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.     

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.     

Structure of compression shock waves in porous elastoplastic materials

The shock-wave structure in a porous elastoplastic material is studied. In a certain range of parameters, the existence of a four-wave structure of a compression shock wave is possible. Regimes in which a reflected shock wave does not appear at all have been found in the problem of shock-wave reflection from a rigid wall. In this case, the entire energy of the incident shock wave transforms to thermal energy due to dissipation induced by the viscous collapse of the pores. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 6, pp. 27–32, November–December, 1998.     

Spectral Stability of Ideal-Gas Shock Layers

Extending recent results in the isentropic case, we use a combination of asymptotic ODE estimates and numerical Evans-function computations to examine the spectral stability of shock-wave solutions of the compressible Navier–Stokes equations with ideal gas equation of state. Our main results are that, in appropriately rescaled coordinates, the Evans function associated with the linearized operator about the wave (i) converges in the large-amplitude limit to the Evans function for a limiting shock profile of the same equations, for which internal energy vanishes at one end state; and (ii) has no unstable (positive real part) zeros outside a uniform ball |λ| ≦ Λ. Thus, the rescaled eigenvalue ODE for the set of all shock waves, augmented with the (nonphysical) limiting case, form a compact family of boundary-value problems that can be conveniently investigated numerically. An extensive numerical Evans-function study yields one-dimensional spectral stability, independent of amplitude, for gas constant γ in [1.2, 3] and ratio ν/μ of heat conduction to viscosity coefficient within [0.2, 5] (γ ≈ 1.4, ν/μ ≈ 1.47 for air). Other values may be treated similarly but were not considered. The method of analysis extends also to the multi-dimensional case, a direction that we shall pursue in a future work.     

Linear problem of a hydrofoil moving under the free surface of a finite-depth fluid

A method of solving the plane linear problem of a steady-state irrotational flow about a body under the free surface of a heavy fluid of finite depth is developed. The boundary-value problem is formulated for a complex perturbed velocity and is reduced to a singular integral equation relative to the intensity of a vortex layer that models the hydrofoil. The kernel of the equation is the exact solution of the corresponding boundary-value problem for a vortex of unit intensity. The equation is solved by the discrete-vortex method. The effect of the parameters of the problem on the hydrodynamic characteristics of the elliptical hydrofoil and the shape of the free surface are estimated numerically. Omsk Division of the Sobolev Institute of Mathematics, Omsk 644099. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 6, pp. 85–90, November–December, 1998.     

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.     

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.     

Formation of a plastic zone in an anisotropic body under loads acting along a crack

The effects of tension and compression along a crack on the plastic zone in a finite anisotropic body under plane strain are studied. The formation pattern for the plastic zone with increasing load is established by numerically solving a boundary-value problem for each of the cases. In particular, a new plastic zone is revealed. It occurs at the crack face under a compressive load of certain magnitude. How this plastic zone interacts with that at the crack tip is established __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 3–19, May 2007.     

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.     

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.     

Impact of a spherical rigid body on the surface of a cavity in a compressible liquid: An axisymmetric problem

The shock interaction of a spherical rigid body with a spherical cavity is studied. This nonstationary mixed boundary-value problem with an unknown boundary is reduced to an infinite system of linear Volterra equations of the second kind and the differential equation of motion of the body. The hydrodynamic and kinematic characteristics of the process are obtained __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 1, pp. 11–19, January 2008.     

Formation of the plastic zone in an anisotropic body with a crack

The influence of the length of a mode I crack on the plastic zone in an anisotropic body under hard loading is studied. The case of a generalized plane stress state is examined. A boundary-value problem is solved numerically to study the behavior of the main plastic zone at the crack tip, the additional plastic zone on the lateral face of the body, and the merged plastic zone Translated from Prikladnaya Mekhanika, Vol. 44, No. 9, pp. 36–52, September 2008.     

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.     

Problem of filling of a spherical cavity in a Kelvin-Voigt medium

This paper is concerned with mathematical modeling and solution of the problem of the collapse of a spherical cavity in a viscoelastic medium under the action of constant pressure at infinity. A differential equation of motion for the cavity boundary is constructed and solved numerically. The existence of three modes of motion of the boundary is established, and a map of these modes in the plane of the determining parameters is constructed. Asymptotic forms of the solutions of the problem for all modes are constructed. The problem of cavity collapse with capillary forces taken into account is formulated and solved. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 93–101, September–October, 2008.     

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.     

Homogenization of the signorini boundary-value problem in a thick plane junction

We consider a mixed boundary-value problem for the Poisson equation in a plane thick junction Ω_ε that is the union of a domain Ω₀ and a large number of ε-periodically located thin rods. The nonuniform Signorini conditions are given on the vertical sides of the thin rods. The asymptotic analysis of this problem is made as ε → 0, i.e., in the case where the number of thin rods infinitely increases and their thickness tends to zero. With the help of the integral identity method, we prove the convergence theorem and show that the nonuniform Signorini conditions are transformed (as ε → 0) into the limiting variational inequalities in the domain that is filled up with thin rods when passing to the limit. The existence and uniqueness of a solution to this nonstandard limit problem are established. The convergence of the energy integrals is proved as well. Published in Neliniini Kolyvannya, Vol. 12, No. 1, pp. 44–58, January–March, 2009.     

Collision of dissimilar blunt elastic bodies: a plane problem

The paper deals with a direct central impact of two infinite cylindrical bodies having differently shaped cross sections and made of different materials. A nonstationary plane problem of elasticity is solved. The contact boundary is moving and determined during the solution. A mixed boundary-value problem is formulated. Its solution has the form of Fourier series. Satisfying mixed boundary conditions gives an infinite system of Volterra equations of the second kind for the unknown coefficients of the series. The basic characteristics of the impact process and their dependence on the physical and mechanical properties of the bodies are determined numerically Translated from Prikladnaya Mekhanika, Vol. 45, No. 2, pp. 36–45, February 2009.     

The plane problem on vertical impact of a rotating circular cylindrical shell against the surface of a liquid

The plane nonsymmetric problem on impact against and immersion into a compressible fluid of a thin electic cylindrical shell is considered. The shell rotates about its axis with a given angular velocity. The boundary-value problem is reduced to an infinite system of integral Volterra equations of the second kind. Translated from Prikladnaya Mekhanika, Vol. 36, No. 5, pp. 103–113, May, 2000.     

Oscillations of an inhomogeneous elastic layer

A method is proposed to analyze wave fields in an elastic layer with elastic properties varying arbitrarily with depth. The method is based on reducing the boundary-value problem to a system of Fredholm integral equations of the second kind, which is then analyzed numerically. Some features of the structure of dispersion sets are analyzed and, in particular, their asymptotes are constructed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 157–164, May–June, 2006.     

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.