【作者】 邱兆山；

【导师】 史宏达；

【作者基本信息】 中国海洋大学 ， 港口、海岸及近海工程， 2003， 硕士

【摘要】 目前广泛应用于潮流计算的二维潮流方程属浅水方程。齐次浅水方程在数学上为拟线性双曲型偏微分方程组。浅水方程在计算机技术发展的推动下已发展了诸多有效的数值计算方法，如有限差分法(FDM)、特征法(MOC)、有限单元法(FEM)、有限体积法(FVM)等。其中有限体积法在70年代开始应用于空气动力学并取得了巨大成功。有学者很早就发现齐次浅水方程组与正压气流欧拉方程组在数学上的一致性。随着空气动力学的发展完善，将空气动力学的理论成果和数值算法应用于计算水动力学已进入实践阶段。本文对有限体积法的研究即是本着这个思路。有限体积法用于积分形式的守恒律方程。它能结合有限差分法与有限单元法的优点，如：计算效率高；可应用于无结构网格(有限元网格)；具有激波捕捉性(shock-capturing property)，可以有效地处理间断问题。 本文从探讨一维和二维浅水方程的特征理论和性质出发，详细阐述了应用于浅水计算的有限体积法一阶及二阶格式(MUSCL途径)的原理及应用方法。应用有限体积法求解浅水方程的关键问题为求解网元界面处的一维黎曼方程，即确定网元边界上的数值通量。精确求解黎曼方程非常繁杂且不必要，所以用有足够精度且简便的近似黎曼解来代替精确的黎曼解。求解近似黎曼解的方法主要有：①Osher格式；②通量向量分裂格式(FVS)；③通量差分裂格式(FDS)等。 本文还以自动生成(Delaunay原理)及编号的三角形网格为例，详细阐述了无结构网格的应用方法。本文采用规则矩形网格编制了模拟一维及二维溃坝问题的测试程序，以检验文中所述方法的正确性和处理间断问题的有效性。最后本文采用无结构三角形网格并采用有限体积法二阶格式编制了模拟大亚湾潮流运动的模拟程序。该模型的模拟结果表明了有限体积法用于潮流计算的有效性，其实现方法也充分体现了有限体积法的上述优势。更多还原

【Abstract】 The widely used 2d equations of tidal flow simulation belong to the shallow water type. When turbulent and dispersive effects are neglected, the shallow water equations become a quasi-linear system of first order hyperbolic partial differential equations. As the development of the computer science, a lot of methods have been used to solve the shallow water equations numerically. The most significant four methods of them are: Finite Difference Method (FDM), Method of Characteristics (MOC), Finite Element Method (FEM) and Finite Volume Method (FVM). In particular, FVM has been extensively studied in recent years, mainly because it appears in some important fields as aeronautical and aerospace engineering. The shallow water equations have the same structure as the compressible Euler Equations, so many theories and techniques of FVM developed in air dynamics have been used to solve the shallow water equations. FVM deals with the conservation form of the equations. It has many advantages over FDS and FEM. FVM combines the simplicity of FDM with the geometric flexibility of FEM. FVM can be applied using an unstructured grid system as FEM. FVM can also capture discontinuities or shocks properly.hi the first part of this paper, Id and 2d shallow water equations’ property and characteristic theory are shown. Then, on the basis of this, the principal and the techniques of using of FVM are discussed. A modern second order method of Monotonic Upstream Schemes for Conservation Laws (MUSCL) approach has also been shown. The key problem of using FVM is to estimate the normal flux through each side of a cell. There are several algorithms to estimate this flux, such as Osher method, Flux Vector Splitting (FVS) method and Flux Difference Splitting (FDS).In this paper, two benchmarks of dam break flow simulation have been designed which use rectangle mesh. The purpose of these benchmarks is to test the correctness and the validity of shock capturing of the model. In the last part, a Dayawan bay tidal flow simulation model, which use triangular mesh and second order scheme has been planed. The results showed that the model is valid to simulate the tidal flow simulation and has significant advantages.更多还原