Sound absorption in a shock wave

It was shown in [1–4] that the reflection of a sound wave or its transmission through a shock front should be accompanied by attenuation or intensification of the wave is regarded as a discontinuity. In accordance with current representations [5, 6], a shock wave includes a viscous shock and a lengthy relaxation zone. Equilibrium is established with respect to translational and rotational degrees of freedom in the viscous shock and with respect to internal degrees of freedom in the relaxation zone. The result of the interaction of the shock and sound waves is determined by the relationship between the length of the sound wave and the width of the shock wave.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 90–94, May–June, 1986.     

Waves of finite amplitude in a hot plasma

A study is made of a plane shock wave of arbitrary strength propagating in a hot rarefied plasma across the magnetic field. The question of the propagation of nonstationary waves of finite but small amplitude under these conditions is examined.Fairly detailed studies have been made of waves of finite amplitude in a cold rarefied plasma. The profile of such waves is formed as the result of nonlinear and dispersion effects, the dispersion effects being caused by electron inertia and plasma anisotropy. If the gas-kinetic pressure of the plasma is taken into account, then dispersion effects appear which are associated with the fact that the Larmor radius of the ions is finite. Stationary waves of small but finite amplitude propagating across the magnetic field in a hot plasma (when the gas-kinetic pressure p is comparable with the magnetic pressure H²/87) have been treated in [1, 2]. In [1] an isolated rarefaction wave was found in a hot plasma, instead of the compression wave characteristic of a cold plasma, and a qualitative picture of the shock wave structure was given. In [2] a study was made of a small-amplitude shock wave with the finite size of the ion Larmor radius taken into account. The present paper investigates the structure of shock waves of arbitrary strength which propagate across the magnetic field in a fairly hot rarefied plasma, and also examines nonstationary waves of finite but small amplitude excited in a plasma by a magnetic piston acting over a limited time interval.Notation p gas-kinetic pressure - H magnetic field - u, v macroscopic velocities along the x and y axes - density - m_e(m_i) mass of electron (ion) - plasma conductivity - _H ion-cyclotron frequency - V_A Alfvèn velocity - c velocity of light - adiabatic exponent - V specific volume - _0e(_0i) electron (ion) plasma frequency - S₀ velocity of sound. In conclusion the author thanks R. Z. Sagdeev and N. N. Yanenko for discussing the paper, and also R. N, Makarov for helping with the numerical computations.     

Investigation of the hypersonic viscous shock layer around blunt bodies in a nonuniform flow

Unseparated viscous gas flow past a body is numerically investigated within the framework of the theory of a thin viscous shock layer [13–15]. The equations of the hypersonic viscous shock layer with generalized Rankine-Hugoniot conditions at the shock wave are solved by a finite-difference method [16] over a broad interval of Reynolds numbers and values of the temperature factor and nonuniformity parameters. Calculation results characterizing the effect of free-stream nonuniformity on the velocity and temperature profiles across the shock layer, the friction and heat transfer coefficients and the shock wave standoff distance are presented. The unseparated flow conditions are investigated and the critical values of the nonuniformity parameter a_k [10] at which reverse-circulatory zones develop on the front of the body are obtained as a function of the Reynolds number. The calculations are compared with the asymptotic solutions [10, 12].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 154–159, May–June, 1987.     

Shock wave structure for a fully ionized plasma

We study planar shock wave structure in a two-temperature model of a fully ionized plasma that includes electron heat conduction and energy exchange between electrons and ions. For steady flow in a reference frame moving with the shock, the model reduces to an autonomous system of ordinary differential equations which can be numerically integrated. A phase space analysis of the differential equations provides an additional insight into the structure of the solutions. For example, below a threshold Mach number, the model produces continuous solutions, while above another threshold Mach number, the solutions contain embedded hydrodynamic shocks. Between the threshold values, the appearance of embedded shocks depends on the electron diffusivity and the electron–ion coupling term. We also find that the ion temperature may achieve a maximum value between the upstream and downstream states and away from the embedded shock. We summarize the methodology for solving for two-temperature shocks and show results for several values of shock strength and plasma parameters in order to quantify the shock structure and explore the range of possible solutions. Such solutions may be used to verify hydrodynamic codes that use similar plasma physics models.     

Stability of Contact Discontinuities for the 1-D Compressible Navier-Stokes Equations

In this paper, we study the large-time asymptotic behavior of solutions of the one-dimensional compressible Navier-Stokes system toward a contact discontinuity, which is one of the basic wave patterns for the compressible Euler equations. It is proved that such a weak contact discontinuity is a metastable wave pattern, in the sense introduced in [24], for the 1-D compressible Navier-Stokes system for polytropic fluid by showing that a viscous contact wave, which approximates the contact discontinuity on any finite-time interval for small heat conduction and then runs away from it for large time, is nonlinearly stable with a uniform convergence rate provided that the initial excess mass is zero. This result is proved by an elaborate combination of elementary energy estimates with a weighted characteristic energy estimate, which makes full use of the underlying structure of the viscous contact wave.     

粘性金属中的正激波稳定性问题

G.H.Miller等把高压金属中的粘性激波作为强间断面处理,解析推论出:在大粘性系数条件下小扰动激波是不稳定的,物质粘性是导致失稳的因素。本文中针对平面正激波,认为高压金属中的粘性激波的物理量是连续变化的,利用线性稳定性理论,用数值解推论出:在有粘性条件下小扰动激波都是稳定的,物质粘性是致稳的因素。指出G.H.Miller等获得错误结论的原因在于:从无粘流动解推出的小扰动边界条件导致粘性激波小扰动增长。给出实验确定的小扰动速度梯度的边界条件,这样既可以把粘性正激波作为强间断面处理,也能够保证粘性正激波的稳定性。     

Interaction of a gasdynamic shock with small disturbances

The stability of shock wave based on the definition of Landau and Lifschitz^[1] is treated in this paper. This is tantamount to solving the problem of interaction of small disturbances with a shock wave. Small disturbances are introduced on both sides of a steady, non-dissipative, plane shock wave. Landau et al.^[1] obtained the stability criterionM ₁>1,M ₂<1 for small disturbances which are travelling in the direction perpendicular to the shock wave. In the present paper, we assume that the small disturbances may be two dimensional, i.e. they may be propagating in the direction inclined to the shock wave. The conclusions obtained are: regardless of whether the incident wave and diverging wave are defined according to the direction of the phase velocity or the group velocity, the shock wave is unstable for some frequencies and longitudinal wave lengths of the disturbances, even if the conditionsM ₁>1,M ₂<1 are fulfilled. Then several experiments are proposed, and the problem of ways to define the incident wave and diverging wave is discussed. The meaning of this problem is illustrated. The same results can be obtained for the steady shock wave in a tube.     

Structure of the heating layer ahead of the front of a strong intensely emitting shock wave

The problem of the structure and brightness of strong shock waves arises in the investigation of such phenomena as the motion of large meteoroids in the atmosphere, optical and electrical discharges, the development of strong explosions, and other similar processes and in the creation of powerful radiation sources based on them. This problem also has a general physics interest. As the propagation velocity of a strong shock wave increases the gas temperature behind its front and the role of emission grow. Part of the radiation emitted by the gas heated and compressed in a shock wave is absorbed ahead of the front, forming the so-called heating layer. The quasisteady structure of a strong intensely emitting shock wave was studied in [1, 2]. In this case a diffusional approximation and the assumption of a gray gas were used to describe the radiation transfer. They introduced the concept of a wave of critical amplitude, when the maximum temperature T_- in the heating layer reaches the temperature T_a determined on the basis of the conservation laws, i.e., from the usual shock adiabat; it is shown that behind a compression shock moving through an already heated gas there is a temperature peak in which the maximum temperature T₊ exceeds T_a. The problem of the quasisteady structure of an emitting shock wave in air of normal density was solved numerically in [3]. The angular distribution of the radiation was approximately taken into account — it was assigned by a simple cosinusoidal law. The spectral effects were taken into account in a multigroup approximation. They introduced 38 spectral intervals, which is insufficient to describe a radiation spectrum with allowance for the numerous lines and absorption bands.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 86–92, September–October, 1978.     

Nonequilibrium viscous shock layer on blunt cones

Some results are given of the numerical investigation into the parameters of the nonequilibrium flow of air in a viscous shock layer in the case of blunt circular cones at zero angle of attack; they are also compared with experimental data obtained during re-entry of ballistic objects into the Earth's atmosphere. The calculations were made with allowance for the nonequilibrium processes of dissociation and ionization, and also vibrational relaxation. The influence of viscosity, heat conduction, and diffusion is taken into account in the complete shock layer. The conditions on the shock wave are posed with allowance for its finite thickness. The characteristic profiles of the velocity, temperature, and electron concentration in the shock layer are given. Good agreement is obtained between the calculated and experimental data on the level and the profiles of the electron concentration. The parameters of the shock layer were determined by a method that is a natural extension of the numerical method of [1] to the case of nonequilibrium flow in a viscous shock layer. Because of this, only the main differences of the method when applied to the calculation of nonequilibrium flows of a multicomponent mixture such as dissociated and ionized air are described.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 15–20, November–December, 1979.     

Dynamics of discontinuity formation in a cavitating liquid layer under shock wave loading

The problem of experimental modeling of discontinuity formation in a cavitating liquid layer under shock wave loading is considered. It is shown that the discontinuity takes the shape of a spherical segment and retains it up to the closure instant. The discontinuity surface becomes covered with a dynamically growing thin boundary layer consisting of bubbles, which transforms to a ring-shaped vortex bubble cluster at the instant of the discontinuity closure, generating a secondary shock wave. Specific features of the structure of the cavitating flow discontinuity arising at loading intensities lower than 0.1 and 5 kJ are discussed.     

Propagation of shock waves in free space

The problem of the exit of a shock wave from an axisymmetric channel and its propagation in a free space occupied by an ideal gas is examined. This problem has been studied earlier in [1], in which the shock wave front was considered planar, as well as in [2], in which the wave front was regarded as a surface of an ellipsoid of revolution. The solutions obtained in these studies assumed the presence of two regions in the wave-front surface: the region of the original shock wave and a region stemming from the decomposition of an infinitesimally thin annular discontinuity of the gas parameters, with the wave intensity over the front surface in each region being considered constant, i.e., the wave character of the process over the front was not considered. In this study a solution will be achieved by the method of characteristics [3–5] of the equations of motion of the shock-wave front, as obtained in [6, 7]. Flow fields are determined for the region immediately adjacent to the shock-wave front for a wide range of shock-wave Mach numbers M _a =1.6–20.0 for = 1.4. On the basis of the data obtained, by introduction of variables connected with the length of the undisturbed zone, as calculated from the channel cross-section along the x axis, together with the pressure transition at the wave front, relationships are proposed which approach self-similarity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 163–166, September–October, 1971.In conclusion, the author thanks S. S. Semenov for his valuable advice on this study.     

Flow of a supersonic electrogasdynamic stream,with formation of an electroconducting gaseous shock layer

Surfaces of strong discontinuity in electrogasdynamics were considered in [1, 2]. An investigation was done for the case when a gas has the properties of a unipolar charged medium on both sides of a surface of discontinuity. However, with sufficiently high supersonic gas flow over bodies the gas becomes electroconducting and acquires the properties of a low-temperature plasma in the compressed layer between the shock wave and the body, because of the temperature increase. Therefore, there is great interest in investigating type S* Shockwaves dividing a unipolar charged medium and a low-temperature plasma. The S* waves separating the uncharged medium and a gas with high electrical activity in the presence of an electrical field were studied in [3]. Below we examine the general properties of S* waves (physicalmodel, relations at the wave, conditions for development, shock adiabats, and polars). We formulate the problem of flow of a supersonic electrogasdynamic stream over bodies, with formation of S* waves. A perturbation method is proposed for solution of the problem, using a small parameter to describe electrogasdynamic interaction. By way of example a complete solution for flow over a wedge is constructed.     

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.     

Stabilization of plasma flow in a transverse magnetic field

Results are given of a theoretical and experimental investigation of the intensive interaction between a plasma flow and a transverse magnetic field. The calculation is made for problems formulated so as to approximate the conditions realized experimentally. The experiment is carried out in a magneto-hydrodynamic (MHD) channel with segmented electrodes (altogether, a total of 10 pairs of electrodes). The electrode length in the direction of the flow is 1 cm, and the interelectrode gap is 0.5 cm. The leading edge of the first electrode pair is at x = 0. The region of interaction (the region of flow) for 10 pairs of electrodes is of length 14.5 cm. An intense shock wave S propagates through argon with an initial temperature T_o = 293 °K and pressure p_o = 10 mm Hg. The front S moves with constant velocity in the region x < 0 and at time t = 0 is at x = 0. The flow parameters behind the incident shock wave are determined from conservation laws at its front in terms of the gas parameters preceding the wave and the wave velocity W_S. The parameters of the flow entering the interaction region are as follows: temperature T ₀ ¹ = 10,000 °K, pressure P ₀ ¹ = 1.5 atm, conduction ₀ ¹ = 3000 ^–1·m^–1, velocity of flow u ₀ ¹ = 3000 m·sec^–1, velocity of sounda ₀ ¹ = 1600 m·sec^–1, degree of ionization = 2%, 0.4. The induction of the transverse magnetic field B = [0, B_y(x), 0] is determined only by the external source. Induced magnetic fields are neglected, since the magnetic Reynolds number Re_m 0.1. It is assumed that the current j = (0, 0, jz) induced in the plasma is removed using the segmented-electrode system of resistance R_e. The internal plasma resistance is R_i = h(A)^–1 (h = 7.2 cm is the channel height; A = 7 cm² is the electrode surface area). From the investigation of the intensive interaction between the plasma flow and the transverse magnetic field in [1–6] it is possible to establish the place x* and time t* of formation of the shock discontinuity formed by the action of ponderomotive forces (the retardation wave R_T), its velocity W_T, and also the changes in its shape in the course of its formation. Two methods are used for the calculation. The characteristic method is used when there are no discontinuities in the flow. When a shock wave R_T is formed, a system of nonsteady one-dimensional equations of magnetohydrodynamics describing the interaction between the ionized gas and the magnetic field is solved numerically using an implicit homogeneous conservative difference scheme for the continuous calculation of shock waves with artificial viscosity [2].Translated from Izvestiya Akademiya Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 112–118, September–October, 1977.     

On the role of heat conduction in the formation of a high-temperature plasma during counter collision of rarefaction waves of solid deuterium

This paper considers the problem of counter collision of rarefaction waves of solid deuterium produced by the simultaneous incidence of two identical shock waves on free surfaces located at a certain distance from each other. The motion of deuterium is described by the equations of one-velocity two-temperature hydrodynamics. The model of electron and ion heat transfer takes into account heat-flux relaxation. The parametric properties of the problem are investigated. It is shown that with decreasing distance between the free surfaces, the maximum temperature of the plasma ceases to depend on this parameter. At moderate distances between the free surfaces, the maximum plasma temperature becomes much lower than the temperature obtained earlier in the problem for the equations of nondissipative hydrodynamics. With increasing pressures in the incident shock wave, the maximum ion temperature increases linearly, reaching a value approximately equal to 160 · 10 ⁶ K at 500 Mbar. In the case of a shock wave with a pressure of 50 Mbar at a gap of 2 mm between the free surfaces of deuterium, the yield of fusion neutrons increases roughly by a factor of 10 compared to the yield of neutrons in the case of no gap.     

Motion of a shock wave through a gas of variable density

A successive approximation method is used to solve the self-similar problem of gas flow accompanying a shock wave propagated through a polytropic gas of variable density. The method is based on a special choice of independent variables and the use of Whitham's approximation [1] as the initial approximation for the motion of the discontinuity. A first approximation for the self-simulation index is calculated which is in good agreement with exact values.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 68–72, September–October 1970.The author wishes to thank S. V. Fal'kovich for suggesting this problem and for his help in the work.     

Numerical solution of the problem of wave diffraction by a wedge

The systematic development of the theory of shock reflection from a solid wall started in [1]. Regular reflection and a three-shock configuration originating in Mach reflection were considered there under the assumption of homogeneity of the domains between the discontinuities and, therefore, of rectilinearity of these latter. The difficulties of the theoretical study include the essential nonlinearity of the process as well as the instability of the tangential discontinuity originating during Mach reflection. Analytic solutions of the problem in a linear formulation are known for a small wedge angle or a weak wave (see [2–4], for example). The solution in a nonlinear formulation has been carried out numerically in [5, 6] for arbitrary wedge angles and wave intensities. Since the wave was nonstationary, the internal flow configuration is difficult to clarify by means of the constant pressure and density curves presented. A formulation of the problem for the complete system of gasdynamics equations in self-similar variables is given in [7] and a method of solution is proposed but no results are presented. Difficulties with the instability of the contact discontinuity are noted. The problem formulation in this paper is analogous to that proposed in [7]. However, a method of straight-through computation without extraction of the compression shocks in the flow field is selected to compute the discontinuous flows. The shocks and contact discontinuities in such a case are domains with abrupt changes in the gasdynamics parameters. The computations were carried out for a broad range of interaction angles and shock intensities. The results obtained are in good agreement with the analytical solutions and experimental results. Information about the additional rise in reflection pressure after the Mach foot has been obtained during the solution.     

Hydrodynamical problems of first-order phase transitions

The two-phase liquid-vapor system in a state of thermodynamic equilibrium is considered. If a shock wave propagates in this medium, during its passage the material undergoes shock compression and transforms into a new equilibrium state. Not only the initial velocity changes in this case, but so does the quantitative composition of the phases. Due to the complication of the process, analytic results have practically not been available so far. Calculations of parameters behind the shock discontinuity were carried out approximately by using various tables and nomograms, restricted basically to only one two-phase system, H₂O. Thus, condensation jumps were treated in [1–4] in two-phase supersonic flows within the single-velocity model and a low content of the liquid phase in the mixture. Using the assumptions mentioned, the various parameters were found at the front of the shock wave by numerical solution of the conservation equations of mass, momentum, and energy at the discontinuity. The thermodynamic parameters are usually given in tabulated form as a function of pressure or temperature for equilibrium conditions of the phases.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 81–87, September–October, 1977.