One class of self-similar flows of a radiating gas

This article discusses self-similar statements of the problem of the motion of a completely radiating and absorbing gas. The field of radiation is assumed to be quasi-steady-state, and the contribution of the radiation to the internal energy, as well as the pressure and the viscosity of the medium, are not taken into account. The presence of local thermodynamic equilibrium is assumed. The absorption coefficient is approximated by a power function of the pressure and the density. Scattering of the radiation is not taken into account. Under these assumptions, there exist self-similar statements of the problem for one-dimensional unsteady-state flows (a strong detonation, the problem of plug-flow, motion under the effect of a radiation source, and others) and two-dimensional steady-state flows (flow in a diffuser, supersonic flow around a wedge or a cone). It is shown that there exists a non steady-state spherically symmetrical flow depending on four parameters; this flow is adiabatic in spite of the presence of radiation. This article is made up of seven sections. It is shown in the first section that the presence of radiation leads to the appearance of new dimensional constants, entering into the equations of the problem. The second section is devoted to self-similar nonsteady-state one-dimensional flows. The third section contains a detailed study of one class of such flows. In a partial case, adiabatic flows of a radiating gas are obtained. In the fourth and fifth sections, a detailed analysis is made of the initial and boundary conditions from the point of view of dimensionality. The sixth section describes self-similar two-dimensional steady-state flows of a radiating-absorbing gas. The seventh section consists of remarks with respect to approximations of the transfer equation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 8–22, July–August, 1970.     

Numerical investigation of unsteady subsonic viscous gas flows in a plane channel with a sudden expansion

The nonlinear processes of development of instability in an unsteady subsonic viscous gas flow in a plane channel with a sudden expansion are investigated numerically with allowance for acoustico-vortex interaction over a broad interval of the characteristic parameters. Effects associated with the acoustic self-excitation of the jet flowing into the wider part of the channel are determined. Approximate relations are obtained for the resonance conditions of self-excitation. The effect of the inlet mean-velocity profiles on the evolution of the flow is estimated. The processes of formation and subsequent interaction of the coherent structures are analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 32–41, July–August, 1988.     

Stability of periodic viscous gas flows

Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 10–16, January–February, 1992.     

Self-similar movement of a viscous gas in a channel

The results presented in [1] refer primarily to dropping liquids for which the influence exerted by the thermal conditions on the flow is related to the temperature dependence of the viscosity. The self-similar flow of a viscous gas in a channel with a linearly increasing wall temperature is examined in this paper. The influence exerted by the Reynolds and Prandtl numbers on heat exchange and the hydrodynamics of the flow is analyzed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 47–54, May–June, 1976.The author wishes to thank A. F. Seleznev for carrying out the calculations and V. N. Shtern for discussing the paper.     

Some basic viscous flows in micropolar fluids

Summary This paper analyzes some basic viscous flows of micropolar fluids. The problems ofCouette andPoiseuille flows between two parallel plates and a rotating fluid with a free surface, are solved using the theory of micropolar fluids. The results are presented graphically and compared with the classical ones, and the differences are discussed.     

Formation of one-dimensional flows of a heat-conducting viscous gas

The article discusses the development of one-dimensional flows in a viscous heat-conducting gas using the example of two flows: 1) the flow arising with the decomposition of a discontinuity of the pressure in the quiescent gas (flow in a shock tube); 2) the flow arising with the application of a constant heat flow at a gassolid interface. For such flows, there has been very little study of the initial stage of the process, right up to the time when nonheat-conducting zones are separated out, described by the Euler equations, as well as dissipation zones of the type of a shock wave or a boundary layer, which can be treated using asymptotic methods [1–3]. With the investigation of the initial stage, the complete solution of the system of Navier—Stokes equations is required. The present article discusses the initial stage of the flows on the basis of a numerical solution of problems 1 and 2. A study is made of the effect of the Prandtl number and of the viscosity coefficient on the behavior of the gas.     

Self-similar turbulent compressible gas flows in a channel

In this study we establish for turbulent compressible gas flow (to within a constant factor) the laws governing the variation of the height (radius) and the static pressure along the length of a planar or axisymmetric channel for which the longitudinal velocity component and gas temperature are functions only of the transverse dimensionless coordinate. In such channels the gas density decrease due to friction is compensated by the increase of the cross-sectional area so that the velocity and temperature profiles remain unchanged at all sections of the channel.The results obtained are a generalization to the gas case of the known laws governing the turbulent flow of an incompressible fluid in a cylindrical channel.The author wished to thank A. P. Byrkin for helpful discussions.     

Instability of self-similar ideal gas flows with shock waves

The results of the numerical simulation of three problems of ideal gas flow with shock waves, which admit self-similar solutions, are presented. These problems are the double Mach-type reflection of a shock from a wedge, the breakdown of a combined discontinuity on a 90° sharp corner, and the outflow of a supersonic jet from an expanding slot. It is shown that for certain input data the self-similar solution may become unstable and is replaced by a fluctuating flow. The reasons for the generation of these fluctuations and their mechanism are discussed. Volgograd. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 166–175, July–August, 1998.     

Some general properties of plane-parallel viscous flows

An expression is obtained for the integral of the equations of plane-parallel viscous incompressible flow expressing conservation of the Bernoulli function along a certain family of lines which, as the viscosity tends to zero, go over into the streamlines. These lines also determine the direction of transfer of the vorticity of the viscous flow.In conclusion the authors wish to thank V. V. Sychev and V. Ya. Neiland for discussion of the results and useful comments.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 176–178, May–June, 1987.     

Aspects of self-similar diffraction gas flows and their numerical simulation

A large number of papers has been devoted to the investigation of the interaction of a plane shock wave with bodies of various geometric shapes, and they have been generalized and classified for a stationary body in [1, 2]. Separate results of experimental and theoretical investigations of the interaction of a shock wave with a wedge, cone, sphere, and cylinder moving with supersonic velocities are contained in [3–9]. Analysis of the available results shows that the features of the unsteady gas flows formed in this case largely depend on the nature of the boundary-value problem that arises for the system of differential gas dynamic equations. The question of the wave structure of the unsteady gas flow and the accuracy of the obtained solution is central to the numerical investigation of the present class of problems. The most characteristic types of unsteady self-similar gas flows that arise on the interaction of a plane shock wave with bodies of a wedge or convex corner type are calculated on the basis of an explicit numerical continuous calculation method of the second order of accuracy. The accuracy of the numerical solutions is discussed on the basis of a comparison with the experimental data. The case of the interaction of a shock wave with the rarefaction wave that arises in a supersonic flow past a convex corner is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 146–152, July–August, 1986.     

Conjugation of the channel and filtration flows of a viscous incompressible fluid

Novosibirsk. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 1, pp. 95–99, January–February, 1995.     

Self-similar channel flow of a viscous gas with heat transfer

We examine self-similar flows of a viscous gas in long, smooth channels with a special heat transfer law at the wall, corresponding to the same Mach number profile at all cross sections.     

Unsteady self-similar viscous flow near a stagnation point

The problem of unsteady viscous incompressible fluid flow near a stagnation point is considered. Self-similar solutions describing plane and axisymmetric flows are constructed.     

Similarity of flows in strongly underexpanded jets of viscous gas

The fulfillment of the conditions formulated in [1] for the similarity of flows in strongly under-expanded jets of a viscous, thermodynamically ideal gas is studied. The limits of applicability of these conditions are established on the basis of exact solutions of the one-dimensional Navier —Stokes equations and experimental investigations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika zhidkosti i Gaza, No. 6, pp. 117–125, November–December, 1978.     

Two-dimensional self-similar flows generated by the interaction between a shock and low-density gas regions

A plane time-dependent flow generated by the interaction between a normal shock and a low-density gas region occupying a quarter of the plane is theoretically investigated. Numerical simulation is performed on the basis of the Euler equations. It is established that after the shock has come in contact with the low-density region two-dimensional self-similar flows of different type can develop. On regular interaction the original shock is refracted on the low-density region with the matching of the accelerated and original shock and the refracted contact discontinuity at a common point. On irregular interaction a complicated flow occurs; it includes curved and oblique shocks, a contact discontinuity with points of inflection, multiple matching points, a high-pressure jet, and a layered vortex. The jet and vortex structures are investigated in detail. The tendency of the gasdynamic structure development with variation in the control parameters of the problem is determined. A simplified, near-analytical technique for estimating the slopes of the main shocks and the gas parameters behind them is proposed.     

On the stability of certain nonparallel flows of a viscous incompressible liquid in a channel

The stability of very simple nonparallel flows of a viscous incompressible liquid in an infinite plane channel described by the exact solutions of the Navier-Stokes equations is studied. Such solutions are realized between two parallel porous plates when the liquid (or gas) is forced in at one wall and drawn out at the same velocity at the other, with a steady flow of liquid along the channel. In this case the transverse velocity component is constant, and the profile of the longitudinal velocity component is independent of the longitudinal con-ordinate x, being an asymmetric function of the transverse coordinate y. A study of the hydrodynamic stability then reduces to the solution of an equation differing from the Orr-Sommerfeld equation by virtue of the presence of additional terms containing the transverse velocity component of the main flow. By numerically solving both this equation and the ordinary Orr-Sommerfeld equation and comparing the corresponding results for various inflowing Reynolds numbers R₀=v₀h/ (v₀ is the inflow velocity, h is the width of the channel), the effect of the nonparallel and asymmetrical nature of such flows on their stability is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 125–129, July–August, 1970.     

Calculation of the deceleration of a viscous supersonic gas flow in a channel

The deceleration of nonuniform viscous supersonic gas flows in planar and axisymmetric channels is investigated. A modification of Prandtl's formula for the turbulent viscosity is proposed in order to take into account the dependence of the mixing length on the value of the axial Mach number. The results of the calculations are compared with known experimental data on the deceleration of a supersonic flow in a subsonic pseudoshock.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 162–166, March–April, 1982.We thank A. N. Sekundov for discussing the work.     

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.     

Steady disturbed viscous incompressible gas flows in a long cylinder with one closed end

There have been many studies of steady disturbed flows in rotating channels. In [1] the steady disturbed flow between two coaxial rotating cylinders was investigated. In [2, 3] the disturbances of Hagen-Poiseuille flow due to rotation of the pipe were considered. In this article other effects: the propagation of disturbances in a long rotating pipe and their interaction with the end face are examined.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 104–112, September–October, 1993.     

Vortex flows of a viscous heat-conducting gas in a slightly divergent tube with volume energy supply

The results of a numerical investigation of viscous vortex flow in a slightly divergent tube with thermal energy supplied to the flow are presented. The initial stage of vortex flow development is considered for two different longitudinal velocity distributions simulating the velocity profiles in jet-like and wake-like vortex flows in the vicinity of the vortex axis. The first type of flow can be considered as a model for the near-axis region of the vortex formed in the flow around a delta wing at incidence. The second type can serve as a model for the near-axis region of the trailing vortex downstream of a high-aspect-ratio wing. The development of the two flows is studied for a constant area tube, a slightly divergent tube, and in the case of thermal energy supply from a volume energy source at a constant wall temperature.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 90–97, September–October, 1996.