不规则对接网格的分区求解计算

采用一种保持通量守恒的不规则对接网格分区求解中交界面耦合条件的计算方法, 结合有限体积法求解了Euler 方程, 无粘通量取用Van Leer 分裂格式, 构造了一种限制器以实现格式的二阶精度和TVD 性质, 并给出数值算例。     

流动变量线性分布的MUSCL格式

本文提出一种线性函数斜率的确定方法并证明用其近似代替网格内流动变量的初始分布,MUSCL格式具有二阶精度。引入单调性限制条件和一定的耗散机制后,该方法适合于Euler坐标系下的计算。本文在处理间断时,用激波关系式代替特征线方程,解决了特征线法在全流场计算中的应用问题。最后,本文给出Emery问题和一个火箭喷流问题的计算结果。     

2阶精度时间后差隐式格式TVD充分条件

非定常流动问题计算中常用到含三个时间层的二阶精度时间后差隐式格式,并且希望构成TVD格式,然而理论上的问题多年一直没有解决。本文找到了解决办法,构造了这种类型的隐式TVD格式,证明了其为TVD的充分条件。理论结果为计算所验证,并表明通常未采取本文对时间差分处理方法的格式尚不具备TVD性质。     

双曲守恒律系统的分段边界层函数方法(PBLM)

本文提出了一个新的高阶Godunov格式。此格式放弃了[9]、[10]中关于参数在格子中满足多项式分布近似及在格子边界上存在间断的假设,直接引入了一个分段边界层型函数分布假设。由于引入的分布函数是单调可微的,因此PBLM格式无需进行如同MUSCL、PPM等格式中的单调性校核。该格式由于不进行单调性修正,在PPM格式中需进行修正而精度降阶的点上仍保持原有精度。对一维激波管的计算表明PBLM格式对激波的展开比PPM格式还要小,计算时间相当。同PPM格式一样,PBLM格式在激波后存在有2%~4%的皮后伪振荡,应加上适当的人工粘性。     

在直角坐标系用NGFM模拟复杂计算域水下高压气泡膨胀问题

采用NGFM(New version of Ghost Fluid Method)处理复杂计算域的固壁边界,用RGFM(Real Ghost Fluid Method)求解气-水界面附近网格节点的状态参数,从而在直角坐标系下对复杂计算域的水下高压气泡膨胀问题进行数值模拟。流场控制方程选用Euler方程,用五阶WENO格式离散空间导数项,二阶Runge-Kutta法离散时间导数项;气-水界面追踪使用Level Set方法,对Level Set方程,用五阶HJ-WENO(Hamilton-Jacobi WENO)和三阶Runge-Kutta法求解。将计算结果与任意坐标系下的结果进行对比,验证了NGFM在笛卡尔网格中处理复杂形状固壁边界的可行性。得到了水下流场压力等值线图、高压气泡的演变过程以及特定点处的压力-时间曲线。计算结果表明,高压气泡在固壁反射激波的作用下,膨胀过程受到抑制;强激波在固壁的反射会导致固壁附近出现大范围的空化流动。     

单个守恒型方程熵耗散格式中熵耗散函数的构造

对于一维单个守恒律方程,文[8]设计了一种非线性守恒型差分格式.此格式为二阶Godunov型的,用的是分片线性重构(reconstruction),重构函数的斜率是根据熵耗散得到的.格式满足熵条件.与传统的守恒格式不同的是此格式在计算过程中不仅用到了数值解还用到了数值熵.在此格式中一个所谓的熵耗散函数起到了很重要的作用,它在每一个网格的计算中耗散熵,以保证格式满足熵条件.文[8]中设计的熵耗散函数比较复杂,并且不是很完善.故数值地分析了在格式的构造中为何应给熵以一定的耗散,及应耗散多少.并且给出了一个新的以数值解的二阶差分作为基本模块的熵耗散函数.最后给出了相应的数值算例.     

求解Euter方程组的一种简单的守恒误差

本文提出了一种解一维不定常Euler方程组的较简单的一阶,二阶守恒差分格式,一阶格式简称为FUDE(First-Order UPwind Difference Scheme for Euler Equations),二阶格式简记为SUDE。本文的格式避免了大的数值振荡及膨胀激波,同时计算量较MacCormack^[1]格式仅略有增加,文中若干数值试验表明本文格式的分辨率是令人满意的。     

二维扩散方程的局部二阶线性保极值节点计算方法及应用

对一般四边形网格设计一种优化的节点控制体, 并构造了一种扩散方程的保极值二阶收敛的局部线性节点计算格式(优化控制体节点格式, VOC格式)。在网格不出现异常节点的情况下, 证明VOC格式是保极值、线性精确和二阶收敛的。而且在均匀的矩形网格上, 修正的逆距离加权格式与VOC格式等价, 从而对间断系数问题也是局部二阶收敛的。VOC格式可以用于单元中心型线性扩散格式和保正格式的节点值计算。数值算例表明对扭曲网格上的间断系数问题, VOC格式是二阶收敛的。采用VOC格式计算节点值的线性九点格式具有线性精确性和二阶收敛性, 采用VOC格式的保正格式也具有二阶收敛性。     

相同尺度下气泡与复杂壁面的耦合特性研究

采用格子Boltzmann方法(LBM)建立了气液固三相耦合的动力学模型,研究了相同尺度下上浮气泡与复杂壁面的相互耦合作用.首先,基于黏性流体理论,通过构建一组格子Boltzmann(LB)方程来描述气液两相的运动,并以LB离散体积力的形式计入了黏性力、表面张力和重力.同时,采用LBM中的Half-way反弹模型与有限差分格式相结合的方式进行固壁边界的处理.然后,利用本文建立的模型,对不同特征尺寸比条件下,气泡与考虑边缘效应的平面固壁和曲面固壁的耦合特性进行了研究.研究发现固壁边界条件以及特征尺寸比对气泡的运动和拓扑结构的变化都具有明显的非线性影响.最后,研究了流体属性对气泡与复杂壁面耦合规律的影响.     

线性双曲型方程新的TVD格式

从双曲型方程的ＴＶＤ条件出发,分析了Ｌａｘ－Ｗｅｎｄｒｏｆｆ格式和Ｗａｒｍｉｎｇ－Ｂｅａｍ格式所存在的ＴＶＤ区间,构造了空间三点和四点新的二阶ＴＶＤ格式,使其在极值点也保持二阶精度。最后给出了新格式与传统ＴＶＤ格式的结果的比较。     

Self-synchronization of nonlinear oscillations in the presence of fluctuations

A statistical mechanical theory is presented for the self-organization of a macroscopic oscillation with the presence of external fluctuations in a system of Van der Pol oscillators coupled through dissipative interactions. Starting from Langevin equations for the Van der Pol oscillators, the static and dynamic characteristics are studied. The threshold condition is given by the relative size between the fluctuation and the interaction. The transitions between synchronous and asynchronous phases are well discussed by a Landau-type equation. The steady state value of the order parameter and the onset time are compared between the theory and the computer experiments and a good agreement is obtained.     

THE QUARK CLUSTER MODEL OF NUCLEAR FORCE:(Ⅱ) QUARK CONFINEMENT POTEPNTIAL AND COLOR VAN DER WAALS FORCE

The color-analog Van der Waals force between two hadrons is studied by the coupling channel resonating group method in the framework of the Gaussian-type quark confinement potential. The problem of the boundary values for the two channel coupling differential equations is changed to the problem of the initial values. The equations are solved numerically by the Gill method. The calculated results show that there is no color Van der Waals force between hadrons in the confinement potential model. This indicates that the confinement potential model not only can describe the internal structure of hadrons but also can be used to calculate the hadron-hadron interactions.     

On the “Mean Field” Interpretation of Burgers' Equation

Fruitful analogies, partially first established by C. M. Newman,(1) between the variables, functions, and equations which describe the equilibrium properties of classical ferro- and antiferromagnets in the Mean Field Approximation (MFA) and those which describe the space-time evolution of compressible Burgers' liquids are developed here for one-dimensional systems. It is shown that the natural analogies are: magnetic field and position coordinate; ferro-/antiferromagnetic coupling constants and negative/positive times; free energy per spin and velocity potential; magnetization and velocity field; magnetic susceptibility and mass density. An unexpected consequence of these analogies is a derivation of the Morette–Van Hove relation. Another novelty is that they necessitate the investigation of weak solutions of Burgers' equation for negative times, corresponding to the Curie–Weiss transition in ferromagnets. This is achieved by solving the “final-value” problem of the homogenous Hamilton–Jacobi equation. Unification of the final- and initial-value problems results in an extended Hopf–Lax variational principle. It is shown that its applicability implies that the velocity potentials at time zero be Lipschitz continuous for the velocity field to be bounded. This is a rather mild condition for the class of physically interesting and functionally compatible velocity potentials, compatible in the sense of satisfying the Morette–Van Hove relation.     

On the Deser-Van Nieuwenhuizen algebraic vierbein gauge

In this note it is shown that for every dynamic vierbein field there is a gauge transformation such that the transformed field satisfies the algebraic gauge condition of Deser and Van Nieuwenhuizen. In this sense there is no Gribov phenomenon for the above hauge condition.     

Initial-value problem for plasma oscillations in a uniform magnetic field

The solution is given of the initial-value problem for the nonrelativistic linearised Vlasov-Maxwell equations describing longitudinal and transverse plasma oscillations in an external uniform magnetic field. The problem is solved for all directions of propagation except normal to the external magnetic field, and the equilibrium distribution is not assumed isotropic. The method of solution is an extension of Van Kampen's eigenfunction expansion technique, already developed considerably by Zelazny and McCure, in which the problem is reduced to the solution of a system of singular integral equations.     

Polarization properties of nonlinear acousto-optical effect in semiconductors

The polarization properties cf second order nonlinear acousto-optical effect in semiconductors are investigated by the symmetry consideration. The scattering tensor of the process is derived in terms of the Clebsh-Gordon coefficients of the crystalline symmetry group. As an example a semiconductor with zinc-blende structure is studied.The author would like to thank Professor Hguyen Van Hieu for suggesting the problem and and for fruitful discussions.     

Dynamics of thin free liquid films stabilized with ionic surfactants

To study the dynamics of free liquid films which are stabilized with ionic surfactants, an electrohydrodynamic theory is developed. The long-range interaction forces, namely the repulsive force arising from the overlap of diffuse electric double layers and the attractive Van der Waals-London force, are described by the Maxwell stress tensor with its related field equations and the Van der Waals potential. The short-range surface force has a normal Laplace component and a tangential surface tension gradient component. The total set of first-order equations and their boundary conditions, which describes the capillary waves on the surfaces and the induced flow motion in the film, has been solved. The dispersion relation for the “squeezing mode” is obtained. The cases of no-slip condition and the long-wavelength limit have been studied in more detail. The testability of the dispersion relation using laser beat spectroscopy is reported.     

Analysis of modified Van der Pol’s oscillator using He’s parameter-expanding methods

In this work, a powerful analytical method, called He’s parameter-expanding methods (HPEM) is used to obtain the exact solutions of non-linear modified Van der Pol’s oscillator. The classical Van der Pol equation with delayed feedback and a modified equation where a delayed term provides the damping are considered. It is shown that one term in series expansions is sufficient to obtain a highly accurate solution, which is valid for the whole solution domain. Comparison of the obtained solution with those obtained using perturbation method shows that this method is effective and convenient to solve this problem. This method introduces a capable tool to solve this kind of non-linear problems.     

Operator Ordering Problem and Coordinate-Free Nature of Geometric Quantization

Operator ordering problem in geometric quantization is investigated. While the quantum operator for p²f(q) given by geometric quantization is found to be ordered via the rule invariant under general coordinate transformations, it is shown that the quantum operators for pⁿ f(q) when n > 2 cannot be well-defined by geometric quantization. This is considered as a natural consequence of the coordinate-free nature of geometric quantization and the Van Hove's theorem.     

On the theory of the Kondo effect in two-dimensional metals

The Kondo effect in a (quasi-)two-dimensional metal is studied. The special feature of the two-dimensionality is the Van Hove singularity in the electron density of states. For the band filling choosen such, that the Fermi level is close to the saddle points of the band spectrum, the Van Hove singularity comes into play and changes the usual Kondo log to the log². It turnes out to be possible to carry out the first order parquet summation and to obtain the conditions for the Kondo antiferromagnetic resonance for an arbitrary geometry of the band spectrum. The connection with the Orthogonality Catastrophe is traced and it is shown, that the weak coupling Kondo problem just corresponds to the intermediate asymptotics of the metal's relaxation in a time-dependent external potential.