Effect of Directional Spreading and Spectral Bandwidth on the Nonlinearity of the Irregular Waves

Transcription

1 Proceeding of the Eighth (1998 International Offhore and Polar Engineering Conference Montréal, Canada, May -9, 1998 Coyright 1998 by The International Society of Offhore and Polar Engineer ISBN (Set; ISBN (Vol. III Effect of Directional Sreading and Sectral Bandwidth on the Nonlinearity of the Irregular Wave Marc Prevoto IFREMER Bret, France ABSTRACT Wherever one tand, dee water, intermediate or hallow water, in extreme condition the nonlinearity in the wave kinematic i large and ha a trong influence on the deign arameter. Simle model of the wave kinematic have been tudied baed on hyothee of narrowband and unidirectionality. Obviouly a real ea i neither narrowbanded nor unidirectional and the width of the ectral denity and the directional reading influence the nonlinear characteritic of the wave (kewne, aymmetrie,... A econd order directional irregular wave model i ued to imulate time erie of the free urface elevation. Baed on a large imulated data bae, a arametric tudy of the influence of the ectral width and directional reading i led on everal wave characteritic enitive to nonlinearitie (kewne, wave and cret height ditribution, teene of maximum cret. Three tyical ituation are analyed which correond to extreme ituation in long and hort fetch wind ea. The validity of the imlified aumtion of narrowband and unidirectionality i then dicued. EYWORDS Wave tatitic, ectral bandwidth, directional reading, nonlinearity, econd order, cret height, kewne. INTRODUCTION Wherever one tand, dee water, intermediate or hallow water, in extreme condition the nonlinearity in the wave kinematic i large and ha a trong influence on the deign arameter. Thi ha been howed clearly for the cret height ditribution in dee water (Nerzic, 199. In a very imle model of a real ea wave, let u ay an unidirectional harmonic wave, the nonlinearity i driven by a teene arameter (the amlitude divided by the wavelength and a dimenionle water deth arameter (the water deth divided by the wavelength. Thi i alo the cae for the narrowband model where the amlitude i relaced by the ignificative height and the wavelength by "a" mean wavelength. Obviouly a real ea i neither narrowbanded nor unidirectional and the width of the ectral denity and the directional reading influence the nonlinear characteritic of the wave (kewne, aymmetry,... In thi tudy we conider uniquely in the kinematic of the wave, the elevation of the free urface. A econd order directional irregular wave model baed on a Stoke exanion and a firt order directional Gauian roce i ued to imulate time erie of the free urface elevation. Baed on a large imulated data bae, a arametric tudy of the influence of the ectral width and directional reading i led on everal wave characteritic enitive to nonlinearitie (kewne, wave and cret height ditribution, teene of maximum cret. Three tyical ituation are analyed which correond to extreme ituation in long and hort fetch wind ea. The validity of the imlified aumtion of narrowband and unidirectionality i then dicued. EFFECT OF DIRECTIONAL SPREADING AND SPECTRAL BANDWIDTH At our knowledge, good literature doe not exit about the effect of the directional reading and ectral bandwidth on the nonlinear characteritic of wave, and articularly none can be found for hallow water. In (Longuet-Higgin, 193 the kewne of the free urface elevation, in dee water, i bounded with a lower bound correonding to the ueroition of two orthogonal longcreted ea and an uer bound correonding to a ingle one:.λ 3, uni λ 3dir, 1.1λ 3, uni (1 with λ 3, uni the kewne in the unidirectional cae. Mot of the tudie about effect of the ectral bandwidth concern the wave height ditribution in a Gauian ea (Longuet-Higgin, 198, (Nae, 1985, conidered a rereentative of the wave height ditribution in nonlinear ea. Tayfun (1983 tudied the nonlinear effect on the ditribution of cret-to-trough wave height but without conidering the directional reading. Effect of the directional reading and ectral bandwidth on the kewne and kurtoi and cumulative ditribution of maximum cret and wave height have been alo tudied in (Stanberg, In that tudy, wave tank meaurement were conidered and he concluded that extreme wave event due to nonlinear modulation are mot ronounced in longcreted wave. Thi remark will be confirmed hereafter 119

2 in the dee water cae. Sea-tate characteritic Three tye of (,, h (water deth configuration have been conidered. A firt one ( =1m, =1, h=1m correond to an offhore long fetch ituation (tyically the North Sea, a econd one ( =m, =9, h=1m correond to an offhore medium fetch ituation (tyically the Aegean ea and the lat one to the reviou one roagated on hallow water ( =m, =9, h=1m. For each of thee ituation the combination of the arameter of the Jonwa model ( = 1,3,1,1 and of the directional ditribution ( = 1,,, were conidered. For each of thee coule (,, 1 time erie have been generated from the dicretized Fourier tranform. We oberve that ome of the (, coule are not really realitic, e.g. ( =1,= which would correond to a readed well. In fact, all the coule have been conidered, the aim of thi tudy being to roceed to a ytematic arametric analyi in order to better undertand the effect of both ea-tate arameter on nonlinear wave characteritic. The different te of the imulation are a follow. Firt, imulation of the firt-order art η 1 (t of the elevation for a given (, coule (eq. (3. Secondly, calculation of the correonding econd-order art η,dir (t, (eq. (8. Finally, in order to comare to a econd-order unidirectional ea-tate elevation with the ame firt-order elevation, calculation of the db( correonding to a unidirectional η 1 (t firt-order art by invere Fourier tranform from: η 1 ( t = dbk ( ex( iωt (13 and econdly, with thee db(, calculation of the econd-order art: η, uni ( t = H ( k 1, k dbk ( 1 db( k ex( i( ω 1 + ω t (1 So we will be able to comare the tatitical arameter of thee three elevation rocee: the linear roce η lin = η 1 (t, the non-linear unidirectional roce η uni = η 1 (t+η,uni (t, the non-linear directional roce η dir = η 1 (t+η,dir (t. Two examle of a wave obtained from thee three elevation rocee are given in figure & 3. We ee here clearly the effect of nonline- SECOND ORDER MODEL The nonlinear model of the free urface elevation of a directional ea which we have ued i a claical econd order model baed on a Stoke exanion and a firt order directional Gauian roce (ee e.g. (Ding, 199. The nonlinear model of the elevation roce i the ueroition of two rocee: η( t = η 1 ( t + η ( t ( The firt order art η 1 of thi model i a directional Gauian roce (ueroition of Airy wave with random hae and amlitude: η 1 ( t = db( ex( iωt (3 with the integration domain defined by: = {(( ω,, ( ω, with k [, ], β [ π, π] } ( the vectorial wavenumber k and the angular frequency ω defined by: ω coβ = gktanhkh, k = k, k = k (5 inβ and with the wavenumber directional ectral denity ( related to db( a Brownian, indeendent increment roce, by: S( dk = E( db( ω, db * ( ω, where E denote the exectation, and with: db( ω, = db * ( ω,,, E( db( ω, = The econd order art can be written a: η ( t = H ( k 1, k db( k 1 db( k ex( i( ω 1 + ω t + More detailed exreion of firt and econd order art are given in aendix. SIMULATED DATA Model of directional ectrum The directional ectrum of the imulated ea-tate ha been baed on a Jonwa ectrum a oint ectrum S(f and the Mituyau (Mituyau, 195 directional ditribution H(f,θ for the directional art. The frequency-angular ectrum i then calculated by: Sfθ (, = Sf (Hfθ (, (9 with H( f, θ = A ( co θ θ (1 π where A( i a normalization factor to enure that Hfθ (, dθ = 1. π H(f,θ i conidered to equal outide [-π/,π/]. The coefficient i frequency-deendent and given by: f f< f f = 5 (11 -- f --- f f f The ectrum i then introduced in (3 and (8 in conidering S( dk = Sfθ (, dfdθ (1 Dicretization The Fourier tranform of equation (3 and (8 have been dicretized in the frequency and angular domain. The frequency domain ha been dicretized in 1 oint with a amling frequency of 1Hz. The angular domain [-π/,π/] ha been divided in 1 equal ector. c τ ( ( (8 A it i exlained in (Chen, 199, the non-linear interaction between long and hort wave are very badly calculated with a hort Stoke exanion and to be taken into account accurately it i better to introduced a modulated wave-mode aroach. Thi aroach furnihe a o-called hybrid model (Zhang, 199. If thi aroach eem necearily in the comutation of the kinematic in the cret, in the cae of free-urface elevation, a more imle method could be ued for the calculation of the non-linear interaction. It conit merely in a truncation of the ectral denity, o removing very long and hort wave. The wave in the frequency band correonding to 1% of the total variance m ( H S = m were removed a illutrated in fig. 1. Fig. 1. Truncated ectrum (H=m, T=9, =1 d (m /Hz * m.1 * m frequency (Hz 1

3 arity and of directional reading, effect that we will find again in the tatitical arameter analyi. In dee water deth a in hallow water deth the nonlinearity, a well known, increae cret elevation. But the directional reading aociated to a econd-order nonlinear model, decreae the cret elevation in dee water deth (fig. even though it increae the cret elevation in hallow water deth (fig. 3. Thee two different behaviour come from the different nonlinearitie concerned in each cae. In dee water deth only a free urface nonlinearity i concerned, but in hallow water nonlinearity i redominantly a bottom effect one. Fig.. Comarion of the hae of a wave, water deth = 1m 8 linear unidirectional directional, = time ( Fig. 3. Comarion of the hae of a wave, water deth = 1m linear unidirectionnal directionnal, = time ( PROCESSING OF THE DATA On each et of 1 time erie, kewne, maximum cret height, maximum wave height and maximum-cret front teene have been comuted. Here cret front teene i defined a: C cf = (15 λ( T with, for the maximum cret height, C and T a defined in figure, and λ(t the dierion relation function between wave eriod and wavelength (ee (Myrhaug, 198. In a econd te the mean kewne and cret front teene have been comuted and a Gumbel law ha been fitted on the two other et of maximum wave characteritic: y a Gy ( = Prob( X N y = ex N ( ex b N with N the number of wave of the time hitory, conidered a contant for a articular. The two arameter of the Gumbel law, the mode a N and the cale b N, have been etimated in uing a Maximum Likelihood Method. All the fit are very good, a well on the maximum cret a on the maximum wave height. Thee reult are ummarized in table 1-5 and commented hereafter. 8 Fig.. Definition of front cret teene 18 time ( ANALYSIS OF STATISTICAL PARAMETERS A indicated reviouly, the kewne ha been calculated emirically on the et of the time erie to be conitent with the analye of the other arameter. But it could have been calculated directly from the directional ectra a rooed in (Longuet-Higgin, 193 or (Ding, 199. The formula are given at the end of the aendix. A it ha been often demontrated, (Stanberg, 1995, (Vinje, 1989, (Vinje, 199, (Nerzic, 199, the mean teene, e.g. defined a = H, i the dominant factor of the nonlinearity in dee water. The three tye of ituation conidered here correond to different S k ( π mean teene ( =1m, =1, h=1m =.1, ( =m, =9, h=1m =. and ( =m, =9, h=1m =.8. Thee difference between the mean teenee are ufficiently low to comare the enitivitie to the directional reading or to the bandwidth. Skewne If we oberve, for the dee water cae, the value of the mean kewne ratio (directional/unidirectional, given in table 1, two tendencie aear clearly. The kewne decreae when the bandwidth increae and when the directional reading increae. But what eem very trange i that for a very weak reading ( = 1 the ratio i higher than unity. Table 1. Skewne ratio, directional/unidirectional = 1m, = 1, deth = 1m = 1 = 3 = 1 = 1 = = = = T A exlanation we recall here the reult given in (Longuet-Higgin, 193. The tranfer function H which enter in the calculation of the kewne (eq. 35 can be written a: H ( k 1, ω 1, k, ω = ( k 1 k 3 / f( η, (1 where i the angle between k 1 and k and η, which indicate the cloene of k 1 and k, i given by: ( k η 1 + k = k ( 1 k 1/ C (18 11

4 In dee and hallow water the effect decreae with the reading. In dee water, the reading can almot comletely cancel the effect of the nonlinearity. In hallow water, the bandwidth ha no effect on the front teene, but the nonlinearity ha a very trong effect on the maxi- The function f( η, i lotted in figure 5. We oberve on thi grah that for very cloe wavenumber (η = 1.1, η = 1.1 the value of the function i higher in the directional cae than in the unidirectional cae, and finally Longuet-Higgin demontrated that f( η, i bounded by the unidirectional cae by:.f( η, f( η, 1.1f( η, (19 Thi relation led him to a imilar relation for the kewne:.λ 3, uni λ 3dir, 1.1λ 3, uni ( In our imulation the ratio i u to 1.1 and never below.1. The ratio. of eq. correond to a very articular condition, two well with erendicular direction but the ratio 1.1 to a more common one, a imle well (e.g. = 1. The higher ratio that we found, correonding to the uer bound in eq., could be exlained by the fact that thi bound wa calculated for infinite water deth and that we conider here wavelength of 3m for water deth of 1m. The finite water deth ituation increae trongly thi contrat between directional and unidirectional cae and exlain our higher value. f(η, 1.8. Fig. 5. Grah of f(η, for variou value of η η = 1 η = 1.1 η = 1.1 η = 1.1 η = 1.5. η = 1.5. η = η = η = η 8= ( ο In table, the value of the kewne for the nonlinear directional cae are lotted. Thee value are all around. for the 1m water deth ituation but very enitive to the directional reading in hallow water. Table. Skewne, directional 1 H = 1m T = 1 h = 1m H = m T h = 1m H = m T % % 3% 3% % % 1% 1% We oberve articularly that the kewne i very high (u to. with the reading ea-tate and cloe to zero in the unidirectional cae. In fact it indicate a change in the wave hae, but, a we will ee hereafter, doe not demontrate the abence of aymmetry between cret and trough height. Mode cret The analyi of the ratio of the mot robable value of the cret height between directional and linear cae (table 3 how a light decreae of the cret with reading, in the dee water cae. Globally, for the 1m water deth, the cret ratio varie from % (g = 1, = to 15% (g = 1, = 1. The value of the ratio are higher for ( =m,, a ea-tate which correond to a higher mean teene. Thi confirm, in dee water, the concluion of (Stanberg, 1995 about the everity of longcreted ea. For the hallow water deth ituation the increae i higher and can reach 35% (g = 1, =. The effect of the bandwidth i ooite in dee water and hallow water. The nonlinear effect i increaed in dee water and decreaed in hallow water. Table 3. Mode cret ratio, directional/linear 1 = 1m = 1 h = 1m = m h = 1m = m % % 1% 1%.5%.5% Mode wave It ha been often mentioned in reviou tudie that, in dee water, wave height are weakly affected by the nonlinearitie. We ee here (table that neither reading nor bandwidth change thi fact. On the contrary, in finite deth (water deth = 1m the effect of the nonlinearitie i comletely different with an increae of u to 18%. In that cae, a high reading diminihe the wave height ratio below 5%. Table. Mode wave ratio, directional/linear 1 = 1m = 1 h = 1m = m h = 1m = m % % 1% 1%.5%.5% Cret front teene The mean maximum-cret front teene follow, in dee water, the ame attern (table 5 a the cret height with an increae of u to 5% in the cae of a unidirectional and Pieron-Mokowitz ectrum model. Table 5. Mean maximum-cret front ratio, directional/linear 1 = 1m = 1 h = 1m = m h = 1m = m % 1% 9% 9% % % 3% 3% 1

5 mum-cret front teene for longcreted ea (increaing u to 13%. Of coure, we have not conidered in thi tudy the different breaking henomena. CONCLUSIONS Taking into account the directional reading and the ectral bandwidth in the nonlinear model of rediction of cret height ha ignificative effect. If in dee water deth the hyothei of unidirectionality i conervative (the longcreted ea i the mot evere, it i not the cae when the deth i ufficiently hallow to affect the nonlinear behaviour of the kinematic. For the wave height and maximum-cret front teene, the hyothei of unidirectionality i alway conervative whatever the water deth. In dee water, the higher the bandwidth i the more everely the nonlinearity change the wave characteritic. In hallow water the bandwidth doe not have very ignificative effect. The nonlinearitie brought into the kinematic in hallow water are different from the dee water ituation, where only the free urface nonlinearity i concerned. Thi change comletely the effect of the directional reading and of the bandwidth with ometime ooite effect comared to the dee water ituation. REFERENCES Chen, L., Zhang, J. (199 "On interaction between intermediate-deth long wave and dee-water hort wave", Offhore Technol. Re. Center Reort, no. A555,. 3. Ding, P.-X., Sun, F., Yu, Z.-W. (199. "Study of the econd-order nonlinear characteritic of ocean wave (1 -- theoretical derivation", Sci. China Ser. B, vol. 3, no. 5, Longuet-Higgin, M.S. (198. "On the ditribution of the height of ea wave: ome effect of nonlinearity and finite band width", J. Geohy. Re., vol. 85, no. C3, Longuet-Higgin, M.S. (193. "The effect of non-linearitie on tatitical ditribution in the theory of ea wave", J. Fluid Mech., vol. 1, Mituyau, H., Taai, F., Suhara, T., Mizuno, S., Ohkuo, M., Honda, T., Rikiihi,. (195. "Obervation of the directional ectrum of ocean wave uing a cloverleaf buoy", Journ. Phyic. Oceanogr., vol. 5,. 5-. Myrhaug, D., jelden, S.P. (198. "Parametric modelling of joint robability denity ditribution for teene and aymmetry in dee water wave", Al. Ocean Re., vol., no.,. -. Nae, A. (1985. "On the ditribution of cret to trough wave height", Ocean Eng., vol. 1, no. 3, Nerzic, R., Prevoto, M. (199. "A Weibull-Stoke Model for the Ditribution of Maximum Wave and Cret Height", Proc. of the th ISOPE Conf., vol. III, Stanberg, C.T. (1995. "Effect from directionality and ectral bandwidth on nonlinear atial modulation of dee-water urface gravity wave train", Proc. th ICCE, vol., Tayfun, M.A. (1983. "Nonlinear effect on the ditribution of cret-to-trough wave height", Ocean Eng., vol. 1, no., Vinje, T. (1989. "The tatitical ditribution of wave height in a random eaway", Alied Ocean Re., vol. 11, no. 3. Vinje, T., Haver, S. (199. "On the non-gauian tructure of ocean wave", Proc. th Intl Conf on the Behaviour of Offhore Structure, vol., Zhang, J., Chen, L., Ye, M., Randall, R.E. (199 "Hybrid wave model for unidirectional irregular wave, Part I. Theory and numerical cheme", Al. Ocean Re., vol. 18, no -3,. -9. APPENDIX: SECOND ORDER MODEL Firt-order art: η 1 ( r, t = db( ex( i( k r ωt (1 with the wavenumber ectral denity S( related to B( a Brownian, indeendent increment roce, by: S( dk = E( db( ω, db * ( ω, and db( ω, = db * ( ω,,, E( db( ω, = where E denote the exectation. ω coβ = gktanhkh, k = k = kd, k = k inβ = {(( ω,, ( ω, with k [, ], β [ π, π] } Second-order art: η ( r, t = with: H ( k 1, ω 1, k, ω db( k 1 db( k ex( i( ϕ 1 + ϕ ϕ 1 = ( k 1 r ω 1 t ϕ = ( k r ω t k 1 coβ 1 k coβ k 1 = k k 1 inβ = 1 k inβ H ( k 1, ω 1, k, ω and c η a contant to enure that E(η = : k c η = S( dk inhkh D i given by: ( ω D( k 1, ω 1, k, ω 1 + ω g ( k 1 k ω 1ω = N( k 1, ω 1, k, ω D( k 1, ω 1, k 1, ω 1 = with: ( (3 ( (5 ( ( (8 (9 (3 (31 (3 (33 Skewne The cumulant are: λ 1 =, λ = S( dk (3 The kewne: λ 3 3 = λ 1 = ( ( ω g 1 + ω D( k 1, ω 1, k, ω + g k 1 k ω 1 ω + ω 1 + ω ω 1 ω g 3 3 ( k 1ω + k ω1 ω 1 ω ( ω 1 + ω N( k 1, ω 1, k, ω N( k 1, ω 1, k, ω = ω 1 ω ( ω + g k + tanh k + h k + = k 1 + k ω + = ω 1 + ω λ 3 = H ( k 1, ω 1, k, ω S( k 1 S( k dk 1 dk + c η (35 (3 13