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.     

Interaction of a centered isentropic compression wave with the bow shock ahead of a body in a rarefied gas

Rarefied gas flow with a centered isentropic compression wave is investigated using direct Monte Carlo simulation of the solution of the Boltzmann equation. For monatomic gas flow the pattern of formation of a suspended compression shock near the geometric center of the compression wave is considered. The flow pattern is compared with the results obtained within the framework of gas dynamics. For a diatomic gas the interference of a centered compression wave with the bow shock ahead of a cylinder is investigated. The dependence of the pressure and the heat transfer to the surface on the Reynolds number and the wave center position relative to the cylinder center is analyzed. The results are compared with those of numerical simulation of the Euler and boundary-layer equations.     

Dynamics of shock waves in a liquid containing gas bubbles

Results are presented of a numerical solution of the Korteweg-de Vries-Burgers equation that describes the propagation and establishment process for a stationary structure to a shock wave in a gas-liquid medium. Data are obtained on the time for the establishment of a stationary structure of a shock wave, propagation velocity, and amplitude oscillations in the front of the shock wave. Experiments are discussed on the basis of the results obtained for the study of shock waves in a liquid containing gas bubbles.     

Delta Shock Wave Solution of the Riemann Problem for the Non-homogeneous Modified Chaplygin Gasdynamics

The motivation of the present study is to derive the solution of the Riemann problem for modified Chaplygin gas equations in the presence of constant external force. The analysis leads to the fact that in some special circumstances delta shock appears in the solution of the Riemann problem. Also, the Rankine–Hugoniot relations for delta shock wave which are utilized to determine the strength, position and propagation speed of the delta shocks have been derived. Delta shock wave solution to the Riemann problem for the modified Chaplygin gas equation is obtained. It is found that the external force term, appearing in the governing equations, influences the Riemann solution for the modified Chaplygin gas equation.

     

Propagation of acoustic waves and shock waves radiated by a sinusoidal pulsation of a sphere during a single period

A spherical sound wave is emitted by a sphere which executes a small sinusoidal pulsation of a single period at high frequency in an inviscid fluid. Nonlinear propagation of the waves is formulated as an initial boundary value problem and is analysed in detail. The governing equation is linear near the sphere, while it is a nonlinear hyperbolic equation in a far field. The nonlinearity has a significant effect there, leading to the formation of two shocks. The exact solution to match the near field solution can easily be obtained for the far field equation. The nonlinear distortion of waveform and the shock formation distance are evaluated from the representation of the solution with strained coordinates. The evolution and nonlinear attenuation of the two shock discontinuities are also examined by making use of the equal-areas rule. In its asymptotic form the entire profile is an N wave with a long tail.     

Effect of surface gas condensation on the velocity of a reflected shock wave

The time-dependent one-dimensional problem of the normal reflection of a shock wave propagating at constant velocity in a gas (vapor) at rest from the plane surface of its condensed phase under steady-state condensation-evaporation conditions on the interphase plane is considered within the framework of the kinetic equation for a monatomic gas with a model collision operator (S-model). The solution is obtained using a conservative second-order finite-difference method. Attention is concentrated on the steady-state regime of the condensation process. The effect of the condensation (evaporation) coefficient on the velocity of the reflected shock wave is studied.     

Supersonic flow of a combustible gas mixture past a sphere

In recent years considerable interest has developed in the problems of steady-state supersonic flow of a mixture of gases about bodies with the formation of detonation waves and slow combustion fronts. This is due in particular to the problem of fuel combustion in a supersonic air stream.In [1] the problem of supersonic flow past a wedge with a detonation wave attached to the wedge apex is solved. This solution is based on using the equation of the detonation polar obtained in [2]-the analog of the shock polar for the case of an exothermic discontinuity. In [3] a solution is given of the problem of cone flow with an attached detonation wave, and [4] presents solutions of the problems of supersonic flow past the wedge and cone with the formation of attached adiabatic shocks with subsequent combustion of the mixture in slow combustion fronts. In the two latter studies two different solutions were also found for the problem of flow past a point ignition source, one solution with gas combustion in the detonation wave, the other with gas combustion in the slow combustion front following the adiabatic shock. These solutions describe two different asymptotic pictures of flow of a combustible gas mixture past bodies.In an experimental study of the motion of a sphere in a combustible gas mixture [5] it was found that the detonation wave formed ahead of the sphere splits at some distance from the body into an ordinary (adiabatic) shock and a slow combustion front. Arguments are presented in [6] which make it possible to explain this phenomenon and in certain cases to predict its occurrence.The present paper presents examples of the calculation of flow of a combustible gas mixture past a sphere with a detonation wave in the case when the wave does not split. In addition, the flow near the point at which the detonation wave splits is analyzed for the case when splitting occurs where the gas velocity behind the wave is greater than the speed of sound. This analysis shows that in the given case the flow calculation may be carried out without any particular difficulties. On the other hand, the calculation of the flow for the case when the point of splitting is located in the subsonic portion of the flow behind the wave (or in the region of influence of the subsonic portion of the flow) presents difficulties. This flow case is similar to the problem of the supersonic jet of finite width impacting on an obstacle.     

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.     

Pressure of an arbitrary acoustical wave on a plane

A study is made of the perturbed flow of a gas, brought about by a weak shock wave, falling on a fixed surface at an arbitrary angle. A solution determining the field of the velocities behind the front of the wave in an initially boundary-value problem with movable boundaries for a three-dimensional wave equation is obtained in the form of a double integral, containing an arbitrarily given function determining the parameters of the gas in the incident wave. The region of integration is a region included within an ellipse, whose relative eccentricity is equal to the sine of the angle of inclination of the front of the incident wave. A formula is obtained for the distribution of the pressure at the plane.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 114–116, January–February, 1975.     

Radiative heat transfer in a hypersonic gas flow generated a spherical source past a plane

An asymptotic solution of radiative gas-dynamic equations for stationary interaction of two hypersonic gas flows emanating from two identical spherical sources is obtained. Under the assumption that the gas in the shock layer is in local thermodynamic equilibrium and volume emission (energy loss for radiation) occurs there, analytical expressions for the distributions of gas-dynamic functions and temperature are derived. The shock wave shape and the radiant flux on the contact plane are examined as functions of problem parameters.     

Focusing of a shock wave in a rarefied gas A numerical study

The process of focusing of a shock wave in a rarefied noble gas is investigated by a numerical solution of the corresponding two dimensional initial–boundary value problem for the Boltzmann equation. The numerical method is based on the splitting algorithm in which the collision integral is computed by a Monte Carlo quadrature, and the free flow equation is solved by a finite volume method. We analyse the development of the shock wave which reflects from a suitably shaped reflector, and we study influence of various factors, involved in the mathematical model of the problem, on the process of focusing. In particular, we investigate the pressure amplification factor and its dependence on the strength of the shock and on the accommodation coefficient appearing in the Maxwell boundary condition modelling the gas-surface interaction. Moreover, we study the dependence of the shock focusing phenomenon on the shape of the reflector, and on the Mach number of the incoming shock. Received 25 May 1998 / Accepted 4 January 2000     

Generalized and exact solutions for oblique shock waves of real gases with application to real air

Calculation of the oblique shock wave of real gases is a difficult and time consuming problem because it involves numerical solution of a set of 10 equations, two of which (i.e., the equation of state and enthalpy function)—if available—are of a very complicated algebraic form. The present work presents a generalized method for calculating oblique shock waves of real gases, based on the Redlich-Kwong equation of state. Also described is an exact method applicable when the exact equation of state and enthalpy function of a real gas are available. Application of the generalized and the exact methods in the case of real air showed that the former is very accurate and at least twenty times faster than the latter. An additional contribution of the study is the derivation of real gas oblique shock wave equations, which are of the same algebraic form as the well known ideal gas normal shock wave relations.     

Shock wave formation as a result of interaction with a weak discontinuity on the boundary of a local subsonic zone

A self-similar solution, which explains the formation of a strong-family shock wave (Mach number behind the wave less than unity) on the sonic line, is obtained for the Tricomi equation of plane potential flow in hodograph variables. A characteristic with a discontinuity of the derivatives of the gas dynamic parameters arrives at the formation (interaction) point, while the characteristic of the other family leaving this point does not contain a singularity. The intensity of the shock wave varies along its generator in accordance with a power law with an exponent close to unity. At the interaction point the discontinuity of the derivatives along the streamline is equal to infinity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 152–158, July–August, 1990.The results were presented at the G. G. Chernyi seminar. The author is grateful to the seminar director and the participants for useful discussions.     

Analytical Solution for Isothermal Flow in a Shock Tube Containing Rigid Granular Material

Analytical solution of shock wave propagation in pure gas in a shock tube is usually addressed in gas dynamics. However, such a solution for granular media is complex due to the inclusion of parameters relating to particles configuration within the medium, which affect the balance equations. In this article, an analytical solution for isothermal shock wave propagation in an isotropic homogenous rigid granular material is presented, and a closed-form solution is obtained for the case of weak shock waves. Fluid mass and momentum equations are first written in wave and (mathematical) non-conservation forms. Afterwards by redefining the sound speed of the gas flowing inside the pores, an analytical solution is obtained using the classical method of characteristics, followed by Taylor’s series expansion based on the assumption of weak flow which finally led to explicit functions for velocity, density and pressure. The solution enables plotting gas velocity, density and pressure variations in the porous medium, which is of high interest in the design of granular shock isolators.     

Interaction of a weak discontinuity wave with a characteristic shock in a dusty gas

In this paper, the evolution of a characteristic shock in a dusty gas is investigated and its interaction with a weak discontinuity wave is studied. The transport equation for the amplitude of the weak discontinuity wave, which is of Bernoulli type, is obtained. The amplitudes of the reflected and transmitted waves after interaction of the weak discontinuity with the characteristic shock are evaluated by using the results of the general theory of wave interaction.      

Investigation of the influence of blowing on the flow in a hypersonic viscous shock layer near the stagnation streamline on a blunt body

In the framework of the approximation of local similarity to the Navier-Stokes equations, an investigation is made of the axisymmetric flow of homogeneous gas in a hypersonic shock layer, this including the region of transition through the shock wave. Boundary conditions, which take into account blowing of gas, are specified on the surface of the body and in the undisturbed flow. A numerical solution to the problem is obtained in a wide range of variation of the Reynolds number and the blowing parameter. Expressions are found for the dependences on the blowing parameter usually employed in boundary layer theory of the coefficients of friction and heat transfer on the surface of the body, which are divided by their values obtained for blowing parameter equal to zero. It is shown that these dependences are universal and the same as the dependences obtained from the solution of the equations of a hypersonic viscous shock layer with modified Rankin-Hugoniot relations across the shock wave and from the solution of the boundary layer equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 199–202, January–February, 1980.     

Nonlinear Wave Structures in a Thermally Relaxing Gas-Liquid Mixture

An evolutionary equation describing a nonlinear wave process in a gas-bubble-liquid mixture is derived. In the mixture interphase heat transfer takes place due to deviation of the gas behavior from adiabatic. Exact partial solutions describing the structures of both shock waves and solitons are given. The mechanism of maximum compression in a shock wave structure propagating in a mixture containing bubbles of a dissolving gas is elucidated. The interval of variation of the input bubble radius on which, as a result of compression, the steady-state wave profile is nonmonotonic is found. A wave profile with an oscillatory structure is shown to exist. Numerical calculations based on the formulas obtained are found to be in fairly satisfactory agreement, at least qualitatively, with the experimental data known to the author.     

Projective Submodel of the Ovsyannikov Vortex

A submodel of the Ovsyannikov vortex with projective symmetry is studied. Integration of the factor system of the submodel reduces to solving a first-order differential equation which is not solved with respect to the derivative. The properties of the solutions of this equation are studied. It is shown that the submodel describes gas ow with a nonstationary source and a nonstationary sink. The problem of the motion of a gas volume between pistons of cylindrical shapes is studied, and its solution with an invariant shock wave is obtained.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 3–16, July–August, 2005.     

Geometrical nonlinear waves in finite deformation elastic rods

IntroductionSomenewphenomenaofnonlinearwavesinthesolidmediumsuchasshockwave ,solitarywaveetc.arepaidmoreattentiontoincreasinglybyresearchersbecausetheytakeonalotofimportantproperties.ItistheoreticallyanalyzedinRefs.[1 -6]thattheformationmechanismsofshockwaveandsolitarywaveintheelasticthinrodsaswellastheirpropagationproperties.TheexistenceofsolitarywaveintheelasticmediumsuchasarodandaplatehasbeenverifiedinRef.[7]byexperiments.Shockwaveandsolitarywavearesteadilypropagatingtraveling_wavesgenerat…