Proceedings of the ASME 2013 32nd International Conference on Ocean, Offshore and Arctic Engineering OMAE2013 June 9-14, 2013, Nantes, France OMAE2013-11405 SECOND-ORDER RANDOM WAVE KINEMATICS AND RESULTING LOADS ON A BOTTOM-FIXED SLENDER MONOPILE Carl Trygve Stansberg MARINTEK Trondheim, Norway Andreas Amundsen MARINTEK Trondheim, Norway Sebastien Fouques MARINTEK Trondheim, Norway Ole David Økland MARINTEK Trondheim, Norway ABSTRACT The importance of including second-order nonlinear random wave kinematics in the numerical prediction of draginduced shear forces and moments, at various levels on a bottom-fixed slender monopile in 40m water depth, is investigated. A vertical circular cylinder of diameter 0.5m is considered, representing typical dimensions of members in jacket type foundations of offshore wind turbines. The focus is here on the wave loads only, and wind and a propeller are therefore not included in this study. In particular, the main focus is on the effects from second-order random wave kinematics on the structural quasi-static time-varying loads due to drag forces in heavy storm wave conditions. Comparisons are made to the traditional use of Airy waves with various ways of stretching. An in-house numerical FEM code developed for structural analysis, NIRWANA, is used for this study. Thus one purpose of the present work is also to verify the implementation of the second-order random waves in the code. The results show significant effects, especially in the wave zone. Extreme crests are around 15% - 20% increased, freesurface extreme particle velocities increase by around 30% - 40%, while the velocities at levels below MWL are, on the other hand, somewhat reduced. The resulting peak shear forces, and in particular the moments, are thereby increased by typically 50% - 100% in the upper parts of the column. At the base the peak shear forces are comparable to the traditional methods, while moments are still somewhat higher. Another effect is the generation of more high-frequency load contributions, which may be important to address further with respect to natural frequencies of such towers. INTRODUCTION Wave loads in harsh weather conditions represent a large part of external forces on offshore wind turbine towers. The importance of including nonlinear wave kinematics in these load predictions is investigated in this paper. Agarwal & Manuel [1] studied such effects on the quasi-static base moments of a 6m diameter stand-alone monopile in a sea state of Hs=5.5m, Tp=11.2s, in 20m water depth. They combined a classical Morison formulation with a second-order random wave model, and found a considerable increase (around 26%) in the maximum moment. It was also found that the largest increase occurs in the drag component; and since the actual case was inertia dominated, one should expect even larger effects on more slender, drag-dominated structures. Furthermore, the work in [1] included considerations on these effects when a turbine is included, and found that this will depend on the natural oscillation period of the total system. The same authors carried out a comprehensive follow-up work in [2] where they also included such effects in coupled simulations and reliability studies as parts of a total, statistical analysis of such wind turbine systems. Here we shall follow an approach similar to that in [1], while with some different details, and on a different stand-alone bottom-fixed monopile structure case consisting of a circular cylinder of diameter 0.5m in 40m water depth, in a very steep sea state Hs=15m, Tp=14s. Here the drag forces will dominate, and therefore we consider only drag forces in this study. The water depth is larger than in [1], while at the same time the wave period is longer, so the effects from the finite depth may be of the same order. We shall focus on a detailed investigation 1 Copyright 2013 by ASME
of the second-order wave contributions in selected extreme wave load events from time-domain simulations in random waves, as compared to use of more traditional wave kinematics models. In addition to base moments, as in [1], we also investigate the particle velocities, the moments and the shear forces at several vertical levels in the region between the bottom and the wave crest. This slender geometry does of course not represent any realistic monotower dimensions, but may rather represent typical dimensions of members in jacket type foundations of offshore wind turbines. A second-order numerical random wave model for arbitrary water depths previously described in [3] is applied. It is of interest to recall that nonlinear wave effects are expected to be more pronounced in finite and shallow water than in deep water, as observed in [3]. Resulting kinematics and loads using second-order random wave kinematics are then compared to those obtained with three more "traditional" kinematics methods (Airy integrated to the mean water level, Airy with vertical extrapolation, and Wheeler strectching). So one purpose of this study is also to benchmark various "traditional" models against each other. An in-house numerical finite-element (FEM) program, NIRWANA, previously used in a variety of different applications such as in [4,5], is used to study these effects. The program was previously developed for the study of waveinduced structural loads and responses on bottom-fixed slender structures in regular and random waves. It is dedicated for nonlinear structural dynamic analysis of e.g. jacket (space frame) structures. One purpose of the present study is actually to verify the implementation of these nonlinear wave effects in NIRWANA. An initial analysis is presented here, where resulting structural loads in given kinematics conditions are observed when keeping the structural element system very stiff, and modelling a quasi-static condition. In this case the observations should simply be equal to the input Morison drag loads integrated over the actual column sections. The ultimate goal is eventually to use this updated tool in complete wind turbine analysis, while since the focus is here on the wave loads only, wind and a propeller are not included in the study presented in this paper. RANDOM WAVE KINEMATICS AND WAVE LOADS Linear (Airy) wave model We formulate the linear random elevation ζ (1) (t) as a random sum of a large number of independent complex harmonic Fourier components N with different frequencies f ω/2π. NF ζ (1) (t i ) = Σ N(f k ) exp(j2πf k t i ) (1) k=-nf The water particle kinematics is, strictly speaking, defined up to the mean water level (MWL) only, with the horizontal velocity u (1) (t i ) formulated as a sum as in eq. (1) but with the Fourier components multiplied by: ( h + z) cosh k G( ω, z) = ω A( ω), z 0 (2) sinh kh where h is the water depth, k is the wave number, and z is the vertical position in consideration. A "vertically extrapolated" Airy model is sometimes used, where one lets the kinematics be constant up to the linear free surface. For further details of the classical Airy wave model we refer to textbooks such as e.g. [6]. Wheeler streching The Wheeler stretching model [7] is based on the same model as in eqs. (1-2), but with the kinematics "stretched" from the MWL to the linear free surface by a change of the vertical coordinate: z ζ z' = ζ 1+ (3) h Second-order random wave model Second-order random wave models for intermediate water depths were originally formulated by Sharma & Dean [8], and slightly reformulated and applied for long-crested waves in Marthinsen & Winterstein [9]. Here we shall use an implementation based upon [9], described in [3,10]. Longcrested waves are assumed. A summary of the method is given below, while more details are given in the references. For the elevation, two second-order correction terms are added: ζ (t i ) = ζ (1) (t i ) + ζ (2+) (t i ) +ζ (2-) (t i ) (4) where ζ (2+) and ζ (2-) are the sum- and difference-frequency terms, respectively. In total, these additional terms may add about 15% - 20% to the linear term in an extreme, steep crest, and the second-order effects increase with decreasing water depths. The validity of the second-order perturbation is reduced when quite shallow conditions are considered, following the Ursell number criterion [6], and it was found in [3] that for severe storm waves with peak periods around 14s, the validity range stops at around 30m 40m. For shallower depths, different formulations should be used for such waves. The second-order horizontal velocity may also be written as a sum of the linear term u (1) (t i ) and second-order contributions: u u tot tot ( 1 ( ) ) ( 2, sum ( ) ) ( 2, diff z = u z + u ( z) + u ) ( z) ; z 0 ( 1) u ( 2, sum ( ) ) ( 2, diff ( ) ) ( ) z = u + z + u 0 + u 0 ; z > 0 z z= 0 0 (5) 2 Copyright 2013 by ASME
where u 0 u (1) (0). For z > 0 this is formally valid up to the linear time-varying free surface level z=ζ (1). See also [3]. The second term in the expression represents a "stretching", which is consistent to second-order as long as the two latter terms are included. Our "stretching" formulation seems to be different from the formulation in [1] where they use Wheeler stretching in combination with the second-order formulation. Slender column wave load model Assuming a slender body formulation, we write for the local wave excitation force on a small vertical column strip with height dz, according to Morrison's equation [11]: df(z,t ) = ¼C M πdρa(z,t)dz + ½C D Dρu(z,t) u(z,t) dz (6) where C M is the inertia coefficient, C D is the drag coefficient, D is the pile diameter, ρ is the water density, and a(z,t) is the water particle acceleration. Notice that in the present study, we shall study the effect from drag loads only, so the first (inertia) term shall be disregarded. Integrated loads on a section between levels Z1 and Z2 of the column height can be written as: Z2 Forces: F(t)= df(z,t) (7) Z1 Z2 Moments around the level Z1: M(t)= zdf(z,t) (8) Z1 Depending on which irregular wave kinematics model that is used, we here define four different types of integrated force formulations: 1) Second-order wave kinematics with loads integrated up to the instantaneous free surface 2) Linear (Airy) wave kinematics integrated up to the mean water level (MWL) 3) Linear (Airy) wave model with the kinematics integrated WL up to the instantaneous free surface, and 4) Wheeler's stretching method: Linear (Airy) wave model with the kinematics "stretched" up to the instantaneous free surface The present drag-force-only model, neglecting the inertia forces, is certainly physically incomplete, while we choose to study this here in order to make a systematic and detailed check on this effect only. In actual windmill (and other) studies to be eventually done, the full Morrison formulation, including the inertia tem, shall be used. STRUCTURAL LOAD MODELLING A response analyses is performed using the program NIRWANA [4,5]. This is a finite element analysis program developed by MARINTEK for nonlinear structural dynamic analysis of fixed offshore structures. Note that in the present application, a quasi-static analysis is run to study the nonlinear wave effects on the load, and to check the implementation of it. The response is solved in the time domain. The formulation includes linear structural geometry and material properties and various formulations for wave kinematics and nonlinear drag force loading based on Morrison's equation, eq. (2). For details of the program we refer to the above references. NIRWANA has been widely used in consulting and research projects related to oil and gas industry and marine technology, such as shown in the given references. The program is currently under further development, being extended by new functionality that enables analyzing dynamic responses of the bottom fixed wind turbines. The present development is focused on implementing the second order wave kinematics model, while several other items are also planned such as a rotating wind turbine and an advanced foundation formulation and development of user interface for structural modeling and results post-processing. The wave-induced load excitation formulations include several choices. Here we shall consider irregular waves and use the four different options defined in the load description above. In addition, a Stokes 5 th order model can also be used (with regular waves), but that is not addressed here. In the present work the loads from the second-order kinematics model are compared to those using the "traditional" formulations. The mono-pile analyzed is representative for load members in a typical jacket foundation for a fixed wind turbine in intermediate water depths. NIRWANA offers the possibility of static (no inertia and damping forces) as well as dynamic simulation of responses from stochastic irregular wave loading. The focus in this paper has been on the validation of the load model, and hence the results shown are from a static analysis. SIMULATION CASE STUDY DEFINITION The structural model for NIRWANA simulations is defined in Table 1. Note that the small diameter of 0.5m means that the slender body assumption should be OK. The choices made for the drag coefficient C D are as recommended in NORSOK [12], and the change at Z=0 is due to marine growth. The mean water level (MWL) is defined at 40m from the bottom. One irregular wave condition is modeled: Hs=15.0m, Tp=14.0s, JONSWAP, Gamma=3, h=40m This is a very heavy storm condition, and is equal to one of the conditions that were previously studied in [3]. The linear wave elevation time series from that study has actually been imported as input for the present study. Thus we can investigate two particular extreme events that were also highlighted in [3], and the results presented below are in fact focusing on those events. 3 Copyright 2013 by ASME
Table 1. NIRWANA monopile input Water depth 40 m H (total height) 60 m Axial stiffness, EA 3.36E+11 N Bending stiffness, EI 1.25E+11 NM^2 Torsional stiffness, GI 4.82E+10 m Hydrodynamic diameter 5.00E-01 m Cd, Z>0 0.65 - Cd, Z<0 1.05 - Number of nodes 81 Boundary condition lower end Fixed Length of wave timeseries 16384 (4.5hrs) s Time interval 0.5 s Fig. 1. Event1 time series samples of wave elevation. -40<Z<-20 Element length 0.5m 1m -20<Z<20 Quasi-static drag-induced structural shear force and moment simulations at various vertical levels are made, using four different formulations for the integrated loads: 1) Second-order wave model 2) Airy wave kinematics integrated up to MWL 3) Airy wave kinematics vertically extrapolated to the free surface 4) Wheeler's stretching method RESULTS WITH DISCUSSION Results for two extreme events are shown in Figs. 1-10. The events are denoted as Event1 and Event2, occurring at t=5975s and t=11432s, respectively. In Figs.1-2, the wave elevation and the horizontal particle velocity at various vertical levels are given for Event1, and the corresponding results for Event2 are given in Figs.3-4. (Note that the apparent abrupt jumps in the velocity time series are just reflecting that the point is passed by the free surface on its way up or down, and flat zero means no water.) Vertical profiles of the simulated horizontal particle velocities for the same two events are presented in Figs. 5-6. We see that the two wave events are quite different: Event1 is a group of very high and long waves, which are clearly influenced by the finite water depth (as we can see e.g. from the secondorder elevation), while Event2 is a shorter but very steep individual wave with a high crest and shallow troughs this event is less influenced by the depth. It is also found that the second-order model predicts 15%-20% higher crests, and 30% - 40% higher velocities at the crest peak, than the other models, while, on the other side, the velocities below MWL are lower. This is also discussed in [3]. Fig. 2. Event1 time series samples of particle velocities at z=+16m, +10m, 0m, -10m and -40m. 4 Copyright 2013 by ASME
Fig. 3. Event2 time series samples of wave elevation. Fig. 4. As Fig. 2, but for Event2. The present use of the Wheeler method needs a comment. In this case, a linear elevation time series is used as input, and the free-surface velocities are seen to be clearly lower than the second-order results. In many cases, however, the Wheeler stretching is used with a measured time series in stead, which includes nonlinear effects. In those cases the free-surface velocity estimates often turn out to be comparable to 2 nd order, and also measured, results, while the velocities at MWL and further down in the fluid are still lower. This must be kept in mind. We interpret that the latter is the model used in [1], in combination with a second-order wave model. The topic has been further investigated in [10]. The bend in the velocity profile for Event1, second-order model, at z=0, is due to the contribution from the second-order sum-frequency potential (which we keep constant for z>0), just below the MWL. In deep water this contribution is zero, while this individual event is a relatively long wave for this water depth. On the other hand, it is not observed in Event2, since this wave cycle is shorter and is almost a deep-water event. Resulting shear forces at the levels +10m, 0m, -10m and - 40m are presented in Figs. 7-8 for the two events. The effects from the second-order kinematics contributions are seen in particular at the upper levels, for both events, with forces 2-3 times higher than vertically extrapolated Airy at z=+10m, and 50% - 70% higher at z=0m. The relative increase is particularly pronounced for Event2, which is a steeper wave event, while Event1 is the one with the highest loads peak. At the base level, the peak forces from the second-order are comparable to those with the vertically extrapolated Airy model. Similarly, results for the moments are given in Figs. 9-10. The effects from Figs. 7-8 are even more pronounced here, and peak moments around z=+10m and z=0m are 3-10 times and 2-3 times higher, respectively, than vertically extrapolated Airy. When the second-order kinematics model is used, the resulting integrated forces and moments are of an even higher nonlinear order, firstly due to the nonlinear drag formulation in eq. (7), but also due to the integration operations in eqs. (6-8). Therefore, as we can see from the plots, the forces and moment time series show clearly nonlinear behaviours when we compare to the use of the linear wave model, with asymmetry and higher peaks. Also the other three "traditional" methods show nonlinear features, especially the vertically extrapolated method, although to a less degree than for the second-order kineamtics. This also leads to more high-frequency contents in the time series for example, the peaks in the force and moment time series appear almost as impact loads of only about 2s-4s duration, depending on the vertical level. This may be important when seen in relation to actual natural frequencies of relevant structures to be applied for offshore wind turbines. For a complete analysis of the relevant physics in context of impactlike loads in steep energetic waves one should perhaps also include effects from the potential-flow ringing excitation phenomenon, which may be quite relevant for slender bodies 5 Copyright 2013 by ASME
Fig. 5. Vertical profile of horizontal particle velocity under crest of Event1. ("Airy" = "Vertically extrapolated Airy" up to z=0). Fig. 6. As Fig. 5, but Event 2. [13,14]. Another mechanism that can add to such effects is wave breaking. The sample maximum forces and moments from the total 4.5 hours simulation are summarized in Fig. 11. It turns out that these maxima are the same as those in Event1. NIRWANA also produces more complete post-processing for subsequent statistical analyses, while that is not addressed in the present paper. A simple analytical calculation has been made based directly on eqs. (6-7) to estimate peak shear forces at different numerical results in Figs. 7 and 9, see Fig. 12. We find that the numerical results are quite similar, but about 5%-7% higher. A plausible part reason for the slight discrepancy is the numerical spatial discretization of the structural model, with an element height of 1m in the upper part. As we can see from Fig. 12, 1m in height at the top corresponds to about 25kN. From an overall judgment we find that the structural model reproduces the quasi-static forces quite well. CONCLUDING REMARKS The influence from second-order effects in random wave kinematics on orbital velocities and resulting structural loads on a slender monopile in intermediate water depth have been investigated by a numerical case study. Thus comparisons have been made to results obtained by "traditional" methods based on Airy wave theory. A severe storm wave condition of Hs=15m, Tp=14s has been modeled. An in-house FEM program, NIRWANA, previously developed for studies of wave-induced structural loads and responses on bottom-fixed platforms, is applied for this study. The present simulations have been done with quasi-static functions only in the structural analysis (neither inertia nor damping forces included), and without wind and propeller, as an initial step in a more complete verification to be done with dynamic functions included. The goal is to apply it for complete studies on bottom-fixed offshore wind turbines in intermediate water depths. 6 Copyright 2013 by ASME
Fig. 7. Event1 time series samples of shear forces at z= +10m, 0m, -10m and -40m. Fig. 8. As Fig. 7, but Event2. 7 Copyright 2013 by ASME
Fig. 9. Event1 time series samples of moments at z= +10m, 0m, -10m and -40m. Fig. 10. As Fig. 7, but Event2. 8 Copyright 2013 by ASME
Fig. 11. Summary of maximum forces and moments. 1=second-order, 2=Airy, 3=Airy w/vertical extrapol., 4=Wheeler Fig. 12. Analytical shear force estimate vs. z, Event1, second-order model. Significant effects are observed from the second-order contributions in the wave zone. While extreme crests are around 15% -20% increased, free-surface extreme particle velocities increase by around 30% - 40%. The resulting peak shear forces at these levels, and in particular the moments, are thereby increased by typically 50% - 200% in the upper parts of the column, and in some cases even more. At levels below MWL the velocities are somewhat lower than for the Airy-based methods. At the base the peak shear forces are comparable to the traditional methods, while moments are still somewhat higher. We also see increased highfrequency load contributions in the loads, with periods around 2s-4s, which may be important to address further with respect to natural frequencies of such towers. The base moment results from the previous work in [1] showed qualitatively similar findings. Here we have primarily addressed, in a deterministic manner, two particular extreme events in a long-duration wave time history. A very slender structure (typically a truss member) was studied, and drag forces only were considered. Follow-up studies will also include more robust statistical analysis for design considerations, as well as inertia forces for completeness. The work has been done also in order to verify the implementation of the second-order wave load function in the NIRWANA structural analysis program. Thus the introduction of nonlinear wave effects in such analyses is a relatively novel technique, and for a robust industry use it is considered quite useful to demonstrate these functions as they are developed and 9 Copyright 2013 by ASME
implemented into different tools worldwide. Comparisons to analytic input load calculations confirm that this works fine in the present tool, within the accuracy that can be expected given the numerical discretization of the element model. [14] Stansberg, C.T. (1997), "Comparing Ringing Loads from Experiments with Cylinders of Different Diameters - An Empirical Study", Proc., BOSS '97, Delft, The Netherlands. ACKNOWLEDGMENTS This work has been carried out as a part of the Nowitech program on offshore wind turbine research. We also wish to thank our MARINTEK colleague Dr. Muk Cheng Ong for his helpful comments and suggestions in the work. REFERENCES [1] Agarwal P. and Manuel L. 2008, "Wave Models for Offshore Wind Turbines", 46 th AIAA Aerospace Sciences Meeting and Exhibit,, Reno, NV, USA. [2] Agarwal, P. and Manuel, L., 2011, "Incorporating Irregular Nonlinear Waves in Coupled Simulation and Reliability Studies of Offshore Wind Turbines", Applied Ocean Research, Vol. 33, pp. 215-227. [3] Stansberg, C.T., 2011, Characteristics of Steep Second- Order Random Waves in Finite and Shallow Water, Paper OMAE2011-50219, Proc., OMAE 2011, Rotterdam, The Netherlands. [4] Karunakaran, D., Haver, S., Bærheim, M. and Spidsøe, N., 2001: "Dynamic behaviour of the Kvitebjørn jacket in the North Sea", Proc. OMAE 2001., OMAE01/OFT-1184, Rio de Janairo, Brazil. [5] Baarholm, G.S., Haver, S., Økland, O.D., 2010, "Combining contours of significant wave height and peak period with platform response distributions for predicting design response". Marine Structures, 23:147-163. [6] Dean, R.G. and Dalrymple, R.A., 1991, Water Wave Mechanics for Engineers and Scientists, World Scientific, Singapore. (Chapter 4). [7] Wheeler, J.D.E., 1970, Method for Calculating Forces Produced by Irregular Waves, Journal of Petroleum Tech., Vol. 249, pp. 359-367. [8] Sharma, J. and Dean, R.G., 1981, Second-Order Directional Seas and Associated Wave Forces, J. Soc. of Petr. Eng., SPE,, pp. 129-140. [9] Marthinsen, T. and Winterstein, S., 1992, On the Skewness of Random Surface Waves, Proc., Vol.3, 2nd ISOPE Conf., San Francisco, Cal., USA, pp. 472-478. [10] Stansberg, C.T., Gudmestad, O.T. and Haver, S.K., 2008, Kinematics under Extreme Waves, ASME Journal of Offshore Mechanics and Offshore Mechanics, Vol. 130, Issue 2. [11] Morison, J.R., O'Brien, M.P., Johnson, J.W. and Schaaf, S.A., (1950), "The Force Exerted by Surface Waves on Piles", Petrol. Trans., AIME, Vol. 189. [12] NORSOK Standard, N-003, Edition 2, (2007) Actions and Action Effects, Norwegian Technology Standards Institution. [13] Faltinsen, O.M., Newman, J.N., and Vinje, T., 1995, Nonlinear Wave Loads on a Slender Vertical Cylinder, Journal of Fluid Mech., Vol. 289, pp. 179-198. 10 Copyright 2013 by ASME