对流扩散方程局部分析解的数值模拟

本文提出了数值求解对流扩散方程的局部解析算法。首先将方程在一个时间步内线性化，然后，将解表示为初始各单元脉冲解的迭加，如此得到的近似解是由初始单元解表达的解析式，具有与一般数值方法(差分、有限元等)完全不同的优势。分别对一维和二维方程的几个典型算例进行了计算，获得了令人满意的数值结果，精度高且没有稳定性的限制，说明本方法具有较强的应用价值。     

恒定条件下物质输移扩散方程的几种解析解

本文从恒定条件下物质输移扩散方程出发，在固体边界上扩散通量F＝－DаC／аn＝0的条件下，进行严密的数学推证，给出恒定条件下物质输移扩散方程的三种解析解，分析了解的合理性，讨论了解的正确性，其结果是令人满意的。对求解许多环境扩散问题具有实际应用价值，也具有重要理论意义。     

二维孤立波的透射与反射

本文应用渐近匹配方法探讨了孤立波遇到二维柱体时的透射与反射问题。远场解由Boussinesq方程制约,用差分方法数值地求解;内场解则从Laplace方程着手,对矩形柱体,分区域按本征函数的展开来表示;内外解经过匹配迭代求解。作为算例,给出了透射和反射的波形以及作用力和力矩随时间变化的曲线。     

在激波作用下任意形状三维物体表面压力分布的计算

本文从线化的波动方程出发,以非定常点源为基本解,求解平面激波绕任意三维形状物体的运动。由于用积分号下的一个阶跃函数来表示变动的积分区域边界,先对速度势的积分表达式求导,给出了压力和物面法向速度的积分表达式。将此二式离散化,由物面边条件确定每一瞬时各块上的源强,使问题得到解决,从而解决了在三维空间中布置非定常奇点的问题。     

对流-反应汇方程特征积分有限差数值解法

求物质对流扩散并具有源或汇作用的经典方程的整体数值解是比较困难的,目前多数采用分步求解的办法.本文在前人工作的基础上发展了特征积分有限差求解方法,这种方法能给出方程的整体数值解.文中仅就无扩散情形下对基本方程的解作了介绍,以表明解法的基本思想,同时,在简单的边界条件下,用所介绍的方法对所提问题进行了数值计算,并与解析解和分步求解法作了比较,结果表明,特征积分有限差解法能给出较好的结果.     

自由跌流的水力特性

本文应用N—S方程式,导出了压力和连续方程的解析式,求解了水流参数的半数值半解析的精确解.其结果表明,精确的理论解和一般的势流理论解的流场计算结果相差很大.本文对前人的试验研究中提出的关于出口断面水流内部是否存在负压问题进行了探讨.本文给出的理论分析结果和实测结果吻合得较好.     

分形油气藏非达西低速渗流时续流和表皮效应的影响

本文建立了分形油气藏非达西低速渗流时续流和表皮效应影响的数学模型，并对定解问题用拉普拉斯变换求解，拉氏空间的特解用格林函数表示，得到了解析解和渐近解。     

正则化长波方程的三次样条差分模拟

本文对正则化长波方程(RLW方程)的时间、空间项分别采用三次样条公式进行离散,所得到的差分方程为三点、三层隐格式,对时间的精度为四阶,对空间的精度为二阶,每个时间步只需求解一个三对角方程组,而三次样条配置法在每个时间步中需要求解三个三对角方程组。本文对单个孤立波的演化进行了计算,参数与文[8]相同,其结果与解析解的最大绝对误差均保持在10~(-3)的量级以内,与文[8]给出的数值方法精度相当。对两、三个孤立波的相互碰撞,本文结果显示出在忽略10~(-3)量级的误差后,碰撞前后的孤立波形状保持不变,无论是一个孤立波的演化,还是两、三个孤立波相互碰撞,当将波形放大100倍后均可以发现有振荡的尾波,这说明这类振荡尾波的产生并非真实的物理现象,而是由数值误差产生的。本文认为仅从工程应用的意义上说,RLW方程具有孤立子性质的结论是可以接受的。     

滑动轴承非线性油膜力的几个理论问题及应用

本文把流体润滑Reynolds方程在Reynolds边界条件下的解转化为自由边值问题，并进而对此自由边值问题等价变分原理进行了系统的归类和证明，为滑动轴承非稳态非线性油膜力的理论研究和寻求其解析或半解析解的方法提供了多渠道的理论依据和应用途径，对于旋转机械的非线性动力分析有重要意义。作为应用提出了一个求解椭圆轴承非稳态非线性油膜力的高精度半解析公式。     

无拉力半空间地基梁静力分析的线性互补解法

本文讨论了无拉力半空间地基梁静力分析的线性互补解法。将地基反力作为未知函数进行近似插值，对梁和半空间地基分别进行解析求解而获得１组地基与梁之间相对竖向位移和反力的线性互补方程。利用Ｌｅｍｋｅ方法求解线性互补方程而得到梁的挠度、转角、弯矩和剪力及地基反力。计算结果表明线性解与非线性解之间的差别是相当明显的。     

非线性波浪绕射问题的数值计算

本文给出非线性波浪绕射问题的计算模型,并用时间步进和边界元法求解。该模型能适用于严重非线性情形,且能满足初始条件和辐射条件。作为对该模型的检验,我们对孤立波绕过穿刺水面的直立圆柱的算例进行了计算,并同Isaacson的解析解、李宝元的差分解进行了比较,结果吻合很好。此外我们还用数值方法同时模拟造波机造波和求该波与直立圆柱的相互作用。结果表明,数值波浪水池能够以尽可能短的暂态过程,模拟波浪与物体的相互作用,提供良好的初始条件。数值波浪水池还能用于任意水深、任意水底形状下的相互作用问题。本文的计算模型和算法具有机时省,计算稳定的优点。     

Corrected SPH methods for solving shallow-water equations

The artificial viscosity in the traditional smoothed particle hydrodynamics(SPH) methodology concerns some empirical coefficients, which limits the capability of the SPH methodology. To overcome this disadvantage and further improve the accuracy of shock capturing, this paper introduces two other ways for numerical viscosity, which are the Lax-Friedrichs flux and the twoshock Riemann solver with MUSCL reconstruction to provide stability. Six SPH methods with different kinds of numerical viscosity are tested against the analytical solution for a 1-D dam break with a wet bed. The comparison shows that the Lax-Friedrichs flux with MUSCL reconstruction can capture the shock wave more accurate than other five methods. The Lax-Friedrichs flux and the artificial viscosity with MUSCL reconstruction are finally both applied to a 2-D dam-break test case in a L-shaped channel and the numerical results are compared with experimental data. It is concluded that this corrected SPH method can be used to solve shallow-water equations well.     

围堰溃决的数值模拟

推导了用于模拟围堰溃决的DIF方程，采用非规则网格有限体积法和显式MacCormack预测-校正数值方法求解该方程，建立了模拟围堰溃决数值模型。实例计算结果表明，本数学模型能合理地模拟天然情况下的河道水流、溃堰前的流场以及围堰溃决后溃堰波的沿程传播。此外，采用在导流洞进口局部区域网格上施加负质量源，出口局部区域网格上施加正质量源间接模拟导流洞过流的方法是合理的。因此，本数学模型可作为工具，为围堰溃决对下游河道两岸城镇的洪灾分析及溃堰预案的制定提供科学依据。     

Dam- Breach Hydrograph Modelling: An Innovative Semi- Analytical Approach

The delineation of the areas exposed to flood hazard caused by a dam existence upstream and its possible failure needs a thorough analysis of the hypothetical dam break incident. The study presented in this paper focusses on the simulation of the dam breach formation and the calculation of the resulting outflow hydrograph using a semi- analytical approach. More specifically the method presented addresses the dam break incident of an embankment dam caused by overtopping. The analysis is based on the assumptions of constant vertical erosion rate for the formation of the breach and the parabolic shape of the breach. Two solutions are presented dependent on whether the capacity of the reservoir is considered prismatic or it is a power function of the water depth in the reservoir. Finally the proposed method is illustrated through the analysis of a hypothetical dam break incident.     

Multi-scale Runge-Kutta_Galerkin method for solving one-dimensional KdV and Burgers equations

In this paper, the multi-scale Runge-Kutta_Galerkin method is developed for solving the evolution equations, with the spatial variables of the equations being discretized by the multi-scale Galerkin method based on the multi-scale orthogonal bases in 0(,)mH a b and then the classical fourth order explicit Runge-Kutta method being applied to solve the resulting initial problem of the ordinary differential equations for the coefficients of the approximate solution. The proposed numerical scheme is validated by applications to the Burgers equation(nonlinear convection-diffusion problem), the Kd V equation(single solitary and 2-solitary wave problems) and the Kd V-Burgers equation, where analytical solutions are available for estimating the errors. Numerical results show that using the algorithm we can solve these equations stably without the need for extra stabilization processes and obtain accurate solutions that agree very well with the corresponding exact solutions in all cases.     

Numerical simulation of wave transformation,breaking and runup by a contravariant fully non-linear Boussinesq equations model

In this paper we propose a new model based on a contravariant integral form of the fully non-linear Boussinesq equations(FNBE) in order to simulate wave transformation phenomena, wave breaking, runup and nearshore currents in computational domains representing the complex morphology of real coastal regions. The above-mentioned contravariant integral form, in which Christoffel symbols are absent, is characterized by the fact that the continuity equation does not include any dispersive term. The Boussinesq equation system is numerically solved by a hybrid finite volume-finite difference scheme. A high-order upwind weighted essentially non-oscillatory(WENO) finite volume scheme that involves an exact Riemann solver is implemented. The wave breaking is represented by discontinuities of the weak solution of the integral form of the non-linear shallow water equations(NSWE). On the basis of the shock-capturing high order WENO scheme a new procedure, for the computation of the structure of the solution of a Riemann problem associated with a wet/dry front, is proposed in order to simulate the run up hydrodynamics in swash zone. The capacity of the proposed model to correctly represent wave propagation, wave breaking, run up and wave induced currents is verified against test cases present in literature. The results obtained are compared with experimental measures, analytical solutions or alternative numerical solutions. The proposed model is applied to a real case regarding the simulation of wave fields and nearshore currents in the coastal region opposite San Mauro Cilento(Italy).     

NUMERICAL SIMULATION OF SPHERICAL, CYLINDRICAL AND AXIAL BUBBLE CLOUDS COLLAPSE

The nonlinear dynamics of a spherical,cylindrical and axial cloud of cavitation bubbles were numerically simulated in order to learn more about the physical phenomena occurring in the cloud cavitation....     

数值水池的一种有效的数值消波方法

本文通过将波高所满足的方程分解成等效的两个x和y方向的一维方程，提出了一种在求解Boussinesq 型方程中进行数值消波的有效方法，在求解方程时采用了一种精度较高的预报校正迭代方法。给出的方法用有实验结果的多向不规则波例子进行了验证。数值结果与实验结果相当吻合。本文的方法解决了波浪仿真中的关键技术：数值消波。     

层状地基中Rayleigh波频散方程改进的解析法

通过引入势函数,推导了Rayleigh波（R波）在层状地基中的频散方程及其改进的解析法;由频散方程的计算特点,在频率一波数域（f—k域）内应用Bisection算法,快速、准确地搜索R波频散方程的根;通过研究不同类型层状地基中R波的高阶频散曲线,修正了解析法不能有效计算软夹层或硬夹层地基的说法;最后,将得到的频散曲线与已有频散曲线作了对比,结果表明本方法能合理地分析R波在层状地基中的频散特性。     

An analysis of dam-break flow on slope

The one-dimensional steep slope shallow water equations are used to model the dam-break flow down a uniform slope with arbitrary inclination, and analytical solutions are derived by the hodograph transformation and the Riemann＇s method in terms of evaluated integrals. An implicit analytical solution is obtained to evaluate the spatio-temporal distributions of dam-break flood hydrographs along the slope. For convenience, the solution for representative wave profiles and velocity distributions is shown in charts. Comparing with the Dressler＇s solution and WES experimental data, the analytical solution is seen reasonable.