A numerical method for the two layer shallow water equations with dry states

A numerical method is proposed for solving the two layer shallow water equations with variable bathymetry in one dimension based on high-resolution f-wave-propagation finite volume methods. The method splits the jump in the fluxes and source terms into waves propagating away from each grid cell interface. It addresses the required determination of the system’s eigenstructure and a scheme for evaluating the flux and source terms. It also handles dry states in the system where the bottom layer depth becomes zero, utilizing existing methods for the single layer solution and handling single layer dry states that can exist independently. Sample results are shown illustrating the method and its handling of dry states including an idealized ocean setting.     

Numerical simulation of wave overtopping of coastal structures using the non-linear shallow water equations

A one-dimensional high-resolution finite volume model capable of simulating storm waves propagating in the coastal surf zone and overtopping a sea wall is presented. The model (AMAZON) is based on solving the non-linear shallow water (NLSW) equations. A modern upwind scheme of the Godunov-type using an HLL approximate Riemann solver is described which captures bore waves in both transcritical and supercritical flows. By employing a finite volume formulation, the method can be implemented on an irregular, structured, boundary-fitted computational mesh. The use of the NLSW equations to model wave overtopping is computationally efficient and practically flexible, though the detailed structure of wave breaking is of course ignored. It is shown that wave overtopping at a vertical wall may also be approximately modelled by representing the wall as a steep bed slope. The AMAZON model solutions have been compared with analytical solutions and laboratory data for wave overtopping at sloping and vertical seawalls and good agreement has been found. The model requires more verification tests for irregular waves before its application as a generic design tool.     

一维全非线性Green-Naghdi水波方程数值模型

建立了求解一维全非线性Green-Naghdi水波方程的中心有限体积/有限差分混合数值格式。采用结构化网格对守恒形式的控制方程进行离散和积分,界面数值通量采用有限体积法计算,剩余项则采用中心有限差分格式求解。其中,采用中心迎风有限体积格式计算控制体界面数值通量,并结合界面变量的线性重构方法,使其在空间上具有四阶精度,通过引入静压重构技术和波浪破碎指标使模型具备处理海岸水-陆动边界及波浪破碎的能力。时间积分则采用具有总时间变差减小(Total Variation Diminishing,TVD)性质的三阶龙格-库塔法进行。应用该模型对孤立波在常水深和斜坡海岸上的传播过程及规则波跨越潜堤传播的实验进行了数值模型研究,数值计算同解析解及实验数据吻合良好。     

Field investigation on the results of non-linear shallow water equations in the Swash Zone

For the study of the cross-shore wave-induced hydrodynamics in the swash zone, a numerical model is developed based on the one-dimensional non-linear shallow water (NSW) equations for prediction of hydrodynamic parameters in the swash zone. In order to evaluate the accuracy of the outputs of the numerical model, the model's predictions in terms of water surface elevations and cross-shore velocities, are compared to field data from full-scale experiments conducted on three sites with different beach slope; mild and steep, several bed particle sizes and under various incident wave conditions. The quantitative and qualitative comparison of the results of the numerical model and the full-scale data reveals that the model can generally predict many aspects of the flow in the surf and swash zone on both types of beach. The accuracy is adequate for application in a sediment transport study. Considering the time-history and probability distribution of water surface elevation, the model is generally more accurate on steep beaches than on the mild beach. The model can adequately simulate the dominant frequency across the beach and saturation of higher frequencies on both mild and steep beaches for various incident wave energy characteristics. With regard to the horizontal (cross-shore) velocity, the sawtooth shape of time-history and negative acceleration of water are well predicted by the model for both mild and steep beaches. Due to the uncertainties in maximum and minimum values of velocity data, clear judgement about the accuracy of the numerical model in this matter was not possible. However, the comparison of the minimum velocities (offshore direction) revealed that the application of friction factors below the range which is suggested by literature best match the data.     

Numerical storm surge model with higher order finite difference method of lines for the coast of Bangladesh

In this study, the method of lines (MOLs) with higher order central difference approximation method coupled with the classical fourth order Runge-Kutta (RK(4,4)) method is used in solving shallow water equations (SWEs) in Cartesian coordinates to foresee water levels associated with a storm accurately along the coast of Bangladesh. In doing so, the partial derivatives of the SWEs with respect to the space variables were discretized with 5-point central difference, as a test case, to obtain a system of ordinary differential equations with time as an independent variable for every spatial grid point, which with initial conditions were solved by the RK(4,4) method. The complex land-sea interface and bottom topographic details were incorporated closely using nested schemes. The coastal and island boundaries were rectangularized through proper stair step representation, and the storing positions of the scalar and momentum variables were specified according to the rules of structured C-grid. A stable tidal regime was made over the model domain considering the effect of the major tidal constituent, M₂ along the southern open boundary of the outermost parent scheme. The Meghna River fresh water discharge was taken into account for the inner most child scheme. To take into account the dynamic interaction of tide and surge, the generated tidal regime was introduced as the initial state of the sea, and the surge was then made to come over it through computer simulation. Numerical experiments were performed with the cyclone April 1991 to simulate water levels due to tide, surge, and their interaction at different stations along the coast of Bangladesh. Our computed results were found to compare reasonable well with the limited observed data obtained from Bangladesh Inland Water Transport Authority (BIWTA) and were found to be better in comparison with the results obtained through the regular finite difference method and the 3-point central difference MOLs coupled with the RK(4,4) method with regard to the root mean square error values.     

基于全和谐型浅水方程的有限体积海啸数值模型开发与应用

数值模拟作为海啸预报的主要研究方法在海啸预警中起着关键作用。本文采用Godunov格式的有限体积方法,使用MUSCL-Hancock格式,并利用HLLC Riemann近似求解器计算单元界面上的流体通量,建立了球坐标系下二阶精度的海啸数值模型。模型所基于的全和谐型浅水方程保证了数值的稳定性,而地形重构方法实现了干湿边界的精准模拟。本文模拟了2015年9月16日智利M_w8.3级地震海啸,通过与智利近岸14个测站和环太平洋20个DART浮标实测数据比较,验证了模型对实际越洋海啸模拟预报的能力。     

Robust source localization in shallow water based on vector optimization

Owing to the multipath effect, the source localization in shallow water has been an area of active interest. However, most methods for source localization in shallow water are sensitive to the assumed model of the underwater environment and have poor robustness against the underwater channel uncertainty, which limit their further application in practical engineering. In this paper, a new method of source localization in shallow water, based on vector optimization concept, is described, which is highly robust against environmental factors affecting the localization, such as the channel depth, the bottom reflection coefficients, and so on. Through constructing the uncertainty set of the source vector errors and extracting the multi-path sound rays from the sea surface and bottom, the proposed method can accurately localize one or more sources in shallow water dominated by multipath propagation. It turns out that the natural formulation of our approach involves minimization of two quadratic functions subject to infinitely many nonconvex quadratic constraints. It shows that this problem (originally intractable) can be reformulated in a convex form as the so-called second-order cone program (SOCP) and solved efficiently by using the well-established interior point method, such as the software tool, SeDuMi. Computer simulations show better performance of the proposed method as compared with existing algorithms and establish a theoretical foundation for the practical engineering application.     

Two-dimensional scour simulations based on coupled model of shallow water equations and sediment transport on unstructured meshes

A two-dimensional scour model based on coupled system of shallow water equations (SWEs) and sediment transport on unstructured mesh is developed. The coupled system of hydrodynamic and morphodynamic equations is solved by finite volume method using Godunov scheme. Roe's approximate Riemann solver is used to calculate the inviscid fluxes. The use of unstructured mesh makes the model applicable to complex domains. However, it is difficult to evaluate the eigenvalues and eigenvectors of the Jacobian matrix in the global coordinate. The method proposed herein to deal with this difficulty is to transform the system into the local coordinate with one of the axes in the same direction as the interface outward normal vector. In the local coordinate system, the Jacobian matrix is simplified and the eigenvalues are analyzed using asymptotic method. Regular expansion breaks down when the flow is near critical. Uniformity of the expansion is achieved by changing the scales. Rotational invariance theorem is used to relate the interfacial fluxes in the global and local coordinate systems. Special treatment of the source term on unstructured grid makes the scheme stable and physically balanced (both mass and momentum). The method proposed in this paper for the eigen-system is very efficient comparing to iterative numerical methods. Results from the test cases show good agreement with the experiments.     

厦门海域浅水三维潮流场动力学模型

基于Casulli的三维浅水模型,改进浅滩处理方法,并入简化的紊流闭合模型,形成完整的海洋动力学基本方程组,改进了紊流闭合模型的求解方法,动力学模拟结果与实测结果符合良好,海域中大量浅滩的干出与淹没的面积和位置与实际情况吻合良好.本模型是厦门海域海洋动力学理论研究中第一个完全的三维斜压潮流场模型,全部程序用FORTRAN语言独立开发和编写.     

辽东湾海域浅水多次波组合压制技术

浅水地震勘探中,由于水层较浅,多次波普遍发育,而且很难压制,如何更好地压制浅水多次波成为了海洋地震资料处理的重要环节。目前来说,依靠单一的多次波压制技术很难将浅水多次波压制彻底,因此本文探索了一种组合压制浅水多次波的思路,首先基于确定性水层多次波压制方法（DWD）将短周期多次波实现压制,然后采用自由表面多次波压制技术（SRME）将近偏移距长周期多次波进行压制,最后再采用高精度Radon变换,将残留的中、远偏移距长周期多次波完成压制。通过辽东湾海域浅水实际资料的多次波压制效果可以看出,组合压制技术实现了对于资料中发育的浅水多次波的良好压制,陷波频率得到较好的恢复,资料品质得到有效改善,信噪比得到有效提高。     

对流扩散方程迎风差分格式的稳定性分析

此文应用Fourier方法和能量不等式研究了常系数和变系数的对流扩散方程在迎风差分格式的情形下数值解的稳定性,所得结果对实际计算有着重要的意义。     

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.     

适用于曲边界的浅水非线性波浪数值模型

实际工程中存在大量的曲边界,因此在曲边界上的计算准确性可以考察出一个数值模型的实用价值。利用Beji的改进型Boussinesq方程建立了一个有限元方法的数值波浪模型。造波方面采用Fenton提出的非线性规则波浪解;在墙边界处,以求解法线方向和切线方向的速度和导数代替求解x、y方向的速度和导数,从而使边界条件直接适用、严格满足,保证了对曲边界计算的准确性。"重开始广义极小残量法"的使用保证了求解方程组的效率和精度,使造波和边界处理方法的有效性和准确性得到了合理地诠释。通过与试验数据、他人数值结果、解析解的比对,显示出该模型计算稳定、结果准确,真正体现出了有限元方法对曲边界适用的优势。     

浅水港口潮汐预报准调和方法的改进

文中提出了浅水港口潮汐预报准调和方法的两个改进方案.一个是在方国洪等(1981)《浅水港口潮汐预报的一个方法》基础上,增加14周/日、16周/日以及与前期潮汐状况有关的浅水准调和项共6项(方案Ⅰ);另一个在是王骥(2001)提出的方案基础上改进的方案,该方案与方案Ⅰ相比,增加了5个准调和项(方案Ⅱ).这两个方案选用相同的57个调和分潮,但分潮的组合及派生出的准调和项有所不同.方案Ⅰ用40项准调和项表示浅水效应,方案Ⅱ则包含长周期、全日、半日周期的准调和项18项.经过大量实测水位资料的分析和预报检验,表明改进后两个方案对浅水港口潮汐的预报精度均较改进前的准调和方法和传统调和法有明显提高,对第一类浅水港(以吴淞港为例),文中所列的8项预报指标都有显著提高,特别是低潮时均方差由26.8min降至15.1min;对第二类浅水港(以成山角为例),高潮时均方差显著减小,由39.6min减至26.7min.     

Practical modified scheme of linear shallow-water equations for distant propagation of tsunamis

A simple but practical numerical model describing a distant propagation of tsunamis is newly proposed by introducing an additional term to the existing modified scheme. The numerical dispersion of the proposed model is manipulated to replace the physical dispersion of the linear Boussinesq equations without any limitation. The new model developed in this study is applied to propagation of a Gaussian hump over a constant water depth and the predicted free surface displacements are compared with available analytical solutions. A very reasonable agreement is observed.     

Allowable span length of submarine pipeline in shallow water

Because of the complex geological conditions of the seabed, submarine pipelines buried beneath the ocean floor become suspended over the seabed under the long-term scour of waves eroding the surrounding sediment. Further, most oil fields were built in offshore areas while the country was developing. This gives the waves seen in shallow water obvious nonlinear features, and the abnormal characteristics of these waves must be considered when calculating their hydrodynamic forces. Particularly under such conditions, these suspended spans of submarine pipelines are prone to damage caused by the action of the external environment load. Such damages and eventual failures may result not only in great property losses but also pollution of the marine environment. The span length of these areas is a key predictive factor in pipeline damages. Therefore, determining the allowable span length for these submarine pipelines will allow future projects to avoid or prevent damage from excessive suspended span lengths. Expressions of the hydrodynamic loads placed on suspended spans of pipeline were developed in this work based on the first-order approximate cnoidal wave theory and Morison equation. The formula for the allowable free span length was derived for the common forms of free spanning submarine pipeline based on the point where maximum bending stresses remain less than the material’s allowable stress. Finally, the allowable free span length of real-world pipelines was calculated for a subsea pipeline project in Bohai Bay. This research shows that, with consideration for the complicated marine environment, existing suspended spans are within allowable length limitations. However, continuing to limit the length of these submarine pipeline spans in the Nanpu oil field will require ongoing attention.     

Euler–Lagrange equations for the spectral element shallow water system

We present the derivation of the discrete Euler–Lagrange equations for an inverse spectral element ocean model based on the shallow water equations. We show that the discrete Euler–Lagrange equations can be obtained from the continuous Euler–Lagrange equations by using a correct combination of the weak and the strong forms of derivatives in the Galerkin integrals, and by changing the order with which elemental assembly and mass averaging are applied in the forward and in the adjoint systems. Our derivation can be extended to obtain an adjoint for any Galerkin finite element and spectral element system.We begin the derivations using a linear wave equation in one dimension. We then apply our technique to a two-dimensional shallow water ocean model and test it on a classic double-gyre problem. The spectral element forward and adjoint ocean models can be used in a variety of inverse applications, ranging from traditional data assimilation and parameter estimation, to the less traditional model sensitivity and stability analyses, and ensemble prediction. Here the Euler–Lagrange equations are solved by an indirect representer algorithm.     

解浅水方程的集中质量有限元方法

本文给出三角形单元集中质量有限元解浅水波方程的方法以及该方法在IBM-PC微机上的应用实例并取得较好结果。     

九州-帕劳海脊两侧深海盆地浅部地层结构特征与分析

由于欧亚板块、澳大利亚板块与太平洋板块运动的作用,菲律宾海盆形成了全球最为复杂的海底地形地貌。以九州-帕劳海脊为界,西菲律宾海盆与帕里西维拉海盆呈现不同的构造走向与地形特征,海脊两侧发育丰富的深海地质现象与地貌特征。参量阵浅地层剖面仪因其差频窄波束的发射特点,能够获得深海海底浅地层的高分辨率剖面。本文利用参量阵浅地层剖面仪在九州-帕劳海脊获取的高分辨率浅地层剖面,分析了横跨海脊东西走向的海底滑坡发育特点、丘状起伏沉积差异、浅层气以及海底流体运移等地质特征,并结合构造与地形地貌特点对上述特征的成因机制进行了探讨,在垂向小时间尺度上完成了对研究区浅部地层结构的特征研究,为该区域的沉积学和地质学研究提供了思路。     

乐清湾港区“浅水深用”开发方案的可行性分析

从建港条件、港池正常回淤和港池骤淤等方面,综合分析了乐清湾港区采取"浅水深用"开发方案的技术可行性。研究表明,乐清湾港区采取"浅水深用"开发建设是可行的,乐清湾港池骤淤可能性不大。对乐清湾港区"浅水深用"开发方案的几个关键技术问题进行了探讨:一是码头前沿线的确定,这不仅直接关系到码头投资和营运成本,还对今后乐清湾其它工程产生较大的影响。运用全寿命周期成本理念,通过对建设投资、疏浚方量、维护成本、营运成本等方面的比较与分析以及对码头前沿线位置进行模型试验研究和多方案比选后认为,码头前沿线采用400 m栈桥方案是最为经济合理的。二是港池大开挖技术方式合理确定。为减少港池回淤,开挖方式应采取非坑槽开挖,港池与深槽连通,南、北码头区港池疏浚连通。三是港池疏浚土资源综合利用。在港池开挖和维护中,要贯彻循环经济理念,加强对港池疏浚废土的综合利用。结合港区围垦陆域工程的实施、港池疏浚吹填和软基处理,对港池疏浚泥进行资源化利用,既解决沿海大量废弃疏浚泥的处理问题,又可避免疏浚泥对环境的污染。四是港池维护措施的制定。乐清湾港区港池开挖后,港池的骤淤量虽不大,但港池开挖后采取维护措施是必要的,应考虑经济合理的备淤深度,以减少开挖方量和维护量。五是固滩护滩措施的制定。考虑到双屿港开闸泄水对港区回淤的影响,对双屿港南侧滩地采取必要的固滩护滩措施,遏制或减少双屿港开闸泄水对滩地泥沙的掀动作用。六是应结合乐清湾港池疏浚与吹填工程,加强现场监测,进一步验证前期科研成果的准确性和可靠性。监测资料分析表明,前期科研成果的准确性和可靠性是有保证的。