Random Waves in the Laboratory – What is Expected for the Extremes? Carl Trygve Stansberg Norwegian Marine Technology Research Institute A/S (MARINTEK) P.O.Box 4125 Valentinlyst, N-7046 TRONDHEIM, Norway [email protected] Abstract. The generation and interpretation of extreme waves in physical model testing is discussed. A list of relevant wave parameters describing the extremes is outlined. A probabilistic approach is considered, with extremes occurring randomly in a wave train synthesized for the test. Statistical reference models based on linear and second-order wave theory are applied. Comparisons to model test results show that the second-order model predicts reasonably well in most cases, although with a slight under-prediction of steep extremes, possibly due to unidirectional wave conditions in the laboratory. Under particular conditions, with narrow-banded unidirectional spectra propagating more than 12 – 15 wavelengths, special modulation effects may occur in energetic wave groups, leading to very high extremes that are clearly beyond second order. This may one possible explanation of “freak waves” observed in the real ocean. The effect is reflected in th 4th order statistical moment (kurtosis), and a prediction formula taking this into account is suggested. 1 Introduction The prediction and reproduction of extreme ocean waves is a complex task, since they are rare events, and therefore hard to observe in the real ocean. Trying to understand all the underlying mechanisms, and the resulting physics, can be confusing, since there may be a number of various conditions leading to the different events actually observed. Ideally, perfect theoretical and physical models should therefore be able to cover a broad range of situations. A discussion of the occurrence and prediction of extreme waves has been given in [1]. Fully nonlinear theoretical models for random extreme waves do still not exist, although there are several theoretical approaches that include essential linear and nonlinear components and properties. Thus the challenge in present day-to-day applications is to sort out which are the most relevant properties to be modelled, and how to model them. This may vary from application to application, but there are also general patterns. In the present paper, the generation and interpretation of wave extremes in physical model testing is discussed. The laboratory generation of waves has been reviewed in [2]. There are still a number of questions to be handled in connection with reproduction and use of extreme wave generation. One of them is: What should we expect – or, in other words, what is our reference? This question may be two-fold: 1) What is required from the application?, and 2) what is actually possible, given the laboratory frame? And furthermore, can we learn something about the wave physics itself from the experiment? Some key words in this process are: Parameters selected for reproduction Input from full scale or theory Methodology (Stochastic vs. deterministic approach; Synthesisation etc.) Basic physics vs. laboratory effects Simplifications Some practical examples from the experience in an offshore model test basin are discussed in the paper, on basis of previous presentations in [3], [4]. Here a stochastic approach is followed, with the synthesisation and physical generation of random storm records (typically of 3-hours duration, full scale). Thus, the extremes occur as random events in the scaled wave field, as the result of the random summation of a large number (thousands) of independent input components. Nonlinear effects observed in the records are then mainly interpreted as results from nonlinear couplings in the actual propagation of the laboratory wave field, although one has to be aware of possible laboratory defined effects. Another approach which has been suggested and applied in the literature, is the design and use of single deterministic, transient wave groups specified with particular extreme value properties [5]. The two different approaches may in certain situations be considered as alternatives to each other, but it is perhaps more fruitful to treat them as complementary, since they are based on quite different background philosophies. The present experimental results are seen in relation to linear and second-order random wave prediction models, with a particular discussion of deviations from the models. Thus one possible way of defining “freak waves” may be considered as waves and crests clearly higher than second-order. A possible connection between such extremes and nonlinear wave grouping is considered. 2 Background: Critical Wave Events and Parameters 2.1 Some Critical Wave Situations in Offshore Engineering The design and operation of FPSO’s in extreme weather exposed areas must take into account the effects from steep and energetic individual wave events. The wave impact on bow and deck structures can be serious, such as the bow slam experienced on the Schiehallion FPSO [6], as well as the water on deck problems reported on Norwegian production vessels [7]. New design tools are being developed as a result of this [8]. The impact loads and possible damages are certainly a combined effect from the wave properties and the interaction with the vessel, but knowledge about the incoming energetic waves is very helpful in the further development in the area. For floating platforms, such as semisubmersibles, TLPs and Spars, the deck clearance (air-gap) is critical. Thus the ability to properly predict the extreme wave crests and their kinematics, in 100-year storms is essential, not only for the prediction of the probability of impact, but also for the prediction of resulting loads. Other direct results from extreme waves interacting with platforms include the ringing problem on TLP’s and GBS’s (see e.g. [9], as well as the possible capsizing of platforms with compliant mooring. Extreme waves or wave groups can also induce particular vessel motions, as a result of particularly large slow-drift forces. For FPSO’s, this may lead to large head angles and, consequently, even larger offset and high nonlinear mooring line loads (static as well as dynamic). Large slow-drift is critical also for the extreme loads of platform moorings. 2.2 Critical Wave Parameters The detailed description of dangerous waves is complex, since the different problems such as described above may depend on different wave properties. A list of possible parameters or characteristics may be as follows: Individual waves Crest height: Wave height: Steepness: Particle velocity: Particle acceleration: Grouping (succeeding waves); Energy envelope: Breaking A max or Hmax or (∂η/∂x) max or Umax (dU/dt)max A max/σ Hmax/σ (kA)max E(t); Group spectrum – relative to linear model Short-term sea state properties Skewness: γ1 = 1/(M σ3 ) ⋅ Σ [ηi -E(η)]3 Kurtosis (grouping parameter): γ2 = 1/(M σ4 ) ⋅ Σ [ηi -E(η)]4 Probability of given extreme levels (1) (2) where η is the elevation, σ is the standard deviation of the elevation record, and M is the number of record samples. In addition, there may also be other parameters relevant for particular problems. 2.3 Extreme Waves: Possible Physical Mechanisms For a proper prediction of extreme and rare waves, it is also important to keep in mind that there may be a range of different physical mechanisms leading to the events. Some of these are: Phase combination of harmonic components Steepness-induced crest increase (“Stokes effects”) Nonlinear self-focusing of energetic wave groups Multi-directional effects Bottom effects (finite water depth; refraction) Current effects (wave-current interaction: refraction) Wind influence Storm age and duration Several storm systems? Thus the description of real cases may be complex. In offshore engineering applications, the first two mechanisms listed are perhaps those with most attention. It may be a reasonable choice to consider them as basic conditions, and then add the effects from the others when appropriate. Later in this paper, effects from nonlinear wave grouping are discussed in particular, on basis of some laboratory results. 3 Specification and Limitations of Laboratory Waves The reproduction of wave conditions in a laboratory must be based upon a chosen specification, which can be essential for the generated extremes. Thus the reproduced conditions will be simplified with regard to some properties, while others are emphasised. Parameters of a specification may include some of the following: In a probabilistic approach: Significant wave height Hs ; Hmo Spectral peak period (o r equivalent) Tp ; T z Spectrum shape (e.g. JONSWAP, 2-peaked etc.) Storm duration Additional requirements? (H max ; A max ; γ1 ; γ2 ; wave grouping; others ?) In a deterministic transient wave approach: Specific properties of single wave (or wave group) Some laboratory simplifications may typically (but not necessarily) be: Uni-directional waves Horizontal bottom or deep water Stationary sea state Mechanical wave generator - no wind influence If “transient wave” : Specific parameters of event Specific laboratory-defined effects: Reflections & diffraction “Parasitic waves” due to imperfect boundary conditions Size & shape of basin / distance from wave generator Repeatability Scale effects (viscous effects; breaking) Synthesization method 4 Proba bilistic Modelling of Linear and Nonlinear Waves Based on the specification, the synthesisation of a random laboratory signal input to the wave-maker is typically made as a linear sum of a large number of independent harmonic components. Nonlinear corrections may also be made [10]. As an example, a 3-hours storm duration may be simulated by inverse Fast Fourier Transform (FFT) with 16000 frequency components. Extreme waves are then a result of this random combination, plus nonlinear interactions during the propagation from the wavemaker to the actual location. The statistical behaviour is observed through parameters like the skewness γ1 and kurtosis γ2 of the wave record, and probability distributions and extremes of the crests and wave heights. The results can then be compared to reference models. In particular there are two models in use: Linear waves, with Gaussian statistics and Rayleigh distributed peaks, and second-order waves [11], [12], with a non-Gaussian correction on the statistics, and with extreme crests deviating from the Rayleigh model. The effect from secondorder contributions on an extreme wave is shown in a numerical example in Fig. 1. One should also take into account the sampling scatter of a finite record [12]. The estimation of extremes from a given 3-hours record can be improved by use of e.g. fitting the tail of the peak distribution to a Weibull distribution, and predict the extreme from that. Fig. 1. Time series sample from numerically generated second-order random wave Based on the reference models, we can derive expectations for the measured statistics, and the extremes in particular. For a simple linear model, the expected skewness and kurtosis are γ1 = 0; γ2 = 3.0, respectively, and extreme crests and wave heights are expected to be Rayleigh distributed with the following commonly used relations: E[A max] ≡ A R = σ [√ (2 ln (M)) + 0.577/√ (2 ln (M))] E[Hmax] = 2 A R (3) (4) In a second-order model, the skewness γ1 increases linearly with the steepness. Models for γ1 and γ2 have been derived in [13]: γ1 = 5.41 (Hm0 / L p ) γ2 – 3 = 3 γ1 2 (5) (6) where L p is the wavelength corresponding to the peak wave period. For extreme crests a simplified formula has been proposed by [14]: E[A max] = A R (1 + ½ kp A R ) (7) where kp is the wave number corresponding to L p , and A R is given in Eq. (3) above. The wave heights are still Rayleigh distributed as in the linear model. The experience from [11] is that the second-order model generally agrees quite well with deep water full scale measurements of crests. Thus the linear model will underpredict extreme crests but not the wave heights. Laboratory measurements in [3] more or less confirm this (see the next chapter), but a slight under-prediction is observed. In special conditions, even higher extremes have been observed [4]. Higherorder models can also predict this [15]. Such events may possibly be seen in relation to full-scale observations of so-called “freak waves”, and will be discussed later in this paper. 5 Observed Nonlinear Behaviour of Random Extremes Observations from a range of model test studies in scales 1:55 – 1:70, with random wave generation in a large wave basin [3], have shown that the largest crest heights deviate systematically from Rayleigh model predictions derived for linear waves. In general, a second-order description fits reasonably well, as concluded from the fullscale study in [11], although it slightly under-predicts the most extreme cases, typically by 5% of the total crests. See Fig. 2. The deviation may be partly due to the fact that most of the results in this figure were obtained with unidirectional waves, while field data are expected to be more or less muliti-directional. There is a also a considerable sampling scatter, as expected from theory. Extreme peak -to-peak wave heights are normally reasonably well predicted by Rayleigh theory. The same results are also reflected in probability distributions (Fig. 3). Fig.2. Measured extreme crests from laboratory tests, compared to second-order and Rayleigh predictions. 3-hours as well as 12 – 18-hours storm duration models (from Stansberg, 2000a) Fig.3a. Probability distributions of crests in 1:55 scaled 18-hours storm model test Fig. 3b. As Fig. 3a, but for wave heights. Under certain conditions, extremes in random wave trains may be observed to be significantly higher than second-order predictions, even for moderately steep wave conditions [4]. This occurs when unidirectional, narrow-banded spectra propagate over large distances, that is, more than about 12-15 wavelengths, in which case higher-order wave group amplification may take place, leading to particularly high crests and wave heights. This is illustrated by an example from a 1:200 scaled laboratory experiment shown in Fig.4. We may interpret it as a so-called “freak wave” event. However, the results in [4] also show that it can be a result of systematic behaviour under these particular conditions. A reasonable physical explanation is the self-focusing casued by amplitude dispersion in energetic wave groups, which can be related to the modulational instabilities commonly referred to as the Benjamin -Feir effect [16]. The physics is studied experimentally in [17]. Results from tests with different scales indicate that the phenomenon is not scale dependent. For bi-chromatic wave trains, observations have been found to agree very well with a higher-order Schrödinger formulation [18]. Probability crest and height distributions from a case where the effect is particularly strong are shown in Fig 5. The difference from Fig. 3 above is clearly seen, also for the peak-to-peak wave heights. The 4th -order statistical moment parameter γ2 (kurtosis) reflects, on an average, the increased groupiness, although it is also statistically unstable [12]. An empirical relation has been derived on basis of the experimental data above, with very Fig. 4. Space and time evolution of energetic wave group into extreme wave – example from model tests. (D = distance from wave-maker, in wavelengths)) long records corresponding to 12, 15, 18 and 36 hour storm models. Thus the kurtosis has been correlated with the corresponding 3-hours extreme crest and wave height estimates Amax , Hmax. The result is shown in Fig. 6, where deviations from the second-order and Rayleigh models (for Amax and Hmax , respectively) are plotted against the kurtosis. The values are normalised by the standard deviation σ of the record. From this, the following simplified formulae are proposed for extreme crests and wave heights, taking into account the second-order term for A max [16] as well as an empirical higher order correction for A max and for Hmax: A max / σ = (A max,R / σ)⋅ (1 + ½ kp A max,R ) + 1.3⋅ (γ2 - 3.0) (8) Hmax /2σ = (Hmax,R /2σ) + (γ2 - 3.25) (9) We sees that for the extreme wave heights, the Rayleigh model overpredicts the measurements when the kurtosis approaches 3.0 (that is, Gaussian waves). This is an expected result in linear waves, due to the de-correlation between crests and neighbouring troughs in a finite-bandwidth spectrum. Fig. 5. Probability distributions of crests and wave heights, after 25 wavelengths Fig. 6. Measured extreme crests and wave heights in 12 – 36 hours storm tests: Deviations from second-order and Rayleigh models, respectively, vs. kurtosis Another, more general alternative to this empirical formula is the Hermite transformation method in [19]), where the extremes are estimated directly on basis of the statistical moments γ1 and γ2 . The kurtosis γ2 will have to be determined for the actual case. Thus there is a task for the future: How do we know when to assume γ2 clearly larger than 3.0 , and how do we predict it? The typical time domain behaviour of the most extreme (“freak”) events is shown in Fig. 7. It normally results from an energetic wave group of 4-6 waves, which after some propagation is focused into a narrower group of of 1-2 waves. Most of the energy is then concentrated in the front wave. Until this point, only little energy has been dissipated from the original wave group. Thus the occurrence of such “freak” events may possibly be caused by the time and space development of wave groups with a duration sufficient to contain a large amount of integrated energy. 6 Conclusions Probabilistic modelling of storm sea states in a laboratory wave basin, with particular emphasis on the resulting extreme wave events, has been discussed and demonstrated. The results are seen in light of what is expected from linear and second Fig. 7. Particularly extreme wave event. order models. Main findings are: An empirically adjusted formula for extremes is suggested. For the crests, this is based on a second-order model plus an empirical correction for kurtosis values larger than 3.0. For the wave heights, a Rayleigh model with a similar kurtosis correction is proposed. The kurtosis is closely connected with the average “groupiness” of the sea state. 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