On the shape of bodies ablating in a supersonic gas stream

When objects move at a high velocity in a dense medium, their front surfaces undergo intense heating. There can then be a strong interaction between the oncoming flow and the surface of the body in which the body is not merely subject to the thermal and force effect of the stream but also significantly changes the flow field itself due to the intense blowing of ablation products from its surface and the change in the geometry of the front surface. Experimental and numerical investigations into the various regimes of strong interaction have established how stable shapes are adopted by bodies ablating in a supersonic gas stream.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 18–21, January–February, 1981.     

One-dimensional problem of a shock in a gas

The self-similar problem of the motion of a cold gas subjected to an instantaneous impulse is considered. A solution is constructed in the neighborhood of the known exact solution for a gas with a specific heat ratio =1.4 [1–4]. An analytic expression is obtained in this neighborhood for the dependence of the index of self-similarity n on the parameter h, which is related to the adiabatic index by h=(+1)/(–1). The results of a numerical calculation of n versus h are compared with the analytic expression.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 74–78, November–December, 1973.     

Two-dimensional transonic vortex flows of an ideal gas

Equations are obtained for two-dimensional transonic adiabatic (nonisoenergetic and nonisoentropic) vortex flows of an ideal gas, using the natural coordinates (=const is the family of streamlines, and =const is the family of lines orthogonal to them). It is not required that the transonic gas flow be close to a uniform sonic flow (the derivation is given without estimates). Solutions are found for equations describing vortex flows inside a Laval nozzle and near the sonic boundary of a free stream.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 105–109, September–October, 1973.     

Slow flow of gas past a strongly heated sphere in the presence of blowing and evaporation from its surface

A solution is obtained to the problem of the simultaneous influence of blowing (evaporation) and large temperature differences on the flow past a sphere and on the force acting on it with allowance for the Burnett thermal stresses in the momentum equation. It is assumed that the Reynolds numbers calculated using the blowing velocity and the velocity of the oncoming flow, respectively, have the order Re_w 1 and Re 1. The temperature difference is determined by the boundary conditions, namely, a constant temperature T_w T on the surface of the sphere (VT/T 1). The problem is solved by the method of matched asymptotic expansions with respect to the small parameter Re. The equations reduce to a system of ordinary differential equations, which are solved numerically by the orthogonal sweep method [1]. It is found that under certain conditions the drag of the sphere can become negative.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 128–134, July–August, 1982.I thank O. G. Fridlender for valuable advice and interest in the work.     

Stability of a perturbed conducting gas flow in a magnetic field for arbitrary magnetic Reynolds numbers

A numerical calculation is made which describes the conversion into a T-layer of a finite perturbation in electrical conductivity imposed on a one-dimensional supersonic flow of a compressible medium for a finite value of the magnetic Reynolds number. The development of the injected perturbation is significantly affected by the magnetic Reynolds number of the unperturbed flow, and to each value of this number there corresponds a particular boundary region in which the perturbation is taken up by the magnetic field into an induced T-layer. The stability is investigated in the linear approximation for a minimal perturbation, and the dispersion equation is solved with allowance for gradients in the unperturbed parameters. It is shown that an overheating instability can arise in the system and lead to the formation of a T-layer.Translated from Zhurnal Prikladnoi Mekhaniki i Teknicheskoi Fiziki, No. 3, pp. 3–9, May–June, 1973.The authors thank L. M. Degtyarev, L. A. Zaklyaz'minskii, and A. P. Favorskii for useful discussions and advice during the completion of this work.     

The problem of a gas bubble in plane ideal fluid flow

We study the problem of two-dimensional fluid flow past a gas bubble adjacent to an infinite rectilinear solid wall.Two-dimensional ideal fluid flow past a gas bubble on whose boundary surface-tension forces act (or a gas bubble bounded by an elastic film) has been studied by several authors. Zhukovskii, who first studied jet flows with consideration of the capillary forces, constructed an exact solution of the problem of symmetric flow past a gas bubble in a rectilinear channel [1]. However, Zhukovskii's solution is not the general solution of the problem; in particular, we cannot obtain the flow past an isolated bubble from his solution. Slezkin [2] reduced the problem of symmetric flow of an infinite fluid stream past a bubble to the study of a nonlinear integral equation. The numerical solution of this problem has recently been found by Petrova [3]. McLeod [4] obtained an exact solution under the assumption that the gas pressure p₁ in the bubble equals the flow stagnation pressure p₀. Beyer [5] proved the existence of a solution to the problem of flow of a stream having a given velocity circulation provided p₁p₀.We examine the problem of two-dimensional ideal fluid flow past a gas bubble adjacent to an infinite rectilinear solid wall. The solution depends on the value of the contact angle . The existence of a solution is proved in some range of variation of the parameters, and a technique for finding this solution is given. The situation in which =1/2 is studied in detail.     

Flow of a sonic free gas stream around a profile

The stream function far from a profile in a two-dimensional sonic free gas jet is constructed. The stream function satisfies the Tricomi equation and is constructed on the , plane by the method of singular integral equations. With unlimited increase of the jet width and by satisfying a certain condition, the stream function transforms to the self-similar solution of Frank [1, 2] and Guderley, which describes an unbounded sonic stream far from the profile. In conclusion, flow of a sonic jet issuing from a duct about a profile is considered.The author wishes to thank S. V. Fal'kovich for valuable suggestions in discussing this article.     

Interaction of an unsteady two‐phase jet with a layer of a disperse medium

Discharge of a twophase jet from a cylindrical channel into a bounded layer of a disperse medium is numerically simulated using the equations of the mechanics of heterogeneous media with allowance for the differences in velocity, temperature, and phase stresses. The effect of separation of the gas phase from the disperse phase in the layer is revealed and verified experimentally. A comparison with a similar process of gas discharge at equal initial pressures shows that in the interaction with the disperse layer, the twophase flow has a longer momentum and direction.     

Transformation and some solutions of the equations of motion of an ideal gas

The equations of one-dimensional and plane steady adiabatic motion of an ideal gas are transformed to a new form in which the role of the independent variables are played by the stream function and the function introduced by Martin [1, 2], It is shown that the function retains a constant value on a strong shock wave (and on a strong shock for plane flows). For one-dimensional isentropic motions the resulting transformation permits new exact solutions to be obtained from the exact solutions of the equations of motion. It is shown also that the one-dimensional motions of an ideal gas with the equation of state p=f(t) and the one-dimensional adiabatic motions of a gas for which p=f() are equivalent (t is time, is the stream function). It is shown that if k=s=–1, m and n are arbitrary (m+n0) and =1, the general solution of the system of equations which is fundamental in the theory of one-dimensional adiabatic self-similar motions [3] is found in parametric form with the aid of quadratures. Plane adiabatic motions of an ideal gas having the property that the pressure depends only on a single geometric coordinate are studied.     

Heat transfer and equilibrium temperature of a sphere in a supersonic rarefied gas flow

Some results are presented of experimental studies of the equilibrium temperature and heat transfer of a sphere in a supersonic rarefied air flow.The notations D sphere diameter - u, , T,,l, freestream parameters (u is velocity, density, T the thermodynamic temperature,l the molecular mean free path, the viscosity coefficient, the thermal conductivity) - T₀ temperature of the adiabatically stagnated stream - T_e mean equilibrium temperature of the sphere - T_w surface temperature of the cold sphere (T_we) - mean heat transfer coefficient - _e air thermal conductivity at the temperature T_e - P Prandtl number - M Mach number     

Motion of large gas bubbles ascending in a liquid

An equation is derived for the ascent velocity of large gas bubbles in a liquid. This velocity is assumed to be governed by the propagation of a wavelike perturbation caused by the bubble in the liquid.Notation w bubble (or drop) velocity - specific gravity - dynamic viscosity - kinematic viscosity - r bubble (or drop) radius - surface tension - coefficient of friction - g gravitational acceleration - D bubble (or drop) diameter - p pressure - c propagation velocity of the wavelike perturbation - wavelength     

Calculation of viscous compressible gas flow past a corner

We consider the problem of calculating the parameters for supersonic viscous compressible gas flow past a corner (angle greater than ). The complete system of Navier-Stokes equations for the viscous compressible gas is solved in the small vicinity Q₁. (characteristic dimensionl~1/R) of the corner point. The conditions for smooth matching of the solution of the Navier-Stokes equations and the solution of the ideal gas or boundary layer equations are specified on the boundary of Q₁. All these solutions are a priori unknown, and the conditions for smooth matching reduce to certain differential equations on the boundary of Q₁. Here account is taken of the interaction of the flows near the wall surface and in the so-called outer region [1].We note that no a priori assumptions are made in Q₁ concerning the qualitative behavior of the solution, in contrast with other studies on viscous flow past a corner (for example, [2–4]).The Navier-Stokes system in Q₁ is solved numerically, using the difference scheme suggested in [5]. This scheme permits obtaining the steady-state solution by the asymptotic method for large Reynolds numbers R, and also has an approximation accuracy adequate to account for the effects of low viscosity and thermal conductivity.     

Penetration depth of a jet injected into an oncoming supersonic flow

The geometrical characteristics of jets injected through an opening in a flat plate into an oncoming supersonic flow have been studied on a number of occasions [1, 3]. However, the results were analyzed under different suppositions about the important dimensionless parameters. In [1], the degree of underexpansion of the jet, characterized by n = p _a /p, was regarded as decisive; in [3], the experimental points were plotted against the relative dynamic head _a u² _a /(u² ) of the jet. In the present paper, dimensional considerations are used to determine the dimensionless parameters which influence the flow field when an injected jet interacts with an oncoming supersonic gas flow. The influence of these determining dimensionless parameters on the depth of penetration of injected jets into a flow was investigated experimentally. It is shown that the relative depth of penetration is determined basically by the relative specific impulse of the jet, the injection angle, and the shape of the blowing nozzle section.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 151–154, July–August, 1980.     

Minimum-drag cascade design for an “impeller-directing vanes-unchoked convergent nozzle” system

The variational problem of designing the slender airfoils of the plane cascade of an impeller-directing vanes-unchoked convergent nozzle system in a supersonic ideal (inviscid and non-heat-conducting) gas flow with a subsonic normal velocity component is solved in the linear approximation. The minimized functional is the axial component of the force acting on the system. The conditions to be met by the optimized airfoils include the specification of the projection of the force applied to the cascade onto the axis running parallel to its front, and the free stream condition. Flows past rather sparse cascades, whose discontinuous characteristics departing from the lower surface of an airfoil, do not impinge on the upper surface of the adjacent airfoil are considered.By analogy with an isolated cascade [1], the upper and lower generators of the optimum airfoils are constructed of straight-line segments. In the calculated example the quadrangular airfoils whose generator segments are inclined in accordance with the double-sided extremum conditions are shown to be optimum.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 162–165, May–June, 1996.     

Rarefied gas flow in a channel with specular‐diffuse boundary conditions for arbitrary knudsen numbers. Onsager relations

The problem of a rarefied gas flow in a channel for arbitrary Knudsen numbers has been solved analytically for the first time in the case where the scattering of gas molecules on the channel walls can be described by speculardiffuse boundary conditions. The mean free path of gas molecules is assumed to be constant, i.e., the collision frequency is proportional to molecular velocity. The gas moves under the action of a streamwise temperature gradient. Exact relations for heat and mass fluxes and for meanmass velocity are obtained. It is shown that the Onsager relations are valid within the entire range of Knudsen numbers in the problem of heat and mass transfer in a channel. The dependence of heat and mass fluxes on the Knudsen number (channel thickness) is analyzed. A comparison with available results is performed.     

Heat Transfer on the Surface on a Spherically Blunted Cone Exposed to a Supersonic Spatial Flow with Injection of a Coolant Gas

The heattransfer processes in a supersonic spatial flow around a spherically blunted cone with allowance for heat overflow along the longitudinal and circumferential coordinates and injection of a coolant gas are studied numerically. The prospects of using highly heatconducting materials and injection of a coolant gas for reduction of the maximum temperatures at the body surface are demonstrated. The solutions of the direct and inverse problems in one, two, and threedimensional formulations for different shell materials are compared. The error of the thinwall method in determining the heat flux on the heatloaded boundary of the body is estimated.     

Supersonic nonequilibrium flow past segmental bodies of a mixture simulating the atmosphere of venus

Calculations of the flow of the mixture 0.94 CO₂+0.05 N₂+0.01 Ar past the forward portion of segmentai bodies are presented. The temperature, pressure, and concentration distributions are given as a function of the pressure ahead of the shock wave and the body velocity. Analysis of the concentration distribution makes it possible to formulate a simplified model for the chemical reaction kinetics in the shock layer that reflects the primary flow characteristics. The density distributions are used to verify the validity of the binary similarity law throughout the shock layer region calculated.The flow of a CO₂+N₂+Ar gas mixture of varying composition past a spherical nose was examined in [1]. The basic flow properties in the shock layer were studied, particularly flow dependence on the free-stream CO₂ and N₂ concentration.New revised data on the properties of the Venusian atmosphere have appeared in the literature [2, 3] One is the dominant CO₂ concentration. This finding permits more rigorous formulation of the problem of blunt body motion in the Venus atmosphere, and attention can be concentrated on revising the CO₂ thermodynamic and kinetic properties that must be used in the calculation.The problem of supersonic nonequilibrium flow past a blunt body is solved within the framework of the problem formulation of [4].Notation V body velocity - shock wave standoff - universal gas constant - ratio of frozen specific heats - hRt/m enthalpy per unit mass undisturbed stream P pressure - density - T temperature - m molecular weight - c_p specific heat at constant pressure - (X) concentration of component X (number of particles in unit mass) - R body radius of curvature at the stagnation point - _j rate of j-th chemical reaction shock layer P V ² pressure - density - TT temperature - mm molecular weight Translated from Izv. AN SSSR. Mekhanika Zhidkosti i Gaza, Vol. 5, No. 2, pp. 67–72, March–April, 1970.The author thanks V. P. Stulov for guidance in this study.     

Analysis of conducting-gas channel flow in a self-induced magnetic field

The flow of a conducting gas in a flat channel is investigated in the hydraulic approximation with allowance for the Hall effect and the differential term in Ohm's law. Although channel flow has been extensively studied (for example, in [1–7]), these factors have not been simultaneously taken into account. It turns out that their presence makes it impossible to introduce a perfectly uniform gas dynamic flow, which complicates the problem. Special attention is paid to the conditions of realization of a smooth transition through the characteristic velocity which, in these circumstances, does not correspond to a gas dynamic Mach number equal to unity.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 1119, January–February, 1993.     

An “ideal equivalent gas method” for the study of shock waves in supersonic real gas flows

Summary A method developed by the author for the systematic study of the thermodynamic and dynamic properties of the gas behind a shock wave is reported.The method is applicable to supersonic flow regimes for which the excitation, dissociation and ionization effects invalidate the usually adopted hypothesis of ideal gas.An Ideal Equivalent Gas, having the ratio of the specific heats _s dependent on Mach number and altitude of flight is postulated.On the basis of the mass, momentum and energy conservation equations, valid through the shock wave, the relations defining the thermodynamic and dynamic state of the gas behind the shock wave are derived. These relations establish an extension of the classic relations valid for the ideal gas and reduce to them identically for _s=.The dependence of the ratio of specific heats _s of the Ideal Equivalent Gas on Mach number and altitude has been established, over a wide range, on the basis of the real gas solutions derived by Huber.

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Sommario Nella presente nota viene esposto un metodo sviluppato dall'autore per lo studio sistematico dello stato termodinamico e dinamico del gas a valle di un'onda d'urto in regime supersonico, allorchè cioè gli effetti dell'eccitazione dei gradi di libertà vibrazionali delle molecole e della loro dissociazione e successiva ionizzazione invalidano l'ipotesi di gas ideale generalmente adottata.Viene definito un gas ideale equivalente avente rapporto dei calori specifici _s funzione del numero di Mach e della quota di volo ed in base alle equazioni di conservazione della massa, della quantità di moto e dell'energia, valide attraverso all'onda d'urto, vengono derivate delle relazioni definenti lo stato termodinamico e dinamico del gas a valle dell'onda d'urto. Tali relazioni costituiscono una estensione delle classiche relazioni dell'urto valide per il gas ideale alle quali si riducono per _s=.La dipendenza del rapporto dei calori specifici _s del gas ideale equivalente, dal numero di Mach e dalla quota è stata stabilita sulla base delle soluzioni ottenute da Huber per il gas reale.

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Effect of hall currents on the flow of a conducting gas at high flow velocities

In [1] the flow of a compressible fluid was examined for the case when the conductivity = with account for the Hall effect. Oates [2] solves the problem of the influence of Hall currents on the flow in an accelerator for channels having a very small ratio of height to length when the velocity component in the direction of the channel height may be assumed to be zero. The problem of the influence of Hall currents on the flow of a conducting gas of finite conductivity is solved below for the case when the gas is accelerated to high velocities ( 50–100 km/sec) with account for the presence of two velocity components.