Nonlinear Wave Crests Research Articles

Extreme wave crest distribution by direct numerical simulations of long-crested nonlinear wave fields

The wave crest height qualification checks are required during the wave calibration before the model test in wave basin. However, the reliable criteria of nonlinear wave crest probability distribution in 3-h duration (full-scale) has not been well established yet. We investigate wave crest-height statistics of long-crested nonlinear wave fields using high-order spectral (HOS) method, which can take the effects of both second-order bound waves and third-order free waves into account. The energy dissipation effects due to wave breaking were included by employing an eddy viscosity model. Sensitivity analyses to the wave breaking onset criterion have been performed. Validation is provided by comparing the obtained numerical results with the available calibration test data. Based on extensive and direct numerical simulations, semi-empirical single realization distributions for wave calibration have been developed through 3-parameter Weibull fitting and systematic regression analyses. Particular attention has been paid to the tail of upper bound of wave crest distributions. The effects of wave steepness and water depth on the maximum wave crest height in 3-h duration have been examined. It is found that with the increase of wave steepness, the extreme wave crest height increases until it reaches a critical value. In addition, for the scale water depth kph < 1.36, the maximum crest height decreases as the water depth increases, while in the opposite case the maximum crest height increases as the water depth increases. Moreover, it is confirmed that that the fourth-order nonlinearity does not have significant effects on the distribution of the wave crest height.

Nonlinear wave crest distribution on a vertical breakwater

The probability distribution of the nonlinear, up to the second order, crest height on a vertical wall is determined under the assumption of finite spectral bandwidth, finite water depth and long-crested waves. The distribution is derived by relying on the Quasi-Deterministic representation of the free surface elevation on the vertical wall. The theoretical results are compared against experimental data obtained by utilizing a compressive sensing algorithm for reconstructing the free surface elevation on the wall. The reconstruction is pursued by starting from recorded wave pressure time histories obtained by utilizing a sequence of pressure transducers located at various levels. The comparison demonstrates an excellent agreement between the proposed distribution and the experimental data, while, notably, the deviation of the crest height distribution from the Rayleigh one is considerable.

Efficient Computation of Nonlinear Crest Distributions for Irregular Stokes Waves

AbstractThis paper concerns the computation of nonlinear crest distributions for irregular Stokes waves, and a numerical algorithm based on the Fast Fourier Transform (FFT) technique has been developed for carrying out the nonlinear computations. In order to further improve the computational efficiency, a new Transformed Rayleigh procedure is first proposed as another alternative for computing the nonlinear wave crest height distributions, and the corresponding computer code has also been developed. In the proposed Transformed Rayleigh procedure, the transformation model is chosen to be a monotonic exponential function, calibrated such that the first three moments of the transformed model match the moments of the true process. The numerical algorithm based on the FFT technique and the proposed Transformed Rayleigh procedure have been applied for calculating the wave crest distributions of a sea state with a Bretschneider spectrum and a sea statewith the surface elevation datameasured at the Poseidon platform. It is demonstrated in these two cases that the numerical algorithm based on the FFT technique and the proposed Transformed Rayleigh procedure can offer better predictions than those from using the empirical wave crest distribution models. Meanwhile, it is found that our proposed Transformed Rayleigh procedure can compute nonlinear crest distributions more than 25 times faster than the numerical algorithm based on the FFT technique.

Space–time long-term statistics of ocean storms

This paper presents an approach for estimating the long-term statistics of random wave crests occurring over a certain space–time domain. Such a problem is relevant in a number of marine engineering applications, as classical analyses based, exclusively, on time domain approaches underestimate wave crest amplitudes associated with a given return period.The return period of a certain wave crest is derived by combining the Equivalent Power Storm model, used in long-term statistical analyses, with the Euler Characteristic (EC) of an excursion set concept, recently applied to the study of sea wave statistics. In this regard, the paper shows that by computing the average EC, the probability distribution of the wave crests can be derived. Specifically, an explicit solution valid for finite crest thresholds can be derived by an approximation of the EC. Thus, removing the limitation of actual solutions, which are valid only for extremely large wave crests. In addition, return period of nonlinear wave crests are derived. In this context, Forristall distribution is adopted and introduced on a heuristic basis in the EC framework.The EC approximation and the reliability of the nonlinear crest distribution is assessed against Monte Carlo data by comparing distributions of maximum wave crests both in a sea state and in a sea storm. Then, return value estimations are discussed for a number of cases.

Probability analysis and parameter estimation for nonlinear relative wave motions on a semi-submersible using the method of LH-moments

Nonlinear relative wave motions and air gap performance greatly affect the operation and security of semi-submersibles in harsh environments. An appropriate air gap is essential to preventing potential impact loads, and accurate prediction for long-return-period events is of great concern. Given the random and nonlinear nature of relative wave motions, a reliable approach for their prediction is a challenging task that warrants further research. In this study, a modified three-parameter Rayleigh distribution model for nonlinear relative wave crests was derived utilising a quadratic transformation of incident waves. The method of LH-moments was used to estimate the unknown parameters from sample data, and the relationship between the parameters and LH-moments was derived. The sample data used for the probability analysis were obtained from a model test programme for a semi-submersible. The first-order parameter of the model was found to decrease with increasing η, whereas the second-order parameter was found to increase. A larger second-order parameter indicates higher nonlinearity of the relative wave motions for a 100-year-return sea state than for a 1-year-return sea state. In general, using LH-moments to estimate the parameters of the probability distribution results in better prediction of large relative wave crests with low probabilities of exceedance, because the estimation of the three parameters in the nonlinear Rayleigh model using LH-moments released the constraint on the linear term.

On long-term statistics of high waves via the equivalent power storm model

Abstract The objective of this paper is to investigate the long-term statistics associated with extreme nonlinear sea wave crests. For this purpose, an equivalent power storm methodology is exploited for modeling the non-stationary time variability of the significant wave height. The method involves the identification of the real storm time histories and a subsequent replacement of each real storm by a fictitious storm that is equivalent to the real one in a statistical sense. Such an approach allows deriving the closed form expression of the return period of a nonlinear wave crest. Numerical results pertain to the calculation of this quantity starting from wave data recorded at the Gulf of Mexico and comparison with alternative methodology recommended in the practice. Next, the paper investigates effects relevant to the design of marine structures in the proximity of a coast. Specifically, the paper considers the effect of the water depth on the return values of the significant wave height and of the maximum expected crest height. The water depth influence relates to a magnification of the return values of wave crest despite the fact that the offshore significant wave height is larger than the near-shore one for a certain value of the return period, due to shoaling-refraction effects.

Crest-height distribution of nonlinear random waves

The wave crest is an important factor for the design of both fixed and floating marine structures. Wave crest height is a dominant parameter in assessing the likelihood of wave-in-deck impact and resultant severe damage. Many empirical and theoretical distribution functions for wave crest heights have been proposed, but there is a lack of agreement between them. It is of significance to develop a better new nonlinear wave crest height distribution model. The progress in the research of wave crest heights is reviewed in this paper. Based on Stokes’ wave theory, an approximate nonlinear wave crest-height distribution formula with simple parameters is derived. Two sets of measured data are presented and compared with various theoretical distributions of wave crests obtained from nonlinear wave models and analysis of the comparison is given in detail. The new crest-height distribution model agrees well with observations. Also, the new theoretical distribution is more accurate than the other methods cited in this paper and has a greater range of applications.

On nonlinear wave groups and crest statistics

We present a second-order stochastic model of weakly nonlinear waves and develop theoretical expressions for the expected shape of large surface displacements. The model also leads to an exact theoretical expression for the statistical distribution of large wave crests in a form that generalizes the Tayfun distribution (Tayfun, J. Geophys. Res., vol. 85, 1980, p. 1548). The generalized distribution depends on a steepness parameter given by μ = λ3/3, where λ3 represents the skewness coefficient of surface displacements. It converges to the Tayfun distribution in narrowband waves, where both distributions describe the crests of all waves well. In broadband waves, the generalized distribution represents the crests of large waves just as well whereas the Tayfun distribution appears as an upper bound and tends to overestimate them. However, the theoretical nature of the generalized distribution presents practical difficulties in oceanic applications. We circumvent these by adopting an appropriate approximation for the steepness parameter. Comparisons with wind-wave measurements from the North Sea suggest that this approximation allows both distributions to assume an identical form with which we can describe the distribution of large wave crests fairly accurately. The same comparisons also show that third-order nonlinear effects do not appear to have any discernable effect on the statistics of large surface displacements or wave crests.

Return period of nonlinear high wave crests

This paper deals with the long‐term statistics for extreme nonlinear crest heights. First, a new analytical solution for the return period R(η), of a sea storm in which the maximum nonlinear crest height exceeds a fixed threshold η, is obtained by applying the ‘Equivalent Triangular Storm’ model and a second‐order crest height distribution. The probability P(ηc max &gt; η∣[0, L]) that maximum nonlinear crest height in the time span L exceeds a fixed threshold is then derived from R(η) solution, assuming that the occurrence of storms with highest crest larger than η is given by a Poisson process. In the applications, both R(η) and P(ηc max &gt; η∣[0, L]) are calculated for some locations. It is shown that narrowband second‐order approach is slightly conservative, with respect to the more general condition of crest distribution for second‐order three‐dimensional waves. Finally, a comparison with Boccotti, Jasper and Krogstad models is presented.

An Approach to Calculate the Probability of Wave Impact on an FPSO Bow

This work aims at characterizing the probability of wave impact and determining the position of impact on an FPSO (floating production storage and offloading platform) bow geometry. In order to determine the instants when impact occurs, an experimental program was performed on a specific bow shape. The bow was instrumented with pressure transducers and the test program, also making use of video recordings, was designed such that it was possible to determine the correlation between undisturbed wave shape and the impact pressure time traces. It has been found that the wave impact at the bow is highly correlated with the local wave steepness, which for very high waves has at least second-order effects. A comparison between the probability distributions of local wave steepness of the experimental undisturbed wave time trace and numerical simulations of second-order wave theory is provided and it confirmed that the latter is very adequate for calculations. The experimental results were further used to determine how the probability of impact varies with free surface vertical velocity. It was found that the significant wave height of the sea state itself does not have significant influence on the result and a regression model was derived for the bow type in the experiments. The proposed model for determining the probability of having an impact is based on combining distributions, adjusted a priori to the numerically generated second-order free surface vertical velocity, and the experimental probability of impact of a known certain seastate and free surface velocity. The analytical description makes it fast and easy to expand to other cases of interest and some example calculations are shown to demonstrate the relative ease of the procedure proposed. The position of the impact is determined by the nonlinear wave crests and the ship motions. The ship motions can be determined based on a linear response to the nonlinear waves considered.

Statistics of nonlinear wave crests and groups

Groups of large nonlinear waves with sharper higher crests can pose hazards to ships, induce harbor resonance and cause wave-overtopping of fixed and floating structures. Past interest in wave groups has mostly been focused on the statistics and modeling of linear wave groups. Studies on nonlinear wave groups are surprisingly few, and address deep water waves only. Here, statistics of nonlinear wave crests and wave-crest groups in deep and transitional water depths are considered, using an appropriate second-order representation for crest heights and the continuous wave-envelope approach. In particular, theoretical expressions describing the statistics of nonlinear wave crests and their groups are posed in the form of a simple second-order transformation of well-known results on linear waves. Predictions from the transformation so posed compare well with nonlinear wave data gathered in the North Sea, and demonstrate that nonlinearities do affect the statistics of large wave crests and their groups significantly.

Statistics of Wave Crests in Storms

The least-upper-bound (lub) theory recently proposed by Tayfun and Al-Humoud as a model for the statistical distribution of nonlinear wave crests is explored further. The theoretical distribution of the lub to the largest crest in N waves is also considered, deriving expressions for the associated mode, mean, and standard deviation. These provide a rational basis for describing the nature and occurrence of freak waves and outliers in observational data. The lub distribution and related statistics are then checked with additional measurements gathered during the peaks of two severe storms in the North Sea. The data comprise two hourly records and coincide with the extremes of more extensive measurements discussed by Forristall. Analyses of both records yield crest height distributions and statistics that compare with the lub model predictions quite well, thus confirming the validity of the lub theory convincingly. The theoretical expressions describing the statistics of the largest crest in N waves are then used to examine the nature of two exceptionally large waves recorded within the two hourly measurements. Although these waves display characteristics in apparent conformity with freak-wave occurrences, they are shown to be simple outliers that can be predicted well within the context of the extreme-value statistics derived here.

Least Upper Bound Distribution for Nonlinear Wave Crests

The exact form of the statistical distribution of nonlinear wave crests is not known. This study proposes and develops a theoretical least upper bound (lub) distribution for large crest heights within the framework of second-order deep-water wave theory. The form of the lub distribution is determined by a simple but specific parameterization of second-order nonlinearities so as to satisfy the theoretical upper-bound on the sea-surface skewness. The implications and various integral properties of the proposed distribution are then examined. Subsequently, these are compared with measurements gathered during Hurricane Camille and also with simulations of the same data recently carried out by Forristall. The comparisons clearly confirm the validity and utility of the proposed lub distribution for large wave crests in deep water.

Distribution of nonlinear wave crests

The distribution of nonlinear wave crests is examined on the basis of a theoretical probability density previously given elsewhere (J. Eng. Mech. 120 (1994) 1009). Certain errors contained in the original theoretical density are corrected, and the corresponding exceedance distribution is derived. The resulting theoretical forms of the probability density and exceedance distribution are then slightly simplified and compared with nonlinear wave data gathered under hurricane conditions. The results indicate that the proposed theoretical forms describe the observed distributions of large wave crests better than the Rayleigh law. However, the quantitative accuracy of the predictions is somewhat poor, as is typical of approximate theories based on Gram–Charlier-type expansions.