Numerical Simulation of Non-Gaussian Wave ...

Proceedings of OMAE2006 25th InternationalDRAFT Conference on Offshore Mechanics and Arctic Engineering June 4-9,Proceedings 2006, Hamburg, Germany of OMAE2006 25th International Conference on Offshore Mechanics and Arctic Engineering June 4-9, 2006, Hamburg, Germany

OMAE2006-92014

OMAE2006-92014 NUMERICAL SIMULATION OF NON-GAUSSIAN WAVE ELEVATION AND KINEMATICS BASED ON TWO-DIMENSIONAL FOURIER TRANSFORM Xiang Yuan Zheng Centre for Ships and Ocean Structures Norwegian University of Science and Technology Trondheim 7491, Norway [email protected]

Torgeir Moan Centre for Ships and Ocean Structures Norwegian University of Science and Technology Trondheim 7491, Norway [email protected]

Ser Tong Quek Civil Engineering Department National University of Singapore 117576, Singapore [email protected]

ABSTRACT The one-dimensional Fast Fourier Transform (FFT) has been extensively applied to efficiently simulate Gaussian wave elevation and water particle kinematics. The actual sea elevation/kinematics exhibit non-Gaussianities that mathematically can be represented by the second-order random wave theory. The elevation/kinematics formulation contains double-summation frequency sum and difference terms which in computation make the dynamic analysis of offshore structural response prohibitive. This study aims at a direct and efficient two-dimensional FFT algorithm for simulating the frequency sum terms. For the frequency difference terms, inverse FFT and FFT are respectively implemented across the two dimensions of the wave interaction matrix. Given specified wave conditions, not only the wave elevation but kinematics and associated Morison force are simulated. Favorable agreements are achieved when the statistics of elevation/kinematics are compared with not only the empirical fits but the analytical solutions developed based on modified eigenvalue/eigenvector approach, while the computation effort is very limited. In addition, the stochastic analyses in both timeand frequency domains show that the near-surface Morison force and induced linear oscillator response exhibits stronger non-Gaussianities by involving the second-order wave effects.

INTRODUCTION In offshore engineering applications, the random sea surface is usually modeled as a stationary Gaussian process by the linear superposition of harmonic wave components [1]. The water particle kinematics, velocity and acceleration, in the fluids follow Gaussian distributions under the linear wave theory. Nevertheless, the actual sea elevation exhibits nonGaussain characteristics, based on numerous field observations and laboratory tests. The wave non-Gaussianities are particularly significant in a severe sea state and in shallow waters, which is a non-negligible factor for the safety considerations of offshore structures. Recently Stansberg [2] reported that, compared with the linear random wave case, the extreme wave crest heights can increase by as much as 20% and the extreme kinematics by about 30%, albeit the energy contribution of the second-order to spectrum is rather small. The early theoretical descriptions of wave nonlinearity using a second-order correction were proposed by LonguetHiggins [3] and Hasselmann [4], and later spanned by Sharma and Dean [5], Huang et al. [6], Tayfun [7], Martinsen and Winterstein [8]. The established second-order models have resulted in many published works on the statistical analysis of the nonlinear random wave elevation, while only very limited literature are available for the kinematics. For this reason, the

1

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authors would utilize Langley’s [9] convenient eigenvalue/eigenvector approach, with some modification introduced. To obtain the more meaningful statistics of wave force and induced structural response, either time- and frequency-domain stochastic analysis can be applied. One more often relies on the time-domain analysis because of the relatively less mathematical complexity involved. Owing to the rapid development of computer technology, the Gaussian wave elevation and kinematics, which is a single summation of linear wave components, can be easily simulated with limited time nowadays. It is true even if the summation is performed by a loop as many years ago and the number of frequency components is large as to serve the central limit theorem for the Gaussian realization. The matrix-vector multiplication technique (as well as the dot-product of two vectors) embedded by modern computer languages has replaced the performance of time-consuming loops and allows one to save more computer efforts [10]. On the other hand, thanks to the Fast Fourier Transform (FFT) algorithm developed in 1960’s, the Gaussian elevation/kinematics can be numerically obtained within a few seconds. The numerical simulation procedure based on one-dimensional inverse transform of Fourier coefficients was presented by Borgman [11] and has been widely applied. It appears not that straightforward when dealing with the nonlinear wave simulations. Attributable to the frequency sum and difference terms that contain double summations over bifrequencies, the computer work dramatically grows. This problem becomes more serious when a large number of components are used to capture the reliable higher-order statistics of wave force and structural response [12]. Also, to remove sampling uncertainty dozens of realizations need to be committed [13,14]. In addition, the total wave force on a slender cylinder of platform, for instance, demands the integration over local Morison forces at various underwater locations. All these impede the implementation of stochastic analysis through time-domain simulations. As the extension of one-dimensional Fourier Transform, a method was proposed by Hudspeth and Chen [15] for simulating the non-Gaussian wave elevation - the Fourier coefficients are corrected in terms of the nonlinear interaction matrix of different frequency components. In reality, the correction on Fourier coefficients involves summations over frequencies still and costs computer time to some extent. Another approach specially designed for the nonGaussian wave simulation in deep waters was suggested by Stansberg [16], which also utilized the 1-dimensional Fourier Transform. In the deep water case, the nonlinear interaction matrix reduces into a much simpler form termed as quadratic transfer function (QTF) [16], which allows the frequency-sum term to be easily calculated with the use of FFT simulation along the secondary diagonal of the QTR matrix. For the more general cases of finite water depth, this study seeks the direct application of two-dimensional Fourier transform to simulating the nonlinear portion of not only wave

elevation but also kinematics. It is particularly important because, though the frequency sum of kinematics vanishes in deep waters, the frequency difference term still exists. The numerical efficiency is self-manifested due to the FFT algorithm, as has been revealed by the bi- and tri-spectral analysis of nonlinear Morison drag effects on a linear structure [12]. The achieved higher-order statistics of wave elevation/kinematics will be compared with the early obtained analytical solutions. For elevation, the comparison will further be conducted against the existent empirical fits. Moreover, the standard time integration method will be used to compute the time history of the displacement of a linear oscillator excited by a local Morison wave force. Comparative studies will be carried out to examine the difference between the cases of linear and second-order nonlinear random wave theories. STATISTICAL ANALYSIS OF ELEVATION/KINEMATICS In this section, we will follow Langley’s approach [9] to analytically obtain the cumulants of non-Gaussian wave elevation/kinematics. The originally proposed approach was for wave elevation only and is applicable to the water particle velocity as well, while for the water particle acceleration, some necessary modifications are introduced herein. Considering the unidirectional wave propagation in a two-dimensional Cartesian plane, the first-order random wave elevation η1, horizontal water particle velocity u1 and acceleration a1 in the water of a finite depth d take the following linear superposition forms [9,11]: N

η1 ( x, t ) = ∑ ( an cos θ n + bn sin θ n )

(1)

n =1

u1 ( x, z, t ) =

a1 ( x, z , t ) =

N

∑ Rn ( z )(an cos θn + bn sin θn )

(2)

n =1

N

∑ ωn Rn ( z )(−an sin θn + bn cos θn )

(3)

n =1

where x is the coordinate for wave propagation direction; z is the coordinate for measuring the submerged location from the still water level (SWL), defined positive upwards; Rn(z) = ωncoshkn(z+d)/sinhknd is the linear transfer function of velocity; ωn is the angular wave frequency by spectral discretization and kn the wave number; ωn and kn are related by the wave dispersion function; θn = -knx+ωnt; N is number of harmonic components; an and bn are Gaussian random variables if η1 is assumed to be stationary and Gaussian [9]. These two random variables have the following properties: E [an2 ] = E [bn2 ] = Sη1 (ωn ) ∆ω

2


(4)


0.00511 rad/s. To avoid high-frequency induced excessive extreme values of waves, the maximum frequency used to simulate second-order portions is determined by a criterion suggested by Stansberg [13], up to around 4~5 times peak wave frequency. The FFT-based computation is so powerful that no more than 10 seconds CPU time is consumed for simulating an individual sequence of wave elevation of duration 1/3 hour. It is much more efficient than the realization using the matrixvector multiplication under the same N and ∆ω – a realization costs roughly 50 minutes. After all, it is known that numerically the 1-dimensional FFT is a process of O(Nlog2N) multiplications that surpasses the standard double-loop approach – a process of O(N2) multiplications. The practical computation shows that the difference of CPU time in these two processes is even much smaller than (log2N)/N. Therefore, the 2-dimensional FFT scheme suggested for second-order wave/kinematics simulation will further outmatches the standard triple-loop approach which, taking into account the symmetries of frequency sum and difference terms, is a O(N2(1+N/2)) process. In Table 1, the higher-order moments of interest, skewness and kurtosis excess, of the surface elevation are presented. The comparative study is carried out among: (a) (b) (c) (d) (e)

It can be observed that all analytically, numerically and empirically obtained skewness and kurtosis excess agree well for the intermediate-depth waters, see (a), (b), (c) and (d) data. The kurtosis excess by Vinje and Haver [21] is almost twice the other three cases because they made a correction by including the contribution of third-order Stokes’ wave effects. By contrast the deep-water statistics appear smaller, which implies the more weakly non-Gaussianity of wave. The further comparison of probability density function can be found in [10]. When the water becomes shallower, the wave non-Gaussianity will be stronger such that employing the deep-water model will result in underestimations. Another important parameter affecting the wave non-Gaussianity was the specified wave height [10]. 0 -10 -20 -30

-50 -60 -70

the analytical solutions the average of FFT simulated results empirical fit by Vinje and Haver [21] empirical fit by Winterstein and Jha [22] deep water analytical solutions

-80

(

0

0.5

1

1.5

2

Fig. 1 Skewness and kurtosis excess of a decay with z 0

Both the empirical fit (c) and (d) were obtained based on infield laser measurements. Different from (c), the fit (d) additionally has the correction considering the finite-waterdepth effects [22]: H −1 λ3 = s 5.45γ −0.084 + {exp[ 7.41( d / L p )1.22 ] −1} LP

Analytical skewness Analytical kurtosis Simulated skewness Simulated kurtosis

z (m) -40

)

(57)

-10 -20 -30 z (m) -40


-50 -60

where Hs is the significant wave height, Lp the wave length at wave peak frequency, γ the peak enhancement factor for JONSWAP spectrum. For the analytical solutions by eigenvalue/eigenvector approach, a limited number of frequency components, say 80, is able to capture the convergent skewness and kurtosis.

-70 -80

-0.4 -0.2

0

0.2

0.4

0.6

0.8

1

1.2

Fig. 2 Skewness and kurtosis excess of u decay with z Table 1. Skewness and kurtosis excess of wave elevation (a)

κ3 0.1892

κ4 0.0545

(b)

0.1867

0.0575

(c)

0.1993

0.1191

(d)

0.1972

0.0535

(e)

0.1668

0.0461

Figures (1 & 2) show the skewness and kurtosis excess of kinematics as a function of location z, analytical versus simulation results. It is obvious to see the good agreement of the two approaches. The non-Gaussian characteristics of u and a are significant in the zone near the still water surface and such strong non-Gaussianities decay rapidly with z. At a deep location, the non-Gaussianities are so weak that both u and a approaches Gaussian - as the result the kurtosis excess of drag

7



K 4u =

2N

2N

∑ 48(λn4 + β n2λn2 )

(23)

n =1

The first two cumulants correspond to mean and variance. Skewness and kurtosis excess are normalized third- and fourthorder cumulants: 3

κ 3u = K3u

(σ 2u ) 2

κ 4u = K 4u

(σ 2u )

a ( z, t ) = a1 ( z , t ) + a2 ( z, t )

(24)

Analogous to u, a can be expressed in matrix notation: a(z,t) = G yT + x [H + L] yT + y [H - L] xT

(25)

where H is a symmetric matrix of which the nmth entry is snsmhnm while L is a skew-symmetric matrix whose nmth entry is snsmlnm. G is a row vector whose nth element is - snωnRn(z). Different from the cases of u and η, the frequency-difference coefficient matrix [H - L] is not symmetric about m and n. Thereby, the following formulation is introduced: y]T + [x y][A] [x y]T

(26)

where matrix A is of the following form: H − L⎤ ⎡ 0 A=⎢ ⎥ 0 ⎥ ⎢⎣ H + L ⎦

(27)

Now A is symmetric such that it has the same property as D: A = P2 Λ2 P2T

(28)

where matrix Λ2 contains the eigenvalues δn of A (n = 1, …, 2N) along the diagonal; P2 is the orthonormal eigenvector matrix. Like u, a takes the sum of N quadratic transformations of independent standard Gaussian variables: a( z,t ) =

matrix

A

has

the

property

that

ground, for the third-order cumulant of a, we have: K 3a =

2N

∑ 8δ n3 + 6ιn2δ n = 0

(30)

n =1

which implies that the associated skewness is not existent. The obtained first four cumulants of kinematics are particularly useful for the polynomial approximation of Morison drag force which is necessary for the frequencydomain stochastic analysis of structural response [10,17]. The cumulant spectral analysis method has been proposed [17] to efficiently estimate the power-, bi & tri-spectra and the associated variance, skewness and kurtosis excess of the total wave force acting on an idealized monopod jack-up platform, as well as the induced deck displacement. To take into account the varying surface induced inundation effects [18,19] for the structural super-harmonic response, the nonlinear portion of kinematics need to be further corrected, especially for the zone above SWL (z > 0). Several wave stretching/extrapolation methods have been suggested for this problem and a comparative study was conducted by Stansberg and Gudmestad [20] against the laboratory measurements in steep waves. On the other hand, without correcting the nonlinear kinematics, the wave inundation effects can be also considered [17,19] by adding an extra wave load around the SWL to the total force which was conventionally integrated from sea bottom to SWL only. In this case, the formulations of kinematics in Eq. (13 & 24) hold validity for z ≤ 0 in this study. FFT SIMULATION OF ELEVATION/KINEMATICS One-dimensional IFFT/FFT simulation In time-domain simulation, the train of linear random wave elevation of overall M number of time steps can be expressed according to Eq. (7):

η1 (th ) =

∑ cn cos(ωnth + φn ),

h =0,...,M -1

(31)

n =0

cn = 2 Sη1 (ωn ) ∆ω

(29)

n =1

where ιn is the nth entry of [0 G]P2; random variable Yn is the nth entry of [x y]P2. Using Eq. (20-23), the fist four cumulants are calculated by replacing βn with ιn and by replacing λn with βn. Considering that matrix A has a zero trace, hence the mean of a is zero as well. In addition, it can be easily found that the

N −1

where th = h∆t , ∆t is the time step; cn is computed as:

2N

∑ ιnYn + δ nYn2

of

2

Since the trace of matrix D is zero, the sum of eigenvalues is zero too, which indicates that u has a zero mean. It is also noted that the kurtosis excess is always positive. The total acceleration is:

a(z,t) = [0 G] [x

eigenvalues

δ n = −δ 2 N +1−n and correspondingly ιn = ι2 N +1−n . On this

(32)

In the deterministic spectral amplitude simulation originally by Rice and Borgman [1,11], the phase φn is a sequence of uniformly distributed random numbers within the interval [0, 2π). Equation (31) can be rewritten as:

4



N −1

η1 (th ) = Re[ ∑ ( cn eiφn )e

i 2π nh M ],

h =0,...,M -1

(33)

n =0

where i is the imaginary unit and Re[ ] denotes the real part of a complex value inside the brackets. While applying the discrete Fourier transform (DFT), M is equal to N, a number usually chosen to be the integer power of 2 for the FFT algorithm. The familiar wave simulation based on inverse Fast Fourier Transform (IFFT) is by the following expression [11]:

η1 (th ) =

N 1 Re[ ∑ Cn e N n =1

i 2π ( n −1)( h −1) N ],

h =1,...,N

1 Sη (ωn ) ∆ω ⋅ eiφn , 2 1

n = 2,...,

N 2

Su1 ( z , ωn ) = Sη1 (ωn ) Rn ( z ) Sa1 ( z , ωn ) = ωn2 Su1 ( z, ωn )

The sequence of acceleration corresponds to the imaginary part of IFFT/FFT data.

(35)

Two-dimensional FFT simulation Fourier Transform is also able to tackle well with the double summations existent in the frequency sum and difference terms of the nonlinear random wave elevation and kinematics. Firstly let us consider the general 2-dimensional Mby-N DFT and inverse DFT pair:

and

F ( h, r ) =

N n = + 2,...., N 2

C N + 2− n = C*n ,

η1 (th ) = Re[ ∑ ( cn e

−iφn

−

)e

i 2π nh N ],

h =0,...,N -1

f ( n, m ) =

(37)

h = 0,..., N − 1 r = 0,..., M − 1

(41)

and (40)

By contrast to the IFFT procedure, the one-sided spectrum is utilized in Eq. (39); for the same number of N, the frequency resolution is two times finer which is at the price of two times coarser time step for the sake of ∆t ·∆f=1/N.

N −1 M −1

∑∑

F ( h, r ) e

h = 0 r =0

i 2π nh i 2π mr N e M ,

n = 0,..., N − 1 (42) m = 0,..., M − 1

f ( n, m) = IFFT2( F ( h, r ))

(43)

which is actually equivalent to separately calculating FFT/IFFT of each dimension of the input matrices. e.g.:

(38)

n = 0,...,N -1 (39)

1 NM

F ( h, r ) = FFT2( f ( n, m))

which means that the sequence of wave elevation is the direct Fourier transform of the coefficients:

η1 (th ) = Re[FFT(Cn )], h,n =0,...,N − 1

f ( n, m )e

i 2π nh i 2π mr − N e M ,

Numerically, matrices F(h,r) and f(n,m) are computed by the 2dimensional FFT/IFFT that is denoted as ‘FFT2’ herein:

n =0

Cn = cn e −iφn = e −iφn 2 Sη1 (ωn ) ∆ω ,

∑∑

−

(36)

Equation (35) shows that the two-sided spectrum is involved and the Nyquist frequency at the symmetry point n= N/2+1. Alternatively, Eq. (33) can be rewritten as: N −1

N −1 M −1 n =0 m = 0

where the asterisk ‘*’ denotes complex conjugate. The discrete sequence of wave elevation is then obtained by synthesize the IFFT of sequence Cn:

η1 (th ) = IFFT(Cn ), h,n =1,...,N

2

(34)

where the complex Fourier coefficients is computed as: Cn = N

To simulate the Gaussian velocity and acceleration, one just needs to replace wave spectrum by velocity and acceleration power spectra respectively. The power spectra of velocity and acceleration at the location z are:

F ( h, r ) = FFTn (FFTm ( f ( n, m ))

(44)

f ( n, m) = IFFTh (IFFTr ( F ( h, r ))

(45)

where, for instance, FFTm ( f ( n, m) indicates the 1dimensional FFT along the second dimension (m for column) of the matrix f(n,m), the result of which is then treated by another 1-dimensional FFT along the first dimension (n for row). Certainly, the order of n and m can be exchanged without affecting the output matrix. Now rewrite the frequency sum term of nonlinear elevation in Eq. (10) in the discrete form:

5



N −1 N −1

η2+ (th ) = Re[ ∑

∑ (cn cm vnm e−iφn e−iφm )e

−

i 2π nh i 2π mh − N e N ],

n = 0 m =0

h = 0,..., N − 1 ⎧ − ⎪ F ( h, r ) = IFFTn (FFTm ( f ( n, m)), r = 0,..., N − 1 ⎪⎪ ⎨ ⎪ − ⎪η2 (th ) = N ⋅ Re[ F ( h, r ) h = r ], h =0,...,N -1 ⎪⎩

(46)

h =0,...,N -1

Compared with Eq. (41), obviously because h = r, this sequence is exactly the real part of the diagonal entries of the 2dimensional FFT output matrix; namely: h = 0,..., N − 1 ⎧ + ⎪ F ( h, r ) = FFT2( f ( n, m)), r = 0,..., N − 1 ⎪⎪ ⎨ ⎪ + ⎪η2 (th ) = Re[ F ( h, r ) h = r ], h =0,...,N -1 ⎪⎩

(47)

where the frequency-sum wave interaction matrix here is: n = 0,..., N − 1 m = 0,..., N − 1

f + ( n, m ) = cn cm vnm e −iφn e −iφm ,

(48)

Similarly, the frequency difference term of elevation is also expressible in the following discrete form: N −1 N −1

η2− (th ) = Re[ ∑

∑ (cn cm wnm

i 2π nh i 2π mh − iφn −iφm e e )e N e N ],

n =0 m = 0

⎡

1 ⎢N ⎣

N −1 ⎛ N −1

∑ ⎜⎜ ∑ ( Ncn cm wnmeiφn e−iφm )e n =0 ⎝ m =0

−

i 2π mh N

RESULTS AND DISCUSSIONS Once the nonlinear kinematics u and a are simulated, the Morison force per unit length on a slender member is also available:

= CM AI a ( z, t ) + C D AD u( z , t ) u( z, t )

(49)

which can be further rewritten as:

η2− (th ) = Re ⎢

where the frequency difference sequence is also the real part of the diagonal elements of the two-step FFT/IFFT output matrix. Unlike the frequency sum term that may rely on the direct application of double FFT, the frequency difference sequence has be to calculated by carrying out FFT and IFFT respectively across the two dimensions (Eq. (53)), but incurring negligible extra computational efforts. The above procedures are applicable to simulating the nonlinear portions of kinematics as well. The sequence of acceleration corresponds to the imaginary part of diagonal entries.

f ( z, t ) = f I + f D

h =0,...,N -1

⎞ i 2π nh ⎤ ⎟e N ⎥ , ⎥ ⎟ ⎠ ⎦

h =0,...,N -1

(53)

(54)

where AI = πρ(Deq)2/4 and AD = ρDeq/2; ρ is water density; Deq is the equivalent diameter of the cylinder; CM and CD, the inertia and drag coefficients. Note that the wave-structure interaction effect which can be alternatively taken into account by adjusting the oscillator damping is not included here because our focus is on the nonlinear random wave effects. Using the standard time integration procedure, the displacement sequence of the following linear oscillator driven by Morison force is obtained:

(50)

MY&& + CY& + KY = f ( z, t )

Denote the frequency-difference wave interaction matrix as: f − ( n, m) = cn cm wnm eiφn e −iφm ,

n = 0,..., N − 1 m = 0,..., N − 1

with the corresponding frequency response function: (51) HYf (ω ) =

then: ⎡ 1 η2− (th ) = N Re ⎢ ⎢N ⎣ h =0,...,N -1

N −1 ⎛ N −1

∑ ⎜⎜ ∑

f − ( n, m)e

n =0 ⎝ m =0

based on Eq. (44 & 45), we have:

−

i 2π mh N

⎞ i 2π nh ⎤ ⎟e N ⎥ , ⎥ ⎟ ⎠ ⎦

(55)

(52)

1 M (ωn2

− ω + 2iξωnω ) 2

(56)

where M is the oscillator mass, ωn is the free-vibration frequency and ξ the damping ratio. We investigate again the previously treated case [10] in which the detailed wave conditions specified for a JONSWAP wave spectrum are: significant wave height of 12.9 meters, water depth of 75 meters and peak wave frequency 0.417 rad/s. Using the second-order random wave model by Sharma and Dean [5], as many as 180 realizations are generated. Each realization utilizes N = 2048 and frequency resolution ∆ω =

6



0.00511 rad/s. To avoid high-frequency induced excessive extreme values of waves, the maximum frequency used to simulate second-order portions is determined by a criterion suggested by Stansberg [13], up to around 4~5 times peak wave frequency. The FFT-based computation is so powerful that no more than 10 seconds CPU time is consumed for simulating an individual sequence of wave elevation of duration 1/3 hour. It is much more efficient than the realization using the matrixvector multiplication under the same N and ∆ω – a realization costs roughly 50 minutes. After all, it is known that numerically the 1-dimensional FFT is a process of O(Nlog2N) multiplications that surpasses the standard double-loop approach – a process of O(N2) multiplications. The practical computation shows that the difference of CPU time in these two processes is even much smaller than (log2N)/N. Therefore, the 2-dimensional FFT scheme suggested for second-order wave/kinematics simulation will further outmatches the standard triple-loop approach which, taking into account the symmetries of frequency sum and difference terms, is a O(N2(1+N/2)) process. In Table 1, the higher-order moments of interest, skewness and kurtosis excess, of the surface elevation are presented. The comparative study is carried out among: (a) (b) (c) (d) (e)

It can be observed that all analytically, numerically and empirically obtained skewness and kurtosis excess agree well for the intermediate-depth waters, see (a), (b), (c) and (d) data. The kurtosis excess by Vinje and Haver [21] is almost twice the other three cases because they made a correction by including the contribution of third-order Stokes’ wave effects. By contrast the deep-water statistics appear smaller, which implies the more weakly non-Gaussianity of wave. The further comparison of probability density function can be found in [10]. When the water becomes shallower, the wave non-Gaussianity will be stronger such that employing the deep-water model will result in underestimations. Another important parameter affecting the wave non-Gaussianity was the specified wave height [10]. 0 -10 -20 -30

-50 -60 -70

the analytical solutions the average of FFT simulated results empirical fit by Vinje and Haver [21] empirical fit by Winterstein and Jha [22] deep water analytical solutions

-80

(

0

0.5

1

1.5

2

Fig. 1 Skewness and kurtosis excess of a decay with z 0

Both the empirical fit (c) and (d) were obtained based on infield laser measurements. Different from (c), the fit (d) additionally has the correction considering the finite-waterdepth effects [22]: H −1 λ3 = s 5.45γ −0.084 + {exp[ 7.41( d / L p )1.22 ] −1} LP


z (m) -40

)

(57)

-10 -20 -30 z (m) -40


-50 -60

where Hs is the significant wave height, Lp the wave length at wave peak frequency, γ the peak enhancement factor for JONSWAP spectrum. For the analytical solutions by eigenvalue/eigenvector approach, a limited number of frequency components, say 80, is able to capture the convergent skewness and kurtosis.

-70 -80

-0.4 -0.2

0

0.2

0.4

0.6

0.8

1

1.2

Fig. 2 Skewness and kurtosis excess of u decay with z Table 1. Skewness and kurtosis excess of wave elevation (a)

κ3 0.1892

κ4 0.0545

(b)

0.1867

0.0575

(c)

0.1993

0.1191

(d)

0.1972

0.0535

(e)

0.1668

0.0461

Figures (1 & 2) show the skewness and kurtosis excess of kinematics as a function of location z, analytical versus simulation results. It is obvious to see the good agreement of the two approaches. The non-Gaussian characteristics of u and a are significant in the zone near the still water surface and such strong non-Gaussianities decay rapidly with z. At a deep location, the non-Gaussianities are so weak that both u and a approaches Gaussian - as the result the kurtosis excess of drag

7



force is rather close to 8.6667 [10,12] and in the practical simulation the linear random wave theory can be used to calculate the distributed Morison forces. It is also interesting to note that, on the contrary to the surface elevation, the velocity skewness is negative, which has been reported by Hu [23] for the deep-water case. This reveals that, the non-Gaussian velocity u, when expanded as the cubic polynomials of Gaussian velocity u1, e.g. [10]: σ u1

+ c2 ( z )(

u1 ( z, t )

σ u1

) + c3 ( z )( 2

u1 ( z, t )

σ u1

)

3

Linear random wave Nonlinear random wave

700

(58)

600

the second-degree polynomial coefficient c2(z) should be negative because the third-order cumulant for skewness estimation is expressed as: 2 K3u = 2c2 ⎡⎢ 3 ( c1 + 6c3 ) + 4c22 + 27c32 ⎤⎥ ⎣ ⎦

800

SYY(ω) (cm2)

u ( z, t ) ≈ c0 ( z ) + c1 ( z )

u1 ( z, t )

second-order nonlinear wave effects, the contribution to the resonance at 2ωp is purely from drag nonlinearity. Thus, apparently the drag nonlinearity becomes stronger due to the wave nonlinearities. This can be also supported by looking into the higher-order statistics of both force and response in Tables 2 & 3.

500 400 300 200

(59)

100 0 0

It turns out that the skewness of Morison drag term is negative as well and the probability distribution of Morison force tends to be left-skewed [10]. Later in Table 3, the linear oscillator displacement will be shown to have a negative skewness too.

1

2

ω/ωp

3

4

5

Fig. 4 Power spectrum of oscillator displacement Table 2 Cumulants of Morison force (z = -3 m)

Linear random wave Nonlinear random wave

6000

2

Sff (ω) (N s)

5000 4000 3000

(A)

κ1 (N) 0

κ2 (N2) 2.0997e+002

κ3 0

κ4 6.7787

(B)

7.0709e-003

2.0867e+002

3.5869e-005

6.3222

(C)

-6.8290e-001

2.5234e+002

-1.5743

11.8701

(D)

-7.0969e-001

2.5059e+002

-1.4994

10.7292

2000

Table 3 Cumulants of oscillator displacement 1000 0 0

1

2

ω/ωp

3

4

5

6

Fig. 3 Power spectrum of Morison force (z = -3 m) A jack-up platform (with the natural frequency of 0.848 rad/s and damping ratio 0.07) considered in a previous study [19] is ideally modeled as a linear oscillator for its excited vibration. We investigate here the response of oscillator when driven by a local Morison force 3 meters below SWL. The assumed oscillator mass is 1000 kg. Figures 3 & 4 respectively show the power spectra of Morison force and oscillator displacement obtained from FFT simulations. Compared with the linear random wave case, the second-order wave effects result in small increase of the amplitude of force spectrum for frequencies greater than peak wave frequency ωp, which however causes the oscillator’s super-harmonic response at 2ωp to be further amplified by about 15% (Fig. 3). Without the

(A)

κ1 (cm) 0

κ2 (cm2) 21.9136

κ3 0

κ4 1.8130

(B)

2.1091e-003

22.3940

2.2279e-003

1.7302

(C)

-9.4938e-002

26.4359

-8.4333e-002

2.8790

(D)

-9.7044e-002

26.8097

-0.11095

2.5084

Here, the first four normalized cumulants (mean, variance, skewness and kurtosis excess) of four cases are compared: (A) Frequency-domain cumulant spectral analysis - linear random wave theory (B) FFT time simulation - linear random wave theory (C) Frequency-domain cumulant spectral analysis second-order nonlinear random wave theory (D) FFT time simulation - second-order nonlinear random wave theory

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Firstly, we can observe that simulation data compare well with the frequency-domain results, especially for mean and variance. For kurtosis, the frequency-domain analysis produces slightly higher estimations. Secondly, variance increases are noticeable, by around 25% for both force and by around 18% for response, attributable to the second-order wave effects. Thirdly, the force and response skewness are no longer zeros, because of the negative skewness of velocity aforementioned. Most importantly, the Morison force exhibits a much stronger nonGaussian behavior under the second-order nonlinear wave theory by a big enlargement (> 80%) of kurtosis excess. The displacement kurtosis excess, much lower that that of force due to the linear filtering effects of oscillator, gains significant increase too. When the water is shallower and the linear structure becomes stiffer, such stronger nonlinearities of force and response due to second-order waves can be found still. With the four moments obtained, Winterstein’s Hermite functional transformation can be applied to the extreme value predictions [13]. The details are not discussed in this study. CONCLUSIONS Based on second-order nonlinear random wave theory, an analytical method is developed in this study to statistically estimate the cumulants of non-Gaussian wave elevation and kinematics in waters of an arbitrary depth. To solve the timeconsuming numerical simulation problem incurred by the double-summations of frequency sum and difference terms, an efficient approach utilizing two-dimensional FFT technique is introduced with detailed procedures. The comparative study shows that the simulated results of wave elevation agree well with not only analytical solutions but empirical fits. The favorable agreement is also observed for kinematics by comparing the simulated and analytical cumulants. In addition, the stochastic response of a linear oscillator driven by a nearsurface Morison force is examined. Again it is found that for both force and response, the simulated statistics compare well with the data obtained by a previously developed frequencydomain method. It is pointed out that, the non-Gaussianity of wave force is greatly strengthened near water surface by involving the second-order wave effects, which causes the response non-Gaussianity to be significantly higher than that resultant from drag nonlinearity only. REFERENCES

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[4] Hasselmann, K., 1962, “On the Nonlinear Energy Transfer in a Gravity-wave Spectrum, Part 1, General Theory,” J. of Fluid Mech. (12), pp. 481-500. [5] Sharma, J. N., and Dean, R. G., 1979, “Development and Evaluation of a Procedure for Simulating a Random Directional Second Order Sea Surface and Associated Wave Forces,” Ocean Eng. Rep. (20), Univ. of Delaware. [6] Huang, N. E., Long, S. R., Tung, C. C., Yuan, Y., 1983, “A Non-Gaussian Statistical Model for Syrface Elevation of Nonlinear Random Wave Fields,” J. of Geophysical Res., 88(C12), pp. 7597-7606. [7] Tayfun, M. A., 1986, “On Narrow-band Representation of Ocean Waves,” J. of Geophysical Res., 91(6), pp. 77437752. [8] Martinsen, T., and Winterstein, S. R., 1992, “On the Skewness of Random Surface Waves,” Proc. 2nd ISOPE Conf., San Francisco, USA, (III), pp. 472-478. [9] Langley, R. S., 1987, “A Statistical Analysis of Non-linear Random Waves. Ocean Eng., 14(5), pp. 389-407. [10] Quek, S. T., Zheng, X. Y., Moan, T., 2005, “Nonlinear Wave Effects on Distributions of Wave Elevation and Morison force,” Proc. 9th ICOSSAR Conf., G. Augusti et al. eds., Millpress, Rotterdam, pp. 3325-3334. [11] Borgman, L. E., 1969, “Ocean Wave Simulation for Engineering Design,” ASCE J. of Waterways and Harbors, 95(4), pp. 557-583. [12] Zheng, X. Y., Liaw, C. Y., 2005, “Response Cumulant Spectral Analysis of Linear Oscillators Driven by Morison Forces,” Applied Ocean Res. 26(3-4), pp. 154-161. [13] Stansberg, C. T., 1998, “Non-Gaussian Extremes in Numerically Generated Second-order Random Waves on Deep Water,” Proc. 8th ISOPE Conf., Montreal Canada, (III), pp. 103-110. [14] Bouyssy, V., Rackwitz, R., 1997, “Polynomial Approximation of Morison Wave Loading,” ASME J. of OMAE, (119), pp. 30-36. [15] Hudspeth R. T., Chen, M. C., 1979, “Digital Simulation of Nonlinear Random Waves,” ASCE J. of Waterways and Harbors, 105(WW1), pp. 67-85. [16] Stansberg, C. T., 1993, “Second-order Numerical Reconstruction of Laboratory Generated Random Waves,” OMAE Conf., (III), pp. 143-151. [17] Zheng, X. Y., Liaw, C. Y., 2004, “Nonlinear Frequencydomain Analysis of Jack-up Platforms,” Int. J. of NonLinear Mech., 39(9), pp. 1519-1534. [18] Tung, C. C., 1995, “Effects of Free Surface Fluctuation on Total Wave Forces on Cylinder,” ASCE J. of Eng. Mech., 121(2), pp. 274-280. [19] Liaw, C. Y., Zheng, X. Y., 2001, “Inundation Effect of Wave Forces on Jack-up Platforms,” Int. J. of Offshore and Polar Eng., 11(2), 87-92. [20] Stansberg, C. T., Gudmestad O. T., 1996, “Nonlinear Random Wave Kinematics Models Verified Against Measurements in Steep Waves,” 15th OMAE Conf., (I), pp. 15-24.

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[21] Vinje, T., Haver, S., 1994, “On the Non-Gaussian Structure of Ocean Waves,” BOSS, (2), pp. 453-480. [22] Winterstein S. R., Jha, A. K., 1995, “Random Models of Second-order Waves and Local Wave Statistics,” Proc. 10th ASCE Eng. Mech. Specialty Conf., pp. 1171-1174. [23] Hu, S-L. J., 1993, “Nonlinear Random Water Wave,” Book: Computational Stochastic Mechanics, Cheng, A. HD., Yang C. Y., eds., Elsevier, pp. 519-543.

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