Nonlinear Random Waves Research Articles

Probability distribution of random wave–current forces

Based on the second-order solutions obtained for the three-dimensional weakly nonlinear random waves propagating over a steady uniform current in finite water depth, the joint statistical distribution of the velocity and acceleration of the fluid particle in the current direction is derived using the characteristic function expansion method. From the joint distribution and the Morison equation, the theoretical distributions of drag forces, inertia forces and total random forces caused by waves propagating over a steady uniform current are determined. The distribution of inertia forces is Gaussian as that derived using the linear wave model, whereas the distributions of drag forces and total random forces deviate slightly from those derived utilizing the linear wave model. The distributions presented can be determined by the wave number spectrum of ocean waves, current speed and the second order wave–wave and wave–current interactions. As an illustrative example, for fully developed deep ocean waves, the parameters appeared in the distributions near still water level are calculated for various wind speeds and current speeds by using Donelan–Pierson–Banner spectrum and the effects of the current and the nonlinearity of ocean waves on the distribution are studied.

Statistical distribution of nonlinear random water wave surface elevation

This study deals with the development of statistical modeling for water wave surface elevation by using a method that combines a dynamic solution with random process statistics. Ocean wave data taken from four NOAA (National Oceanic and Atmospheric Administration) buoys moored in the northeast Pacific were used to validate the model. The results indicated that the nonlinear probability density distribution of ocean wave surface elevation derived from the model described the measurements much better than Gaussian distribution and Longuet-Higgins distribution.

Simulation of nonlinear random finite depth waves coupled with an elastic structure

Time accurate computation of random nonlinear shallow-water waves is required for the design of offshore wind turbines, and also in a multitude of other areas. Assuming the fluid to be inviscid and irrotational, nonlinear finite amplitude waves in a liquid body of finite depth are computed based on potential theory, i.e. the velocity potential must satisfy the Laplace equation at every instant in time. The free surface boundary conditions give a partial differential equation for the velocity potential and the free surface elevation on the free boundary. Similar coupling conditions are obtained at the boundary in contact with the deformable body. During the computation, the repeated solution of Laplace’s equation can be obtained by solving a boundary integral equation with the BEM and a fast multipole solver. The incident wave field is given by a stochastic wave process far from the structure, where the scattered (diffracted and radiated) wave field is comparatively small. This is finally coupled to another fluid description at the structure, which has a fully nonlinear description of both incident and scattered wave. The effect of the interacting flexible structure is included via a coupling computation. The chosen software architecture allows a modular coupling concept reflected in the algorithmically strong coupling of the various fluid domains, the structure, and the soil. As a numerical example, an offshore wind turbine is simulated, which has an additional fluid interaction with the stochastic wind field.

On non-linear very large sea wave groups

The paper deals with the non-linear effects for sea wave groups. Boccotti's quasi-determinism theory, which is exact to the first-order in a Stokes expansion, gives the mechanics of sea wave groups when either a very high crest (first formulation of the theory—‘New wave’), or a large crest-to-trough wave height (second formulation of the theory) occurs. In this paper, the quasi-determinism theory, in both formulations, is extended to the second-order, by obtaining the expressions of free surface displacement and velocity potential, as a function of wave spectrum. Finally it is shown that analytical predictions are in good agreement with both field data and data of Monte Carlo simulations of non-linear random waves.

Fully dispersive nonlinear water wave model in curvilinear coordinates

A vertically integrated fully dispersive nonlinear wave model is expressed in curvilinear coordinates with non-orthogonal grids for the simulation of broad-banded nonlinear random water waves in regions of arbitrary geometry. The transformation is performed for both dependent and independent variables, hence an irregular physical domain is converted into a rectangular computational domain with contravariant velocities. Use of contravariant velocity components as dependent variables ensures easy and accurate satisfaction of the wall condition for lateral enclosures surrounding a physical domain, such as a coastal area, channel, or harbor. The numerical scheme is based on finite-difference approximations with staggered grids which results in implicit formulations for the momentum equations and a semi-explicit formulation for the continuity equation. Linear long wave propagation in a channel of varying cross-section and linear random wave propagation in a circular channel are presented as test cases for comparisons with the corresponding analytical solutions. Cnoidal and Stokes waves in a circular channel are also simulated as examples to nonlinear wave propagation within curved walls.

Statistical nonlinear acoustics: First studies and current state

Statistical physics of high-intensity noise waves has, in principle, a long-term history. Nonlinear interactions between random and regular waves studied in the 1930s were devoted to sound absorption in solids caused by energy transfer from coherent phonons (signal) to Debye’s thermal quantum (noise). However, exhaustive studies of nonequilibrium nonlinear random wave processes have been carried out only in the 1970s, after experimental and theoretical nonlinear acoustics reached an advanced stage of development. David Blackstock, one of the founders of modern nonlinear acoustics, performed pioneering experiments on broadband noise spectra distortion and signal–noise interactions in tubes and outdoors. These experiments confirmed existing data and formed the fundamental basis for both future studies and applications. Along with temporal statistics, spatial ones were considered by D. T. Blackstock during his studies of nonlinear diffraction caused by orifices and screens with irregular edges. These results are also of great significance for applications. A brief overview of the first studies in statistical nonlinear acoustics is given, as well as of new results obtained in recent years.

Drag force on a circular cylinder in unseparated laminar high Reynolds number nonlinear random oscillatory flow

The stochastic properties of the drag force maxima on a circular cylinder subjected to nonlinear random waves are investigated. Unseparated laminar high Reynolds number flow is considered. A simplified approach based on second order Stokes waves is presented, including the sum-frequency effect only. It is demonstrated how a drag force formula valid for regular linear waves can be used to find the cumulative distribution function of individual drag force maxima for nonlinear irregular waves. Here the Wang (1968) drag force coefficient is used.

Forecasting of higher order spectral quantities of nonlinear random waves over surf zone

Accurate modeling of energy dissipation mechanisms due to wave breaking in the frequency domain is an overdue task hampered by the facts that wave breaking is highly localized in time. On the other hand, a spatially localized eddy viscosity formulation based on the mixing length hypothesis is capable of providing accurate predictions of wave skewness and asymmetry in the surf zone. In this study, the structure of dissipation term is re-examined using the data of Kraus et al. (1992) and Buhr Hansen and Svendsen (1979) from the viewpoint of time dependent Boussinesq equations. Based on these results, the structure of frequency domain dissipation model over the surf zone is deduced. It turns out that dissipation due to wave breaking should be biased strongly toward higher frequencies in spectral calculations, and that a quadratic dependence on frequency within the dissipation coefficient comes closest to matching the desired structure of the breaking term. The prediction of skewness by numerical model with twenty harmonics involved is promising whereas the asymmetry in the surf zone is somewhat under-predicted.

Laminar Bottom Friction Beneath Nonlinear Random Waves

An approach by which the stochastic properties of the bed shear stresses can be derived from the irregular nonlinear wave motion outside the laminar bottom boundary layer is presented. It is demonstrated how bed friction formulas valid for regular second order Stokes waves can be used to find the cumulative distribution function of individual shear stress maxima for nonlinear irregular waves. The friction factor for nonlinear random waves is also determined. An example is given, and an extension to nonlinear random waves plus current flow is also suggested.

Probability density functions for non-linear random waves and responses

Non-linear waves have been a research topic for a long time, but have not yet entered common ocean engineering practice. In this article the problem of calculating the probability density function (pdf) for non-linear waves is investigated. The ‘saddle point approximation’ is applied to a second-order wave model to estimate the pdf. The performance of the approximation is compared with the performances of the more commonly used methods for different types of input spectra. In addition, a non-parametric transformation method, with extrapolation of the level crossing intensity, is proposed to estimate the pdf. One simulation and three different data sets are used to investigate the performance of the model.

Spectral analysis of Stokes waves

The spectral properties of Stokes waves are shown in this paper by theoretical and numerical methods. This is done by expressing wave profiles and velocities of water particles as nonlinear combinations of the first order component of wave profiles. Under the assumption of the first order wave profiles being zero mean Gaussian processes, the relationship between autocorrelation functions of wave profiles and velocities of water particles and the first order component of wave profiles is established using the nonlinear spectral analysis. The spectral densities of nonlinear random waves, the velocities and accelerations of water particles are then obtained. Numerical computations are carried out to analyze the effect of fundamental parameters of waves. The results indicate that wave height is the most sensible parameter to the root mean squares related and wave depth is the least sensible one of all.

Bedload Sediment Transport Rate by Nonlinear Random Waves

An approach by which the stochastic properties of the net bedload transport rate can be derived from the irregular nonlinear wave motion outside the bottom boundary layer is presented. It is demonstrated how bedload transport formulas valid for regular nonlinear waves can be used to find the probability distribution function of the net bedload transport rate for nonlinear (asymmetric) irregular waves. Here the bedload transport formula for second order Stokes waves proposed by Damgaard et al. (1996) combined with the Soulsby (1997) wave friction factor have been chosen to illustrate the method. The Madsen and Grant (1976) bedload transport formula combined with the same friction factor has also been included for comparison. Generally, it is recommended that a stochastic approach should be used rather than using the rms values in an otherwise deterministic approach.

A method of studying nonlinear random field of surface gravity waves by direct numerical simulation

A method of studying nonlinear random fields of surface gravity waves by direct numerical simulation is developed by combining the “high-order spectral method” of West et al. (J. Geophys. Res. 92 (1987) 11,803–11,824) and the complex amplitude function b(k) introduced by Zakharov (J. Appl. Mech. Tech. Phys. (Engl. Transl.) 2 (1968) 190–194) in his Hamiltonian formalism of the water wave problem. By computing the rate of change of b(k) for a given wave field by both the high-order spectral method and Zakharov's equation, it is confirmed that the high-order spectral method, when the order of nonlinearity is appropriately chosen, gives exactly the same result as Zakharov's equation within a much shorter computation time. Methods of evaluating the frequency spectrum, directional spectrum and the component phase speed are given. Results of some preliminary computations are also presented to prove the great potentialities of the present approach as a tool for the examination of various characteristics of nonlinear random wave fields with continuous spectra, such as ocean waves.

Probability distribution of random wave forces in weakly nonlinear random waves

Based on the second-order random wave theory, the joint statistical distribution of the horizontal velocity and acceleration is derived using the characteristic function expansion method. From the joint distribution and the Morison equation, the theoretical distributions of drag forces, inertia forces and total random wave forces are determined. The distribution of inertia forces is Gaussian as that derived using the linear wave model, whereas the distributions of drag forces and total random forces deviate slightly from those derived utilizing the linear wave model. It is found that the distribution of wave forces depends solely on the frequency spectrum of sea waves associated with the first order approximation and the second order wave–wave interaction.

Modeling the floating-body motion in nonlinear random waves based on vertically integrated wave equations

An efficient numerical model, based on vertically integrated nonlinear-wave equations, is proposed for the analysis of wave-induced floating-body motion in nonlinear random waves. Wave equations adopted here are derived by using a multi-term coupling technique, which favorably represents the vertical structure of the fluid motion with plural vertical-dependence functions. A nonlinear wave-body interaction model, given also in vertically integrated form, is incorporated into the wave equations. The result of fundamental examinations, conducted via comparisons between the present FEM-based numerical model and other analytical and numerical models, demonstrates sufficient reliability and desirable performance of the model proposed.

On the joint distribution of surface slopes for the fourth order nonlinear random sea waves

Based upon the nonlinear model of Longuet-Higgins the joint distribution of wave surface slopes is theoretically investigated. It is shown that in the fourth order approximation, the distribution is given by truncated Gram-Charlier series. The types of wave-wave coupling interactions are related to the order of approximation to nonlinearity of sea surface, which eventually defines the truncated term of the Gram-Charlier series. For each order approximation, the coefficients in the series are modified comparatively to the corresponding ones for the previous order approximation. If the nonlinear effect of the kurtosis is considered, the wave surface must be as accurate at least as to the third order approximation, and with regard to skewness, the wave surface must be as accurate at least as to the fourth order approximation.

A spectral model for unidirectional nonlinear wave propagation over arbitrary depths

A weakly-nonlinear and dispersive wave equation recently developed by the authors is used for formulating a spectral-type unidirectional wave propagation model describing spectral transformations of narrow-band waves travelling over arbitrary depths. The essential characteristics of the model equation are recapitulated first and then the spectral domain representation in terms of spatially varying harmonic amplitudes is presented. The resulting evolution equations are used to simulate the experiments concerning harmonic generation in shallow water and nonlinear random wave transformations over a submerged bar. Furthermore, the spectral model predictions are compared with the field measurements in nearshore with satisfactory results.

Statistical distribution of surface elevation for the fourth order nonlinear random sea waves

Based upon the nonlinear model of random sea waves, the statistical distribution of wave surface elevation exact to the fourth order is derived as the truncated Gram-Charlier series,by calculating directly each order moment. The phenomenon found by Huanget al. that the agreement between observed data and investigated series deteriorates much more when the series is kept to λ8 is explained. The effect of the approximation order on the truncation of series and the determination of coefficients is investigated. For themth order approximation, the derived series is truncated atH3m-3 with the absence of H3m-4, and the coefficients ofH3m-3 andH3m-6 are connected by a simple algebraic relation.

A fully dispersive weakly nonlinear model for water waves

A fully dispersive weakly nonlinear water wave model is developed via a new approach named the multiterm–coupling technique , in which the velocity field is represented by a few vertical–dependence functions having different wave–numbers. This expression of velocity, which is approximately irrotational for variable depth, is used to satisfy the continuity and momentum equations. The Galerkin method is invoked to obtain a solvable set of coupled equations for the horizontal velocity components and shown to provide an optimum combination of the prescribed depth–dependence functions to represent a random wave–field with diversely varying wave–numbers. The new wave equations are valid for arbitrary ratios of depth to wavelength and therefore it is possible to recover all the well–known linear and weakly nonlinear wave models as special cases. Numerical simulations are carried out to demonstrate that a wide spectrum of waves, such as random deep water waves and solitary waves over constant depth as well as nonlinear random waves over variable depth, is well reproduced at affordable computational cost.

Stochastic Benjamin-Ono equation and its application to the dynamics of nonlinear random waves.

The stochastic Benjamin-Ono equation is introduced, which models the propagation of nonlinear random waves in a two-layer fluid system with and without uneven bottom topography. In the case of the flat bottom, the effect of the external random flow field on the evolution of both soliton and periodic wave is investigated. In particular, the mean value and the correlation function of these nonlinear wave fields are calculated exactly under the assumption that the flow obeys the Gaussian stochastic process with a white noise. It is found that in the limit of large time, the mean value of an algebraic soliton approaches a Gaussian wave packet whereas that of a periodic wave is represented by Jacobi's theta function. In the case of the uneven bottom, a perturbation analysis is performed to evaluate the mean value of an algebraic soliton under the influence of random change of bottom topography. The large time asymptotic of the soliton is shown to exhibit a Gaussian wave packet with a small amount of the phase shift caused by the interaction between the soliton and the random bottom topography.