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Calculating crest statistics of shallow water nonlinear waves based on standard spectra and measured data at the Poseidon platform
Ocean Engineering 87 (2014) 16–24
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Calculating crest statistics of shallow water nonlinear waves based on standard spectra and measured data at the Poseidon platform Yingguang Wang n Department of Naval Architecture and Ocean Engineering, School of Naval architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
art ic l e i nf o
a b s t r a c t
Article history: Received 2 July 2013 Accepted 22 May 2014
This paper introduces a new approach for the calculation of the wave crest distribution of shallow water nonlinear waves by utilizing a Transformed Rayleigh method where the second order nonlinear wave model is incorporated. The proposed Transformed Rayleigh method is based on a deterministic and time instantaneous functional transformation. It is very efficient and accurate and can be used for engineering purposes. Meanwhile, the correction for the effect of bottom on the wave spectrum, a procedure that is frequently overlooked by some researchers, has been integrated into the Transformed Rayleigh method. The proposed new approach has been first applied for calculating the wave crest height exceedance probabilities of sea states with standard JONSWAP spectra corresponding to different water depths, and the calculation results have been favorably validated against Monte Carlo simulation results. Meanwhile, the wave crest height exceedance probabilities of these sea states obtained from using the Jahns and Wheeler finite depth wave crest height distribution model have also been included for comparison purpose. It is found that in all cases the Transformed Rayleigh method can offer better predictions than the Jahns and Wheeler model. The Transformed Rayleigh method is then applied to calculate the wave crest height exceedance probabilities of a combined sea state with a bimodal Torsethaugen spectrum corresponding to a water depth of 25 m, and its accuracy and efficiency are again favorably validated by using Monte Carlo simulations. Finally, the Transformed Rayleigh method is applied to predict the wave crest height exceedance probabilities of a sea state with the surface elevation data measured at the Poseidon platform in the Japan Sea, and its accuracy and efficiency have been once again convincingly substantiated. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Crest height Nonlinear waves Transformed Rayleigh method Standard spectra Measured data Poseidon platform
1. Introduction Accurate prediction of the wave crest height exceedance probabilities is of vital importance to the design of both offshore platforms and offshore wind turbines. An important parameter regarding the structural safety of an offshore platform is the height from the still water level to the lowest deck level of the platform, and the value of this parameter is commonly denoted to be a stillwater airgap distance in the offshore engineering literature (Sweetman and Winterstein, 2001). In order to choose a proper value for this still-water airgap in the design process for the platform, accurately predicting the wave crest height distribution at the location of the offshore platform is inevitably required. On an offshore wind turbine, there are some parts of the turbine structure (such as the blades) which are not designed to be exposed to hydrodynamic loads. These parts shall be positioned at a height
n
Tel.: þ 86 21 34206514; fax: þ86 21 34206701. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.oceaneng.2014.05.012 0029-8018/& 2014 Elsevier Ltd. All rights reserved.
with a minimum clearance relative to the expected value of the highest crest elevation with a specific recurrence period. This also requires the designer to accurately calculate the wave crest height exceedance probabilities at the location of the offshore wind turbine. This paper concentrates on the calculation of the wave crest height distribution of shallow water nonlinear waves. The motivation for carrying out this research is twofold. First, at present offshore wind turbines are usually installed in shallow water areas. The author of this article has studied the papers written by some wind turbine researchers (such as Colwell and Basu (2009), Trumars et al. (2005), Shi et al. (2013); to name just a few). These researchers were found to have overlooked the correction for the effect of bottom on the wave spectrum during their offshore wind turbine analysis. As pointed by Graber and Madsen (1988), shoaling of waves as they propagate in water of slowly varying depth and dissipation through bottom friction will affect the evolution of the wave field. This author has thus developed a particular interest in figuring out the negative impact on the wave crest prediction when an uncorrected wave spectrum is used for the offshore wind turbine analysis. Second, it is well known that in the ocean engineering
Y. Wang / Ocean Engineering 87 (2014) 16–24
Nomenclature h Hs
Φ η k
ω
crest height g the acceleration of gravity significant wave height velocity potential surface elevation wave number angular frequency
literature there exist some empirical wave crest distribution models (such as those in Tayfun (1980), Huang et al. (1983), Kriebel and Dawson (1991) (1993), Forristall (2000), etc.) that can be conveniently used for engineering purposes. However, the previous research work of this author's research group (Wang and Xia, 2012) has pointed out that “these empirical wave statistics formulas in the existing literature should be used with caution. Although these formulas are generally very concise and easy to use, sometimes they predict wave statistics results deviating considerably from the true ones”. Therefore, there is a strong motivation for this author to develop a computationally efficient method for calculating the shallow water wave crest distribution that is more accurate than the empirical wave crest distribution models. Summarizing the research work of the author of this article, this paper introduces a new approach for the calculation of the wave crest distribution of shallow water waves by utilizing a Transformed Rayleigh method where the second order nonlinear wave model is incorporated. The proposed Transformed Rayleigh method is a deterministic and time instantaneous functional transformation that is very efficient and accurate and can be used for engineering purposes. Meanwhile, the correction for the effect of bottom on the wave spectrum, a procedure that is frequently overlooked by some researchers, has been integrated into the Transformed Rayleigh method. The proposed new approach will be first applied for calculating the wave crest height exceedance probabilities of sea states with standard JONSWAP spectra corresponding to different water depths, and the results will be validated against Monte Carlo simulation results. Meanwhile, the wave crest height exceedance probabilities of these sea states obtained from using the Jahns and Wheeler finite depth wave crest height distribution model will also be included for comparison purpose. The Transformed Rayleigh method will also be applied to calculate the wave crest height exceedance probabilities for a combined sea state with a bimodal Torsethaugen spectrum corresponding to a water depth of 25 m, and its accuracy and efficiency will again be validated by using Monte Carlo simulations. Finally, the Transformed Rayleigh method will be applied to predict the wave crest height exceedance probabilities of a sea state with the surface elevation data measured at the Poseidon platform in the Japan Sea, and its accuracy and efficiency will once again be substantiated. This paper begins in Section 2 by introducing the knowledge of the second order nonlinear irregular waves. It continues in Section 3 by elucidating the theoretical background of the proposed Transformed Rayleigh method, and in Section 4 the method is applied to calculate the wave crest height exceedance probabilities of shallow water nonlinear waves based on standard spectra and measured data at the Poseidon platform. These results are compared with those from Monte Carlo simulations and from empirical distribution models, with concluding remarks provided in Section 5.
2. The second order nonlinear irregular waves The small oscillation amplitude ocean waves in a sufficiently deep sea are considered to be a Gaussian random process (Ochi,
L g ^η m
s2η
m3 m4 A Tp
17
wavelength the acceleration of gravity mean variance skewness kurtosis wave amplitude spectral peak period
1998). The wave crest distributions in a Gaussian random sea are generally regarded as to obey the Rayleigh probability law (Longuet-higgins, 1952; Chakrabarti, 1987) as shown in Eq. (1): " 2 # h PðAc o hÞ ¼ 1 exp 8 ð1Þ Hs In Eq. (1) h is the crest height, Hs is the significant wave height. In the Gaussian sea model the individual cosine wave trains superimpose linearly (add) without interaction. Therefore, the model is also called the linear sea model. However, it is known that as the water depths decrease, the non-linearities of sea waves become more and more relevant, and the ocean surface elevation process deviates significantly from the Gaussian assumption, i.e. the observed surface process is positively skewed with higher crests and shallower troughs than expected under the Gaussian assumption (Ochi, 1998). Meanwhile, it should be pointed out that wave nonlinearities are not only due to bottom effects (shallow water conditions). They can also be present in deep water conditions, as found by Casas-Prat and Holthuijsen (2010), for instance. In the following we briefly introduce the theoretical background of the second order nonlinear irregular waves. The fluid region is described using three-dimensional Cartesian coordinates (x, y, z), with x and y the horizontal coordinates, and z the vertical coordinate – with the positive z-direction opposing the direction of the gravitational acceleration. Time is denoted by t. The free surface is located at z¼ η(x, y, t). If the fluid is assumed to be ideal, incompressible and inviscid, and the fluid motion irrotational, so that the velocity potential Φðx; y; z; tÞ exists, then for constant water depth d the potential Φðx; y; z; tÞ and the surface elevation η(x, y, t) are determined by the following boundary value problem (see, for example, Toffoli et al., 2006): ∇2 Φ ¼ 0
ð2Þ
∂Φ 1 þ ð∇ΦÞ2 þ gz ¼ 0 ∂t 2
at
∂η ∂Φ ∂η ∂ Φ ∂ η ∂Φ þ þ ¼0 ∂t ∂x ∂x ∂y ∂y ∂z ∂Φ ¼0 ∂z
at
z ¼ d
z ¼ ηðx; y; tÞ at
z ¼ ηðx; y; tÞ
ð3Þ ð4Þ ð5Þ
Eq. (2) is the Laplace equation and Eqs. (3)–(5) are respectively the dynamic free surface boundary condition, the kinematic free surface boundary condition, and the bottom boundary condition. Solutions of system (2)–(5) can be sought using the following expansion (see, e.g., Toffoli et al., 2006): ( Φ ¼ Φð1Þ þ Φð2Þ þ … Φðn þ 1Þ ηðn þ 1Þ where ¼ ðnÞ ¼ OðεÞ ð6Þ η η ¼ ηð1Þ þ ηð2Þ þ … ΦðnÞ Here ε is a small parameter in expansion (6) and it is typically proportional to the wave steepness ξ ¼ kA, where k ¼2π/L (L ¼wavelength) is the wave number, and A is the wave amplitude which is equal to half the wave height. For an irregular sea state
18
Y. Wang / Ocean Engineering 87 (2014) 16–24
For deep water waves the quadratic transfer function Eðωn ; ωm Þ simplifies to:
characterized by a certain spectral density function SðωÞ where ω is the angular frequency, it is straightforward to show that a first order solution of system (2)–(5) takes the following form ( see, e. g., Marthinsen, 1992): igAn cosh kðz þdÞ iðωn t kn xÞ e cosh kd N 2 ωn
N
Φð1Þ ðx; tÞ ¼ Re ∑ n¼
N
ηð1Þ ðx; tÞ ¼ Re ∑
n ¼ N
Eðωn ; ωm Þ ¼
ð8Þ
N
^ η ¼ η ¼ ∑ A2n Eðωn ; ω n Þ m n¼1
An 2 n¼1 2 N
s2η ¼ ðη ηÞ2 ¼ ∑
m3 ¼ ðη ηÞ3 =sη 3 " N 3 ¼ ∑ An 2 Am 2 ðEðωn ; ωm Þ þ Eðωn ; ω m ÞÞ 2sη 3 n;m ¼ 1 N 1 ∑ A4n ðEðωm ; ωm Þ þ Eðωm ; ω m ÞÞ 2 n¼1 m4 ¼ ½ð4m3 Þ=32
ð10Þ
N
∑
n ¼ N m ¼ N
N
ηð2Þ ðx; tÞ ¼ Re ∑
N
∑
n ¼ N m ¼ N
s2η ¼ m3 ¼
2
ð11Þ
An Am Eðωn ; ωm Þeiðωn t kn xÞ eiðωm t km xÞ 4
ð12Þ
n ¼ 12
Z
1 0
6
s2η
SðωÞdω
Z
1 0
Z
1 0
½Eðω1 ; ω2 Þ þ Eðω1 ; ω 2 ÞSðω1 ÞSðω2 Þdω1 dω2
ðg 2 kn km =2ωn ωm Þ ð1=4Þðω2n þ ω2m þ ωn ωm Þ þ g 2 =4ðωn km þ ωm kn Þ=ðωn ωm ðωn þ ωm ÞÞ ðωn þ ωm Þ ðgkn þ km Þ=ðωn þ ωm Þtanhððkn þ km ÞdÞ 2
2
ð23Þ ð24Þ
2
ðgkn km =ωn ωm Þ ð1=2gÞðωn 2 þ ωm 2 þ ωn ωm Þ þ ðg=2Þðωn km þ ωm kn Þ=ðωn ωm ðωn þ ωm Þ 1 ðgkn þ km Þ=ðωn þ ωm Þ2 tanhððkn þ km ÞdÞ
ð22Þ
ð13Þ
where β1 and β2 are the empirical coefficients. β1 ¼ 4.37 and β2 ¼ 0.57 are recommended by Haring and Heideman (1978). However, the previous research work of this author's research group has shown that such kinds of empirical formulas will
The Kroenecker delta (δ n;m ¼ 1 if nþ m ¼0, zero otherwise) is introduced to avoid a singular Pðωn ; ωm Þ. The quadratic transfer function Eðωn ; ωm Þ is given by the following equation:
Eðωn ; ωm Þ ¼
ð19Þ
For nonlinear irregular waves, the wave crests will become higher and steeper, and the troughs of the nonlinear waves will become shallower and flatter. Obviously, the Rayleigh distribution which is good for predicting the crests of linear Gaussian waves will underestimate the crests of nonlinear irregular waves. An empirical correction to the Rayleigh model of the crest height distribution was suggested four decades ago by Jahns and Wheeler (1972) ( ) 2 h h h hZ0 ð25Þ PðAc o hÞ ¼ 1 exp 8 1 β1 β2 Hs d d
In the above two equations, Pðωn ; ωm Þ and Eðωn ; ωm Þ are referred to as quadratic transfer functions. The quadratic transfer function Pðωn ; ωm Þ is given by the following equation:
Pðωn ; ωm Þ ¼ ð1 δ n;m Þ
ð18Þ
ð20Þ
m4 ¼ ½ð4m3 Þ=32
An gkn t sinh 2kn h
eiðωn t kn xÞ eiðωm t km xÞ þ ∑
ð17Þ
0
iAn Am coshðkn þkm Þðz þ dÞ Pðωn ; ωm Þ coshðkn þkm Þd 2 N
ð16Þ
Letting N-1 and passing to a continuous representation lead to the following expressions. Z 1 ^η¼2 Eðω1 ; ω 1 ÞSðω1 Þdω1 ð21Þ m
where g and d are the acceleration of gravity and water depth, respectively. Shallow water wave data usually does not follow the linear Gaussian model. The linear Gaussian sea model can be corrected by including quadratic terms. Following Langley (1987) and Marthinsen (1992) the quadratic corrections are given by the following equations: N
ð15Þ
The leading order mean, variance, skewness and kurtosis of the surface elevation ηðx; tÞ are given by (see e.g., Marthinsen, 1992):
where Δω ¼ ωc =N and ωc is the upper cut off frequency beyond which the power spectral density function SðωÞ may be assumed to be zero for either mathematical or physical reasons. In Eqs. (7) and (8), the individual frequencies, ωn and wave numbers, kn are functionally related through the so-called linear dispersion relation:
Φð2Þ ðx; tÞ ¼ Re ∑
1 2 ω ω2m 2g n
ηðx; tÞ ¼ ηð1Þ ðx; tÞ þ ηð2Þ ðx; tÞ
as N tends to infinity. In Eqs. (7) and (8), Re denotes the real part of the complex number, and for each elementary sinusoidal wave An denotes its complex valued amplitude, ωn the angular frequency, and kn the wave number. Since ηð1Þ should be a real valued field, we need to assume that ω m ¼ ωm , k m ¼ km . If ηð1Þ is assumed to be stationary and Gaussian, then the complex amplitudes An are also Gaussian distributed, that is, An ¼ sn ðU n iV n Þ, where Un and Vn are independent zero mean and variance one Gaussian random variables, and s2n is the energy of waves with angular frequencies ωn and ωn. The mean square amplitudes are related to the wave spectrum SðωÞ by the following equation: 2 ð9Þ E½An ¼ 2Sðjωn jÞΔω
ω2n ¼ gkn tanhðkn dÞ
Eðωn ; ωm Þ ¼
where ωn and ωm are positive and satisfies the same relation as in the linear model. Finally, by combining Eqs. (8) and (12) the wave surface elevations for the second order nonlinear irregular waves can be written as follows:
ð7Þ
An iðωn t kn xÞ e 2
1 2 ðω þ ω2m Þ; 2g n
2
ðgkn km =2ωn ωm Þ þ ð1=2gÞðωn 2 þ ωm 2 þ ωn ωm Þ
ð14Þ
Y. Wang / Ocean Engineering 87 (2014) 16–24
sometimes predict wave crest distributions that differ considerably from the true ones (Wang and Xia, 2012). Therefore, the author of this article has proposed a Transformed Rayleigh method in order to calculate the crest distributions of nonlinear irregular waves more accurately and efficiently. In Section 3 of this article the theoretical background of this Transformed Rayleigh method will be elucidated.
3. The proposed Transformed Rayleigh method From Section 2 we know that ηð0; tÞ (corresponding to x ¼ 0 in Eq. (16)) is the height of the sea level at a fixed point as a function of time t. Here, we write ηðtÞ for ηð0; tÞ so that we can simplify the notation. We also use the so called mean down crossing wave to define a wave, i.e. the wave is considered as a part of a function between the consecutive down crossings of the mean sea level. Assume ηðtÞ crosses a mean sea level un a finite number of times, which are denoted by ti, 0 o t 1 o t 2 o :::;. The crest of the ith wave, M i n , is defined as the global maximum of ηðtÞ during the interval t i ot o t i þ 1 . For a specific nonlinear sea state, ηðtÞ will be a non-Gaussian random process. Here we will use the simplest (but widely effective) model for a non-Gaussian sea where ηðtÞ is expressed as a function of a stationary zero mean Gaussian process η~ ðtÞ with variance one (V½η~ ð0Þ ¼ 1), i.e. (Wang and Xia, 2013)
ηðtÞ ¼ Gðη~ ðtÞÞ;
dG 4 0; dη
Gð0Þ ¼ 0:
ð26Þ
We can see that the above formula is a deterministic and time instantaneous functional transformation. Note that once the distribution of crests in η~ ðtÞ is computed, then the corresponding wave crest distribution in ηðtÞ is obtained by simple transformations involving only the inverse of G, which we shall denote by gn. Actually we shall use the function gn to define the transformation instead of G, and use the relation η~ ðtÞ ¼ g n ðηðtÞÞ between the real sea data ηðtÞ and the transformed data η~ ðtÞ. In our research, we have tried to determine the G-function (or gn-function which is the inverse of G) directly from the cumulative distribution functions of η and η~ . For this purpose, we need to estimate the cumulative distribution functions (or probability density functions f η ðηÞ or f η~ ðη~ Þ) from moments because usually in an offshore engineering design project the only information we have on the waves is the wave power spectral density function. From Winterstein (1988) we have the following relationships:
μη ðuÞ f η ðηÞ ¼ ^ η Þ f η ðm ^ ηÞ μðm
ð27Þ
μη ðuÞ ðG ðuÞÞ ¼ exp ^ ηÞ 2 μðm 1
2
!
n
ðg ðuÞÞ ¼ exp 2
2
2 n z^ d exp n dz 2
! ¼ ð 1Þ H n ðz^ Þexp
hn ¼
in which μη ðuÞ denotes the level up-crossing rate of the level μ by ^ η denotes the mean value of ηðtÞ. It is convenient to ηðtÞ; and m ^ η Þ=sη in which sη consider the standardized variable z^ ¼ ðηðtÞ m denotes the standard deviation of ηðtÞ. Note that z^ ¼ η~ ðtÞ. Then the non-Gaussian probability density function of z can be expressed as the following Gram–Charlier series: 2 ffiexp z^2 f ðz^ Þ ¼ p1ffiffiffiffi 2π h i 4 þ 30 H 6 ðz^ Þ þ ::: 1 þ m3!3 H 3 ðz^ Þ þ m44! 3H 4 ðz^ Þ þ m5 5!10m3 H 5 ðz^ Þ þ m6 15m 6! ! 2 N z^ 1 ¼ pffiffiffiffiffiffiexp ð29Þ 1 þ ∑ hn H n ðz^ Þ 2 2π n¼1 in which the Hermite polynomials of degree n, denoted by H n ðz^ Þ, is defined as a function which satisfies the relationship given by
2 z^ 2
!
1 E½Hen ðz^ Þ n!
ð30Þ
ð31Þ
We can then use the following formula for calculating the Gfunction (or gn-function which is the inverse of G) (Winterstein, 1988): ^ η þ k^ sη ½z^ þ h~ 3 ðz^ 1Þ þ h~ 4 ðz^ 3z^ Þ þ ::: GðzÞ ¼ m 2
3
ð32Þ
The coefficients h~ n control the shape of the standardized distribution, while k^ is a scaling factor. The coefficients h~ n may be related to the Hermite moments hn in Eq. (31) by applying a Hermite polynomial to Eq. (32) and take expectations. For N ¼4 moments, the solution is (Winterstein et al., 1994): m3 h~ 3 ¼ ð1 0:015jm3 j þ 0:3m3 2Þ=ð1 þ 0:2ðm4 3ÞÞ 6
ð33Þ
0:8 h~ 4 ¼ 0:1ðð1 þ 1:25ðm4 3ÞÞ1=3 1Þð1 1:43m23 =ðm4 3ÞÞð1 0:1m4 Þ
ð34Þ We can clearly see that the accuracy of the G-function calculated according to Eq. (32) depends on how many moments are kept during the calculation. The more moments are kept, the more accurate the G-function will be. We can also notice that in order to calculate the G-function according to Eq. (32), we need to know the values of the skewness, kurtosis, mean and standard deviation of the second order nonlinear random waves. These values can be calculated using Eqs. (21)–(24) in this paper. Next, we turn to the relation between the level up-crossing rate μη ðuÞ and the probability distribution of the crest height F Mn ðuÞ. As in Wang and Xia (2013), we have that
0rzru
ð28Þ
n
In Eq. (29), the parameters m3, m4, m5, m6, etc. are the moments of the standardized variable z^ and these moments can be calculated based on the wave power spectral density functions. m3 is the skewness, and m4 is the kurtosis. Now it seems that by combining Eqs. (27)–(30) and solving them we can obtain the Gfunction (or gn-function which is the inverse of G). However, the Gram–Charlier series can behave erratically, yielding multimodal and even negative probability densities. Here we consider to use a procedure as specified in Winterstein (1988) to calculate the G-function (or gn-function which is the inverse of G). This precedure is based on the notion of a “Hermite moment” which is the coefficient hn in Eq. (29).
1 F Mn ðuÞ r min
!
19
μη ðuÞ ^ ηÞ μðm
ð35Þ
Combining Eqs. (28) and (35) we obtain the following relationship: ! ðG 1 ðuÞÞ2 F Mn ðuÞ Z1 exp ð36Þ 2 for all u Z0, and G 1 ðuÞ is defined by Eq. (32). Eq. (36) can also be written in a form of ! ðG 1 ðuÞÞ2 F Mn ðuÞ Z1 exp ð37Þ 2m0 where m0 is the zero-order spectral moment of the stationary zero mean and variance one Gaussian process η~ ðtÞ, m0 ¼ s2η~ ¼ 1. We notice that the exponent in the right part of Eq. (37) is the Rayleigh probability that the linear crest exceeds the predefined threshold u. That is why this model is called a Transformed Rayleigh method.
20
Y. Wang / Ocean Engineering 87 (2014) 16–24
4. Application examples, calculation results and discussions 4.5 4
2
S(w) [m s / rad]
In this section we utilize the Transformed Rayleigh method to calculate the crest statistics of shallow water nonlinear waves based on standard spectra and measured data at the Poseidon platform. The proposed new approach is first applied for calculating the wave crest height exceedance probabilities of sea states with standard JONSWAP spectra corresponding to different water depths (15 m, 25 m, 35 m, 45 m, 135 m). The JONSWAP formulation is based on an extensive wave measurement program known as the Joint North Sea Wave Project carried out in 1968 and 1969 along a line extending over 160 km into the North Sea from Sylt Island. The spectrum represents wind-generated seas with fetch limitation (Ochi, 1998). The mathematical expression of the infinite water depth JONSWAP frequency spectrum with a significant wave height Hs and spectral peak period Tp is as follows: ! ð2πω=T 1Þ2 g2 H2 5 2π 4 exp 2δp2 SðωÞ ¼ 5:061 5 4s ð1 0:287 lnðγ ÞÞexp γ ω Tp 4 ωT p
Spectral density
2.5 2 1.5 1
0
0.4
Inp the ffiffiffiffiffiffiffiffiabove formula ωd is a dimensionless frequency defined by ω d=g and kd is the wave number associated with the linear dispersion relation
ω2 ¼ gkd tanhðkd dÞ
ð40Þ
where g and d are the acceleration of gravity and water depth, respectively. In formula (39), Sc ðωÞ represents the finite water depth spectrum, and SðωÞ represents the infinite water depth spectrum. Fig. 1 shows the obtained corrected JONSWAP spectra with a significant wave height Hs ¼4.0 m, a spectral peak period Tp ¼10.0 s, and γ ¼3.3 corresponding to different water depths (15 m, 25 m, 35 m, 45 m, 135 m). It should be pointed out that the figure of a JONSWAP spectrum with a water depth of 135 m is almost identical to that of a JONSWAP spectrum with an infinite water depth. Fig. 2 shows the calculation results of the wave crest height exceedance probabilities for a sea state with a JONSWAP spectrum with Hs ¼4.0 m, Tp ¼10.0 s and γ ¼3.3 for a water depth of 15 m. It should be clarified that in this figure X is the deviation from the mean water level. Fig. 2 shows the probability of the wave crest height, X, exceeding the range of possible deviations in surface elevation from the mean water level, x. The solid green line in Fig. 2 represents the Monte Carlo Simulation (MCS) results of the wave crest height exceedance probabilities of this shallow water nonlinear sea state. In the process of the MCS simulation, a wave elevation time series consisting of 2,000,000 points (enough to reduce the simulation variance) were first generated based on the
0.7
0.8
0.9
1
1.1
1.2
The probability of X exceeding x
100 10-1 10-2
Monte Carlo Simulation Transformed Rayleigh method Un-modified Jahns and Wheeler model Modified Jahns and Wheeler model
10-3 10-4
In shallow water, in order to consider the correction for the effect of bottom on the wave spectrum, a “corrected wave spectrum Sc ðωÞ” is obtained by multiplying the original “standard JONSWAP spectrum SðωÞ” by a function ϕðωd Þ that ranges between 0 and 1 according to the similarity law of Buows et al. (1985) " # ðkðω; dÞÞ 3 ð∂=∂ωÞkðω; dÞ Sc ðωÞ ¼ SðωÞϕ½ωd ¼ SðωÞ ðkðω; 1ÞÞ 3 ð∂=∂ωÞkðω; 1Þ " # ðkd Þ 3 ð∂=∂ωÞkd ¼ SðωÞ ð39Þ ðk1 Þ 3 ð∂=∂ωÞk1
0.6
Fig. 1. JONSWAP spectra with Hs ¼ 4.0 m, Tp ¼ 10.0 s and γ¼ 3.3 for different water depths.
δ ¼ 0:09 if ω Z 2π =T p pffiffiffiffiffiffi γ ¼ expð3:484ð1 0:1975ð0:036 0:0056T p = H s ÞT 4p =H2s ÞÞ
0.5
Frequency [rad/s]
1-F(x)
δ ¼ 0:07 if ω o 2π =T p
h=15m h=25m h=35m h=45m h=135m
3
0.5
ð38Þ where
3.5
0
0.5
1
1.5
2
2.5
x (m) Fig. 2. Wave crest height exceedance probabilities for a sea state with JONSWAP spectrum with Hs ¼ 4.0 m, Tp ¼ 10.0 s and γ ¼3.3 for a water depth of 15 m. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
corrected spectrum by summation of sine functions with random phase angles uniformly distributed in the range of [0, 2π]. The wave crest height time series were then extracted from these 2,000,000 wave elevation points. After extracting these time series, exact Epanechnikov kernel density estimates were performed for obtaining the probability density function of the wave crest heights. Then cumulative trapezoidal numerical integration was carried out on the above probability density function in order to get the probability distribution (F) of the wave crest heights. Finally, the wave crest height exceedance probabilities were obtained based on the probability distribution by utilizing the formula P ¼1 F. The above MCS results are utilized as the criteria against which the accuracy of the results from the transformed Rayleigh method or from the empirical crest distribution models is checked. In Fig. 2, the red solid line represents the results of the wave crest height exceedance probabilities obtained from utilizing the Transformed Rayleigh method. In applying the Transformed Rayleigh method, the mean, variance, skewness and kurtosis of the surface elevation were first calculated based on the corrected JONSWAP spectrum by using Eqs. (21)–(24). Then the g*-function (which is the inverse of the G-function) was calculated by using Eqs. (32)–(34). We can see that the Transformed Rayleigh method gave the closest predictions of the results than those predicted using the other methods. In Fig. 2, the small blue “*” represents a wave crest height exceedance probability predicted by using the original Jahns and Wheeler model (i.e. by using Eq. (25)) in which the value of Hs was taken to be 4.0 m. We can notice that the original Jahns and Wheeler model gave poorer predictions in the wave crest height region of about [0.5 m, 2.5 m] than the Transformed Rayleigh method did. In Fig. 2, the solid blue line represents the wave crest height exceedance probabilities
Y. Wang / Ocean Engineering 87 (2014) 16–24
1-F(x)
10-1 Monte Carlo Simulation Transformed Rayleigh method Un-modified Jahns and W heeler model Modified Jahns and Wheeler model
10-2
10
-3
1-F(x)
10-1 10-2
10
Monte Carlo simulation Transformed Rayleigh method Un-modified Jahns and W heeler model Modified Jahns and Wheeler model
-3
10-4
0
0.5
1
1.5
2
Fig. 5. Wave crest height exceedance probabilities for a sea state with JONSWAP spectrum with Hs ¼ 4.0 m and Tp ¼10.0 s for a water depth of 45 m.
The probability of X exceeding x
100 10-1 10-2
10
Monte Carlo simulation Transformed Rayleigh method Un-modified Jahns and W heeler model Modified Jahns and Wheeler model
-3
10-4
0
0.5
1
1.5
2
Fig. 6. Wave crest height exceedance probabilities for a sea state with JONSWAP spectrum with Hs ¼ 4.0 m and Tp ¼10.0 s for a water depth of 135 m.
S þ ðωÞ ¼ ∑ SJþ ðω; H s;i ; ωp;i ; λi ; N i M i ; αi Þ i¼1
0.5
1
1.5
2
2.5
x (m) Fig. 3. Wave crest height exceedance probabilities for a sea state with JONSWAP spectrum with Hs ¼ 4.0 m and Tp ¼ 10.0 s for a water depth of 25 m.
The probability of X exceeding x
100
where i¼ 1 means the first peak and i ¼2 means the second peak, and SJ þ ð Þ is the JONSWAP spectrum defined by the following equation: ðω=ω 1Þ2 g2 M ωp N exp 2ps2 þ ð42Þ SJ ðωÞ ¼ α M exp λ N ω ω
s ¼ 0:07 if ω o ωp
10
1-F(x)
ð41Þ
where
-1
10-2
10
2.5
x(m)
2
10-4
0
2.5
x (m)
can conclude that it is more advantageous to apply the Transformed Rayleigh method to predict the wave crest height exceedance probabilities of the “more shallow water” nonlinear sea states. As another example, the Transformed Rayleigh method was utilized to calculate the wave crest height exceedance probabilities for a combined swell and wind sea state characterized by a Torsethaugen bimodal spectrum. The Torsethaugen bimodal spectrum is expressed as follows:
The probability of X exceeding x
100
The probability of X exceeding x
100
1-F(x)
calculated by utilizing a “corrected” Jahns and Wheeler model (i.e. the value of Hs is taken to be four times the standard deviation of the surface elevation based on the corrected JONSWAP spectrum). We find that the “corrected” Jahns and Wheeler model gave much better predictions than the original Jahns and Wheeler model. However, the Transformed Rayleigh method still gave slightly better predictions in the wave crest height region of about [1.5 m, 2.5 m] than the “corrected” Jahns and Wheeler model did. Finally, it is mentioned that in this case, the time series simulation with 2,000,000 wave elevation points and the post statistical processing for getting the continuous green line in Fig. 2 took about 486 s on a desktop computer. By comparison, it took less than 2 s to obtain the results of the wave crest height exceedance probabilities represented by the red solid line in Fig. 2 by applying the Transformed Rayleigh method. The accuracy and efficiency of the Transformed Rayleigh method for calculating the crest height distributions of shallow water nonlinear waves can thus be validated. We have carried out similar calculations for the corrected JONSWAP spectra with a significant wave height Hs ¼4.0 m, a spectral peak period Tp ¼10.0 s, and γ ¼3.3 corresponding to other water depths (25 m, 35 m, 45 m, 135 m) and obtained similar results. Our calculation results are summarized in Figs. 3–6. We can see that the tendencies of the four lines in Fig. 3 (or in Fig. 4, or in Fig. 5, or in Fig. 6) are similar to those of the four lines in Fig. 2. If we look more closely at Figs. 2–6 and compare them, we can find that in Fig. 6 the differences between the results from the original Jahns and Wheeler model and the Transformed Rayleigh method are the smallest, while in Fig. 2 the corresponding differences are the biggest. Meanwhile, we can notice that in all cases the Transformed Rayleigh method had predicted slightly better results than the corrected Jahns and Wheeler model had. Therefore, we
21
s ¼ 0:09 if ω Z ωp
Monte Carlo Simulation Transformed Rayleigh method Un-modified Jahns and W heeler model Modified Jahns and W heeler model
-3
M¼5; N ¼4.
α 5:061
10-4
H 2s T 4p
ð1 0:287 lnðλÞÞ pffiffiffiffiffiffi
0
0.5
1
1.5
2
2.5
x(m) Fig. 4. Wave crest height exceedance probabilities for a sea state with JONSWAP spectrum with Hs ¼ 4.0 m and Tp ¼ 10.0 s for a water depth of 35 m.
λ ¼ expð3:484ð1 0:1975ð0:036 0:0056T p = Hs ÞðT 4p =H2s ÞÞ In the above JONSWAP spectrum Hs is significant wave height and Tp is spectral peak period. The values of the parameters H s1 ; ωp1 ; λ1 ; α1 H s2 ; ωp2 ; λ2 ; α2 in the Torsethaugen bimodal
22
Y. Wang / Ocean Engineering 87 (2014) 16–24
Spectral density 2
S(w) [m s / rad]
1.5 Original Torsethaugen spectrum Modified Torsethaugen spectrum
2
1
0.5
0 0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
Frequency [rad/s]
Fig. 7. A shallow water (d ¼25 m) Torsethaugen spectrum with Hs ¼4 m and Tp ¼ 7 s (solid pink line) and the corresponding infinite water depth spectrum (black line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The probability of X exceeding x
100
1-F(x)
10-1 10-2
10
Monte Carlo simulation Transformed Rayleigh method Un-modified Jahns and W heeler model Modified Jahns and Wheeler model
-3
10-4
0
0.5
1
1.5
2
2.5
3
3.5
x(m) Fig. 8. Wave crest height exceedance probabilities for a sea state with Torsethaugen spectrum with Hs ¼4.0 m and Tp ¼ 7.0 s for a water depth of 25 m. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
spectrum can all be obtained if the values of Hs and Tp are given (For details, please refer to: Brodtkorb (2004)). A Torsethaugen bimodal spectrum with Hs ¼4 m and Tp ¼7 s has been plotted in Fig. 7 and is shown as a black line. In order to study the shallow water effects, a “modified Torsethaugen spectrum” is obtained by multiplying the original Torsethaugen spectrum by a function ϕðωd Þ that ranges between 0 and 1 according to the similarity law of Buows et al. (1985): # " # " ðkðω; dÞÞ 3 ð∂=∂ωÞkðω; dÞ ðkd Þ 3 ð∂=∂ωÞkd ϕ½ωd ¼ ð43Þ ¼ ðkðω; 1ÞÞ ¼ 3 ð∂=∂ωÞkðω; 1Þ ðk1 Þ 3 ð∂=∂ωÞk1 Inp the ffiffiffiffiffiffiffiffiabove formula ωd is a dimensionless frequency defined by ω d=g and kd is the wave number associated with the linear dispersion relation:
ω2 ¼ gkd tanhðkd dÞ
ð44Þ
where g and d are the acceleration of gravity and water depth, respectively. In Fig. 7 the solid pink line represents the obtained “modified Torsethaugen spectrum” with a water depth of 25 m (i.e. d ¼25 m). Fig. 8 shows our calculation results of the wave crest height exceedance probabilities for a sea state with the above “modified Torsethaugen spectrum” with Hs ¼ 4.0 m and Tp ¼ 7.0 s. The solid green line in Fig. 8 represents the Monte Carlo simulation results of the wave crest height exceedance probabilities of this shallow water nonlinear sea state. In the process of the MCS simulation, a wave elevation time series consisting of 2,000,000 points were first generated based on the modified Torsethaugen spectrum by
summation of sine functions with random phase angles uniformly distributed in the range of [0, 2π]. The wave crest height time series were then extracted from these 2,000,000 wave elevation points. Finally, post statistical processing were conducted for getting the wave crest height exceedance probabilities based on these wave crest height time series. In Fig. 8, the red solid line represents the results of the wave crest height exceedance probabilities obtained from utilizing the Transformed Rayleigh method (by applying Eqs. (21)–(24) and Eqs. (32)–(37)). In Fig. 8, the small blue “*” represents a wave crest height exceedance probability predicted by using the original Jahns and Wheeler model (i.e. by using Eq. (25)) in which the value of Hs was taken to be 4.0 m. We can find that the original Jahns and Wheeler model gave poorer predictions in the wave crest height region of about [1 m, 3.5 m] than the Transformed Rayleigh method did. In Fig. 8, the solid blue line represents the wave crest height exceedance probabilities calculated by utilizing a “corrected” Jahns and Wheeler model (i.e. the value of Hs was taken to be four times the standard deviation of the surface elevation based on the “modified Torsethaugen spectrum”). We can find that the “corrected” Jahns and Wheeler model gave much better predictions than the original Jahns and Wheeler model did, and the predictions from using the “corrected” Jahns and Wheeler model and from using the Transformed Rayleigh method are almost identical. Finally, it is mentioned that in this case, the Monte Carlo simulation and the post statistical processing for getting the continuous green line in Fig. 8 took about 114 s on a desktop computer. By comparison, it took less than 2 s to obtain the results of the wave crest height exceedance probabilities represented by the red solid line in Fig. 8 by applying the Transformed Rayleigh method. The accuracy and efficiency of the Transformed Rayleigh method for calculating the crest height distributions of shallow water nonlinear waves are again readily validated. Finally, the Transformed Rayleigh method was applied to predict the wave crest height exceedance probabilities of a sea state with the surface elevation data measured at the Poseidon platform in the Japan Sea. Poseidon was located 3 km off the coast of Yura in the Yamagata prefecture, in the Japan Sea during the measurements. The water depth at the Poseidon platform is 41 m. Poseidon is a full scale test structure that consists twelve legs with footings which support the upper structure. The upper structure is mainly composed of box-typed girders around four sides. The instrumentation house is arranged on the upper deck for power supply and data acquisition (Yago et al., 1991). The surface elevation data at the platform were measured with three ultrasonic wave gauges located at the sea floor from 8:12 a.m. on November 24, 1987 to 7:57 a.m. on November 25, 1987, and three sets of data each consisting of 85,547 wave elevation points were obtained. Fig. 9 shows the first 300 wave elevation points time series measured with the 3# ultrasonic wave gauge (Data source: http://code.google.com/p/wafo/downloads/detail?name=wafo25. 7z. These data are used for academic research only and not for commercial purposes). We can clearly notice that the crests are higher and the troughs are shallower in this time series. Fig. 10 shows the power spectrum corresponding to the 85547 wave elevation points measured with the 3# ultrasonic wave gauge at the Poseidon platform. Fig. 11 shows the calculated wave crest height exceedance probabilities in this study. The solid green line in Fig. 11 represents the results of the wave crest height exceedance probabilities directly obtained from the 85547 wave elevation points measured with the 3# ultrasonic wave gauge at the Poseidon platform. For obtaining these results, the wave crest height time series were first extracted from these 85,547 wave elevation points. After extracting these time series, exact Epanechnikov kernel density estimates were performed for obtaining the probability density function of the wave crest
Y. Wang / Ocean Engineering 87 (2014) 16–24
Surface elevation from mean water level (MWL).
3 - +2
2 1 0 -1
Standard Deviation
Distance from MWL.(m)
4
-2 - -2 -3 0
50
100
150
200
250
Time (sec)
Fig. 9. The first 300 wave elevation points time series measured at the Poseidon platform.
Spectral density 4
2
S(w) [m s / rad]
3.5 3 2.5
identical predictions of the results to those directly obtained from the measured data. In Fig. 11, the small blue “*” represents a wave crest height exceedance probability predicted by using the Jahns and Wheeler model (i.e. by using Eq. (25)) in which the value of Hs was taken to be four times the standard deviation of the measured data. We can notice that the Jahns and Wheeler model gave poorer predictions in the wave crest height region of about [1.5 m, 5 m] than the Transformed Rayleigh method did. It should be noted that in this case (the Poseidon data) the un-modified Jahns and Wheeler model and the modified Jahns and Wheeler model are the same, i.e. the two models are identical. Obviously the Hs value in this case should not be modified further (because it is based on the measured data), i.e. in this case we do not need to have a modified Jahns and Wheeler model. Finally, it is mentioned that in this case, it took less than 2 s to obtain the results of the wave crest height exceedance probabilities represented by the solid red line in Fig. 11 by applying the Transformed Rayleigh method. The accuracy and efficiency of the Transformed Rayleigh method for calculating the crest height distributions of shallow water nonlinear waves have been once again convincingly validated.
2 1.5
5. Concluding remarks
1 0.5 0
0
0.5
1
1.5
2
2.5
Frequency [rad/s] Fig. 10. The spectrum corresponding to the measured data at the Poseidon platform.
The probability of X exceeding x 100 10-1
1-F(x)
23
Measured Poseidon data Transformed Rayleigh method Jahns and Wheeler model
10-2
10
-3
10-4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
X(m)
Fig. 11. Wave crest height exceedance probabilities for the spectrum corresponding to the measured data at the Poseidon platform. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
heights. Then cumulative trapezoidal numerical integration was carried out on the above probability density function in order to get the probability distribution (F) of the wave crest height. Finally, the wave crest height exceedance probabilities were obtained based on the probability distribution by utilizing the formula P ¼1 F. In Fig. 11, the solid red line represents the results of the wave crest height exceedance probabilities obtained from utilizing the Transformed Rayleigh method. In applying the Transformed Rayleigh method, the mean, variance, skewness and kurtosis of the surface elevation were first calculated based on the spectrum in Fig. 10 by using Eqs. (21)–(24). Then the gn-function (which is the inverse of the G-function) was calculated by using Eqs. (32)–(34). We can see that the Transformed Rayleigh method gave almost
This article proposed a new approach for calculating the wave crest distribution of shallow water nonlinear waves by utilizing a Transformed Rayleigh method where the second order nonlinear wave model is incorporated. The essence of the proposed new approach is based on a deterministic and time instantaneous functional transformation, and the detailed procedures and formulas for obtaining this transformation have been specified in this paper. The proposed new approach has been applied for calculating the wave crest height exceedance probabilities of sea states with a standard JONSWAP spectrum, with a bimodal Torsethaugen spectrum and with the surface elevation data observed at the Poseidon platform. It has been demonstrated that in all cases the Transformed Rayleigh method can offer better predictions than those using the empirical wave crest distribution models. Another advantage of using the proposed new approach is that the correction for the effect of bottom on the wave spectrum, a procedure that was frequently overlooked by some researchers, has been integrated into the Transformed Rayleigh method. The accuracy and efficiency of the proposed new approach have also been validated by using Monte Carlo simulations. The research results gained from this study demonstrate that the proposed Transformed Rayleigh method can be used for engineering purposes.
Acknowledgment The work reported in this article is supported by the funding of an independent research project from the Chinese State Key Laboratory of Ocean Engineering (Grant no. GKZD010038). Special thanks are due to the two anonymous reviewers of this article for their useful comments that have led to the improved quality of this paper.
References Brodtkorb, P.A., 2004. The Probability of Occurrence of Dangerous Wave Situations at Sea (Ph.D. thesis). Norwegian University of Science and Technology, NTNU, Trondheim, Norway.
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Buows, E., Gunther, H., Rosenthal, W., Vincent, C.L., 1985. Similarity of the wind wave spectrum in finite depth water: 1 spectral form. J. Geophys. Res. 90 (C1), 975–986. Casas-Prat, M., Holthuijsen, L.H., 2010. Short-term statistics of waves observed in deep water. J. Geophys. Res. 115, C09024 (20 pp.). Chakrabarti, S.K., 1987. Hydrodynamics of Offshore Structures. Computational Mechanics Publications Inc., Billerica. Colwell, S., Basu, B., 2009. Tuned liquid column dampers in offshore wind turbines for structural control. Eng. Struct. 31, 358–368. Forristall, G.Z., 2000. Wave crest distributions: observations and second-order theory. J. Phys. Oceanogr. 30 (8), 1931–1943. Graber, H.C., Madsen, O.S., 1988. A finite-depth wind-wave model. Part 1: model description. J. Phys. Oceanogr. 18, 1465–1483. Haring, R.E., Heideman, J.C., 1978. Gulf of Mexico rare wave return periods. Offshore Technology Conference, vol. 3230, Houston. Huang, N.E., Long, S.R., Tung, C.C., Yuen, Y., Bliven, L.F., 1983. A non-Gaussian statistical model for surface elevation of nonlinear random wave fields. J. Geophys. Res. Am. Geophys. Union, 88; , pp. 7597–7606. Jahns, H.O., Wheeler, J.D., 1972. Long term wave probabilities based on hindcasting of severe storms. Offshore Technology Conference, vol. 1590, Houston, U. S. A. Yago, Kiyokazu, Ohkawa, Yutaka, Saito, Masakatsu, Sekita, Kinji, 1991. At sea experiment of a floating offshore structure, Part 5, Measurement of stress range distributions for fatigue analysis. J. Soc. Naval Archit. Jpn. 170, 609–618 (in Japanese). Kriebel, D.L., Dawson, T.H., 1991. Nonlinear effects on wave groups in random seas. J. Offshore Mech. Arctic Eng. 113, 142–147. Kriebel, D.L., T.H. Dawson, 1993. Nonlinearity in wave crest statistics. In: Proceedings of the Second International Symposium on Ocean Wave Measurement and Analysis New Orleans, LA, ASCE, pp. 61–75.
Langley, R.S., 1987. A statistical analysis of non-linear random waves. Ocean Eng. 14 (5), 389–407. Longuet-higgins, M.S., 1952. On the statistical distribution of the heights of sea waves. J. Mar. Res. 11, 245–266. Marthinsen, T., 1992. On the Statistics of Irregular Second Order Waves. Report No. RMS-ll. Standford University. Ochi, M.K., 1998. Ocean Waves, the Stochastic Approach. Cambridge University Press, Cambridge. Shi, W., Park, H., Chung, C., Baek, J., Kim, Y., Kim, C., 2013. Load analysis and comparison of different jacket foundations. Renew. Energy 54, 201–210. Sweetman, B., Winterstein, S.R., 2001. Airgap prediction: use of second order diffraction and multi-column models. In: Proceedings of the 11th International Offshore and polar Engineering Conference. Stavanger, Norway, pp. 390–397. Tayfun, M.A., 1980. Narrow-band nonlinear sea waves. J. Geophys. Res. 85, 1548–1552. Toffoli, A., Onorato, M., Monbaliu, J., 2006. Wave statistics in unimodal and bimodal seas from a second-order model. Eur. J. Mech. B-Fluids 25, 649–661. Trumars, J.M.V., Tarp-Johansen, N.J., Krogh, T., 2005. Fatigue loads on offshore wind turbines due to weakly non-linear waves. In: Proceedings of the 24th International Conference on Offshore Mechanics and Arctic Engineering (OMAE 2005). Halkidiki, Greece. Wang, Y.G., Xia, Y.Q., 2012. Simulating mixed sea state waves for marine design. Appl. Ocean Res. 37, 33–44. Wang, Y.G., Xia, Y.Q., 2013. Calculating nonlinear wave crest exceedance probabilities using a Transformed Rayleigh method. Coast. Eng. 78, 1–12. Winterstein, S.R., 1988. Nonlinear vibration models for extremes and fatigue. J. Eng. Mech. 114 (10), 1772–1790. Winterstein, S.R., Ude, T.C., Kleiven, G., 1994. Springing and slow drift responses: predicted extremes and fatigue vs. simulation. In: Proceedings of the 7th International behaviour of Offshore structures (BOSS), vol. 3, pp. 1–15.
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Calculating crest statistics of shallow water nonlinear waves based on standard spectra and measured data at the Poseidon platform Yingguang Wang n Department of Naval Architecture and Ocean Engineering, School of Naval architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
art ic l e i nf o
a b s t r a c t
Article history: Received 2 July 2013 Accepted 22 May 2014
This paper introduces a new approach for the calculation of the wave crest distribution of shallow water nonlinear waves by utilizing a Transformed Rayleigh method where the second order nonlinear wave model is incorporated. The proposed Transformed Rayleigh method is based on a deterministic and time instantaneous functional transformation. It is very efficient and accurate and can be used for engineering purposes. Meanwhile, the correction for the effect of bottom on the wave spectrum, a procedure that is frequently overlooked by some researchers, has been integrated into the Transformed Rayleigh method. The proposed new approach has been first applied for calculating the wave crest height exceedance probabilities of sea states with standard JONSWAP spectra corresponding to different water depths, and the calculation results have been favorably validated against Monte Carlo simulation results. Meanwhile, the wave crest height exceedance probabilities of these sea states obtained from using the Jahns and Wheeler finite depth wave crest height distribution model have also been included for comparison purpose. It is found that in all cases the Transformed Rayleigh method can offer better predictions than the Jahns and Wheeler model. The Transformed Rayleigh method is then applied to calculate the wave crest height exceedance probabilities of a combined sea state with a bimodal Torsethaugen spectrum corresponding to a water depth of 25 m, and its accuracy and efficiency are again favorably validated by using Monte Carlo simulations. Finally, the Transformed Rayleigh method is applied to predict the wave crest height exceedance probabilities of a sea state with the surface elevation data measured at the Poseidon platform in the Japan Sea, and its accuracy and efficiency have been once again convincingly substantiated. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Crest height Nonlinear waves Transformed Rayleigh method Standard spectra Measured data Poseidon platform
1. Introduction Accurate prediction of the wave crest height exceedance probabilities is of vital importance to the design of both offshore platforms and offshore wind turbines. An important parameter regarding the structural safety of an offshore platform is the height from the still water level to the lowest deck level of the platform, and the value of this parameter is commonly denoted to be a stillwater airgap distance in the offshore engineering literature (Sweetman and Winterstein, 2001). In order to choose a proper value for this still-water airgap in the design process for the platform, accurately predicting the wave crest height distribution at the location of the offshore platform is inevitably required. On an offshore wind turbine, there are some parts of the turbine structure (such as the blades) which are not designed to be exposed to hydrodynamic loads. These parts shall be positioned at a height
n
Tel.: þ 86 21 34206514; fax: þ86 21 34206701. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.oceaneng.2014.05.012 0029-8018/& 2014 Elsevier Ltd. All rights reserved.
with a minimum clearance relative to the expected value of the highest crest elevation with a specific recurrence period. This also requires the designer to accurately calculate the wave crest height exceedance probabilities at the location of the offshore wind turbine. This paper concentrates on the calculation of the wave crest height distribution of shallow water nonlinear waves. The motivation for carrying out this research is twofold. First, at present offshore wind turbines are usually installed in shallow water areas. The author of this article has studied the papers written by some wind turbine researchers (such as Colwell and Basu (2009), Trumars et al. (2005), Shi et al. (2013); to name just a few). These researchers were found to have overlooked the correction for the effect of bottom on the wave spectrum during their offshore wind turbine analysis. As pointed by Graber and Madsen (1988), shoaling of waves as they propagate in water of slowly varying depth and dissipation through bottom friction will affect the evolution of the wave field. This author has thus developed a particular interest in figuring out the negative impact on the wave crest prediction when an uncorrected wave spectrum is used for the offshore wind turbine analysis. Second, it is well known that in the ocean engineering
Y. Wang / Ocean Engineering 87 (2014) 16–24
Nomenclature h Hs
Φ η k
ω
crest height g the acceleration of gravity significant wave height velocity potential surface elevation wave number angular frequency
literature there exist some empirical wave crest distribution models (such as those in Tayfun (1980), Huang et al. (1983), Kriebel and Dawson (1991) (1993), Forristall (2000), etc.) that can be conveniently used for engineering purposes. However, the previous research work of this author's research group (Wang and Xia, 2012) has pointed out that “these empirical wave statistics formulas in the existing literature should be used with caution. Although these formulas are generally very concise and easy to use, sometimes they predict wave statistics results deviating considerably from the true ones”. Therefore, there is a strong motivation for this author to develop a computationally efficient method for calculating the shallow water wave crest distribution that is more accurate than the empirical wave crest distribution models. Summarizing the research work of the author of this article, this paper introduces a new approach for the calculation of the wave crest distribution of shallow water waves by utilizing a Transformed Rayleigh method where the second order nonlinear wave model is incorporated. The proposed Transformed Rayleigh method is a deterministic and time instantaneous functional transformation that is very efficient and accurate and can be used for engineering purposes. Meanwhile, the correction for the effect of bottom on the wave spectrum, a procedure that is frequently overlooked by some researchers, has been integrated into the Transformed Rayleigh method. The proposed new approach will be first applied for calculating the wave crest height exceedance probabilities of sea states with standard JONSWAP spectra corresponding to different water depths, and the results will be validated against Monte Carlo simulation results. Meanwhile, the wave crest height exceedance probabilities of these sea states obtained from using the Jahns and Wheeler finite depth wave crest height distribution model will also be included for comparison purpose. The Transformed Rayleigh method will also be applied to calculate the wave crest height exceedance probabilities for a combined sea state with a bimodal Torsethaugen spectrum corresponding to a water depth of 25 m, and its accuracy and efficiency will again be validated by using Monte Carlo simulations. Finally, the Transformed Rayleigh method will be applied to predict the wave crest height exceedance probabilities of a sea state with the surface elevation data measured at the Poseidon platform in the Japan Sea, and its accuracy and efficiency will once again be substantiated. This paper begins in Section 2 by introducing the knowledge of the second order nonlinear irregular waves. It continues in Section 3 by elucidating the theoretical background of the proposed Transformed Rayleigh method, and in Section 4 the method is applied to calculate the wave crest height exceedance probabilities of shallow water nonlinear waves based on standard spectra and measured data at the Poseidon platform. These results are compared with those from Monte Carlo simulations and from empirical distribution models, with concluding remarks provided in Section 5.
2. The second order nonlinear irregular waves The small oscillation amplitude ocean waves in a sufficiently deep sea are considered to be a Gaussian random process (Ochi,
L g ^η m
s2η
m3 m4 A Tp
17
wavelength the acceleration of gravity mean variance skewness kurtosis wave amplitude spectral peak period
1998). The wave crest distributions in a Gaussian random sea are generally regarded as to obey the Rayleigh probability law (Longuet-higgins, 1952; Chakrabarti, 1987) as shown in Eq. (1): " 2 # h PðAc o hÞ ¼ 1 exp 8 ð1Þ Hs In Eq. (1) h is the crest height, Hs is the significant wave height. In the Gaussian sea model the individual cosine wave trains superimpose linearly (add) without interaction. Therefore, the model is also called the linear sea model. However, it is known that as the water depths decrease, the non-linearities of sea waves become more and more relevant, and the ocean surface elevation process deviates significantly from the Gaussian assumption, i.e. the observed surface process is positively skewed with higher crests and shallower troughs than expected under the Gaussian assumption (Ochi, 1998). Meanwhile, it should be pointed out that wave nonlinearities are not only due to bottom effects (shallow water conditions). They can also be present in deep water conditions, as found by Casas-Prat and Holthuijsen (2010), for instance. In the following we briefly introduce the theoretical background of the second order nonlinear irregular waves. The fluid region is described using three-dimensional Cartesian coordinates (x, y, z), with x and y the horizontal coordinates, and z the vertical coordinate – with the positive z-direction opposing the direction of the gravitational acceleration. Time is denoted by t. The free surface is located at z¼ η(x, y, t). If the fluid is assumed to be ideal, incompressible and inviscid, and the fluid motion irrotational, so that the velocity potential Φðx; y; z; tÞ exists, then for constant water depth d the potential Φðx; y; z; tÞ and the surface elevation η(x, y, t) are determined by the following boundary value problem (see, for example, Toffoli et al., 2006): ∇2 Φ ¼ 0
ð2Þ
∂Φ 1 þ ð∇ΦÞ2 þ gz ¼ 0 ∂t 2
at
∂η ∂Φ ∂η ∂ Φ ∂ η ∂Φ þ þ ¼0 ∂t ∂x ∂x ∂y ∂y ∂z ∂Φ ¼0 ∂z
at
z ¼ d
z ¼ ηðx; y; tÞ at
z ¼ ηðx; y; tÞ
ð3Þ ð4Þ ð5Þ
Eq. (2) is the Laplace equation and Eqs. (3)–(5) are respectively the dynamic free surface boundary condition, the kinematic free surface boundary condition, and the bottom boundary condition. Solutions of system (2)–(5) can be sought using the following expansion (see, e.g., Toffoli et al., 2006): ( Φ ¼ Φð1Þ þ Φð2Þ þ … Φðn þ 1Þ ηðn þ 1Þ where ¼ ðnÞ ¼ OðεÞ ð6Þ η η ¼ ηð1Þ þ ηð2Þ þ … ΦðnÞ Here ε is a small parameter in expansion (6) and it is typically proportional to the wave steepness ξ ¼ kA, where k ¼2π/L (L ¼wavelength) is the wave number, and A is the wave amplitude which is equal to half the wave height. For an irregular sea state
18
Y. Wang / Ocean Engineering 87 (2014) 16–24
For deep water waves the quadratic transfer function Eðωn ; ωm Þ simplifies to:
characterized by a certain spectral density function SðωÞ where ω is the angular frequency, it is straightforward to show that a first order solution of system (2)–(5) takes the following form ( see, e. g., Marthinsen, 1992): igAn cosh kðz þdÞ iðωn t kn xÞ e cosh kd N 2 ωn
N
Φð1Þ ðx; tÞ ¼ Re ∑ n¼
N
ηð1Þ ðx; tÞ ¼ Re ∑
n ¼ N
Eðωn ; ωm Þ ¼
ð8Þ
N
^ η ¼ η ¼ ∑ A2n Eðωn ; ω n Þ m n¼1
An 2 n¼1 2 N
s2η ¼ ðη ηÞ2 ¼ ∑
m3 ¼ ðη ηÞ3 =sη 3 " N 3 ¼ ∑ An 2 Am 2 ðEðωn ; ωm Þ þ Eðωn ; ω m ÞÞ 2sη 3 n;m ¼ 1 N 1 ∑ A4n ðEðωm ; ωm Þ þ Eðωm ; ω m ÞÞ 2 n¼1 m4 ¼ ½ð4m3 Þ=32
ð10Þ
N
∑
n ¼ N m ¼ N
N
ηð2Þ ðx; tÞ ¼ Re ∑
N
∑
n ¼ N m ¼ N
s2η ¼ m3 ¼
2
ð11Þ
An Am Eðωn ; ωm Þeiðωn t kn xÞ eiðωm t km xÞ 4
ð12Þ
n ¼ 12
Z
1 0
6
s2η
SðωÞdω
Z
1 0
Z
1 0
½Eðω1 ; ω2 Þ þ Eðω1 ; ω 2 ÞSðω1 ÞSðω2 Þdω1 dω2
ðg 2 kn km =2ωn ωm Þ ð1=4Þðω2n þ ω2m þ ωn ωm Þ þ g 2 =4ðωn km þ ωm kn Þ=ðωn ωm ðωn þ ωm ÞÞ ðωn þ ωm Þ ðgkn þ km Þ=ðωn þ ωm Þtanhððkn þ km ÞdÞ 2
2
ð23Þ ð24Þ
2
ðgkn km =ωn ωm Þ ð1=2gÞðωn 2 þ ωm 2 þ ωn ωm Þ þ ðg=2Þðωn km þ ωm kn Þ=ðωn ωm ðωn þ ωm Þ 1 ðgkn þ km Þ=ðωn þ ωm Þ2 tanhððkn þ km ÞdÞ
ð22Þ
ð13Þ
where β1 and β2 are the empirical coefficients. β1 ¼ 4.37 and β2 ¼ 0.57 are recommended by Haring and Heideman (1978). However, the previous research work of this author's research group has shown that such kinds of empirical formulas will
The Kroenecker delta (δ n;m ¼ 1 if nþ m ¼0, zero otherwise) is introduced to avoid a singular Pðωn ; ωm Þ. The quadratic transfer function Eðωn ; ωm Þ is given by the following equation:
Eðωn ; ωm Þ ¼
ð19Þ
For nonlinear irregular waves, the wave crests will become higher and steeper, and the troughs of the nonlinear waves will become shallower and flatter. Obviously, the Rayleigh distribution which is good for predicting the crests of linear Gaussian waves will underestimate the crests of nonlinear irregular waves. An empirical correction to the Rayleigh model of the crest height distribution was suggested four decades ago by Jahns and Wheeler (1972) ( ) 2 h h h hZ0 ð25Þ PðAc o hÞ ¼ 1 exp 8 1 β1 β2 Hs d d
In the above two equations, Pðωn ; ωm Þ and Eðωn ; ωm Þ are referred to as quadratic transfer functions. The quadratic transfer function Pðωn ; ωm Þ is given by the following equation:
Pðωn ; ωm Þ ¼ ð1 δ n;m Þ
ð18Þ
ð20Þ
m4 ¼ ½ð4m3 Þ=32
An gkn t sinh 2kn h
eiðωn t kn xÞ eiðωm t km xÞ þ ∑
ð17Þ
0
iAn Am coshðkn þkm Þðz þ dÞ Pðωn ; ωm Þ coshðkn þkm Þd 2 N
ð16Þ
Letting N-1 and passing to a continuous representation lead to the following expressions. Z 1 ^η¼2 Eðω1 ; ω 1 ÞSðω1 Þdω1 ð21Þ m
where g and d are the acceleration of gravity and water depth, respectively. Shallow water wave data usually does not follow the linear Gaussian model. The linear Gaussian sea model can be corrected by including quadratic terms. Following Langley (1987) and Marthinsen (1992) the quadratic corrections are given by the following equations: N
ð15Þ
The leading order mean, variance, skewness and kurtosis of the surface elevation ηðx; tÞ are given by (see e.g., Marthinsen, 1992):
where Δω ¼ ωc =N and ωc is the upper cut off frequency beyond which the power spectral density function SðωÞ may be assumed to be zero for either mathematical or physical reasons. In Eqs. (7) and (8), the individual frequencies, ωn and wave numbers, kn are functionally related through the so-called linear dispersion relation:
Φð2Þ ðx; tÞ ¼ Re ∑
1 2 ω ω2m 2g n
ηðx; tÞ ¼ ηð1Þ ðx; tÞ þ ηð2Þ ðx; tÞ
as N tends to infinity. In Eqs. (7) and (8), Re denotes the real part of the complex number, and for each elementary sinusoidal wave An denotes its complex valued amplitude, ωn the angular frequency, and kn the wave number. Since ηð1Þ should be a real valued field, we need to assume that ω m ¼ ωm , k m ¼ km . If ηð1Þ is assumed to be stationary and Gaussian, then the complex amplitudes An are also Gaussian distributed, that is, An ¼ sn ðU n iV n Þ, where Un and Vn are independent zero mean and variance one Gaussian random variables, and s2n is the energy of waves with angular frequencies ωn and ωn. The mean square amplitudes are related to the wave spectrum SðωÞ by the following equation: 2 ð9Þ E½An ¼ 2Sðjωn jÞΔω
ω2n ¼ gkn tanhðkn dÞ
Eðωn ; ωm Þ ¼
where ωn and ωm are positive and satisfies the same relation as in the linear model. Finally, by combining Eqs. (8) and (12) the wave surface elevations for the second order nonlinear irregular waves can be written as follows:
ð7Þ
An iðωn t kn xÞ e 2
1 2 ðω þ ω2m Þ; 2g n
2
ðgkn km =2ωn ωm Þ þ ð1=2gÞðωn 2 þ ωm 2 þ ωn ωm Þ
ð14Þ
Y. Wang / Ocean Engineering 87 (2014) 16–24
sometimes predict wave crest distributions that differ considerably from the true ones (Wang and Xia, 2012). Therefore, the author of this article has proposed a Transformed Rayleigh method in order to calculate the crest distributions of nonlinear irregular waves more accurately and efficiently. In Section 3 of this article the theoretical background of this Transformed Rayleigh method will be elucidated.
3. The proposed Transformed Rayleigh method From Section 2 we know that ηð0; tÞ (corresponding to x ¼ 0 in Eq. (16)) is the height of the sea level at a fixed point as a function of time t. Here, we write ηðtÞ for ηð0; tÞ so that we can simplify the notation. We also use the so called mean down crossing wave to define a wave, i.e. the wave is considered as a part of a function between the consecutive down crossings of the mean sea level. Assume ηðtÞ crosses a mean sea level un a finite number of times, which are denoted by ti, 0 o t 1 o t 2 o :::;. The crest of the ith wave, M i n , is defined as the global maximum of ηðtÞ during the interval t i ot o t i þ 1 . For a specific nonlinear sea state, ηðtÞ will be a non-Gaussian random process. Here we will use the simplest (but widely effective) model for a non-Gaussian sea where ηðtÞ is expressed as a function of a stationary zero mean Gaussian process η~ ðtÞ with variance one (V½η~ ð0Þ ¼ 1), i.e. (Wang and Xia, 2013)
ηðtÞ ¼ Gðη~ ðtÞÞ;
dG 4 0; dη
Gð0Þ ¼ 0:
ð26Þ
We can see that the above formula is a deterministic and time instantaneous functional transformation. Note that once the distribution of crests in η~ ðtÞ is computed, then the corresponding wave crest distribution in ηðtÞ is obtained by simple transformations involving only the inverse of G, which we shall denote by gn. Actually we shall use the function gn to define the transformation instead of G, and use the relation η~ ðtÞ ¼ g n ðηðtÞÞ between the real sea data ηðtÞ and the transformed data η~ ðtÞ. In our research, we have tried to determine the G-function (or gn-function which is the inverse of G) directly from the cumulative distribution functions of η and η~ . For this purpose, we need to estimate the cumulative distribution functions (or probability density functions f η ðηÞ or f η~ ðη~ Þ) from moments because usually in an offshore engineering design project the only information we have on the waves is the wave power spectral density function. From Winterstein (1988) we have the following relationships:
μη ðuÞ f η ðηÞ ¼ ^ η Þ f η ðm ^ ηÞ μðm
ð27Þ
μη ðuÞ ðG ðuÞÞ ¼ exp ^ ηÞ 2 μðm 1
2
!
n
ðg ðuÞÞ ¼ exp 2
2
2 n z^ d exp n dz 2
! ¼ ð 1Þ H n ðz^ Þexp
hn ¼
in which μη ðuÞ denotes the level up-crossing rate of the level μ by ^ η denotes the mean value of ηðtÞ. It is convenient to ηðtÞ; and m ^ η Þ=sη in which sη consider the standardized variable z^ ¼ ðηðtÞ m denotes the standard deviation of ηðtÞ. Note that z^ ¼ η~ ðtÞ. Then the non-Gaussian probability density function of z can be expressed as the following Gram–Charlier series: 2 ffiexp z^2 f ðz^ Þ ¼ p1ffiffiffiffi 2π h i 4 þ 30 H 6 ðz^ Þ þ ::: 1 þ m3!3 H 3 ðz^ Þ þ m44! 3H 4 ðz^ Þ þ m5 5!10m3 H 5 ðz^ Þ þ m6 15m 6! ! 2 N z^ 1 ¼ pffiffiffiffiffiffiexp ð29Þ 1 þ ∑ hn H n ðz^ Þ 2 2π n¼1 in which the Hermite polynomials of degree n, denoted by H n ðz^ Þ, is defined as a function which satisfies the relationship given by
2 z^ 2
!
1 E½Hen ðz^ Þ n!
ð30Þ
ð31Þ
We can then use the following formula for calculating the Gfunction (or gn-function which is the inverse of G) (Winterstein, 1988): ^ η þ k^ sη ½z^ þ h~ 3 ðz^ 1Þ þ h~ 4 ðz^ 3z^ Þ þ ::: GðzÞ ¼ m 2
3
ð32Þ
The coefficients h~ n control the shape of the standardized distribution, while k^ is a scaling factor. The coefficients h~ n may be related to the Hermite moments hn in Eq. (31) by applying a Hermite polynomial to Eq. (32) and take expectations. For N ¼4 moments, the solution is (Winterstein et al., 1994): m3 h~ 3 ¼ ð1 0:015jm3 j þ 0:3m3 2Þ=ð1 þ 0:2ðm4 3ÞÞ 6
ð33Þ
0:8 h~ 4 ¼ 0:1ðð1 þ 1:25ðm4 3ÞÞ1=3 1Þð1 1:43m23 =ðm4 3ÞÞð1 0:1m4 Þ
ð34Þ We can clearly see that the accuracy of the G-function calculated according to Eq. (32) depends on how many moments are kept during the calculation. The more moments are kept, the more accurate the G-function will be. We can also notice that in order to calculate the G-function according to Eq. (32), we need to know the values of the skewness, kurtosis, mean and standard deviation of the second order nonlinear random waves. These values can be calculated using Eqs. (21)–(24) in this paper. Next, we turn to the relation between the level up-crossing rate μη ðuÞ and the probability distribution of the crest height F Mn ðuÞ. As in Wang and Xia (2013), we have that
0rzru
ð28Þ
n
In Eq. (29), the parameters m3, m4, m5, m6, etc. are the moments of the standardized variable z^ and these moments can be calculated based on the wave power spectral density functions. m3 is the skewness, and m4 is the kurtosis. Now it seems that by combining Eqs. (27)–(30) and solving them we can obtain the Gfunction (or gn-function which is the inverse of G). However, the Gram–Charlier series can behave erratically, yielding multimodal and even negative probability densities. Here we consider to use a procedure as specified in Winterstein (1988) to calculate the G-function (or gn-function which is the inverse of G). This precedure is based on the notion of a “Hermite moment” which is the coefficient hn in Eq. (29).
1 F Mn ðuÞ r min
!
19
μη ðuÞ ^ ηÞ μðm
ð35Þ
Combining Eqs. (28) and (35) we obtain the following relationship: ! ðG 1 ðuÞÞ2 F Mn ðuÞ Z1 exp ð36Þ 2 for all u Z0, and G 1 ðuÞ is defined by Eq. (32). Eq. (36) can also be written in a form of ! ðG 1 ðuÞÞ2 F Mn ðuÞ Z1 exp ð37Þ 2m0 where m0 is the zero-order spectral moment of the stationary zero mean and variance one Gaussian process η~ ðtÞ, m0 ¼ s2η~ ¼ 1. We notice that the exponent in the right part of Eq. (37) is the Rayleigh probability that the linear crest exceeds the predefined threshold u. That is why this model is called a Transformed Rayleigh method.
20
Y. Wang / Ocean Engineering 87 (2014) 16–24
4. Application examples, calculation results and discussions 4.5 4
2
S(w) [m s / rad]
In this section we utilize the Transformed Rayleigh method to calculate the crest statistics of shallow water nonlinear waves based on standard spectra and measured data at the Poseidon platform. The proposed new approach is first applied for calculating the wave crest height exceedance probabilities of sea states with standard JONSWAP spectra corresponding to different water depths (15 m, 25 m, 35 m, 45 m, 135 m). The JONSWAP formulation is based on an extensive wave measurement program known as the Joint North Sea Wave Project carried out in 1968 and 1969 along a line extending over 160 km into the North Sea from Sylt Island. The spectrum represents wind-generated seas with fetch limitation (Ochi, 1998). The mathematical expression of the infinite water depth JONSWAP frequency spectrum with a significant wave height Hs and spectral peak period Tp is as follows: ! ð2πω=T 1Þ2 g2 H2 5 2π 4 exp 2δp2 SðωÞ ¼ 5:061 5 4s ð1 0:287 lnðγ ÞÞexp γ ω Tp 4 ωT p
Spectral density
2.5 2 1.5 1
0
0.4
Inp the ffiffiffiffiffiffiffiffiabove formula ωd is a dimensionless frequency defined by ω d=g and kd is the wave number associated with the linear dispersion relation
ω2 ¼ gkd tanhðkd dÞ
ð40Þ
where g and d are the acceleration of gravity and water depth, respectively. In formula (39), Sc ðωÞ represents the finite water depth spectrum, and SðωÞ represents the infinite water depth spectrum. Fig. 1 shows the obtained corrected JONSWAP spectra with a significant wave height Hs ¼4.0 m, a spectral peak period Tp ¼10.0 s, and γ ¼3.3 corresponding to different water depths (15 m, 25 m, 35 m, 45 m, 135 m). It should be pointed out that the figure of a JONSWAP spectrum with a water depth of 135 m is almost identical to that of a JONSWAP spectrum with an infinite water depth. Fig. 2 shows the calculation results of the wave crest height exceedance probabilities for a sea state with a JONSWAP spectrum with Hs ¼4.0 m, Tp ¼10.0 s and γ ¼3.3 for a water depth of 15 m. It should be clarified that in this figure X is the deviation from the mean water level. Fig. 2 shows the probability of the wave crest height, X, exceeding the range of possible deviations in surface elevation from the mean water level, x. The solid green line in Fig. 2 represents the Monte Carlo Simulation (MCS) results of the wave crest height exceedance probabilities of this shallow water nonlinear sea state. In the process of the MCS simulation, a wave elevation time series consisting of 2,000,000 points (enough to reduce the simulation variance) were first generated based on the
0.7
0.8
0.9
1
1.1
1.2
The probability of X exceeding x
100 10-1 10-2
Monte Carlo Simulation Transformed Rayleigh method Un-modified Jahns and Wheeler model Modified Jahns and Wheeler model
10-3 10-4
In shallow water, in order to consider the correction for the effect of bottom on the wave spectrum, a “corrected wave spectrum Sc ðωÞ” is obtained by multiplying the original “standard JONSWAP spectrum SðωÞ” by a function ϕðωd Þ that ranges between 0 and 1 according to the similarity law of Buows et al. (1985) " # ðkðω; dÞÞ 3 ð∂=∂ωÞkðω; dÞ Sc ðωÞ ¼ SðωÞϕ½ωd ¼ SðωÞ ðkðω; 1ÞÞ 3 ð∂=∂ωÞkðω; 1Þ " # ðkd Þ 3 ð∂=∂ωÞkd ¼ SðωÞ ð39Þ ðk1 Þ 3 ð∂=∂ωÞk1
0.6
Fig. 1. JONSWAP spectra with Hs ¼ 4.0 m, Tp ¼ 10.0 s and γ¼ 3.3 for different water depths.
δ ¼ 0:09 if ω Z 2π =T p pffiffiffiffiffiffi γ ¼ expð3:484ð1 0:1975ð0:036 0:0056T p = H s ÞT 4p =H2s ÞÞ
0.5
Frequency [rad/s]
1-F(x)
δ ¼ 0:07 if ω o 2π =T p
h=15m h=25m h=35m h=45m h=135m
3
0.5
ð38Þ where
3.5
0
0.5
1
1.5
2
2.5
x (m) Fig. 2. Wave crest height exceedance probabilities for a sea state with JONSWAP spectrum with Hs ¼ 4.0 m, Tp ¼ 10.0 s and γ ¼3.3 for a water depth of 15 m. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
corrected spectrum by summation of sine functions with random phase angles uniformly distributed in the range of [0, 2π]. The wave crest height time series were then extracted from these 2,000,000 wave elevation points. After extracting these time series, exact Epanechnikov kernel density estimates were performed for obtaining the probability density function of the wave crest heights. Then cumulative trapezoidal numerical integration was carried out on the above probability density function in order to get the probability distribution (F) of the wave crest heights. Finally, the wave crest height exceedance probabilities were obtained based on the probability distribution by utilizing the formula P ¼1 F. The above MCS results are utilized as the criteria against which the accuracy of the results from the transformed Rayleigh method or from the empirical crest distribution models is checked. In Fig. 2, the red solid line represents the results of the wave crest height exceedance probabilities obtained from utilizing the Transformed Rayleigh method. In applying the Transformed Rayleigh method, the mean, variance, skewness and kurtosis of the surface elevation were first calculated based on the corrected JONSWAP spectrum by using Eqs. (21)–(24). Then the g*-function (which is the inverse of the G-function) was calculated by using Eqs. (32)–(34). We can see that the Transformed Rayleigh method gave the closest predictions of the results than those predicted using the other methods. In Fig. 2, the small blue “*” represents a wave crest height exceedance probability predicted by using the original Jahns and Wheeler model (i.e. by using Eq. (25)) in which the value of Hs was taken to be 4.0 m. We can notice that the original Jahns and Wheeler model gave poorer predictions in the wave crest height region of about [0.5 m, 2.5 m] than the Transformed Rayleigh method did. In Fig. 2, the solid blue line represents the wave crest height exceedance probabilities
Y. Wang / Ocean Engineering 87 (2014) 16–24
1-F(x)
10-1 Monte Carlo Simulation Transformed Rayleigh method Un-modified Jahns and W heeler model Modified Jahns and Wheeler model
10-2
10
-3
1-F(x)
10-1 10-2
10
Monte Carlo simulation Transformed Rayleigh method Un-modified Jahns and W heeler model Modified Jahns and Wheeler model
-3
10-4
0
0.5
1
1.5
2
Fig. 5. Wave crest height exceedance probabilities for a sea state with JONSWAP spectrum with Hs ¼ 4.0 m and Tp ¼10.0 s for a water depth of 45 m.
The probability of X exceeding x
100 10-1 10-2
10
Monte Carlo simulation Transformed Rayleigh method Un-modified Jahns and W heeler model Modified Jahns and Wheeler model
-3
10-4
0
0.5
1
1.5
2
Fig. 6. Wave crest height exceedance probabilities for a sea state with JONSWAP spectrum with Hs ¼ 4.0 m and Tp ¼10.0 s for a water depth of 135 m.
S þ ðωÞ ¼ ∑ SJþ ðω; H s;i ; ωp;i ; λi ; N i M i ; αi Þ i¼1
0.5
1
1.5
2
2.5
x (m) Fig. 3. Wave crest height exceedance probabilities for a sea state with JONSWAP spectrum with Hs ¼ 4.0 m and Tp ¼ 10.0 s for a water depth of 25 m.
The probability of X exceeding x
100
where i¼ 1 means the first peak and i ¼2 means the second peak, and SJ þ ð Þ is the JONSWAP spectrum defined by the following equation: ðω=ω 1Þ2 g2 M ωp N exp 2ps2 þ ð42Þ SJ ðωÞ ¼ α M exp λ N ω ω
s ¼ 0:07 if ω o ωp
10
1-F(x)
ð41Þ
where
-1
10-2
10
2.5
x(m)
2
10-4
0
2.5
x (m)
can conclude that it is more advantageous to apply the Transformed Rayleigh method to predict the wave crest height exceedance probabilities of the “more shallow water” nonlinear sea states. As another example, the Transformed Rayleigh method was utilized to calculate the wave crest height exceedance probabilities for a combined swell and wind sea state characterized by a Torsethaugen bimodal spectrum. The Torsethaugen bimodal spectrum is expressed as follows:
The probability of X exceeding x
100
The probability of X exceeding x
100
1-F(x)
calculated by utilizing a “corrected” Jahns and Wheeler model (i.e. the value of Hs is taken to be four times the standard deviation of the surface elevation based on the corrected JONSWAP spectrum). We find that the “corrected” Jahns and Wheeler model gave much better predictions than the original Jahns and Wheeler model. However, the Transformed Rayleigh method still gave slightly better predictions in the wave crest height region of about [1.5 m, 2.5 m] than the “corrected” Jahns and Wheeler model did. Finally, it is mentioned that in this case, the time series simulation with 2,000,000 wave elevation points and the post statistical processing for getting the continuous green line in Fig. 2 took about 486 s on a desktop computer. By comparison, it took less than 2 s to obtain the results of the wave crest height exceedance probabilities represented by the red solid line in Fig. 2 by applying the Transformed Rayleigh method. The accuracy and efficiency of the Transformed Rayleigh method for calculating the crest height distributions of shallow water nonlinear waves can thus be validated. We have carried out similar calculations for the corrected JONSWAP spectra with a significant wave height Hs ¼4.0 m, a spectral peak period Tp ¼10.0 s, and γ ¼3.3 corresponding to other water depths (25 m, 35 m, 45 m, 135 m) and obtained similar results. Our calculation results are summarized in Figs. 3–6. We can see that the tendencies of the four lines in Fig. 3 (or in Fig. 4, or in Fig. 5, or in Fig. 6) are similar to those of the four lines in Fig. 2. If we look more closely at Figs. 2–6 and compare them, we can find that in Fig. 6 the differences between the results from the original Jahns and Wheeler model and the Transformed Rayleigh method are the smallest, while in Fig. 2 the corresponding differences are the biggest. Meanwhile, we can notice that in all cases the Transformed Rayleigh method had predicted slightly better results than the corrected Jahns and Wheeler model had. Therefore, we
21
s ¼ 0:09 if ω Z ωp
Monte Carlo Simulation Transformed Rayleigh method Un-modified Jahns and W heeler model Modified Jahns and W heeler model
-3
M¼5; N ¼4.
α 5:061
10-4
H 2s T 4p
ð1 0:287 lnðλÞÞ pffiffiffiffiffiffi
0
0.5
1
1.5
2
2.5
x(m) Fig. 4. Wave crest height exceedance probabilities for a sea state with JONSWAP spectrum with Hs ¼ 4.0 m and Tp ¼ 10.0 s for a water depth of 35 m.
λ ¼ expð3:484ð1 0:1975ð0:036 0:0056T p = Hs ÞðT 4p =H2s ÞÞ In the above JONSWAP spectrum Hs is significant wave height and Tp is spectral peak period. The values of the parameters H s1 ; ωp1 ; λ1 ; α1 H s2 ; ωp2 ; λ2 ; α2 in the Torsethaugen bimodal
22
Y. Wang / Ocean Engineering 87 (2014) 16–24
Spectral density 2
S(w) [m s / rad]
1.5 Original Torsethaugen spectrum Modified Torsethaugen spectrum
2
1
0.5
0 0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
Frequency [rad/s]
Fig. 7. A shallow water (d ¼25 m) Torsethaugen spectrum with Hs ¼4 m and Tp ¼ 7 s (solid pink line) and the corresponding infinite water depth spectrum (black line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The probability of X exceeding x
100
1-F(x)
10-1 10-2
10
Monte Carlo simulation Transformed Rayleigh method Un-modified Jahns and W heeler model Modified Jahns and Wheeler model
-3
10-4
0
0.5
1
1.5
2
2.5
3
3.5
x(m) Fig. 8. Wave crest height exceedance probabilities for a sea state with Torsethaugen spectrum with Hs ¼4.0 m and Tp ¼ 7.0 s for a water depth of 25 m. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
spectrum can all be obtained if the values of Hs and Tp are given (For details, please refer to: Brodtkorb (2004)). A Torsethaugen bimodal spectrum with Hs ¼4 m and Tp ¼7 s has been plotted in Fig. 7 and is shown as a black line. In order to study the shallow water effects, a “modified Torsethaugen spectrum” is obtained by multiplying the original Torsethaugen spectrum by a function ϕðωd Þ that ranges between 0 and 1 according to the similarity law of Buows et al. (1985): # " # " ðkðω; dÞÞ 3 ð∂=∂ωÞkðω; dÞ ðkd Þ 3 ð∂=∂ωÞkd ϕ½ωd ¼ ð43Þ ¼ ðkðω; 1ÞÞ ¼ 3 ð∂=∂ωÞkðω; 1Þ ðk1 Þ 3 ð∂=∂ωÞk1 Inp the ffiffiffiffiffiffiffiffiabove formula ωd is a dimensionless frequency defined by ω d=g and kd is the wave number associated with the linear dispersion relation:
ω2 ¼ gkd tanhðkd dÞ
ð44Þ
where g and d are the acceleration of gravity and water depth, respectively. In Fig. 7 the solid pink line represents the obtained “modified Torsethaugen spectrum” with a water depth of 25 m (i.e. d ¼25 m). Fig. 8 shows our calculation results of the wave crest height exceedance probabilities for a sea state with the above “modified Torsethaugen spectrum” with Hs ¼ 4.0 m and Tp ¼ 7.0 s. The solid green line in Fig. 8 represents the Monte Carlo simulation results of the wave crest height exceedance probabilities of this shallow water nonlinear sea state. In the process of the MCS simulation, a wave elevation time series consisting of 2,000,000 points were first generated based on the modified Torsethaugen spectrum by
summation of sine functions with random phase angles uniformly distributed in the range of [0, 2π]. The wave crest height time series were then extracted from these 2,000,000 wave elevation points. Finally, post statistical processing were conducted for getting the wave crest height exceedance probabilities based on these wave crest height time series. In Fig. 8, the red solid line represents the results of the wave crest height exceedance probabilities obtained from utilizing the Transformed Rayleigh method (by applying Eqs. (21)–(24) and Eqs. (32)–(37)). In Fig. 8, the small blue “*” represents a wave crest height exceedance probability predicted by using the original Jahns and Wheeler model (i.e. by using Eq. (25)) in which the value of Hs was taken to be 4.0 m. We can find that the original Jahns and Wheeler model gave poorer predictions in the wave crest height region of about [1 m, 3.5 m] than the Transformed Rayleigh method did. In Fig. 8, the solid blue line represents the wave crest height exceedance probabilities calculated by utilizing a “corrected” Jahns and Wheeler model (i.e. the value of Hs was taken to be four times the standard deviation of the surface elevation based on the “modified Torsethaugen spectrum”). We can find that the “corrected” Jahns and Wheeler model gave much better predictions than the original Jahns and Wheeler model did, and the predictions from using the “corrected” Jahns and Wheeler model and from using the Transformed Rayleigh method are almost identical. Finally, it is mentioned that in this case, the Monte Carlo simulation and the post statistical processing for getting the continuous green line in Fig. 8 took about 114 s on a desktop computer. By comparison, it took less than 2 s to obtain the results of the wave crest height exceedance probabilities represented by the red solid line in Fig. 8 by applying the Transformed Rayleigh method. The accuracy and efficiency of the Transformed Rayleigh method for calculating the crest height distributions of shallow water nonlinear waves are again readily validated. Finally, the Transformed Rayleigh method was applied to predict the wave crest height exceedance probabilities of a sea state with the surface elevation data measured at the Poseidon platform in the Japan Sea. Poseidon was located 3 km off the coast of Yura in the Yamagata prefecture, in the Japan Sea during the measurements. The water depth at the Poseidon platform is 41 m. Poseidon is a full scale test structure that consists twelve legs with footings which support the upper structure. The upper structure is mainly composed of box-typed girders around four sides. The instrumentation house is arranged on the upper deck for power supply and data acquisition (Yago et al., 1991). The surface elevation data at the platform were measured with three ultrasonic wave gauges located at the sea floor from 8:12 a.m. on November 24, 1987 to 7:57 a.m. on November 25, 1987, and three sets of data each consisting of 85,547 wave elevation points were obtained. Fig. 9 shows the first 300 wave elevation points time series measured with the 3# ultrasonic wave gauge (Data source: http://code.google.com/p/wafo/downloads/detail?name=wafo25. 7z. These data are used for academic research only and not for commercial purposes). We can clearly notice that the crests are higher and the troughs are shallower in this time series. Fig. 10 shows the power spectrum corresponding to the 85547 wave elevation points measured with the 3# ultrasonic wave gauge at the Poseidon platform. Fig. 11 shows the calculated wave crest height exceedance probabilities in this study. The solid green line in Fig. 11 represents the results of the wave crest height exceedance probabilities directly obtained from the 85547 wave elevation points measured with the 3# ultrasonic wave gauge at the Poseidon platform. For obtaining these results, the wave crest height time series were first extracted from these 85,547 wave elevation points. After extracting these time series, exact Epanechnikov kernel density estimates were performed for obtaining the probability density function of the wave crest
Y. Wang / Ocean Engineering 87 (2014) 16–24
Surface elevation from mean water level (MWL).
3 - +2
2 1 0 -1
Standard Deviation
Distance from MWL.(m)
4
-2 - -2 -3 0
50
100
150
200
250
Time (sec)
Fig. 9. The first 300 wave elevation points time series measured at the Poseidon platform.
Spectral density 4
2
S(w) [m s / rad]
3.5 3 2.5
identical predictions of the results to those directly obtained from the measured data. In Fig. 11, the small blue “*” represents a wave crest height exceedance probability predicted by using the Jahns and Wheeler model (i.e. by using Eq. (25)) in which the value of Hs was taken to be four times the standard deviation of the measured data. We can notice that the Jahns and Wheeler model gave poorer predictions in the wave crest height region of about [1.5 m, 5 m] than the Transformed Rayleigh method did. It should be noted that in this case (the Poseidon data) the un-modified Jahns and Wheeler model and the modified Jahns and Wheeler model are the same, i.e. the two models are identical. Obviously the Hs value in this case should not be modified further (because it is based on the measured data), i.e. in this case we do not need to have a modified Jahns and Wheeler model. Finally, it is mentioned that in this case, it took less than 2 s to obtain the results of the wave crest height exceedance probabilities represented by the solid red line in Fig. 11 by applying the Transformed Rayleigh method. The accuracy and efficiency of the Transformed Rayleigh method for calculating the crest height distributions of shallow water nonlinear waves have been once again convincingly validated.
2 1.5
5. Concluding remarks
1 0.5 0
0
0.5
1
1.5
2
2.5
Frequency [rad/s] Fig. 10. The spectrum corresponding to the measured data at the Poseidon platform.
The probability of X exceeding x 100 10-1
1-F(x)
23
Measured Poseidon data Transformed Rayleigh method Jahns and Wheeler model
10-2
10
-3
10-4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
X(m)
Fig. 11. Wave crest height exceedance probabilities for the spectrum corresponding to the measured data at the Poseidon platform. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
heights. Then cumulative trapezoidal numerical integration was carried out on the above probability density function in order to get the probability distribution (F) of the wave crest height. Finally, the wave crest height exceedance probabilities were obtained based on the probability distribution by utilizing the formula P ¼1 F. In Fig. 11, the solid red line represents the results of the wave crest height exceedance probabilities obtained from utilizing the Transformed Rayleigh method. In applying the Transformed Rayleigh method, the mean, variance, skewness and kurtosis of the surface elevation were first calculated based on the spectrum in Fig. 10 by using Eqs. (21)–(24). Then the gn-function (which is the inverse of the G-function) was calculated by using Eqs. (32)–(34). We can see that the Transformed Rayleigh method gave almost
This article proposed a new approach for calculating the wave crest distribution of shallow water nonlinear waves by utilizing a Transformed Rayleigh method where the second order nonlinear wave model is incorporated. The essence of the proposed new approach is based on a deterministic and time instantaneous functional transformation, and the detailed procedures and formulas for obtaining this transformation have been specified in this paper. The proposed new approach has been applied for calculating the wave crest height exceedance probabilities of sea states with a standard JONSWAP spectrum, with a bimodal Torsethaugen spectrum and with the surface elevation data observed at the Poseidon platform. It has been demonstrated that in all cases the Transformed Rayleigh method can offer better predictions than those using the empirical wave crest distribution models. Another advantage of using the proposed new approach is that the correction for the effect of bottom on the wave spectrum, a procedure that was frequently overlooked by some researchers, has been integrated into the Transformed Rayleigh method. The accuracy and efficiency of the proposed new approach have also been validated by using Monte Carlo simulations. The research results gained from this study demonstrate that the proposed Transformed Rayleigh method can be used for engineering purposes.
Acknowledgment The work reported in this article is supported by the funding of an independent research project from the Chinese State Key Laboratory of Ocean Engineering (Grant no. GKZD010038). Special thanks are due to the two anonymous reviewers of this article for their useful comments that have led to the improved quality of this paper.
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