【作者】 柏禄海；

【导师】 金生；

【作者基本信息】 大连理工大学 ， 水力学及河流动力学， 2013， 博士

【摘要】 本文以近似Riemann解为基础,建立了一、二维带源项浅水方程的高精度、高分辨率守恒型模型。应用特征分解和迎风处理源项的方法,保证守恒型计算格式的“和谐性”。通过改进的底坡计算方法及守恒型通量限制方式,较好地解决了干底Riemann解处理动边界和总质量守恒等问题,确保了计算基本无质量误差。建立了耦合的二维守恒型浅水水质模型,采用修正浓度的方法,使浓度计算结果合理,保证水体和污染物浓度同时保持数值质量守恒。本文主要工作如下：1.以Roe格式的近似Riemann解为基础,应用通量限制法和MUSCL型坡度限制法分别建立了求解一维非平底浅水流动方程的高分辨率Godunov格式,同时采用改进的两步Runge-Kutta方法获得时间二阶精度。为了使方程左边的对流项与方程右边的底坡源项始终保持“和谐”,对源项采用迎风特征分解,以满足守恒性。讨论了使用不同通量限制器时,对流量和水位计算结果的影响。2.逐点插入法可以对任意给定复杂边界的二维区域按照需求进行加密网格,本文设计了一种基于Bowyer-Watson逐点插入算法的Delaunay网格生成方法,为二维浅水方程计算提供了有力的保证。运用不需添加辅助点的边界恢复方法,在初始网格生成时就满足Delaunay网格准则。在加点的次序和选择加点位置上使用了一种比较合理的机制,使得生成的加密网格比较规则。在加密点的插入问题上,通过三角形网络的拓扑关系,利用两点是否位于一条直线的异侧和拓扑关系来解决这一问题,提高了新点插入的效率,使生成加密网格的时间大大缩短。3.提出了一种处理带有干湿界面的基本无质量误差的高分辨率守恒型数值格式。采用非结构化网格的有限体积法,对底坡源项采用特征分解保证“和谐性”,摩擦力源项采用半隐式格式增加格式的稳定性。通过改进底坡的计算方法及守恒型通量限制方式,解决了计算过程中干湿界面(如露滩问题)处理不当引起的非静水解、出现负水深单元、质量不守恒等诸多问题。在此基础上,构造了一种基于MUSCL格式的PLCD (Project Limited Central Difference)方法,通过对底坡源项进行相应的数值修正以满足淹没情况下的静水问题,保证高分辨率格式的“和谐性”,使格式的时空均达到二阶精度。4.基于本文构造的MUSCL格式的新型PLCD方法,建立了耦合的高分辨率二维守恒型浅水水质模型。采用修正浓度的方法,使浓度计算结果合理且满足守恒性。考虑到对流扩散方程的稳定性由对流和扩散共同决定、时间步长由Peclet数和CFL数共同决定等问题,讨论了不同扩散系数影响下的显式和隐式求解扩散项技术的差异。由于隐式计算扩散项方法为无条件稳定,时间步长完全由对流项决定,能够有效避免时间步长过小的问题,同时具有较小的耗散性。数值实验表明该模型计算的浓度解具有较好的稳定性和光滑性,保证水体和污染物浓度同时保持数值质量守恒,能够较好地模拟水流水质问题。5.进一步讨论了MUSCL格式的几种梯度算子(LCD, Durlofsky, PLCD, MLG, MLG-Wierse)在水流水质问题数值计算中的性能,并且通过一些经典算例将一阶及其它二阶格式进行了比较分析。数值实验表明PLCD格式、MLG格式和MLG-Wierse格式三种格式计算结果最佳,且新构造的PLCD格式在计算量少于MLG和MLG-Wierse两种格式,可以作为一种高精度、高分辨率的格式应用于水流水质问题的实际模拟。更多还原

【Abstract】 Based on the finite volume method and Roe’s approximate Riemann solver, the One-dimensional and Two-dimensional conservative high-resolution shallow water models with source terms are developed in this study.The discretization of the bed slope source terms is done following an upwind approach to protect the scheme harmonious. It is shown that the numerical technique of improving bed slope and limited flux style can exactly reproduce steady state of still water and enable the model to achieve zero numerical errors. A fully conservative form applied to a coupled system of two-dimensional water flow and solute motion is presented. This model ensures a global conservation and positive values of both water level and solute concentration. The main work of this article is as following:1. A technique has been investigated for extending Roe’s finite volume method to second-order spatial and temporal accurate MUSCL-type slope-limiting approach in order to simulate shallow water flow over uneven beds. The discretization of the bed slope source terms is done following an upwind approach to ensure convective term from left side of the equation and bottom slope source term from right side of the equation to preserve the "humanity" of conservative numerical scheme. The objective of this study is to compare the performances corresponding to different variables of reconstruction to determine whether there exists all optimal approach.2. A Delaunay mesh generation method based on Bowyer-Watson method is designed for a complex region. Based on the initial mesh, a edge recovery algorithm by no adding Steiner points is presented. A new technique considering the order and the location of insertion points is established to generate high quality mesh. Many methods are adopted to improve the quality and efficiency of the mesh.3. A wetting/drying condition (WDC) for unsteady shallow water flow leading to zero mass error is presented. The WDC has been incorporated into a cell-centred finite volume method based on Roe’s approximate Riemann solver on unstructured grids. The discretization of the bed slope source terms was done following all upwind approach and the semi-implicit treatment was used for the friction source terms. It is shown that the numerical technique of improving bed slope and limited flux style can exactly reproduce steady state of still water and enable the model to achieve zero numerical errors in unsteady flow over configurations with strong variations on bed slope. A new PLCD method based on two-dimensional MUSCL-type finite volume schemes is developed. Numerical results are shown which demonstrate the effectiveness of the WDC in flood propagation and dam break flows over real complex geometries and bottom slope variation.4. A fully conservative form applied to a coupled system of two-dimensional water flow and solute motion is presented. This model corrects solute concentration to keep the results reasonable. This paper discusses the difference of diffusion terms between explicit and implicit discretization. The centered discretization of the diffusion terms is in an implicit way in this model. Numerical experiments show that this model has good stability and smoothness and ensures conservation of the quality of water and solute concentration. Therefore, it could be used for the simulation of solute transport.5. A PLCD (Project Limited Central Difference) technique has been investigated based on MUSCL-type slope-limiting approach. The performances corresponding to different variables of reconstruction (LCD, Durlofsky, PLCD, MLG and MLG-Wierse) are compared. Numerical results indicate that PLCD, MLG and MLG-Wierse are efficient and the new PLCD computation expense of the new constructed PLCD scheme is much less than MLG and MLG-Wierse.更多还原