【作者】 赵张益；

【导师】 李世森； 张庆河；

【作者基本信息】 天津大学 ， 港口、海岸及近海工程， 2010， 硕士

【摘要】 浅水方程在海洋、河流、气候模拟、海洋生物学和环境等众多领域有着广泛的应用,例如风暴潮、海啸和洪水的预测,泥沙和污染物的输运等。本文在一维浅水方程组的基础上建立了二阶Runge-Kutta司断有限元数值模型,并成功应用于复杂地形条件下间断浅水流动的数值模拟。本文的主要内容和结论如下：(1)从守恒形式的浅水方程组出发,对求解区域进行单元剖分,并在单元上对方程进行积分。在模型的空间离散过程中,跨单元边界处采用HLL和HLLC的近似Riemann解算子形式的数值流向量,将非线性方程转化为线性方程；在时间离散过程中,采用具有TVD性质的三阶Runge-Kutta时间离散方法。(2)模型建立过程中,为了抑制间断处的非物理伪振荡,采用了间断检测器结合斜率限制器或稳定算子的方法。考虑了对底坡源项的离散处理,使模型能够处理复杂地形条件下包含强间断的浅水流动问题,并建立了一种均衡格式的Runge-Kutta间断有限元数值模型。(3)将本文建立的Runge-Kutta间断有限元数值模型对浅水流动中的一些经典或存在解析解的算例,例如理想化溃坝问题、稳定水体微小扰动问题、跨临界流问题、水跃、潮波以及涌潮等问题,进行了数值模拟。而且在模拟过程中考虑了非棱柱体渠道、不连续底坡等实际情况。所有计算结果与解析解吻合较好,在间断处不含非物理的伪振荡,从而验证了模型的正确性与适用性。更多还原

【Abstract】 The shallow water equations have a wide range of applications in the ocean, river, climate modeling, marine biology and environmental engineering.Related issues include storm surges, tsunamis and floods prediction, sediment and contaminant transport, etc.A second-order Runge-Kutta discontinuous Galerkin finite element model is built based on one-dimensional shallow water equations, and is applied to numerical simulation of discontinuous shallow flows on irregular bottom topography successfully. The main contents and conclusions are as follows:(1) The model is based on conservation forms of the shallow water equations. The computational domain is partitioned into a set of elements, and then the equations are integrated over the element. For the space discretization in the model, the HLL or HLLC approximate Riemann solver as the numerical flux at interfaces of elements is employed. For the time discretization, a third-order Runge-Kutta scheme which is TVD, is applied.(2) In the model, a slope limiter or stabilization operator only around discontinuities using discontinuity detector is used in order to alleviate spurious oscillations near discontinuities. A discretization of the source terms is applied, so the model can solve the discontinuous shallow water flows with complex bed topography. So a well-balanced discontinuous Galerkin finite element model is constructed finally.(3) The model is applied to numerical simulation of some classic discontinuous flow examples to prove its validity and applicability. The examples include idealized dam-break problem, a small perturbation of a steady-state water, transcritical flow over a hump, hydraulic jump, tidal waves and tidal bores. When solving discontinuous shallow water flow, the situations of non-prismatic channels and discontinuous bed topography can be considered. All the computations are good agreement with the analytical solutions, and the results exclude non-physical spurious oscillations near the shock.更多还原