Dynamics of shock waves in a liquid containing gas bubbles

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

Coupled elastic waves in uniform isotropic media

This article deals with a certain type of wave in an infinite elastic medium. In contrast to ordinary longitudinal and transverse waves, the amplitude of the type of wave in question depends sinusoidally on the coordinates of a plane which is transverse to the direction of propagation of the wave, i.e., the wave is actually a packet of travelling and stationary waves. Longitudinal waves of this type are always coupled with transverse waves, while transverse waves of the given type may be coupled with longitudinal waves or another transverse wave or may exist as a single wave in the form of a packet containing a travelling wave and a stationary wave. The coupled waves have two phase velocities, which depend on the mechanical properties of the medium, the frequency of vibration, and the wave numbers of the stationary waves. Coupled surface waves in an elastic medium are more general in character than Rayleigh waves; they exhibit dispersion, and they can be used to explain certain seismological observations made during earthquakes—the complete absence of vertical displacements in some cases and the frequent occurrence of horizontal displacements parallel to the wave front. Allowing for the coupling of elastic waves in a layer leads to a more general characteristic equation than the equation obtained in the Rayleigh-Lamb problem. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 9, pp. 19–28, September, 1999.     

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.     

Wave front analysis of a plane compressional pulse scattered by a cylindrical elastic inclusion

The plane-strain problem of a stress pulse striking an elastic circular cylindrical inclusion embedded in an infinite elastic medium is treated. The method used determines dominant stress singularities that arise at wave fronts from the focusing of waves refracted into the interior. It is found that a necessary and sufficient condition for the existence of a propagating stress singularity is that the incident pulse has a step discontinuity at its front. The asymptotic wave front behavior of the first few P and SV waves to focus are determined explicitly and it is shown that the contribution from other waves are less important. In the exterior, it is found that in most composite materials the reflected waves have a singularity at their wave front which depends on the angle of reflection. Also the wave front behavior of the first few singular transmitted waves is given explicitly.The analysis is based on the use of a Watson-type lemma, developed here, and Friedlander's method [5]. The lemma relates the asymptotic behavior of the solution at the wave front to the asymptotic behavior of its Fourier transform on time for large values of the transform parameter. Friedlander's method is used to represent the solution in terms of angularly propagating wave forms. This method employs integral transforms on both time and θ, the circumferential coordinate. The θ inversion integral is asymptotically evaluated for large values of the time transform parameter by use of appropriate asymptotics for Bessel and Hankel functions and the method of stationary phase. The Watson-type lemma is then used to determine the behavior of the solution at singular wave fronts.The Watson-type lemma is generally applicable to problems which involve singular loadings or focusing in which wave front behavior is important. It yields the behavior of singular wave fronts whether or not the singular wave is the first to arrive. This application extends Friedlander's method to an interior region and physically interprets the resulting representation in terms of ray theory.     

Localised knife waves in a structured interface

We consider a Mode III lattice with an interface layer where the dynamic crack growth is caused by a localised sinusoidal wave. In the wave–fracture scenario, the ‘feeding wave’ (here also called the knife wave) delivers energy to the moving crack front, while the dissipative waves carry a part of this energy away from the front. The questions addressed here are:

• What are the conditions of existence of the localised knife wave?

• What is the lower bound of the amplitude of the feeding wave, which supports the crack propagation, for a given deformational fracture criterion?

• How does the crack speed depend on the amplitude of the feeding wave?

• What are the dissipative waves? How much energy is irradiated by these waves and what is the total dissipation?

• What are the conditions of existence of the steady-state regime for the propagating crack?

We consider analytically two established regimes: the steady-state regime, where the motion of neighbouring masses (along the interface) differs only by a constant shift in time, and an alternating-strain regime, where the corresponding amplitudes differ by sign. We also present the numerical simulation results for a model of a high-contrast interface structure. Along with the energy of the feeding and dissipative waves, an energy radiated to the bulk of the lattice is identified.

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.     

Motion of a nonisothermal adsorbable mixture with enhanced values of the concentrations through a porous medium

The equations of motion of a nonisothermal adsorbable mixture with enhanced values of the concentrations of the components in the case of infinitely large coefficients of heat and mass transfer reduce to a hyperbolic quasilinear system of equations. The invariant solutions of this system are analyzed. Convexity conditions are obtained under which a traveling-wave regime is realized in the porous medium. A system of equations is found for determining the concentrations of the adsorbable components of the mixture when a self-similar regime of dispersing waves is realized. For the case of finite values of the coefficients of heat and mass transfer, expressions are given for the width of the stationary front in the traveling-wave regime.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 76–86, September–October, 1980.     

Effect of continuous variation of the density of a liquid on the waves generated by moving surface pressures

This study investigates the plane linear problem of steady-state internal waves in an ideal incompressible liquid with nonuniform density. The waves are generated by surface pressures applied in a bounded region which moves at constant velocity. It is assumed that the density in the unperturbed state varies continuously with depth, remaining constant in the upper and lower layers and varying according to an exponential law in the middle layer. The problem may be regarded, in particular, as a hydrodynamic model for the study of internal waves produced by a cyclone moving over the surface of the ocean. Analogous investigations for a homogeneous liquid were carried out in [1–3]; internal waves for a liquid with the above-mentioned law of density variation but with stationary pressure changes which are periodic with respect to time were studied in [4]. Problems analogous to the one considered here, both for exponential variation of density in the entire layer and for the case of a nonuniform layer near the surface, were investigated in [5, 6]. An analysis of non-linear waves of the steady-state type with arbitrary distribution of vorticity and density with respect to depth was carried out in [7, 8].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 55–62, November–December, 1973.     

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.     

Turbulent bore in a supercritical flow over an irregular bed

Breaking waves in a free-surface homogeneous fluid flow in the neighborhood of a local variation in the channel depth are studied experimentally and theoretically. The structure of both a steady-state hydraulic jump generated by a local obstacle in the channel and an unsteady wave configuration consisting of two turbulent bores in the problem of lock failure is studied. Using the turbulent bore model [1], analytic profiles of breaking waves are obtained and the time-dependent problem is numerically investigated and compared with experimental data. It is shown that the model [1] with a hydrostatic pressure distribution over the depth adequately describes both the location and the structure of the steady-state and unsteady wave fronts.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, 2005, pp. 62–70. Original Russian Text Copyright © 2005 by Gusev and Lyapidevskii.     

Waves in a heavy two-layer liquid excited by an oscillating sphere

An approximate solution of an initial-boundary-value problem appropriate for the semiaxist>0 (t is time) is constructed for a system of integrodifferential equations which describes the waves excited in an initially stationary unbounded heavy two-layer fluid by a vertically oscillating sphere located at a distance from the interface that is significantly greater than its radius. The shape of the steady-state wave is found by passing to the limit as time increases indefinitely. The wave resistance experienced by the sphere during the transient process and in the steady-state regime is studied as a function of frequency. Nizhnii Novgorod. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 120–133, March–April, 1998.     

Breaking of waves of limiting amplitude over an obstacle

Homogeneous heavy fluid flows over an uneven bottom are studied in a long-wave approximation. A mathematical model is proposed that takes into account both the dispersion effects and the formation of a turbulent upper layer due to the breaking of surface gravity waves. The asymptotic behavior of nonlinear perturbations at the wave front is studied, and the conditions of transition from smooth flows to breaking waves are obtained for steady-state supercritical flow over a local obstacle. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 3–11, May–June, 2006.     

Unsteady and nonequilibrium airflow near a stagnation line

The propagation of shock waves in a medium with a nonuniform distribution of the parameters is the subject of recently published research [1–3]. The present paper deals with the problem of the gas flow ahead of the forward point of a blunt body moving at supersonic speed in air with variable parameters. The chemical reaction processes behind the shock front are taken into account. As a result of numerical calculations by the method of characteristics with isolation of the forward shock the time-dependent position of the shock front and the distributions of the composition and gas dynamic parameters in the shock layer are found. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 170–172, November–December, 1986.     

Internal streamwise action of saw-tooth sound waves on turbulent jets

The results of an experimental investigation of the acoustic field produced by turbulent subsonic jets under internal acoustic excitation are presented. It is shown that under the action of saw-tooth finite-amplitude waves the turbulent jets can radiate Mach waves into the ambient medium due to compact acoustic disturbances traveling along the jet at a velocity greater than the speed of sound in the surrounding space.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, 2004, pp. 153–158. Original Russian Text Copyright © 2004 by Pimshtein.     

Nonsteady-state plasma flows behind a moving leading front

The nonsteady-state motion of a conducting and optically transparent gas separated from a rarefied stationary medium by a front S which passes through the zone of a nonuniform magnetic field is considered. Boundary conditions at S are proposed for several interaction mechanisms of S with the rarefied medium. Solutions are obtained in linear approximation. The solutions of a general system of equations are found by means of numerical methods. The sharply nonmonotonic nature of the motion of S in the nonuniform magnetic field is established.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 156–162, July–August, 1973.In conclusion, the author thanks A. B. Vatazhin for useful discussions.     

Spatially polarized structure of detonation waves ionizing a gas

Additional relationships must be used [1–3], in addition to those following from the main integral laws, in describing ionizing detonation waves, exactly as for ionizing shocks. These additional relationships are obtained from the requirement for the existence of wave structure. The structure of detonation waves ionizing a gas in an oblique magnetic field was investigated in [1, 2]. The case of a plane-polarized structure was considered, when the velocity vector and the magnetic field lie in a plane passing through the normal to the front. The structure of ionizing detonation waves is studied in this paper for the case when the wave is spatially polarized and both transverse magnetic field components vary in the structure. It is considered that the magnetic viscosity and a quantity reciprocal to the chemical reaction rate are much greater than the remaining dissipative coefficients in the layer representing the structure. Conditions for the existence of such a spatial structure are clarified. Plane-polarized ionizing detonation waves whose structure is not planar are also considered. When the characteristic length of magnetic field dissipation is much greater or much less than the characteristic length of the chemical reaction, the additional relationships assuring the existence of structure are written down explicitly or are investigated qualitatively.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 166–169, November–December, 1976.     

Asymptotic analysis of stationary distribution of the front of a two-stage consecutive exothermic reaction in a condensed medium

An approximate theory of the stationary distribution of the plane front of a two-stage exothermic consecutive chemical reaction in a condensed medium is developed in the article. The method of joined asymptotic expansions is used in constructing the solutions. The ratio of the sum of the activation energies of the reactions to the final adiabatic combustion temperature is a parameter of the expansion. The characteristic limiting states of the stationary distribution of the wave corresponding to different values of the parameters figuring in the problem are shown. Approximate analytical expressions for the wave velocity and distribution of concentrations are obtained for each of the states.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 75–87, January–February, 1973.     

Diffraction of stationary cylindrical and plane sound waves by a cylinder in a viscous heat-conducting medium

The diffraction of cylindrical and plane stationary sound waves by a cylinder in a viscous heat-conducting medium is analyzed. A solution of this problem is given in [1] without regard for heat conduction.Tula. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 147–154, May–June, 1972.     

Production of periodically modulated laser-driven blast waves in a clustering gas

To study hydrodynamic behavior on thin shell high Mach number blast waves, experiments have been performed in which spatially tailored shock waves have been launched in a gas of clusters using an intense 35 fs laser pulse. The target medium was first modified by destroying clusters in specific locations using a spatially modulated laser focus. Under subsequent intense laser irradiation, the efficient absorption properties of the remaining clustered regions compared to those regions with no clusters led to a pattern of hot and cold plasma resulting in a cylindrical blast wave with a periodic modulation imprinted on the shock front. This technique may provide a method for studying thin shell instabilities in strongly radiative blast waves.