Blowing of a foreign gas in a hypersonic viscous shock layer

We consider the problem of a hypersonic viscous flow of a nonreactive mixture of ideal gases around smooth thick bodies in the framework of a two-layer model of a thin shock layer for moderately small Reynolds numbers. We investigate the effect of blowing of a foreign gas through a permeable surface in the bow region of a spherical blunt body. We introduce a transformation of variables that gives a number of important advantages in the numerical solution of the problem under consideration. The problem of mass blowing from the surface of a body into a boundary layer has an extensive literature. The effect of blowing for moderately small Reynolds numbers has been considerably less studied [1–5], and in the majority of papers on this question either the critical point of a blunt body or the blowing of a gas homogeneous with the gas in the incoming flow is investigated.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 110–116, July–August, 1972.     

Hypersonic viscous shock layer in a swirling gas flow on a permeable surface

A hypersonic swirling flow of viscous compressible gas past rotating axisymmetric blunt bodies is considered, its velocity vector being parallel to the axis of rotation of the body. The body surface is assumed permeable, while, in the general case, the gas is not injected (drawn off) along the normal to the body surface. An analytic solution of the problem, valid at small Reynolds numbers, is found in the first approximation of the integral method of successive approximations. On the basis of the results of the numerical solution, obtained in a wide range of variation of the determining parameters of the problem, we investigate the influence of the swirling of the free-stream flow, the angular velocity of rotation of the body, the Reynolds number and the injection (suction) parameter on the structure of the compressed layer, and the coefficients of friction and heat transfer on the body surface. The influence of the swirling of the flow on the nature of the asymptotic behavior of the viscous shock layer equations at large Reynolds numbers is studied. It is shown that the presence of a nonzero peripheral component for the velocity vector of the gas in the shock layer can lead to a qualitative change in the nature of the flow. Deceased Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 27–37, November–December, 1986. The authors thank G. G. Chernyi for his useful discussion of the results of the work.     

Investigation of stationary flow of viscous gas in a thin three-dimensional shock layer

A three-dimensional shock layer near the blunt surface of a fairly smooth body is analyzed asymptotically. Equations of the first approximation are obtained and justified in various cases of the limit 1, 0, ( – 1)^–1M ^-2 0. These equations are simplified for the flow near the stagnation point of a body with double curvature and near the blunt leading edge of a sweptback wing. The results of some calculations are given and compared with the results of [17, 18] in the case of axisymmetric flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 115–126, September–October, 1980.     

Model of a viscous nonequilibrium shock layer with a thin shock wave

It is shown that the concept of a viscous shock layer with boundary conditions specified in a thin shock wave is unsuitable for analyzing the flow of a chemically reacting gas, even in the case of high Reynolds numbers; it may produce a finite error when determining the parameters of the shock layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 175–178, September–October, 1973.     

Asymptotic solutions of the two-dimensional equations for a thin viscous shock layer in a rarefied gas

Two-dimensional hypersonic rarefied gas flow around blunt bodies is investigated for the continuum to free-molecular transition regime. In [1], as a result of an asymptotic analysis, three rarefied gas flow regimes, depending on the relationship between the problem parameters, were detected and one of these regimes was investigated. In the present study, asymptotic solutions of the thin viscous shock layer equations at small Reynolds numbers are obtained for the other two flow regimes. Analytical expressions for the heat transfer, friction and pressure coefficients are obtained as functions of the incident flow parameters and the body geometry and temperature. As the Reynolds number tends to zero, the values of these coefficients approach their values in free-molecular flow. The scaling parameters of hypersonic rarefied gas flow around bodies are determined for different regimes. The asymptotic solutions are compared with the results of direct Monte Carlo simulation.     

Unsteady three-dimensional viscous shock layer in nonuniform gas flow in the neighborhood of a stagnation point

The combined influence of unsteady effects and free-stream nonuniformity on the variation of the flow structure near the stagnation line and the mechanical and thermal surface loads is investigated within the framework of the thin viscous shock layer model with reference to the example of the motion of blunt bodies at constant velocity through a plane temperature inhomogeneity. The dependence of the friction and heat transfer coefficients on the Reynolds number, the shape of the body and the parameters of the temperature inhomogeneity is analyzed. A number of properties of the flow are established on the basis of numerical solutions obtained over a broad range of variation of the governing parameters. By comparing the solutions obtained in the exact formulation with the calculations made in the quasisteady approximation the region of applicability of the latter is determined. In a number of cases of the motion of a body at supersonic speed in nonuniform media it is necessary to take into account the effect of the nonstationarity of the problem on the flow parameters. In particular, as the results of experiments [1] show, at Strouhal numbers of the order of unity the unsteady effects are important in the problem of the motion of a body through a temperature inhomogeneity. In a number of studies the nonstationary effect associated with supersonic motion in nonuniform media has already been investigated theoretically. In [2] the Euler equations were used, while in [3–5] the equations of a viscous shock layer were used; moreover, whereas in [3–4] the solution was limited to the neighborhood of the stagnation line, in [5] it was obtained for the entire forward surface of a sphere. The effect of free-stream nonuniformity on the structure of the viscous shock layer in steady flow past axisymmetric bodies was studied in [6, 7] and for certain particular cases of three-dimensional flow in [8–11].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 175–180, May–June, 1990.     

Rotation-vibration-dissociation interaction in a multicomponent nonequilibrium viscous shock layer

A new, simple and physically adequate method of calculating vibrationally nonequilibrium dissociation constants is proposed on the basis of a dissociation model which takes into account the equilibrium excitation of the rotational degrees of freedom of the molecules and the nonequilibrium excitation of vibrational quantum states. This rotation-vibration-dissociation interaction model contains only the indeterminacy associated with the indeterminacy of the experimental data on the interaction potentials and the collision cross sections of the components. In the case of thermodynamic equilibrium the model gives values of the dissociation constants close to those generally accepted. The use of this model in multicomponent nonequilibrium total viscous shock layer calculations gives values for the shock detachment distance within 5% of the experimental values. The indeterminacy in the values of the vibrational energy lost by air molecules during dissociation and recovered during recombination does not lead to serious errors in the macrocharacteristics of the flow. The nonequilibrium excitation of vibrational degrees of freedom proves to be not so important in computing the macrocharacteristics of the flow as previously assumed and the existing algorithms for calculating chemically nonequilibrium flows on the assumption of thermodynamic equilibrium can be used with satisfactory accuracy for calculating the values of the heat flux, the position of the shock wave, and the temperature and pressure in the shock layer for partially dissociated and ionized air.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 166–180, November–December, 1994.     

Study of the turbulent flow of a gas over long blunt bodies in a viscous shock layer

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 1, pp. 69–75, January–February, 1993.     

Hypersonic three-dimensional viscous shock layer in a nonuniform gas flow in the neighborhood of the stagnation point

Three-dimensional hypersonic viscous gas flow past smooth blunt bodies in the presence of injection or suction is considered. The effect of the nonuniformity of the approach stream on the shock-wave standoff, the flow structure and the friction and heat transfer coefficients is investigated with reference to the examples of flow from a supersonic spherical source and flow of the far wake type. It is shown that this effect depends importantly on the Reynolds number, the nature of the nonuniformity and the shape of the body.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 136–145, November–December, 1987.     

Nonequilibrium viscous shock layer on blunt cones

Some results are given of the numerical investigation into the parameters of the nonequilibrium flow of air in a viscous shock layer in the case of blunt circular cones at zero angle of attack; they are also compared with experimental data obtained during re-entry of ballistic objects into the Earth's atmosphere. The calculations were made with allowance for the nonequilibrium processes of dissociation and ionization, and also vibrational relaxation. The influence of viscosity, heat conduction, and diffusion is taken into account in the complete shock layer. The conditions on the shock wave are posed with allowance for its finite thickness. The characteristic profiles of the velocity, temperature, and electron concentration in the shock layer are given. Good agreement is obtained between the calculated and experimental data on the level and the profiles of the electron concentration. The parameters of the shock layer were determined by a method that is a natural extension of the numerical method of [1] to the case of nonequilibrium flow in a viscous shock layer. Because of this, only the main differences of the method when applied to the calculation of nonequilibrium flows of a multicomponent mixture such as dissociated and ionized air are described.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 15–20, November–December, 1979.     

Calculation of a nonequilibrium viscous shock layer with allowance for coupled heat transfer

An investigation is made into the characteristics of coupled heat and mass transfer using the theory of a nonequilibrium viscous shock layer in the case of an axisymmetric blunt body moving along a given trajectory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 146–153, March–April, 1984.     

Multicomponent chemically-reacting turbulent viscous shock layer on a catalytic surface

Turbulent flows past blunt bodies at high supersonic speeds are mainly investigated within the framework of the boundary layer model. However, even at large Reynolds numbers owing to the strong entropy gradient on the lateral surface it becomes necessary to take boundary layer corrections into account in the higher approximations [1]. The use of viscous shock layer theory makes it possible to obtain fairly accurate results over a broad interval of variation of the Reynolds numbers without organizing iterations with respect to vorticity and displacement thickness. The nonequilibrium nature of both homogeneous and heterogeneous catalytic reactions is taken into account. The results obtained are compared with the experimental data [2, 3]. Previously, in [4, 5] turbulent flow was investigated within the framework of viscous shock layer theory in the case of equilibrium homogeneous reactions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 144–149, March–April, 1989.     

Flow around stepped bodies with strong compression in a shock layer

This article discusses plane and axisymmetric flows of a nonviscous ideal gas around bodies of stepped form, forming with a Mach number M= and an adiabatic indexN1. The greatest amount of attention is paid to the case where there is no Newtonian free layer, but the shock layer is detached at great distances from the nose of the body.Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 4, pp. 104–112, July–August, 1973.     

Movement of gas bubbles in a liquid layer in the presence of diffusion and chemical reactions

The problem of a bubbling reactor, in which gas and liquid are mixed by the passage of bubbles of gas through a liquid layer, is discussed. We give the results of a numerical solution of the system of equations describing the processes occurring in the reactor in the case where there are no chemical reactions, and also in the case where chemical reactions take place at constant temperature.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 131–135, January–February, 1971.The authors thank L. A. Chudov for advice and interest in the work.