Collapse of Shock Waves upon Their Interaction with a Local Energy Source

Numerical analysis is performed of the interaction of a shock wave with a local energy source and the wake behind it. It is shown that for specified shock-wave intensity and flow parameters there is a threshold value of the energy release starting with which the shock wave collapses. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 2, pp. 3–7, March–April, 2006.     

Existence of a new type of irregular shock-wave interaction

Several types of plane shock-wave interactions [1–5], realized in various cases of practical importance, are now known. In [7] the possibility of the existence of a new type of shock-wave interaction was demonstrated by numerically solving the axisymmetric boundary-value problem. Here, the corresponding two-dimensional boundary-value problem of the interaction between a shock wave and a plane shock is numerically studied, a theoretical basis is obtained for the region of existence of the shock-wave interaction detected, and its principal properties are investigated. Moscow, Dnepropetovsk. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 181–183, September–October, 1988.     

Shock-wave structure in the near zone upon explosion of spatial charges in air

Profiles and values of pressure in shock waves are determined for the case of spherical, linear, and spatial charges, such as a coil of a bulk spiral and plane annular coils and Archimedes’ spiral of various lenths, exploded in air. In the case of explosion of rings and spirals, a complex wave structure in the form of a sequence of several shock waves is registered near the charges along the spiral axes; a weaker attenuation of shock waves with distance and pressure amplitudes two to three times higher than in the case of a spherical charge of the same mass are observed. It was found that an increase in the length of a plane spiral does not lead to an increase in the maximum pressure in the shock wave at distances of the order of several pitches of the spiral from its plane. With distance from spatial charges of different shape but identical mass, the pressure values in the shock-wave fronts coincide and tend asymptotically to the parameters of a spherical explosion with a significant increase in the duration of a wave packet generated by the spatial charge. Dependences for evaluation of shock-wave pressure amplitudes in the near zone of the explosion are presented. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 5, pp. 81–90, September–October, 2000.     

Shock-wave structure in a dispersing plasma

The shock-wave structure in an isotropic plasma is studied in the scale of the Debye shielding radius. The boundary condition at “infinity” is due to the dissipation mechanism of ions reflected from the ion wavefront. The case of the absence of dissipation corresponds to a collisionless shock wave. It is known that there is a critical Mach number above which the laminar shock wave is impossible. In addition, two types of boundary conditions which occur if collisions in an equilibrium high- and low-ionized plasma are taken into account are considered. The reflected ions are assumed to be scattered by electrons in the first case, and the momentum of the reflected ions is assumed to be transferred to uncharged particles in the second case. The critical Mach number of a shock wave does not exist under the conditions of collision dissipation of a flux of reflected ions. St. Petersburg State University, St. Petersburg 198904. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 3, pp. 14–21, May–June, 1998.     

Quasitransverse shock waves in elastic media with an internal structure

We consider the structure of small-amplitude quasitransverse shock waves in a weakly anisotropic elastic medium which possesses an internal structure generating the wave dispersion. The dispersion is modeled by introducing terms with higher derivatives into the equations of the theory of elasticity, and the dissipation is represented by viscous terms. In one of the two possible cases treated below, the requirement that the discontinuity structure exist leads to a set of admissible discontinuities of complex structure. A considerable part of the shock adiabat consists of a set of short portions and separate points, the number of which increases as the viscosity decreases. This complex set of admissible discontinuities is the general case where the dispersion in the shock-wave structure is sufficiently strong. Steklov Institute of Mathematics, Russian Academy of Sciences, Moscow 117526. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 174–180, March–April, 1999.     

Effect of Wave Formation during Shock-Wave Compaction of Powders

The problem of shock-wave compaction of a metal powder enclosed in a metal container with a transverse partition is solved. A model of wave formation on the partition and in the compact adjacent to the partition is proposed; the model is based on the loss of strength in the powder due to collapsing of pores and to development of instability of the partition being compressed in the shock wave. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 1, pp. 119–130, January–February, 2006.     

Formation conditions for bubble suspensions upon shock-wave loading of liquids

The conditions of development of bubble cavitation in liquid media upon shock-wave loading are found. It is s shown that, for the development of an unbounded cavitation, the bubbles should grow to certain critical sizes sufficient for their transition to a nonequilibrium state owing to the elastic energy transferred by a rarefaction wave to a liquid sample (at the stage of unloading). In contrast to low-viscosity liquids, in high-viscosity ones (such as glycerin) these conditions cannot be satisfied for any really attainable parameters of shock waves. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 2, pp. 53–63, March–April, 2000.     

Differential characteristics of the flow field in a plane overexpanded jet in the vicinity of the nozzle lip

A method of theoretical investigation of the flow field in a two-dimensional (plane-parallel or axisymmetric) overexpanded jet of an ideal perfect gas in the vicinity of the nozzle lip is described. The changes in curvature of the shock wave emanating from the lip, as well as the shock-wave intensity and flow parameters behind the shock are analyzed as functions of the Mach number, pressure ratio in the plane jet, and ratio of specific heats of the gas. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 72–83, May–June, 2006.     

The shock-wave structure in a two-velocity mixture of compressible media with different pressures

The problem of the shock-wave structure in a mixture of two compressible media with different velocities and pressures of components is considered. The problem is reduced to solving a boundary-value problem for two ordinary differential equations that describe the velocity relaxation and pressure equalization of the components. Using methods of the qualitative theory of dynamic systems on a plane, the existence and uniqueness of four types of waves are shown: (a) fully dispersed waves; (b) frozen-dispersed waves; (c) dispersed-frozen waves; (d) frozen waves of two-front configuration. A chart of solutions of the corresponding flow types is constructed in the plane of the following parameters: the initial velocity of the mixture and the initial volume concentration of one of the components. The numerical calculations conducted illustrate the obtained analytical structures of the shock wave. It is shown that the results obtained using the suggested mathematical model are in agreement with experimental data on the dependence of the velocity of the dispersed shock wave on the equilibrium pressure behind the shock-wave front for a mixture of silica sand and water. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 2, pp. 10–19, March–April, 1998.     

One method of producing a high-temperature dense plasma

This paper considers the interaction between an absolutely rigid wall or a steel plate and the rarefaction wave arising in solid deuterium when a 30–150 GPa shock wave arrives at the free surface. It is shown that, in the entropy trace near the wall or interface with the plate, a high-temperature plasma arises, in which a thermonuclear fusion is possible, at least, for shock-wave pressures above 70 GPa. The dimension of the plasma region and the time of its establishment are proportional to the distance between the free surface and the wall. Estimates of the proportionality coefficients are given. It is noted that, in this case, unlike in other methods of high-temperature plasma generation, the time of existence of the plasma may not depend on the sound velocity in it. It is shown that, by using a conical solid-state target wit an exit hole, the shock-wave pressure in solid deuterium can be increased from 10 to 100 GPa. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 15–24, May–June, 2009.     

Shock-wave-initiated lifting of particles from a cavity

The dynamics of particles of the disperse phase in a turbulent gas flow in planar shock waves sliding along a solid surface with a trapezoid cavity is examined numerically. Lifting of particles from the cavity walls is calculated in the approximation of a rarefied gas suspension. It is shown that the intensity of the transient shock wave and the initial positions of particles have a significant effect on the particle-lifting properties. The height of particle lifting is found to nonmonotonically depend on the initial streamwise coordinate and shock-wave Mach number. It is shown that zones of aggregation and subtraction of particles may be formed at the cavity bottom. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 24–34, January–February, 2007.     

The quasi-stationary structure of radiating shock waves. I. The one-temperature fluid

We calculate the quasi-stationary structure of a radiating shock wave propagating through a spherically symmetric shell of cold gas by solving the time-dependent equations of radiation hydrodynamics on an implicit adaptive grid. We show that this code successfully resolves the shock wave in both the subcritical and supercritical cases and, for the first time, we have reproduced all the expected features – including the optically thin temperature spike at a supercritical shock front – without invoking analytic jump conditions at the discontinuity. We solve the full moment equations for the radiation flux and energy density, but the shock wave structure can also be reproduced if the radiation flux is assumed to be proportional to the gradient of the energy density (the diffusion approximation), as long as the radiation energy density is determined by the appropriate radiative transfer moment equation. We find that Zel'dovich and Raizer's (1967) analytic solution for the shock wave structure accurately describes a subcritical shock but it underestimates the gas temperature, pressure, and the radiation flux in the gas ahead of a supercritical shock. We argue that this discrepancy is a consequence of neglecting terms which are second order in the minimum inverse shock compression ratio [, where is the adiabatic index] and the inaccurate treatment of radiative transfer near the discontinuity. In addition, we verify that the maximum temperature of the gas immediately behind the shock is given by , where is the gas temperature far behind the shock. Received 21 September 1998/ Accepted 2 February 1999     

3D Steady and Unsteady Bifurcations in a Shock-wave/Laminar Boundary Layer Interaction: A Numerical Study

The principal objective of this paper is to study some unsteady characteristics of an interaction between an incident oblique shock wave impinging a laminar boundary layer developing on a plate plane. More precisely, this paper shows that some unsteadiness, in particular the low frequency unsteadiness, originate in a supercritical Hopf bifurcation related to the dynamics of the separated boundary layer and not necessarily to the coherent structures resulting from the turbulent character of the boundary layer crossing the shock wave. Numerical computations of a shock-wave/laminar boundary-layer interaction (SWBLI) have been compared with a classical test case (Degrez test case) and both two-dimensional and three-dimensional (3D) unsteady Navier–Stokes equations are numerically solved with an implicit dual time stepping for the temporal algorithm and high order AUSM+ scheme for the spatial discretization. A parametric study on the oblique shock-wave angle has been performed to characterize the unsteady behaviour onset. Finally, discussions and assumptions are made about the origin of the 3D low frequency unsteadiness.     

Dynamics and radiation of a single bubble under conditions of abnormal compressibility of a bubbly liquid

An equation is proposed for the pulsation of a single cavity in an abnormally compressible bubbly liquid which is in pressure equilibrium and whose state is described by the Lyakhov equation. In the equilibrium case, this equation is significantly simplified. Numerical analysis is performed of the bubble dynamics and acoustic losses (the profile and amplitude of the radiation wave generated on the bubble wall from the side of the liquid). It is shown that as the volumetric gas concentration k₀ in the equilibrium bubbly medium increases, the degree of compression of the cavity by stationary shock wave decreases and its pulsations decrease considerably and disappear already at k₀ = 3%. In the compression process, the cavity asymptotically reaches an equilibrium state that does not depend on the value of k₀ and is determined only by the shock-wave amplitude. The radiation wave takes the shape of a soliton whose amplitude is much smaller and whose width is considerably greater than the corresponding parameters in a single-phase liquid. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 51–57, May–June, 2007.     

Detonation Initiation by Shock Reflection from an Orifice Plate

Results from an experimental investigation of the interaction of a “non-ideal” shock wave and a single obstacle are reported. The shock wave is produced ahead of an accelerated flame in a 14 cm inner-diameter tube partially filled with orifice plates. The shock wave interacts with a single larger blockage orifice plate placed 15–45 cm after the last orifice plate in the flame acceleration section of the tube. Experiments were performed with stoichiometric ethylene–oxygen mixtures with varying amounts of nitrogen dilution at atmospheric pressure and temperature. The critical nitrogen dilution was found for detonation initiation. It is shown that detonation initiation occurs if the chemical induction time based on the reflected shock state is shorter than the time required for an acoustic wave to traverse the orifice plate upstream surface, from the inner to the outer diameter. The similarity between the present results and those obtained from previous investigators looking at detonation initiation by ideal shock reflection produced in a shock tube indicates that the phenomenon is not sensitive to the detailed structure of the shock front but only on the average shock strength.This paper is based on work that was presented at the 20th International Colloquium on the Dynamics of Explosions and Reactive Systems, Montreal, Canada, July 31–August 5, 2005.     

Resonant phenomena in a transonic flow around an airfoil with pulsed periodic energy supply

Interaction of a pulsed periodic source of energy with a closing shock wave arising near airfoils in transonic flight is studied. The evolution of the shock-wave structure of the flow around a symmetric airfoil is examined by solving two-dimensional unsteady gas-dynamic equations, and a resonant mechanism of interaction is found, which leads to considerable (by an order of magnitude) reduction of the wave drag of the airfoil.     

Nonlinear effects caused by pulsed periodic supply of energy in the vicinity of a symmetric airfoil in a transonic flow

Changes in the structure of a transonic flow around a symmetric airfoil and a decrease in the wave drag of the latter, depending on the energy-supply period and on localization and shape of the energy-supply zone, are considered by means of the numerical solution of two-dimensional unsteady equations of gas dynamics. Energy addition to the gas ahead of the closing shock wave in an immediate vicinity of the contour in zones extended along the contour is found to significantly reduce the wave drag of the airfoil. The nature of this decrease in drag is clarified. The existence of a limiting frequency of energy supply is found. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 64–71, May–June, 2006.     

Special and extreme triple shock-wave configurations

Triple shock-wave configurations in steady supersonic flows of an inviscid perfect gas are considered. Triple configurations with special intensities of shock waves and extreme ratios of various flow parameters behind these configurations are determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 4, pp. 39–53, July–August, 2006.     

Gradients at a curved shock in reacting flow

The inviscid equations of motion for the flow at the downstream side of a curved shock are solved for the shock–normal derivatives. Combining them with the shock–parallel derivatives yields gradients and substantial derivatives. In general these consist of two terms, one proportional to the rate of removal of specific enthalpy by the reaction, and one proportional to the shock curvature. Results about the streamline curvature show that, for sufficiently fast exothermic reaction, no Crocco point exists. This leads to a stability argument for sinusoidally perturbed normal shocks that relates to the formation of the structure of a detonation wave. Application to the deflection–pressure map of a streamline emerging from a triple shock point leads to the conclusion that, for non–reacting flow, the curvature of the Mach stem and reflected shock must be zero at the triple point, if the incident shock is straight. The direction and magnitude of the gradient at the shock of any flow quantity may be written down using the results. The sonic line slope in reacting flow serves as an example. Extension of the results – derived in the first place for plane flow – to three dimensions is straightforward. Received 12 February 1997 / Accepted 10 June 1997     

Start shock waves and the vortex structures arising in the formation of jets

Experimental studies were made of the state behind a shock wave arising in front of a pulse jet in formation, and the development of the vortex structure of the jet itself. A comparison was made of the experimental axial density distribution in the jet in the region between the front of the gas and the primary shock wave with the solutions to the problem of an expanding spherical piston and the nonself-similar problem of a point explosion. Moscow Translated from Izvestiya, Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 146–150, September–October, 1988.