanics, 132(2):190–200, 2006.

[11] C. E. Kees, I. Akkerman, M. W. Farthing, and Y. Bazilevs.A conservative level set method suitable for variable-order approximations and unstructured meshes. Journal ofComputational Physics, 230:4536–4558, 2011.

[12] Stanley Osher and James a. Sethian. Fronts propagatingwith curvature dependent speed: Algorithms based onHamilton-Jacobi formulations. Journal of ComputationalPhysics, (1988):12–49, 1988.

[13] Johannes Palm, Claes Eskilsson, Guilherme Moura Paredes,and Lars Bergdahl. Coupled mooring analysis of floatingwave energy converters using CFD. International Journalof Marine Energy, 16:83–99, 2016.

[14] L Qian, D M Causon, D M Ingram, and C G Mingham.Cartesian Cut Cell Two-Fluid Solver for Hydraulic FlowProblems. Jounal of Hydraulic Engineering, (September):688–696, 2003.

[15] Iradj Tadjbakhsh and Joseph B Keller. Standing SurfaceWaves of Finite Amplitude. (December), 1959.

[16] Tayfun E Tezduyar. Finite Element Methods for FlowProblems with Moving Boundaries and Interfaces. Com-putational Methods in Engineering, 8(September 2000):83–130, 2001.

[17] Onno Ubbink. Numerical prediction of two fluid systemswith sharp interfaces. PhD thesis, 1997.