Mass,momentum and total excess energy transported by a weak planeN wave

A weakly nonlinear plane acoustic wave is emitted into an ideal gas of semi-infinite extent from an infinite plate by its sinusoidal motion of single period. The wave develops into anN wave in the far field, as long as the energy dissipation is negligible everywhere except for discontinuous shock fronts. The third-order effects at shock fronts are evaluated, i.e., the generation of reflected acoustic wave as a result of the interaction of shock and expansion wave and the production of entropy by the energy dissipation at shock fronts. Consideration of these effects enables one to estimate the whole mass, momentum and total excess energy (sum of the kinetic energy and excess of internal energy over an initial undisturbed value) transported by theN wave to the accuracy of third order of wave amplitude. It is shown that the mass and total excess energy transported by theN wave increase and the momentum decreases to asymptotic limits as the wave propagates. The result shows good agreement with a numerical result obtained by solving the Euler equations with a high-resolution TVD upwind scheme.     

Discrete element simulation of shock wave propagation in polycrystalline copper

The implementation of the characteristic of compressive plasticity into the Discrete Element Code, DM2, while maintaining its quasi-molecular scheme, is described. The code is used to simulate the shock compression of polycrystalline copper at 3.35 and 11.0 GPa. The model polycrystal has a normal distribution of grain sizes, with mean diameter 14 μm, and three distinct grain orientations are permitted with respect to the shock direction; 〈1 0 0〉, 〈1 1 0〉, and 〈1 1 1〉. Particle velocity dispersion (PVD) is present in the shock-induced flow, attaining its maximum magnitude at the plastic wave rise. PVD normalised to the average particle velocity of and are yielded for the 3.35 and 11.0 GPa shocks, respectively, and are of the same order as those seen in the experiment. Non-planar elastic and plastic wave fronts are present, the distribution in shock front position increasing with propagation distance. The rate of increase of the spread in shock front positions is found to be significantly smaller than that seen in probabilistic calculations on nickel polycrystals, and this difference is attributed, in the main, to grain interaction. Reflections at free surfaces yield a region of tension near to the target free surface. Due to the dispersive nature of the shock particle velocity and the non-planarity of the shock front, the tensile pressure is distributed. This may have implications for the spall strength, which are discussed. Simulations reveal a transient shear stress distribution behind the shock front. Such a distribution agrees with that put forward by Lipkin and Asay to explain the quasi-elastic reloading phenomenon. Simulation of reloading shocks show that the shear stress distribution can give rise to quasi-elastic reloading on the grain scale.     

Regular reflection of plane detonation waves from an elastic half-space

An effective method for the approximate solution of the Eq. [1] for the intensity of a reflected shock wave in the case of oblique incidence of a detonation wave on an elastic half-space is described; the elastic half-space is described by a certain specific form of the equation of state. Formulas relating the front and particle velocities behind the transmitted wave front to physical parameters are derived. Values of the wave intensity and other quantities determined with the aid of a Ural-2 computer are cited.The author of [1, 2] investigated the regular reflection of shock waves from the boundary between two bodies. In the present paper we solve the analogous problem in the case of oblique incidence of a detonation wave on an elastic half-space. The detonation wave deforms the elastic half-space, which assumes the position OK₁ (Fig. 1) forming the angle to the initial direction KO of the halfspace boundary. We assume that the acoustic stiffness of the halfspace is larger than the acoustic stiffness of the explosive. In this case, both reflected wave 2 and transmitted wave 3 are shock waves [3]. Let us denote the velocities of propagation of the detonation, reflected, and transmitted waves by U_i(i=1, 2, 3), respectively; let the pressure be p_i and let the density bep _i(i=0, 1, 2, 3, 4). The quantities U₁, ₁, ₀, and ₄ are given. We determine the intensities of waves 2 and 3, their velocities of propagation, and the angles ₂, ₃ and . The parameters are constant within each of the domains a, b, c, d, and e. In domains a and e the medium is stationary, i.e., u₀=u₄ =0. The basic equations of the problem express the conditions at the wave fronts and the dynamic and kinematic relationships.     

A gaseous shock wave theory of the galactic spiral structure

In this paper, the galactic spiral structure is studied by the galactic shock wave of interstellar gas with self-gravitation. The perturbed gravitation of stars is not a necessary condition for the existence of such shock. It is proved first of all that there exists solution of local shock wave even if the perturbed gravitation is absent. The condition |₀|> is required for such solution. The spiral structure can only be explained by the shock solution when the difference of density between the regions of arm and interarm is larger. The grand design of shock wave with self-gravitation is obtained by the iterative method. The features of shock wave can be analyzed qualitatively in the velocity plane for a special perturbed gravitation which is used to simulate the self-gravitation of interstellar gas. As the mass distribution in proto- galactic disk is irregular initially, the grand design of the galactic shock wave was developed by the processes of winding, growth of instability and overlapping of waves. Hence, it gives a complete figure about the origin, evolution and persistance. A lot of observed phenomena and classificational features of the galactic spiral structure can be explained by adopting these ideas.     

Investigation of a “hanging” shock wave near a generating point

The hodograph method is used to plot a hanging shock wave in the plane nonequilibrium supersonic flow of an ideal gas. This paper considers the general case of an analytical solution in the plane of the hodograph at the point of generation of the shock wave. A type of limiting line is established which makes it possible to plot a shock wave (it is found that the shock wave may not extend over the whole flow, with a convolution in the physical plane).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 30–37, November–December, 1971.     

Reflection of a strong shock wave from a sphere and a cylinder

We examine the flow on the axis in the vicinity of the stagnation point for reflection of a strong plane shock wave (with uniform parameters behind the wave) from a sphere and a circular cylinder whose generators are parallel to the incident wave front.The small parameter method [1, 2] is used to obtain, in closed form, relations which define the time variation of the velocity profile, pressure, enthalpy, and reflected shock wave standoff.As the time t , these relations reduce to the known formulas [3, 4] which define the steady flow on the axis for the flow behind the incident shock wave about a body, if account is taken of the leading terms containing the small parameter.The solution is extended to the case in which account for equilibrium dissociation and ionization is necessary.Comparison of the results with measurement [5] of the reflected shock wave distance from a sphere, as a function of time, shows satisfactory agreement.     

Measurement of parameters of a gas suspension behind a shock wave in foam

A study has been made of the propagation of a shock wave in dry polyhedral foam with cell diameter 1 cm. The experiments were made in a shock tube in the range of Mach numbers M < 1.4 of the shock wave. The interaction of the shock wave with the foam was photographed. This established that the destruction of the foam by the shock wave leads to the formation of a gas-droplet flow behind the shock front. To determine the parameters of the suspension, the flow was probed by He-Ne lasers with different radiation wavelengths. The spectral-transparency method was used to find the modal diameter of the droplets of the gas suspension and the volume concentration of the droplets in the flow. The modal diameter of the droplets was 2m, and the volume concentration of the droplets decreased downstream.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 134–141, May–June, 1993.     

Interaction of shock waves with a combined discontinuity in two-phase media. 2. Nonequilibrium approximation

Interaction of a shock wave and a motionless combined discontinuity separating two twocomponent mixtures with different initial volume concentrations is studied on the basis of numerical simulation of unsteady processes. The calculations were performed using a modified method of coarse particles and a highaccuracy TVD difference scheme adapted to calculation of twophase flows. Flow parameters determined by analytical dependences coincide with those obtained by numerical simulation at large times of the process. Upon interaction of the shock wave and the combined discontinuity, the type of the transient or reflected shock wave may coincide with or differ from the type of the incident shock wave. The possibility of existence of a pressure difference at the combined discontinuity boundary, which was earlier predicted analytically, is confirmed.     

The evolution laws of dilatational spherical and cylindrical weak nonlinear shock waves in elastic non-conductors

The theory of singular surfaces yields a set of coupled evolution equations for the shock amplitude and the amplitudes of the higher order discontinuities which accompany the shock. To solve these equations, we use perturbation methods with a perturbation parameter characterising the initial shock amplitude. It is shown that for decaying shock waves, if the accompanying second order discontinuity is of order one, the straightforward perturbation procedure yields uniformly valid solutions, but if the accompanying second order discontinuity is of order , the method of multiple scales is needed in order to render the perturbation solutions uniformly valid with respect to the distance of travel. We also construct shock wave solutions from modulated simple wave solutions which are obtained with the aid ofHunter & Keller's Weakly Nonlinear Geometrical Optics method. The two approaches give exactly the same results within their common range of validity. The explicit evolution laws thus obtained enable us to see clearly how weak nonlinear curved shock waves are attenuated because of the effects of geometry and material nonlinearity, and on what length scale these effects are most pronounced. Communicated by C. C. Wang     

Theory of anomalous modes of regular reflection of shock waves

It is well known [1, 2] from numerical calculations of the reflection of a shock wave for a diatomic gass that in some cases regular reflection is accompanied by higher pressures than the pressure of normal reflection (anomalous modes of regular reflection). A theory explaining this phenomenon is presented in this paper. It is shown that if the adiabatic exponent is larger than some critical value, then for any shock wave intensity there exists a finite range of angles of incidence for which anomalous reflection modes occur. If the adiabatic exponent is smaller than this critical value, anomalous reflection occurs only for shock waves whose intensity is smaller than some characteristic value dependent on the adiabatic exponent. Explicit formulas are obtained which relate the angle and pressure of reflection of a shock wave to the initial parameters of the problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 117–125, September–October, 1973.The author thanks V. A. Belokon' for stimulating discussions.     

Influence of vibration-dissociation coupling on heat transfer and hypersonic drag

A numerical investigation is carried out within the framework of the multicomponent total viscous shock layer model [6, 7], according to which when Re 100 the flow near a blunt body can be divided into a shock wave zone and a viscous shock layer. At the inner edge of the shock wave the generalized Rankine-Hugoniot relations are imposed, and in the shock layer the complete system of viscous shock layer equations is solved with allowance for vibrational relaxation and nonequilibrium dissociation and ionization reactions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 141–151, May–June, 1990.     

Evolution of the Diffusion Mixing Layer of Two Gases upon Interaction with Shock Waves

A mathematical model of mechanics of a twovelocity twotemperature mixture of gases is developed. Based on this model, evolution of the mixing layer of two gases with different densities under the action of shock and compression waves is considered by methods of mathematical simulation in the onedimensional unsteady approximation. In the asymptotic approximation of the full model, a solution of an initialboundary problem is obtained, which describes the formation of a diffusion layer between two gases. Problems of interaction of shock and compression waves with the diffusion layer are solved numerically in the full formulation. It is shown that the layer is compressed as the shock wave traverses it; the magnitude of compression depends on shockwave intensity. As the shock wave passes from the heavy gas to the light gas, the mixing layer becomes overcompressed and expands after shockwave transition. The wave pattern of the flow is described in detail. The calculated evolution of the mixinglayer width is in good agreement with experimental data.     

Incidence of plane supersonic jet on a plane at an arbitrary angle

A solution method is described and results are presented of numerical calculations for the problem of determining the subsonic part of the flow with incidence of a plane uniform supersonic jet on a plane at an arbitrary angle, which corresponds to the flow regime with a a detached shock wave. For the problem solution we use the method of integral relations in the first approximation in a polar coordinate system. The calculation results (pressure distribution on the plate, shock wave shape, and velocity gradient magnitude at the stagnation point) are presented for Mach numbers of 5 and 20, in a range of incidence angles from 0° to 35°, and also for M=3 for an incidence angle 0° (angles measured from normal to the plate).In conclusion the authors wish to thank G. I. Taganov for guidance in this work.     

Numerical modeling of flow with chemical reactions in a strong shock wave with approximate allowance for translational nonequilibrium

The effect of translational nonequilibrium on the course of chemical reactions in a shock wave is studied using the beam–gas model extended to the case of a multicomponent gas. For Arrhenius reactions of general form with collisions between beam and gas molecules, a modified expression of reaction rate is obtained that takes into account the relative motion of the two media. A procedure for numerical solution of the problem is considered, and calculation results for a shock wave in a dissociating air at an oncoming flow velocity of 6000 m/sec are given.     

Macro-mechanical modeling of blast-wave mitigation in foams. Part II: reliability of pressure measurements

A phenomenological study of the process occurring when a plane shock wave reflected off an aqueous foam column filling the test section of a vertical shock tube has been undertaken. The experiments were conducted with initial shock wave Mach numbers in the range $1.25\le {M}_\mathrm{s} \le 1.7$ and foam column heights in the range 100–450 mm. Miniature piezotrone circuit electronic pressure transducers were used to record the pressure histories upstream and alongside the foam column. The aim of these experiments was to find a simple way to eliminate a spatial averaging as an artifact of the pressure history recorded by the side-on transducer. For this purpose, we discuss first the common behaviors of the pressure traces in extended time scales. These observations evidently quantify the low frequency variations of the pressure field within the different flow domains of the shock tube. Thereafter, we focus on the fronts of the pressure signals, which, in turn, characterize the high-frequency response of the foam column to the shock wave impact. Since the front shape and the amplitude of the pressure signal most likely play a significant role in the foam destruction, phase changes and/or other physical factors, such as high capacity, viscosity, etc., the common practice of the data processing is revised and discussed in detail. Generally, side-on pressure measurements must be used with great caution when performed in wet aqueous foams, because the low sound speed is especially prone to this effect. Since the spatial averaged recorded pressure signals do not reproduce well the real behaviors of the pressure rise, the recorded shape of the shock wave front in the foam appears much thicker. It is also found that when a thin liquid film wet the sensing membrane, the transducer sensitivity was changed. As a result, the pressure recorded in the foam could exceed the real amplitude of the post-shock wave flow. A simple procedure, which allows correcting this imperfection, is discussed in detail.     

Propagation of Quasiacoustic Pulses in an Elastoplastic Medium

Expressions for the velocity of a plastic shock wave and phase velocity of longitudinal waves in an elastoplastic medium with hardening are obtained in a quasiacoustic approximation. An analytical solution of the problem of shockpulse attenuation is constructed. A special feature of the amplitude of the attenuating plastic shock wave is that it reaches the amplitude of the elastic precursor in a finite time, whereas in hydrodynamics, the amplitude of a quasiacoustic shock pulse tends to zero asymptotically.     

Role of the absorption of radiation ahead of a shock wave with hypersonic flow around a blunt body

A study was made of conditions at the front of a strong shock wave taking account of the absorption of leading radiation. Emphasis is laid on the role of the dimensionless parameters which arise under these circumstances, and an evaluation is made of the values of these parameters for a number of practically important cases involving the entry of blunt bodies into dense layers of the Earth's atmosphere. Calculations are carried out to determine the composition and the parameters of the flow of molecular nitrogen entering into the shock wave, and conclusions are drawn with respect to the general problem of hypersonic flow around a blunt body, taking radiation into account. In an investigation of the flow of a hypersonic stream of air around a blunt body, taking account of radiation, it is necessary to have some idea of how the radiation leaving the zone of the shock wave reacts with the oncoming flow of cold air. The importance of taking this reaction into account is indicated by the results of observation of the reentry of spacecraft into dense layers of the atmosphere [1], and by existing experimental data on strong shock waves [2]. This reaction is bound up with the fact that the absorption of intense short-wave radiation from the shock wave in cold air leads to photodissociation and photoionization of the molecules of air, i.e., to an actual increase in the enthalpy of the air. Some of the general questions of the structure of a very strong direct shock wave, taking account of the absorption of radiation leading the wave front, have been investigated in [3],Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 40–47, November–December, 1970.     

Initial stage of reflection of a plane shock wave from a cylinder,sphere, and ellipsoid of revolution

The stage of regular reflection of a plane shock wave from a blunt body (cylinder, sphere, and ellipsoid of revolution) is considered. At the point of intersection of the reflected shock wave and the surface of the body, analytic expressions are found for the derivative of the Mach number of the wave with respect to the time, the curvature of the wave, the normal derivatives of the density and the pressure, and the derivative of the Mach number along the wave front. It is shown that the flow has a singularity at = _* < _** (s** is the limiting angle [1] of regular reflection of a shock wave from a rigid surface). The distribution of the parameters in the region between the reflected shock wave and the surface of the body is found up to terms of third order in the time. The density distribution behind the reflected shock wave was measured experimentally, and also the shape of the reflected wave at different instants of time.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 94–100, March–April, 1980.     

Attenuation of shock waves propagating in polyurethane foams

Shock wave attenuation in polyurethane foams is investigated experimentally and numerically. This study is a part of research project regarding shock propagation in polyurethane foams with high-porosities = 0.951 ~ 0.977 and low densities of ρc = 27.6 ~55.8 kg/m³. Sixty Millimeter long cylindrical foams with various cell numbers and foam insertion condition were installed in a horizontal shock tube of 50 mm i.d. and 5.4 mm in length. Results of pressure measurements in air/foam combination are compared with CFD simulation solving the one-dimensional Euler equations. In the case of a foam B fixed on shock tube wall, pressures at the shock tube end wall increases relatively slowly comparing to non-fixed foam, free to move and a foam A fixed on shock tube wall. This implies that elastic inertia hardly contributes to pressure build up. Pressures behind a foam C fixed on shock tube wall decrease indicating that shock wave is degenerated into compression wave. Dimensionless impulse and attenuation factor decrease as the initial cell number increases. The momentum loss varies depending on cell structure and cell number.     

Notes on gas–dynamic design of supersonic flying vehicles

It is shown that the lift–to–drag ratio of a thin delta wing is significantly lower than the lift–to–drag ratio of an infinitely long swept plate with an identical lift force. The effect of sweep on a finite wing may be used by excluding disturbances from the leading edge of the wing via introducing a hardened stream surface (wedge) and increasing the wing length. A three–shock waverider is proposed for choosing the optimal parameters. The sharp wedge may be avoided by replacing planar shock waves by a cylindrical shock wave upstream of the blunted wedge. If the leading edge of the wedge is not parallel to the rib that is a source of the expansion wave, a plate with zero wave drag, generating a lift force, may be obtained behind this rib. The system of regularly intersecting shock waves may be applied to design a forward–swept wing.