Propagation of solitary waves over a semi-infinte underwater trough

Self-similar solutions describing the incidence of a uniform solitary wave on a semi-infinite linear trough are obtained on the basis of the nonlinear ray method [1]. Previously, in investigating the incidence of a wave on a trough [2] the conditions at the discontinuities present in the solutions were derived on the assumption that they are of low intensity. In the present study the use of the conditions at the discontinuities obtained by investigating soliton interaction [3–5] has made it possible to construct a series of new solutions and take into account wave reflection effects and the formation of a shadow zone beyond the trough. The types of solutions that occur are established in terms of the relations between the wave parameters and the relative depth of the trough. To ensure that self-similar solutions exist for all values of the parameters it was necessary to introduce a type of discontinuity not previously encountered [5–7].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 102–107, July–August, 1987.The author wishes to thank A. G. Kulikovskii and A. A. Barmin for discussing the work.     

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.     

Sensitivity of a supersonic boundary layer to acoustic disturbances

The results obtained by the authors in [1] are extended to the case of arbitrary angles of incidence of the external wave. This is not a trivial generalization, since the acoustic scattering undergoes a qualitative change. It is possible to distinguish two excitation channels: the first is connected with the diffraction of the acoustic wave by the spatial inhomogeneity resulting from the displacing action of the boundary layer, and the second with the presence of concentrated acoustic field sources associated with the scattering of the wave at the leading edge. The latter makes the principal contribution to the initial amplitude of the unstable modes when the angles of incidence of the sound are substantially different from zero. At low angles of incidence there is a singularity which can be revealed by introducing narrow intervals in the neighborhood of the limiting values of the wave numbers, where the two excitation channels are approximately equivalent. It is possible to obtain composite expressions for the initial amplitudes of the unstable modes uniformly valid for all angles of incidence of the acoustic wave.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 40–47, January–February, 1992.     

Reflection of a sound wave by an evaporating surface

The reflection of a sound wave traveling through saturated vapor by the flat surface of the condensed phase is considered. It is shown that the intensity of the reflected wave is less than in the absence of evaporation and depends nonmonotonically on the angle of incidence, the position of the minimum being determined by the coefficient of condensation and independent of the viscous and heat conducting properties of the gas. The effect can be used to measure the condensation coefficient. The structure of the layers that arise near the surface is considered. A new type of acoustic flow with a velocity of the order of the amplitude of the speed of sound is found.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. A, 149–156, July–August, 1988.     

Localized Coherent Structures in the Boundary Layer

A Blasius laminar boundary layer and a steady turbulent boundary layer on a flat plate in an incompressible fluid are considered. The spectral characteristics of the Tollmien—Schlichting (TS) and Squire waves are numerically determined in a wide range of Reynolds numbers. Based on the spectral characteristics, relations determining the three–wave resonance of TS waves are studied. It is shown that the three–wave resonance is responsible for the appearance of a continuous low–frequency spectrum in the laminar region of the boundary layer. The spectral characteristics allow one to obtain quantities that enter the equations of dynamics of localized perturbations. By analogy with the laminar boundary layer, the three–wave resonance of TS waves in a turbulent boundary layer is considered.     

Effect of periodic action on boundary layer stability

The effect of a wave traveling over the surface and suction-blowing in the form of a traveling wave on boundary layer stability and laminarturbulent transition is investigated. The perturbation parameters are assumed not to be related to the parameters of the Tollmien-Schlichting wave. The parameters corresponding to an increase in the critical Reynolds number by a factor of 2–2.5 are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 109–115, May–June, 1988.The author is grateful to V. A. Kuparev for supplying the program for calculating the stability of the boundary layer.     

On One Contact Problem in Fracture Mechanics for a Normally Incident Tension–Compression Wave

The contact interaction of the faces of a penny-shaped crack in a three-dimensional space is studied for the case of normal incidence of a harmonic tension–compression wave. The problem is solved by the method of boundary integral equations. The dependence of the mode I stress intensity factor on the wave number is studied. The solution is compared with the results obtained for a penny-shaped crack when the contact interaction is neglected.     

Reflection of a Dam–Break Wave at a Vertical Wall. Numerical Modeling and Experiment

Results of the experimental study and numerical modeling of the reflection of a dam–break wave at the vertical end wall of a channel are given. A wave forms with distance from a partition creating the initial level difference of the liquid. It is shown that a numerical calculation based on the Zheleznyak—Pelinovskii nonlinear dispersion model satisfactorily describes the height of the splash–up, the amplitude of reflected waves, and the wave velocity in front of the wall for smooth and dam–break waves. It is also shown that, for smooth and weakly breaking (without significant entrainment of air) incoming waves, the experimental values of the height of the splash–up at the wall agree well with relevant experimental and calculated data for solitary waves.     

Asymptotic of the flow upon shock incidence on a wedge cavity

The asymptotic of the motion originating because of shock incidence on a wedge cavity in a metal is investigated as the wave amplitude tends to zero. It has been shown in [1] that the flow is hence divided into two domains. The principal term governing the flow in the first domain agrees with the acoustic approximation. The flow in the second domain is described by incompressible fluid equations in the principal term. Determination of the flow in the second domain is reduced herein to the solution of a singular nonlinear integral equation. A numerical solution is found for a series of values of the cavity aperture.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 129–138, May–June, 1972.     

Plane-parallel supersonic flow with a discontinuity

This article considers a plant-parallel supersonic flow, with a shock wave terminating within the flow; the shock wave is regarded as a distortion. A line of discontinuity is located ahead of the shock wave in the supersonic zone. The problem is solved by the method of indeterminate coefficients.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 95–100, July–August, 1970.The authors thank S. V. Fal'kovich for his valuable advice and for his evaluation of the results obtained.     

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.     

Study of three-dimensional wave structure of nonstationary gas outflow from a planar sonic nozzle

Results are reported of an experimental study of the three-dimensional structure of nonstationary gas outflow from a planar nozzle. Outflow of a heated shock wave in a nitrogen tube at different moments of time from the start of outflow (0–1 msec) in two mutually perpendicular directions is considered. A scheme for reconstructing the flow at different outflow stages is proposed. The dimensions of the Riemann wave are found to oscillate.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 41–45, January–February, 1976.     

Self-similar solutions describing the propagation of solitary waves above underwater ridges and troughs

The nonlinear ray method [1] is used to investigate the propagation of solitary waves over an uneven bottom. In the process of nonlinear evolution of the wave front, singular points develop in it; these are treated in the given model as discontinuities [2, 3]. In contrast to earlier studies, it is not assumed here that the intensity of the discontinuity is weak. Boundary conditions at the discontinuities are introduced on the basis of the results of Miles and Bakholdin [4–6], and this makes it possible to take into account the energy loss at a discontinuity and the effects of wave reflection and construct a number of new self-similar solutions for the propagation of a wave above a ridge and trough. The main attention is devoted to considering how the type of solution depends on the parameters of the wave and the relief. For certain values of the parameters, the self-similar solution of the encounter of a homogeneous wave with a ridge is not unique. The reason for this is the singularity of the relief at the end of the ridge. A numerical investigation has therefore also been made of the encounter of a wave with a ridge having a smooth relief at its end. For an under-water trough and a ridge—trough system, self-similar solutions with complete or partial reflection or transmission of the wave energy into the trough are found. A reflected wave can also arise from an encounter with a ridge.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 137–144, September–October, 1985.I thank A. G. Kulikovskii and A. A. Barmin for their interest in the work and for valuable comments made as the paper was being prepared for press.     

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.     

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.     

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.     

Dynamics of cavitational demolition in the interaction of a shock wave with a free surface

The problem of cavitation in the tension wave associated with the reflection of a shock wave from a free fluid surface is considered. A method of calculating the cavitation zone dynamics which makes it possible to determine the structure of the cavitation front, including for large space scales, is developed. A procedure for determining the dispersity of the fragment-drops of dispersed fluid, which takes into account the initial size distribution of the cavitation nuclei and the parameters of the incident shock wave, is proposed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 73–80, November–December, 1992.     

Pressure of an arbitrary acoustical wave on a plane

A study is made of the perturbed flow of a gas, brought about by a weak shock wave, falling on a fixed surface at an arbitrary angle. A solution determining the field of the velocities behind the front of the wave in an initially boundary-value problem with movable boundaries for a three-dimensional wave equation is obtained in the form of a double integral, containing an arbitrarily given function determining the parameters of the gas in the incident wave. The region of integration is a region included within an ellipse, whose relative eccentricity is equal to the sine of the angle of inclination of the front of the incident wave. A formula is obtained for the distribution of the pressure at the plane.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 114–116, January–February, 1975.     

Effect of the Conductance and Thickness of a Conducting Plate on the Signal from a Material–Velocity Inductive Transducer

The perturbation problem of the magnetic field of a constant–current turn located above a conducting plate set into motion by a plane shock wave with a rectangular profile is considered. It is shown that not only the velocity of the plate but also its dynamic conductivity can be determined on the basis of the electromotive force of induction recorded by means of the turn. For the case where the conductance of the plate is known for both the conducting half–space and for a plate whose thickness is comparable with the skin–layer thickness, approximatecalculated dependences for the velocity of the plate are obtained. A comparison with experimental data and the clarification of the calculated dependences allows one to conclude that the approaches proposed can be used for determining the conductance of metals in shock–wave processes.     

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.