From deterministic regular waves to a random field. From a determinstic regular wave to a deterministic irregular solution
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1 Classiicatio: Iteral Status: Drat z ξ(x,y, w& x w u u& h Particle ositio From determiistic regular waves to a radom ield Sverre Haver, StatoilHydro, Jauary 8 From a determistic regular wave to a determiistic irregular solutio Solutio o liearized goverig equatios (i.e. boudary equatios are alied at the mea ree surace ad terms o order higher tha are eglected: ω ξ ( x, φ si( ωt kx ξ si( ωt kx g Real ocea waves excet swell do ot look like this. They look less regular! Let us assume that the sea surace reeat ater T ad use Fourier. The we ca write the observed surace as sum o siusoidals. ξ ( π acos t + b T π si t T Each comoet is a solutio to the liearized equatios, i.e. the sum is also a solutio to the liearized equatios. we have a solutio that looks more reasoable.
2 A radom solutio By itroducig: b ξ a + b ad θ arcta a The sum ca be writte: ξ ( ξ cos( ωt θ, requecy resolutio (ω i+ ω i is: This solutio is a determistic solutio. How could we establish a more geeral solutio? Let us itroduce the hase as a radom variable beig uiormly distributed betwee ad π, while keeig our origial values or amlitude ξ, i.e.: ( ξ cos( ω t I order to obtai oe realizatio, o (, we geerate dieret hases, θ,,. By geeratig a ew set o hases we will obtai a ew realizatio o the radom rocess, ( a.s.o.. Θ ote: Frequecy resolutio is deied by legth o time widow, T. 3 Δω π T We have a solutio rom which we ca geerate a iiite amout o ossible realizatios. The observed time history we started with is oe o the may ossible realizatios. 4 Probabilistic structure o our solutio I Solutio: ( ξ cos( ωt Θ What is the distributio o ( t? Observatio: t is a sum o may ideedet radom comoets. I o o the comoets domiates over the other, we will accordig to the cetral limit theorem have: ( ξ; ξ ex π ( ( t is Gaussia with zero mea ad variace equal to the sum o the comoet variaces. The variace o a harmoic wave is give by: The variace o t thus becomes: ( ξ Observatio: As ar as we do ot chage the comoet amlitudes durig the time widow, the rocess variace will be costat.
3 5 Descritio o a Gaussia surace rocess The oit distributio o (t i ad (t is a two-dimesioal Gaussia: i ( ξi, ξ ex π ρi ξ i ρi ( ξ ξ ξ i ρ + i Where: (, Cov i ρ i E[ ( t ( t ] r ( τ i, τ t i -t r (τ ideedet o absolute time, i rocess is statioary. A Gaussia stochastic rocess is i ricile described by a oit distributio o its values at a large umber o times, t, t,., t. The arameters o this oit model ca be determied rom r (τ. A statioary Gaussia rocess is i a statistical sese comletely characterized by the autocorrelatio uctio r (τ. Questio: How ca the variace be determied rom the auto-correlatio uctio? 6 Wave sectrum I ractise it is more commo to work with the Fourier trasorm o auto-correlatio uctio which is the wave sectrum. The oe-sided wave sectrum reads: iωτ s ( ω e r ( τ dτ ; ω π r i ( τ r ( τ e ωτ s ( ω dω; τ We kow rom questio o revious slide that: r ( From exressio or the auto-correlatio above, we see that: s ( ω dω Imortat result!! I we have exaded the rocess o legth T i comoet, the variace o the th comoet reads: s ( ω Δω where Δω π T 3
4 7 Direct estimatio o sectrum rom a wave history We kow rom a revious slide that the variace o the th comoet also is give by: ξ Settig the two exressios or the variace equal, ad solve with resect to the wave sectrum, gives: ξ s ( ω Δω The Fourier amlitides, ξ, are tyically calculated rom a time history o ξ(, usig a Fast Fourier Trasorm techique. Simulatig a realizatio o X( corresodig to a kow wave sectrum 8 The radom rocess is give by: ( ξ cos( ωt Θ From revious slide we have: ξ s ( ω Δω ( s ( ω Δω cos( ω t Θ I we ow geerate m sets o radom hases, we get m realizatios o the rocess. They will all look dieret, but wave wave heights ad wave eriods will o course show some similarity. I we calculate the wave sectrum rom these simulated time histories, we will id that they all roduce (as exected the iut sectrum, i.e. a smooth sectrum i iut sectrum is smooth. I we estimate the wave sectrum rom a observed time history o legth T, we will id that the estimated sectrum is o very irregular shae. I our simulatios shall relect the same behaviour, we will also have to itroduce the Fourier amlitude,, as a radom variable. I that case it should be itroduced as a Rayleigh variable. 4
5 9 Stadard models or wave sectrum Fully develoed wid sea: Pierso-Moskowitz (PM s ( { 4. t 4 } 4.35 h t 5 ex 5 s J Growig wid sea: JOSWAP sectrum (J PM s (.35h s t 5 ex.5 4 (.87 lγ γ ex.5 T Pure swell sea: JOSWAP sectrum with γ e.g. -5 Combied sea (Wid sea lus swell: Torsethauge sectrum (T Torsethauge sectrum suerosistio o two Joswa tye sectra Wid Sea I sectrum give i ad oe would reer to have i w, oe ca id the latter by: ω s ( ω s ( π π Swell Sea Torsethauge versus JOSWAP Hs.m, T.s Hs.m, T s 3 5 Sectral Desity (m***s 5 Torsethauge (4 JOSWAP Frequecy (Hz Sectral Desity (m***s Torsethauge(4 JOSWAP Frequecy (Hz Hs.m, T 4.s Sectral Desity (m***s Torsethauge (4 JOSWAP Frequecy (Hz 5
6 Rage o validity or sectral models (Figure is meat to give a qualitative idicatio. Oe should kow which sea states that are o maor imortace whe selectig sectral model. -year cotour or h s ad t Secod order surace rocess The Gaussia surace model is a aroximatio. It is i agreemet with liear wave theory. A secod order model is give by: ξ ( t ξ ( t ( t G + ξ Comariso Gaussia rocess ad secod order sum rocess. I ricile there will also be a dierece requecy comoet. Gaussia rocess Secod order correctio ξ ( Re c m m c + [ h i( ωm + ω t e + h i( ω t m ω e ] m m I roblem uder cosideratio is sesitive to the detail surace elevatio, a o-gaussia model should be adoted. Surace elevatio is straight orward, the corresodig kiematics are more challegig. 6
7 3 Crest heigts o Gaussia ad Scod order rocesses Cumulative robability Rayleigh.3 Jahs ad Wheeler. Forristall. Witerstei Crest height (m Distributio uctio or the 3-hour maximum crest height or a sea state with h s 8.m ad t 7s (**(-4 aual robability sea states. (Forristall Secod order 7
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