Decay of a plane shock wave in a two-parameter medium with arbitrary equation of state

Special curves, called shock polars, are frequently used to determine the state of the gas behind an oblique shock wave from known parameters of the oncoming flow. For a perfect gas, these curves have been constructed and investigated in detail [1]. However, for the solution of problems associated with gas flow at high velocities and high temperatures it is necessary to use models of gases with complicated equations of state. It is therefore of interest to study the properties of oblique shocks in such media. In the present paper, a study is made of the form of the shock polars for two-parameter media with arbitrary equation of state, these satisfying the conditions of Cemplen's theorem. Some properties of oblique shocks in such media that are new compared with a perfect gas are established. On the basis of the obtained results, the existence of triple configurations in steady supersonic flows obtained by the decay of plane shock waves is considered. It is shown that D'yakov-unstable discontinuities decompose into an oblique shock and a centered rarefaction wave, while spontaneously radiating discontinuities decompose into two shocks or into a shock and a rarefaction wave.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 147–153, November–December, 1982.     

不同状态方程对固体介质中斜激波反射的影响

相比气体,固体介质在高压下的状态方程更为复杂,形式也多种多样.现有关于固体介质中激波反射的理论研究,一般直接采用某种状态方程,缺乏对采用不同状态方程得到的结果的对比.本项工作采用激波极曲线的理论分析方法,选择4种不同组合形式的状态方程(一次冲击激波采用线性的冲击波速度与粒子速度关系式,二次冲击激波采用Gr(u|¨)neisen状态方程;一次冲击和二次冲击激波均采用冲击波速度与粒子速度关系式:一次冲击激波采用线性冲击波速度与粒子速度关系式,二次冲击激波采用刚性气体状态方程;以及一次冲击激波和二次冲击激波均采用刚性气体状态方程),研究固体介质中的斜激波反射,比较了采用不同组合形式的状态方程对反射激波波后压力的影响.利用量纲分析方法讨论了简化状态方程达到较高精度的条件.此外,用ANSYS/LS-DYNA软件,对激波极曲线理论给出的结果进行了验证.本项工作可为固体介质中激波反射问题状态方程的选取提供一定的指导.     

A correlation of the isentropic exponents of real gases

In previous publications, three isentropic exponents, k_pv, k_Tv, k_pT, have been introduced, which when used in place of the classical isentropic exponent k = c_p/c_v in the ideal gas isentropic change equations, the latter can describe very accurately the isentropic change of real gases. The present work provides a general method for determining the values of k_pv, k_Tv, k_pT within the ranges of reduced pressure p_r = 0 to 10 and of reduced temperature T_r = 1 to 3.5, thus allowing the calculation of the isentropic flow of those real gases for which no detailed thermodynamic data are available. The accuracy obtained is limited only by the accuracy of the generalized Lee-Kesler equation of state, which is employed in the method developed.     

An implicit characteristic-flux-averaging method for the euler equations for real gases

A formulation of an implicit characteristic-flux-averaging method for the unsteady Euler equations with real gas effects is presented. Incorporation of a real gas into a general equation of state is achieved by considering the pressure as a function of density and specific internal energy. The Ricmann solver as well as the flux-split algorithm are modified by introducing the pressure derivatives with respect to density and internal energy. Expressions for calculating the values of the flow variables for a real gas at the cell faces are derived. The Jacobian matrices and the eigenvectors are defined for a general equation of state. The solution of the system of equations is obtained by using a mesh-sequencing method for acceleration of the convergence. Finally, a test case for a simple form of equation of state displays the differences from the corresponding solution for an ideal gas.     

A nonuniform shock wave subjected to an oblique magnetic field

Summary A family of exact solutions for the one dimensional, unsteady, non-isentropic flow of an ideal, inviscid, perfectly conducting, compressible fluid subjected to an oblique magnetic field, is connected by a nonuniform shock wave to gas at rest in which the density distribution is nonuniform. The method of solution is inverse in the sense that the shock path is determined by the usual jump conditions, and then the density distribution in front of the shock is obtained in parametric form in terms of the equation of the shock locus.     

Airflow behind strong shock front with account for nonequilibrium ionization and radiation

We consider the gas state behind a shock wave front in air with a velocity v10 km/sec. Nonequilibrium ionization and radiative transport are taken into account. We take into consideration the real air spectrum — the numerous lines, bands, and continuua. Account for the radiation leads to an integrodifferential system of equations for which a solution method is developed. As a result we obtain the gas parameter profiles behind the shock wave, which are affected by the relaxation processes and radiative cooling. The calculations were made for v=10–16 km/sec and a pressure p=10^–5–10^–2 atm ahead of the front.In order to obtain realistic results, we consider only the gas layer bounded by the shock and a surface parallel to it. It is assumed that the gas bounded by these planes is not irradiated from without. In this formulation still another defining parameter appears—the distancel between the planes. The calculations were made forl=1–100 cm.     

Weakly nonlinear magnetosonic waves and structure of low-intensity discontinuities in magnetohydrodynamics

Mathematical techniques are proposed which make it possible to reduce the system of magnetohydrodynamic equations for a viscous heat-conducting gas with finite electric conductivity and a general equation of state to the model Burgers equation. On the basis of this equation the structure of weakly nonlinear magnetohydrodynamic shock waves is studied. In particular, the width of the shock wave is estimated.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 43–48, May–June, 1993.     

‘Generalized Burgers' equation’ for nonlinear viscoelastic waves

This paper deals with nonlinear longitudinal waves in a viscoelastic medium in which the viscoelastic relaxation function has the form K(t) = const. t^-v (0<v<1). This sort of slow relaxation may be more appropriate for polymers than the often used exponential relaxation. For a far field evolution of unindirectional waves, a “generalized Burgers' equation” is obtained, which is of a form with the second derivative in the usual Burgers' equation replaced by the derivative of real order 1 + v. The steady shock solution and self-similar pulse solution to this equation are discussed. In both cases numerical solutions are presented and analytic results are obtained for the asymptotic behaviors of the solutions. It is found that both shock and pulse solutions rise exponentially, but in their tails they have slow, algebraic decay.     

Integral inequalities in the dynamics of a gravitating gas

In the framework of Newtonian mechanics, a study is made of the spherically symmetric problem of the adiabatic motion of a gravitating perfect gas in the presence of a shock wave produced by inhomogeneous gravitational collapse or a point explosion. The method of Golubyatnikov [1, 2] is used to construct a system of integro—differential inequalities that determine, in particular, the law of motion of the shock wave if the initial state of the gas is known. The investigated examples include some self-similar and nearly self-similar solutions to the problem of the gravitational contraction of dust with the formation of a strong shock wave, possibly with the release of energy; the self-similar problem of a point explosion in a gas at rest; and also the problem of the equilibrium of a gas sphere for =4/3 and arbitrary distribution of the entropy. In these cases, the inequalities reduce to algebraic relations and can be solved numerically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 169–173, September–October, 1980.     

Evolution of cylindrical waves of finite amplitude diverging in heated gas

The behavior of weak cylindrical and spherical waves of finite amplitude in a dissipative gas close to the wave front is described by a generalized Burgers equation [1]. The construction of various types of solution of this equation for large Reynolds numbers is known [1–3]. For the evolution of diverging perturbations in heated gas, a study of this equation in the region Re < 1, where Re is the effective Reynolds number at the initial time, is of interest. The direct application of the method of successive approximations to this problem is restricted by the condition Re 1, and becomes more and more difficult as the Reynolds number grows and the form of the initial wave becomes more complex. This paper describes in explicit form the construction of an approximate solution of the Cauchy problem for the generalized Burgers equation in the case of cylindrical symmetry in the region Re < 1. The initial wave selected is the arbitrary perturbation represented by a function which is absolutely integrable on the real axis. An integral estimate of the error as a function of Re is given. The question of how the structure of the solution corresponds to the Cole-Hopf transformation is discussed. All the treatment can easily be extended to the spherically symmetric case.Translated from Izvestlya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 150–153, July–August, 1985.     

NONLINEAR WAVES AND PERIODIC SOLUTION IN FINITE DEFORMATION ELASTIC ROD

A nonlinear wave equation of elastic rod taking account of finite deformation, transverse inertia and shearing strain is derived by means of the Hamilton principle in this paper. Nonlinear wave equation and truncated nonlinear wave equation are solved by the Jacobi elliptic sine function expansion and the third kind of Jacobi elliptic function expansion method. The exact periodic solutions of these nonlinear equations are obtained, including the shock wave solution and the solitary wave solution. The necessary condition of exact periodic solutions, shock solution and solitary solution existence is discussed.     

有限变形弹性杆中的非线性波及周期解

利用Ham ilton变分原理,导出了计及有限变形和横向Possion效应的弹性杆中非线性纵向波动方程.利用Jacob i椭圆正弦函数展开和第三类Jacob i椭圆函数展开法,对该方程和截断的非线性方程进行求解,得到了非线性波动方程的两类准确周期解及相应的孤波解和冲击波解,讨论了这些解存在的必要条件.     

Dynamics of a flat cavity in a low-viscosity liquid

The problem of the motion of a cavity in a plane-parallel flow of an ideal liquid, taking account of surface tension, was first discussed in [1], in which an exact equation was obtained describing the equilibrium form of the cavity. In [2] an analysis was made of this equation, and, in a particular case, the existence of an analytical solution was demonstrated. Articles [3, 4] give the results of numerical solutions. In the present article, the cavity is defined by an infinite set of generalized coordinates, and Lagrange equations determining the dynamics of the cavity are given in explicit form. The problem discussed in [1–4] is reduced to the problem of seeking a minimum of a function of an infinite number of variables. The explicit form of this function is found. In distinction from [1–4], on the basis of the Lagrauge equations, a study is also made of the unsteady-state motion of the cavity. The dynamic equations are generalized for the case of a cavity moving in a heavy viscous liquid with surface tension at large Reynolds numbers. Under these circumstances, the steady-state motion of the cavity is determined from an infinite system of algebraic equations written in explicit form. An exact solution of the dynamic equations is obtained for an elliptical cavity in the case of an ideal liquid. An approximation of the cavity by an ellipse is used to find the approximate analytical dependence of the Weber number on the deformation, and a comparison is made with numerical calculations [3, 4]. The problem of the motion of an elliptical cavity is considered in a manner analogous to the problem of an ellipsoidal cavity for an axisymmetric flow [5, 6]. In distinction from [6], the equilibrium form of a flat cavity in a heavy viscous liquid becomes unstable if the ratio of the axes of the cavity is greater than 2.06.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 15–23, September–October, 1973.The author thanks G. Yu. Stepanov for his useful observations.     

Solution to the Riemann problem for drift-flux model with modified Chaplygin two-phase flows

In this paper, we concern about the Riemann problem for compressible no-slip drift-flux model which represents a system of quasi-linear partial differential equations derived by averaging the mass and momentum conservation laws with modified Chaplygin two-phase flows. We obtain the exact solution of Riemann problem by elaborately analyzing characteristic fields and discuss the elementary waves namely, shock wave, rarefaction wave and contact discontinuity wave. By employing the equality of pressure and velocity across the middle characteristic field, two nonlinear algebraic equations with two unknowns as gas density ahead and behind the middle wave are formed. The Newton–Raphson method of two variables is applied to find the unknowns with a series of initial data from the literature. Finally, the exact solution for the physical quantities such as gas density, liquid density, velocity, and pressure are illustrated graphically.     

Reflection of an Oblique Shock Wave in a Reacting Gas with a Finite Relaxation–Zone Length

Reflection of an oblique shock wave in a reacting gas with a finite length of the chemical–reaction zone is studied. Shock polars for an arbitrary heat release behind the oblique shock wave are constructed. Transition criteria from regular to Mach reflection and back are obtained. It is shown that transition criteria are significantly changed if the reaction–zone length is taken into account.     

关于超音速流中运动激波若干问题的研究

研究了在无粘完全气体流中的运动激波 ,讨论了激波运动速度D和来流速度U对激波后气流参数的影响 ,包括对激波后的总焓比值和总压比值以及对流转角的影响。计算结果表明它们不同于通常静止激波下所得到的结果。该内容涉及到超音速射流与障碍物或空腔体相互作用时出现的失稳状态下激波的振动和空腔体底部的反常加热问题。     

梁中非线性弯曲波传播特性的研究

研究了梁中的非线性弯曲波的传播特性，同时考虑了梁的大挠度引起的几何非线性效应和 梁的转动惯性导致的弥散效应，利用Hamilton变分法建立了梁中非线性弯曲波的波动方程. 对该方程进行了定性分析，在不同的条件下，该方程在相平面上存在同宿轨道或异宿轨道， 分别对应于方程的孤波解或冲击波解. 利用Jacobi椭圆函数展开法，对该非线性方程进行 求解，得到了非线性波动方程的准确周期解及相对应的孤波解和冲击波解，讨论了这些解存 在的必要条件，这与定性分析的结果完全相同. 利用约化摄动法从非线性弯曲波动方程中导 出了非线性Schr\"{o}dinger方程，从理论上证明了考虑梁的大挠度和转动惯性时梁中存在 包络孤立波.     

A generalized auxiliary equation method and its applications

In this paper, a generalized auxiliary equation method with the aid of the computer symbolic computation system Maple is proposed to construct more exact solutions of nonlinear evolution equations, namely, the higher-order nonlinear Schrödinger equation, the Whitham–Broer–Kaup system, and the generalized Zakharov equations. As a result, some new types of exact travelling wave solutions are obtained, including soliton-like solutions, trigonometric function solutions, exponential solutions, and rational solutions. The method is straightforward and concise, and its applications are promising.