Directional spread parameter at intermediate water depth

Transcription

1 Directional spread parameter at intermediate water depth V. Sanil Kumar a,*, M.C. Deo b, N.M. Anand a, K. Ashok Kumar a a Ocean Engineering Division, National Institute of Oceanography, Dona Paula, Goa , India b Civil Engineering Department, Indian Institute of Technology, Bombay, India Abstract The characteristics of directional spread parameters at intermediate water depth are investigated based on a cosine power 2s directional spreading model. This is based on wave measurements carried out using a Datawell directional waverider buoy in 23 m water depth. An empirical equation for the frequency dependent directional spreading parameter is presented. Directional spreading function estimated based on the Maximum Entropy Method is compared with those obtained using a cosine power 2s parameter model. A set of empirical equations relating the directional spreading parameter corresponding to the peak of wave spectrum to other wave parameters like significant wave height and period are obtained. It shows that the wave directional spreading at peak wave frequency can be related to the non-linearity parameter, which allows estimation of directional spreading without reference to wind information. Keywords: Spreading function; Directional spectra; Maximum entropy method 1. Introduction A reliable estimation of the directional wave properties at a particular location is a necessary prerequisite in design and operation of coastal or offshore structures. The directional wave characteristics can be conveniently studied through a directional

2 wave spectrum which represents distribution of wave energies over various wave frequencies and directions. A number of methods are available to estimate directional spectrum from the measurements made by a moored buoy (Longuet-Higgins et al., 1963; Borgman, 1982; Isobe et al., 1984; Kobune and Hashimoto, 1986; Kuik et al., 1988). The simplest among them considers representation of directional spectrum as a product of unidirectional spectrum and a directional spreading function. The directional spreading function can be modelled using a variety of parametric models (Pierson et al., 1955; Cote et al., 1960; Longuet-Higgins et al., 1963; Donelan et al., 1985). No single model however is universally accepted due to the idealization involved or due to site specificity associated with particular formulations (Niedzwecki and Whatley, 1991). The cosine power 2s model, originally proposed by Longuet-Higgins et al. (1963), is very popular due to its proven generality The cosine power 2s model The cosine power 2s model is as follows; D(f, ) G(s)cos 2s [( m )/2] (1) where G(s) 22s 2 (s 1) 2 (2s 1) 1 (s 1) (2) 2 (s 0.5) and D(f, ) is directional spreading function, f is wave frequency, is wave direction, m is mean wave direction, is gamma function and s is spreading parameter. Here the value of the parameter s controls directional spreading around the mean wave direction. The spreading parameter is required to be estimated form measured data. There are several ways to do so. Representing the spreading function into a Fourier series (Cartwright, 1963) showed that s can be related either to the First order Fourier coefficients a o, a 1 and b 1, or second order ones, viz., a 2 and b 2. Accordingly two estimates of s viz., s 1 and s 2 result and they are as follows. s 1 r 1 and s 1 r 2 1 3r 2 (1 14r 2 r 2 1 2(1 r 2 ) where r 1 a2 1 b 2 1 a o 2) 1/2 (3) and r 2 a2 2 b 2 2 a o (4) Fourier coefficients in turn are estimated from the auto, co and quad spectrum of the collected buoy signals. These signals pertained to the vertical motion and two horizontal translations, viz., north south and east west motions of the buoy. Mitsuyasu et al. (1975) showed that maximum spreading parameter corresponding to the peak frequency of wave spectrum can be determined from the nondimensional parameter of wave age, which is the ratio of wave phase speed to wind speed. Accordingly a value of 10 was recommended for wind waves, 25 for swell with short decay and 75 for swell with long decay distance (Goda, 1985).

3 Wang (1992) showed that value of s can be related to the wave length associated with peak frequency of the spectrum, determined from the linear dispersion relationship, and the significant wave height. Studies to correlate the spreading parameter with basic and measurable wave parameters need continuation owing to the incomplete knowledge available in this regard. Present study is therefore aimed at knowing how the value of s can be related to significant wave height H s, average zero crossing period T z and water depth d. This has been achieved by measuring and analysing wave data from a waverider buoy as shown below Data measurement and analysis Wave measurements were carried out using Datawell directional waverider buoy (Stephen and Kollstad, 1991) at a location along the west coast of India where the water depth was 23 m during June to September The sampling interval was s and the data were recorded for 20 minutes duration at every 3 hr interval. The data analysis is carried out by using the technique proposed by Kuik et al. (1988) wherein the characteristic parameters of directional spreading function at each frequency are obtained directly from Fourier coefficients a 0, a 1, b 1, a 2 and b 2 without any assumption of model. The directional spreading function parameters were obtained from the measured data as below: mean wave direction, m arctan(b 1 /a 1 ) (5) principal wave direction, p 0.5 arctan(b 2 /a 2 ) (6) directional width, 2(1 m 1 ) (7) mean spreading angle, k (8) arctan 0.5b2 1(1 a 2 ) a 1 b 1 b 2 0.5a 2 1(1 a 2 ) a 2 1 b 2 1 where m 1, m 2 and n 2 are centered Fourier coefficients estimated from Fourier coefficients, a o, a 1, b 1, a 2 and b 2 (Kuik et al., 1988). Among the above parameters the mean and principal wave directions were defined by Longuet-Higgins et al. (1963). Directional width is an index of directional spreading which is the root mean square spread about mean wave direction valid for a narrow directional distribution. To get an estimation of directional spreading, Goda et al. (1981) defined mean spreading angle ( k ). The mean spreading angle varies from 0 to /2. Both these parameters will be zero for unidirectional waves. Derivation of the directional wave parameters requires information on wave properties like significant wave height, wave length etc. These can be calculated

4 using wave theories. Wave theories may be of linear and non-linear type and it becomes necessary to distinguish between them for the application concerned. Swart and Loubser (1978) proposed a parameter F c defined below, relating it to significant wave height (H s ), wave period (T z ) and water depth (d) to assist the evaluation of the appropriate range of various wave theories. H F c d s T z d 5/2 g where, g is acceleration due to gravity. For linear sinusoidal waves F c is small and its value increases with the departure of wave shape from the sinusoidal form. Wave non-linearity is further related to the rate of energy dispersion through the Ursell (1953) parameter given below. U H sl 2 d 3 (10) where L is the wave length. Goda (1983) proposed the parameter which in deep water simplifies to wave steepness. H s L coth3 (kd) (11) where k is wave number 2 /L. The sharpness or flatness in the shape of the uni-directional wave spectrum is judged through the spectral peakedness parameter, Q p (Goda, 1970). Spectra with sharper peaks will have larger values of the peakedness parameter. Q p 2 m f[s(f)] 2 df (12) where S(f) is the spectral density corresponding to frequency f and m 0 is zeroth moment of energy spectrum about origin given below. m 0 f 0 S(f)df (12a) 0 (9) 2.1. Maximum entropy method (MEM) The entropy M (entropy) corresponding to the probability density function of distribution function, D(f, ) was defined by Kobune and Hashimoto (1986) as:

5 2 M(entropy) D(f, )ln D(f, )d (13) The directional distribution function D(f, ) is determined by maximising the above defined entropy. Lygre and Krogstad (1986) proposed an alternative maximum entropy method. However Kim et al. (1993) confirmed better applicability of the model proposed by Kobune and Hashimoto (1986) than other methods when applied to actual measured wave data. 3. Results and discussion 3.1. H s T z relationship The variation of zero crossing wave period (T z ) and wave period corresponding to maximum spectral energy (T p ) with significant wave height (H s ) during the observation period of June to September 1996 is shown in Fig. 1. H s varied from 0.7 to Fig. 1. Variation of zero crossing wave period and period corresponding to maximum spectral energy with significant wave height.

6 5m,T z ranged from 4 to 9 s while T p changed from 6 to 16 s. H s seemed to be well correlated with T z (which is an average value), rather than T p (which is a single unique quantity). By fitting non-linear regressions the relationships between H s, T z and T p comes out as in Fig. 1. The regression coefficient between H s and T z is 0.74 and between H s and T p is The 95% confidence limits for the fit is shown in the figure as dotted line. The low regression coefficient between H s and T p is due to the fact that most of the waves recorded are swells. As swell moves away from the storm that created it, H s decreases relatively quickly, while T p changes relatively slowly Mean and principal directions The comparison of mean and principal wave directions corresponding to the maximum spectral energy shows that these parameters for the peak frequency are almost identical with a regression coefficient of 0.99 (Fig. 2). This indicates that at peak frequency the directional distribution is unimodal. Their magnitudes varied from 219 to 319 with respect to north Spreading angle and directional width The study shows that mean spreading angle, k obtained by Eq. (8) and directional width, obtained by Eq. (7) have fairly similar values (Fig. 3). Goda et al. (1981); Fig. 2. Correlation of mean and principal wave direction corresponding to maximum spectral energy.

7 895 Fig. 3. Correlation of mean spreading angle and directional width. Benoit (1992); Besnard and Benoit (1994) also observed similar results. In the present case the mean spreading angle turns out to be a little smaller than the directional width. The maximum spreading parameter s 1 was found to greatly reduce with an increase in the directional width, and it followed the following empirical relation (Fig. 4). s 1 max (14) 3.4. Wave steepness The steepness of the measured waves were determined using the wave period corresponding to maximum spectral energy. It was also obtained from the average zero crossing wave period (the underlying wave theory used was linear). When these two quantities were compared with each other (Fig. 5), it was found that the former value was almost half of the later. This can be expected since the spectral peak period (that goes into the denominator when we calculate the wave steepness) is much higher than the average zero crossing period (see Fig. 1).

8 Fig. 4. Variation of maximum spreading parameter s 1 with directional width Spreading and other wave parameters Based on the collected data it is found that the value of non-linearity parameter, F c varied from 3 to 35 and the Ursell parameter varied from 0.05 to 4.4 (Fig. 6). An attempt was made to relate the spreading parameter s 1 obtained from Eq. (3) corresponding to maximum spectral energy with (i) wave steepness obtained from wave period corresponding to the maximum spectral energy, (ii) wave steepness derived based on zero crossing wave period, (iii) peakedness parameter, (iv) Ursell parameter, (v) non-linearity parameter and (vi) parameter. The result is shown in Fig. 6 which also indicates the empirical equations derived by relating the spreading parameter with the above wave parameters. Even though these relationships are not non-dimensional they can be used to derive an unknown parameter involved from the known one. The figure shows that the scatter is much larger for wave steepness based on a wave period corresponding to the maximum spectral energy and on a zero crossing wave period with a regression coefficient of 0.29 and 0.31 respectively. It is found that the maximum spreading parameter can be reasonably well represented by the non-linearity parameter and Ursell parameter with regression coefficients of 0.65 each. It appears that the consideration of direct wave steepness causes departure in the resulting spreading parameters as compared to its consideration along with water depth. These equations are valid for the peak energy frequency, f p. To obtain

9 897 Fig. 5. Variation of wave steepness obtained from peak period with wave steepness obtained from zero crossing wave period. spreading parameters at any wave frequency, f, use of relative frequency f/f p K was made (Wang, 1992). This gave the transformed empirical relationships as shown below: s F c K s U K where K (f/f p ) b with b 5 for f f p and b 2.5 for f > f p. The lower and upper limit of the frequency considered are and 0.6 Hz. These are based on the data measuring system Differences in the spreading parameter values Eqs. (15) and (16) are useful in getting the spreading parameter from characteristic wave parameters. It would be of interest to know how far such values differ from a standard spreading parameter s 1. The maximum value of the spreading parameters s 2, s 3 and s 4 are compared in Fig. 7 with the corresponding s 1 value. It shows that the values of s 1 are smaller than s 2 which is consistent with the observations (15) (16)

10 Fig. 6. Variation of maximum spreading parameter, s 1 with wave steepness, peakedness parameter, non-linearity parameter, Ursell parameter and parameter.

11 Fig. 7. Variation of maximum spreading parameters s 2, s 3 and s 4 with s

12 of Cartwright (1963); Mitsuyasu et al. (1975); Hasselmann et al. (1980); Tucker (1987); Wang (1992). The difference is attributed to the noise in the system (Tucker, 1989) along with the limitation of the resolution of the buoy data. A better estimate of the spreading parameter s in such a case would be an average of s 1 and s 2 (Cartwright, 1963). The effect of noise on s 1 and s 2 by numerical simulation was studied by Ewing and Laing (1987) who concluded that s 1 is more sensitive than s 2 to noise and hence s 2 be used in preference to s 1. The value of the spreading parameter s 3 estimated from Eq. (15) is found to be fairly comparable with s 1. This is more or less true with s 4 also. It shows that s 1 generally provides an envelope to other spreading parameters. A two dimensional spectral model is generally represented by considering the product of a unidirectional spectrum and a directional distribution function. The directional distribution function was also evaluated in the present studies by using the Maximum Entropy Method that involves maximization of Eq. (13). The value of other spreading functions were evaluated using the cosine power model of Eq. (1). The spreading parameter s used in Eq. (1) were chosen as s 1, and s 3 as per Eqs. (3) and (15). Fig. 8 shows comparison between D(f, ) derived using all above methods for 6 data sets. Some investigators, e.g. Chaplin et al. (1993) in the past assumed a constant spreading parameter, s 2, and then derived D(f, ). It may be seen that the constant spreading parameter model (s 2) yields a very flat curve while the MEM based distribution is quite steep. The directional spreading function estimated based on spreading parameter s 3 provides an average variation and hence can be recommended for use. It is to be noted that here the difference in MEM and other models at peak may be due to the fact that MEM overpredicts peak (Brissette and Tsanis, 1994). The regression coefficient estimated between the spreading function based on MEM and cos 2s1 and between MEM and cos 2s3 are shown in Fig. 9. It shows that the correlation is good for both the cases. The variation of the error parameter, which is essentially the ratio of the volume of the summation of the error between two functions over the total energy, proposed by Brissette and Tsanis (1994), between the spreading functions estimated based on MEM and cos 2s1 and between MEM and cos 2s3 are shown in Fig. 10. It indicates that since the expression for s 3 involves only the significant wave height, zero crossing wave period and water depth, the spreading function based on s 3 can be used for practical application. In the model based on s 3 the mean wave direction is an input and this has to be obtained for each location. In the present case the mean wave direction obtained from the measurement is taken as input since there was not much correlation between the mean wave and the wind direction measured during the study period. It is also to be noted that the directional spreading function recommended will not model the multi-modality if present in the observed data. However most of the designs are based on extreme conditions and at higher sea level states, the tendency of directional spectra to exhibit bi-modality is significantly reduced.

13 Fig. 8. Directional spreading function based on (a) MEM ( ) (b) cos 2s1 (---)and(c)cos 2s3 ( ) with frequency and direction for 6 data sets. 901

14 Fig. 9. Variation of regression coefficient between spreading functions based on (a) cos 2s1 and MEM (b) cos 2s3 and MEM. Fig. 10. Variation of error parameter between spreading functions based on (a) cos 2s1 and MEM (b) cos 2s3 and MEM.

15 Conclusions 1. The mean wave direction is more or less similar to the principal wave direction at the peak wave frequency. 2. The mean spreading angle is found to be slightly lower than the directional width. 3. The maximum spreading parameter, s, undergoes sudden reduction in its value with the increase in the value of directional width and this can be expressed by an empirical Eq. (14). 4. Steepness of the significant wave calculated based on the peak energy period was almost half of the same obtained through the use of an average zero crossing period. 5. Empirical equations have been derived to determine the spreading parameter s from the given values of (i) non-linearity parameter and (ii) Ursell parameter. These are shown in Fig The variation of the spreading parameters calculated as above with the wave frequency showed that the spreading parameter s determined from the first order Fourier coefficients provides an enveloping curve to all other spreading parameter variations. 7. The directional distribution function based on the cosine power model wherein the spreading parameter involved is evaluated using the non-linearity parameter can be recommended for practical use as it provides an averaged distribution. Acknowledgements The authors would like to thank the Department of Science and Technology, New Delhi, for funding the project titled Directional wave modelling, Director, National Institute of Oceanography, Goa for encouragement and support and the staff of the Ocean Engineering Division, National Institute of Oceanography, Goa for the support and help during the data collection program. NIO Contribution number References Benoit, M., Practical comparative performance survey of methods used for estimating directional wave spectra from heave-pitch-roll data. Proceedings of Conference on Coastal Engineering, pp Besnard, J.C., Benoit, M., Representative directional wave parameters Review and comparison on numerical simulation. International symposium: Waves Physical and numerical modeling, IAHR, pp Borgman, L.E., Maximum entropy and data adaptive procedures in the investigation of ocean waves. Second workshop on maximum entropy and Bayesian methods in Applied statistics, Laramie.

16 Brissette, F.P., Tsanis, K., Estimation of wave directional spectra from pitch-roll buoy data. Journal of Waterway, Port, Coastal and Ocean Engineering 120, Cartwright, D.E., The use of directional spectra in studying the output of a wave recorder on a moving ship. In: Ocean Wave Spectra. Prentice Hall, New York, pp Chaplin, J.R., Subbaiah, K., Irani, M.B., Effects of wave directionality on the in-line loading of a vertical cylinder. Proceedings of the Third International Offshore and Polar Engineering Conference, pp Cote, L.J., Davis, O., Markes, W., McGough, R.J., Mehr, E., Pierson, W.J., Ropek, J.F., Stephenson, G., Vetter, R.C., The directional spectrum of a wind generated sea as determined from data obtained by the Stereo Wave Observation Project, Meteor. Pap. 2(6), New York University, College of Engineering. Donelan, M.A., Hamilton, J., Hui, W.H., Directional spectra of wind generated waves. Philosophical Transactions of Royal Society, London A315, Ewing, J.A., Laing, A.K., Directional spectra of seas near full development. Journal of Physical Oceanography 17, Goda, Y., Numerical experiments on waves statistics with spectral simulation, report. Port and Harbour Research Institute, Japan 9, Goda, Y., A unified nonlinearity parameter of water waves, report. Port, Harbour, Research Institute, Japan 22, Goda, Y., Random seas and design of maritime structures. University of Tokyo press. Goda, Y., Miura, K., Kato, K., On-board analysis of mean wave direction with discus buoy. Proceedings International Conference on wave and wind directionality Application to the design of structures. Paris, pp Hasselmann, D.E., Dunkel, M., Ewing, J.A., Directional wave spectra observed during JONSWAP Journal of Physical Oceanography 10, Isobe, M., Kondo, K., Horikawa, K., Extension of the MLM for estimating directional wave spectrum. Symposium on Description and Modelling of Directional Seas, Technical University, Denmark, A 6, pp Kim, T., Lin, L., Wang, H., Comparisons of directional wave analysis methods. Proceedings WAVES 93, British Columbia, Canada, pp Kobune, K., Hashimoto, N., Estimation of directional spectra from the maximum entropy principle. Proceedings 5th International Offshore and Arctic Engineering, Tokyo, Japan, I, pp Kuik, A.J., Vledder, G., Holthuijsen, L.H., A method for the routine analysis of pitch and roll buoy wave data. Journal of Physical Oceanography 18, Longuet-Higgins, M.S. Cartwright, D.E., Smith, N.D., Observations of the directional spectrum of sea waves using motions of a floating buoy. In: Ocean Wave Spectra. Prentice Hall, New York, pp Lygre, A., Krogstad, H.E., Maximum Entropy Estimation of the Directional distribution in ocean wave spectra. Journal of Physical Oceanography 16, Mitsuyasu, H., Tasai, F., Suhara, T., Mizuno, S., Ohkusu, M., Honda, T., Rikiishi, K., Observations of the directional spectrum of ocean waves using a cloverleaf buoy. Journal of Physical Oceanography 5, Niedzwecki, J.M., Whatley, C.P., Comparative study of some directional sea models. Ocean Engineering 18 (12), Pierson, W.J., Neuman, G., James, R.W., Practical methods for observing and forecasting ocean waves by means of wave spectra and statistics. Pub. No US Naval Hydrographic Office. Stephen, F.B., Kollstad, T., Field trials of the directional waverider. Proceedings of the First International Offshore and Polar Engineering Conference, III, pp Swart, D.H., Loubser, C.C., Vocoidal theory for all non-breaking waves. Proc. 16th coast. Engrg Conf., vol. 1, pp Tucker, M.J., Directional wave data: The interpretation of the spread factors. Deep Sea Research 3 (4), Tucker, M.J., Interpreting directional data from large pitch-roll-heave buoys. Ocean Engineering 16,

17 Ursell, F., The long wave paradox in the theory of gravity waves. Proceedings Cambridge Phil. Society 46, Wang, D.W., Estimation of wave directional spreading in severe seas. Proceedings of the Second International Offshore and Polar Engineering Conference, III, pp