Gas-dynamic shock waves in a relaxing gas

St. Petersburg State University, St. Petersburg 199004. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 36, No. 3, pp. 92–97, May–June, 1995.     

Sonic gas flow around a nonsymmetric profile

A particular solution of the transonic equations describing two-dimensional flow of an ideal gas is obtained for nonsymmetric flow around a certain profile. The aerodynamic characteristics of the profile are determined.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 76–81, September–October, 1984.The authors are grateful to S. V. Fal'kovich for the useful discussion of the results.     

Second order viscosity approximation for a relaxing gas

Summary If in a moving gas a thermodynamic relaxation process takes place the thermodynamic equilibrium pressure differs from the negative normal stress. To a first approximation the difference is determined by the bulk viscosity of the gas. An already known expression for the bulk viscosity is derived under rather general assumptions about the gas. Furthermore, the difference between thermodynamic equilibrium pressure and normal stress is calculated to a second order approximation. Application of the result to the calculation of shock structure in a relaxing gas shows oscillatory behavior of the upstream flow variables for higher shock strengths. This corresponds to the same behavior of solutions obtained by applying the Burnett-approximation in the kinetic theory of gases to the shock structure problem.

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Übersicht Wenn in einem strömenden Gas thermodynamische Relaxationsprozesse ablaufen, weicht der thermodynamische Gleichgewichtsdruck von der negativen Normalspannung im Gas ab. In erster Näherung wird diese Abweichung durch die Druckzähigkeit des Gases bestimmt. Im folgenden wird ein schon bekannter Ausdruck für die Druckzähigkeit unter allgemeinen Voraussetzungen für das Gas hergeleitet. Außerdem wird die Abweichung zwischen Druck und Normalspannung in nächsthöherer Näherung berechnet. Die Anwendung des Ergebnisses auf die Berechnung der Stoßstruktur in einem relaxierenden Gas zeigt, daß ebenso wie bei Anwendung der Burnett-Näherung der kinetischen Gastheorie auf dieses Problem Oszillationen in den Anströmgrößen für stärkere Stöße auftreten.

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Herrn Professor Dr. H. Görtier aus Anlaß der Vollendung des 60. Lebensjahres gewidmet.

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Diese Arbeit wurde während eines Gastaufenthaltes des Verfassers am Department of Engineering and Applied Science der Yale-University, New-Haven (Conn.) geschrieben.     

Similarity method for imploding strong shocks in a non-ideal relaxing gas

Self-similar solutions arise naturally as special solutions of system of partial differential equations (PDEs) from dimensional analysis and, more generally, from the invariance of system of PDEs under scaling of variables. Usually, such solutions do not globally satisfy imposed boundary conditions. However, through delicate analysis, one can often show that a self-similar solution holds asymptotically in certain identified domains. In the present paper, it is shown that self-similar phenomena can be studied through use of many ideas arising in the study of dynamical systems. In particular, there is a discussion of the role of symmetries in the context of self-similar dynamics. We use the method of Lie group invariance to determine the class of self-similar solutions to a problem involving plane and radially symmetric flows of a relaxing non-ideal gas involving strong shocks. The ambient gas ahead of the shock is considered to be homogeneous. The method yields a general form of the relaxation rate for which the self-similar solutions are admitted. The arbitrary constants, occurring in the expressions for the generators of the local Lie group of transformations, give rise to different cases of possible solutions with a power law, exponential or logarithmic shock paths. In contrast to situations without relaxation, the inclusion of relaxation effects imply constraint conditions. A particular case of the collapse of an imploding shock is worked out in detail for radially symmetric flows. Numerical calculations have been performed to determine the values of the self-similarity exponent and the profile of the flow variables behind the shock. All computations are performed using the computation package Mathematica.     

Weakly disturbed steady transonic flows of a vibrationally relaxing gas

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 2, pp. 48–58, March–April, 1992.     

Evolution of finite perturbations in a viscoelastic relaxing liquid with gas bubbles

The propagation of one-dimensional perturbations in a viscoelastic relaxing liquid containing gas bubbles is investigated within the framework of the homogeneous model of the medium when the wavelength of the perturbation is much larger than the distance between the bubbles and the bubble radius. The evolution of stationary and nonstationary waves is investigated analytically and with the use of numerical integration; shock waves are also investigated. The results are compared with the behavior of perturbation waves in a Newtonian liquid with gaseous inclusions. The models of the gas-liquid medium [1, 2] are generalized to the case when the liquid phase is a viscoelastic liquid, for example, a weak aqueous solution of polymers. The propagation of longwave perturbations of finite amplitude in such a mixture is investigated using the technique developed in [3].     

One dimensional steepening of waves in non-ideal relaxing gas

In this paper, we studied the behavior of different modes of wave propagation and breaking of wave front by employing the theory of singular surfaces in a plane and radially symmetric flow of a non-ideal relaxing gas. The one dimensional steepening of waves is considered and the transport equation for the jump discontinuity of velocity gradient is obtained. The effects of relaxation and van der Waals excluded volume of the medium on the jump discontinuity of velocity gradient are analyzed.     

Propagation of small disturbances in a relaxing and radiating diatomic gas in vibrational nonequilibrium

In radiation gasdynamical problems, where the primary object of investigation is a moving gas, the influence of radiation on the parameters of the gas flow is usually neglected to avoid overcomplication of the problem. The growth and behavior of initial disturbances in a scattering, radiating, absorbing, viscous, heat-conducting gas characterized by local thermodynamic equilibrium has been investigated previously [1]. However, for low pressures (p10^–4 to 10^–3 technical atm) and fairly high temperatures of the active molecular degrees of freedom (T10³ to 3·10³K) the distribution of the molecules among the vibrational levels can differ markedly from the equilibrium distribution due to the or der-of-magnitude closeness of the vibrational relaxation time _c associated with collisions and the radiative deactivation time * of excited molecules [2, 3]. We now analyze normal modes in a vibrationally nonequilibrium medium with allowance for radiation scattering in the vibrational-rotational band. We formulate a dispersion relation and discuss some limiting cases.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 168–171, September–October, 1976.The author is grateful to V. I. Kruglov, Yu. V. Khodyko, and M. A. El'yashevich for their interest and discussions.     

Free nonlinear oscillations of a thermally relaxing spherical gas bubble in an incompressible liquid

The quasi-adiabatic regime of free oscillation of a bubble in the presence of irreversible interphase heat transfer between the bubble and the ambient liquid is studied. On the basis of simplified model equations of a rarefield bubble mixture, a nonlinear-oscillation equation of the relaxation type is obtained. In constructing an exact particular solution of this equation, the heat transfer law associated with bubble compression is established. For studying the harmonic oscillations, the Krylov-Bogolyubov-Mitropol’skii asymptotic method is used. It is shown that, for a small bubble, the viscosity and heat transfer effects are of the same order. For a small bubble, the influence of these effects on the formation of the natural-oscillation frequency, which is small in the linear approximation, may be significant in the nonlinear formulation. For a large bubble, the influence of these effects is negligible in both approximations. For the approximate solution of the nonlinear equation, a uniformly valid second-order expansion is constructed.     

Periodic dislocation motion in a relaxing medium

We consider periodic helical dislocation motion in a para-elastic medium under variable external stresses. The para-elastic properties of the medium are determined by the short-range-order parameters between atoms of the different components of the alloy. The solution of the nonlinear dislocation equation of motion is obtained in four different regions of the amplitude-frequency space. The conditions are indicated under which dislocation motion is viscous and is in the nature of breakaway from the polarization atmosphere.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 79–85, November–December, 1970.     

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.     

Nonlinear dispersive waves in a relaxing medium

The dispersion of nonlinear waves in a relaxing medium is analysed by making use of the evolution equations for longitudinal waves. The dispersion relations are obtained and the behaviour of the waves compared to those that arise when they are governed by the well-known Korteweg-de Vries (KdV) and Benjamin-Bona-Mahony (BBM) equations that describe unidirectional motion and also by the time regularized long wave (TRLW) equation that describes bi-directional motion. The nonlinear steady wave solutions are obtained. The general mathematical model used throughout this paper is obtained by the theory of nonlinear elasticity with weak relaxation effects (standard viscoelasticity). A further generalization using a four-element model is also discussed briefly.     

Mixing of plane laminar relaxing gas jets with radiation taken into account

The influence of a chain pumping reaction mechanism in a diffusion-type continuous-action chemical amplifier on an RF molecule is investigated. The actual flow picture is simulated by boundary-layer equations taking account of coherent radiation. It is shown that the efficiency of a chemical amplifier in a mode with partial fluorine dissociation is sufficiently high under the condition of the presence of a line with 0 v5.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 137–142, May–June, 1978.The authors are grateful to B. S. Aleksandrov, Yu. V. Lapin, and M. Kh. Strelets for numerous useful discussion.