Calculating crest statistics of shallow water nonlinear waves based on standard spectra and measured data at the Poseidon platform

Accurate prediction of the wave crest height exceedance probabilities is of paramount importance to the design of coastal structures, offshore structures and ships. Firstly, the wave crest height assessment, including specifying sound safety limits for overtopping hazards, is important for all kinds of coastal structures such as seawalls, dikes and breakwaters. Secondly, in the case of offshore structures such as offshore drilling platforms, offshore oil production platforms or wind turbines supported by floating platforms, their deck elevations are usually designed to maintain an adequate air gap so that the impact of the highest wave crests on the underside of the deck structures can be prevented. Finally, for a ship in the ocean, the occurrence of green water on deck, the wave slamming on the bow flare, and the extreme vessel roll motion are all dependent on the extreme wave crests.

Waves in the real world are nonlinear. In the literature, there exist some empirical models of wave crest height distributions of nonlinear random waves. However, the previous research work of Dr. Yingguang Wang’s research group has shown that such kind of empirical formulas will sometimes predict wave characteristic distributions that differ considerably from the true ones. In order to overcome the weakness of the empirical approach for predicting the wave crest height distributions, Monte Carlo Simulation (MCS) can be performed through the superposition of harmonics based on a wave spectrum. The MCS starts with first generating the first order linear wave time histories with a specific wave power spectral density function, for each of these the MCS numerical algorithm then evaluates the full set of second order corrections. Unfortunately, in a typical ocean engineering simulation project, usually there are a huge number of second order corrections that need to be evaluated. Therefore, using MCS for wave crest height prediction will become very time consuming, laborious and unsuitable for practical implementations.

Dr. Yingguang Wang has developed a pioneering method called “Transformed Rayleigh method” for the calculation of wave crest height distribution of shallow water nonlinear waves. The second order nonlinear wave model has been seamlessly integrated into Dr. Yingguang Wang’s “Transformed Rayleigh method”. Meanwhile，the correction for the effect of bottom on the wave spectrum, a procedure that is frequently overlooked by some researchers, has also been incorporated into the Transformed Rayleigh method. This is achieved through utilizing a  “corrected wave spectrum” that is obtained by multiplying the original “standard wave spectrum”  by a function that ranges between 0 and 1 according to a similarity law. In this regard, Dr. Yingguang Wang’s outstanding research work is a leap into the next generation, greatly advancing our understanding of the applications of the similarity law, a fundamental law in the field of ocean engineering. Dr. Yingguang Wang’s new Transformed Rayleigh method is based on a deterministic and time instantaneous functional transformation. In his research, Dr. Yingguang Wang has found that the Gram-Charlier series representation of the non-Gaussian probability density function of the wave elevation process can behave erratically, yielding multimodal and even negative probability densities. Therefore, the time instantaneous  functional transformation should never be determined from the probability density functions of the Gaussian and the non-Gaussisn wave elevation processes. Based on this rigorous discernment, Dr. Yingguang Wang innovatively applied a Hermite model for the time instantaneous  functional transformation. More specifically, in the Hermite transformation model the transformation is chosen to be a monotonic cubic polynomial, calibrated such that the first four moments of the transformed model match the moments of the true process. The first four moments (the values of the mean, standard deviation, skewness and kurtosis) of the second order nonlinear random waves can be calculated using closed-form equations based on the information of the wave spectrum and quadratic transfer functions. After the time instantaneous functional transformation is obtained, the level up-crossing rate of a specific sea level by the nonlinear wave surface elevations at a fixed point can subsequently be calculated. The nonlinear wave crest height probability distribution can finally be obtained by utilizing a closed-form formulation after the level up-crossing rates are calculated. The above-mentioned detailed procedures for the Transformed Rayleigh method have been executed in a computer code, which can be conveniently utilized to predict the nonlinear wave crest height distributions for real world ocean engineering projects.

The Transformed Rayleigh method is Dr. Yingguang Wang’s original scientific contribution of major significance in the field of ocean engineering. He has achieved a mighty advance in the theory of nonlinear random ocean waves, which is exceptionally rigorous and challenging. This new Transformed Rayleigh method has been applied for calculating the wave crest height exceedance probabilities of sea states with standard JONSWAP spectra, with a bimodal Torsethaugen spectrum and with the surface elevation data measured at the Poseidon platform. It is demonstrated that in all these cases the Transformed Rayleigh method can offer much better predictions than those from using the empirical wave crest height distribution models. Meanwhile, it is found that the Transformed Rayleigh method can compute nonlinear wave crest height distributions 57 times faster than the Monte Carlo Simulation method can do. In summary, Dr. Yingguang Wang’s groundbreaking Transformed Rayleigh method can calculate nonlinear wave crest height distributions substantially more accurately and efficiently than the traditional methods can do, and this is enormously beneficial to the design of various kinds of coastal structures, offshore structures and ships.