Internal generation of waves: Delta source function method and source term addition method

In this study, we investigate two internal wave generation methods in numerical modeling of time-dependent equations for water wave propagation, i.e., delta source function method and source term addition method, the latter of which has been called the line source method in literatures. We derive delta source functions for the Boussinesq-type equations and extended mild-slope equations. By applying the fractional step splitting method, we show that the delta source function method is equivalent to the source term addition method employing the energy velocity. This suggests that the energy velocity should be used rather than the phase velocity for the transport of incident wave energy in the source term addition method. Finally, the performance of the delta source function method is verified by accurately generating nonlinear cnoidal waves as well as linear waves for horizontally one-dimensional cases.     

Internal solitary waves shoaling onto a shelf: Comparisons of weakly-nonlinear and fully nonlinear models for hyperbolic-tangent stratifications

In this study the evolution of internal solitary waves shoaling onto a shelf is considered. The results of high resolution two-dimensional numerical simulations of the incompressible Euler equations are compared with the predictions of several weakly-nonlinear shoaling models of the Korteweg–de Vries family including the Gardner equation and the cubic regularized long wave (or Benjamin–Bona–Mahoney) equation. Wave models in both physical x–t space and in s–x space are considered where s is a commonly used characteristic time variable. The effects of rotation, background currents and damping are ignored. The Boussinesq and rigid lid approximations are also used. The shoaling internal solitary waves generally fission into several waves. Reflected waves are negligible in the cases considered here. Several hyperbolic tangent stratifications are considered with and without a critical point. Among the equations in x–t space the cubic regularized long wave equation gives the best predictions. The Gardner equation in s–x space gives the best predictions of the shape of the leading waves on the shelf, but for many stratifications it predicts a propagation speed that is too large.     

Sea ice evolution and internal wave generation due to a tidal jet in a frozen sea

The purpose of this study is to investigate the influence of tidal currents on sea ice in Spitsbergen fjords which may cause rapid decrease of the ice thickness due to erosion and melting of the ice. The effect was studied in-situ near the narrow channel connecting the Van Mijen Fjord and Lake Vallunden. The strong jet-like tidal currents in the strait driven by semidiurnal tide continue into the lake preventing ice freezing along a narrow strip during high tide and relatively warm weather. Understanding the formation of open water regions or regions with thin ice is important for the safe transportation on ice. We estimate conditions and representative time over which strong tidal current influences ice thickness along a narrow strip in solid ice. Changes of tidal phase and decrease in air temperature influence freezing of the strip in one-two days. While the tidal flow leaves the strait it overflows a shallow bar and generates internal lee waves propagating downslope and mixing the water. Tidal forcing of internal waves was measured using pressure gauges and by scanning of the ice surface during flood and ebb phases. Internal waves were measured using three types of CTD instruments and an ADCP current meter. The generation of wave packets occurs every tidal cycle when the current flows into the lake, but no generation occurs during the ebb phase of the tide because the currents over the bar slope are low. Parameters of internal waves are estimated. Model simulations confirm generation of internal wave train by the tidal current descending downslope.     

Internal wave generation in an improved two-dimensional Boussinesq model

A set of Boussinesq-type equations with improved linear frequency dispersion in deeper water is solved numerically using a fourth order accurate predictor-corrector method. The model can be used to simulate the evolution of relatively long, weakly nonlinear waves in water of constant or variable depth provided the bed slope is of the same order of magnitude as the frequency dispersion parameter. By performing a linearized stability analysis, the phase and amplitude portraits of the numerical schemes are quantified, providing important information on practical grid resolutions in time and space. In contrast to previous models of the same kind, the incident wave field is generated inside the fluid domain by considering the scattered wave field in one part of the fluid domain and the total wave field in the other. Consequently, waves leaving the fluid domain are absorbed almost perfectly in the boundary regions by employment of damping terms in the mass and momentum equations. Additionally, the form of the incident regular wave field is computed by a Fourier approximation method which satisfies the governing equations accurately in water of constant depth. Since the Fourier approximation method requires an Eulerian mean current below wave trough level or a net mass transport velocity to be specified, the method can be used to study the interaction of waves and currents in closed as well as open basins. Several computational examples are given. These illustrate the potential of the wave generation method and the capability of the developed model.     

用ＯｐｅｎＦＯＡＭ 实现数值水池造波和消波

基于OpenFOAM求解器interDyMFoam,开发实现数值水池造波(包括推板和摇板造波)和阻尼消波。所编写的造波边界条件,可依据实验造波理论将各种造波模式植入其中,从而实现各种波的数值造波。首先进行了线性波的数值造波实验,通过结合阻尼消波段的应用,造波水池可提供稳定的线性波。还进行了瞬时极限波和有限振幅的数值造波实验,与实验数据或同类数值结果吻合很好,进一步验证了的数值造波和消波方式的可靠性。     

非静压波浪模型无反射造波

非静压波浪模型是一种近年来得到迅速发展且有望在工程计算中获得广泛应用的波浪模型。为了在非静压波浪模型中实现无反射造波,在连续性方程中添加了质量源项进行域内源造波,并采用阻尼消波方法进行数值消波。数值算例结果表明,所建立的无反射造波方法可以较好地避免非静压波浪数学模型中的二次反射,实现长时间稳定波浪模拟。     

基于物理造波和主动消波的黏性数值波浪水槽开发

基于推板造波理论和摇板造波理论,在Open FOAM平台上采用重叠网格技术建立黏性数值波浪水槽,并使用一种结合SIMPLE算法和PISO算法的PIMPLE算法对数值模型进行求解。利用开发的数值模型通过数值收敛性测试和网格独立性测试分别重点研究了时间步长、库朗数和网格尺寸对数值精度和计算效率的影响。并对比研究了此数值模型分别嵌入层流模型和湍流模型的计算精度和计算效率。实现的规则波和二阶有限振幅波与理论结果和试验结果吻合,验证了此黏性数值波浪水槽的造波和主动消波功能。基于二维数值波浪水槽,进一步研究了三维数值造波,数值计算结果与理论结果吻合良好。研究结果不仅验证了重叠网格在二维和三维两相流体域中求解运动物体与流场交互的可靠性和正确性,而且为使用此黏性数值波浪水槽解决更复杂的海洋工程问题提供了依据。     

Internal generation of waves on an arc in a rectangular grid system

This paper presents a technique to generate waves at oblique angles in finite difference numerical models in a rectangular grid system by using internal generation technique [Lee, C., Suh, K.D., 1998. Internal generation of waves for time-dependent mild-slope equations. Coast. Eng. 34, 35–57.] along an arc-shaped line source. Tests were made for four different types of wave generation layouts. Quantitative experiments were conducted under the following conditions: the propagation of waves on a flat bottom, the refraction and shoaling of waves on a planar slope, and the diffraction of waves to a semi-infinite breakwater. Numerical experiments were conducted using the extended mild-slope equations of Suh et al. [Suh, K.D., Lee, C., Park, W.S., 1997. Time-dependent equations for wave propagation on rapidly varying topography. Coast. Eng. 32, 91–117.]. The fourth layout type consisting of two parallel lines connected to a semicircle showed the best solutions, especially for a small grid size. This technique is useful for the numerical simulation of irregular waves with broad-banded directional spectrum using conventional spectral wave models for the reasonable estimation of bottom friction and wave-breaking.     

Validation of a new generation system for bottom boundary layer beneath solitary wave

Understanding of sea bottom boundary layer characteristics, especially bottom shear stress acting on the sea bed, is an important step needed in sediment transport modeling for practical application purposes. In the present study, a new generation system for bottom boundary layer under solitary wave is proposed. Applicability of this system is examined by comparing measured and numerical solution velocities. Moreover, transitional behavior from laminar to turbulence was investigated. It is concluded that the critical Reynolds number in the experiments shows good agreement with DNS result of Vittori and Blondeaux (2008) and laboratory data of Sumer et al. (2010), indicating validity of the generation system. Since the present generation system enables continuous measurement to obtain ensemble averaged quantities, it can be effectively utilized for future experimental studies on solitary wave boundary layers, including sediment transport experiments with movable bed.     

A finite difference method for effective treatment of mild-slope wave equation subject to non-reflecting boundary conditions

A numerical model for coastal water wave motion that includes an effective method for treatment of non-reflecting boundaries is presented. The second-order one-way wave equation to approximate the non-reflecting boundary condition is found to be excellent and it ensures a very low level of reflection for waves approaching the boundary at a fairly wide range of the incidence angle. If the Newman approximation is adopted, the resulting boundary condition has a unique property to allow the free propagation of wave components along the boundary. The study is also based on a newly derived mild-slope wave equation system that can be easily made compatible to the one-way wave equation. The equation system is theoretically more accurate than the previous equations in terms of the mild-slope assumption. The finite difference method defined on a staggered grid is employed to solve the basic equations and to implement the non-reflecting boundary condition. For verification, the numerical model is then applied to three coastal water wave problems including the classical problem of plane wave diffraction by a vertical circular cylinder, the problem of combined wave diffraction and refraction over a submerged hump in the open sea, and the wave deformation around a detached breakwater. In all cases, the numerical results are demonstrated to agree very well with the relevant analytical solutions or with experimental data. It is thus concluded that the numerical model proposed in this study is effective and advantageous.     

浪流作用下系泊船舶撞击力和系缆力试验研究

以某25万吨矿石码头工程为例,分别进行了单流、单浪和浪流共同作用下,系泊船舶撞击力和系缆力试验。研究了不同水位、不同船舶载度、不同浪流夹角,单流、单浪和浪流共同作用时对船舶撞击力和系缆力的影响。分析了该码头工程护舷和缆绳的布置情况,为工程设计提供了依据。     

Investigation of the generation and propagation of low frequency internal waves: A case study for the east coast of India

SAR (Synthetic Aperture Radar) images in the northwest Bay of Bengal indicate the existence of internal waves and their occurrence and intensity is topography dependent as indicated by in-situ data and satellite information. To complement and comprehend the observations, a three-dimensional Princeton Ocean Model is applied to study the generation and propagation of internal waves. The model domain is configured with a variable curvilinear grid and the input fields comprise bathymetry, initial temperature and salinity, wind stress, air-sea heat flux and tidal information. The numerical investigation indicated a predominant activity of internal waves in the north, and the rationale is three-fold. The first one could be the stable stratification due to fresh water discharge from head-bay major river system, secondly, the significant magnitude/range of the tides and finally, the bathymetry in the coastal waters off Paradip is about 12% shallower compared to that of Visakhapatnam and further south. The cumulative effect of these causes the predominance of internal waves in the north. The core of the energy is essentially in the low-frequency range and the model is able to simulate semi-diurnal and diurnal components reasonably well up to 6 h frequency (0.162 cph).     

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.     

波浪与结构物作用分析的一种高阶边界元方法--自由项和柯西主值积分的直接数值计算

采用直接数值计算方法计算了势流问题高阶边界元方法中的自由项系数和柯西主值积分,建立了波浪与结构物作用的一种高阶边界元方法.通过算例研究了物体表面上固角系数的计算精度和不同网格剖分、不同阶高斯积分点对柯西主值积分的影响.对截断圆柱上的波浪作用力与解析解做了对比,发现本方法具有很高的计算精度,随网格的加密迅速收敛于解析解.     

有效吸收和传递波浪能的锚泊系统的设计制作

将传统定位浮标加以改进,可以形成一种能有效吸收和传递波浪能的锚泊系统。此系统的最上端是漂浮着的海面浮标,海面浮标下端连着锚泊钢缆,锚泊钢缆下端到海底之间又依次连有张紧锤、储链和重物锚块。锚泊系统中的海面浮标随波浪同步上下起伏,从而带动与其连在一起的由张紧锤绷直的锚泊钢缆上下振动,于是海面浮标吸收波浪能并由引导缆向下传递。20 m长的储链可以保证整个装置在涨潮、退潮的极限水深情况下,仍旧可以有效地传递海表面波浪能;合适的重物锚块可以起到为整个系统定位的功能。最后给出了一个实际应用波浪能的例子。设计制作的锚泊系统可以有效吸收和传递波浪能,为波浪能的利用提供了一种行之有效的方法。     

Calibration and verification of a spectral wind–wave model for Lake Okeechobee

A spectral wind wave model SWAN (Simulation WAves Nearshore) that represents the generation, propagation and dissipation of waves was applied to Lake Okeechobee. This model includes the effects of refraction, shoaling, and blocking in wave propagation. It accounts for wave dissipation by whitecapping, bottom friction, and depth-induced wave breaking. The wave–wave interaction effect also is included in this model. Measurements of wind and wave heights were made at different stations and different time periods in Lake Okeechobee. Significant wave height values were computed from the recorded data. The correlation between wind stress and significant wave height also was analyzed. A 6-day simulation using 1989 data was conducted for model calibration. Another 6-day simulation using 1996 data was conducted for model verification. The simulated significant wave heights were found to agree reasonably well with measured significant wave heights for calibration and verification periods. Agreement between observed and simulated values was based on graphical comparisons, mean, absolute and root mean square errors, and correlation coefficient. Comparisons showed that the model reproduced both general observed trends and short term fluctuations.     

Effects of mooring line drag damping on response statistics of vessels excited by first- and second-order wave forces

The drag-induced damping in a mooring cable due to combined first- and second-order wave excited motion of a moored vessel has been determined by statistical linearisation. A dynamic stiffness approach developed elsewhere is used to deal with the dynamics of the mooring cables. The power spectral densities of low- and wave-frequency responses are obtained which clearly show the influence of mooring line damping. The non-Gaussian probability density functions (pdf) and expected crossing rates of vessel responses and dynamic cable tensions are determined using the Kac–Seigert technique, and the influence of drag damping is highlighted.     

An experimental model application of wavescreen: Dynamic pressure, water particle velocity, and wave measurements

This paper analyses the results of an application of a piled wavescreen. Experimental measurements were undertaken in the laboratory conditions for a given structural configuration under the attack of regular and irregular waves. Dynamic pressure distribution along and around the inclined piles was obtained employing pressure transducers. Using these data, in-line dynamic wave forces acting on piles were also determined. Water particle (orbital) velocities were measured at seaward and landward of the wavescreen using two acoustic Doppler velocimeters (ADV) simultaneously. Furthermore, wave data were collected using resistance type wave gauges at the seaward and landward of the structure. Based on those data, wave attenuation performance of the wavescreen was explored for two different depth values. Findings showed that piled wavescreen can provide effective shore protection as an environmentally friendly coastal structure.     

散货船及油船垂向波浪载荷短期预报经验公式

应用WALCS软件对6艘散货船及6艘油船进行了短期波浪载荷直接计算,依据计算结果分析了这两种船型在航速为0、迎浪、均匀满载及隔舱满载工况下的船舯垂向弯矩、L/4(L为船长)处垂向剪力、3L/4处垂向剪力的若干规律。根据这些规律,总结出了一套针对散货船及油船的垂向波浪载荷短期预报经验公式。依据此经验公式,在缺乏完备图纸资料的情况下,可以快速估计航行于海上的受损船舶的短期载荷情况,为评估其安全状态提供数值依据,为事故船东方做出安全可行的处理决策提供优质服务。     

Numerical and simplified methods for the calculation of the total horizontal wave force on a perforated caisson with a top cover

The VOF method and the k–ε model, combined with the equation of state of air at constant temperature, have been used to calculate the total horizontal wave force caused by monochromatic waves acting on a perforated caisson with a top cover. From comparison of various parameters, such as the total horizontal force, the pressure difference on the front wall, the pressure on the back wall and the pressure on the top cover, between the numerical results and test data, it can be seen that the numerical results agree well with the test data. It is concluded that the method described in this paper can be utilized to calculate wave forces acting on perforated caissons with a top cover in the case of nonovertopping, nonbreaking waves. A simplified method to calculate the total horizontal force has been developed, based on test data, using a least-squares method. A comparison between the numerical results and the values calculated from the simplified equations shows good agreement. Therefore the simplified equations can be used in engineering applications to evaluate the total horizontal force on a perforated caisson with a top cover.