Numerical solution of the incompressible Navier-Stokes equations with an upwind compact difference scheme

A new finite difference method for the discretization of the incompressible Navier-Stokes equations is presented. The scheme is constructed on a staggered-mesh grid system. The convection terms are discretized with a fifth-order-accurate upwind compact difference approximation, the viscous terms are discretized with a sixth-order symmetrical compact difference approximation, the continuity equation and the pressure gradient in the momentum equations are discretized with a fourth-order difference approximation on a cell-centered mesh. Time advancement uses a three-stage Runge-Kutta method. The Poisson equation for computing the pressure is solved with preconditioning. Accuracy analysis shows that the new method has high resolving efficiency. Validation of the method by computation of Taylor's vortex array is presented.

State Vector: A New Approach to Prediction of the Failure of Brittle Heterogeneous Media and Large Earthquakes

The concept of state vector stems from statistical physics, where it is usually used to describe activity patterns of a physical field in its manner of coarsegrain. In this paper, we propose an approach by which the state vector was applied to describe quantitatively the damage evolution of the brittle heterogeneous systems, and some interesting results are presented, i.e., prior to the macro-fracture of rock specimens and occurrence of a strong earthquake, evolutions of the four relevant scalars time series derived from the state vectors changed anomalously. As retrospective studies, some prominent large earthquakes occurred in the Chinese Mainland (e.g., the M 7.4 Haicheng earthquake on February 4, 1975, and the M 7.8 Tangshan earthquake on July 28, 1976, etc) were investigated. Results show considerable promise that the time-dependent state vectors could serve as a kind of precursor to predict earthquakes.

Perturbational Finite Volume Method For The Solution of 2-D Navier-Stokes Equations on Unstructured and Structured Colocated Meshes

Based on the first-order upwind and second-order central type of finite volume( UFV and CFV) scheme, upwind and central type of perturbation finite volume ( UPFV and CPFV) schemes of the Navier-Stokes equations were developed. In PFV method, the mass fluxes of across the cell faces of the control volume (CV) were expanded into power series of the grid spacing and the coefficients of the power series were determined by means of the conservation equation itself. The UPFV and CPFV scheme respectively uses the same nodes and expressions as those of the normal first-order upwind and second-order central scheme, which is apt to programming. The results of numerical experiments about the flow in a lid-driven cavity and the problem of transport of a scalar quantity in a known velocity field show that compared to the first-order UFV and second-order CFV schemes, upwind PFV scheme is higher accuracy and resolution, especially better robustness. The numerical computation to flow in a lid-driven cavity shows that the under-relaxation factor can be arbitrarily selected ranging from 0.3 to 0. 8 and convergence perform excellent with Reynolds number variation from 102 to 104.

Simulation of Free-Surface Flow in a Tank Using the Navier-Stokes Model and Unstructured Finite Volume Method

Modelling free-surface flow has very important applications in many engineering areas such as oil transportation and offshore structures. Current research focuses on the modelling of free surface flow in a tank by solving the Navier-Stokes equation. An unstructured finite volume method is used to discretize the governing equations. The free surface is tracked by dynamically adapting the mesh and making it always surface conforming. A mesh-smoothing scheme based on the spring analogy is also implemented to ensure mesh quality throughout the computaiton. Studies are performed on the sloshing response of a liquid in an elastic container subjected to various excitation frequencies. Further investigations are also carried out on the critical frequency that leads to large deformation of the tank walls. Another numerical simulation involves the free-surface flow past as submerged obstacle placed in the tank to show the flow separation and vortices. All these cases demonstrate the capability of this numerical method in modelling complicated practical problems.

Modification of Bertrand's theorem and extended Runge-Lenz vector

It is shown that for a particle with suitable angular moments in the screened Coulomb potential or isotropic harmonic potential, there still exist closed orbits rather than ellipse, characterized by the conserved aphelion and perihelion vectors, i.e. extended Runge-Lenz vector, which implies a higher dynamical symmetry than the geometrical symmetry O-3. The closeness of a planar orbit implies the radial and angular motional frequencies are commensurable.

扩散抛物化Navier-Stokes方程数值解法评述

20世纪60年代末期在边界层理论基础上发展起来的各种简化Navier-Stokes(N-S)方程(统称为扩散抛物化N-S方程)及其算法,较为彻底地解决了无黏流及黏流的相互干扰问题,并为高雷诺数大型复杂黏性流场的数值模拟开辟了新的途径.本文将系统地评述这一领域的主要成果,包括各种简化N-S模型的优缺点;数学奇性及正则化方法;代表性的数值解法以及最近几年的新进展.

Divergence-free vector-field and reduction

By the Lie symmetry group, the reduction for divergence-free vector-fields (DFVs) is studied, and the following results are found. A n-dimensional DFV can be locally reduced to a (n - 1)-dimensional DFV if it admits a one-parameter symmetry group that is spatial and divergenceless. More generally, a n-dimensional DFV admitting a r-parameter, spatial, divergenceless Abelian (commutable) symmetry group can be locally reduced to a (n - r)-dimensional DFV.

Navier-Stokes方程二阶速度滑移边界条件的检验

对微尺度气体流动，Navier-Stokes方程和一阶速度滑移边界条件的结果与实验数据相比，在滑移区相互符合，在过渡领域则显著偏离，为改善Navier-Stokes方程在过渡领域的表现，有些研究者尝试引入二阶速度滑移边界条件，如Cercignani模型，Deissler模型和Beskok-Karniadakis模型．以微槽道气体流动为例，将Navier-Stokes方程在不同的二阶速度滑移模型下的结果与动理论的直接模拟Monte Carlo（DSMC）方法和信息保存（IP）方法以及实验数据进行比较．在所考察的3种具有代表性的二阶速度滑移模型中，Cercignani模型表现最好，其所给出的质量流率在Knudsen数为0．4时仍与DSMC和IP结果相符；然而，细致比较表明，Cercignani模型给出的物面滑移速度及其附近的速度分布在滑流区和过渡领域的分界处（Kn=0．1）已明显偏离DSMC和IP的结果．

高雷诺数流动的控制方程体系和扩散抛物化Navier-Stokes方程组的意义和用途

在计算机发达的时代,高雷诺(Re)数绕流计算中有无必要使用简化NS方程组,本文讨论这个问题.主要内容如下:(1)高Re数绕流包含3种基本流动:所有方向对流占优流动、所有方向对流扩散竞争流动和部分方向对流占优部分方向对流扩散竞争流动(简称干扰剪切流动),3个基本流动的特征彼此不同且在流场中所占领域大小彼此相差悬殊,NS方程区域很小,它们的最简单控制方程组Euler、Navier-Stokes(NS)和扩散抛物化(DP)NS方程组的数学性质彼此不同,因此利用Euler-DPNS-NS方程组体系分析计算高Re数绕流流动就是一个合乎逻辑的选择,该法与利用单一NS方程组的常用方法可以彼此检验和补充.(2)流体之间以及流体与外界的动量、能量和质量交换,流态从层流到湍流的演化主要发生在干扰剪切流动中,干扰剪切流及其最简单控制方程--DPNS方程组具有基础意义;DPNS方程组笔者在1967年已提出.(3)诸简化NS方程组:DPNS、抛物化(P)NS、薄层(TL)NS、黏性层(VL)NS方程组的发展、相互关系,它们的历史贡献和今后的用途;它们的数学性质均为扩散抛物型,但它们包含的黏性项彼此有所不同;从流体力学角度来看,它们中只有DPNS方程组能够准确描述干扰剪切流动.提出把诸简化NS方程组统一为DPNS方程组的建议.(4)干扰剪切流--DPNS方程组与无干扰剪切流--边界层方程组之间的关系以及进一步研究干扰剪切流的意义.

Navier-Stokes(NS)方程组差分计算中的物理和网格尺度效应及NS方程组的简化

对于高Re数流动计算,在通常二阶精度NS差分格式和网格数条件下,存在某些粘性项落入修正微分方程截断误差项的问题。这类NS方程组计算实际是计算某种简化NS方程组,而且重复计算误差物理粘性项既浪费机时和内存,误差积累又会对数值解产生不可预测的影响。避免上述缺陷的办法一个是提高NS差分格式的精度 ,另一个是丢掉可能落入截断误差项的物理粘性项,把NS方程组简化为广义NS方程组。广义NS计算避免了误差物理粘性项误差积累对数值解的不可知影响,又可节省内存和机时,对高Re数流体工程计算很有好处。利用广义NS方程组计算超声速绕前向和后向台阶流动的结果表明:广义NS方程组与NS方程组的数值结果很好相符。

Comparison Between Lurr And State Vector Analysis Before Strong Earthquakes In Southern California Since 1980

There are seven strong earthquakes with M >= 6.5 that occurred in southern California during the period from 1980 to 2005. In this paper, these earthquakes were studied by the LURR (Load/Unload Response Ratio) method and the State Vector method to detect if there are anomalies before them. The results show that LURR anomalies appeared before 6 earthquakes out of 7 and State Vector anomalies appeared before all 7 earthquakes. For the LURR method, the interval between maximum LURR value and the forthcoming earthquake is 1 to 19 months, and the dominant mean interval is about 10.7 months. For the State Vector method, the interval between the maximum modulus of increment State Vector and the forthcoming earthquake is from 3 to 27 months, but the dominant mean interval between the occurrence time of the maximum State Vector anomaly and the forthcoming earthquake is about 4.7 months. The results also show that the minimum valid space window scale for the LURR and the State Vector is a circle with a radius of 100 km and a square of 3 degrees 3 degrees, respectively. These results imply that the State Vector method is more effective for short-term earthquake prediction than the LURR method, however the LURR method is more effective for location prediction than the State Vector method.

关于三维Navier－Stokes方程的粘性项计算

借助于张量分析和张量计算，在贴体曲线坐标系下本文讨论了不同求解变量导致了粘性项个数上的重大差异和不同大小的计算量，并提出了便于粘性计算的最佳形式。文中借助于有限体积离散技术，通过引进两个对称辅助矩阵［Ａ］和［Ｂ］，使粘性项的计算量大大减少，这对完成三维粘性流的数值计算具有重要的指导意义。借助于上述方法，本文完成了某型真实进气道两种工况的三维粘性Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程计算（即Ｍ∞＝３．０，α＝０°和设计工况Ｍ∞＝２．６５，α＝０°），获得了满意的全场结果；对于Ｍ∞＝２．６５的设计工况，同实验数据作了比较，符合良好。由于本文的方法明显的减少了粘性项的计算量且节省了大量内存，以致于使三维流场的Ｎ－Ｓ求解能在普通微机上进行。

The vector fields admitting one-parameter spatial symmetry group and their reduction

For a n-dimensional vector fields preserving some n-form, the following conclusion is reached by the method of Lie group. That is, if it admits an one-parameter, n-form preserving symmetry group, a transformation independent of the vector field is constructed explicitly, which can reduce not only dimesion of the vector field by one, but also make the reduced vector field preserve the corresponding ( n - 1)-form. In partic ular, while n = 3, an important result can be directly got which is given by Me,ie and Wiggins in 1994.

简化Navier-Stokes(SNS)方程在二维层流边界层分离点邻域的特性

文中证明了本文第二作者提出的简化Navier-Stokes(SNS)方程在层流边界层分离点数学上为正则.Davis和Голвачев-Куэьмин-Попов 提出的SNS方程在分离点为数学奇异.进而论证了文献[2,3]的SNS方程在层流边界层分离点的奇异阶.最后给出了Navier-Stokes方程、上述两种SNS方程以及边界层方程在分离点邻域特性的比较.

简化Navier-Stokes方程的基本形式

通过考察各种典型粘性敏感流区的结构,分析粘性扩散项的量级大小层次,本文确立了简化Navier-Stokes方程的基本形式,并推诸一般附体坐标系,为简化Navier-Stokes方程的理论研究和一般应用提供了基础。