The Second Order Contribution to Wave Crest Amplitude Random Simulations and NewWave

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1 The Secon Orer Contribution to Wave Crest Amplitue Ranom Simulations an NewWave Thomas A.A. Acock Department of Engineering Science University of Oxfor Parks Roa Oxfor Unite Kingom Scott Draper School of Civil Environmental an Mining Engineering an Centre for Offshore Founation Systems University of Western Australia 5 Stirling Highway Perth Australia ABSTRACT Estimating the probability of a wave crest exceeing a given threshol is of funamental importance in offshore engineering. At present secon-orer theory is use with the Forristall istribution typically being use for this calculation. In this paper we investigate several points in connection with this approach. Firstly we examine some of the sensitivities involve in eriving secon orer statistics incluing the high frequency cut-off the irectional spreaing an spectral banwith. The paper then examines how the secon orer contribution varies for waves of ifferent shapes. We investigate whether the NewWave can be use to preict the secon orer contribution to a wave-crest but fin that the variation in wave shape is too great for this to be practical. KEY WORDS: Secon orer wave crest; Forristall istribution; Ranom ocean waves; NewWave INTRODUCTION One of the key problems in offshore engineering is calculating the short-term probability of a wave-crest of given magnitue occurring at a point. A common use for this is in air-gap calculations for offshore platforms. The air-gap problem is complicate an this paper only consiers one part of the calculation: the secon-orer contribution to the wave-crest height. There is an ongoing ispute as to whether physics other than that given by secon-orer theory is neee for escribing the evolution of nonbreaking wave-crests over a timescale in which a large wave-group forms an then isperses. Waves for which aitional physics are important are terme rogue waves (see reviews by Kharif & Pelinovsky (); Dysthe et al. (9); Acock & Taylor (14)). Current esign practice assumes that physics beyon secon orer is not significant for practical offshore calculations an assumption which appears to be generally consistent with fiel measurements. The stanar statistical moel that is use to escribe the short-term istribution of wave-crests is the Forristall istribution (Forristall ). This istribution was erive by fitting a probability istribution to the results of numerical simulations of ranom waves using seconorer theory. Forristall erive two istributions: a D an a D. The D is only applicable to laboratory stuies. The D istribution was base on simulations of JONSWAP spectra with a number of ifferent peak enhancement factors an with a constant frequency epenent irectional spreaing function. The Forristall istribution is straightforwar to apply to practical engineering calculations an seems to agree satisfactorily with most measurements an laboratory tests. The istribution is known to prouce inaccurate results in some situations. For instance if there is significant wave-breaking then the istribution will over-preict the magnitue of wave-crests. Conversely in crossing seas the set-own uner a large wave preicte by secon-orer theory can become a set-up (Christou et al. 9) although the changes to the magnitue of the secon orer sum term mean that this oes not always increase the overall crest height. These variations are not capture in the Forristall istribution as this only consiers one form of irectional spreaing. This paper examines some etails an sensitivities of the calculation of secon orer wave crest statistics. We also examine whether it is possible to avoi the computational emans of simulating ranom waves by using the secon orer structure from a NewWave wavegroup to calculate the secon orer correction. Secon orer ranom simulations are also use in other applications such as calculating velocities an loas (Alberello et al. 14). THEORY Secon orer correction There is no change to the ispersion equation that governs the propagation of wave components in secon orer theory. Thus the free surface can be escribe by a number of free or linear waves which propagate as escribe by the linear ispersion equation an boun waves which are a correction to the free surface profile an which are a function only of the linear wave components. This paper uses stanar secon orer theory. For ranom irectional linear waves this theory was first given in Dean & Sharma (1981) although unfortunately this contains a number of typos. The results

2 presente in Forristall () an Dalzell (1999) are consistent (although formulate ifferently). As highlighte in Dalzell (1999) there is an inconsistency when a linear component interacts with itself in finite water epth an a further assumption has to be mae. For ranom simulations this is generally not an issue as the self-interaction term is small compare to the interactions of ifferent waves. In this paper we have set the value of the interaction kernel to zero for such interactions. Following Dalzell we present the secon orer contribution for the interaction of two waves (which is easy to generalise to N interacting waves). Let us assume that the linear wave components have amplitue a j an phase φ j which is given by the components natural frequency multiplie by time plus some arbitrary phase shift. The linear waves are then given by η linear = a j (1) It is convenient to split the secon orer contribution into sum an ifference terms. These have frequencies that are the sum an ifference of the interacting linear components. The sum terms are given by η + = j=1 cosϕ j j=1 j a i a j B p cos( ϕ i +ϕ j ) () where B p is a given by a complicate expression an is a function of the interacting frequencies an the water epth (Dalzell 1999) given in the appenix. Similarly the ifference terms are given by η = a i a j cos(ϕ i ϕ j ) where again the interaction kernel is given in the appenix with the reaer also referre to Dalzell (1999) an Forristall (). We emonstrate the form of the secon orer correction to the free surface by plotting the results of fully non-linear simulations carrie out by Gibbs & Taylor (5). The simulations are of a focuse NewWave wave-group with an amplitue ak =.1 an a wrappe normal irectional spreaing of 15 starting perios before linear focus. The wave-group is split into linear an secon orer constituents using the metho escribe in Taylor et al. (5) an by stanar spectral filtering. Figure 1 shows the linear an secon orer waves at linear focus emonstrating the secon orer sum terms being in phase with a large crest whilst the ifference terms are out of phase. The sum an ifference terms are in excellent agreement with those preicte by secon orer theory (Acock 9). Ranom waves The funamental assumption unerlying much offshore wave theory is that the linear waves at ifferent frequencies are uncorrelate with each other an that the free surface is a ranom Gaussian process. For a given spectrum simulating ranom waves can be one numerically. In this paper we follow the general approach escribe in Tucker et al. (1984). Once ranom linear waves have been generate these can be written as given in equation 1 an the secon orer correction to the free surface may be calculate from equations () an () for a given irectional istribution of energy. () (m) + (m) linear (m) 4 Linear waves Time (s) Secon orer sum waves Time (s) Secon orer ifference waves Time (s) Figure 1 Example of secon orer contribution to the free surface from fully non-linear potential flow simulations base on a NewWave wavegroup. In calculating a secon orer ranom time series it is necessary to curtail the calculation of the interactions at some high frequency cutoff. For instance in his seminal paper Forristall () states: The calculations escribe in this paper were typically truncate [at] four or five times the peak frequency. One reason for this truncation is computational. Calculating a irectionally sprea secon orer sea is computer intensive an for practical purposes many simulations may be require in orer to erive robust statistics. However there is a more significant reason why this is an issue. To emonstrate let us consier the interaction between two components with angular frequency ω! an ω!. In this example we consier waves travelling in the same irection but the general result is the same for waves travelling in ifferent irections. Taking ω! =.5 ra s -1 the magnitue of the interactions B! an B! for ifferent values of ω! are plotte in Figure for eep water (actually m) an intermeiate to shallow water ( m). Both interaction terms increase in magnitue as the ifference in frequency increases. The value of B! increases at the square of the ifference in frequencies whilst the increase in B! is more complicate but similar in leaing orer term. For a JONSWAP spectrum the high frequency tail of the amplitue ecays with ω -. Thus the ecay in the magnitue of the secon orer terms is very slow. SENSITIVITY TO HIGH FREQUENCY CUT-OFF

3 Figure Different components of a simulate secon orer spectrum. Linear black; secon orer ifference magenta; secon orer sum green. Interaction coefficient B p (=) B P (=) (=) (=) / 1 Figure Magnitue of secon orer interaction kernel for co-linear waves The issue can be seen by examining the spectra of a ranom wave simulation. A JONSWAP spectrum is use with γ =. with a low frequency cut-off at.5ω! an a high frequency cut-off at 4ω!. The irectional spreaing of the sea-state is given by the Ewans (1998) istribution. Figure shows the omniirectional spectra of the ifferent constituents of the secon-orer sea state. The secon orer ifference spectra shows a slow ecrease in amplitue between zero frequency an the artificial cut-off cause by curtailing the linear spectrum at ω! - 4ω! an ecays to zero at ω! - 4ω!. Similarly the secon orer sum term increases between.5ω! an ω!. It then ecays slowly until the point at which the linear peak is no longer taking part in the interactions. The key result is that prior to the change-point cause by curtailing the linear spectrum the ecay of both secon orer sum an secon orer ifference terms is slow. Power (m s) ω/ω p This behavior is not physically correct. This is a known problem perhaps first escribe in general terms by Barik & Webber (1977). The physics of the anomaly is resolve in Janssen (9) where it is shown that linear an thir orer terms will interact to cancel much of the upper tail of the spectrum. There remains the problem of where to curtail the spectrum for practical engineering calculations. In practice the calculation of extreme wave-crests in ranom simulations is not overly sensitive to the precise cut-off point. The reason for this is that a crest will ten to occur where the highest energy parts aroun the peak of the spectrum are in phase with the low energy components at higher frequencies less likely to be in phase at the crest. This means that the secon orer terms ue to interactions between components in the peak (that are correctly given by secon orer theory) will generally be correlate giving the require secon orer correction. However the interactions between the peak an the tail of the spectrum are likely to not have a phase that leas to them being correlate with the crest an so these introuce relatively small errors into the estimate of the secon orer correction at the extreme crest. In this paper we use.5ω! as a suitable cut-off. It shoul be note however that for high-cycle fatigue calculations on objects near the surface the cut-off point will be more significant. INFLUENCE OF SPECTRAL SHAPE ON SECOND ORDER CONTRIBUTION Directional spreaing The magnitue of the secon orer contribution will be epenent on the spectral shape. Whilst it woul be possible to investigate this by running ranom simulations it is easier to analyse the problem using eterministic wave-groups. In this stuy we use NewWave wave groups. These are the expecte shape of a large wave-group in a linear ranom sea-state. A NewWave in the time omain focusing at t = is given by equation (4) an an example is shown in Figure 1. The erivation is given in Lingren (197) an it was brought into offshore engineering practice by Tromans et al. (1991). It has been shown to agree with fiel measurements in numerous stuies (e.g. Jonathan & Taylor 1997) an to be vali up until the point where waves are ominate by shallow water wave breaking (Haniffah 1). It may be note that a spectrum erive from the time history of a NewWave ecays rather quicker than that of the spectrum S(ω) which was use to erive the NewWave. Thus there is no problem with high frequency cut-off location η(t) = n S( ω n )cos ω n t n S( ω n ) ( ) Except for in very shallow water all sea states are irectionally sprea. The egree of irectional spreaing will vary epening on the type of storm. In realistic sea-states irectional spreaing is strongly epenent on frequency but in this investigation we assume a constant irectional spreaing across all frequencies. We use a wrappe normal irectional spreaing an calculate the magnitue of the secon orer contribution uner a large crest for ifferent withs of irectional spreaing. In this we use a constant omni-irectional frequency spectrum given by a JONSWAP spectrum with γ =. water epth 4 m an a unit NewWave (i.e. a crest amplitue of 1m). (4)

4 Figure 4 shows the magnitue of the sum an ifference contributions uner the crest. The magnitue of both terms reuces as irectional spreaing is increase. It is noticeable the magnitue of the secon orer ifference term although smaller ecreases by a relatively greater amount which is one reason why Acock & Taylor (9) use this to estimate irectional spreaing from a measure Eulerian timeseries. What is remarkable is that the changes almost cancel out so that the combine extra elevation is insensitive to the egree of irectional spreaing. This result is consistent across ifferent water epths until the water is sufficiently shallow that the whole approach in this paper breaks own. shows plots of the D free surface aroun two large waves with linear crest amplitues greater than significant wave height H! generate ranomly from the same unerlying spectrum. The ifference in the general shape an particularly the broaness of the crest is marke Sum term Difference term Combine Figure 4 Secon orer contribution to wave crest height normalize by amplitue uner a NewWave group as a function of irectional sprea σ in egrees. A more realistic spreaing function is that given by Ewans (1998). If this is use then the secon-orer sum contribution is.44 an the ifference -.61 giving a combine secon-orer contribution of.184. This is broaly consistent with the frequency inepenent spreaing estimates. We conclue that for following sea-states (i.e. ones where the waves are going in a narrowly confine irection) an accurate knowlege of spreaing is not crucial in etermining secon-orer amplitues implying that the Forristall istribution can be applie. The relative inepenence from irectional spreaing also suggests that any nonlinear reuction in spreaing uner a large wave (Gibbs & Taylor 5; Acock et al. 1) can be ignore when calculating secon orer statistics. Of course as note in the introuction in crossing sea states there can be a significant ifference in the secon orer contribution to crest amplitue. Spectral banwith The above simulations consiere a constant spectral with escribe by the JONSWAP spectral enhancement factor γ. Similar simulations of secon-orer NewWaves were carrie out but showe that secon orer sum an secon orer ifference contributions change negligibly for values of γ between 1 an 5. VARIATION OF SECOND ORDER CONTRIBUTION IN RANDOM SIMULATIONS In a real sea-state large waves that are of practical interest to engineers show a significant variation in shape although the average profile is given by the theoretical NewWave form (Tucker 1999). Figure 5 Figure 5 Examples of two large waves rawn from numerical simulations of ientical sea-states with ifferent shapes. Vertical axis free surface elevation normalize by H s. Mean wave irection in x irection The ifference in the shape of the wave will lea to variations in the magnitue of the secon orer contribution to the crest elevation. It is not immeiately obvious to the Authors whether the variation in the shape of an extreme wave event will lea to a small or large change in the secon orer contribution. If the magnitue of the secon orer contribution were strongly epenent only on the amplitue of the wave then knowlege of the linear wave amplitue statistics woul immeiately allow one to etermine secon orer statistics greatly simplifying esign. We investigate how secon orer contributions vary by running 1 simulations of three hours of waves an recoring the secon orer contribution to the largest (linear) wave observe in each simulation. We have use a JONSWAP spectrum with γ =. an a Ewans spreaing function. The magnitue of the linear component the secon orer sum an secon orer ifference components were recore. Figure 6 plots the magnitue of the secon orer components against the linear crest amplitue. If we consier a given linear amplitue of wave we can see that there is a significant variation in both secon orer terms. Despite the large waves consiere in these simulations there is no obvious sign that the absolute magnitue of the variation in the size of the secon orer terms reuces for larger waves as woul be the case if the largest waves were all close to the NewWave in form. It can also be observe that the total secon orer term (sum + ifference) shows slightly less variation than the sum term alone suggesting that to some extent a larger sum term than the mean is correlate with a larger (negative) ifference term than the mean; hence some of the variation cancels out.

5 .5. + a /H s a/hs this to calculate the secon orer correction. This is straightforwar computationally quick an as note above oes not suffer from ambiguity in the high frequency cut-off. It woul also be straightforwar to exten it to thir or higher orer which woul be computationally ifficult for ranom wave simulations. Figure 6 shows the sum ifference an combine amplitue preicte by the NewWave approach. The approach shows goo general agreement with the ranom wave simulations although it appears to slightly unerestimate the mean size of the secon orer contribution. However this approach obviously oes not capture the variation in wave shapes in a realistic sea state. In estimating the probability of an extreme crest it is vital to capture this variation an so the NewWave approach cannot be use for this purpose. 4 x 1 VARIATION IN SECOND ORDER CONTRIBUTION IN FIELD DATA a /Hs a/hs a /H s. Examining the variation in secon orer contribution in fiel ata is problematic as sea-states o not remain stationary long enough to observe sufficient extreme waves to buil up a robust picture of what is happening. In examining fiel ata it is consierably easier to separate the low frequency ifference terms compare to the high frequency sum terms as there are generally few other waves present in the low frequency part of the spectrum. Despite the lack of ata analysis of the low frequency waves present in North Sea ata (Acock & Taylor 9) an hurricane ata (Santo et al 1) oes inicate that large waves of similar magnitue occurring a short time from each other will have very ifferent secon orer contributions..15 CONCLUSIONS a/h s Figure 6 Magnitue of secon orer contribution to wave crest amplitue from ranom wave simulations (ots) an NewWave estimation (continuous line) as a function of linear crest amplitue. Top secon orer sum; mile secon orer ifference; bottom combine (sum + ifference) It is also noticeable than espite this simulation being in a following sea a few large waves have a positive secon orer ifference term (i.e. they have a set-up uner the crest). This occurs when much of the highly irectionally sprea high frequency tail is in phase with the wave crest. There are however only a hanful of set-ups that are of equal magnitue to the expecte set-own as was observe at the famous Draupner wave recore in the North Sea (Walker et al. 5; Acock et al. 11). This paper investigates the contribution secon orer boun waves make to the amplitue of the crests of large ocean waves. We note than when computing these the point chosen for the high-frequency cut-off can influence the results ue to the limitations of secon orer theory. The extra amplitue uner a wave crest is remarkably inepenent of irectional spreaing as the changes in sum an ifference contributions to the amplitue roughly cancel out. The magnitue of the secon orer contribution is strongly epenent on the local timehistory of the wave an not just on linear crest amplitue. This leas to consierable variation in the magnitue of the secon orer contribution across waves of similar amplitue. This has implication both for eriving crest statistics an for analysis of fiel measurements. Calculating the magnitue of the secon orer contribution using NewWave gives a reasonable first estimate but oes not capture the important variations in secon orer contribution ue to the ranom nature of waves. ACKNOWLEDGEMENTS USING NEWWAVE TO ESTIMATE THE SECOND ORDER CORRECTION SD woul like to acknowlege the support of the Lloy s Register Founation. Lloy s Register Founation invests in science engineering an technology for public benefit worlwie. A number of Authors have at various times trie to estimate the secon orer correction from the local properties of the waves without carrying out full secon orer simulations (e.g. Kriebel & Dawson 1991). These have typically assume that the linear waves are a slowly moulate sine wave an have calculate a local secon orer correction by assuming that locally the linear wave is a Stokes wave. This obviously oes not account for the secon orer ifference wave or for finite irectional spreaing. To account for these we coul instea assume that the free surface aroun an extreme event is given by NewWave an use REFERENCES Acock TAA (9). Aspects of wave ynamics an statistics on the open ocean DPhil thesis University of Oxfor Trinity Term. Acock TAA Gibbs RG an Taylor PH (1). The non-linear evolution an approximate scaling of irectionally sprea wave-groups on eep water Proceeings of Royal Society A 9: Acock TAA an Taylor PH (9). Estimating ocean wave

6 irectional spreaing from an Eulerian surface elevation time-history Proceeings of Royal Society A 465: Acock TAA an Taylor PH (14). The physics of anomalous ("rogue") ocean waves Reports on Progress in Physics Acock TAA Taylor PH Yan S Ma QW an Janssen PAEM (11). Di the Draupner wave occur in a crossing sea? Proceeings of Royal Society A 467:4-1. Alberello A Chabchoub A Gramsta O Babanin A an Toffoli A (14). Statistics of Wave Orbital Velocities in Ranom Directional Sea States 19th Australasian Flui Mechanics Conference Melbourne Australia 8-11 December Barrick DE an Weber BL (1977). On the nonlinear theory for gravity waves on the ocean s surface. Part. Interpretation an applications Journal of Physical Oceanography 7:11 1. Dalzell JF (1999). A note on finite epth secon-orer wave wave interactions Applie Ocean Research 1(): Dysthe K Krogsta HE an Müller P (8). "Oceanic rogue waves" Annual Review of Flui Mechanics 4:87-1. Ewans KC (1998). Observations of the irectional spectrum of fetchlimite waves Journal of Physical Oceanography 8 (): Forristall GZ (). "Wave crest istributions: Observations an secon-orer theory" Journal of Physical Oceanography (8): Gibbs RG an Taylor PH (5). "Formation of walls of water in fully nonlinear simulations" Applie Ocean Research 7(): Haniffah MR (1) Wave Evolution on Gentle Slopes - Statistical Analysis an Green-Naghi Moelling DPhil thesis University of Oxfor Michaelmas Term. Janssen PAEM (9). "On some consequences of the canonical transformation in the Hamiltonian theory of water waves" Journal of Flui Mechanics 67:1-44. Jonathan P an Taylor PH. (1997). "On irregular nonlinear waves in a sprea sea" Journal of Offshore Mechanics an Arctic Engineering 119(1):7-41. Kharif C an Pelinovsky E (). "Physical mechanisms of the rogue wave phenomenon" European Journal of Mechanics-B/Fluis.6:6-64. Kriebel DL an Dawson TH (1991) Nonlinear effects on wave groups in ranom seas Journal of Offshore Mechanics an Arctic Engineering 11: Lingren G. (197). Some properties of a normal process near a local maximum Ann. Math. Statist. 41: Santo H Taylor PH Eatock Taylor R an Choo YS (1). Average Properties of the Largest Waves in Hurricane Camille Journal of Offshore Mechanics an Arctic Engineering 15(1):116. Sharma JN an Dean RG (1981). Secon-orer irectional seas an associate wave forces Society of Petroleum Engineering Journal 1: Taylor PH Walker DAG Eatock Taylor R an Hunt AC (5). On the estimation of irectional spreaing from a single wave staff Ocean Waves Measurement an Analysis 5th Int. Symposium Waves Mari. Tromans PS. Anaturk AR an Hagemeijer P (1991). "A new moel for the kinematics of large ocean waves-application as a esign wave." The First International Offshore an Polar Engineering Conference. International Society of Offshore an Polar Engineers. Tucker MJ (1999). The Shape Perio an Wavelength of High Storm Waves Unerwater Technology (4): Tucker MJ Challenor PG an Carter DJT (1984). "Numerical simulation of a ranom sea: a common error an its effect upon wave group statistics." Applie Ocean Research 6(): Walker DAG Taylor PH an Eatock Taylor R (5). "The shape of large surface waves on the open sea an the Draupner New Year wave" Applie Ocean Research 6(): 7-8. APPENDIX This interaction kernels escribe the magnitue of the secon orer ifference interaction between two interacting components in equations an. The frequency an wavenumber of the linear components ω an k an relate by the linear ispersion relationship. The angle between components is θ an the water epth is. The plus an minus terms B p an respectively are B p = α β pγ p δ p an = α + β mγ m where α = ω 1 g δ m +ω + ε pζ p δ p + ε mζ m δ m β p = ω " 1ω g 1 cos( θ) % ' $ # tanh( k 1 )tanh( k ) & γ p +ω ) + g k 1 + k + k δ p +ω ) g k 1 + k + k ε p = ω 1 +ω g ζ p = sinh ω 1 ( k 1 ) + ω sinh k ' β m = ω! 1ω g 1+ cos( θ) $ & # " tanh( k 1 )tanh( k ) % γ m ω ) + g k 1 k k δ m ω ) g k 1 k k ε m = ω 1 ω g ζ m = sinh ω 1 ( k 1 ) ω sinh k ( ). &