水槽中浅水非线性长波传播的 Boussinesq 数值模拟

浅水非线性长波传播变形中会产生波-波相互作用,为较好地模拟这种现象,在非交错网格下建立了近似在阶完全非线性的高阶 Boussinesq 数值模型.数值模型中采用了混合 4 阶 Adams- Bashforth -Moulton 格式和内部造波技术.数值计算了非线性长波在波浪水槽中的传播变形,计算结果与相关实验数据吻合较好,验证了该数值模型实用性.     

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。     

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。     

波浪与水流共同作用下的流速场

本文讨论了波浪与稳定流相互作用的二元流条件下的流速场问题。分析表明由于波浪与水流相互作用的结果水流的垂线流速分布变得更为均匀,这一结果已经实验验证。结果还表明对于波流共同作用下的水平流速场可以应用波浪与水流二者水平流速值的叠加原理。数值计算的结果表明,对于波面高程和水平流速值,由非线性波理论所得的结果较线性波理论的结果为好。作者认为,在波流共同作用条件下,在工程实用上为了计算水平流速场,当相对水深 d/L 2大于0.1时,司采用斯托克斯三阶波或五阶波理论,当相对水深 d/L_2小于0.1时,司采用椭圆余弦波理论(d 为水深,L_2为静水中的波长)。     

交错网格下的Boussinesq水波数值模型

针对一组近似到二阶完全非线性,四阶色散的Boussinesq方程,在交错网格下建立了数学模型.计算中时间层不交错,模型的求解利用混合四阶Adams-Bashforth-Moulton格式的有限差分法.数值模拟了波浪在潜堤上的演化过程,再现了波浪的浅化、反射以及非线性波能量传递等现象.对数值计算结果采用Friouer变换...     

基于Boussinesq水波模型的聚焦波模拟

基于最高导数为3阶的单层Boussinesq方程,建立了聚焦波的时域波浪计算模型.数值模型求解采用了预报-校正的有限差分法.对于时间差分格式,预报和校正分别采用3阶Adams-Bashforth格式和4阶Adams-Moulton格式.首先,针对不同水深条件下水槽中传播的强非线性波进行模拟,并将数值结果与流函数的数值解...     

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。     

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。     

非线性波传播的新型数值模拟模型及其实验验证

以一种新型的Boussinesq型方程为控制方程组,采用五阶Runge-Kutta-England格式离散时间积分,采用七点差分格式离散空间导数,并通过采用恰当的出流边界条件,从而建立了非线性波传播的新型数值模拟模型.通过对均匀水深水域内波浪传播的数值模拟说明,模型能较好地模拟大水深水域和强非线性波的传播.通过设置不同的入射波参数来进行潜堤地形上波浪传播的物理模型实验,并将数值解与物理模型实验结果进行了比较.     

坡度、水深和波高对孤立波分裂影响作用的数值研究

基于有限差分法建立高阶Boussinesq方程的一维数值模型,时间步进上采用三阶预报、四阶校正格式。在验证数值模型适用性的基础上研究了坡度、水深和波高对孤立波分裂位置、主峰和次峰波高大小的影响,证明了孤立波的分裂与非线性特征有关。坡度缓,非线性演化的时间长,孤立波更易分裂,但坡度变缓不会明显增强波浪非线性特征,从而对主、次峰波高影响不大;入射波高大、水深浅(深水水深或浅水水深)的孤立波非线性特征更强,波形更尖锐,孤立波更易发生分裂,且主、次峰波高也越大。     

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.     

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.     

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.     

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.     

On the internal wave generation in Boussinesq and mild-slope equations

The literature contains empirical knowledge on whether the wave celerity or the group velocity should be used in the line source function for internal wave generation for at given set of Boussinesq or mild-slope equations. Theoretical derivations that confirm and explain these empirical findings are devised. For Boussinesq equations with, e.g. Padé[2,2]-type of dispersion relation some procedures for internal wave generation are affected by their excitation of an evanescent mode. This has some undesirable consequences, but the evanescent-mode excitation can be avoided by the use of an “internal flux boundary”.     

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.     

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 higher order Boussinesq equations

On the basis of the higher order Boussinesq equations derived by the author (1999), a new form of higher order Boussinesq equations is developed through replacing the depth-averaged velocity vector by a new velocity vector in the equations in order to increase the accuracy of the linear dispersion, shoaling property and nonlinear characteristics of the equations. The dispersion of the new equations is accurate to a [4/4] Pade expansion in kh. Compared to the previous higher order Boussinesq equations, the accuracy of quadratic transfer functions is improved and the shoaling property of the equations have higher accuracy from shallow water to deep water.