Regular reflection of a magnetohydrodynamic shock wave from a conducting wall

In two-dimensional supersonic gasdynamics, one of the classical steady-state problems, which include shock waves and other discontinuities, is the problem concerning the oblique reflection of a shock wave from a plane wall. It is well known [1–3] that two types of reflection are possible: regular and Mach. The problem concerning the regular reflection of a magnetohydrodynamic shock wave from an infinitely conducting plane wall is considered here within the scope of ideal magnetohydrodynamics [4]. It is supposed that the magnetic field, normal to the wall, is not equal to zero. The solution of the problem is constructed for incident waves of different types (fast and slow). It is found that, depending on the initial data, the solution can have a qualitatively different nature. In contrast from gasdynamics, the incident wave is reflected in the form of two waves, which can be centered rarefaction waves. A similar problem for the special case of the magnetic field parallel to the flow was considered earlier in [5, 6]. The normal component of the magnetic field at the wall was equated to zero, the solution was constructed only for the case of incidence of a fast shock wave, and the flow pattern is similar in form to that of gasdynamics. The solution of the problem concerning the reflection of a shock wave constructed in this paper is necessary for the interpretation of experiments in shock tubes [7–10].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 102–109, May–June, 1977.The author thanks A. A. Barmin, A. G. Kulikovskii, and G. A. Lyubimov for useful discussion of the results obtained.     

Interaction of rarefaction waves with wet foam

The interaction of rarefaction waves of different shapes with wet water foams is studied experimentally. It is found that the observed values of the pressure are greater, while the surface velocity is lower than the corresponding values predicted by the pseudogas model. The foam breakdown starts as the pressure decreases by 0.3 atm relative to the initial pressure. During downstream propagation of the rarefaction-wave leading edge the propagation velocity decreases.Using of water-based foams as effective screens for damping blast waves in different technological processes has caused considerable interest in studying wave propagation in such systems. The pressure wave dynamics in a foam have been investigated in much detail, both experimentally and theoretically [1–3]. However, the interaction of rarefaction waves with foam has practically never been studied, although it was mentioned in [4] that the unloading phase following the compression wave phase is one of the factors defining the damaging action of blast waves. Besides blast-wave damping, rarefaction wave propagation takes place if such waves are used to breakup foam in oil-producing wells [5].Below, the interaction of rarefaction waves of different shapes with wet water foams is studied. The vertical shock tube described in detail in [3] was used in these experiments.Brest. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 76–82, March–April, 1995.     

Shock wave in a gas-liquid medium

The propagation of weak shock waves and the conditions for their existence in a gas-liquid medium are studied in [1]. The article [2] is devoted to an examination of powerful shock waves in liquids containing gas bubbles. The possibility of the existence in such a medium of a shock wave having an oscillatory pressure profile at the front is demonstrated in [3] based on the general results of nonlinear wave dynamics. It is shown in [4, 5] that a shock wave in a gas-liquid mixture actually has a profile having an oscillating pressure. The drawback of [3–5] is the necessity of postulating the existence of the shock waves. This is connected with the absence of a direct calculation of the dissipative effects in the fundamental equations. The present article is devoted to the theoretical and experimental study of the structure of a shock wave in a gas-liquid medium. It is shown, within the framework of a homogeneous biphasic model, that the structure of the shock wave can be studied on the basis of the Burgers-Korteweg-de Vries equation. The results of piezoelectric measurements of the pressure profile along the shock wave front agree qualitatively with the theoretical representations of the structure of the shock wave.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 65–69, May–June, 1973.     

Investigation of surfaces of discontinuity in perfectly conducting magnetizing media

Relationships on discontinuities in magnetizing perfectly conducting media in a magnetic field are investigated. The magnetic permeabilities before and after the discontinuity are assumed to be constant, but unequal, quantities. It is shown that shocks of two kinds, fast and slow, are possible in the formulation under consideration in the hydrodynamics of magnetizing media, as in magnetic hydrodynamics: It is shown that the entropy decreases on the rarefaction shocks diminishing the magnetic permeability, but can grow on the rarefaction shocks increasing the magnetic permeability, but such waves are not evolutionary. The relationships on discontinuities in the mechanics of a continuous medium are written down in general form in [1] with the electromagnetic field, polarization, and magnetization effects taken into account. Relationships on discontinuities in the ferrohydrodynamic and elec trohydrodynamic approximations were written down in [2] and [3–5], respectively, for the cases when the magnetic permeability and dielectric permittivity of the medium ahead of and behind the discontinuity are arbitrary functions of their arguments and are identical. A system of relationships on discontinuities propagated into a magnetizing perfectly conducting medium is investigated in this paper. The method proposed in [6] is used in the investigation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 104–110, January–February, 1976.We are grateful to A. A. Barmin for discussing the paper and for valuable remarks.     

Equations of hydrodynamics for wind wave circulation on a shelf

In order to predict the propagation of an impurity and water quality on a shelf it is necessary to know the water mass dynamics and the water exchange. However, the hydrodyamics of the shelf zone differ considerably from those of the open expanses of seas and lakes owing to the steepness of the bottom, the complex structure of the shoreline, the major role of wind waves, and their breaking [1]. In [2, 3] the importance of surface waves and their breaking for inshore flows was demonstrated and the equations of hydrodynamics, averaged over the depth, were derived. For regions of the shelf remote from the shoreline it is also necessary to take into account the interaction of waves with the bottom and with essentially three-dimensional flows. In this note the equations of hydrodynamics are derived for wind wave flows averaged over the wave period in the threedimensional formulation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1-, pp. 174–176, January–February, 1987.     

Transmission of waves in a liquid through absorbing layers and subsequent interaction of the waves with an obstacle

The propagation of waves in porous media is investigated both experimentally [1, 2] and by numerical simulation [3–5]. The influence of the relaxation properties of porous media on the propagation of waves has been investigated theoretically and compared with experiments [3, 4]. The interaction of a wave in air that passes through a layer of porous medium before interacting with an obstacle has been investigated with allowance for the relaxation properties [5]. In the present paper, in which the relaxation properties are also taken into account, a similar investigation is made into the interaction with an obstacle of a wave in a liquid that passes through a layer of a porous medium before encountering the obstacle.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 53–53, March–April, 1983.     

Numerical simulation of shock wave intersections

A large number of papers, generalized and classified in [1, 2], have been devoted to unsteady gas flows arising in shock wave interaction. Experimental results [3–5] and theoretical analysis [6–9] indicate that the most interesting and least studied types of interaction arise in cases when there are several shock waves. At the same time, nonlinear effects, which depend largely on the nature of the shock wave intersections, become appreciable. Regions of existence of different types, of plane shock wave intersections have been analyzed in [10–13]. It has been shown that in a number of cases the simultaneous existence of different types of intersections is possible. The aim of the present paper is to study unsteady shock wave intersections in the framework of a numerical solution of the axisymmetric boundary-value problem that arises in the diffraction of a plane shock wave on a cone in a supersonic gas flow. Flow regimes that augment the experimental data of [3–5] and the theoretical analysis of [9] are considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 134–140, September–October, 1986.     

Analysis of some supersonic nonsymmetric nozzles

Supersonic nonsymmetric plane nozzles, which are characterized mainly by centered compression and rarefaction waves, were constructed earlier in [1]. An intensive compression wave is undesirable in a number of cases, because of the possibility of boundary-layer detachment on the nozzle walls, for instance. Hence, some constraints on the pressure gradient, for example, the condition that the pressure does not grow at the walls, must be included in the formulation of the problem. The presence of this last condition distinguishes the results in [1] only by the fact that the centered compression waves in the nozzles are replaced by constant parameter sections. Considered below under identical conditions are nonsymmetric nozzles with compression waves, with constant parameter sections, and, also, known nozzles with a straight lower wall for comparison.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 154–156, May–June, 1976.The author is grateful to A. N. Kraiko for his interest and attention to the research and for useful discussions.     

Nonstationary waves in the continuously stratified flow of a fluid of finite depth

A theoretical investigation is made of the development of linear two-dimensional waves in a continuously stratified flow of an ideal incompressible fluid. The waves are generated by pressures that are independent of time and that are applied at time t=0 to a bounded region on the free surface of an initially undisturbed flow. The stationary internal waves generated by such a disturbance have been investigated in [1–3]. The nonstationary waves in a continuously stratified fluid that are generated by initial disturbances or periodic surface pressures applied to the entire free surface have been studied in [4–7] in the absence of a slow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 87–93, November–December, 1976.     

Propagation of nonlinear body waves in a magnetic tube

The propagation of slow symmetrical small-amplitude body waves in a cylindrical magnetic tube is investigated on the basis of the nonlinear equation obtained in [3, 4]. The breaking of periodic disturbances of a certain type in a finite time is numerically demonstrated. It is noted that the equation in question does not have solutions in the form of solitary waves.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 81–84, January–February, 1990.The author is grateful to M. S. Ruderman for formulating the problem and to V. B. Baranov for his interest in the work.     

Flow of a fluid across an obstacle with breaking of the wave front

The phenomenon of breaking of surface gravity waves on the sea in a coastal zone has much in common with a steady hydraulic jump. When surface waves break a vortex forms which rolls down along the leading slope of the wave, as in the case of the formation of a hydraulic jump. Up to the present time a number of theoretical flow models have been proposed for the region in which a wave front breaks. Many of these studies are discussed in [1, 2]. However, the questions of the origin and dynamics of the vortex and its effect on the flow remain open. With the object of studying these questions in greater detail, the results are given below of numerical and experimental simulation of a hydraulic jump. The results of numerical calculations of the shape of the free surface and the velocity profiles in the various sections agree fairly well with the experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 103–106, May–June, 1985.     

Strong explosion in a combustible gas mixture

Assume that a planar, cylindrical, or spherical point explosion takes place in a combustible mixture of gases. As a result of the explosion a strong shock wave develops and triggers chemical reactions with the release of heat. The solution of the problem for the case in which the thickness of the heat release zone is neglected (the infinitely thin detonation wave model) was obtained in [1–3].It was emphasized in [4] that these solutions can be considered only as asymptotic solutions for time and distance scales which are large in comparison with the scales which are characteristic for the chemical reactions, and under the assumption that as the overdriven detonation wave which is formed in the explosion is weakened by the rarefaction waves it does not degenerate into an ordinary compression shock. Here the question remains open of the possibility of obtaining such asymptotic solutions with account for finite chemical-reaction rates.In conclusion the authors wish to thank E. Bishimov for carrying out most of the computations for this study.     

One-dimensional linear waves with axial and central symmetries in saturated porous media

The features of propagation of one-dimensional monochromatic waves and dynamics of weak perturbations with axial and central symmetries in liquid-saturated porous medium are investigated. Non-stationary interaction forces and viscoelastic skeleton characteristics are taken into account. The research is carried out within the two-velocity, two-stress tensor model by applying methods of multiphase media mechanics. The system of equations is solved numerically by applying Fast Fourier Transform (FFT) algorithm. The influence of geometry of the process on wave propagation behavior is studied.It is shown that the initial pressure perturbation splits into two waves: fast (deformational) wave and slow (filtrational) one. Each of them is followed by the balance wave: that is, rarefaction wave after compression wave and compression wave after rarefaction wave; at that slow wave and balance one following fast wave may interfere.     

Simple stationary waves in an inclined magnetic field

One component of the solution to the problem of flow around a corner within the scope of magnetohydrodynamics, with the interception or stationary reflection of magnetohydrodynamic shock waves, and also steady-state problems comprising an ionizing shock wave, is the steady-state solution of the equations of magnetohydrodynamics, independent of length but depending on a combination of space variables, for example, on the angle. The flows described by these solutions are called stationary simple waves; they were considered for the first time in [1], where the behavior of the flow was investigated in stationary rotary simple waves, in which no change of density occurs. For a magnetic wave, of parallel velocity, the first integrals were found and the solution was reduced to a quadrature. The investigations and the applications of the solutions obtained for a qualitative construction of the problems of streamline flow were continued in [2–8]. In particular, problems were solved concerning flow around thin bodies of a conducting ideal gas. The general solution of the problem of streamline flow or the intersection of shock waves was not found because stationary simple waves with the magnetic field not parallel to the flow velocity were not investigated. The necessity for the calculation of such a flow may arise during the interpretation of the experimental results [9] in relation to the flow of an ionized gas. In the present paper, we consider stationary simple waves with the magnetic field not parallel to the flow velocity. A system of three nonlinear differential equations, describing fast and slow simple waves, is investigated qualitatively. On the basis of the pattern constructed of the behavior of the integral curves, the change of density, magnetic field, and velocity are found and a classification of the waves is undertaken, according to the nature of the change in their physical quantities. The relation between waves with outgoing and incoming characteristics is explained. A qualitative difference is discovered for the flow investigated from the flow in a magnetic field parallel to the flow velocity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 130–138, September–October, 1976.The author thanks A. A. Barmin and A. G. Kulikovskii for constant interest in the work and for valuable advice.     

Propagation of modulated nonlinear waves in a relaxing gas-liquid mixture

The propagation of nonstationary weak shock waves in a chemically active medium is essentially dispersive and dissipative. The equations for short-wavelength waves for such media were obtained and investigated in [1–4]. It is of interest to study quasimonochromatic waves with slowly varying amplitude and phase. A general method for obtaining the equations for modulated oscillations in nonlinear dispersive media without dissipation was proposed in [5–8]. In the present paper, for a dispersive, weakly nonlinear and weakly dissipative medium we derive in the three-dimensional formulation equations for waves of short wavelength and a Schrödinger equation, which describes slow modulations of the amplitude and phase of an arbitrary wave. The coefficients of the equations are particularized for the considered gas-liquid mixture. Solutions are obtained for narrow beams in a given defocusing medium as well as linear and nonlinear solutions in the neighborhood of a diffraction beam. A solution near a caustic for quasimonochromatic waves was found in [9].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 133–143, January–February, 1980.     

Propagation of shock waves in a medium with exponentially varying density

The results of Raizer [1], Hays [2], and Chernous'ko [3] are generalized to-the case of self-similar propagation of shock waves in a gas with exponentially varying density and constant pressure. A solution is found by the method of successive approximations. The zero-order approximation coincides with the Whitham method [4]. The first-order approximation is in good agreement with numerical calculations in [2]. The non-selfsimilar motion of a weak shock wave is investigated in the framework of linear theory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 48–54, November–December, 1970.     

Rarefied gas flow past a sphere with injection

Hypersonic rarefied gas flow over the windward face of a sphere is considered in the presence of distributed injection from the surface of the body. A similar problem was previously solved in [1–3] within the framework of continuum mechanics and in [4] on the basis of model kinetic equations. In the present study the calculations were carried out using the Monte Carlo method of direct statistical modeling [5, 6]. The injected gas was the same as the free-stream gas. A simple monatomic gas model with a rigid sphere interaction potential was employed. The reflection of the molecules from the surface of the body was assumed to be diffuse with total energy accommodation. The calculation procedure using weighting factors is described in [7]. The influence of injection on the mechanical and thermal effect of the gas flow on the body is investigated for various degrees of rarefaction of the medium and injection rates.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 175–179, July–August, 1990.     

Relaxation of overdriven detonation waves with finite reaction rate

A numerical study is made of the interaction of a detonation wave having finite reaction velocity with a rarefaction wave of different intensity which approaches it from the rear, for the Zeldovich-Neumann-Doring (ZND) model with a single irreversible reaction A B. It is found that, for a fixed value of the parameter characterizing the initial supercompression (depending on the activation energy and the heating value of the mixture), the considered interaction leads either to a gradual relaxation of the detonation wave and its transition to the Chapman-Jouguet (CJ) regime, or to the development of undamped oscillations.Interest in the problems of detonation and supersonic combustion has increased in recent years. This is associated with the appearance and development of new experimental and theoretical techniques; it is also associated with the further development of air-breathing reaction engines, and other practical requirements. The present state of detonation theory is reflected in the survey [1].It has been established [2] that the detonation wave in gases nearly always has a complex nonuniform structure. Transverse disturbances are observed under a wide range of conditions and differ both in amplitude and wavelength. At the same time, behind the detonation leading front there is a region of uncompletely burned gas corresponding to the effective ignition induction period [3]. In spinning detonation the induction period is significantly longer than the heat release period and transverse detonation waves traveling in the induction zone of the head wave appear [3, 4]. Such a secondary detonation wave is free of transverse disturbances. The same is true of the detonation waves observed in the wake behind a body moving at high speed in a combustible medium [5] or in a gas which has been preheated by a shock wave [6].Although it is possible, under favorable conditions, to study in detail the system of discontinuities accompanying detonation, information on the extensive zones in which heat release takes place is scarce, the mechanism of detonation wave autonomy (in particular, the role of the rarefaction zone behind the wave) is not entirely clear, and the fact that, in spite of the complex structure, an autonomous detonation propagates with the CJ velocity calculated on the basis of one-dimensional theory has not yet been explained.In studying the nonlinear phenomena associated with the finite reaction rate it is quite acceptable to investigate only the simple one-dimensional detonation model, with which it is convenient to restrict ourselves to a single effective chemical reaction. This model is particularly reasonable since, in certain cases, the real detonation is virtually one-dimensional.The question of the stability of the one-dimensional detonation wave to disturbances of its structure has been examined by several authors [7–13]. The use of computers makes possible the direct computation of flows with heat release and the study of their properties. This method has been used in [11–13] to study the stability problem for a detonation wave with respect to finite disturbances.In the present paper we present a numerical study of the interaction of a detonation wave having finite chemical reaction rate with a rarefaction wave of different intensity approaching it from the rear for the ZND model with a single irreversible reaction A B. It is found that for a fixed value of the parameter characterizing the difference between detonation and the CJ waves, depending on the activation energy E and the mixture heating value Q_m, the interaction in question leads either to a gradual relaxation of the detonation wave and its transition to the CJ regime (this relaxation may be accompanied by decaying oscillations) or to the appearance of undamped oscillations (the unstable regime). The parameters E and Q_m affect the wave stability differently: with increase of Q_m, the wave is stabilized; with increase of E, it is destabilized. The boundary between the stable and unstable detonation wave propagation regimes is found. This boundary has a weak dependence on the rarefaction wave intensity. Estimates and calculated examples show that the amplitude of the unstable wave oscillations is finite and that the average detonation propagation velocity is close to the CJ velocity computed for the given heating value Qm.The author wishes to thank G. G. Chernyi for his guidance and L. A. Chudov for advice on computational questions.     

The influence of a uniformly compressed floating elastic plate on the development of three-dimensional waves in a homogeneous fluid

A study is made in the linear formulation of the influence of a uniformly compressed floating elastic plate on the unsteady three-dimensional wave motion of a homogeneous fluid of finite depth. Waves are excited by a region of normal stresses which moves on the surface of the plate. Three-dimensional flexural-gravity waves were studied in [1, 2] without allowance for compressing forces. Plane waves under conditions of longitudinal compression were considered in [3, 4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 78–83, November–December, 1984.     

Effect of viscosity on the propagation of nonlinear Alfvén waves along a tangential magnetohydrodynamic discontinuity in an incompressible fluid

It is proposed to consider the propagation of surface waves along a tangential magnetohydrodynamic discontinuity in the particular case where the fluid velocities on both sides of the interface are equal to zero. In [1] it was shown that waves called surface Alfvén waves may be propagated along the surface separating a semi-infinite region without a field from a region with a uniform magnetic field. The linear theory of surface Alfvén waves in a compressible medium was considered in [2]. In [3] the damping of surface Alfvén waves as a result of viscosity and heat conduction was investigated. The propagation of low-amplitude nonlinear surface Alfvén waves in an incompressible fluid in the absence of dissipative processes is described by the integrodifferential equation obtained in [4]. By means of a numerical solution of this equation it was shown that a perturbation initially in the form of a sinusoidal wave will break. The breaking time was determined. In this paper the equation derived in [4] is extended to the case of a viscous fluid. It is shown that the equation obtained does not have steady-state solutions. The propagation of periodic disturbances is investigated numerically. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 94–104, November–December, 1986. The author wishes to thank L. S. Fedorov for assisting with the calculations.