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。     

用含高色散性Boussinesq方程模拟潜堤上的波浪变形

基于一种高阶Boussiensq方程(刘忠波等,2004),采用预报-校正格式的有限差分法对该方程进行了数值离散,建立了数值模型。针对动量方程中三阶项的差分形式,采用了迎风格式和五点格式。通过数值模拟常水深下不同周期波浪传播变形,指出迎风格式在计算小周期波浪时存在的问题。为进一步验证数值模型的适用性,模拟了淹没潜堤上的传播变形。从数值结果与实验值的对比结果上看,该数值模型能较好地模拟波浪变形,可用于模拟实际中的波浪场问题。     

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.     

新型Boussinesq方程的进一步改善

基于传统的Ｂｏｕｓｓｉｎｅｓｑ方程,通过速度变换及在方程中加入高阶项的方法,给出了进一步改善线性色散关系及变浅作用系数的方程,其线性色散关系及变浅作用系数的精度均为Ｏ（β８）（β＝ｈ０／Ｌ,Ｌ为特征波长,ｈ０为特征水深）．     

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.     

Boussinesq水波方程新型数值解法

为建立高效的Boussinesq类水波数值模型,提出了一种新型的、基于有限差分和有限体积方法的混合数值格式。针对守恒形式的一维控制方程,在等间距矩形控制体内对其进行积分并离散,采用有限体积方法计算界面数值通量,剩余源项采用有限差分方法计算。其中,采用MUSTA格式并结合高精度状态插值方法计算控制体界面数值通量。时间积分则采用具有TVD性质的三阶龙格-库塔多步积分法进行。除验证模型外,重点对MUSTA格式和广泛使用的HLL格式进行了比较。结果表明,MUSTA格式可用于Boussinesq类水波方程数值求解,综合考虑数值精度、计算效率、程序编制和实际应用这几个方面,其较HLL格式更具有优势。     

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.     

基于双层Boussinesq方程的三维波浪数值模型及其验证

Liu等给出的最高导数为2的双层Boussinesq水波方程具有较好的色散性和非线性,基于该方程建立了有限差分法的三维波浪数值模型。在矩形网格上对方程进行了空间离散,采用高阶导数近似方程中的时、空项,时间积分采用混合4阶Adams-Bashforth-Moulton的预报—校正格式。模拟了深水条件下的规则波传播过程,计算波面与解析结果吻合较好,反映出数值模型能很好地刻画波面过程及波面处的速度变化;在kh=2π条件下可较为准确获得沿水深分布的水平和垂向速度,这与理论分析结果一致。最后,利用数值模型计算了规则波在三维特征地形上的传播变形,数值结果和试验数据吻合较好;高阶非线性项会对波浪数值结果产生一定的影响,当波浪非线性增强,水深减少将产生更多的高次谐波。建立的双层Boussinesq模型对强非线性波浪的演化具有较好的模拟精度。     

Boussinesq方程波浪数学模型的应用

介绍了Ｂｏｕｓｓｉｎｅｓｑ 方程的推导过程和发展过程,基于深水和缓变地形的色散关系,建立了Ｂｏｕｓｓｉｎｅｓｑ方程的波浪数学模型。该模型可以产生波浪,模拟吸收边界和不同反射率的反射边界。该模型可用于研究深水和浅水地区波浪的浅水变形、折射、绕射和反射     

适合中等水流的Boussinesq方程

推导了含量阶为O(ε1/2)的瞬变非均匀流的Boussinesq水波方程,讨论了该量阶水流对流场速度和压力分布的影响,采用了Crank-Nicolson格式的预估-校正有限差分法对该方程进行了数值求解.把数值结果与无水流情况的实验结果进行了对比,验证了该方程和数值计算方法的有效性,与经典的Boussinesq方程和含量阶为O(1)的瞬变非均匀流的Boussinesq水波方程的计算结果进行了比较,考察了该方程的适用范围.     

Alternative forms of the higher-order Boussinesq equations: Derivations and validations

An alternative form of the Boussinesq equations is developed, creating a model which is fully nonlinear up to O(μ⁴) (μ is the ratio of water depth to wavelength) and has dispersion accurate to the Padé [4,4] approximation. No limitation is imposed on the bottom slope; the variable distance between free surface and sea bottom is accounted for by a σ-transformation. Two reduced forms of the model are also presented, which simplify O(μ⁴) terms using the assumption ε = O(μ^2/3) (ε is the ratio of wave height to water depth). These can be seen as extensions of Serre's equations, with dispersions given by the Padé [2,2] and Padé [4,4] approximations. The third-order nonlinear characteristics of these three models are discussed using Fourier analysis, and compared to other high-order formulations of the Boussinesq equations. The models are validated against experimental measurements of wave propagation over a submerged breakwater. Finally, the nonlinear evolution of wave groups along a horizontal flume is simulated and compared to experimental data in order to investigate the effects of the amplitude dispersion and the four-wave resonant interaction.     

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.     

Comparison between characteristics of mild slope equations and Boussinesq equations

Boussinesq-type equations and mild-slope equations are compared in terms of their basic forms and characteristics. It is concluded that linear mild-slope equations on dispersion relation are better than non-linear Boussinesq equations. In addition, Berkhoffexperiments are computed and compared by the two models, and agreement between model results and available experimental data is found to be quite reasonable, which demonstrates the two models＇ capacity to simulate wave transformation. However they can deal with different physical processes respectively, and they have their own characteristics.     

Essential properties of Boussinesq equations for internal and surface waves in a two-fluid system

Boussinesq equations describing motions of internal waves in a two-fluid system with the presence of free surface are theoretically derived, and the associated essential properties are examined in this study. Eliminating the dependence on the vertical coordinate from all variables, four equations constitute the Boussinesq model with two flexible parameters, z_u and z_l, which indicate the specific elevations, respectively, in the upper and lower fluids. Similar to the Boussinesq model for a single-layer fluid, z_u and z_l are determined by matching the linear dispersion relation with Lamb's solution. This determines the optimal model. In the analysis stage, this problem is classified into two cases, the thicker-upper-layer case and the thicker-lower-case case, to avoid the possible divergence of wave properties as the thickness ratio grows. Since there exist two modes of motions that may be excited, cases of both modes are separately analyzed. Linear characteristics including the amplitude ratios and normalized particle velocities are analyzed. Second-order harmonic waves are examined to validate nonlinear behaviors of present model. Results of linear and nonlinear investigations show that the present model indeed extends the applicable range of traditional Boussinesq equations.     

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.     

Numerical model for solving Bousiinesq-type equations: comparison and validation

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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.