Investigation on parametrically excited motions of point absorbers in regular waves. Kevin Tarrant & Craig Meskell Trinity College, Dublin Published in Ocean Engineering 111: 67-81 (2016) http://dx.doi.org/10.1016/j.oceaneng.2015.10.041 Abstract. Free floating objects such as a self-reacting wave energy converter (WEC), may experience a condition known as parametric resonance. In this situation, at least two degrees of freedom become coupled when the incident wave has a frequency approximately twice the pitch or roll natural frequency. This can result in very large amplitude motion in pitch and/or roll. While classic linear theory has proven sufficient for describing small motions due to small amplitude waves, a point absorber WEC is often designed to operate in resonant conditions, and therefore, exhibits significant nonlinear responses. In this paper, a time-domain nonlinear numerical model is presented for describing the dynamic stability of point absorbers. The pressure of the incident wave is integrated over the instantaneous wetted surface to obtain the nonlinear Froude-Krylov excitation force and the nonlinear hydrostatic restoring forces, while first order diffraction-radiation forces are computed by a linear potential flow formulation. A numerical benchmark study for the simulation of parametric resonance of a specific WEC - the Wavebob - has been implemented and validated against experimental results. The implemented model has shown good accuracy in reproducing both the onset and steady state response of parametric resonance. Limits of stability were numerically computed showing the instability regions in the roll and pitch modes. Investigation on parametrically excited motions of point absorbers in regular waves Kevin Tarranta, Craig Meskella,∗ aSchool of Engineering, Trinity College Dublin, Dublin 2, Ireland Abstract Free floating objects such as a self reacting wave energy converter (WEC), may experience a condition known as parametric resonance. In this situation, at least two degrees of freedom become coupled when the incident wave has a frequency approximately twice the pitch or roll natural frequency. This can result in very large amplitude motion in pitch and/or roll. While classic linear theory has proven sufficient for describing small motions due to small amplitude waves, a point absorber WEC is often designed to operate in res- onant conditions, and therefore, exhibits significant nonlinear responses. In this paper, a time-domain nonlinear numerical model is presented for describ- ing the dynamic stability of point absorbers. The pressure of the incident wave is integrated over the instantaneous wetted surface to obtain the non- linear Froude-Krylov excitation force and the nonlinear hydrostatic restoring forces, while first order diffraction-radiation forces are computed by a linear potential flow formulation. A numerical benchmark study for the simula- tion of parametric resonance of a specific WEC—the Wavebob—has been implemented and validated against experimental results. The implemented model has shown good accuracy in reproducing both the onset and steady state response of parametric resonance. Limits of stability were numerically computed showing the instability regions in the roll and pitch modes. Keywords: Wave energy converter; Point absorber; Parametric resonance; Nonlinear equations ∗Corresponding author. Tel: +353 1 8961455,Fax: +353 1 6795554 Email addresses: [email protected](Kevin Tarrant), [email protected](Craig Meskell) Preprint submitted to Journal Of Ocean Engineering October 12, 2015 1. Introduction The inevitable depletion of fossil fuels, together with the ongoing issue of global warming, has shifted opinion towards the imperative of address- ing climate change by developing sustainable energy resources (IPCC, 2014). Ocean wave energy is one such renewable source which has the potential to be a significant contributor to this development for many regions in the world. Wave energy has the highest energy density among renewable energy sources (Clement et al., 2002), has high availability and has good predictabil- ity. However, it is important to understand the difficulties which wave power developments face. Due to the nature of sea waves, a wave energy converter (WEC) hastooperateunder awiderangeofexcitationfrequencies andampli- tudes. It also has to be capable of withstanding extreme weather conditions. There are also challenges in efficiently coupling the irregular, low frequency motion of sea waves with an electrical generator. Nevertheless, several con- figurations of WECs have been proposed and tested over the years varying greatly in both design and technology (Thorpe, 1999). Among the wide va- riety of wave energy devices, floating point absorbers have attracted a lot of attention due to their ability to respond to the incident wave climate and their ability to take the wave at any direction. Point absorbers are oscil- lating bodies whose horizontal dimensions are small in comparison with the representative wavelength. A number of different numerical techniques are used to model WECs. In manycases,frequency-domainpotentialtheorybasedrepresentations(Bosma et al., 2012) are utilized due to their simplicity and computational efficiency. While classic linear theory has proven sufficient for describing small motions duetosmallamplitudewaves, WECsareoftensubject tomanynonlinearities such as those from power-take-off (PTO) systems, complex mooring arrange- ments and nonlinear responses due to large amplitude motions. In this case, the linear model may not be able to accurately represent the actual system dynamics. As a result, time-domain models of WECs are often employed to take these nonlinear characteristics into account (Gilloteaux et al., 2007a; Guerinel et al., 2011). This paper focuses on the time-domain simulations of a specific floating point absorber WEC—the Wavebob. It is well known that floating structures can be subject to nonlinear un- stable motions when heave resonance produces large heave motions even with small wave excitation. The excitation energy input to the heave or pitch mode may be transferred into the roll motion due to nonlinear cou- 2 pling among these modes. This can result in large amplitude roll motions as well as heave and pitch motions due to a phenomenon known as parametric resonance. While the Mathieu and Hill equations date to the 19th century, the in- vestigation of parametric roll in floating structures came later. Nonetheless, there are numerous examples of studies which deal with the phenomenon (for example Eatock Taylor and Knoop , 1982). More recently, studies into the field of parametric resonance of offshore structures focused on the para- metric roll of ships (where the requirements for suitable numerical models of parametric roll were addressed) due to accidents involving loss of cargo to container ships as a result of large roll angles due to parametric roll (France et al., 2003). Numerous mathematical models for describing the dynamics of parametric roll of ships have been proposed over the years, the most com- mon method based on a Mathieu-type (or Mathieu-Duffing type) 1-DOF roll equation to describe the onset of heavy roll motion in regular longitudinal seas (Shin et al., 2004; Umeda et al., 2003). It is widely believed that one of the main factors driving parametric resonance in ships is due to the time varying water-plane area when a wave crest or trough is amidships, an effect which is accentuated for ships with large bow flare and stern overhang. This may not, however, be the case for parametric resonance of point absorbers which are characterized by a large draught in comparison to the diameter of the body at the water surface. There is little work published in the literature on parametric resonance of point absorbers, however, insight into the physics behind parametric motion of such devices may be gained by observing para- metricinstabilities ofsparplatformsandvertical cylinders which have similar geometric properties to point absorbers. Investigations into the occurrence of parametric resonance of spar plat- forms have been published (e.g. Haslum and Faltinsen, 1999; Hong et al., 2005; Rho and Choi, 2002) in which the unstable motions were attributed to Mathieu type instabilities associated with sinusoidal variations of the meta- centric height appearing in the roll/pitch hydrostatic restoring force. De- pending on the values of the coefficients in the Mathieu equation, the solu- tion yields either a bounded or unbounded solution. It was assumed in each of these works that the variation in submerged volume and wave passage effects are not important in the analysis of parametric motion. Neves et al. (2008) demonstrated analytically that in the case of structures with vertical walls, simple harmonic parametric excitation does not come from pure hydro- static terms as in the case of ships but instead from wave passage effects only. 3 They showed that the hydrostatic contribution to parametric excitation of roll (or pitch) due to heave is composed of a bi-harmonic and a time indepen- dent term. Liu et al. (2010) and Li et al. (2011) present numerical examples of parametric instabilities of spar platforms with their models focusing on heave-pitch coupling. This paper investigates a different occurrence of parametric resonance than that which occurs in ships or spars, namely one that occurs in point absorbers involving relative motion between two floating bodies which are connected by a PTO. It is not clear what the effect of the relative motion and the level of PTO damping has on parametric resonance. In order to clarify some of these issues, a fully coupled numerical model is used to pre- dict the motions of Wavebob in the time-domain. The hydrostatic restoring forcesandFroude-Krylovexcitationforcesareconsiderednonlinearsincethey are computed on the instantaneous wetted surface due to the incident wave profile while the flow model and first order diffraction and radiation forces are computed by a linear potential flow formulation. The effects of linear PTO damping, wave excitation frequency and wave height on parametric motion are investigated while comparisons with experimental results are also presented for the Wavebob device in regular waves. 2. Wave energy converter description The WEC designed by Wavebob is an axi-symmetric, self-reacting point absorber that primarily operates in the heave mode. It consists of two con- centric floating buoys: a torus and a float-neck-tank (FNT). Fig. 1 shows a 1:4 scale Wavebob model at sea. In this figure, the torus can be clearly seen as the outer floating body while the FNT is positioned inside the torus with a small gap called a moonpool separating the two bodies. The PTO system can be seen attached to the top of the structure. A schematic representation of the device is shown in Fig. 2. It can be seen that the only relative mo- tion between the two bodies is in the vertical (heave) direction. Due to the different mass and hydrodynamic properties, the torus is characterized by a high natural frequency, while the FNT acts as a high-inertia body with a low natural frequency. This results in both buoys responding with different amplitudes and phases when excited by ocean waves, thus creating relative motion between them. The torus and FNT are linked via the PTO unit which transforms the available energy from relative motion to useful electri- calpower. A catenary mooring system is generally used for this type of WEC 4 to prevent the device from drifting. 3. Wave energy converter numerical model The motion of each body is characterised by six oscillatory modes of motion as outlined in Table 1. The two bodies oscillate relative to each other only in the heave direction, while for the other modes of oscillation the two bodies are rigidly connected. Thus, from a modelling point of view, the device can be seen as a system of two bodies with seven degrees of freedom correspondingtothreerotations(roll,pitch,yaw)andtwotranslations(surge, sway), plustwoadditionaltranslationsrepresentingtheheavemotionsofeach body. Mode No. Mode Name Mode No. Mode Name 1 Torus surge 7 FNT surge 2 Torus sway 8 FNT sway 3 Torus heave 9 FNT heave 4 Torus roll 10 FNT roll 5 Torus pitch 11 FNT pitch 6 Torus yaw 12 FNT yaw Table 1: 12 modes of oscillatory motion for the torus (body A) & FNT (body B) Thecoordinatesystems usedtodefinethesystemdynamicsareillustrated inFig.2. Anearthfixed frame O xyz is assumed to beinertial with theO xy o o plane lying on the still water surface; it defines the trajectory of the body as well as the incident of the waves. The body fixed frames of the torus (body A) and the FNT (body B) are denoted O xyz and O xyz respectively and A B have their origin at the centre of gravity of the respective bodies; they define the angular movement of each individual body with respect to the earthfixed frame. An inertial translating frame O xyz is fixed to the equilibrium state h with its vertical axes passing through the centre of gravity of the torus. The O xyz frameisnotfixedtothebodyandtheO z axisisalwaysperpendicular h h to the still water level; it defines the translational movement of each body. The equation of motion for the 7 degree of freedom system can be ex- pressed as Mξ¨+F = F +F (1) c h ext where 5 • ξ = [ξ ,ξ ,ξ ,ξ ,ξ ,ξ ,ξ ]T is the motion vector of the system. ξ and 1 2 3 9 4 5 6 3 ξ are the torus and FNT heave motions respectively; 9 • M ∈ R7×7 is the mass matrix; • F ∈ R7 is the vector of Coriolis forces; c • F ∈ R7 is the vector of pressure forces due to fluid-structure interac- h tions; • F ∈ R7 is the vector of external forces acting on the system from the ext PTO and the mooring loads. Expressing Eq. (1) in matrix form we get ⎡ ⎤⎡ ⎤ ¨ m 0 0 0 0 m z 0 ξ1 ⎢ t t g ⎥⎢ ⎥ ⎢⎢ 0 mt 0 0 −mtzg 0 0 ⎥⎥⎢⎢ξ¨2⎥⎥ ⎢ ⎥⎢¨⎥ ⎢ 0 0 mA 0 0 0 0 ⎥⎢ξ3⎥ ⎢ ⎥⎢ ⎥ ⎢ 0 0 0 m 0 0 0 ⎥⎢ξ¨⎥ ⎢ B ⎥⎢ 9⎥ ⎢⎢⎢ 0 −mtzg 0 0 Ixxt 0 0 ⎥⎥⎥⎢⎢⎢ξ¨4⎥⎥⎥ ⎣m z 0 0 0 0 I 0 ⎦⎣ξ¨⎦ t g yyt 5 0 0 0 0 0 0 Izzt ξ¨6 ⎡ ⎤ ⎡ ⎤ ⎢mt(−ξ˙2ξ˙6)+mA(ξ˙3ξ˙5)+mB(ξ˙9ξ˙5)⎥ ⎢Fh,1 +Fext,1⎥ ⎢⎢ mt(ξ˙1ξ˙6)−mA(ξ˙3ξ˙4)−mB(ξ˙9ξ˙4) ⎥⎥ ⎢⎢Fh,2 +Fext,2⎥⎥ ⎢⎢ mA(−ξ˙1ξ˙5 +ξ˙2ξ˙4) ⎥⎥ ⎢⎢Fh,3 +Fext,3⎥⎥ ⎢ ⎥ ⎢ ⎥ +⎢ m (−ξ˙ ξ˙ +ξ˙ ξ˙ ) ⎥ = ⎢F +F ⎥ (2) ⎢ B 1 5 2 4 ⎥ ⎢ h,9 ext,9⎥ ⎢⎢⎢ (Izzt −Iyyt)ξ˙5ξ˙6 ⎥⎥⎥ ⎢⎢⎢Fh,4 +Fext,4⎥⎥⎥ ⎣ (Ixxt −Izzt)ξ˙6ξ˙4 ⎦ ⎣Fh,5 +Fext,5⎦ (Iyyt −Ixxt)ξ˙4ξ˙5 Fh,6 +Fext,6 Here, m is the totalmass of thesystem while m andm arethe mass of the t A B individual bodies. I ,I and I are the total moments of inertia of the xxt yyt zzt two bodies about the respective axes with I = I due to symmetry. z is xxt yyt g the vertical coordinate of the centre of gravity of the system. The numerical subscripts on the right hand side of the equation refers to the degree of freedom, i.e., F +F is the summation of the pressure forces due to fluid- h,1 ext,1 structure interactions and the external forces in the surge direction (mode 1). 6 3.1. Fluid-structure interaction Intheanalysis ofthemotionresponse ofthedevice inwaves, itisassumed that the fluid is incompressible, inviscid and irrotational. In addition, a drag force has been included in the numerical model to approximate the energy lossesduetoviscosity. Thecontributionfromtheforcesduetofluid-structure interactions can be expressed as F = F +F +F +F +F (3) h r d FKd FKs Drag where • F ∈ R7 is the vector of linear radiation induced hydrodynamic forces; r • F ∈ R7 is the vector of linear diffraction excitation forces; d • F ∈ R7 is the vector of nonlinear dynamic Froude-Krylov excitation FKd forces; • F ∈ R7 is the vector of nonlinear static Froude-Krylov restoring FKs forces; • F ∈ R7 is the vector of nonlinear drag force. Drag Thelinear radiationanddiffractionforcesarecomputed usingWAMIT which is a frequency-domain hydrodynamic software package with multi-body func- tionality. Special care was taken when analysing the effect of the moonpool 1 in WAMIT. It is known that resonant interactions can occur in the moon- pool which are manifested as extreme amplifications of the hydrodynamic loading (Mavrakos and Chatjigeorgiou, 2009). Since linear potential theory generally overpredicts the amplitude of the resonances, an effective way to apply a damping factor to the moonpool in WAMIT was implemented by replacing the physical free surface of the moonpool by a lid, or heaving pis- ton as outlined by Newman (2004). Since the lid is free to heave, without external constraints, it acts just like the free surface. With moderate choices of damping to the lid heave mode, the resonant modes are attenuated. As the moonpool cross-section is small, only the heave mode was relevant within the range of wave periods considered. 1The small gap between the torus and the FNT is known as a moonpool 7 3.1.1. Radiation forces Hydrodynamic radiation forces occur due to an oscillating body radiating waves away from itself, which in turn exerts reaction forces on the oscillating body. These forces are treated independently to excitation forces and are analyzed under the premise that no incident waves exist. The radiated waves generated by the floating bodies at any given time will persist indefinitely. These generated waves affect the fluid pressure field and hence the body force of the floating body at all subsequent times. This situation introduces memory effects which can be expressed mathematically by a convolution integral (Cummins, 1962) according to (cid:8) t F = −Aξ¨ − K (t−τ)ξ˙(τ)dτ (4) r r 0 in which A ∈ R7×7 is the constant infinite frequency added mass matrix representing the inertia of the fluid that is put in motion by the body when oscillating and K ∈ R7×7 is the matrix of radiation impulse response func- r tions. Duetothecomplexityincalculatingtheaddedmassandimpulseresponse functions, a frequency-domain hydrodynamic software package with multi- body functionality (such as WAMIT) may be used to obtain the added mass and damping coefficients. For a two-body system, the frequency dependent added mass coefficient matrix is given by A(ω) ∈ R12×12 while the frequency dependent hydrodynamic damping coefficient matrix is given by B(ω) ∈ R12×12. InordertorelatethematricesAandK inEq.(4)totheirfrequency- r domain counterparts, it is necessary to use the approach given by Ogilvie (2010), such that (cid:8) 1 ∞ A(ω) = A− K (τ)sin(ωτ)dτ (5) r ω 0 (cid:8) ∞ B(ω) = K (τ)cos(ωτ)dτ (6) r 0 Eq. (5) must be valid for all ω, hence, we choose ω = ∞ implying that A = A(∞) (7) Eq. (6) is rewritten using the inverse Fourier transform giving (cid:8) 2 ∞ K (t) = B(ω)cos(ωt)dω (8) r π 0 8 Therefore, using the approach by Ogilvie, frequency dependent hydrody- namic added mass and damping coefficients can be transformed into fre- quency independent added mass coefficients and radiation impulse response functions for describing the behaviour of a body and the fluid in the time domain. 3.1.2. Diffraction forces The diffraction problem is studied on the body while it is kept fixed in a regular wave field. The presence of the body in the fluid results in diffraction of the incident wave system and the addition of a disturbance to the incident wave potential associated with the scattering effect of the body. The diffraction force for the currently considered WEC is defined to act at the reference centre of the body. Due to the large geometry of the body, the WEC is affected by the incident wave before it reaches the reference centre. To account for this acausal nature of the excitation force, a convolution product is used to represent the diffraction force as (cid:8) +∞ F = K (t−τ)η(τ)dτ (9) d d −∞ where K ∈ R7×7 is the matrix of diffraction impulse response functions and d η(x,t) is the free surface elevation of the incident wave at the prescribed reference point in the inertial frame. Newman (1977) states that for longitu- dinal head waves, the equation of surface elevation according to linear Airy theory is defined as η(x,t) = A cos(kx+ω t) (10) w e where A is the wave amplitude, k is the wave number defined as: k = w 2π/λ ≈ 1.56 T2 (deep water), T is the wave period, λ is the wavelength and ω is the wave encounter frequency. e 3.1.3. Nonlinear Froude-Krylov excitation force The Froude-Krylov excitation forces and moments are the loads intro- duced by the pressure field generated by undisturbed waves. In linear theory, the pressure is integrated over the surface of the body at the mean water level with the body in its equilibrium position. This method is suitable for small amplitude waves, however, when the device undergoes large amplitude motions due to resonance conditions and/or large amplitude waves for exam- ple, there can be a significant change in wetted surface area and wave profile 9