Kinematics of overturning solitary waves and their relations to breaker types

Numerical simulations using a full-nonlinear BIM (Boundary Integral Method) potential-theory wave model are carried out to study the internal velocity and acceleration fields of an solitary wave overturning on a reef with vertical face (submerged breakwater) and their relation to breaker type. The simulations make it clear that the jet size normalized by the incident wave height is uniquely governed by the crown height of the reef, while the jet shape is similar and independent of the size. Further, they reveal that the overall internal kinematics of overturning waves is clearly related to the jet size. As the jet size increases and the breaker type changes from spilling to plunging, the kinematics thus become increasingly different from those of steady waves. Water particles with the greatest velocities or accelerations within the wave converge towards the jet. After the breaking, both of the velocities and accelerations almost simultaneously reach extreme values near locations beneath the jet. Some of the extreme values are closely related to the breaker type and can be uniquely determined by substituting the breaker type index into the regression equations suggested here.     

极限波浪运动特性的非线性数值模拟

利用时域高阶边界元方法建立了模拟极限波浪运动的完全非线性数值模型,其中自由水面满足完全非线性自由水面条件.采用半混合欧拉-拉格朗日方法追踪流体瞬时水面,运用四阶Runge-Kutta方法更新下一时间步的波面和速度势,同时应用镜像格林函数消除水槽两个侧面和底面上的积分.研究中利用波浪聚焦的方法产生极限波浪,并且在水槽中开展了物理模型实验,将测点试验数据与数值结果进行了对比,两者吻合得很好.对极限波浪运动的非线性和流域内速度分布进行了研究.     

Flow kinematics of focused wave groups on a plane beach in the U.K. Coastal Research Facility

Measurements are presented of the water particle kinematics of focused wave groups generated in the U.K. Coastal Research Facility. Single and repeated wave groups are considered at normal and 20° incidence to a 1:20 plane beach. The single focused wave groups model extreme transient events without the complication of reflections during the data acquisition process. A symmetry-based separation of harmonics method is used to interpret the water particle kinematics at the point of focus. Although the largest component is linear, there are also considerable second order kinematics terms (both low frequency and high frequency). Away from the free surface, the 2nd order difference contribution to the kinematics is a return current opposed to the direction of wave advance. For repeated wave groups, the measured kinematics confirms the presence of a low frequency free wave, followed by higher frequency waves of the main group and trailing higher order harmonic waves. In the breaker and surf zones, there is also evidence of the saw-tooth behaviour of broken waves, followed by scatter due to breaker-induced turbulence. Pulsatile wave breaking of repeated wave groups at oblique incidence is found to drive a longshore current.     

Simulation of directional waves in a numerical basin by a desingularized integral equation approach

A numerical solution is developed to investigate the generation and propagation of small-amplitude water waves in a semi-infinite rectangular wave basin. The three-dimensional wave field is produced by the prescribed “snake-like” motion of an array of segmented wave generators located along the wall at one end of the tank. The solution technique is based on the boundary element approach and uses an appropriate three-dimensional Green function which explicitly satisfies the tank-wall boundary conditions. The Green function and its derivatives which appear in the integral equation formulation can be shown to be slowly convergent when the source and field points are in close proximity. Therefore, when computing the velocity potentials on the wave generators, the source points are chosen outside the fluid domain, thereby ensuring the rapid convergence of these functions and rendering the integral equations non-singular. Numerical results are shown which illustrate the influence of the various wavemaker and basin parameters on the generated wave field. Finally, the complete wave field produced by the diffraction of oblique waves by a vertical circular cylinder in a basin is presented.     

Application of desingularized approach to water wave propagation over three-dimensional topography

A numerical approach based on desingularized boundary element method and mixed Eulerian–Lagrangian formulation [Zhang et al., 2006. Wave propagation in a fully nonlinear numerical wave tank: a desingularized method. Ocean Engineering 33, 2310–2331] is extended to solve the water wave propagation over arbitrary topography in a three-dimensional wave tank. A robust damping layer applicable for regular and irregular incident waves is employed to minimize the outgoing wave reflection back into the wave tank. Numerical results on the propagation of regular and irregular incident waves over the flat bottom and linear incident waves over an elliptical shoal show good concurrence with the corresponding analytical solutions and experimental data.     

Effects of high-order nonlinear interactions on unidirectional wave trains

Numerical simulations of gravity waves with high-order nonlinearities in two-dimensional domain are performed by using the pseudo spectral method. High-order nonlinearities more than third order excite apparently chaotic evolutions of the Fourier energy in deep water random waves. The high-order nonlinearities increase kurtosis, wave height distribution and H_max/H_1/3 in deep water and decrease these wave statistics in shallow water. Moreover, they can generate a single extreme high wave with an outstanding crest height in deep water. High-order nonlinearities (more than third order) can be regarded as one cause of freak waves in deep water.     

An explicit approximation to the wavelength of nonlinear waves

An explicit and concise approximation to the wavelength in which the effect of nonlinearity is involved and presented in terms of wave height, wave period, water depth and gravitational acceleration. The present approximation is in a rational form of which Fenton and Mckee's (1990, Coastal Engng 14, 499–513) approximation is reserved in the numerator and the wave steepness is involved in the denominator. The rational form of this approximation can be converted to an alternative form of a power-series polynomial which indicates that the wavelength increases with wave height and decreases with water depth. If the determined coefficients in the present approximation are fixed, the approximating formula can provide a good agreement with the wavelengths numerically obtained by Rienecker and Fenton's (1981, J. Fluid Mech. 104, 119–137) Fourier series method, but has large deviations when waves of small amplitude are in deep water or all waves are in shallow water. The present approximation with variable coefficients can provide excellent predictions of the wavelengths for both long and short waves even, for high waves.     

Steepness and asymmetry of extreme waves and the highest waves in deep water

Traditional wave steepness s=H/L does not define steep asymmetric waves in a random sea uniquey. Three additional parameters characterising single zero-downcross waves in a time series are crest front steepness, vertical asymmetry factor and horizontal asymmetry factor. Results for steepness and asymmetry from zero-downcross analysis of wave data obtained from full scale measurements in deep water on the Norwegian continental shelf in 58 time series are presented. The analysis demonstrates clearly the asymmetry of both “extreme waves” and the highest waves. The period and height of the highest waves are also given together with their correlation to spectral parameters. The measured maximum wave heights are also compared with predicted values of maximum wave heights showing good agreement.     

Nonlinear progressive waves in water of finite depth — an analytic approximation

An analytical solution using homotopy analysis method is developed to describe the nonlinear progressive waves in water of finite depth. The velocity potential of the wave is expressed by Fourier series and the nonlinear free surface boundary conditions are satisfied by continuous mapping. Unlike the perturbation method, the present approach is not dependent on small parameters. Thus solutions are possible for steep waves. Furthermore, a significant improvement of the convergence rate and region is achieved by applying Homotopy-Padé Approximants. The calculated wave characteristics of the present solution agree well with previous numerical and experimental results.     

Estimation of scour around submarine pipelines with Artificial Neural Network

The process of scour around submarine pipelines laid on mobile beds is complicated due to physical processes arising from the triple interaction of waves/currents, beds and pipelines. This paper presents Artificial Neural Network (ANN) models for predicting the scour depth beneath submarine pipelines for different storm conditions. The storm conditions are considered for both regular and irregular wave attacks. The developed models use the Feed Forward Back Propagation (FFBP) Artificial Neural Network (ANN) technique. The training, validation and testing data are selected from appropriate experimental data collected in this study. Various estimation models were developed using both deep water wave parameters and local wave parameters. Alternative ANN models with different inputs and neuron numbers were evaluated by determining the best models using a trial and error approach. The estimation results show good agreement with measurements.     

Shoaling Surface Gravity Waves Cause a Force and a Torque on the Bottom

Freely propagating surface gravity waves are observed to slow down and to stop at a beach when the bottom has a relatively gentle upward slope toward the shore and the frequency range of the waves covers the most energetic wind waves (sea and swell). Essentially no wave reflection can be seen and the measured reflected energy is very small compared to that transmitted shoreward. One consequence of this is that the flux of the wave’s linear momentum decreases in the direction of wave propagation, which is equivalent to a time rate of change of the momentum. It takes a force to cause the time rate of change of the momentum. Therefore, the bottom exerts a force on the waves in order to decrease the momentum flux. By Newton’s third law (action equals reaction) the waves then impart an equal but opposite force to the bottom. In shallow (but finite) water depths the wave force per unit bottom area is calculated, for normal angle of incidence to the beach, to be directly proportional to the square of the wave amplitude and to the bottom slope and inversely proportional to the mean depth; it is independent of the wave frequency. Constants of proportionality are: 1/4, the fluid density and the acceleration of gravity. Swell attenuation near coasts and some characteristics of sand movement in the near-shore region are not inconsistent with the algebraic structure of the wave force formula. Since the force has a depth variation which is significantly faster than that of the dimensions of the particle orbits in the vertical direction, the bottom induces a torque on the fluid particles that decreases the angular momentum flux of the waves. By an extension of Newton’s third law, the waves also exert an equal but opposite torque on the bottom. And because the bottom force on the waves exists over a horizontal distance, it does work on the waves and decreases their energy flux. Thus, theoretically, the fluxes of energy, angular and linear momentum are not conserved for shoaling surface gravity waves. Mass flux, associated with the Stokes drift, is assumed to be conserved, and the wave frequency is constant for a steady medium.     

Cross-shore hydrodynamics within an unsaturated surf zone

This paper concerns the hydrodynamics induced by random waves incident on a steep beach. New experimental results are presented on surface elevation and kinematic probability density functions, cross-shore variation in wave heights, the fraction of broken waves and velocity moments. The surf zone is found to be unsaturated at incident wave frequencies, with a significant proportion of the incident wave energy remaining at the shoreline in the form of bores. Wave heights in both the outer and inner surf zones are best described by a full Rayleigh distribution [Thornton, E.B., Guza, R.T., 1983. Transformation of wave height distribution. J. Geophys. Res. 88, 5925–5938], rather than a truncated Rayleigh distribution as used by Battjes and Janssen (1978) [Battjes, J.A, Janssen, J.P., 1978. Energy loss and setup due to breaking of random waves. Proc. 16th Int. Conf. Coastal Eng. ASCE, New York, pp. 569–588]. A new parametric wave transformation model is outlined which provides explicit expressions for the fraction of broken waves and the energy dissipation rate within the surf zone. On steep beaches, the model appears to offer improved predictive capabilities over the original Battjes and Janssen model. Cross-shore variations in the velocity variance and velocity moments are best described using Linear Gaussian wave theory, with less than 20% of the velocity variance in the inner surf zone due to low frequency energy.     

On the Fifth-Order Stokes Solution for Steady Water Waves

This paper presents a universal fifth-order Stokes solution for steady water waves on the basis of potential theory. It uses a global perturbation parameter, considers a depth uniform current, and thus admits the flexibilities on the definition of the perturbation parameter and on the determination of the wave celerity. The universal solution can be extended to that of Chappelear (1961), confirming the correctness for the universal theory. Furthermore, a particular fifth-order solution is obtained where the wave steepness is used as the perturbation parameter. The applicable range of this solution in shallow depth is analyzed. Comparisons with the Fourier approximated results and with the experimental measurements show that the solution is fairly suited to waves with the Ursell number not exceeding 46.7.     

南海灾害性波浪基本特征研究

本文基于1991-2016年全球卫星高度计融合数据对南海灾害性波浪基本特征进行了分析，根据灾害性波浪诱发天气类型不同，将其分为"台风浪"和"非台风浪"。依此主线，对两类波浪在南海不同海域的特征进行了研究，并提出了用于定量研究两类波浪强度关系的台风浪权重系数（W），得到了两类波浪在南海相对强弱关系的分布规律，量化研究了南海灾害性波浪的特征。本文以卫星高度计波高数据为样本进行了极值分析，得到了南海重现期波浪要素整体分布规律，研究发现W值大小与广义极值曲线类型显著相关。     

浅水极限波浪几何特征的实验研究

该文通过物理模型实验,对浅水区域内的波浪在破碎前极限状态下的几何特征进行了研究。实验基于JONSWAP谱对不规则波浪进行模拟,通过对波群中出现的单体极限波浪进行捕捉并对波形进行测量而得到研究样本。为了考察底坡因素对极限波浪几何特征的影响,实验共考虑了3组大小分别为β=1/15、1/30以及1/45的地形坡度。统计结果表明,在实验所采用的坡度范围内,当地波高与水深对近岸极限波浪的影响最为显著,随着水深与波高因素变化,极限波浪的几何特征也出现明显的改变。坡度因素对极限波陡和偏度的影响很小,可以被忽略,但是对不对称度参数的影响相对比较明显,坡度越陡,不对称程度越剧烈。最后,通过参数化,本文给出了极限波浪几何特征变化的经验公式。     

Water surface resonance in the L-shaped channel of seawater exchange breakwater

The resonance period of the L-shaped channel in the caisson is predicted analytically for the seawater exchange breakwater of “Applicability Study of the Seawater Exchange Breakwater (1). Korea Ministry of Maritime Affairs and Fisheries (in Korean) (1999a)”. Hydraulic experiments are conducted for a composite breakwater with a rear reservoir that is one of the seawater exchange breakwaters developed by them. For regular waves, the water surface elevation in the channel and the flow rate through the breakwater are measured. For irregular waves, the flow rate through the breakwater and the reflection coefficient on the breakwater are measured. The resonant maximum values in both the surface elevation and the flow rate, and the resonant minimum values in the reflection coefficient are all at wave periods slightly longer than analytically predicted ones. The measured resonance period for irregular waves is closer to the predicted one than for regular waves. If the resonance period of the L-shaped channel is fitted to the dominant period of incident waves, there would be high efficiency of seawater exchange between inside and outside the harbor.     

Modeling of propagation and transformation of transient nonlinear waves on a current

A novel theoretical approach is applied to predict the propagation and transformation of transient nonlinear waves on a current. The problem was solved by applying an eigenfunction expansion method and the derived semi-analytical solution was employed to study the transformation of wave profile and the evolution of wave spectrum arising from the nonlinear interactions of wave components in a wave train which may lead to the formation of very large waves. The results show that the propagation of wave trains is significantly affected by a current. A relatively small current may substantially affect wave train components and the wave train shape. This is observed for both opposing and following current. The results demonstrate that the application of the nonlinear model has a substantial effect on the shape of a wave spectrum. A train of originally linear and very narrow-banded waves changes its one-peak spectrum to a multi-peak one in a fairly short distance from an initial position. The discrepancies between the wave trains predicted by applying the linear and nonlinear models increase with the increasing wavelength and become significant in shallow water even for waves with low steepness. Laboratory experiments were conducted in a wave flume to verify theoretical results. The free-surface elevations recorded by a system of wave gauges are compared with the results provided by the nonlinear model. Additional verification was achieved by applying a Fourier analysis and comparing wave amplitude spectra obtained from theoretical results with experimental data. A reasonable agreement between theoretical results and experimental data is observed for both amplitudes and phases. The model predicts fairly well multi-peak spectra, including wave spectra with significant nonlinear wave components.     

The BCI criterion for the initiation of breaking process in Boussinesq-type equations wave models

The Breaking Celerity Index (BCI) is proposed as a new wave breaking criterion for Boussinesq-type equations wave propagation models (BTE).The BCI effectiveness in determining the breaking initiation location has been verified against data from different experimental investigations conducted with incident regular and irregular waves propagating along uniform slope [Utku, M. (1999). “The Relative Trough Froude Number. A New Criteria for Wave Breaking”. Ph.D. Dissertation, Dept. of Civil and Enviromental Engineering, Old Dominion University, Norfolk, VA; Gonsalves Veloso dos Reis, M.T.L. (1992). “Characteristics of waves in the surf zone”. MS Thesis, Department of Civil Engineering, University of Liverpool., Liverpool; Lara, J.L., Losada, I.J., and Liu, P.L.-F. (2006). “Breaking waves over a mild gravel slope: experimental and numerical analysis”. Journal of Geophysical Research, VOL 111, C11019] and barred beaches [Tomasicchio, G.R., and Sancho, F. (2002). “On wave induced undertow at a barred beach”. Proceedings of 28th International Conference on Coastal Engineering, ASCE, New York, 557–569]. The considered experiments were carried out in small-scale and large-scale facilities. In addition, one set of data has been obtained by the use of the COBRAS model based upon the Reynolds Averaged Navier Stokes (RANS) equations [Liu, P.L.-F., Lin, P., Hsu, T., Chang, K., Losada, I.J., Vidal, C., and Sakakiyama, T. (2000). “A Reynolds averaged Navier–Stokes equation model for nonlinear water wave and structure interactions”. Proceedings of Coastal Structures ‘99, Balkema, Rotterdam, 169–174; Losada, I.J., Lara, J.L., and Liu, P.L.-F. (2005). “Numerical simulation based on a RANS model of wave groups on an impermeable slope”. Proceedings of Fifth International Symposium WAVES 2005, Madrid].Numerical simulations have been performed with the 1D-FUNWAVE model [Kirby, J.T., Wei, G., Chen, Q., Kennedy, A.B., and Dalrymple, R.A. (1998). “FUNWAVE 1.0 Fully Nonlinear Boussinesq Wave Model Documentation and User's Manual”. Research Report No CACR-98-06, Center for Applied Coastal Research, University of Delaware, Newark]. With regard to the adopted experimental conditions, the breaking location has been calculated for different trigger mechanisms [Zelt, J.A. (1991). “The run-up of nonbreaking and breaking solitary waves”. Coastal Engineering, 15, 205–246; Kennedy, A.B., Chen, Q., Kirby, J.T., and Dalrymple, R.A. (2000). “Boussinesq modeling of wave transformation, breaking and run-up. I: 1D”. Journal of Waterway, Port, Coastal and Ocean Engineering, 126, 39–47; Utku, M., and Basco, D.R. (2002). “A new criteria for wave breaking based on the Relative Trough Froude Number”. Proceedings of 28th International Conference on Coastal Engineering, ASCE, New York, 258–268] including the proposed BCI.The calculations have shown that BCI gives a better agreement with the physical data with respect to the other trigger criteria, both for spilling and plunging breaking events, with a not negligible reduction of the calculation time.     

A locally nonlinear interpretation of PUV measurements

A time domain method is presented for analyzing simultaneous measurements of pressure and the horizontal components of velocity obtained beneath irregular multidirectional wave fields. This new method differs from the usual linear directional analyses applied to PUV data in two important aspects. First, the essential nonlinearity of the measured waves is not sacrificed to achieve a solution. Therefore, predictions of sea surface elevation and directional kinematics throughout the water column accurately portray the actual nonlinear character of the waves. Second, the analysis method is `local' in that it can be applied to segments of PUV time series much shorter than an individual wave. The viability of the locally nonlinear methodology developed in this paper is proven by demonstrating agreement with higher-order theoretical steady waves. Predictions of sea surface elevation and wave kinematics are also made using actual measurements from PUV instruments at two ocean sites off the west coast of the United States.