Two sets of higher-order Boussinesq-type equations for water waves

Based on the classical Boussinesq model by Peregrine [Peregrine, D.H., 1967. Long waves on a beach. J. Fluid Mech. 27 (4), 815–827], two parameters are introduced to improve dispersion and linear shoaling characteristics. The higher order non-linear terms are added to the modified Boussinesq equations. The non-linearity of the Boussinesq model is analyzed. A parameter related to h/L₀ is used to improve the quadratic transfer function in relatively deep water. Since the dispersion characteristic of the modified Boussinesq equations with two parameters is only equal to the second-order Padé expansion of the linear dispersion relation, further improvement is done by introducing a new velocity vector to replace the depth-averaged one in the modified Boussinesq equations. The dispersion characteristic of the further modified Boussinesq equations is accurate to the fourth-order Padé approximation of the linear dispersion relation. Compared to the modified Boussinesq equations, the accuracy of quadratic transfer functions is improved and the shoaling characteristic of the equations has higher accuracy from shallow water to deep water.     

Higher order Boussinesq equations

A new form of Boussinesq-type equations accurate to the third order are derived in this paper to improve the linear dispersion and nonlinearity characteristics in deeper water. Fourth spatial derivatives in the third order terms of the equations are transformed into second derivatives and present no difficulty in numerical computations. With the increase in accuracy of the equations, the nonlinear and dispersion characteristics of the equations are of one order of magnitude higher accuracy than those of the classical Boussinesq equations. The equations can serve as a fully nonlinear model for shallow water waves. The shoaling property of the equations is also of high accuracy through shallow water to deep water by introducing an extra source term into the second order continuity equation. An approach to increase the accuracy of the nonlinear characteristics of the new equations is introduced. The expression for the vertical distribution of the horizontal velocities is a fourth order polynomial.     

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。     

A note on linear dispersion and shoaling properties in extended Boussinesq equations

A set of optimum parameter α is obtained to evaluate the linear dispersion and shoaling properties in the extended Boussinesq equations of [Madsen and Sorensen, 1992 and Nwogu, 1993], and [Chen and Liu, 1995]. Optimum α values are determined to produce minimal errors in each wave property of phase velocity, group velocity, or shoaling coefficient relative to the analytical one given by the Stokes wave theory. Comparisons are made of the percent errors in phase velocity, group velocity, and shoaling coefficient produced by the Boussinesq equations with a different set of optimum α values. The case with a fixed value of α = −0.4 is also presented in the comparison. The comparisons reveal that the optimum α value tuned for a particular wave property gives in general poor results for other properties. Considering all the properties simultaneously, the fixed value of α = −0.4 may give overall accuracies in phase velocity and shoaling coefficient for all the types of Boussinesq equations selected in this study.     

一种新型二阶全非线性Boussinesq方程及其数值验证

通过改进二阶全非线性 Boussinesq 波浪方程中的色散项,得到了一组没有改变原方程的数学形式但适用于更大变化水深的新方程,其色散性能和变浅性能都比原方程有了很大改进,所适用的水深范围更大,能更好地描述从深水到近岸浅水处的波浪传播;并基于新方程建立了波浪数值模型,通过模拟波浪从浅水到深水的传播变形来验证新方程的有效性.     

Simplified higher-order Boussinesq equations: I. Linear simplifications

In this paper, we derive and test simplified higher-order Boussinesq equations, i.e., higher-order Boussinesq equations which only show lower-order terms. Simplifications are performed linearly for flat beds and slopes of O(∇h). With proper coefficient choice, dispersion and shoaling properties are found to be good, while interior fluid velocities show relatively greater error at high wavenumbers.The resulting sets of equations are found to be variants of already-existing equations, which may be easily modified to improve performance. The new equations have dispersion identical to previous results but significantly improved shoaling.     

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.     

Boussinesq–Green–Naghdi rotational water wave theory

Using Boussinesq scaling for water waves while imposing no constraints on rotationality, we derive and test model equations for nonlinear water wave transformation over varying depth. These use polynomial basis functions to create velocity profiles which are inserted into the basic equations of motion keeping terms up to the desired Boussinesq scaling order, and solved in a weighted residual sense. The models show rapid convergence to exact solutions for linear dispersion, shoaling, and orbital velocities; however, properties may be substantially improved for a given order of approximation using asymptotic rearrangements. This improvement is accomplished using the large numbers of degrees of freedom inherent in the definitions of the polynomial basis functions either to match additional terms in a Taylor series, or to minimize errors over a range. Explicit coefficients are given at O(μ²) and O(μ⁴), while more generalized basis functions are given at higher order. Nonlinear performance is somewhat more limited as, for reasons of complexity, we only provide explicitly lower order nonlinear terms. Still, second order harmonics may remain good to kh ≈ 10 for O(μ⁴) equations. Numerical tests for wave transformation over a shoal show good agreement with experiments. Future work will harness the full rotational performance of these systems by incorporating turbulent and viscous stresses into the equations, making them into surf zone models.     

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 fast convolution approach to the transformation of surface gravity waves: Linear waves in 1DH

The transformation of irrotational surface gravity waves in an inviscid fluid can be studied by time stepping the kinematic and dynamic surface boundary conditions. This requires a closure providing the normal surface particle velocity in terms of the surface velocity potential or its tangential derivative. A convolution integral giving this closure as an explicit expression is derived for linear 1D waves over a mildly sloping bottom. The model has exact linear dispersion and shoaling properties. A discrete numerical model is developed for a spatially staggered uniform grid. The model involves a spatial derivative which is discretized by an arbitrary-order finite-difference scheme. Error control is attained by solving the discrete dispersion relation a priori and model results make a perfect match to this prediction. A procedure is developed by which the computational effort is minimized for a specific physical problem while adapting the numerical parameters under the constraint of a predefined tolerance of damping and dispersion error. Two computational examples show that accurate irregular-wave transformation on the kilometre scale can be computed in seconds. Thus, the method makes up a highly efficient basis for a forthcoming extension that includes nonlinearity at arbitrary order. The relation to Boussinesq equations, mild-slope wave equations, boundary integral equations and spectral methods is briefly discussed.     

Fully Nonlinear Boussinesq-Type Equations with Optimized Parameters for Water Wave Propagation

For simulating water wave propagation in coastal areas, various Boussinesq-type equations with improved properties in intermediate or deep water have been presented in the past several decades. How to choose proper Boussinesq-type equations has been a practical problem for engineers. In this paper, approaches of improving the characteristics of the equations, i.e. linear dispersion, shoaling gradient and nonlinearity, are reviewed and the advantages and disadvantages of several different Boussinesq-type equations are compared for the applications of these Boussinesq-type equations in coastal engineering with relatively large sea areas. Then for improving the properties of Boussinesq-type equations, a new set of fully nonlinear Boussinseq-type equations with modified representative velocity are derived, which can be used for better linear dispersion and nonlinearity. Based on the method of minimizing the overall error in different ranges of applications, sets of parameters are determined with optimized linear dispersion, linear shoaling and nonlinearity, respectively. Finally, a test example is given for validating the results of this study. Both results show that the equations with optimized parameters display better characteristics than the ones obtained by matching with padé approximation.     

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.     

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 phase-averaged Boussinesq model with effective description of carrier wave group and associated long wave evolution

From the phase-resolving improved Boussinesq equations (Beji and Nadaoka, Ocean Engineering 23 (1996) 691), a phase-averaged Boussinesq model for water waves is derived by more effectively describing carrier wave groups and accompanying long wave evolution with less CPU time. Linear shoaling characteristics of carrier wave equations are investigated and found to agree exactly with the analytical expression obtained from the constancy of energy flux for the improved Boussinesq equations themselves, showing that the present model equations are the results of a consistent derivation procedure regarding energy considerations. Numerical simulations of the derived equations for the single wave group and narrow-banded random waves show the validity of the present model and its high performance, especially on the CPU time.     

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。     

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.     

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。     

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.