Investigation of the evolutionarity of discontinuities in binary mixture flows through a porous medium

The discontinuity surfaces (shock waves) that arise in nonisothermal carbon dioxide-water binary mixture flows through a porous medium are considered. In the plane of determining parameters the discontinuity adiabats are investigated and their evolutionarity diagrams are plotted. It is shown that one of the adiabat branches corresponds to the displacement fronts at which there are no temperature jumps and phase transitions and the other branch to temperature jumps and phase transition fronts. The adiabat branches may intersect at a point that corresponds to the Jouguet point for the parameters both ahead of and behind the finite-amplitude jump. It is shown that in the neighborhood of this double Jouguet point the adiabat behavior differs from the classical adiabat behavior at single Jouguet points.     

Shock adiabats in a mixture of gas and particles

Shock waves in a mixture of a gas and incompressible drops or particles are considered. We construct the shock adiabat connecting the states in front of and behind a discontinuity, on which the processes of interaction of the phases are assumed to be frozen. It follows from analysis of this adiabat that when particles are present in the gas pressure discontinuities of infinite intensity are impossible, which distinguishes this adiabat from the Hugoniot adiabat in gas dynamics.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 163–167, September–October, 1979.I thank V. V. Gogosov, and also V. A. Naletov and G. A. Shaposhnikov for assistance in the work and R. I. Nigmatulin for valuable comments.     

Eulerian formulation of transport equations for three-dimensional shock waves in simple elastic solids

The propagation of a three-dimensional shock wave in an elastic solid is studied. The material is assumed to be a simple elastic solid in which the Cauchy stress depends on the deformation gradient only. It is shown that the growth or decay of a discontinuity ψ depends on (i) an unknown quantity φ^? behind the shock wave, (ii) the two principal curvatures of the shock surface, (iii) the gradient on the shock surface of the shock wave speeds and (iv) the inhomogeneous term which depends on the motion ahead of the shock surface and vanishes when the motion ahead of the shock surface is uniform. If a proper choice is made of the propagation vectorb along which the growth or decay of the discontinuity is measured, the dependence on item (iii) can be avoided. However,b assumes different directions depending on the choice of discontinuity ψ with which one is concerned and the unknown quantity φ^? behind the shock wave on which one chooses to depend. As in the case of one-dimensional shock waves, the growth (or decay) of one discontinuity may not be accompanied by the growth (or decay) of other discontinuities. A universal equation relating the growth or decay of discontinuities in the normal stress, normal velocity and specific volume is also presented.     

Transformations of graphite and boron nitride in shock waves

Peculiarities of shock adiabat of graphite are attributed to the graphite–diamond transformation. However only a very small amount of diamond can be recovered from pure shocked graphite with a density approaching the theoretical value. In order to interpret this fact, accessible data concerning the behaviour of graphite under static and dynamic load have been analysed. An additional peculiarity of the shock adiabat of graphite has been found at 12 GPa by analysing compressibility data. It has been attributed to shearing in the basal planes that paves the way for deformation of the planes. An isotherm of cold compression of graphite can be constructed on the basis of the results from theoretical modelling published in the literature. Another isotherm, fitting experimental data, has been proposed. An isotherm for graphitic boron nitride has been also proposed. The isotherms have been used in the interpretation of the peculiarities of shock adiabats. It has been shown that the so-called “mixed-phase” region is an apparent compressibility curve. Energy evaluations based on the isotherms have proved that the peculiarities of the shock adiabat of graphite correspond to the formation of hexagonal instead of cubic diamond. Similarly the formation of the wurtzite modification of BN is responsible for the peculiarities of the shock adiabat of BN. Literature data concerning the mechanism of the polymorphous transformations of graphite and BN in shock waves have been reviewed. On the basis of proposed isotherms of cold compression, the activation energy has been appraised and an equation of kinetics proposed. The equation has been analysed by comparing results of theoretical modelling and accessible experimental data. Received 11 March 1993 / Accepted 15 September 1993     

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.     

Investigation of pressure-discontinuity system with one and two triplets,taking into account equilibrium physicochemical transformations of air

It is known that the interaction of pressure discontinuities preceding the projecting elements of a supersonic object considerably increase the pressure in the interaction region [1–3]. Existing methods of estimating this excess pressure at the leading edge of the projecting element are based on the calculation of the configuration of pressure-discontinuity intersections with two or one triple points for a perfect gas with a constant adiabatic modulus . The calculation reduces to the successive solution of two transcendental equations for the determination of the angles of slope of the discontinuities at the node points [2, 4]. The present paper states the formulation of the problem and results of flow calculations in pressure-discontinuity configurations with triple points, taking into account the equilibrium dissociation of air. The Predvoditelev approximation is used to calculate the thermodynamic function of the pressure p, as proposed in [5]. The formulation of the problem is considered for the calculation of the flow taking into account the equilibrium dissociation of air in the interference region of pressure discontinuities with two and one triple points — interactions of types I and II, according to the classification of [4]. Some results of the computer solution of the resulting system of equations are given both for a flow of cold unperturbed air (the interaction region w of the leading shock wave of an object with its projecting elements) and for a flow of hot dissociating air (the interaction region O with the boundary-layer breakaway region at the surface of the supersonic object). It is shown that, both in region w and in region O, the relative pressure is considerably affected not only by the velocity and the angle of the incident pressure discontinuity but also by the density of the incoming flow (the flight altitude of the object). Depending on this parameter, the relative pressure in the interaction region may be less or more than the pressure calculation for a perfect gas with = 1.4 to analogous flow conditions. The results obtained indicate the need to take account of the real properties of air in determining the mechanical and thermal loads in the interaction region of the pressure discontinuities at the surface of projecting elements of a hypersonic object.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 111–116, September–October, 1978.     

A theoretical Taylor–Galerkin model for first‐order hyperbolic equation

This paper presents a class of Taylor–Galerkin (TG) finite‐element models for solving the first‐order hyperbolic equation which admits discontinuities. Five parameters are introduced for purposes of controlling stability, monotonicity and accuracy. In this paper, the total variation diminishing concept and the theory of M‐matrix are applied to construct a monotonic TG model for capturing discontinuities. To avoid making the scheme overly diffusive, we apply a flux‐corrected transport (FCT) technique of Boris and Book to overcome the difficulty with anti‐diffusive flux. In smooth flow regions, our strategyof developing the temporal and spatial high‐order TG finite‐element model is based on modified equation analysis. In regions where discontinuity is encountered, we resort to two dispersively more accurate models to make the prediction accuracy as high as that obtained in smooth cases. These models are developed using the entropy‐increasing principle and the theory of group velocity. Guided by this theory, a slower group velocity should be used ahead of the shock. To avoid a train of post‐shocks, free parameters should be chosen properly to obtain a group velocity which takes on a larger value than the exact phase velocity. In this paper, we also apply the entropy‐increasing principle to determine free parameters introduced in the finite‐element model. Under the entropy‐increasing requirement, it is mandatory that coefficients of the even and odd derivative terms shown in the modified equation should change signs alternatively in order to avoid non‐physical wiggles. Several benchmark problems have been investigated to confirm the integrity of these proposed characteristic models. Copyright © 2003 John Wiley & Sons, Ltd.     

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].     

Shock waves in a mixture of materials. Hydrodynamic approximation

We formulate equations of motion and of hydrodynamics for the calculation of shock adiabats of a mixture of condensed materials under very high pressures, the assumption being made that strength properties can be neglected. Use is made of the general principles for constructing models of interacting continuous media [1–3] systematically presented in [4]. In our calculations we involve the difference of the pressures in the component materials of the mixture. In this regard we invoke the following conditions: the condition of proportional (with respect to mass) shock increase of energy; the consistency condition requiring equality of particle velocities and shock velocity in the individual particles; and, finally, also the condition of proportionality of the pressures in the individual phases. We present numerical calculations for mixtures of tungsten and paraffin and also for mixtures of aluminum and epoxy resin. Our calculations agree with experimental data and also with calculations made upon specifying the equality of the phase pressures [5–8]; they are also in agreement with calculations made in accordance with the additive rule (see [9–11]).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 113–126, July–August, 1976.     

Moment equations in the kinetic theory of gases and wave velocities

We consider the evolution system for N-moments of the Boltzmann equation and we require the compatibility with an entropy law. This implies that the distribution function depends only on a single scalar variable which is a polynomial in . It is then possible to construct the generators such that the system assumes a symmetric hyperbolic form in the main field. For an arbitrary we prove that the systems obtained maximise the entropy density. If we require that the entropy coincides with the usual one of non-degenerate gases, we obtain an exponential function for , which was already found by Dreyer. From these results the behaviour of the characteristic wave velocities for an increasing number of moments is studied and we show that in the classical theory the maximum velocity increases and tends to infinity, while in the relativistic case the wave and shock velocities are bounded by the speed of light. Received June 5, 1997     

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.     

An Exact Riemann Solver for Multicomponent Turbulent Flow

The main objective of this paper is to provide some adequate way to compute the non-conservative hyperbolic system which describes a multicomponent turbulent flow. The model is written for an isentropic gas. The exact solution of the Riemann Problem (RP) associated to the hyperbolic system is exhibited. It is composed of constant states separated by rarefaction waves, or shock waves and a contact discontinuity.

The selection of the admissible part of the shock curve is obtained using an entropy criterion. This entropy is the total energy of the system. Thanks to the latter, one may compute the exact solution of the Riemann problem, assuming genuinely non linear fields contain sufficiently weak shocks.     

Vanishing Viscosity Method for Transonic Flow

A vanishing viscosity method is formulated for two-dimensional transonic steady irrotational compressible fluid flows with adiabatic constant . This formulation allows a family of invariant regions in the phase plane for the corresponding viscous problem, which implies an upper bound uniformly away from cavitation for the viscous approximate velocity fields. Mathematical entropy pairs are constructed through the Loewner–Morawetz relation via entropy generators governed by a generalized Tricomi equation of mixed elliptic–hyperbolic type, and the corresponding entropy dissipation measures are analyzed so that the viscous approximate solutions satisfy the compensated compactness framework. Then the method of compensated compactness is applied to show that a sequence of solutions to the artificial viscous problem, staying uniformly away from stagnation with uniformly bounded velocity angles, converges to an entropy solution of the inviscid transonic flow problem. Dedicated to Constantine M. Dafermos on the Occasion of His 65th Birthday     

Viscous limits for piecewise smooth solutions to systems of conservation laws

In this paper we study the zero dissipation problem for a general system of conservation laws with positive viscosity. It is shown that if the solution of the problem with zero viscosity is piecewise smooth with a finite number of noninteracting shocks satisfying the entropy condition, then there exist solutions to the corresponding system with viscosity that converge to the solutions of the system without viscosity away from shock discontinuities at a rate of order as the viscosity coefficient goes to zero. The proof uses a matched asymptotic analysis and an energy estimate related to the stability theory for viscous shock profiles.     

Entropy and Global Existence for Hyperbolic Balance Laws

This paper presents a general result on the existence of global smooth solutions to hyperbolic systems of balance laws in several space variables. We propose an entropy dissipation condition and prove the existence of global smooth solutions under initial data close to a constant equilibrium state. In addition, we show that a system of balance laws satisfies a Kawashima condition if and only if its first-order approximation, namely the hyperbolic-parabolic system derived through the Chapman-Enskog expansion, satisfies the corresponding Kawashima condition. The result is then applied to Bouchuts discrete velocity BGK models approximating hyperbolic systems of conservation laws.     

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.     

Oblique interaction of alfvén and contact discontinuities in magnetohydrodynamics

A general method of solving problems of the interaction of stationary discontinuities is proposed. The problem of the oblique incidence of an Alfvén plane-polarized discontinuity on a contact discontinuity is examined in the general formulation. A solution is constructed numerically over the entire range of variation of the governing parameters. A number of effects associated with the magnetohydrodynamic nature of the interaction are explored. For example, the formation in space of sectors in which the density falls by several orders (almost to a vacuum) is detected. The solutions obtained are of interest, for example, for investigating the interaction between Alfvén discontinuities in the solar wind and the magnetopause, plasmopause and other inhomogeneities whose boundary can be approximated by a contact discontinuity [13–15].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 131–142, January–February, 1990.     

Dispersion relation in the limit of high frequency for a hyperbolic system with multiple eigenvalues

The results of a previous paper (Muracchini et al., 1992) are generalized by considering a hyperbolic system in one space dimension with multiple eigenvalues. The dispersion relation for linear plane waves in the high-frequency limit is analyzed and the recurrence formulas for the phase velocity and the attenuation factor are derived in terms of the coefficients of a formal series expansion in powers of the reciprocal of frequency. In the case of multiple eigenvalues, it is also verified that linear stability implies λ$\lambda$-stability for the waves of weak discontinuity. Moreover, for the linearized system, the relationship between entropy and stability is studied. When the nonzero eigenvalue is simple, the results of the paper mentioned above are recovered. In order to illustrate the procedure, an example of the linear hyperbolic system is presented in which, depending on the values of parameters, the multiplicity of nonzero eigenvalues is either one or two. This example describes the dynamics of a mixture of two interacting phonon gases.     

The evolution laws of dilatational spherical and cylindrical weak nonlinear shock waves in elastic non-conductors

The theory of singular surfaces yields a set of coupled evolution equations for the shock amplitude and the amplitudes of the higher order discontinuities which accompany the shock. To solve these equations, we use perturbation methods with a perturbation parameter characterising the initial shock amplitude. It is shown that for decaying shock waves, if the accompanying second order discontinuity is of order one, the straightforward perturbation procedure yields uniformly valid solutions, but if the accompanying second order discontinuity is of order , the method of multiple scales is needed in order to render the perturbation solutions uniformly valid with respect to the distance of travel. We also construct shock wave solutions from modulated simple wave solutions which are obtained with the aid ofHunter & Keller's Weakly Nonlinear Geometrical Optics method. The two approaches give exactly the same results within their common range of validity. The explicit evolution laws thus obtained enable us to see clearly how weak nonlinear curved shock waves are attenuated because of the effects of geometry and material nonlinearity, and on what length scale these effects are most pronounced. Communicated by C. C. Wang