Numerical Simulation of Non-Gaussian Wave Elevation and Kinematics Based on Two-Dimensional Fourier Transform

2006 ◽  
Author(s):  
Xiang Yuan Zheng ◽  
Torgeir Moan ◽  
Ser Tong Quek
Keyword(s):  
Fourier Transform ◽  
Wave Interaction ◽  
Wave Theory ◽  
Two Dimensions ◽  
Second Order ◽  
Interaction Matrix ◽  
Random Wave ◽  
Two Dimensional ◽  
Near Surface ◽  
Wave Elevation

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 time-and 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.

Non-Gaussian Random Wave Simulation by Two-Dimensional Fourier Transform and Linear Oscillator Response to Morison Force

2007 ◽  
Vol 129 (4) ◽  
pp. 327-334 ◽  
Author(s):  
Xiang Yuan Zheng ◽  
Torgeir Moan ◽  
Ser Tong Quek
Keyword(s):  
Fourier Transform ◽  
Time Domain ◽  
Two Dimensions ◽  
Second Order ◽  
Linear Oscillator ◽  
Interaction Matrix ◽  
Random Wave ◽  
Two Dimensional ◽  
Near Surface ◽  
Non Gaussian

The one-dimensional fast Fourier transform (FFT) has been applied extensively to simulate Gaussian random wave elevations and water particle kinematics. The actual sea elevations/kinematics exhibit non-Gaussian characteristics that can be represented mathematically by a second-order random wave theory. The elevations/kinematics formulations contain frequency sum and difference terms that usually lead to expensive time-domain dynamic analyses of offshore structural responses. 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 forward FFT are implemented, respectively, across the two dimensions of the wave interaction matrix. Given specified wave conditions, the statistics of simulated elevations/kinematics compare well with not only the empirical fits but also the analytical solutions based on a modified eigenvalue/eigenvector approach, while the computational effort of simulation is very economical. In addition, the stochastic analyses in both time domain and frequency domain show that, attributable to the second-order nonlinear wave effects, the near-surface Morison force and induced linear oscillator response are more non-Gaussian than those subjected to Gaussian random waves.

TWO-DIMENSIONAL NONLINEAR OPTICS SPECTROSCOPY: SIMULATIONS AND EXPERIMENTAL DEMONSTRATION

1996 ◽  
Vol 05 (03) ◽  
pp. 465-476 ◽  
Author(s):  
L. LEPETIT ◽  
G. CHÉRIAUX ◽  
M. JOFFRE
Keyword(s):  
Fourier Transform ◽  
Nonlinear Response ◽  
Two Dimensions ◽  
Second Order ◽  
Phase Matching ◽  
Two Dimensional ◽  
Experimental Demonstration ◽  
Spectral Interferometry ◽  
Dimensional Measurement ◽  
A New Technique

We propose a new technique, using femtosecond Fourier-transform spectral interferometry, to measure the second-order nonlinear response of a material in two dimensions of frequency. We show numerically the specific and unique information obtained from such a two-dimensional measurement. The technique is demonstrated by measuring the second-order phase-matching map of two non-resonant nonlinear crystals.

Scattering of Waves by Two-Dimensional Circular Obstacles in Finite Water Depths

1979 ◽  
Vol 23 (01) ◽  
pp. 32-42 ◽  
Author(s):  
Robert A. Naftzger ◽  
Subrata K. Chakrabarti
Keyword(s):  
Wave Interaction ◽  
Wave Theory ◽  
Finite Depth ◽  
Wave Forces ◽  
Linear Wave ◽  
Two Dimensional ◽  
Linear Wave Theory ◽  
Dimensional Object ◽  
Limiting Case ◽  
Water Depths

The wave forces on a fixed two-dimensional object submerged in water of finite depth are obtained under the assumptions of linear wave theory. The far-field characteristics of the wave interaction with the object are also examined. The boundary-value problem for the wave potential is formulated in terms of Green's theorem, and the resulting integral equation is solved numerically. Results for a submerged and half-submerged circular cylinder and a bottom-seated half cylinder are presented. In the limiting case of infinite depth the numerical results compare quite well with known solutions.

On higher-order wave theory for submerged two-dimensional bodies

1969 ◽  
Vol 38 (2) ◽  
pp. 415-432 ◽  
Author(s):  
Nils Salvesen
Keyword(s):  
Free Surface ◽  
Wave Theory ◽  
Order Theory ◽  
Wave Drag ◽  
Higher Order ◽  
Second Order ◽  
Two Dimensional ◽  
Third Order ◽  
Free Surface Waves ◽  
Body Shapes

The importance of non-linear free-surface effects on potential flow past two-dimensional submerged bodies is investigated by the use of higher-order perturbation theory. A consistent second-order solution for general body shapes is derived. A comparison between experimental data and theory is presented for the free-surface waves and for the wave resistance of a foil-shaped body. The agreement is good in general for the second-order theory, while the linear theory is shown to be inadequate for predicting the wave drag at the relatively small submergence treated here. It is also shown, by including the third-order freesurface effects, how the solution to the general wave theory breaks down at low speeds.

scholarly journals Comparison of Extreme Offshore Structural Response from Two Alterna-tive Stretching Techniques

2013 ◽  
Vol 7 (1) ◽  
pp. 273-281 ◽  
Author(s):  
N.I. Mohd Zaki ◽  
M.K. Abu Husain ◽  
G. Najafian
Keyword(s):  
Wave Theory ◽  
Surface Elevation ◽  
Random Wave ◽  
Surface Zone ◽  
Offshore Structure ◽  
Near Surface ◽  
Water Particle ◽  
Wave Kinematics ◽  
Water Particle Kinematics ◽  
Extreme Responses

Linear random wave theory (LRWT) has successfully explained most properties of real sea waves with the ex-ception of some nonlinear effects for surface elevation and water particle kinematics. Due to its simplicity, it is frequently used to simulate water particle kinematics at different nodes of an offshore structure from a reference surface elevation record; however, predicted water particle kinematics from LRWT suffer from unrealistically large high-frequency compo-nents in the vicinity of mean water level (MWL). To overcome this deficiency, a common industry practice for evaluation of wave kinematics in the free surface zone consists of using linear random wave theory in conjunction with empirical techniques (such as Wheeler and vertical stretching methods) to provide a more realistic representation of near-surface wave kinematics. It is well known that the predicted kinematics from these methods are different; however, no systematic study has been conducted to investigate the effect of this on the magnitude of extreme responses of an offshore structure. In this paper, probability distributions of extreme responses of an offshore structure from Wheeler and vertical stretching methods are compared. It is shown that the difference is significant; consequently, further research is required to deter-mine which method is more reliable.

scholarly journals Variance Measures for Symmetric Positive (Semi-) Definite Tensors in Two Dimensions

2021 ◽  
pp. 3-22
Author(s):  
Magnus Herberthson ◽  
Evren Özarslan ◽  
Carl-Fredrik Westin
Keyword(s):  
Tensor Product ◽  
Fourth Order ◽  
Equivalence Problem ◽  
Elasticity Tensor ◽  
Two Dimensions ◽  
Second Order ◽  
Two Dimensional ◽  
Order Tensor ◽  
Symmetry Properties ◽  
Fourth Order Tensor

AbstractCalculating the variance of a family of tensors, each represented by a symmetric positive semi-definite second order tensor/matrix, involves the formation of a fourth order tensor $$R_{abcd}$$ R abcd . To form this tensor, the tensor product of each second order tensor with itself is formed, and these products are then summed, giving the tensor $$R_{abcd}$$ R abcd the same symmetry properties as the elasticity tensor in continuum mechanics. This tensor has been studied with respect to many properties: representations, invariants, decomposition, the equivalence problem et cetera. In this paper we focus on the two-dimensional case where we give a set of invariants which ensures equivalence of two such fourth order tensors $$R_{abcd}$$ R abcd and $$\widetilde{R}_{abcd}$$ R ~ abcd . In terms of components, such an equivalence means that components $$R_{ijkl}$$ R ijkl of the first tensor will transform into the components $$\widetilde{R}_{ijkl}$$ R ~ ijkl of the second tensor for some change of the coordinate system.

The Effect of Different Methods of Simulating Water Particle Kinematics on the 100-Year Responses

2016 ◽  
Author(s):  
N. I. Mohd Zaki ◽  
M. K. Abu Husain ◽  
N. Abdullah Shuhaimy ◽  
G. Najafian
Keyword(s):  
Wave Theory ◽  
Random Wave ◽  
Offshore Structure ◽  
Near Surface ◽  
Water Particle ◽  
Wave Kinematics ◽  
Realistic Representation ◽  
Frequency Components ◽  
Offshore Industry ◽  
Water Particle Kinematics

Linear random wave theory (LRWT) is frequently used to simulate water particle kinematics at different nodes of an offshore structure from a reference surface elevation record. However, it is well known that LRWT leads to water particle kinematics with exaggerated high-frequency components in the vicinity of mean water level (MWL). A number of empirical techniques have been suggested to provide a more realistic representation of near surface wave kinematics. The empirical techniques popular in the offshore industry include Wheeler stretching, linear extrapolation, delta stretching, and vertical stretching. Each of these methods is intended to calculate sensible kinematics above the MWL, yet they have been found to differ from one another in the results yielded. In this paper, two new methods of simulating water particle kinematics are introduced. In this study, the values of 100-year responses derived from different methods of simulating wave kinematics are compared.

Freak Waves Within the Third Order Model

2010 ◽  
Author(s):  
Xiang Yuan Zheng ◽  
Torgeir Moan
Keyword(s):  
Wave Train ◽  
Wave Interaction ◽  
Physical Nature ◽  
Wave Theory ◽  
Higher Order ◽  
Random Wave ◽  
Occurrence Probability ◽  
Order Model ◽  
Freak Waves ◽  
Wave Models

After the New Year Wave was recorded in January 1995, considerable works have been devoted to explore the definition, physical nature and occurrence probability of freak waves. Within the frame of classical wave theories, the linear and 2nd-order random wave models have been chosen most often in the numerical simulations to study the occurrence of freak waves. This paper employs the 3rd-order random wave theory in simulation to investigate the effects of higher-order wave interaction on the freak wave occurrence. The New Year Wave is used as the case study herein. Its crest is 18.5 m, a criterion applied to categorize the simulated wave trains. To efficiently simulate a wave train with complicated wave interactions, a 1D FFT method is suggested. It is pointed out that the high values of skewness and kurtosis excess of New Year Wave cannot be captured by the 2nd-order model. The 2nd-order model is more suitable for reproducing events of high crests. Extreme events need a higher-order wave model. This work extends the study by Prevosto and Bouffandeau (2002). With totally 75,000,000 waves simulated the occurrences of freak waves of crest > 18.5 m are compared among linear, 2nd- and 3rd-order wave models. The comparative study also includes the predictions by Forristall (2000), Prevesto (2000) and the one derived based on Mori and Janssen (2004). The simulation once again reveals the inadequacy of the 2nd-order model to generate freak waves. The occurrence probability of freak waves under the 3rd-order model is more than 6 times higher than that under the 2nd-order.

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