An explicit finite difference model for simulating weakly nonlinear and weakly dispersive waves over slowly varying water depth

In this paper, a modified leap-frog finite difference (FD) scheme is developed to solve Non linear Shallow Water Equations (NSWE). By adjusting the FD mesh system and modifying the leap-frog algorithm, numerical dispersion is manipulated to mimic physical frequency dispersion for water wave propagation. The resulting numerical scheme is suitable for weakly nonlinear and weakly dispersive waves propagating over a slowly varying water depth. Numerical studies demonstrate that the results of the new numerical scheme agree well with those obtained by directly solving Boussinesq-type models for both long distance propagation, shoaling and re-fraction over a slowly varying bathymetry. Most importantly, the new algorithm is much more computationally efficient than existing Boussinesq-type models, making it an excellent alternative tool for simulating tsunami waves when the frequency dispersion needs to be considered.     

Applicability of numerical models to nonlinear dispersive waves

The applicability of three different wave-propagation models in nonlinear dispersive wave fields has been investigated. The numerical models tested here are based on three different wave theories: a fully nonlinear potential theory, a Stokes second-order theory, and a Boussinesq-type theory with an improved dispersion relation. Physical experiments and computations were conducted for wave evolutions during passage over a submerged shelf under various wave conditions. As expected, the fully nonlinear solutions agree better with the measurements than do the other solutions. Although the second-order solution has sufficient accuracy for smaller-amplitude wave cases, the truncation after the third harmonics causes significant discrepancies in wave form for larger waves. In addition, the second-order model markedly overestimates the first- and second-harmonic amplitudes in transmitted waves. The Boussinesq model provides excellent predictions of wave profile over the shelf even in larger wave cases. However, this model also overestimates the magnitudes of several higher harmonics in transmitted waves. These facts may indicate that energy transfer from bound components into free waves in these higher harmonics cannot be accurately evaluated by the Boussinesq-type equations.     

Influence of currents on weakly nonlinear topographic waves

In the Boussinesq approximation, we consider trapped topographic waves in an inhomogeneous current directed along isobaths. The influence of the current on the dispersion properties of trapped topographic waves in the Norwegian Sea is studied. We determine the mean currents and nonoscillatory (on the time scale of the waves) density corrections induced by the waves due to their nonlinearity. It is shown that the influence of currents is significant in the short-wave region. Its influence leads to a decrease in the wavelength for the constant period of waves, whereas the mean current caused by nonlinearity noticeably varies, especially in the bottom layer.     

强非线性和色散性Boussinesq方程数值模型检验

采用同位网格有限差分法,建立了强非线性和色散性Boussinesq方程数值计算模型。以稳恒波Fourier近似解给定入射波边界条件,对均匀水深深水和浅水域不同非线性的行进波、缓坡地形上深水至浅水域的浅水变形波、以及缓坡和陡坡地形上的波浪水槽实验进行了数值计算,并将计算结果与解析解、解析数值解以及实验值进行了较为详细的比较,从而检验了模型的色散性、非线性以及不同底坡下非线性波的浅水变形性能。     

Note on conservation equations for nonlinear surface waves

Euler's equations of motion in conjunction with the dynamic boundary condition are manipulated to obtain exact (and approximate) alternative momentum equations for nonlinear irrotational surface waves. The Airy and Boussinesq equations are re-derived as demonstrative examples. A fully nonlinear version of the improved Boussinesq equations is presented as a new application of the proposed equations. Further use of the equations in developing depth-integrated wave models, which are not necessarily restricted to finite depths, is also pointed out.     

一维全非线性Green-Naghdi水波方程数值模型

建立了求解一维全非线性Green-Naghdi水波方程的中心有限体积/有限差分混合数值格式。采用结构化网格对守恒形式的控制方程进行离散和积分,界面数值通量采用有限体积法计算,剩余项则采用中心有限差分格式求解。其中,采用中心迎风有限体积格式计算控制体界面数值通量,并结合界面变量的线性重构方法,使其在空间上具有四阶精度,通过引入静压重构技术和波浪破碎指标使模型具备处理海岸水-陆动边界及波浪破碎的能力。时间积分则采用具有总时间变差减小(Total Variation Diminishing,TVD)性质的三阶龙格-库塔法进行。应用该模型对孤立波在常水深和斜坡海岸上的传播过程及规则波跨越潜堤传播的实验进行了数值模型研究,数值计算同解析解及实验数据吻合良好。     

Interaction of dispersive water waves with weakly sheared currents of arbitrary profile

A set of depth-integrated equations describing combined wave–current flows is derived and validated. To account for the effect of turbulence induced by interactions between waves and currents with arbitrary horizontal vorticity, new additional stress terms are introduced. These stresses are functions of a parameter b that relates the relative importance of wave radiation stress and bottom friction stress to the wave–current interaction. To solve the equations, a fourth-order MUSCL-TVD scheme with an approximate Riemann solver is adopted. As a first-order check of the model, the Doppler shift effect and wave dispersion over linearly sheared currents are analytically shown to be retained appropriately in the equation set. The model results are then validated through comparisons with three experimental data sets. First, based on the experiments of Kemp and Simons (1982, 1983), a reasonable functional form of b is estimated. Second, simulations examining the propagation of a weakly dispersive wave over a depth-uniform or linearly sheared current are performed. Finally, the model is applied to a more complex configuration where bichromatic waves interact with spatially varying currents. Simulated results indicate that the model is capable of predicting nearshore interactions of waves with currents of arbitrary vertical structure. One of the unique properties of the developed model is its ability to assimilate an external current field from any source, be it from a circulation model or an observation, and predict the interaction of a nonlinear and dispersive wave field with that current.     

完全非线性深水波的数值模拟

基于势流理论,并结合深水波质点运动从水面向下呈e指数衰减的特性,建立了完全非线性数值变深水槽模型,通过实时模拟活塞式造波机运动来产生波浪.采用时域高阶边界元法进行模拟,利用混合欧拉-拉格朗日方法和四阶Runge-Kutta方法追踪流体瞬时水面,应用镜像格林函数消除了水槽两个侧面的积分,在水槽末端布置人工阻尼层来消除反射...     

高阶非线性完全频散性波浪数值模型及应用

为更精确地模拟强非线性完全频散性波浪的传播,采用长波上非线性重力表面波传播高阶数学模型,综合参考此模式已有的研究成果,建立了一个高达五阶的完全频散性非线性数值模型。应用该五阶模式对斜坡地形、潜堤地形及正弦沙链地形进行模拟计算,并与已有的实验资料进行对比,结果显示五阶模式较低阶模式模拟结果的精度上有了明显提高,模拟波形与实验结果吻合度良好,证明高阶模式更适用于高频散高非线性波浪传播的数值模拟。     

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.     

A Boussinesq-type method for fully nonlinear waves interacting with a rapidly varying bathymetry

New equations are derived for fully nonlinear and highly dispersive water waves interacting with a rapidly varying bathymetry. The derivation is an extension of a recent high order Boussinesq type formulation valid on a mildly sloping bottom. It is based on a series expansion from a rapidly spatially varying expansion level and the resulting general velocity formulation is given as a triple-summation of terms involving high derivatives of this expansion level. For practical implementation, it is necessary to simplify and truncate this general formulation and we do this by assuming that the expansion level (but not the bathymetry) is slowly varying in space. On this basis, the general expressions are simplified to include first and second derivatives of the expansion level and up to fifth-derivatives of the velocity variables. With this new approach, the accuracy of the dispersion relation can locally deteriorate, and we provide a guideline for using this technique within acceptable accuracy bounds. Numerical results are given for the linear reflection from a plane shelf, a Gaussian shaped trench, and a symmetric trench with sloped transitions. Furthermore, we simulate the linear class I and class II Bragg scattering from an undular sea bottom. The computations are verified against measurements, theoretical solutions and numerical models from the literature. Finally, we make a detailed investigation of nonlinear class III Bragg scattering and results are given for the sub-harmonic and super-harmonic interactions with the sea bed. We provide a new explanation and a prediction of the resulting downshift/upshift of the peak reflection/transmission as a function of wave steepness.     

弱非线性斯托克斯波在非平整海底上传播的新型抛物型方程

由于缓坡方程计算量大和其本身的缓坡假定而在实际应用中受到了限制,故对斯托克斯波在非平整海底(适用于缓坡和陡坡地形)上传播的Liu和Dingemans的三阶演化方程进行抛物逼近,得到一个新的非线性抛物型方程,它能够包含同类方程未曾考虑的二阶长波效应.通过数值计算结果与Berkhoff等人的经典实验数据的比较,证明所提出的抛物型模型理论具有较高的精度.     

Interactions between fully nonlinear water waves and cylinder arrays in a wave tank

Fully nonlinear interactions between water waves and vertical cylinder arrays in a numerical tank are studied based on a finite element method (FEM). The three-dimensional (3D) mesh is constructed through an extension of a 2D Delaunay surface grid along the vertical line. The velocity potential is obtained by solving a linear matrix system of FEM, and a difference scheme is then used to calculate the velocity on the free surface to track its movement. Waves and hydrodynamic forces are obtained for both bottom mounted and truncated cylinders. The simulations have provided many results to show the nature of mutual interference between cylinders in arrays and its effects on waves and forces at the nearly trapped mode frequency. The effect of the tank wall on waves and forces has been investigated, and the nonlinear features of waves and forces have also been discussed.     

完全非线性波浪与二维固定结构物作用的IGN-BEM耦合分析模型

为高效准确地对完全非线性波浪与二维固定结构物的相互作用进行模拟分析,建立了二维完全非线性时域耦合模型。耦合模型将计算域划分为靠近结构物的内域和远离结构物的外域,每个区域均采用满足完全非线性自由水面边界条件的波浪模型进行求解。在内域使用Laplace方程描述流体运动并采用高阶边界元法（BEM）对其进行求解;而在没有结构物的外域,波浪运动的控制方程为Irrotational Green-Naghdi（IGN）方程并采用有限元法（FEM）对其进行求解。内域和外域通过一段重叠区域进行耦合,从而实现模型间变量的传递。首先利用耦合模型分别对规则波的传播、直墙前立波的生成以及相关物理模型试验进行模拟,数值结果与精确解和试验结果的良好吻合验证了耦合模型耦合方式的合理性以及处理非线性问题的准确性;然后使用耦合模型模拟分析了波浪与固定结构物间的相互作用,并将结果与线性解析解以及完全非线性BEM模型的结果进行了对比分析,进一步证明了耦合模型的正确性与高效性。     

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.     

A depth-integrated model for weakly dispersive, turbulent, and rotational fluid flows

A set of weakly dispersive Boussinesq-type equations, derived to include viscosity and vorticity terms in a physically consistent manner, is presented in conservative form. The model includes the approximate effects of bottom-induced turbulence, in a depth-integrated sense, as a second-order correction. Associated with this turbulence, vertical and horizontal rotational effects are captured. While the turbulence and horizontal vorticity models are simplified, a model with known physical limitations has been derived that includes the quadratic bottom friction term commonly added in an ad hoc manner to the inviscid equations. An interesting result of this derivation is that one should take care when adding such ad hoc models; it is clear from this exercise that (1) it is not necessary to do so – the terms can be included through a consistent derivation from the viscous primitive equations – and (2) one cannot properly add the quadratic bottom friction term without also adding a number of additional terms in the integrated governing equations. To solve these equations numerically, a highly accurate and stable model is developed. The numerical method uses a fourth-order MUSCL-TVD scheme to solve the leading order (shallow water) terms. For the dispersive terms, a cell averaged finite volume method is implemented. To verify the derived equations and the numerical model, four cases of verifications are given. First, solitary wave propagation is examined as a basic, yet fundamental, test of the models ability to predict dispersive and nonlinear wave propagation with minimal numerical error. Vertical velocity distributions of spatially uniform flows are compared with existing theory to investigate the effects of the newly included horizontal vorticity terms. Other test cases include comparisons with experiments that generate strong vorticity by the change of bottom bathymetry as well as by tidal jets through inlet structures. Very reasonable agreements are observed for the four cases, and the results provide some information as to the importance of dispersion and horizontal vorticity.     

基于四阶完全非线性Boussinesq水波方程二维波浪传播数值模型

建立基于四阶完全非线性Boussinesq水波方程的二维波浪传播数值模型。采用Kennedy等提出的涡粘方法模拟波浪破碎。在矩形网格上对控制方程进行离散,采用高精度的数值格式对离散方程进行数值求解。对规则波在具有三维特征地形上的传播过程进行了数值模拟,通过数值模拟结果与实验结果的对比,对所建立的波浪传播模型进行了验证。同时,为了考察非线性对波浪传播的影响,给出和上述模型具有同阶色散性、变浅作用性能但仅具有二阶完全非线性特征的波浪模型的数值结果。通过对比两个模型的数值结果以及实验数据,讨论非线性在波浪传播过程中的作用。研究结果表明,所建立的Boussinesq水波方程在深水范围内不但具有较精确的色散性和变浅作用性能,而且具有四阶完全非线性特征,适合模拟波浪在近岸水域的非线性运动。     

Hydroelastic analysis of fully nonlinear water waves with floating elastic plate via multiple knot B-splines

Hydroelastic analysis of fully nonlinear water waves with the floating elastic plate is a hard mission. Especially, the behavior of the wave would be more complex when water wave encounter the floating elastic plate. In this paper, the meshless numerical method is devoted to solve such a problem. Fundamental solution method is applied to approximate the velocity potential in the fluid domain. When the water wave encounters the plate, the wave function would not be enough smooth in the edge of plate compared to the other points. Hence, to analyze numerically the behavior of wave, the solution space should include the basis functions that are not enough smooth in the edge of plate. Moreover, to decrease computational cost significantly, the basis functions had better to have local compact support. The multiple knot B-spline basis functions are suitable that contain both properties. The number of repeated knots, the degree of B-spline and the spatial points are challengeable that are discussed in this paper. The results are in good agreement with those obtained from other numerical works.     

数值波浪水槽中完全非线性浅水波仿真特性研究

应用基于势流理论的时域高阶边界元方法,建立一个完全非线性的三维数值波浪水槽,通过实时模拟推板造波运动的方式产生波浪。通过混合欧拉-拉格朗日方法和四阶Runge-Kutta方法更新自由水面和造波板的瞬时位置。利用所建模型分别模拟了有限水深波和浅水波,与试验结果、相关文献结果和浅水理论结果吻合较好,且波浪能够稳定传播。系统地讨论造波板的运动圆频率、振幅和水深等对波浪传播和波浪特性的影响,并对波浪的非线性特性进行分析,研究发现造波板运动频率、运动振幅以及水深均将对波浪形态和波浪非线性产生显著影响。结果为真实水槽造波机的运动控制以及波浪生成试验提供了依据,便于实验室设置更合理的参数来准确模拟不同条件下的波浪。     

Efficient computation method for two-dimensional nonlinear waves

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