一维活塞问题激波解的整体存在性

本文用改进的Glimm格式的方法,研究一维活塞问题当活塞的运动速度是一个常数的扰动时含有激波的弱解的存在性.对波的相互作用以及扰动波在主激波和活塞上的反射作出了精确的估计,在对主激波的强度不加限制的情况下证明了激波解的整体存在性.     

一维活塞问题激波解的整体存在性

本文用改进的Glimm格式的方法,研究一维活塞问题当活塞的运动速度是一个常数的扰动时含有激波的弱解的存在性.对波的相互作用以及扰动波在主激波和活塞上的反射作出了精确的估计,在对主激波的强度不加限制的情况下证明了激波解的整体存在性.     

压差方程Chapman-Jouguet燃烧模型爆轰波与激波的相互作用

研究了气体动力学压差方程Chapman-Jouguet(CJ)燃烧模型爆轰波与激波的相互作用.给出了该CJ燃烧模型的几类基本波线:激波线、疏散波线、强爆轰波线和CJ爆轰波线.通过研究该CJ燃烧模型的初值为三片常状态的一类初值问题,并利用相平面分析的方法构造出该问题的整体分片光滑解,得到了压差方程CJ燃烧模型爆轰波与激波相互作用的结果.进一步地,得到了对应燃烧Riemann问题解的初值扰动稳定性.     

一种在双驱动激波管和激波风洞中实现运动激波与头激波斜相互作用的新方法

本文在讨论和分析了国外现有的运动激波与头激波斜相互作用的两大类实验方案的基础上,提出并实现了在双驱动激波管和激波风洞中形成运动激波与头激波斜相互作用的新方法.这种方法不仅可以获得双波(指运动激波与头激波,下同.)斜相互作用所需要的平面的运动激波,而且可以同时得到双波斜相互作用条件下试验模型表面瞬态压力曲线和流场照片.这种方法还可以用于研究在运动激波前有气流情况下,运动激波在尖劈或尖锥表面规则反射(Regular Reflection)与Mach反射(Mach Reflection)之间的转变.在测试技术方面,本文还提出了一种改进方法,用于测量运动激波的激波Mach数.     

运动气流的二维激波动力学方程组

本文把原来适用于波前为静止气体的激波动力学方法,推广到波前为运动气流的情况中。导出了以激波马赫数和波面方向角表示的波前为均匀流的二维激波动力学方程组和波前为非均匀流的二维激波动力学方程组。文中还简单介绍了运动气流的激波动力学方程组在“双波干扰”中的应用。     

一维Chaplygin气体欧拉方程组中波的相互作用

研究一维Chaplygin气体欧拉方程组中波的相互作用.方程组的波包含接触间断和在密度变量以及内能变量上同时具有狄拉克函数的狄拉克激波.根据这些波的不同组合,问题被分成了7种情形.通过详细地构造每种情形的整体解,获得了各种波相互作用的完整结果.特别地,对于一类初值,两个接触间断相互作用后,产生了一个狄拉克激波;然而,对于另外一类初值,一个狄拉克激波与一个接触间断相互作用后,狄拉克激波消失.这些都是波相互作用中非常特别的现象.     

非齐次二维Burgers方程的非自相似黎曼解的奇性结构

该文研究了二维非齐次Burgers方程Riemann问题的激波解和稀疏波解之间相互作用的全局奇性结构及其演化,其中初值被两个相离的圆隔开并分成三片常数.首先得到了由初值间断发出的激波解和稀疏波解的表达式;其次,讨论了这些激波和稀疏波的相互作用,并发现了一些新现象,其与齐次情形相比,激波和稀疏波能一直相互作用,相互作用的时间没有使得结构发生改变的临界值;最后构造了非自相似解的全局结构,并发现了有别于齐次情形的渐近行为,即基本波区域的直径是有界的.     

广义Chaplygin气体磁流体力学方程组的Riemann问题

利用特征分析和相平面分析的方法,由Rankine-Hugoniot条件和稳定性条件,构造性地得到了一维等熵广义Chaplygin气体磁流体力学方程组的Riemann解的存在唯一性.同时,详细研究了疏散波曲线和激波曲线的性质.     

延拓Chaplygin气体黎曼解的流逼近极限过程中的集中性研究(英文)

本文研究了延拓Chaplygin气体的黎曼解在流逼近极限过程中的集中现象和狄拉克激波的两种形成机制问题.利用相平面分析法和广义特征分析法,构造出了延拓Chaplygin气体的整体黎曼解,并获得了两个结果:当压力消失时,延拓Chaplygin气体的包含两个激波的解收敛到输运方程的狄拉克激波解;当压力项趋近于广义Chaplygin压力项时,延拓Chaplygin气体的包含两个激波的解收敛到广义Chaplygin气体的狄拉克激波解.结论推广到了延拓Chaplygin气体.     

四个对激波与涡旋有高分辨率的计算格式的比较*

近十年来,计算非定常无粘可压气体力学Euler方程组的高分辨率差分格式有显着进展.本文选择四个近年受到重视的格式,用一较复杂的二维不定常问题作进一步的考验.所选算例为平面激波遇矩形障碍物初始阶段的绕射与反射.在挡板头部有两个尖角点,角点附近流场参量变化剧烈,会有中心稀疏波和集中涡的出现,要模拟好它们,就要求格式有较好的适应性.本文选择特殊的激波马赫数M_s=2.068,使静止坐标系下激波后流速恰为声速,并沿中心稀疏波区从角点发出的一条曲线也有这一现象,以考察各格式在方程组某一特征值恰为零时的计算特点,因零特征值可以使某些格式局部受损.计算结果的图形显示可表明四个格式在激波分辨率,格式粘性、膨胀波的计算、模拟非定常集中涡产生过程的能力等方面的性质.     

Surface waves in micropolar thermoelasticity

Free surface waves of arbitrary form in a homogeneous and isotropic linear micropolar thermoelastic half-space with stress-free plane boundary are investigated. It is found that all physical quantities associated with the waves are derivable from two scalar functions and that there exist two families of waves in general. One of these is the classical thermoelastic wave modified under the influence of the microelastic field and the other is a new surface wave not encountered in classical elasticity. The waves are not necessarily plane waves and even when these are assumed to propagate in a fixed direction parallel to the boundary, unlike in classical elasticity, the problem is not one of plane strain. Explicit expressions for the displacement vector, microrotation vector and the temperature are obtained and the nature of deformation has been analysed. Several earlier results are deduced as particular cases of the more general results obtained here.     

On the acceleration of shock waves and concentration of energy

By a series of simple examples related to exact solutions of problems in gas dynamics and magnetohydrodynamics, possible mechanisms of acceleration of shock waves and concentration of energy are elucidated. The acceleration of a shock wave is investigated in the problem of motion of a plane piston at a constant velocity in the case when the initial density of the medium drops in the presence of constant counterpressure. It is shown that in this situation a “blow-up” regime is induced by a shock wave going to infinity in finite time even for limited work of the piston. A simple spherically symmetric solution with a converging shock wave is constructed and shown to lead to the concentration of energy. A general method for solving one-dimensional non-self-similar problems related to matching the equilibrium state to a motion with homogeneous deformation on a shock wave is discussed; this method leads to a solution in quadratures.     

The decay of an arbitrary initial discontinuity in an elastic medium

The solution of the problem of the decay of an arbitrary discontinuity in elastic theory is studied. It is assumed that a plane boundary separates an elastic homogeneous, non-heat-conducting medium into two half-spaces with different elastic properties and densities. Each of the media possesses an arbitrary kind of homogneous initial strain (stress) and velocity. In the sequel the stresses and velocities of the media are assumed to be continuous at the boundary. This results in the formation of a system of plane selfsimilar waves (simple and shock), which propagate in each of the half-spaces. The problem is solved under the assumption of weak non-linearity and anisotropy of the materials. This permits an approximate evaluation of the stress and strain at the contact discontinuity. After this the problem on the decay of an arbitrary initial discontinuity is reduced to two problems on the sudden change of load on a half-space boundary, which are solved independently for each of the media.     

Characteristic wave fronts in magnetohydrodynamics

The influence of magnetic field on the process of steepening or flattening of the characteristic wave fronts in a plane and cylindrically symmetric motion of an ideal plasma is investigated. This aspect of the problem has not been considered until now. Remarkable differences between plane, cylindrical diverging, and cylindrical converging waves are discovered. For instance, when the adiabatic index γ is 2, the magnetic field does not affect the behaviour of plane waves, but does affect cylindrical waves. As the field strength increases, the time t_c taken for the shock formation varies monotonically for plane waves, while for cylindrical waves, in some situations t_c exhibits a unique minimum for diverging waves and a unique maximum for converging waves. For cylindrical converging waves, a shock formation takes place if and only if, γ and the field strength are restricted to certain finite intervals. Moreover, t_c is bounded in all cases except for cylindrical diverging waves. The discontinuity in the velocity gradient at the wave front is shown to satisfy a Bernoulli-type equation. The discussion of the solutions of such equations reported in the literature is shown to be incomplete, and three general theorems are established.     

Stability of viscous shock waves associated with a system of nonstrictly hyperbolic conservations laws

We introduce a simple model of two conservation laws which is strictly hyperbolic except for a degenerate parabolic line in the state space. Besides classical shock waves, it also exhibits overcompressive, marginal overcompressive, and marginal undercompressive shock waves. Our purpose is to study the behavior of the corresponding viscous waves, in particular the manner in which these waves are stable. There are several basic differences between classical shock waves and other types of shock waves. A perturbation of an overcompressive shock wave gives rise to a new wave. Monotone marginal overcompressive waves behave distinctly from the nonmonotone ones. Analytical techniques used in our study include characteristic-energy and weighted-energy methods, and nonlinear superposition through time-invariants. Although we carry out our analysis for a simple model, the general phenomena would hold for overcompressive waves which occur in other physical models.     

The harmonic oscillations of a viscoelastic rod of triangular cross-section

Two exact solutions of the plane strain problem of the harmonic oscillations of a viscoelastic rod, the cross-section of which is a right triangle, are proposed. Either the normal displacement and the shear stress or the shear displacement and the normal stress of the side surface of the rod are given. Six dimensionless parameters which affect the dynamic deformation process are derived. Two parameters characterize the contribution of the viscous properties with respect to the elastic properties, two others define the logarithmic decrement of the longitudinal and shear harmonic waves, and two other parameters affect the wavelength of the corresponding wave and the velocity of motion of the wave front of these waves. The velocities of both types of waves and their wavelengths turn out to be greater than the velocities and wavelengths of the corresponding elastic waves. It is shown that, for certain values of the viscosity and the oscillation frequency, pseudo-resonance frequencies are possible which are higher than the resonance frequencies for an elastic medium.     

The diffraction of plane elastic waves by a delaminated rigid inclusion when there is smooth contact in the delamination region

A solution of the problem of the diffraction of harmonic elastic waves by a thin rigid strip-like delaminated inclusion in an unbounded elastic medium, in which the conditions for plane deformation are satisfied, is proposed. We mean by a delaminated inclusion an inclusion, one side of which is completely bonded to the elastic medium, while the second does not interact in any way with it, or this interaction is partial. It is assumed that the conditions for smooth contact are satisfied in the delamination region. The method of solution is based on the use of previously constructed discontinuous solutions of the equations describing the vibrations of an elastic medium under plane deformation conditions. The problem therefore reduces to solving a system of three singular integral equations in the unknown stress and strain jumps at the inclusion. An approximate solution of the latter enabled formulae to be obtained that are convenient for numerical realization when investigating the stressed state in the region of the inclusion and its displacements when acted upon by incident waves.     

有限变形弹性杆中三种非线性弥散波

在一维弹性细杆拉压、扭转和弯曲波的经典线性理论基础上,分别计入有限变形和弥散效应,借助Hamilton变分原理,由统一的方法导出了3种非线性弥散波的演化方程．对3种演化方程进行了定性分析．结果表明,这些方程在相平面上存在同宿轨道或异宿轨道,分别相应于孤波解或冲击波解．根据齐次平衡原理,用Jacobi椭圆函数展开对这些演化方程进行了求解,在一定的条件下它们均可能存在孤立波解或冲击波解,这与方程的定性分析完全一致．     

The property of inner symmetry in optimum multilayer structures under the action of elastic waves

The general properties of laminar, linearly elastic, locally isotropic bodies which are energetically optimum for a specified inclined incidence of a plane wave are investigated. It is shown that, in the case of special supplementary conditions of the type that the frequencies of the maximum attainable value of the energy characteristic should be present in the required spectrum or that the angle of incidence is sufficiently small, the problem reduces to an alternate configuration of no more than two of the given set of layers which are identical both in their physical properties and in their width. These properties were established earlier for the propagation of electromagnetic and purely longitudinal waves.     

Globally Hyperbolic Regularization of Grad's Moment System

In this paper, we propose a globally hyperbolic regularization to the general Grad's moment system in multidimensional spaces. Systems with moments up to an arbitrary order are studied. The characteristic speeds of the regularized moment system can be analytically given and depend only on the macroscopic velocity and the temperature. The structure of the eigenvalues and eigenvectors of the coefficient matrix is fully clarified. The regularization together with the properties of the resulting moment systems is consistent with the simple one‐dimensional case discussed in ¹. In addition, all characteristic waves are proven to be genuinely nonlinear or linearly degenerate, and the studies on the properties of rarefaction waves, contact discontinuities, and shock waves are included. © 2014 Wiley Periodicals, Inc.