Radiating hydrogen flow in axisymmetric nozzles

A considerable number of studies published in recent years have been devoted to the study of gas in channels and pipes. In view of the complexity of the question and the lack of analytic techniques, individual aspects of the problem are generally considered. The determination of the radiant field characteristics in regions of simple geometric form filled with a stationary radiating-absorbing medium has been carried out in several studies. The articles [1–3] are devoted to the calculation of the radiant field and the temperature field for a given flow of a perfect inviscid nonheat-conducting radiating gas with constant absorption coefficient. The flow is assumed to be irrotational [1, 2] or nearly potential [3]. The authors investigated the accuracy of the solution obtained with the aid of various approximate methods and found that the diffusion approximation yields a small error in calculating the radiation density field and the values of the radiant thermal fluxes for a quite broad class of wall reflecting properties. We may note also [4, 5], in which a calculation is made of one-dimensional steady flow of a viscous heat-conducting radiating perfect gas with constant transport coefficients.In [1–5] the absorption coefficient is considered constant. This assumption simplifies the solution process considerably, since as the independent variables we can take the corresponding optical thicknesses. The study [3] contains a remark that the calculation method proposed there may be used with a variable absorption coefficient. However, this possibility was not used in the calculations presented.For a constant absorption coefficient these studies yield a rather complete analysis of the methods for solving two-dimensional problems in geometrically simple regions in the absence of mechanical motion and one-dimensional problems with motion. They contain results obtained for the exact integral or integrodlfferential equations and present an analysis of the approximate methods. The study [3] considers broader possibilities of solving two-dimensional problems (using the Monte-Carlo method), but the flow is assumed known ahead of time.In the following we present a method for calculating the two-dimensional equilibrium flow of an inviscid non-heat-conducting radiating gas with variable absorption coefficient. As an example, we consider the flow of radiating-absorbing hydrogen in axisymmetric nozzles. It is assumed that the radiation is gray and is in local thermodynamic equilibrium. The transport equation is considered in the diffusion approximation. The nozzles examined have a semi-infinite cylindrical inlet section. The initial gas flow in the cylindrical section is supersonic. In the solution process we determine the radiation density field and all the flow parameters within the nozzle.The author wishes to thank Yu. D. Shmyglevskii for his continued interest in this study.     

A self- similar solution of a shock propagation in a mixture of a non-ideal gas and small solid particles

Similarity solutions are obtained for unsteady, one-dimensional self-similar flow behind a strong shock wave, driven by a moving piston, in a dusty gas. The dusty gas is assumed to consist of a mixture of small solid particles and a non-ideal gas, in which solid particles are continuously distributed. It is assumed that the equilibrium flow-condition is maintained and variable energy input is continuously supplied by the piston. Solutions are obtained under both the isothermal and adiabatic conditions of the flow-field. The spherical case is worked out in detail to investigate to what extent the flow-field behind the shock is influenced by the non-idealness of the gas in the mixture as well as by the mass concentration of the solid particles, by the ratio of density of the solid particles to the initial density of the mixture and by the energy input due to moving piston. A comparison is also made between isothermal and adiabatic cases.     

Calculation of laminar flow of compressible gas in the presence of heat transfer in two-dimensional curvilinear channels

Numerical solution of the complete system of Navier-Stokes equations is used to investigate laminar (Re ? 1000) subsonic flows of a compressible gas in the presence of heat transfer (cooled walls) in two-dimensional channels containing a bend section (for different curvature parameters). The appearance of closed separation regions of the flow on the channel walls, their deformation as the parameters of the problem are changed, and the loss of pressure are studied. The sections of the channel walls with maximal and minimal heat fluxes are determined, and the connection between these sections and the separation regions is elucidated.     

The influence of magnetic field upon the collapse of a cylindrical shock wave

In the present paper, the problem of propagation of collapsing cylindrical shock wave in an ideal gas permeated by a transverse magnetic field with infinite electrical conductivity is investigated. Here it is assumed that the medium ahead of the shock front is uniform and at rest. Also, its counter pressure concerning the motion of the wave front is neglected. This problem admits a self similar solution of second kind. The similarity exponent has been computed by solving a nonlinear eigenvalue problem and integrating numerically the self-similar equations for various values of adiabatic heat exponent and Cowling number. Numerical computations have been performed to determine the flow field behind the shock wave. The influence of magnetic field strength and adiabatic heat exponent on the flow parameters for various cases is presented.     

Unsteady interaction of uniformly vortex flows

The problem of the decay of an arbitrary discontinuity (the Riemann problem) for the system of equations describing vortex plane-parallel flows of an ideal incompressible liquid with a free boundary is studied in a long-wave approximation. A class of particular solutions that correspond to flows with piecewise-constant vorticity is considered. Under certain restrictions on the initial data of the problem, it is proved that this class contains self-similar solutions that describe the propagation of strong and weak discontinuities and the simple waves resulting from the nonlinear interaction of the specified vortex flows. An algorithm for determining the type of resulting wave configurations from initial data is proposed. It extends the known approaches of the theory of one-dimensional gas flows to the case of substantially two-dimensional flows. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 5, pp. 55–66, September–October, 1998.     

Flow over blunt cones during entry into an atmosphere of carbon dioxide and nitrogen

An investigation has been made of hypersonic flow over spherically blunted cones in an atmosphere consisting of carbon dioxide and nitrogen. Local thermodynamic equilibrium is assumed in the shock layer. Account is taken of viscosity, diffusion, heat conduction, and radiative energy transport. The problem is solved using equations for dynamics of a viscous radiating gas without isolating inviscid flow and boundary-layer regions in the shock layer. The selectivity of the radiation is allowed for by using a two-stage approximation for the spectral dependence of the absorption coefficient, obtained by processing detailed data on absorption cross sections. The solution is found by a flow establishment method. Results are presented for flow over blunt cones with different semiangles.     

Hydrodynamic stability of evaporation fronts in porous media

The stability of phase transition fronts in water flows through porous media is considered. In the short-wave approximation a linear stability analysis is carried out and a sufficient condition of hydrodynamic instability of the phase discontinuity is proposed. The problem of injection of a water-vapor mixture into a two-dimensional mixture-saturated formation is solved and its numerical solution is compared with an exact solution of the corresponding one-dimensional self-similar problem. It is discovered that, instead of the unstable discontinuities in the one-dimensional formulation, in the two-dimensional case a lengthy mixing zone with a characteristic scale that increases self-similarly with time is formed.     

Unsteady Self-Similar Flow of a Viscous Incompressible Fluid in a Plane Divergent Channel

The problem of two-dimensional unsteady flow of a viscous incompressible fluid in a sector-like domain is considered. Initially a strictly radial flow is imposed, which makes it possible to seek solutions within the class of self-similar flows. A numerical method based on mixed finite-difference and spectral spatial discretization is developed, making it possible to find the self-similar solution efficiently. The process of development and establishment of the steady Hamel-Jeffery and Moffatt flows is modeled mathematically.     

Effect of a Standing Acoustic Wave on the Development of Large-Scale Convection in a Horizontal Layer

The effect of a standing acoustic wave on the development of long-wave convective perturbations in a horizontal layer with thermally insulated boundaries is investigated. The main two-dimensional flow is determined. A nonlinear amplitude equation with spatially-periodic coefficients is derived for investigating the stability of the main flow and secondary convection flows in the neighborhood of the stability threshold. The intensity of the acoustic field is assumed to be low. It is shown that the acoustic action leads to destabilization of the layer. Plane and three-dimensional perturbations are critical at large and small Prandtl numbers, respectively. Nonlinear one-dimensional steady-state solutions of the amplitude equation are obtained and their stability is investigated.     

Distribution of radiant thermal flux over the surface of a sphere for hypersonic flow of a nonviscous radiating gas past the sphere

In the present study using the Newtonian approximation [1] we obtain an analytical solution to the problem of flow of a steady, uniform, hypersonic, nonviscous, radiating gas past a sphere. The three-dimensional radiative-loss approximation is used. A distribution is found for the gasdynamic parameters in the shock layer, the withdrawal of the shock wave and the radiant thermal flux to the surface of the sphere. The Newtonian approximation was used earlier in [2, 3] to analyze a gas flow with radiation near the critical line. In [2] the radiation field was considered in the differential approximation, with the optical absorption coefficient being assumed constant. In [3] the integrodifferential energy equation with account of radiation was solved numerically for a gray gas. In [4–7] the problem of the flow of a nonviscous, nonheat-conducting gas behind a shock wave with account of radiation was solved numerically. To calculate the radiation field in [4, 7] the three-dimensional radiative-loss approximation was used; in [5, 6] the self-absorption of the gas was taken into account. A comparison of the equations obtained in the present study for radiant flow from radiating air to a sphere with the numerical calculations [4–7] shows them to have satisfactory accuracy.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 44–49, November–December, 1972.In conclusion the author thanks G. A. Tirskii and É. A. Gershbein for discussion and valuable remarks.     

Hypersonic flow past a flat body in a regime of intense radiative heat transfer

The paper is devoted to the investigation of hypersonic flow regimes in which radiative transfer plays a significant part. A numerical solution is obtained to the two-dimensional steady problem of hypersonic flow past a flat thermally insulated body of an inviscid radiating gas with allowance for radiative transfer of energy in the approximation of radiative thermal conductivity. It is noted that a heated region is formed around the body, its dimensions exceeding by an order of magnitude those of the body itself; the temperature is effectively equalized, and the gas velocity is close to the velocity of the oncoming flow. Heated gas flows past the body at a moderate Mach number (M ～ 3–6). A thin region of strongly compressed gas is formed directly in front of the body.     

Effect of the form of the electrodes on the current distribution in a magnetogasdynamic channel

A solution is given to the plane problem of the flow of a conducting gas across a homogeneous magnetic field in a magnetogasdynamic channel taking account of the Hall effect at small magnetic Reynolds numbers. The channel is formed by two long electrodes, and the cross section of the channel varies slightly and periodically along the gas flow. It is assumed that the electromagnetic forces are small. It is shown that the current distribution in the channel is nonuniform to a consider able degree and that inverse currents can form at the electrodes, with both subsonic and supersonic flows of the conducting gas. Transverse motion of the gas, due to a change in the cross section of the channel, leads to an increase of Joule energy losses. In [1] the current distribution was obtained in a flat channel formed by infinite dielectric walls, with the flow of a steady-state stream of plasma through the channel across a homogeneous magnetic field. With interaction between the flow and the magnetic field, closed current loops develop in the channel.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 26–33, November–December, 1970.     

Some self-similar viscous gas flows in a channel

We consider the class of self-similar axisymmetric and two-dimensional laminar flows of a viscous gas in a long channel with smooth contour, in which the longitudinal component of the velocity and the gas temperature are functions of a single dimensionless transverse coordinate. Such flows correspond to exponential (axisymmetric flow) or linear (two-dimensional flow) increase of the radius or height of the channel and corresponding exponential or hyperbolic decrease of the static pressure along the channel.     

Limiting regimes of self-similar gas flows with account for finite chemical-reaction rate

One-dimensional unsteady flows of a combustible gas mixture with account for the finite chemical-reaction rate were studied in [1]. The conditions for self-similarity of such flows were indicated, mathematical formulation of the problem was given, and several numerical calculations were carried out.The authors pointed out the necessity for conducting additional studies, since they were not able to obtain numerically, by means of passages to the limit, self-sustaining detonation waves propagating with the Chapman-Jouguet (CJ) velocity.In this article we point out the reason why it was not possible to reach the CJ regime in [1], and a qualitative analysis is made, by means of the results of [2], of the system of equations describing the self-similar flows of a gas with finite chemical-reaction rate, and the passage to the limit is made to the self-sustaining CJ detonation waves in the presence of chemical reactions. It is also shown that the problem of unsteady flows of a combustible mixture of gases with finite chemical-reaction rate is analogous to the problem of the flow of a gas heated by radiation, examined in [3].In conclusion the authors wish to thank I. V. Nemchinov and A. G. Kulikovskii for discussions of this study.     

Self-similar solutions for energy liberation in a gas stream

By using dimensional analysis some possible kinds of nonstationary and stationary gas flows with energy liberation which result in self-similar problems are investigated. The cases of energy liberation in a gas at rest and in uniform supersonic and hypersonic streams are examined. The gas is assumed inviscid and perfect. Results of a computation of some hypersonic self-similar gas motions are presented. Three classes of self-similar gas motions have been well studied at this time: the strong explosion, the power-law flow caused by the expansion of a plane, cylindrical, or spherical piston [1], and conical flow (including combustion and detonation waves [2–4]). Some new self-similar motions caused by energy liberation on certain lines, surfaces, or in volumes will be examined below.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 106–113, November–December, 1974.     

Self-similar solutions for unsteady flow behind an exponential shock in an axisymmetric rotating dusty gas

Similarity solutions are obtained for one-dimensional unsteady isothermal and adiabatic flows behind a strong exponential cylindrical shock wave propagating in a rotational axisymmetric dusty gas, which has variable azimuthal and axial fluid velocities. The shock wave is driven by a piston moving with time according to an exponential law. Similarity solutions exist only when the surrounding medium is of constant density. The azimuthal and axial components of the fluid velocity in the ambient medium are assumed to obey exponential laws. The dusty gas is assumed to be a mixture of small solid particles and a perfect gas. To obtain some essential features of the shock propagation, small solid particles are considered as a pseudo-fluid; it is assumed that the equilibrium flow conditions are maintained in the flow field, and that the viscous stresses and heat conduction in the mixture are negligible. Solutions are obtained for the cases when the flow between the shock and the piston is either isothermal or adiabatic, by taking into account the components of the vorticity vector. It is found that the assumption of zero temperature gradient results in a profound change in the density distribution as compared to that for the adiabatic case. The effects of the variation of the mass concentration of solid particles in the mixture \(K_p\) , and the ratio of the density of solid particles to the initial density of the gas \(G_a\) are investigated. A comparison between the solutions for the isothermal and adiabatic cases is also made.     

On the existence of self-similar solutions of the equations governing unsteady flow through a porous medium

In this paper the conditions for the existence of self-similar solutions of the equations governing unsteady flows through a porous medium are presented and discussed. The first two sections deal with the case of non-Newtonian fluids of power-law behavior; the third section analyzes the case of non-Darcy gas flows. The boundary and initial conditions occuring currently in a large class of fluid mechanics problems, of practical interest in engineering, are considered.     

Development of pulsation regimes in one-dimensional unsteady MHD flows with switching off of the electrical conductivity

This paper investigates the gas flow in an electromagnetic field when the conductivity, being a function of the thermodynamic gas parameters, vanishes during the flow (switching off of the conductivity). In the case of steady supersonic flows in an expanding nozzle it was first shown analytically [1] and then confirmed by numerical experiment [2] that stable steady flow is not possible for all the problem parameters (for example, the values of the magnetic field at the exit). Instead of a steady flow a periodic regime is realized when narrow regions of conducting gas with currents flowing through them detach from the conducting region and propagate down the channel. In these papers the conductivity was assumed to be a function of only the temperature, such that for T T* (T) = 0. In [3, 4] the flows of conducting gas in the channels were calculated both with the given dependence of the gas conductivity on the temperature and on the basis of a three-component model by means of the Saha equation. At the same time, the development of periodic regimes in the flow in the nozzle was observed in both cases, but the mechanism of the origin of the current layers was not explained. The self-similar problem of the withdrawal of a nonconducting piston from a half-space occupied by a conducting gas with a magnetic field was investigated in [5] in a linear formulation. At the same time, regions of the problem parameters (the velocity of the piston and the magnetic field on it) were found when, in spite of the self-similar formulation of the problem, there is no self-similar solution. At the same time, regions exist where several solutions are possible. The possibility of the formation of isothermal rarefaction zones with low electrical conductivity when the Joule heating is balanced by the cooling of the gas on expansion (Butler waves) [6] was not taken into account in this paper, since they are unstable with respect to superheating. However, in the case of flow in a nozzle it was shown [2] that precisely the development of instabilities in these zones leads to the formation of the periodic regime. In the present paper the solution of the self-similar problem is constructed in a nonlinear formulation. The reason for the occurrence of regions in which the solution is multiply valued, which is associated with the process of arrival at self-similar boundary conditions, is explained. It is shown that a quasiperiodic regime can arise in the solution, occurring, in particular, in the regions of the problem parameters where there is no self-similar solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 115–122, July–August, 1986.     

Planar jet flow of a radiating gas

Propagation of a laminar jet of a thermodynamically equilibrious gray gas is examined in the boundary layer theory approximation. The one-dimensional radiative energy transport is accounted for in the P ₁ approximation of the spherical harmonic method. Numerical solution of the problem is made under additional simplifying assumptions for various values of the radiation parameters to illustrate the radiation effect. The method and the computational scheme used are applicable to the study of complex jet flows of a radiating gas.The author thanks Yu. P. Lun'kin for his assistance in posing the problem and for his continued interest in the study.