coupled dynamic simulations of offshore wind turbines...

Coupled dynamic simulations of offshore wind turbines using linear, weakly and fullynonlinear wave models: the limitations of the second-order wave theory

E. Marino1, C. Lugni2, G. Stabile1, C. Borri11Dept. Civil and Environmental Engineering, University of Florence, Via di S. Marta 3, 50139 Florence, Italy

2CNR-INSEAN, National Research Council, Via di Vallerano 139, 00128 Rome, Italy2AMOS, Centre for Autonomous Marine Operations and Systems, NTNU, NO-7491 Trondheim, Norway

ABSTRACT: The present work investigates the dynamic response of a fixed–bottom offshore wind turbine subjected to the combinedwind-waves action employing different nonlinear wave kinematic models. Linear, 2nd-order and fully nonlinear models are imple-mented in the hydrodynamic module of a global hydro-aero-servo-elastic solver. All the wave models are based on the potential flowassumption. A first study of the structural response is performed in regular waves with increasing steepness considering the turbineboth in parked condition and in power production. A more realistic simulation is then carried out with irregular waves and turbulentwind. Hydrodynamic loads associated to the three wave models are coupled with aerodynamic loads acting on the rotor of a 5-MWwind turbine. Hydro-aero-elastic calculations are performed using the NREL software FAST. The paper shows that from wave steep-ness ka = 0.1 on the 2nd-order model becomes inaccurate. It underestimates the structural loads as well as the resonant oscillationsof the tower caused by the higher-order components.

KEYWORDS: Offshore Wind Turbines; Nonlinear waves; Structural dynamics; Springing.

1 INTRODUCTION

The development of more accurate simulation tools capable ofcapturing the effects of complex environmental conditions onlarge multi-megawatt wind turbines are strongly required withinreliability–based design framework. A novel numerical packagecapable of predicting the nonlinear loads acting on offshore windturbines (OWTs) exposed to nonlinear sea states has been re-cently proposed in [1, 2, 3]. The model reproduces the structuralresponse by coupling a fully nonlinear wave kinematic solver,[4], [5], [6], [7], [8], [9] and [10], with a hydro-aero-elastic simu-lator of the entire system [11]. The numerical approach proposedin [1, 2, 3] proved to be computationally very efficient due to adomain decomposition strategy. The fully nonlinear numericalwave solver is invoked only on special sub-domains where non-linear waves are expected, wheres on the remaining parts of thespace-time domain the linear wave theory is assumed. Recently,in [12] it has been shown that 2nd-order wave contributions havesignificant effects on the loads assessment of OWT. However,in [13] we observed that, even for moderate sea-states, the 2nd-order wave theory may miss important effects on the structuralresponse.

In the present paper, we integrate the linear, 2nd-order andfully nonlinear (without domain decomposition) wave models ina fully coupled hydro-aero-elastic solver and compare the struc-tural response in order to have a clearer picture of the effects thathigher-order contributions have as the wave steepness increases.It is performed first a study in regular waves with the turbine bothin parked and operating conditions, then an extreme sea state isreproduced with irregular waves.

The paper is structured as follows: in Section 2 and Sec-tion 3 the Fully Nonlinear (FNL) and 2nd-order wave modelsare briefly recalled. Section 4 summarizes the global dynamicmodel and the main features of the baseline wind turbine usedin this study are listed. In Section 5 the effects of the three wavemodels on the system response are presented and discussed. Fi-nally, in Section 6 the main conclusions are drawn.

2 FULLY NONLINEAR IRREGULAR WAVES MODEL

2.1 Mathematical and numerical formulation

The 2D problem governing the nonlinear propagation of gravitywaves is formulated within the potential-flow assumption. For an

inviscid fluid in irrotational flow, the potential function Φ(t, p)describes the velocity field at time t in each point p ∈ Ω(t). Foran incompressible fluid, Laplace’s equation is valid in the wholedomain

∇2Φ = 0 ∀p ∈ Ω(t) (1)

The 2D domain Ω(t) is bounded by four boundaries: inflowΓi1(t), rigid bottom Γb, outflow Γi2(t), and free-surface Γf (t)(see Figure 1). On each of them, suitable boundary conditionshave to be enforced. In particular, the impermeability condition∇Φ · n̄ = 0 is imposed on the bottom Γb, while the continuitywith the linear wave kinematics has to be ensured on Γi1(t) andΓi2(t); finally, nonlinear kinematic and dynamic boundary con-ditions must be imposed on Γf (t)

Dr̄

Dt= ∇Φ , ∀p ∈ Γf (2)

DΦ

Dlt= − pa

ρw− gη− 1

2∇Φ · ∇Φ , ∀p ∈ Γf (3)

where r̄ is the position vector of the water particle p and η thefree surface elevation. pa and ρw denote the atmospheric pres-sure and the water density, respectively.

Figure 1: Two–dimensional domain of the time-dependingLaplace equation.

2.2 Numerical formulation

The solution of the fully nonlinear time-depending free-surfaceproblem is based on a two-step Mixed Eulerian-Lagrangian

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014Porto, Portugal, 30 June - 2 July 2014

A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)ISSN: 2311-9020; ISBN: 978-972-752-165-4

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(MEL) approach, which splits the problem into a kinetic andtime evolution sub-parts.

In the Eulerian step, at a given time instant, the solution of theBoundary Value Problem (BVP) for the Laplace equation withmixed Dirichlet and Neumann boundary conditions is reformu-lated in an integral representation of the velocity potential (usingGreen’s second identity). The boundary integral equation is dis-cretized in space by means of quadratic isoparametric boundaryelements [14]. We refer to [1, 15] for additional details.

The Lagrangian step (time evolution) consists in the time in-tegration of Eq.s (2) and (3) where the potential and the free-surface profile at the new time step t + dt are provided. Thefourth–order Runge-Kutta (RK4) algorithm is used for the timeintegration scheme.

3 SECOND ORDER NONLINEAR IRREGULAR WAVES

The 2nd-order wave theory is implemented starting from thesecond order velocity potential. To solve Laplace’s equationwith nonlinear boundary conditions a perturbation approach isused [16]. The free surface elevation η(t) is described as a sumof a first–order term η1(t) and a second–order term η2(t). Thefirst–order contribution comes from the sum of all the wave com-ponents used to discretize the wave spectrum

η1(t) =N∑

m=1

Am cos(ωmt− φm) (4)

The 2nd-order term is obtained as sum of differences and sumsof frequencies as follows

η2(t) = η−2 (t) + η+2 (t) (5)

η±2 (t) =N∑

m=1

N∑n=1

[AmAn{B±mn cos((Ψm ±Ψn)}] (6)

where Ψm = ωmt− φm and Ψn = ωnt− φn. Am, An, ωm andωn are, respectively, the amplitude and the circular frequencyof the m–th and n–th wave components. B−

mn and B+mn are the

second order transfer functions [16]. Similarly, the second–ordervelocity potential is obtained as a sum of a first order and a sec-ond order term Φ(t) = Φ1(t) + Φ2(t). Φ2(t) is defined by dif-ferences and sums of frequencies: Φ2 = Φ−

2 +Φ+2 , where

Φ±2 =

1

4

N∑m=1

N∑n=1

[AmAng2

ωmωn

cosh(k±mn(h+ z))

cosh(k±mnh)·

D±mn

(ωm ± ωn)sin(ψm ±ψn)

] (7)

In the above equation k±mn = |km ± kn| and D±mn are transfer

functions (not reported here, see [12]). The horizontal particlevelocity and the horizontal particle acceleration may be obtainedwith the second order velocity potential by taking the gradient ofΦ.

3.1 Numerical implementation

The numerical implementation of the 2nd–order theory (as wellas the first order theory) may be efficiently performed using theInverse Fast Fourier Transform (IFFT). First–order terms, thatinvolve single summations, require the definition of the first–order coefficients, see [12].

Using the IFFT, in order to get a result compatible with theperiod of the simulation desired, a correct discretization of thewave spectrum is needed. Let Tsim and N be the simulationtime and the number of samples, respectively. The time incre-ment is Δt = Tsim/(N − 1), while the time vector is definedas tp = pΔt with p = 0,1,2...,N − 1. The increment of the

wave circular frequency used to discretize the wave spectrumis Δω = 2π/(ΔtN) and the frequency vector is defined asωm = mΔω with m = 0,1,2, . . . ,N − 1. In this way the IFFTreturns a correct result compatible with the simulation time Tsimand with the number of samples N .

Note that before using the IFFT to compute the 2nd–orderterms, Eq. (5) is rewritten as

η±2 (tp)=R

[N∑

m=1

N∑n=1

[X±mnexp(−i(ωm±ωn)tp)]

](8)

with

X±mn = AmAnB

±mn exp(−i(φm ± φn)) (9)

The second order wave kinematics components may be eval-uated using the same procedure and changing only the expres-sion of the Fourier coefficients. For a complete description ofthe model and for the details of the numerical implementationwe refer to [12].

4 GLOBAL SOLVER

Aero-hydro-servo-elastic computations are carried withFAST [11]. FAST is based on a combined modal and multibodydynamics formulation and can be used to model the rigid andflexible bodies of a wind turbine. Aerodynamic loads actingon the blades are calculated by means of the Blade-ElementMomentum theory using the software AeroDyn [17], [18].Hydrodynamic loads are computed within the FAST by meansof the Morison equation [19]. These loads are made of twocontributions: the viscous and the inertial terms. In addition tothe nonlinear kinematics, we point out that the square of thewater velocity in the viscous term also introduces higher-ordercomponents; wheres, the inertial term is purely affected by thenonlinear wave kinematics. Turbulent wind is generated withTurbSim [20].

The turbine model used in this study is the 5-MW ReferenceWind Turbine for Offshore System Development [21], whosemain characteristics are listed in Table 1. The diameter and thewall thickness vary linearly with the tower height. The base di-ameter of 6 m is equal to the diameter of the monopile. A waterdepth of 20 m is considered in all the simulations presented inthe next section.

Table 1: Key properties of the NREL 5–MW Baseline Wind Tur-bine.

Rating Power 5 MWRotor Orientation, Configuration Upwind, 3 BladesRotor, Hub Diameter 126 m, 3 mHub Height 90 mCut-In, Rated, Cut-Out Wind Speed 3 m/s, 11.4 m/s, 25 m/sCut-In, Rated Rotor Speed 6.9 rpm, 12.1 rpmRated Tip Speed 80 m/sRotor Mass 110 tNacelle Mass 240 tTower Mass 347 460 tPile length, diameter 30 m, 6 mTower top diameter, wall thickness 3.87 m, 0.019 mPile wall thickness, total weight 0.060 m, 187.90 t

5 EFFECTS OF WAVES MODELS ON THE TURBINERESPONSE

5.1 Regular waves

We first address the case of regular waves. Regular waves aregenerated with circular frequency ω = 0.6283 rad/s (T = 10 s)and increasing steepness ka from 0.05 to 0.3 with increment of0.05. Simulations have been performed considering two config-urations: i) parked condition, that is the rotor idles with bladespitched to feather (at a pitch angle of 90◦) and no wind is

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

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blowing; ii) power production, that is rotor speed of 12.1 rpmwith blades pitched to 17.5◦ and a hub-height constant wind of20 m/s is generated. The second configuration is meant to repro-duce, at least in terms of global damping, a realistic case whenthe turbine is operating [1]. A total simulation time of 30T isconsidered for each condition.

Figures 2 shows the time series of the wave elevation(WaveElev), tower-base fore-aft bending moment (TwrBsMyt)and tower-top fore-aft deflection (TTDspFA) corresponding tothe linear, 2nd-order and FNL wave kinematics when the tur-bine is parked and in power production. The wave steepness iska = 0.20.

The maximum wave elevations obtained with the linear, 2nd-order and FNL models are 3.86 m, 4.80 m and 5.7 m, respec-tively. In the parked case, linear and 2nd-order wave kinematicsprovide similar results for the tower-base shear force (not shownin figure) and bending moment. In contrast, the FNL produces anincrease of about 20% on TwrBsFxt and 50% on TwrBsMyt. Themaximum tower motion increases dramatically from approxima-tively 2 cm (linear and 2nd–order) to 10 cm (FNL). A clear low-frequency component of 0.02 Hz is observed. This contributionis much more relevant in the FNL case because it is associatedwith the difference frequency 3f − fn, where f = 0.1 Hz is thewave frequency and fn = 0.28 Hz is the first natural bendingfrequency of the tower. Conversely, linear and 2nd-order mod-els give a very small contribution at the frequency 3f , which ismainly related to the viscous drag contribution in the Morisonequation.

When the blades rotate under a constant wind speed, TwrB-sFxt (not reported here) and TwrBsMyt maximum values in-crease both of about 15%. Due to the rotor thrust caused bythe wind action, the tower top oscillates around a mean valueof 22.7 cm. The FNL minimum peak is 15.5 cm while the lin-ear and 2nd-order minimum peaks are 19 cm. We observe thatthe low-frequency oscillation, characterizing the tower motionin parked condition, almost disappears in the present case.

5.1.1 Power Spectral DensitiesThe power spectral densities (PSDs) of the wave elevation,tower-base fore-aft bending moment and tower-top fore-aft de-flection corresponding to the linear, 2nd-order and FNL wavekinematics when the turbine is parked (upper group) and inpower production (lower group) are shown in Figure 3. The wavesteepness increases from 0.05 to 0.30. For ka = 0.05 no differ-ences exist in the PSDs of the structural loads by using the threedifferent models for the wave kinematic as well as by changingthe operating conditions.

In the PSD of the sea surface elevation process, the domi-nant peak occurs at 0.1 Hz (since T = 10 s). As the steepnessincreases, peaks gradually appear at 2f and 3f , that is at 0.2 Hzand 0.3 Hz.

The system response has a main peak at 0.1 Hz. At ka = 0.05(Figure 3(a)), the nonlinear load contributions are negligible asa consequence of the almost linear forcing. Increasing the ka,the nonlinearities in the structural loads become relevant. Anevident peak at the natural frequency of the structure, i.e. atfn = 0.28 Hz, appears in the parked case, while it vanishes inthe power production. This behavior, studied in [1], is related tothe aeroelastic damping induced by the rotation of the blades.However, we anticipate that the peak in the TTDspFA PSD at fn(see Figure 3(a) parked case) is order of magnitudes smaller thanthe one that would be caused by the nonlinear wave kinematics(see Section 5.2).

Still in parked case, as the steepness increases, more and moreenergy is provided at 3f , therefore, as confirmed in Figures 3(b)-3(d) (see PSDs of TTDspFA) the peak moves gradually from fnto 3f , that is from 0.28 Hz to 0.3 Hz.

When the turbine is in power production (see Figure 3 lowerpanels) the transient behavior dissipates quite soon and the sys-tem responds at the loading frequencies. We only observe anaugmentation of the energy at the frequencies 2f and 3f as

the steepness increases. Note that to facilitate the comparisonsamong the figures, the minimum frequency is 0.05 Hz, so thatthe PSD at 0.02 Hz is not shown in these plots.

5.1.2 Amplitudes vs. wave steepness

The response amplitudes at given frequencies are plotted againstthe increasing steepnesses in Figure 4. The figure shows howthe amplitudes of TwrBsFxt, TwrBsMyt and TTDspFA at 0.1,0.2 and 0.3 Hz are influenced by the steepness of the incidentwave system. No substantial differences between the parked (notshown here) and the power production configurations are ob-served.

At 0.1 Hz the amplitudes are overestimated by the linear andweakly nonlinear wave models (see Figure 4(a)). At 0.2 Hz thelinear and 2nd-order models give the same results, while the FNLmodel captures a significant increase of the amplitudes for all thethree response channels (Figure 4(b)).

The estimated amplitudes of TwrBsFxt, TwrBsMyt and TTD-spFA at 0.2 Hz are the same when the linear or 2nd-order modelsare used. This happens because at 2f wave kinematics nonlin-earities are negligible with respect to the 2nd-order componentsintroduced by the square of the velocity in the viscous term ofMorison’s equation. Thus, both linear and nonlinear wave mod-els undergo the same growth as the steepness increases. On thecontrary, the FNL model shows a remarkable increase due tothe interaction of the 3rd-order components (3f − f ), see Fig-ure 4(b).

At 0.3 Hz, see Figure 4(c), the 2nd-order model differs fromthe linear one due to the interaction 2f + f , however it under-estimates the actual increase of the amplitudes due to the wavesteepening. We point out that after ka = 0.2 the growth rate ofthe FNL curve becomes smaller. This may be justified by thefrequent use of smoothing and regridding in the FNL numericalsolver to avoid wave breakings. Above ka = 0.2, the solutionmay result not as much accurate as below 0.2.

5.2 Irregular waves

We consider an extreme sea state characterized by Hs = 9 m,Tp = 11.8 s and a mean hub-height wind speed U = 33 m/s [22].Given the wind speed above the cut-out level, the turbine is set inparked condition. A total simulation time of 300 s is reproduced.

Figure 5 shows time series of the hub-height longitudinal windvelocity (WindVxi) (top panel), the wave elevations (WaveElev)(second panel from the top) and the system response: tower-baseshear (TwrBsFxt) and bending moment (TwrBsMyt) and, in thebottom panel, the tower-top fore-aft displacement (TTDspFA).

Figure 6 shows the estimated power spectral density functionsof WaveElev, TwrBsFxt, TwrBsMyt, and TTDspFA. In the sur-face elevation PSD the dominant peak occurs at 0.085 Hz (sinceTp = 11.8 s). At this frequency, the PSD of the tower-base fore-aft shear force (TwrBsFxt) is overestimated by the linear and2nd-order wave models. Since the turbine is parked, the domi-nant loads are hydrodynamic; thus, the tower-base shear forcemainly reflects the effect of these hydrodynamic forces.

Moreover, due to the large lumped mass at the top of the tower(with an associated large moment arm), the tower-base bend-ing moment PSD follows the tower-top motion, with a smalleramount of energy associated with hydrodynamic loading at fre-quencies around 0.85 and 0.17 Hz. Peaks in the PSDs of thetower-base bending mode (TwrBsMyt) and tower-top fore-aftdeflection (TTDspFA) exhibits significant amplifications at thetower fundamental frequency when the FNL model is used.These amplifications are justified by the springing-like vibra-tions clearly visible also in the time series of Figure [?]. In thiscase the structure is excited by a spectrum of wave componentscausing the excitation of its natural frequency. In the previouscase of regular waves, we observed a peak at 0.3 Hz (see for in-stance Figure 3(d)) that here is not present. Higher-order wavecomponents (captured by the FNL model) cause a very strong


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amplification of the response at 0.28 Hz. The structure experi-ences resonant vibrations such that the PSD at 0.28 Hz is oneorder of magnitude larger than in the case of regular waves. Forthis reason the peaks at the sea fundamental frequencies are verymuch attenuated in the PSD of TwrBsMyt and no longer visiblein the TTDspFA PSD, see Figure 6.

6 CONCLUSIONS

A comparison of the effects of linear, weakly and fully nonlin-ear regular and irregular wave models on the dynamic responseof a fixed-bottom offshore wind turbine has been presented. Inregular waves, for very small wave steepnesses, the structure re-sponds with the same amplitudes at the main wave frequencyregardless of the wave theory used. As the steepness increases,the structure responds also at the higher-order loading frequen-cies. In this case, differences between linear, 2nd-order and FNLmodels become dramatically relevant. From ka = 0.01 on, thelinear and 2nd-order theories become inaccurate. The structureresponds more and more at the loading frequencies of f , 2f and3f so that the PSDs between the parked and operating conditionassume the same shape.

The structural response under a severe sea state with irreg-ular nonlinear waves shows that when the turbine is parked, itbecomes extremely sensitive to resonant vibrations. The nonlin-ear kinematics excites the first natural frequency of the towercausing a very high peak in the tower motion PSD. This peak,associated with springing-like phenomena, is remarkably under-estimated (more that 50% less) by both the linear and 2nd-orderwave models.

A more comprehensive study aimed at establishing the effectson the long-term design loads as well as at assessing the accumu-lated fatigue damage (considering other sea states) when linear,2nd-order and FNL models are employed will follow. The largedifferences between the FNL and 2nd-order models is expectedto have great importance on these structural aspects.

ACKNOWLEDGMENTS

This work has been partially funded by the Centre of ExcellenceAMOS, NTNU, Norway and partially by the Flagship ProjectRITMARE The Italian Research for the Sea coordinated by theItalian National Research Council and funded by the Italian Min-istry of Education, University and Research within the NationalResearch Program 2011-2013.

REFERENCES

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[2] E. Marino, C. Borri, and U. Peil, “A fully nonlinear wave-model to account for breaking wave impact loads on off-shore wind turbines,” Journal of Wind Engineering and In-dustrial Aerodynamics, vol. 99, pp. 483–490, 2011.

[3] E. Marino, C. Borri, and C. Lugni, “Influence of wind-waves energy transfer on the impulsive hydrodynamicloads acting on offshore wind turbines,” Journal ofWind Engineering and Industrial Aerodynamics, vol. 99,pp. 767–775, 2011.

[4] T. F. Ogilvie, “Nonlinear high-froude-number free surfaceproblems,” J. of Engineering Mathematics, vol. 1, no. 3,pp. 215–235, 1967.

[5] M. S. Longuett-Higgins and E. D. Cokelet, “The deforma-tion of steep surface waves on water. i a numerical method

of computation.,” in Proc. R. Soc. London A, vol. 350,pp. 1–26, 1976.

[6] O. M. Faltinsen, “Numerical solutions of transient nonlin-ear free–surface motion outside or inside moving bodies,”in Proc. of 2nd Int. Conf. Num. Ship Hydr., 1977.

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[10] T. Nakayama, “A computational method for simulatingtransient motions of an incompressible inviscid fluid with afree surface,” International Journal for Numerical Methodsin Fluids, vol. 10, no. 6, pp. 683–695, 1990.

[11] J. Jonkman and M. Buhl, “FAST User’s Guide,” Tech. Rep.NREL/EL-500-38230, National Renewable Energy Labo-ratory, Golden, CO, 2005.

[12] P. Agarwal and L. Manuel, “Incorporating irregular nonlin-ear waves in coupled simulation and reliability studies ofoffshore wind turbines,” Applied Ocean Research, vol. 33,no. 3, pp. 215 – 227, 2011.

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[14] C. Brebbia and J. Dominguez, Boundary Elements, An In-troductory Course. WIT Press, Boston, Southamton, 1998.

[15] E. Marino, An Integrated Nonlinear Wind-Waves Model forOffshore Wind Turbines. FUP, Firenze University Press,2011.

[16] J. Sharma and R. Dean, “Development and evaluation of aprocedure for simulating arandomdirectional second-ordersea surface and associatedwave forces.,” Tech. rep. oceanengineering report no. 20, 1979.

[17] P. J. Moriarty and A. C. Hansen, “AeroDyn Theory ManualAeroDyn Theory Manual,” Tech. Rep. December, NationalRenewable Energy Laboratory, 2005.

[18] D. J. Laino and A. Craig Hansen, “AeroDyn User’s Guide,”tech. rep., National Renewable Energy Laboratory, 2002.

[19] J. R. Morison, M. P. O’Brien, J. W. Johnson, and S. A.Schaaf, “The force exerted by surface wave on piles,”Petroleum Transactions (American Institute of Mining En-gineers), vol. 189, pp. 149–154, 1950.

[20] B. Jonkman, “TurbSim User’s Guide: Version 1.50,” Tech.Rep. NREL/TP-500-46198, National Renewable EnergyLaboratory, Golden, CO, 2009.

[21] J. Jonkman, S. Butterfield, W. Musial, and G. Scott, “Def-inition of a 5-mw reference wind turbine for offshore sys-tem development,” tech. rep., NREL, 2009.

[22] O. M. Faltinsen, Sea Loads on Ships and Ocean Structures.Cambridge University, 1990.


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−4.3E+000

−7.1E−001

2.9E+000

6.5E+000

WaveE

lev[m

]

−3.1E+004

−9.9E+003

1.2E+004

3.3E+004

TwrB

sMyt[kNm]

120 140 160 180 200 220 240 260 280−1.5E−001

−6.0E−002

2.6E−002

1.1E−001

t [s]

TTDspFA

[m]

FNL 2nd-order Linear

7.0E+003

2.7E+004

4.7E+004

6.7E+004

TwrB

sMyt[kNm]

120 140 160 180 200 220 240 260 2801.5E−001

1.9E−001

2.3E−001

2.7E−001

t [s]

TTDspFA

[m]

Figure 2: Time series of the wave elevation (first panel from top), tower-base fore-aft bending moment (TwrBsMyt) and tower-topfore-aft deflection (TTDspFA) for the parked (second and third panels from top) and operation (forth and fifth panels from top),corresponding to the linear (blu dotted line), 2nd-order (green dashed line), FNL with (red solid line) wave kinematic models with ka0.20.


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0

2

4

6

8

10

12

14

WaveE

levPSD

[m2/Hz]

0

0.5

1

1.5

2

2.5

3

x 108

TwrB

sMytPSD

[(kNm)2/Hz]

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

1

2

3

4

5

6

7

8

x 10−4

TTDspFA

PSD

[m2/Hz]


0

10

20

30

40

50

WaveE

levPSD

[m2/Hz]

0

2

4

6

8

10

12

x 108

TwrB

sMytPSD

[(kNm)2/Hz]

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.2

0.4

0.6

0.8

1

x 10−3

TTDspFA

PSD

[m2/Hz]


0

50

100

150

200

WaveE

levPSD

[m2/Hz]

0

1

2

3

4

5

x 109

TwrB

sMytPSD

[(kNm)2/Hz]

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.01

0.02

0.03

0.04

TTDspFA

PSD

[m2/Hz]


0

100

200

300

400

500

WaveE

levPSD

[m2/Hz]

0

2

4

6

8

10

12

x 109

TwrB

sMytPSD

[(kNm)2/Hz]

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.01

0.02

0.03

0.04

0.05

0.06

TTDspFA

PSD

[m2/Hz]


Parked case.

0

0.5

1

1.5

2

2.5

3

x 108

TwrB

sMytPSD

[(kNm)2/Hz]

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

1

2

3

x 10−4

Hz

TTDspFA

PSD

[m2/Hz]

(a) ka = 0.05.

0

2

4

6

8

10

12

x 108

TwrB

sMytPSD

[(kNm)2/Hz]

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.2

0.4

0.6

0.8

1

1.2x 10

−3

Hz

TTDspFA

PSD

[m2/Hz]

(b) ka = 0.10.

0

1

2

3

4

5

x 109

TwrB

sMytPSD

[(kNm)2/Hz]

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

Hz

TTDspFA

PSD

[m2/Hz]

(c) ka = 0.20.

0

2

4

6

8

10

12

x 109

TwrB

sMytPSD

[(kNm)2/Hz]

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Hz

TTDspFA

PSD

[m2/Hz]

(d) ka = 0.30.

Power production.

Figure 3: Spectra of the wave elevation (first row), tower base fore-aft bending moment and tower-top fore-aft deflection for the parked(second and third rows) and operation (forth and fifth rows) conditions corresponding to the linear (blu dashed dotted), second–order(green dashed), FNL with (red solid) wave kinematic models with increasing ka.


3608

0

500

1000

1500

2000

2500

3000

Response Amplitudes at 0.1 Hz

TwrB

sFxtAmplitude[kN]

0

0.5

1

1.5

2

2.5

3x 10

4

TwrB

sMytAmplitude[kNm]

0.05 0.1 0.15 0.2 0.25 0.30

0.005

0.01

0.015

0.02

0.025

0.03

TTDispFA

Amplitude[m

]

ka

L 2nd FNL

(a) Amplitudes at 0.1 Hz.

0

200

400

600

800

1000

1200

1400


TwrB

sFxtAmplitude[kN]

0

0.5

1

1.5

2

2.5x 10

4

TwrB

sMytAmplitude[kNm]

0.05 0.1 0.15 0.2 0.25 0.30

0.005

0.01

0.015

0.02

0.025

0.03

0.035

TTDispFA

Amplitude[m

]

ka

L 2nd FNL

(b) Amplitudes at 0.2 Hz.

0

50

100

150

200

250

300

350

400


TwrB

sFxtAmplitude[kN]

0

1000

2000

3000

4000

5000

6000

7000

8000

TwrB

sMytAmplitude[kNm]

0.05 0.1 0.15 0.2 0.25 0.30

0.01

0.02

0.03

0.04

0.05

0.06

TTDispFA

Amplitude[m

]

ka

L 2nd FNL

(c) Amplitudes at 0.3 Hz.

Figure 4: Amplitudes vs. wave steepness of tower-base shear force and bending moment (TwrBsFxt, TwrBsMyt) and tower-top fore-aft deflection (TTDspFA) corresponding to the linear (blu with stars), 2nd-order (green with circles) and FNL with (red with crosses)wave kinematic models at different frequencies with the blades rotating under a steady hub-height wind speed (constant pitch).


3609

1.7E+001

2.8E+001

3.8E+001

4.9E+001

WindVxi[m

/s]

−6.5E+000

−2.0E+000

2.5E+000

7.1E+000

WaveE

lev[m

]

−3.5E+003

−9.4E+002

1.6E+003

4.2E+003

TwrB

sFxt[kN]

−4.9E+004

−8.9E+003

3.1E+004

7.2E+004

TwrB

sMyt[kNm]

50 100 150 200 250−2.3E−001

−5.9E−002

1.1E−001

2.9E−001

t [s]

TTDspFA

[m]


Figure 5: Time series of the hub-height longitudinal wind velocity (WindVxi), wave elevation (WaveElev), tower base fore-aft shearforce (TwrBsFxt), tower base fore-aft bending moment (TwrBsMyt) and tower-top fore-aft deflection (TTDspFA), corresponding tothe linear (blu dotted), 2nd-order (green dashed), FNL (red solid) wave kinematic models.

0

10

20

30

40

50

60

70

80

90

PSD

[m2/Hz]

WaveElev

0

5

10

15x 10

6

PSD

[(kN)2/Hz]

TwrBsFxt

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

2

4

6

8

10

x 109

Hz

PSD

[(kNm)2/Hz]

TwrBsMyt

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Hz

PSD

[m2/Hz]

TTDspFA


Figure 6: PSDs of the wave elevation (WaveElev), tower base fore-aft shear force (TwrBsFxt), tower base fore-aft bending moment(TwrBsMyt) and tower-top fore-aft deflection (TTDspFA), corresponding to the linear (blu dotted), 2nd-order (green dashed line),FNL (red solid line) wave kinematic models.


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coupled dynamic simulations of offshore wind turbines...

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