A New Approach to High-Order Boussinesq-Type Equations with Ambient Currents

A new approach to high-order Boussinesq-type equations with ambient currents is presented. The current velocity is assumed to be uniform over depth and of the same magnitude as the shallow water wave celerity. The wave velocity field is expressed in terms of the horizontal and vertical wave velocity components at an arbitrary water depth level. Linear operators are introduced to improve the accuracy of the kinematic condition at the sea bottom. The dynamic and kinematic conditions at the free surface are expressed in terms of wave velocity variables defined directly on the free surface. The new equations provide high accuracy of linear properties as well as nonlinear properties from shallow to deep water, and extend the applicable range of relative water depth in the case of opposing currents.

A set of Boussinesq-type equations for interfacial internal waves in two-layer stratified fluid

Many new forms of Boussinesq-type equations have been developed to extend the range of applicability of the classical Boussinesq equations to deeper water in the Study of the surface waves. One approach was used by Nwogu （1993. J. Wtrw. Port Coastal and Oc. Eng. 119, 618-638） to improve the linear dispersion characteristics of the classical Boussinesq equations by using the velocity at an arbitrary level as the velocity variable in derived equations and obtain a new form of Boussinesq-type equations, in which the dispersion property can be optimized by choosing the velocity variable at an adequate level. In this paper, a set of Boussinesq-type equations describing the motions of the interracial waves propagating alone the interface between two homogeneous incompressible and inviscid fluids of different densities with a free surface and a variable water depth were derived using a method similar to that used by Nwogu （1993. J. Wtrw. Port Coastal and Oc. Eng. 119, 618-638） for surface waves. The equations were expressed in terms of the displacements of free surface and density-interface, and the velocity vectors at arbitrary vertical locations in the upper layer and the lower layer （or depth-averaged velocity vector across each layer） of a two-layer fluid. As expected, the equations derived in the present work include as special cases those obtained by Nwogu （1993, J. Wtrw. Port Coastal and Oc. Eng. 119, 618-638） and Peregrine （1967, J. Fluid Mech. 27, 815-827） for surface waves when the density of the upper fluid is taken as zero.

Higher-order Boussinesq-type equations for interfacial waves in a two-fluid system

Interracial waves propagating along the interface between a three-dimensional two-fluid system with a rigid upper boundary and an uneven bottom are considered. There is a light fluid layer overlying a heavier one in the system, and a small density difference exists between the two layers. A set of higher-order Boussinesq-type equations in terms of the depth-averaged velocities accounting for stronger nonlinearity are derived. When the small parameter measuring frequency dispersion keeping up to lower-order and full nonlinearity are considered, the equations include the Choi and Camassa＇s results （1999）. The enhanced equations in terms of the depth-averaged velocities are obtained by applying the enhancement technique introduced by Madsen et al. （1991） and Schaffer and Madsen （1995a）. It is noted that the equations derived from the present study include, as special cases, those obtained by Madsen and Schaffer （1998）. By comparison with the dispersion relation of the linear Stokes waves, we found that the dispersion relation is more improved than Choi and Camassa＇s （1999） results, and the applicable scope of water depth is deeper.

The consistent Riccati expansion and new interaction solution for a Boussinesq-type coupled system

Starting from the Davey-Stewartson equation, a Boussinesq-type coupled equation system is obtained by using a variable separation approach. For the Boussinesq-type coupled equation system, its consistent Riccati expansion （CRE） solvability is studied with the help of a Riccati equation. It is significant that the soliton--cnoidal wave interaction solution, expressed explicitly by Jacobi elliptic functions and the third type of incomplete elliptic integral, of the system is also given.

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 pad6 approximation.

Determination of fractional energy loss of waves in nearshore waters using an improved high-order Boussinesq-type model

Fractional energy losses of waves due to wave breaking when passing over a submerged bar are studied systematically using a modified numerical code that is based on the high-order Boussinesq-type equations.The model is first tested by the additional experimental data,and the model's capability of simulating the wave transformation over both gentle slope and steep slope is demonstrated.Then,the model's breaking index is replaced and tested.The new breaking index,which is optimized from the several breaking indices,is not sensitive to the spatial grid length and includes the bottom slopes.Numerical tests show that the modified model with the new breaking index is more stable and efficient for the shallow-water wave breaking.Finally,the modified model is used to study the fractional energy losses for the regular waves propagating and breaking over a submerged bar.Our results have revealed that how the nonlinearity and the dispersion of the incident waves as well as the dimensionless bar height(normalized by water depth) dominate the fractional energy losses.It is also found that the bar slope(limited to gentle slopes that less than 1:10) and the dimensionless bar length(normalized by incident wave length) have negligible effects on the fractional energy losses.

Shallow-water sloshing motions in rectangular tank in general motions based on Boussinesq-type equations

Based on the highly accurate Boussinesq-type equations in terms of velocity potential, the shallow-water sloshing in a two-dimensional rectangular tank is studied. The rectangular tank in harmonic sway, heave and roll motions with small excitation amplitudes is considered. The total velocity potential is divided into two parts： the particular solution and the remaining part to be determined by the Boussinesq-type equations. The Stokes-Joukowski potential is adopted in the particular solution for the roll excitation motion. The comparisons of the numerical results indicate that the shallow-water sloshing motions in a rectangular tank can be predicted well based on the Boussinesq-type equations.

An extended form of Boussinesq-type equations for nonlinear water waves

An extended form of Boussinesq-type equations with improved nonlinearity is presented for the water wave propagation and transformation in coastal areas. To improve the nonlinearity of lower order Boussinesq-type equations,without including higher order derivative terms, an extended form of Boussinesq-type equations is derived by a generalized method. The Stokes-type analysis of the extended equations shows that the accuracy of the second order and third order nonlinear characteristics is improved greatly. The numerical test is also carried out to investigate the practical performance of the new equations under different wave conditions. Better agreement with experimental data is found in the regions of strong nonlinear wave-wave interaction and harmonic generation.

Generalized Cauchy Matrix Approach for Lattice Boussinesq-Type Equations

The authors generalize the Cauchy matrix approach to get exact solutions to the lattice Boussinesq-type equations:lattice Boussinesq equation,lattice modified Boussinesq equation and lattice Schwarzian Boussinesq equation.Some kinds of solutions including soliton solutions,Jordan block solutions and mixed solutions are obtained.

两种高阶非线性偏微分方程组守恒律的构造

利用Ibragimov新守恒定理和对称-共轭对称‘对’方法构造了3阶形变Boussinesq型方程、耦合KdV方程等2个高阶非线性偏微分方程组的局部守恒律,并对所求得的守恒律进行比较,揭示了2种方法在求解高阶非线性偏微分方程组守恒律中的等价关系。

双层Boussinesq水波方程速度公式的修正

为提高Boussinesq水波方程中的速度精度,以最高空间导数为2的双层Boussinesq方程为研究对象,提出增加带有常系数的三阶项以修正速度公式.适用水深在0<kh<8(k为波数,h为静水深)范围内,以方程的水平速度和垂向速度与Stokes线性波速度解析解的积分误差最小为目标,优化系数取值.在1%误差内,改进公式水平速度和垂向速度的适用水深kh分别为7.34和7.83,均比原计算公式适用范围大.利用数值模型对稳态波和聚焦波演化进行计算,将最大波峰下的水平速度分别与流函数解析解和试验结果进行对比,发现改进后的吻合程度更高,验证了改进公式的有效性.研究表明,改进公式的速度精度有较大幅度提高,该方法可为其他Boussinesq模型的速度场改进提供重要参考.

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。

High-Order Models of Nonlinear and Dispersive Wave in Water of Varying Depth with Arbitrary Sloping Bottom

High-order models with a dissipative term for nonlinear and dispersive wave in water of varying depth with an arbitrary sloping bottom are presented in this article. First, the formal derivations to any high order of mu(= h/lambda, depth to deep-water wave length ratio) and epsilon(= a/h, wave amplitude to depth ratio) for velocity potential, particle velocity vector, pressure and the Boussinesq-type equations for surface elevation eta and horizontal velocity vector (U) over right arrow at any given level in water are given. Then, the exact explicit expressions to the fourth order of mu are derived. Finally, the linear solutions of eta, (U) over right arrow, C (phase-celerity) and C-g (group velocity) for a constant water depth are obtained. Compared with the Airy theory, excellent results can be found even for a water depth as large as the wave legnth. The present high-order models are applicable to nonlinear regular and irregular waves in water of any varying depth (from shallow to deep) and bottom slope (from mild to steep).

具阻尼项的“坏”的Boussinesq型方程的Cauchy问题(英文)

研究一类具阻尼项的"坏"的Boussinesq型方程utt-uxx-2kuxxt-αuxxxx=β(un)xx的Cauchy问题,其中k,α为大于零的实数,β是实数,n≥2是整数。在关于初值的适当假设下,证明了Cauchy问题存在一个整体光滑解u∈C∞((0,T];H∞(R))∩C([0,T];H1(R))∩C1([0,T];H-1(R))对任何T>0。

Boussinesq型方程对三维非线性波传播的数值模拟

该文建立了适宜于相对水深和复杂地形的非线性波传播的三维数学模型 ,模型包括任意水深点流场与波动净压力场的求解。在均匀水深条件下就不同波高、水深及波长的组合对推进波的波面、流速场和波动净压力场进行了全面数值模拟。将计算结果与解析解或物模实验结果进行对比 ,表明模型能较好地模拟各种组合下波浪传播的波动特征以及复杂地形上非线性波的传播。

数值水池的一种有效的数值消波方法

本文通过将波高所满足的方程分解成等效的两个 x和 y方向的一维方程 ,提出了一种在求解 Boussinesq型方程中进行数值消波的有效方法 ,在求解方程时采用了一种精度较高的预报校正线迭代方法。给出的方法用有实验结果的多向不规则波例子进行了验证。数值结果与实验结果相当吻合。本文的方法解决了波浪仿真中的关键技术

非线性波传播的新型数值模拟模型及其实验验证 引入变换速度变量

以一种新型的含变换速度变量的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.对均匀水深水域内波浪传播的数值模拟,说明在引入变换速度后进一步增大了模型的水深适用范围.对潜堤地形上波浪传播的数值模拟说明,在引入变换速度后进一步提高了模型的数值模拟精度.

一类具弱阻尼的奇性扰动Boussinesq型方程的渐近性质

采用位势井方法研究了一类具弱阻尼的奇性扰动Boussinesq型方程的初边值问题,只要初值属于某一个位势井,初始能量是正的但适当小,得到了相应初边值问题整体广义解具有指数衰减性质.

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.

Tests and Applications of An Approach to Absorbing Reflected Waves Towards Incident Boundary

If the upstream boundary conditions are prescribed based on the incident wave only, the time-dependent numerical models cannot effectively simulate the wave field when the physical or spurious reflected waves become significant. This paper describes carefully an approach to specifying the incident wave boundary conditions combined with a set sponge layer to absorb the reflected waves towards the incident boundary. Incorporated into a time-dependent numerical model, whose governing equations are the Boussinesq-type ones, the effectiveness of the approach is studied in detail. The general boundary conditions, describing the down-wave boundary conditions are also generalized to the case of random waves. The numerical model is in detail examined. The test cases include both the normal one-dimensional incident regular or random waves and the two-dimensional oblique incident regular waves. The calculated results show that the present approach is effective on damping the reflected waves towards the incident wave boundary.