【作者】 刘刚；

【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， 2009， 硕士

【摘要】 现代浅水流动数值模拟的一个主要方向是利用齐次浅水方程和Euler方程在数学形式上的相似性,借用计算气体动力学的高性能算法,并结合浅水流动的特殊性建立适合与模拟溃坝、涌潮等有间断或弱间断纯在的流动数值模拟。本文在前人研究的基础上,采用高性能格式有限体积方法,建立了一套基于非结构化网格(三角形)的二维浅水流动数值模拟。在模型的空间离散过程中,本文应用迎风有限体积方法,建立了二维带源项浅水的高精度、高分辨率非结构化网格模型,并成功应用于复杂地形下间断流和溃坝波的数值模拟。采用非结构化网格技术,以Roe格式的近似Riemann解为基础,建立了二维带源项浅水方程的通量平衡Godunov求解格式。提出了特征分解和迎风处理源项的方法,平衡了非平底时界面通量,保证了非平底坡浅水方程计算的和谐性、增加了格式的稳定性。方程中通量梯度项与源项的平衡,使模型可以适合复杂地形下浅水流动问题和间断问题的求解,并最终建立了和谐的Roe-Upwind格式的有限体积模型。动边界是浅水模拟中一个关键性难题。本文提出了一种处理带有干湿界面的非恒定浅水流动的无质量误差方法。采用适当的干湿界面处理技术以满足静水问题,同时对有干湿界面的复杂地形的非恒定流达到无质量误差。将本文建立的非结构网格Roe迎风格式的有限体积模型对间断水流中一些经典的或有解析解的算例,比如Stoker问题、二维局部溃坝问题、倾斜水跃问题、二维非平底溃坝问题以及有激波混合流问题等,进行数值模拟。所有计算结果符合其物理意义,与解析解吻合较好,在间断附近陡峭,不含非物理的伪振荡,验证了本文模型的正确性与适用性。模型也成功应用于实际水流、溃坝的数值模拟中,验证了该格式具有相容性好、物理意义更为清晰,编程易于实现等优点。所有研究成果表明,本文建立的数学模型具有较好的水流模拟性能,具有广泛的应用前景。更多还原

【Abstract】 One of the primary trends of the modern shallow water flows simulation is to make use of the mathematic similarity between the homogeneous shallow water equation and Euler equation,as well the high performance algorithm of computational aerodynamics such as Osher,Roe,FVS, HLL,HLLC,ENO and WENO to simulate flow that contains discontinuity or weak discontinuity such as dam-break and bore.The quotations will be adjusted in accordance with the particularity of shallow water flows.Based on the research of other scholars,this paper uses the finite volume method with high performance schemes to two-dimensional shallow water equation and builds a mathematical model that can stimulate two-dimensional shallow water flows on unstructured grids(triangular).In the space discreteness of the model,this paper applies the finite volume framework,the two-dimensional unstructured grid high-precision,high-resolution model is presented with nonlinear shallow water equations with source terms,and successfully applied to discontinuous flow and the dam-break flow of the numerical simulation with complicated geometry and topography.Based on the unstructured grid,the Roe’s approximate Riemann solver is used for the computation of inviscid numerical flux functions,well-balanced between the Godunov scheme and the two-dimensional model is presented for the shallow water equation with source terms.The source terms are decomposed in the characteristic directions,which keep the flux balance at the interface and protected the scheme harmonious.Balancing flux gradients and source terms in equations makes the model can solve the shallow water flows and discontinuous flows with variable depth.So the harmonious Roe-Upwind finite volume model is constructed finally.Moving boundary shallow water simulation is a key problem.A wetting-drying condition for unsteady shallow water flow in two-dimensional leading to zero numerical error in mass conservation is presented in this work.It is shown that this numerical technique reproduces exactly steady state of still water and enables to achieve zero numerical errors in unsteady flow over configurations with strong variations on bed slope.Numerical results are shown which demonstrate the effectiveness of the wetting-drying condition in flood propagation and dam break flows over real complex geometries and bottom slope variation. The 2D,Roe-Upwind finite volume model is applied to numerical simulation of some classic discontinuous flow examples to prove the validity and applicability.The examples include Stoker problem,2D dam-break problem,oblique hydraulic jump problem and mixed flow problem.All the computed results are good agreement the analytic solutions,and present steep jump without non-physical oscillations near the shock.Model also successfully applied to the actual water flow or dam-break flow,the better to verify the results.All results show that the model is efficient,accurate,stable,and can be used widely.更多还原