Irregular reflection of a strong shock wave from a thin wedge

The problem of irregular reflection of a strong shock wave from a rigid wall has been studied [1–3] mainly within the framework of the linear theory. It has been found that near the front of a shock wave there exist a region of large gradients of gasdynamic parameters in which the linear theory is no longer valid [4]. In the present paper we consider the nonlinear problem of Mach reflection when there is interaction between a shock wave of high intensity and a thin wedge. The solution of the problem is constructed on the assumption that the ratio of densities along the front of the impinging shock wave is small [5, 6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 55–61, September–October, 1974.In conclusion, the authors wish to express their gratitude to A. A. Grib for his interest in the subject and to L. A. Rumyantsev for his help in carrying out the calculations.     

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.     

Normal interaction of shock waves in the presence of vibrational relaxation

The effect of nonequilibrium physicochemical processes on the flow resulting from the normal collision and reflection of shock waves is studied by the example of nonequilibrium excitation of molecular oscillations in nitrogen. It is shown that the thermal effect of vibrational relaxation is small and the problem can be linearized around a known solution [1]. A similar approach to the solution of the problem of flow around a wedge and certain one-dimensional non-steady-state problems was used earlier in [2–4]. The solution of these problems was constructed in an angular domain, bounded by the shock wave and a solid wall (or the contact surface) and was reduced to a well-known functional equation [6]. The solution of this problem, because of the presence of two angular domains divided by a tangential discontinuity, reduces to a functional equation of more general form than in [6]. The results are obtained in finite form. In the special case of shocks of equal intensity, the normal reflection parameters are obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 90–96, July–August, 1976.     

Reflection of a plane shock wave from a rigid concave wall

We present the results of an experimental study of the reflection of a plane stationary shock wave with Mach number in the range 1.21–1.35 from a rigid cylindrical concave wall. The experiments were carried out in a shock tube. In experimental shock tube technology the reflection of a shock wave from a rigid wall is often used for obtaining high temperatures [1]. This circumstance is associated with the fact that the temperature behind the reflected wave is significantly higher than that behind the incident wave.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 33–39, July–August, 1970.     

Investigation of the properties of a shock wave arising in a supersonic flow of plasma,passing through a transverse magnetic field

In a flow of plasma, set up by an ionizing shock wave and moving through a transverse magnetic field, under definite conditions there arises a gasdynamic shock wave. The appearance of such shock waves has been observed in experimental [1–4] and theoretical [5–7] work, where an investigation was made of the interaction between a plasma and electrical and magnetic fields. The aim of the present work was a determination of the effect of the intensity of the interaction between the plasma and the magnetic field on the velocity of the motion of this shock wave. The investigation was carried out in a magnetohydrogasdynamic unit, described in [8]. The process was recorded by the Töpler method (IAB-451 instrument) through a slit along the axis of the channel, on a film moving in a direction perpendicular to the slit. The calculation of the flow is based on the one-dimensional unsteady-state equations of magnetic gasdynamics. Using a model of the process described in [9], calculations were made for conditions close to those realized experimentally. In addition, a simplified calculation is made of the velocity of the motion of the above shock wave, under the assumption that its front moves at a constant velocity ahead of the region of interaction, while in the region of interaction itself the flow is steady-state.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 86–91, January–February, 1975.     

Numerical simulation of flow during compression of cylindrical samples by a glancing detonation wave

The parameters of shock waves created in cylindrical samples of various materials during detonation of explosive charges surrounding them have been determined experimentally [1–3]. It was established that in a number of materials the reflection of a conical shock wave from the symmetry axis of a sample leads to the formation of a Mach triple shock-wave configuration which gives rise to a complex flow pattern in the region beyond the shock waves. Analytic study of irregular reflection is a complex problem. Solutions obtained under various assumptions about the nature of the flow are presented in papers reviewed in [4]. In the present paper, axisymmetric flow of detonation products (DP) and sample material in the region adjacent to the detonation front is determined from the solution of a two-dimensional, time-dependent problem in gasdynamics by the finite-difference method [5].     

Transition from regular reflection to mach reflection when a shock wave interacts with a wall in a two-phase gas—liquid medium

A study is made of the transition from regular reflection to Mach reflection when a plane moderately strong or weak shock wave interacts with a wall in a two-phase gas—liquid medium. An equilibrium model that differs from the model of Parkin et al. [1] by the introduction of the adiabatic velocity of sound is used to investigate shock wave reflection in the complete range of gas concentrations. For the reflection of weak shock waves, nonlinear asymptotic expansions [2] are used. In the limiting cases, the results agree with those already known for single-phase media [2, 3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 190–192, September–October, 1983.     

Regular reflection of a shock wave from a solid wall in a stream of fuel mixture

The two-dimensional stationary problem of regular reflection of a shock wave from a plane solid wall in a fuel gas mixture is examined in the case when the mixture is ignited at the intersection of the incident wave with the wall and a flame front is formed behind the reflected shock wave. The shock waves and the flame front are considered plane surfaces of discontinuity. The fuel mixture and the reaction products are considered perfect, inviscid, and non-heat-conducting gases.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 161–163, July–August, 1978.     

Initial stage of the collision of blast waves

The collision of two blast waves is analyzed for the case of variable parameters of the gas behind the wave front and wave reflection at a plane, a cylindrical, and a spherical obstacle. The reflection of a blast wave from a nonmoving obstacle is investigated in detail. The problem of the collision of two shock waves with constant parameters behind the front is solved both in the symmetrical case (reflection from a nonmoving wall) and in the case of waves of different amplitudes by a system of algebraic relations for the compression shocks. The reflection of a strong point-source spherical shock wave from a wall has been treated in [1, 2]. The present article examines the initial stage of wave collision for an arbitrary distribution of the parameters behind the front.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 41–48, September–October, 1971.The authors are grateful to V. P. Korobeinikov for a discussion of the results and to V. P. Kolgan for furnishing the numerical solutions.     

Shock strengthening in a wedge-shaped cavity

The unsteady problem of the entry of a shock wave of arbitrary intensity into a wedge-shaped cavity is examined. An exact solution of the non-linear problem of reflection of a plane wave from a nonplanar wall is found for certain cavity angles. Numerical wave focusing calculations are carried out for arbitrary cavity angles. A single scaling law is obtained for gas flows with waves of moderate and high intensity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 123–129, September–October, 1987.     

Sporadic interaction of a strong shock wave with a thin cone

The axisymmetrical problem concerning Mach reflection of an intense shock wave from the surface of a thin cone is considered in a formulation similar to that of [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 117–123, November–December, 1976.The authors express their appreciation to A. A. Grib for discussion of the work and for useful comments.     

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.     

Optimal shapes of bodies with attached shock waves

We consider the problem of finding the shape of two-dimensional and axisymmetric bodies having minimal wave drag in a supersonic perfect gas flow. The solution is sought among bodies having attached shock waves. The limitations on the body contour are arbitrary: these constraints may be body dimensions, volume, area, etc. Such problems with arbitrary isoperimetric conditions may be solved by the method suggested in [1, 2]. This method involves the use of the exact equations of gasdynamics which describe the flow as additional constraints. This method was developed further in [3–6] in the solution of several problems.The author wishes to thank V. M. Borisov, A. N. Kraiko and Yu. D. Shmyglevskii for their interest in this study.     

Theory of regular and mach reflection of shock waves in a two-phase gas-liquid medium

An equilibrium model of a two-phase gas-liquid medium, with allowance for the proportion, density, and compressibility of the components, and with a difference from [1] in that the adiabatic velocity of sound is introduced, has been used in order to study the regular and Mach (elementary theory) reflection of a shock wave of moderate intensity from a solid wall throughout the whole range of gas proportions. A complicated nonmonotonic variation has been found for the pressure on the wall behind the reflected wave, the angle of reflection, and the angle of departure of the triple point as functions of the gas proportion, the angle of incidence, and the intensity of the incident wave. In particular, it is shown that oblique reflection for moderate and low gas contents leads to the formation of a stronger reflected shock wave than does normal reflection. The effect of the gas proportion on the position of the boundary between the regions of regular and Mach reflection has already been studied in [2]. The results are described of serial calculations of the parameters of reflection for an air-water mixture, and these results agree fairly well for normal reflection with the known experimental data [3] for low and moderate gas contents. In the limiting case, the results agree with the known results for single-phase media [4, 5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 188–190, March–April, 1985.     

Reflection of a spherical blast wave from a planar surface

Numerous authors have carried out rather extensive studies in the last twenty to thirty years of the problem of the interaction of shock and blast waves with obstacles in their paths. Owing to the complexity of the problem, they assumed certain limiting cases for the shock wave interactions in which the parameters behind the shock wave were usually taken to be constants. The first wave diffraction studies involving variable parameters behind the front were presented in [1, 2], wherein a development of the theory of “short waves” (blast waves at a substantial distance from the center of an explosion) and their reflection from a planar surface was given. The theory of short waves assumes that the jump in pressure at the wave front and the region over which the parameters vary are small. The problem concerning reflection of a blast wave from a surface was also considered in [3, 4], wherein a solution in the region behind the reflected wave was obtained at initial times. The initial stage in the reflection of a blast wave from a planar, cylindrical, or spherical surface (the one-dimensional case) was studied in [5]. In this paper we investigate the interaction of a spherical blast wave, resulting from a point explosion, with a planar surface; we consider both regular and non-regular reflection stages. In solving this problem we use S. L. Godunov's finite-difference method. We obtain numerical solutions for various values of the shock strength at the instant of its encounter with the surface. We present the pressure fields in the flow regions, the pressure distribution over the surface at various instants of time, and the trajectories of the triple point. The parameter values at the front of the reflected wave are compared with results obtained from the theory of regular reflection of shock waves.     

Unsteady reflection of a shock wave from a body with a cylindrical recess

In a number of cases of supersonic flow past bodies with recesses pulsations in the flow arise [1–3]. Experiments [4, 5] indicate that stabilization of the steady supersonic flow past the body with a recess on which a shock wave is incident takes place after a series of oscillations of the bow wave. Numerical calculation of the interaction of a supersonic jet with a cylindrical cavity [6] reveals that damped pressure pulsations arise inside the cavity if the jet is homogeneous, and undamped pulsations it is inhomogeneous. The authors explain the damping of the pulsations by the influence of artificial viscosity. This paper investigates experimentally and theoretically (by numerical methods) the oscillations of the bow shock wave and the parameters of the flow behind it in the case of unsteady reflection of a shock wave from a body with a cylindrical recess turned towards the flow. The problem is posed as follows. A plane shock wave with constant parameters impinges on a cylinder with a cavity. The unsteady flow originating from this interaction is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 199–202, September–October, 1984.     

On regular reflection of detonation waves

The boundaries of regular reflection of detonation waves by a rigid wall are calculated. It is assumed that detonation is initiated at the point of reflection when a shock wave is incident on the wall at a finite angle in a gas fuel mixture, the detonation propagating instantaneously along the reflected front.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 178–180, March–April, 1983.     

Gas content factor in the interaction of shock waves of different intensity in a two-phase gas-liquid medium

The transition from regular to Mach interaction is investigated in connection with the interaction of two plane weak or moderate shock waves of different intensity in a two-phase gas-liquid medium over the entire range of gas contents. A nonmonotonic dependence of the transition limit and the flow parameters on the gas content is detected. The investigation extends the results of [1] corresponding to the reflection of a shock wave from a wall. At intermediate gas contents in the case of opposing shock waves, analogous to the normal reflection of a shock wave from a solid wall, the results are in agreement with [2]. In the case of weak shock waves non-linear asymptotic expansions [3] are employed. In the extreme cases of single-phase media the results coincide with the findings of [3, 4]. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 172–174, November–December, 1986.     

Formation of one-dimensional flows of viscous heat-conducting gas for small parameter perturbations

The formation of unsteady one-dimensional flows is studied, using the solution of the problem of reflection of a normally incident plane shock wave from a heat-conducting wall as an example. The process is considered for low intensities of the incident wave, behind which the gas temperature hardly differs from the initial wall temperature. The flows with complicated internal structures that arise are investigated on the basis of Navier-Stokes equations linearized near the initial state. An analytic solution of the problem describing the discontinuous structure of the reflected flow is constructed, which can serve as a test in the numerical solution of the original nonlinearized Navier-Stokes equations. The influence of the Prandtl numbers, the specific heat ratio, and dissipative and other factors is considered. The features of the effects of viscosity, thermal conductivity, and accommodation on the formation of flows and ideal (inviscid, nonheat-conducting) and dissipative zones are traced. It is shown that the solution of the linearized system agrees with the solution for asymptotic flow regimes.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 105–111, September–October, 1986.     

Three-wave resonance and averaged equations of the interaction of two waves in media described by a cubic Schrödinger equation

A numerical solution of a cubic Schrödinger equation established the phenomenon of Mach reflection of Stokes waves from a vertical wall [1]. In [2], this phenomenon was interpreted as the resonance interaction of three waves. This was based on the derivation of averaged equations of the interaction of two waves, the region of interaction of the incident and reflected waves being treated as a self-similar solution of these equations. The present paper establishes the possibility of describing these solutions by the relations of three-wave resonance; the mathematical significance of the resonance as splitting into two waves is revealed; and the properties of the averaged system are investigated.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 107–116, January–February, 1992.I thank A. G. Kulikovskii and A. A. Barmin for helpful discussions and also colleagues at the Mechanics Section of the V. A. Steklov Mathematics Institute of the USSR Akademy of Sciences for critical comments that stimulated interest in the paper.