适合复杂地形的高阶Boussinesq水波方程

针对海底坡度较大(量阶为O(1))或海底曲率较大的复杂地形,建立了一个新型高阶Boussinesq水波方程.该方程可用于研究海底存在若干相互平行沙坝引起的Bragg反射问题.方程的水平速度沿水深的分布为四次多项式,色散性和变浅作用性能的精度比经典Boussinesq方程高了一阶.方程在浅水水域可以是完全非线性的.     

潜堤上破碎波浪传播变形的数值模型及其验证

为模拟潜堤上破碎波浪传播时产生能量的耗散这一特性,在改进的具有四阶色散的Boussinesq水波方程中中入二阶紊动粘性项,建立了考虑波浪破碎的水波数学模型.在非交错网格下建立了有限差分数值模型,并利用三阶Adams-Bash forth格式预报、四阶Adams-Mouton格式校正对数值模型进行求解.通过数值试验,模拟...     

关于Boussinesq型水波方程理论和应用研究的综述

Boussinesq型方程是研究水波传播与演化问题的重要工具之一,本文就1967-2018年常用的Boussinesq型水波方程从理论推导和数值应用两个方面进行了回顾,以期推动该类方程在海岸(海洋)工程波浪水动力方向的深入研究和应用。此类方程推导主要从欧拉方程或Laplace方程出发。在一定的非线性和缓坡假设等条件下,国内外学者建立了多个Boussinesq型水波方程,并以Stokes波的相关理论为依据,考察了这些方程在相速度、群速度、线性变浅梯度、二阶非线性、三阶非线性、波幅离散、速度沿水深分布以及和(差)频等多方面性能的精度。将Boussinesq型水波方程分为水平二维和三维两大类,并对主要Boussinesq型水波方程的特性进行了评述。进而又对适合渗透地形和存在流体分层情况下的Boussinesq型水波方程进行了简述与评论。最后对这些方程的应用进行了总结与分析。     

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。     

适用于沙坝上Bragg反射的二阶Boussinesq方程数学模型及其数值验证

在二阶 Boussinesq 方程基础上,通过引入含水深导数项对该方程进行了理论上的改进,使得该方程在应用于无限沙坝 Bragg反射问题时与理论解析解在更大范围内符合.基于该改进的高阶 Boussinesq 方程,在非交错网格下建立了混合 4 阶的Adams-Bashforth- Moulton 格式的数学模型.将数值模型应用到有限个连续沙坝上波浪传播变形问题的数值模拟中,通过两点法给出数值波浪反射系数,将这些反射系数与已有的实验数据进行对比,对比表明改进后的模型计算出的反射系数与实验结果吻合更好,这验证了本文理论改进的有效性.     

二阶全非线性Boussinesq方程的二维数值模拟研究

在非交错网格下采用有限差分法首次对一组非线性精确至O(μ2)阶的全非线性Boussinesq方程数学模型进行了二维数值模拟分析.首先通过在方程的非线性项中引入缓坡假定,考察了其对模型数值精度的影响;其次,在模型中对二阶非线性项采用不同精度,考察了其对模型数值结果的影响.数值模拟结果表明,所建立的二阶完全非线性Boussinesq方程二维数值模型具有良好的适用性,模型非线性项中引入缓坡假定以及在二阶非线性项选用不同的精度对数值模拟结果影响不明显.     

加强的适合复杂地形的水波方程及其一维数值模型验证

在他人给出的方程的基础上,通过在其动量方程中引入含4个参数的公式,推导出了加强的适合复杂地形的水波方程,新方程的色散、变浅作用以及非线性均比原来适合复杂地形的方程有了改善:色散关系式与斯托克斯线性波的Padé(4,4)阶展开式一致;变浅作用在相对水深(波数乘水深)不大于6时与解析解符合较好;非线性在相对水深不大于1.05时保持在5%的误差之内.基于该方程,在非交错网格下建立的时间差分格式为混合4阶Adams-Bashforth-Moulton的一维数值模型,并在数值计算中利用了五对角宽带解法.数值模拟了潜堤上波浪传播变形,并将数值计算结果与实验结果进行了对比,验证了该数值模型是合理的.     

二维非线性浅水波的数值模拟

研究工作的目的在于建立一个能够模拟二维潮流、洪水波(长波)和浅水波浪(短波)的综合数学模型.基本模型建立在非线性的Boussinesq方程基础之上.本文主要讨论浅水波浪即短波的数值模拟.模型可以考虑必要的外力项,如柯氏力、风应力、大气压力和底摩阻力等.针对Boussinesq方程提出了一个全隐的二维差分格式,讨论了人工开边界的处理方法.模型被用来计算了突然扩张渠槽中的环流和单突堤后的水波绕射,取得了满意的结果.     

改进的波流相互作用Boussinesq方程

对于波流相互作用的Boussinesq方程,为了考虑水流作用所引起的平均水面变化,通过引入新的计算速度,使方程色散关系中的水深为考虑水流影响后的实际水深,并使方程色散精度达到了Padé[4,4]。通过计算潜堤强水流和裂流两种背景水流流场情况下的波浪运动特征,验证了该模型和数值方法的正确性。     

含强水流高阶Boussinesq水波方程

采用摄动法并利用已建立的纯波情况下高阶Boussinesq方程,建立了可以考虑强水流与波浪相互作用的高阶Boussinesq方程.水流速度与波浪群速具有相同量级,且随时间和空间的变化尺度远大于波浪周期和波长.方程色散性近似到[4/4]阶Pade展开,对浅水情况方程可以是完全非线性的,可适用于波流相互作用的强非线性问题.通过将水流存在时波长和波幅的结果与一阶斯托克斯波结果对比,讨论了具有不同近似程度的3种含波流相互作用的Boussinesq方程的适用性.     

A Numerical Model for Nonlinear Wave Propagation on Non-uniform Current

On the basis of the new type Boussinesq equations (Madsen et al.,2002),a set of equations explicitly including the effects of currents on waves are derived.A numerical implementation of the present equations in one dimension is described.The numerical model is tested for wave propagation in a wave flume of uniform depth with current present.The present numerical results are compared with those of other researchers.It is validated that the present numerical model can reasonably reflect the nonlinear influences of currents on waves.Moreover,the effects of inputting different incident boundary conditions on the calculated results are studied.     

Numerical Models of Higher-Order Boussinesq Equations and Comparisons with Laboratory Measurement

Nonlinear water wave propagation passing a submerged shelf is studied experimentally and numerically. The applicability of two different wave propagation models has been investigated. One is higher-order Boussinesq equations derived by Zou (1999) and the other is the classic Boussinesq equations. Physical experiments are conducted, three different front slopes (1:10, 1:5 and 1:2) of the shelf are set up in the experiment and their effects on wave propagation are investigated. Comparisons of numerical results with test data are made, the model of higher-order Boussinesq equations agrees much better with the measurements than the model of the classical Boussinesq equations. The results show that the higher-order Boussinesq equations can also be applied to the steeper slope case although the mild slope assumption is employed in the derivation of the higher order terms of higher order Boussinesq equations.     

One-dimensional numerical models of higher-order Boussinesq equations with high dispersion accuracy

Abstract-Nonlinear water wave propagation passing a submerged shelf is studied experimentally andnumerically. The applicability of the wave propagation model of higher-order Boussinesq equations de-rived by Zou(2000, Ocean Engneering, 27, 557～575) is investigated. Physical experiments areconducted; three different front slopes (1:10, 1:5 and 1:2) of the shelf are set-up in the experimentand their effects on the wave propagation are investigated. Comparisons of the numerical results withtest data are made and the higher-order Boussinesq equations agree well with the measurements since thedispersion of the model is of high accuracy. The numerical results show that the good results can also beobtained for the steep-slope cases although the mild-slope assumption is employed in the derivation of thehigher-order terms in the higher-order Boussinesq equations.     

完全非线性波浪破碎模型沿岸流数值模拟

探讨一种基于完全非线性Boussinesq方程的波浪破碎模型在沿岸流计算中的应用问题。针对控制方程中的完全非线性项对沿岸流成长过程的影响进行了深入讨论。数学模型计算结果表明,完全非线性项有使平均流局部化的作用;通过数模实验还发现,垂向高阶涡度项可以有效抑制破波区外回流;运用Visser的实验室沿岸流实测资料从沿岸流速度、波高和平均水位几方面对所提模型进行了验证,并给出了紊动参数的计算结果。     

Numerical Study of Influence of Currents on Nonlinear Focusing Waves

In this paper,a numerical model for nonlinear wave propagation in currents is formulated by a set of enhanced fully nonlinear Boussinesq equations with ambient currents.This model is verified by comparison with the published results.Then the influence of currents on nonlinear focusing waves is studied by use of the numerical model.It is found that the effect of currents on the surface elevations at the focal location is negligible.Following currents can augment the maximum crest of focusing wave while decre...     

具有精确色散性的非线性波浪水平二维数学模型

建立具有色散性的水平二维非线性波浪方程,方程的非线性近似到了三阶。方程以波面升高和自由表面速度势表达的微分-积分型数学方程,给出方程的数值求解方法和算例,对方程积分项的处理给出了计算方法。计算结果与Boussinesq方程模型和缓坡方程模型的对应计算结果进行了对比。     

高阶Boussinesq类方程数值求解及试验验证

基于二阶非线性与色散的Boussinesq类方程,采用改善的Crank-Nicolson方法对不同情况下淹没潜堤上的波浪传播进行数值模拟。高阶方程与传统、改进型的Boussinesq方程计算结果进行比较,高阶方程的计算结果与实验吻合得更好。表明该高阶Boussinesq方程能够精确预测变水深、强非线性的复杂波况,可用于实际近岸海域波浪问题的计算。     

A temporally second-order accurate Godunov-type scheme for solving the extended Boussinesq equations

A numerical scheme for solving the class of extended Boussinesq equations is presented. Unlike previous schemes, where the governing equations are integrated through time using a fourth-order method, a second-order Godunov-type scheme is used thus saving storage and computational resources. The spatial derivatives are discretised using a combination of finite-volume and finite-difference methods. A fourth-order MUSCL reconstruction technique is used to compute the values at the cell interfaces for use in the local Riemann problems, whilst the bed source and dispersion terms are discretised using centred finite-differences of up to fourth-order accuracy. Numerical results show that the class of extended Boussinesq equations can be accurately solved without the need for a fourth-order time discretisation, thus improving the computational speed of Boussinesq-type numerical models. The numerical scheme has been applied to model a number of standard test cases for the extended Boussinesq equations and comparisons made to physical wave flume experiments.     

A new form of generalized Boussinesq equations for varying water depth

A new set of equations of motion for wave propagation in water with varying depth is derived in this study. The equations expressed by the velocity potentials and the wave surface elevations include first-order non-linearity of waves and have the same dispersion characteristic to the extended Boussinesq equations. Compared to the extended Boussinesq equations, the equations have only two unknown scalars and do not contain spatial derivatives with an order higher than 2. The wave equations are solved by a finite element method. Fourth-order predictor–corrector method is applied in the time integration and a damping layer is applied at the open boundary for absorbing the outgoing waves. The model is applied to several examples of wave propagation in variable water depth. The computational results are compared with experimental data and other numerical results available in literature. The comparison demonstrates that the new form of the equations is capable of calculating wave transformation from relative deep water to shallow water.     

A Compact Explicit Difference Scheme of High Accuracy for Extended Boussinesq Equations

Presented here is a compact explicit difference scheme of high accuracy for solving the extended Boussinesq equations.For time discretization,a three-stage explicit Runge-Kutta method with TVD property is used at predicting stage,a cubic spline function is adopted at correcting stage,which made the time discretization accuracy up to fourth order;For spatial discretization,a three-point explicit compact difference scheme with arbitrary order accuracy is employed.The extended Boussinesq equations derived by Beji and Nadaoka are solved by the proposed scheme.The numerical results agree well with the experimental data.At the same time,the comparisons of the two numerical results between the present scheme and low accuracy difference method are made,which further show the necessity of using high accuracy scheme to solve the extended Boussinesq equations.As a valid sample,the wave propagation on the rectangular step is formulated by the present scheme,the modelled results are in better agreement with the experimental data than those of Kittitanasuan.