不规则地形上浅水模拟平衡性的实现

天然河道地形复杂而无规律,采用守恒型浅水方程进行水流运动数值模拟时往往会出现不平衡现象。本文在任意四边形网格的基础上采用Roe方法对守恒型浅水方程进行离散,针对连续方程离散后所出现的不平衡性,从原方程及离散方法的物理实质出发提出了局部水位法,使该问题得以解决;参照数值通量构造方法采用有限体积法构造了底坡项的离散方式,消除了动量方程底坡项离散后可能出现的虚假流动现象,同时保证了物理量的守恒性。将模型应用于松花江佳木斯河段的水流模拟中,其计算结果表现出了良好的守恒性、收敛性和平衡性。     

变深度浅水域中非定常船波

以Green—Naghdi(G—N)方程为基础,采用波动方程／有限元法计算船舶经过变深度浅水域时非定常波浪特性．把运动船舶对水面的扰动作为移动压强直接加在Green-Naghdi方程里,以描述运动船体和水面的相互作用．以Series60 CB＝0．6船为算例,给出自由面坡高,波浪阻力在船舶经过一个水下凸包时变化规律,并与浅水方程的结果进行了比较．计算结果表明,当船舶经过凸包时,波浪阻力先增加,后减少,并逐渐趋于正常．同时发现,当船速小于临界速度时(Fr＝√gh＜1．0),G—N方程给出的船后尾波比浅水方程的结果明显,波浪阻力也比浅水方程的结果有所提高,频率散射必须考虑．当船速大于临界速度时(Fr＝√gh＞1．0),G—N方程的计算结果与浅水方程差别不大,频率散射的影响可以忽略．     

用格子Boltzmann方程模拟浅水波问题

提出了用格子Ｂｏｌｔｚｍａｎｎ方程（ＬＢＥ）模拟浅水波问题的方法．通过无粘气体运动方程与浅水波方程的比较,确定了ＬＢＥ方法中平衡态的形式,使宏观方程与浅水波方程一致．计算了二维浅水波的一个问题,数值结果与精确解做了比较．     

用离散速度方法计算浅水长波方程

用离散速度法计算浅水波方程，将空气动力学方程和浅水波方程作了比较，用Nadiga提出的近平衡流动方法模拟浅水波方程的连续和间断解。计算了一维的溃坝波问题和Thacker提出的连续解问题，结果与精确解作了比较，并且计算了水流跃过障碍物的问题。     

复杂地形网格生成研究

在研究羽流扩散的过程中,根据实验中提供的复杂地形数据,采用一系列的格式转化,将其变为流体力学计算软件Fluent和其建模软件Gambit能够读取的数据,并生成相应的计算网格,为计算复杂地形上的羽流扩散创造了条件。     

平坦地形条件下改进RPF的TAN方法

针对地形辅助导航系统中递归地形匹配方法在平坦地形条件下位置估计鲁棒性差的问题,提出了一种基于集合卡尔曼滤波和正则化粒子滤波（RPF）的地形匹配方法。首先分别以航行器的水平位置分量和多波束声纳的高程测量值作为地形匹配系统的状态量和观测量,然后采用基于投影的方案补偿航行器姿态变化导致的测深误差,最后利用集合卡尔曼滤波器更新RPF中的条件建议分布以实现递归地形匹配。通过船载湖试数据评估了改进RPF在不同初始匹配位置误差条件下的地形匹配跟踪性能,结果表明：所提地形匹配滤波器能始终保持有界的定位误差,位置跟踪精度和置信区间估计性能较高,在10 m分辨率的先验数字地形图中地形匹配误差均值小于2个网格。     

三维复杂流动的间断有限元方法模拟

本文基于三维可压缩Euler方程,采用基于Runge-Kutta时间离散的间断有限元方法(RKDG方法),对三维前台阶、三维Riemann问题和球Riemann等问题进行了模拟。结果表明,本文的RKDG方法能够在很少的网格内清晰地捕捉到三维复杂流场中的激波和接触间断;同时,将球Riemann问题中z=0.4平面压强沿到对称轴距离的分布与文献中的近似精确解相比,吻合较好,这也验证了本文的RKDG方法不仅能够进行三维复杂流场的定性描述,也能够应用于三维复杂流场的定量计算。     

改进的地形熵算法在地形辅助导航中的应用

地形辅助导航是水下航行器导航技术的一个发展新方向,但它并不能够在任何地形区域都可以工作,比如在平坦区域的导航效果很差。通过计算不同区域的水深标准差,选择地形特征独特的区域作为适配区域。基于熵的算法对于地形复杂区域的匹配分析是快速有效的,但传统的地形熵算法匹配精度不高,本文引入了地形差异熵的概念并对其进行改进,在选定的地形区域使用MATLAB软件进行了仿真研究。仿真结果表明,改进的地形熵算法在选定的地形区域位置误差在250m左右,可以满足水下航行器的导航要求。     

求解浅水波方程的半离散中心迎风方法

将Jin's的界面方法应用到求解双曲守恒型方程的半离散中心迎风方法中,给出了一种新的求解浅水波方程的半离散中心迎风差分方法。对于源项,不是采用传统的单元均值而是采用单元界面处的值来近似,使所得格式对稳定态的求解是均衡的。且已证明所给的二阶精度的求解格式保持水深的非负性,这一特性使其能够较好的处理干河床问题。使用该方法产生的数值粘性(与O(Δ2r-1)同阶)要比交错的中心格式小(与O(Δx2r/Δt)同阶),而且由于数值粘性与时间步长无关,从而时间步长可根据稳定性需要尽可能的小,因此适用于稳定态的求解。     

无网格法在三维复杂超音速流场模拟中的应用研究

对最小二乘无网格方法在含复杂外形三维超音速流场中的应用进行了研究.选用分解法求解采用最小二乘法得到的对称方程组,针对最小二乘无网格方法的计算特点生成近似正交均匀分布的离散点,对B1AC2R常规导弹超音速流场采用最小二乘无网格方法进行了无粘数值模拟,计算了B1AC2R常规导弹在不同攻角下的轴向力、法向力及俯仰力矩系数,并将数值结果与实验结果进行了比较.结果表明,最小二乘无网格方法在求解含复杂外形超音速流场时具有较高的准确度,将其应用于三维含复杂外形超音速流场的模拟是完全可行的.     

Numerical solution of shallow water magnetohydrodynamic equations with non-flat bottom topography

The governing equations of shallow water magnetohydrodynamics describe the dynamics of a thin layer of nearly incompressible and electrically conducting fluids for which the evolution is nearly two-dimensional with magnetic equilibrium in the third direction. A high-resolution central-upwind scheme is applied to solve the model equations considering non-flat bottom topography. The suggested method is an upwind biased non-oscillatory finite volume scheme which doées not require a Riemann solver at each time step. To satisfy the divergence-free constraint, the projection method is used. Several case studies are carried out. For validation, a gas kinetic flux vector splitting scheme is also applied to the same model.     

A Newton multigrid method for steady-state shallow water equations with topography and dry areas

A Newton multigrid method is developed for one-dimensional (1D) and two-dimensional (2D) steady-state shallow water equations (SWEs) with topography and dry areas. The nonlinear system arising from the well-balanced finite volume discretization of the steady-state SWEs is solved by the Newton method as the outer iteration and a geometric multigrid method with the block symmetric Gauss-Seidel smoother as the inner iteration. The proposed Newton multigrid method makes use of the local residual to regularize the Jacobian matrix of the Newton iteration, and can handle the steady-state problem with wet/dry transition. Several numerical experiments are conducted to demonstrate the efficiency, robustness, and well-balanced property of the proposed method. The relation between the convergence behavior of the Newton multigrid method and the distribution of the eigenvalues of the iteration matrix is detailedly discussed.     

Multidimensional upwind schemes for the shallow water equations

A multidimensional discretisation of the shallow water equations governing unsteady free-surface flow is proposed. The method, based on a residual distribution discretisation, relies on a characteristic eigenvector decomposition of each cell residual, and the use of appropriate distribution schemes. For uncoupled equations, multidimensional convection schemes on compact stencils are used, while for coupled equations, either system distribution schemes such as the Lax–Wendroff scheme or scalar schemes may be used. For steady subcritical flows, the equations can be partially diagonalised into a purely convective equation of hyperbolic nature, and a set of coupled equations of elliptic nature. The multidimensional discretisation, which is second-order-accurate at steady state, is shown to be superior to the standard Lax–Wendroff discretisation. For steady supercritical flows, the equations can be fully diagonalised into a set of convective equations corresponding to the steady state characteristics. Discontinuities such as hydraulic jumps, are captured in a sharp and non-oscillatory way. For unsteady flows, the characteristic equations remain coupled. An appropriate treatment of the coupling terms allows the discretisation of these equations at the scalar level. Although presently only first-order-accurate in space and time, the classical dam-break problem demonstrates the validity of the approach. © 1998 John Wiley & Sons, Ltd.     

A slope modification method for shallow water equations

A slope modification method is proposed for non-oscillatory schemes based on the Lax-Friedrich solver. The modified scheme is proved to be total-variation-diminishing (TVD) and second-order accurate. Application of the scheme to the shallow water equations produces sharp profiles for shocks and achieves high accuracy in the smooth regions of the solution.     

A time-splitting method for the three-dimensional shallow water equations

In this paper we describe a time-splitting method for the three-dimensional shallow water equations. The stability of this method neither depends on the vertical diffusion term nor on the terms describing the propagation of the surface waves. The method consists of two stages and requires the solution of a sequence of linear systems. For the solution of these systems we apply a Jacobi-type iteration method and a conjugate gradient iteration method. The performance of both methods is accelerated by a technique based on smoothing. The resulting method is mass-conservative and efficient on vector and parallel computers. The accuracy, stability and computational efficiency of this method are demonstrated for wind-induced problems in a rectangular basin.     

Adaptive finite element method for analysis of pollutant dispersion in shallow water

A finite element method for analysis of pollutant dispersion in shallow water is presented. The analysis is divided into two parts ： （ 1 ） computation of the velocity flow field and water surface elevation, and （2） computation of the pollutant concentration field from the dispersion model. The method was combined with an adaptive meshing technique to increase the solution accuracy, as well as to reduce the computational time and computer memory. The finite element formulation and the computer programs were validated by several examples that have known solutions. In addition, the capability of the combined method was demonstrated by analyzing pollutant dispersion in Chao Phraya River near the gulf of Thailand.     

基于非结构网格求解二维浅水方程的高精度有限体积方法

采用HLL格式,在三角形非结构网格下采用有限体积离散,建立了求解二维浅水方程的高精度的数值模型.本文采用多维重构和多维限制器的方法来获得高精度的空间格式以及防止非物理振荡的产生,时间离散采用三阶Runge-Kutta法以获得高阶的时间精度.基于三角形网格,底坡源项采用简单的斜底模型离散,为保证计算格式的和谐性,对经典的HLL格式计算的数值通量中的静水压力项进行了修正.算例证明本文提出的方法的和谐性并具有高精度的间断捕捉能力和稳定性.     

A transport approach to the convolution method for numerical modelling of linearized 3D circulation in shallow seas

A new method for solving the linearized equations of motion is presented in this paper, which is the implementation of an outstanding idea suggested by Welander: a transport approach to the convolution method. The present work focuses on the case of constant eddy viscosity and constant density but can be easily extended to the case of arbitrary but time-invariant eddy viscosity or density structure. As two of the three equations of motion are solved analytically and the main numerical ‘do-loop’ only updates the sea level and the transport, the method features succinctness and fast convergence. The method is tested in Heaps' basin and the results are compared with Heaps' results for the transient state and with analytical solutions for the steady state. The comparison yields satisfactory agreement. The computational advantage of the method compared with Heaps' spectral method and Jelesnianski's bottom stress method is analysed and illustrated with examples. Attention is also paid to the recent efforts made in the spectral method to accelerate the convergence of the velocity profile. This study suggests that an efficient way to accelerate the convergence is to extract both the windinduced surface Ekman spiral and the pressure-induced bottom Ekman spiral as a prespecified part of the profile. The present work also provides a direct way to find the eigenfunctions for arbitrary eddy viscosity profile. In addition, mode-trucated errors are analysed and tabulated as functions of mode number and the ratio of the Ekman depth to the water depth, which allows a determination of a proper mode number given an error tolerance.