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.     

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。     

Finite-amplitude analysis of some Boussinesq-type equations

Recent progress in formulating Boussinesq-type equations includes improved features of linear dispersion and higher-order nonlinearity. Nonlinear characteristics of these equations have been previously analysed on the assumption of weak nonlinearity, being therefore limited to moderate wave height. The present work addresses this aspect for an important class of wave problems, namely, regular waves of permanent form on a constant depth. Using a numerical procedure which is valid up to the maximum wave height, permanent-form waves admitted by three sets of advanced Boussinesq-type equations are analysed. Further, the characteristics of each set of the Boussinesq-type equations are discussed in the light of those from the potential theory satisfying the exact free-surface conditions. Phase velocity, amplitude dispersion, harmonic amplitudes (namely, second and third) and skewness of surface profile are shown over a two-parameter space of frequency and wave height.     

A GPU accelerated Boussinesq-type model for coastal waves

This study presents an efficient Boussinesq-type wave model accelerated by a single Graphics Processing Unit (GPU). The model uses the hybrid finite volume and finite difference method to solve weakly dispersive and nonlinear Boussinesq equations in the horizontal plane, enabling the model to have the shock-capturing ability to deal with breaking waves and moving shoreline properly. The code is written in CUDA C. To achieve better performance, the model uses cyclic reduction technique to solve massive tridiagonal linear systems and overlapped tiling/shared memory to reduce global memory access and enhance data reuse. Four numerical tests are conducted to validate the GPU implementation. The performance of the GPU model is evaluated by running a series of numerical simulations on two GPU platforms with different hardware configurations. Compared with the CPU version, the maximum speedup ratios for single-precision and double-precision calculations are 55.56 and 32.57, respectively.     

Experimental and numerical investigation of viscous effects on solitary wave propagation in a wave tank

Two-dimensional depth-averaged Boussinesq-type equations were presented with the consideration of slowly varying bathymetry and effects of bottom viscous boundary layer. These Boussinesq-type equations were written in terms of the horizontal velocity components evaluated at an arbitrary elevation in the water depth and the free surface displacement. The leading order effects of the bottom boundary layer were represented by a convolution integral in the depth-integrated continuity equation. To test the validity of the theory, a set of laboratory experiments was performed to measure the viscous damping and shoaling of a solitary wave propagating in a wave tank. The time histories of the free surface profiles were measured at several locations along the centerline of the flume. To compare these laboratory data with theoretical results, the two-dimensional Boussinesq-type equations were integrated across the wave tank, resulting in a set of one-dimensional equations, while the side-wall boundary layers were properly considered. The agreement between the experimental data and numerical results was very good. The bottom shear stress formula was also given and its impact on the sediment transport rate was discussed.     

Linear shoaling in Boussinesq-type wave propagation models

This work focuses on linear shoaling performance of low order Boussinesq-type equations. It is shown that the linear shoaling errors can be important in well known equations in the literature. New sets of coefficients are presented for three well known sets of equations. The sets are found so as to minimize a global linear error that includes celerity and shoaling errors. Finally, a new set of enhanced bilayer low order equations is presented, with much improved linear behavior (errors in wave celerity and wave amplitude below 1% up to kh = 20). For completeness, the equations are written in their fully nonlinear version, and the nonlinear coefficients are also given.     

A weakly dispersive edge wave model

We derive a general linear, weakly dispersive, Boussinesq-type equation that can be used to study edge waves on beaches with slow cross-shore variation of the depth and the alongshore current. The equation is more accurate than the non-dispersive shallow water equations and simpler than the fully dispersive elliptic mild slope equation (especially for a non-zero alongshore current). The improved performance of the new Boussinesq-type model is demonstrated using analytic solutions for edge waves on a plane beach with zero alongshore current.     

Numerical modeling of nonlinear water waves over heterogeneous porous beds

The transformation of nonlinear water waves over porous beds is studied by applying a numerical model based on Chen's [2006. Fully nonlinear Boussinesq-type equations for waves and currents over porous beds. Journal of Engineering Mechanics, 132:2, 220–230] Boussinesq-type equations for highly nonlinear waves on permeable beds. The numerical model uses a high-order time-marching solution and fourth-order finite-difference schemes for discretization of first-order spatial derivatives to obtain a computational accuracy consistent with the model equations. By forcing the wave celerity and spatial porous-damping rate of the linearized model to match the exact linear theory for horizontal porous bed over a prescribed range of relative depths, the values of the model parameters are optimally determined. Numerical simulations of the damped wave propagation over finite-thickness porous layer demonstrate the accuracy of both the numerical model and governing equations, which have been shown by prior theoretical analyses to be accurate for both nominal and thick porous layers. These simulations also elucidate on the significance of the higher-order porous-damping terms and the influence of the hydraulic parameters. Application of the model to the simulation of the wave field around a laboratory-scale submerged porous mound provides a measure of its capability, as well as useful insight into the scaling of the porous-resistance coefficients. For application to heterogeneous porous beds, the assumption of weak spatial variation of the porous resistance is examined using truncated forms of the governing equations. The results indicate that the complete set of Boussinesq-type equations is applicable to porous beds of nonhomogeneous makeup.     

A numerical study of swash flows generated by bores

The dynamic processes of bore propagation over a uniform slope are studied numerically using a 2-D Reynolds Averaged Navier–Stokes (RANS) solver, coupled to a non-linear k − ε turbulence closure and a volume of fluid (VOF) method. The dam-break mechanism is used to generate bores in a constant depth region. Present numerical results for the ensemble-averaged flow field are compared with existing experimental data as well as theoretical and numerical results based on non-linear shallow water (NSW) equations. Reasonable agreement between the present numerical solutions and experimental data is observed. Using the numerical results, small-scale bore behaviors and flow features, such as the bore collapse process near the still-water shoreline, the ‘mini-collapse’ during the runup phase and the ‘back-wash bore’ in the down-rush phase, are described. In the case of a strong bore, the evolution of the averaged turbulence kinetic energy (TKE) over the swash zone consists of two phases: in the region near the still-water shoreline, the production and the dissipation of TKE are roughly in balance; in the region farther landwards of the still-water shoreline, the TKE decay rate is very close to that of homogeneous grid turbulence. On the other hand, in the case of a weak bore, the bore collapse generated turbulence is confined near the bottom boundary layer and the TKE decays at a much slower rate.     

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.     

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.     

A Numerical Model of Nonlinear Wave Propagation in Curvilinear Coordinates

For the simulation of the nonlinear wave propagation in coastal areas with complex boundaries,a numerical model is developed in curvilinear coordinates. In the model,the Boussinesq-type equations including the dissipation terms are employed as the governing equations. In the present model,the dependent variables of the transformed equations are the free surface elevation and the utility velocity variables,instead of the usual primitive velocity variables. The introduction of utility velocity variables which...     

Moving shoreline boundary condition for nearshore models

The paper develops and analyzes two fully nonlinear boundary conditions that incorporate the motion of the shoreline in nonlinear time domain nearshore models. A moving shoreline essentially means the computational domain is changing with the solution of the flow. The problem is solved in two steps. The first is to establish an equation that determines the motion of the shoreline based on the local momentum balance. The second is to develop and implement into a shoreline model the capability of accommodating a changing computational domain. The two models represent two different ways of addressing this step: one is to track the position of the shoreline in a fixed grid by establishing a special shoreline point which generally is not a fixed grid point. The second is by a coordinate transformation that maps the changing domain onto a fixed domain and solves the basic equations in the mapped domain. The two shoreline conditions are tested against three known solution for nonlinear shoreline motion. Two are the 1-D solutions to the nonlinear shallow water (NSW) equations by Carrier and Greenspan [J. Fluid Mech. 4 (1958) 97], one representing the response to a transient change in the offshore water level, the other the motion due to a periodic standing wave, both on slopes steep enough to allow full reflection. The third is the 2-D horizontal (2DH) computational solution by Zelt [Coast. Eng. 15 (1991) 205] for the run-up of a solitary wave on a cusped beach. In all cases, both models are shown to behave well and give high accuracy results for suitably chosen grid and time spacings.     

Hydraulic Modeling of A Curtain-Walled Dissipater by the Coupling of RANS and Boussinesq Equations

A hybrid numerical method for the hydraulic modeling of a curtain-walled dissipater of reflected waves from breakwa-ters is presented. In this method, a zonal approach that combines a nonlinear weakly dispersive wave (Boussinesq-type equation) method and a Reynolds-Averaged Navier-Stokes (RANS) method is used. The Boussinesq-type equation is solved in the far field to describe wave transformation in shallow water. The RANS method is used in the near field to re-solve the turbulent boundary layer and vortex flows around the structure. Suitable matching conditions are enforced at the interface between the viscous and the Boussinesq region. The Coupled RANS and Boussinesq method successfully resolves the vortex characteristics of flow in the vicinity of the structure, while unexpected phenomena like wave re-reflection are effectively controlled by lengthening the Boussinesq region. Extensive results on hydraulic performance of a curtain-walled dissipater and the mechanism of dissipation of reflected waves     

Discussion of ‘Moving shoreline boundary condition for nearshore models’ by Prasad and Svendsen

The equations for shoreline motions are derived and the difference between the derived equations and those derived by Prasad and Svendsen [Prasad and Svendsen, 2003, Moving shoreline boundary condition for nearshore models Coastal Engineering, 49 :239-261.] is discussed. An alternative approach to treat the moving shoreline boundary condition is proposed based on the new derivation.     

采掘坑位置对珊瑚礁海岸波浪传播变形影响试验

为研究珊瑚礁坪上采掘坑位置变化对珊瑚礁海岸波浪传播变形的影响, 本文通过物理模型试验测试了采掘坑在不同位置和无坑情况下一系列不规则波工况的波浪特征。结果表明, 随着采掘坑位置朝岸线附近移动直至无坑时, 岸线附近的短波波高逐渐减小; 采掘坑的存在减弱了岸线附近的低频长波波高, 当采掘坑位于岸线附近时, 长波波高还受到局部水深增加的影响而进一步减弱。采掘坑从礁缘移动至岸线附近直到无坑时, 岸线附近的增水逐渐增大, 这种趋势在礁坪水深较大时更为明显。通过相干函数分析, 证明了礁坪上低频长波是由于短波群破碎点的移动而产生, 采掘坑位置的变化对低频长波的产生无明显影响; 通过传递函数分析, 验证了礁坪上的低频长波存在一阶共振放大效应, 采掘坑的存在减弱了这种放大效应, 当坑位于礁坪中间和岸线附近时, 这种减弱效应更为显著。     

Application of a Boussinesq model for the computation of breaking waves: Part 2: Wave-induced setdown and setup on a submerged coral reef

Based on a set of Boussinesq-type equations with improved linear frequency dispersion characteristics in deeper water, the present paper incorporates the simplified effect of spilling wave breaking into the equations. The analysis is restricted to a single horizontal dimension but the method can be extended to include the second horizontal dimension. Inside the surf zone the vertical variation of the horizontal velocity profile is assumed to be composed of an (initially unknown) organised velocity component below the roller and a surface roller travelling with the wave celerity. This leads to a new set of equations which is capable of simulating the transformation of waves before, during and after wave breaking. The model is calibrated and verified by comparison with several wave flume measurements. The results show that the model produces sound physical results.     

堆积海岸平衡岸线轮廓的解析

对堆积海岸平衡岸线轮廓的三个表达式进行了分析和对比,其中平衡岸线轮廓表达式在一定程度上包含了潮流作用对岸线塑造的影响,适用于对凹岸和沙质海岸岸线轮廓的分析,同时也适用于对凸岸和淤泥质海岸岸线轮廓的分析,因而较基本平衡岸线形态方程和B.A.保保夫堆积岸弧方程适用范围更广。利用平衡岸线轮廓表达式,对舟山群岛金塘山和册子山附近水道的两段岸线轮廓进行了分析,结果表明:两岸段均为堆积岸弧,且理论弧长与实测弧长、理论半径与实测半径的值接近,说明理论岸线与实际岸线形态吻合得较好,同时,金塘山西部岸段较册子山西部岸段岸线形态更趋稳定,但两岸线均未达到稳定状态,目前仍处于演变过程中。     

Generation of waves in Boussinesq models using a source function method

A method for generating waves in Boussinesq-type wave models is described. The method employs a source term added to the governing equations, either in the form of a mass source in the continuity equation or an applied pressure forcing in the momentum equations. Assuming linearity, we derive a transfer function which relates source amplitude to surface wave characteristics. We then test the model for generation of desired incident waves, including regular and random waves, for both one and two dimensions. We also compare some model results with analytical solution and available experiment data.