Lagrangian description of transport equations for shock waves in three-dimensional elastic solids

A set of transport equations for the growth or decay of theamplitudes of shock waves along an arbitrary propagation directionin three-dimensional nonlinear elastic solids is derived using theLagrangian coordinates.The transport equations obtained showthat the time derivative of the amplitude of a shock wave alongany propagation ray depends on (i) an unknown quantity immediatelybehind the shock wave,(ii) the two principal curvatures of theshock surface,(iii) the gradient taken on the shock surface ofthe normal shock wave speed and (iv) the inhomogeneous term.whichis related to the motion ahead of the shock surface.vanisheswhen the motion ahead of the shock surface is uniform.Severalchoices of the propagation vector are given for which the tran-sport equations can be simplified.Some universal relations,which relate the time derivatives of various jump quantities toeach other but which do not depend on the constitutive equationsof the material,are also presented.     

Evolution of weak shocks in one dimensional planar and non-planar gasdynamic flows

Asymptotic decay laws for planar and non-planar shock waves and the first order associated discontinuities that catch up with the shock from behind are obtained using four different approximation methods. The singular surface theory is used to derive a pair of transport equations for the shock strength and the associated first order discontinuity, which represents the effect of precursor disturbances that overtake the shock from behind. The asymptotic behaviour of both the discontinuities is completely analysed. It is noticed that the decay of a first order discontinuity is much faster than the decay of the shock; indeed, if the amplitude of the accompanying discontinuity is small then the shock decays faster as compared to the case when the amplitude of the first order discontinuity is finite (not necessarily small). It is shown that for a weak shock, the precursor disturbance evolves like an acceleration wave at the leading order. We show that the asymptotic decay laws for weak shocks and the accompanying first order discontinuity are exactly the ones obtained by using the theory of non-linear geometrical optics, the theory of simple waves using Riemann invariants, and the theory of relatively undistorted waves. It follows that the relatively undistorted wave approximation is a consequence of the simple wave formalism using Riemann invariants.     

Inviscid vortex structures in the shock layers of conical flows around V wings

The applicability of the criteria of existence of inviscid vortex structures (vortex Ferri singularities) is studied in the case in which a contact discontinuity of the corresponding intensity proceeds from the branching point of the λ shock wave configuration accompanying turbulent boundary layer separation under the action of an inner shock incident on the leeward wing panel. The calculated and experimental data are analyzed, in particular, those obtained using the special shadow technique developed for visualizing supersonic conical streams in nonsymmetric, Mach number 3 flow around a wing with zero sweep of the leading edges and the vee angle of 2π /3. The applicability of the criteria of existence of inviscid vortex structures is established for contact discontinuities generated by the λ shock wave configuration accompanying turbulent boundary layer separation realized under the action of a shock wave incident on the leeward wing panel. Thus, it is established that the formation of the vortex Ferri singularities in a shock layer is independent of the reason for the existence of the contact discontinuity and depends only on its intensity.     

Interaction Between a Slow Magnetohydrodynamic Shock Wave and a Tangential Discontinuity

The self-similar problem of the oblique interaction between a slow MHD shock wave and a tangential discontinuity is solved within the framework of the ideal magnetohydrodynamic model. The constraints on the initial parameters necessary for the existence of a regular solution are found. Various feasible wave flow patterns are found in the steady-state coordinate system moving with the line of intersection of the discontinuities. As distinct from the problems of interaction between fast shock waves and other discontinuities, when the incident shock wave is slow the state ahead of it cannot be given and must to be determined in the process of solving the problem. As an example, a flow in which the slow shock wave incident on the tangential discontinuity is generated by an ideally conducting wedge located in the flow is considered. The basic features of the developing flows are determined.     

Dynamic steady crack growth in elastic-plastic solids—propagation of strong discontinuities

The near-tip field of a mode I crack growing steadily under plane strain conditions is studied. A key issue is whether strong discontinuities can propagate under dynamic conditions. Theories which impose rather restrictive assumptions on the structure of an admissible deformation path through a dynamically propagating discontinuity have been proposed recently. Asymptotic solutions for dynamic crack growth, based on such theories, do not contain any discontinuities. In the present work a broader family of deformation paths is considered and we show that a discontinuity can propagate dynamically without violating any of the mechanical constitutive relations of the material. The proposed theory for the propagation of strong discontinuities is corroborated by very detailed finite element calculations. The latter shows a plane of strong discontinuity emanating from the crack tip (with its normal pointing in the direction of crack advance) and moving with the tip. Elastic unloading ahead of and/or behind the plane of discontinuity and behind the crack tip have also been observed.The numerical investigation is performed within the framework of a boundary layer formulation whereby the remote loading is fully specified by the first two terms in the asymptotic solution of the elasto-dynamic crack tip field, characterized by K₁, and T. It is shown that the family of near-tip fields, associated with a given crack speed, can be arranged into a one-parameter field based on a characteristic length, L_g, which scales with the smallest dimension of the plastic zone. This extends a previous result for quasi-static crack growth.     

Weak discontinuities in a chemically reacting atmosphere

Summary The growth and decay of a weak discontinuity headed by a singular surface of arbitrary shape in three dimensions is investigated in a chemically reacting atmosphere, in the absence of dissipative mechanisms such as viscosity, diffusion and heat conduction. The combined effects of the disequilibrium due to the chemical reaction and a wave front curvature on the propagation of discontinuities have been examined and discussed. It has been observed that the chemical disequilibrium, with its Arrhenius rate dependence, causes the compression wave to steepen more swiftly that it does in an inert atmosphere. The critical values of the initial discontinuity, and time for shock formation, in cases of diverging and converging waves, have been determined.     

Growth of discontinuities in fluids with internal relaxation

The growth equation for weak discontinuities headed by wave fronts of arbitrary shape in a relaxing gas flow is derived along the orthogonal trajectories of the wave fronts. An explicit criteria for the growth and decay of weak discontinuities along their orthogonal trajectories is given. It is concluded that the internal relaxation processes in the flow as well as the wave front curvature both have a stabilizing effect on the tendency of the wave surface to grow into a shock in the sense that they cause the shock formation time to increase.     

Global Uniqueness of Transonic Shocks in Two-Dimensional Steady Compressible Euler Flows

We prove that for the two-dimensional steady complete compressible Euler system, with given uniform upcoming supersonic flows, the following three fundamental flow patterns (special solutions) in gas dynamics involving transonic shocks are all unique in the class of piecewise C ¹ smooth functions, under appropriate conditions on the downstream subsonic flows: (i) the normal transonic shocks in a straight duct with finite or infinite length, after fixing a point the shock-front passing through; (ii) the oblique transonic shocks attached to an infinite wedge; (iii) a flat Mach configuration containing one supersonic shock, two transonic shocks, and a contact discontinuity, after fixing a point where the four discontinuities intersect. These special solutions are constructed traditionally under the assumption that they are piecewise constant, and they have played important roles in the studies of mathematical gas dynamics. Our results show that the assumption of a piecewise constant can be replaced by some weaker assumptions on the downstream subsonic flows, which are sufficient to uniquely determine these special solutions. Mathematically, these are uniqueness results on solutions of free boundary problems of a quasi-linear system of elliptic-hyperbolic composite-mixed type in bounded or unbounded planar domains, without any assumptions on smallness. The proof relies on an elliptic system of pressure p and the tangent of the flow angle w = v/u obtained by decomposition of the Euler system in Lagrangian coordinates, and a newly developed method for the L ^∞ estimate that is independent of the free boundaries, by combining the maximum principles of elliptic equations, and careful analysis of the shock polar applied on the (maybe curved) shock-fronts.     

Impact of the interplanetary magnetic field on thewave flow pattern in the neighborhood of the Earth’s bow shock under sharp variations in the solar wind dynamic pressure

The impact of the interplanetary magnetic field on transformation and disintegration of the Earth’s bow shock into a system of magnetohydrodynamic (MHD) shock waves, rotational discontinuities and rarefaction waves under the action of abrupt variations in the solar wind dynamic pressure is simulated in the three-dimensional non-plane-polarized formulation within the framework of the ideal magnetohydrodynamic model using the solution of the MHD Riemann problem of breakdown of an arbitrary discontinuity. This discontinuity arises when a contact discontinuity, on which the solar wind density increases or decreases suddenly and which travels together with the solar wind, impinges on the Earth’s bow shock and propagates along its surface. The interaction pattern is constructed in the quasisteady- state formulation as a mosaic of exact solutions obtained on computer using an original MHD Riemann solver. The wave flow patterns are found for all elements of the surface of the bow shock as functions of their latitude and longitude for various jumps in the density on the contact discontinuity and characteristics parameters of the solar wind and interplanetary magnetic field at the Earth’s orbit. It is found that when the solar wind dynamic pressure increases, a fast MHD shock wave, that first penetrates into the magnetosheath, is always formed. When the solar wind dynamic pressure decreases, the influence of the interplanetary magnetic field can lead to the development of the leading fast MHD shock wave in certain zones on the surface of the Earth’s bow shock. The solution obtained can be used to interpret measurements on spacecraft in the solar wind at the libration point and in the neighborhood of the Earth’s magnetosphere.     

The behavior of induced discontinuities behind curved shocks in isotropic linear elastic materials

In this paper we examine the behavior of the induced discontinuities behind curved longitudinal and transverse shock waves in isotropic linear elastic materials. It is shown that in either case the governing differential equation of the induced discontinuity differs from that of the shock amplitude. The latter depends linearly on the second fundamental form of the shock surface and exhibits purely geometrical effects. The former, however, depends non-linearly on the second fundamental form of the shock surface, and on the shock amplitude. These terms are dominant for a strong shock and their effects diminish as the shock weakens. In particular, the governing differential equation for an acceleration wave is obtained in the limit as the shock amplitude vanishes. The results obtained are quite unexpected, and they demonstrate the complex evolutionary behavior of mechanical waves due to geometrical considerations alone.     

Interference of stationary and non-stationary shock waves

The mathematical model of a gasdynamic discontinuity is used in the area of study concerning gas flows with large gradients of gasdynamic functions. Gasdynamic functions before and after the discontinuity meet non-linear algebraic equations called the dynamic compatibility conditions on the discontinuities. Different modes of shock wave structures forming as a result of regular or irregular interference of the incoming discontinuities of different types are described. Ranges of the initial flow parameters definition such that either shock wave structures of different modes take place or interference equations have no solutions are determined. Most attention is given to arbitrary triple shock-wave configurations. Their classification is proposed. Differential characteristics of the steady flow are studied. The notion “differential characteristics” includes first derivatives of the fundamental gasdynamic parameters with respect to natural coordinates and curvatures of the discontinuities surfaces. Effect of unsteadiness on the triple-shock configuration is examined. Some problems arising at creation of complete local theory of steady and propagating gasdynamic discontinuities interference are formulated.     

Effect of incident shock wave strength on the decay of Richtmyer–Meshkov instability-introduced perturbations in the refracted shock wave

The effect of incident shock wave strength on the decay of interface introduced perturbations in the refracted shock wave was studied by performing 20 different simulations with varying incident shock wave Mach numbers (M ~ 1.1? 3.5). The analysis showed that the amplitude decay can be represented as a power law model shown in Eq.7, where A is the average amplitude of perturbations (cm), B is the base constant (cm^?(E?1), S is the distance travelled by the refracted shockwave (cm), and E is the power constant. The proposed model fits the data well for low incident Mach numbers, while at higher mach numbers the presence of large and irregular late time oscillations of the perturbation amplitude makes it hard for the power law to fit as effectively. When the coefficients from the power law decay model are plotted versus Mach number, a distinct transition region can be seen. This region is likely to result from the transition of the post-shock heavy gas velocity from subsonic to supersonic range in the lab frame. This region separates the data into a high and low Mach number region. Correlations for the power law coefficients to the incident shock Mach number are reported for the high and low Mach number regions. It is shown that perturbations in the refracted shock wave persist even at late times for high incident Mach numbers.     

MHD simulation of the distribution of the gasdynamic parameters and magnetic field behind the Earth’s bow shock under sharp variations in the solar wind dynamic pressure

The distributions of the gasdynamic parameters (density, pressure, and velocity) and the magnetic field behind the Earth’s bow shock (on the outer boundary of the magnetosheath) generated under sharp variations in the solar wind dynamic pressure are found in the three-dimensional non-planepolarized formulation with allowance for the interplanetary magnetic field within the framework of the ideal magnetohydrodynamic model using the solution to the MHD Riemann problem of breakdown of an arbitrary discontinuity. Such a discontinuity which depends on the inclination of an element of the bow shock surface arises when a contact discontinuity traveling together with the solar wind and on which the solar wind density and, consequently, the dynamic pressure, increases or decreases suddenly impinges on the Earth’s bow shock and propagates along its surface initiating the development of to six waves or discontinuities (shocks). The general interaction pattern is constructed for the entire bow shock surface as a mosaic of exact solutions to the MHD Riemann problem obtained on computer using an original software (MHD Riemann solver) so that the flow pattern is a function of the angular surface coordinates (latitude and longitude). The calculations are carried out for various jumps in density on the contact discontinuity and characteristics parameters of the solar wind and interplanetary magnetic field at the Earth’s orbit. It is found that there exist horseshoe zones on the bow shock in which the increase in the density and the magnetic field strength in the fast shock waves or their reduced decrease in the fast rarefaction waves penetrating into the magnetosheath and arising as a result of sharp variation in the solar wind dynamic pressure is superposed on significant drop in the density and growth in the magnetic field strength in slow rarefaction waves. The distributions of the hydrodynamic parameters and the magnetic field can be used to interpret measurements carried out on spacecraft in the solar wind at the libration point and orbiters in the neighborhood of the Earth’s magnetosphere.     

Investigation of the problem of the decay of an arbitrary two-dimensional discontinuity

The problem of the decay of an arbitrary two-dimensional discontinuity in a gas has been investigated analytically and numerically. When the boundary of the original discontinuity is a straight line with a break, the problem is reduced to the interaction of two one-dimensional arbitrary discontinuities. It is shown that the difference between the two-dimensional and one-dimensional solutions has a maximum when the decay of the the discontinuity is accompanied by the formation of shock waves that interact with each other and with other gas-dynamic discontinuities. The effect of the angle of deviation of the original boundary of the discontinuity on the two-dimensionality of the flow is estimated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 159–164, March–April, 1989.     

The propagation of initially plane waves in nonhomogeneous viscoelastic media

In this study, the propagation of an initially plane wave in a linear isotropic nonhomogeneous viscoelastic medium, where the nonhomogeneity varies transversely to the direction of propagation, is investigated. For this purpose, first the propagation of waves in a linear isotropic viscoelastic medium of arbitrary inhomogeneity is studied by employing the notion of singular surfaces. The characteristic equation governing wave velocities, and the growth and decay equations describing the change of the strength of the discontinuity as the wave front moves are obtained.In the second part of this work, the propagation of initially plane waves is studied for three types of inhomogeneities by employing the findings established in the first part. The first kind of inhomogeneity considered is of axisymmetrical type where the wave propagation velocity depends on the radial coordinate only, increasing linearly up to a certain radial distance and remaining constant thereafter. The second kind is also axisymmetrical with a wave velocity distribution decreasing linearly till a given value of the radial coordinate. In the third one, the wave velocity is assumed to vary linearly over a given interval along a certain coordinate axis only, which is perpendicular to the direction of propagation, and remain constant outside. The ray and wave front analyses are carried out and the decay or growth of stress and velocity discontinuities are studied for each of the three cases.     

Propagation of discontinuity waves in complex rheological media with viscous properties

The propagation of discontinuity waves of various order in rheological media is examined. It is assumed that the region of discontinuity of values can be represented by an intermediate layer of infinitesimal thickness. By means of this representation, results can be obtained for a rather wide class of continuous media with viscous properties, which generalize Duhem's results. The first integrals of the laws of momentum and energy conservation are obtained, which hold inside the intermediate layer at a shock wave.It is shown that when viscosity elements are introduced in a special way into the rheological model of a continuous medium, discontinuity waves of any order are propagated in the medium, and that at the surface of a strong discontinuity in a heat-conducting medium, the temperature is continuous. Additional conditions for strain discontinuities at the viscosity elements are obtained. For certain inclusions of the viscosity elements into the rheological model discontinuity waves do not propagate; instead there is merely a weak discontinuity surface which acts as an interface between the flow region of the continuous medium and the region in the state of rest. Contact discontinuities can occur in any continuous medium.The possible existence of a geometrical discontinuity surface in a viscous gas was examined first by Duhem [1]. He established that singluar strong-discontinuity surfaces cannot take place in a viscous gas. However, if one assumes that the velocity and temperature are continuous in the passage through a singular surface, only contact discontinuities are possible [2].     

Evolutionary behavior of induced discontinuities behind one dimensional shock waves in non-linear elastic materials

The governing differential equation of induced discontinuities behind one dimensinal shock waves in non-linear elastic materials has been derived. This equation depends, in particular, on the shock amplitude itself. Therefore, its solution depends on the solution of the governing equation of the shock amplitudes which, in turn, depend on the induced discontinuities. It is shown in the special case pertaining to a first-order approximation that there exists a critical shock amplitude S _c such that the evolutionary behavior of the induced discontinuities depends on the relative magnitudes of the shock amplitudes and S _c. However, in the special case pertaining to a second-order approximation the evolutionary behavior of the induced discontinuities is monotone.     

Critical distances for the formation of strong discontinuities in fluid-saturated solids

An increase in the stiffness of a solid in compression is known to lead to the steepening of the profiles of compression waves and, as a consequence, to the formation of strong discontinuities from continuous waves propagating in the solid. In this paper, the critical distance required for a continuous wave to turn into a shock wave is calculated from the evolution equation for a weak discontinuity (acceleration wave) propagating into a quiescent region. Infinite growth of the amplitude of an acceleration wave in a finite time signifies the transition to a strong discontinuity. Relations between the critical distances for plane, cylindrical and spherical waves are established. Numerical examples are presented for a particular case of the pressure-dependent stiffness typical of granular solids such as sand or soil, with emphasis placed on the influence of a small amount of free gas in the pore fluid.     

Application of unsteady mixing equations to some aerodynamic problems

Tangential discontinuities [1] are introduced in solving several transient and steady-state problems of gas dynamics. These discontinuities are unstable [2] as a result of the effects of viscosity and thermal conductivity. Therefore it is advisable to replace the tangential discontinuity by a mixing region and account for its interaction with the inviscid flows, establishing on the boundaries of this region the conditions of vanishing friction stress and equality of the velocity and temperature components to the corresponding velocity and temperature components of the inviscid flows. This formulation improves the accuracy of the solution of such problems by posing them as problems with irregular reflection and intersection of shock waves [1].The consideration of the interaction of unsteady turbulent mixing regions with the inviscid flow also permits the formulation of several problems in which the effects of viscosity lead to complete rearrangement of the flow pattern (the lambda-configuration) with the interaction of the reflected shock wave with the boundary layer in the shock tube [3,4], the formation of zones of developed separation ahead of obstacles, etc.).In this connection, §1 presents an analysis of the self-similar solutions of the unsteady turbulent mixing equations (a corresponding analysis of the laminar mixing equations which coincide with the boundary layer equations is presented in [1]). It is shown that these self-similar solutions describe, along with the several problems noted above, the problems of the formation of steady jets and mixing zones in the base wake.As an example, §2 presents, within the framework of the proposed schematization, an approximate solution of the problem of the interaction of a shock wave reflected from a semi-infinite wall with the boundary layer on a horizontal plate behind the incident shock wave. The results obtained are applied to the analysis of reflection in a shock tube. Computational results are presented which are in qualitative agreement with experiment [3, 4].