On the Statistical Properties Measured at the Japan Sea

Transcription

1 On the Statistical Properties Measured at the Japan Sea TE LE.' 1'VflITEIT - 'z cum voor.;crn9chanica Archef Mc!z!wcg, E8 CD Deft liirofui Yoshioto, Shigeo Ohiatsu Ship Research Institute, Ministry of Transport. Japan Deformation of some deck plates of "POSPI- DON" has been observed and this phenomenon has been investigating. Photo i shows an example of the deformation of the bottom of the deck plate. A possible explanation of the deformation is that waves may hit the bottom of' the plate. In the first place, wave amplitude statistics has been investigated, especially in stormy sea states. As a result, the measured wave data in stormy sea states reveal that high wave crests occur more frequently than the predicted by the Rayleigh distribution. The non-linear theory, developed by Hinenol>, resulting in a non-rayleigh distribution of wave crest and trough amplitudes has been used to predict the highest 1/n amplitude and the extreme statistics in stormy sea state. In this paper, a comparison between the measurements and the non-linear theory is presented. 1. Wave data Photo i Example of the deformation of the bottom of the deck plate The analysis has been carried out using the 6 continuous data when low atmospheric pressures L.A.P. ) in winter or a typhoon approached to the experimental site. The data have been divided into 54 runs of 048 seconds each. Table i shows statistical values of its highest significant wave height H113 for each runs where TFI1/3' H and TFax are the significant wave period, the maximum wave height and the maximum wave period respectively. Table 1 Statistical values of the wave data File Length(h) H1/3(m) T}11/3(s) H3x(m) Tax(5) D Typhoon D ii L.A. P. D li. 646 L. A. P L.A.P. D L. A. P. D L. A. P.. Analysis Fig.1 shows the definition sketch of the wave crest amplitude A and the wave trough amplitude At. A is defined to represent the zero up-crossing wave crest height and At represents the zero up-crossing wave trough excursion. Fig. shows the relation between the wave steepness and the average values of the highest 1/n amplitudes divided by the standard Fig. i Definition sketch of wave amplitudes i

2 deviation of the surface elevation, where A11 is an average value of the highest 1/n c-est amplitudes and At11, is an average value of the highest 1/n trough amplitudes. The wave steepiess a is defined as a = H113/(gT/) (1) where g T acceleration due to the gravity zero up-crossing mean period. 3 l. <.1 o i: *ave Steepness (H1/3/1.56Tz*) (a) Al/l and Ati,a < <.. 1 Fig '07 Wave Steepness (H1/3/1. S6Tz") td) Aol/lo and Relation between the wave steepness a and the average values of the highest 1/n amplitudes The solid line shows the expected values of the highest 1/n amplitude predicted by the Rayleigh distribution (linear theory). It is clearly seen that the measured A11 increases as the wave steepness increases, while the measured A111 become smaller, whereas the measured becomes larger than the prediction by Rayleigh distribution, especially for the data on high steepness condi t ions. Fig. 3 also shows the relation between the wave steepness and the maximum amplitudes divided by the standard deviation where is the maximum crest amplitude and Atax is the maximum, trough amplitude. The solid line shows the expected value of the maximum amplitude predicted by the following formula derived by Cartwright and Longuet-Higgins. E[Aax/a]

3 <5 - b N. 43 D + o- 4th Nov. ' th Dec. '87 JtJa D0 c o -:: ódudqj D + 0o1 9 -f-d U +.0,i_ D++o_ p i1 N:10000 N000 D++ + \N *+ + * + :OO Wave Steepness (HI/a/i. 56Tz)a Fig.3 Relation between the wave steepness a and the maximum wave amplitudes E [Amax/o] = [(lnn)'1+ a/(lnn)] () m 87/1/17 17th Dec. '87 Probe-1 Wave Probe- H,14.6O (m) H,,3:7.47 (m) A...,/cr:4.811 where N is the number of zero up-crossing and the measured N distributes around It is seen that the measured A< occurs more frequently than the expected value predicted by the equation () for N = 10,000. In the linear theory, the expected value of Ataax becomes the same value as that of Acax, but there is no case for the measured Atmax 8 to exceed the expected value for N 10, 000. Fig. 4 shows examples of recorded waves with high Acsax/ Fig. 3 and they indicate that there are high crest-trough dissymnmetries. These results suggest the increased occurrence of high - crests and flattened troughs caused by non-linearities in random sea. In order to account for the in observed non-linear effects, a non-linear probability density function, developed by Hineno0, 10 s resulting distribution of wave crest and trough amplitudes has been used to predict the the highest 1/n amplitudes and the maximum amplitude. Wave Probe th Nov. '87 87/11/ }L: (m) H,,3:5.50 (m) :6.51 Fig. 4 Examples of recorded waves line(s) omo Tine(s) 90 in a non-rayleigh expected values of 3. Non-linear probability density function Hinenol> developed a non-linear probability density function of wave amplitudes by using the Vinje' s nethod3 based on the narrow-band assumption. It includes the effect of the second-order wave and is given as 3

4 P(z) = exp(-z/k0). [ -K10/K0-K1/K0K0+K 30/K0+z/K0+ (K10+K 1/K0-K30/K0) z/k0+k30/6k04z4 ] K10 = f S()G(, -)d, 0 = fs(jg1( da. K0 f S() G1( d K1 =ff( +)SC1)S() (G1(1)G1(-)G(1. )+G1(-1)G1(-)G(1, )}d1d K33 = 3f/s Li) S L) (G1 (-) G1 (-) G L1. ') G (-1)G1 (-)G' Li. ) } d1d ( Limits on integrals, -toare omitted in this paper ) where P(z) : probability density function circular frequency S() : double side wave spectrum G1() linear response amplitude operator quadratic response amplitude operator * : complex conjugate Equation (3) shows that the non-linear probability density function can be calculated with the response amplitude operators ( RAO ) and wave spectrum. Before the theoretical predictions by the equation (3) are compared with the measured data, the effect of spectrum width and water depths on the probability density function has been investigated by numerical simulations. 4. Nuaerical Si.ulation Numerical simulation has been carried out using four input wave spectra and the quadratic RAO both in shallow water and in deep water. The input wave spectra are based on the following modified J0NSAP spectra4> with the significant wave height H113 = 6 um and the peak frequency f = Hz. Fig. 5 shows the input wave spectra, giving the spectral peak enhancement factor = 1,, 4, and 6 respectively to estimate the effect of spectrum width. exp[- (fpf)/afp] SLj(f) = -1 S (f) = (i/3 f, ) (f/fy4exp(-/3(f/fy6) = 0.07 ( f f ) (f> f F where S(f) : modified J0NS'AP spectrum coefficient to equate the area of S1(f) with that of S1(f) F1 f peak frequency of wave spectrum 'y : peak enhancement factor IO m'sec Spectra for Siriletica 'Y o Fig. 5 Freqoency(flz) Input wave spectra for simulation 4

5 RAO The linear RAO is simply G1 = 1, while the following expressions5 are found for the quadratic in shallow water. GC, )=K(3coth3Kh-cothKh)exp(-iK. x)/ G(. - ) = o. G (i 1) = Kexp(-iKt. x)/. { () (1+cothK1h. cothkh) + (1cothK1h+cothKh) cothkh} /(gk1- () cothkh) G (ei. -) =lcexp(-ilç. x)/. ((t)+o1 (l-cothk1h. cothkh) - (1cothK1h+cothKh) cothkth} /(gk- () cothkth) where K wave number h : water depth I' 1' K4=K1+K, IC=K1-k (5) We can (5) N also obtain the quadratic RAO in deep water by taking the limit h -> in the equation RayIeqh 0.8 crest Distribution f Deep 0.7 Crest Distribution ( Shallow Rayleiqh 0. 1 L z//r , Trouqfl Ditribut,o, ( Deo [0.5 ' [Q.5 4 z//r'5 (a) Deep water 0 (b) Shallow water Fig.6 Probability density distribution of wave amplitudes, wherejo defined by the equation (3) equals to the standard deviation of the surface elevations. z/1k y= r=4 -- Fig. 6 shows examples of the effect of spectrum width and water depths on the probability density function for crest and trough amplitudes through the above expressions from (3) to (5). The water depth in the shallow water case equals to the water depth of the experimental site 43 in ). Fig. 7 shows the expected values of the highest 1/n amplitude and the maximum amplitude predicted by the results shown in Fig.6. It is found that the water depth has a great effect on both the peak and the tail of the distribution leading to the increased probabilities of high wave crest amplitudes, while the effect of spectrum width is small. lt is also found that the equation (4) gives the negative tail probabilities for trough amplitudes and the expected values of in sallow water cannot be shown due to the inadequate tail behavior. This suggests that there is a possibility that it produces anomalous results and an alternative statistical model has to be adopted. 5

6 Crest (with shallow effect) 3. 5 Crest dee. water 0 N =6.0 o /n 0 (a) Expected values of Al/ and A11 <4.5 W4 Crest (with shallow effect) Crest (deep water) 3. 5 Trou.h dee. water - 7_4 r= -- y= N 300 (b) Expected values of and Atsax Fig.7 Expected values of the highest 1/n aniplitude and the inaxinium amplitude, whereft0 defined by the equation (3) equals to the standard deviation of the surface elevations. 5. Couparison between non-linear theory and ucasured data Bearing the features of the probability density function developed by Hineno in our mind, the measured data have been compared with the predictions by the non-linear theory. 60 m'sec L.ce Spectra FkaLLuQ frequency (Hz) Fig.8 Input wave spectra for the each wave steepness 6

7 , H, Fig.8 shows the input wave spectra with the values of the parameters given by Table, where and f are average values of the measured data in the range of the wave steepness. The quadratic RAO in shallow water is used and the water depth is 43 in. Table Parameters of Modified JONSYiAP spectra Steepness. H1, In Fig. 9, the measured data are compared to the expected values of Ad/fl predicted by the nonlinear theory, where the expected values of Ati,ì cannot be shown due to the negative tail probabilities for trough amplitudes. lt is found that the theoretical curves for the non-linear theory increase with the increase of the wave steepness a. the measured data has also the sanie tendency. However, there is a slight difference between the theoretical predictions by the nonlinear theory and the measured data. lt may come from the inadequate tail behavior of the probability density function developed by Hineno b c b \-.-Q oç + ++J - U i _: Non-Iinear(nd) t. o o.b3 005 Wave Steepness (111/3/1. 56Tz.) a (a) Expected values of ACl/ Non-tinear(nd o < JyI +++ o ++ + r w1i?; 'O id o. B _.yleigh hi Wave Steepness (111/3/1. 56Tzi') cr (b) Expected values of Aii10 Fig. 9 Expected values of Aii 7

8 In Fig. 10, the measured data are compared to the expected values of Acax predicted by the nonlinear theory for N = 00, 1000, 000, and 10,000. It is found that the non-linear theory shows about lo X increase of the expected values as compared to the Rayleigh distribution. However, we must note that there are still several cases exceeding the theoretical predictions by the nonlinear theory for N =10,000. The Yasuda's proposal6 that non-linear interactions higher than third order cause the large Acm can be considered a possible explanation of this phenomena. It may suggest the necessity to develop a new nonlinear theory including non-linearities of thirc order I I I 0.07 wave Steepness (H1/3/1. 5STz*+) Q Fig. 10 Expected values of 6. Conclusion The conclusions from the present investigation are summarized as follows - The measured Ad/fl increases as the wave steepness increases, while the measured become smaller, whereas the measured Au becomes larger than the prediction by Rayleigh distribution, especially for the data on high steepness conditions. - The non-linear probability density function includes the effect of second order wave has been used to predict the highest 1/n amplitude and the extreme statistics in stormy sea state. As a result, the expected values predicted by the non-linear theory increase with the increase of the wave steepness a. the measured data has also the same tendency. However, the non-linear probability density function gives the negative tail probabilities for trough amplitudes. - The water depth has a great effect on both the peak and the tail of the distribution leading to the increased probabilities of high wave crest amplitudes, while the effect of spectrum width is small. The prediction method presented in this paper are clearly not valid for the maximum amplitude, especially AC(. Furthermore, an alternative statistical model has to be adopted due to the limitation of the non-linear probability density function developed by Hineno. After making the above problems clear, this paper will be submitted to the OCEAN 93 Conference. 8

9 The References 1)Hineno, M. : A calculation of the Statistical distribution of the Maxima of Non-linear responses in irregular Waves, Journal of the Society of Naval Architects of Japan, Vol. 156, Dec. 1984, ( n Japanese )Cartwright & Longuet-Higgins, M.S : Statistical distribution of the Maxima of a random function, Proceedings of The Royal Society A, Vol.37, )Vinje, T. : On the Calculation of Maxima of Non-linear Wave Forces and Wave induced Motion, international Shipbuilding Progress, Vol.3, No.68, )Yoshimoto, H. et.al. : At-sea Experiment of a Floating Offshore Structure -Characteristics of Directional Wave Spectra Measured at Japan Sea-, Journal of the Society of Naval Architects of Japan, Vol.168, )Hamada, T. : The Secondary interaction of Surface Wave, Report of Port and Harbor Technical Research institute, Report No. 10, 1965, pp )Yasuda, T. et.al. : Non-linear Effects on the wave Height Distribution of Unidirectional irregular Waves, Proceeding of Japan Society of Civil Engineers, No.443/-18, 199, pp.83-9, in Japanese 9

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