Effect of the shape of a blunt nose on the drag coefficient of a body in a hypersonic rarefied gas flow

The effect of the shape of a blunt nose of a body located in a hypersonic rarefied gas flow on the field of flow and on the aerodynamic characteristics is studied in the example of flow round ellipsoids of revolution at a zero angle of attack. The problem of the flow in the transition regime is solved on the basis of numerical analysis of the model kinetic Bhatnagar—Gross—Krook (BGK) equation for a monatomic gas. The good agreement of the results of the numerical calculations with the experimental data in a broad range of Mach numbers has shown [1, 2] that the numerical solution of the model kinetic equations is a reliable and effective means for studying flow problems. In the case when the problem is posed of determining the laws of the purely force interaction of a flow with the body, sufficiently good accuracy is given by the use of the model BGK equation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 190–192, March–April, 1985.     

Transverse flow of a rarefield gas around a plate

A method of characteristics is developed for the numerical solution of kinetic equations in the case of fully developed plane-parallel motions of a gas. A solution is given to the problem of flow around a plate arranged normal to an oncoming supersonic flow. The results of the calculations are in satisfactory agreement with existing experimental data.Moscow. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 6, pp. 107–113, November–December, 1972.     

Asymptotic solution of the euler equations in the shock layer on a blunt body in nonuniform flow with gas injection from the body surface

Supersonic nonuniform gas flow over blunt bodies without surface injection has previously been investigated by both numerical [1–3] and experimental [3] methods. The processes of surface vaporization under the influence of an intense heat flux, artificial gas injection and surface combustion [4] are all worthy of study. The problem of the interaction between a nonuniform supersonic flow and a body in the presence of intense gas injection from the surface is examined and an analytical solution is constructed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 126–134, November–December, 1989.     

Plane Couette flow of binary gaseous mixture in the whole range of the Knudsen number

The Couette flow of binary gaseous mixtures is studied on the basis of the McCormack model of the Boltzmann equation, which was solved numerically by the discrete velocity method. The calculations were carried out for three mixtures of noble gases: neon–argon, helium–argon, and helium–xenon. The stress tensor and bulk velocity of both species were calculated for several values of the gas rarefaction in the range from 0.01 to 40 for three values of the molar concentrations: 0.1,0.5 and 0.9. The numerical solution together with an analytical solution based on the slip boundary condition cover the whole range of the gas rarefaction. It was showed that the Couette flow is weakly affected by the intermolecular interaction law.     

Flow around the afterbody of a plate in a longitudinal gas stream with transverse injection from the lateral surface

The flow around the afterbody of a plate of finite thickness in a supersonic gas stream is investigated on the basis of a numerical solution of the time-dependent Navier-Stokes equations for a compressible viscous heat-conducting gas. The change in the flow pattern with the onset of transverse slot injection from the body surface in the vicinity of the base section is studied. For constant supersonic injection, both steady and unsteady flow regimes could be obtained depending on the values of certain relevant parameters.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 157–163, March–April, 1996.     

A spatial laminar boundary layer on a streamline in conical external flow of a uniform gas in the absence of sources and sinks

Theoretical study of a three-dimensional laminar boundary layer is a complex problem, but it can be substantially simplified in certain particular cases and even reduced to the solution of ordinary differential equations.One such particular case is the flow of a compressible gas on a streamline in conical external flow. The case is of considerable practical importance because the local heat fluxes may take extremal values on such lines.Such flow, except for the conical case, has been examined [1–4], and an approximate method has been given [1] on the basis of integral relationships and a special form for the approximating functions. A numerical solution has been given [2, 3] for such flow around an infinite cylinder. It was assumed in [1–3] that the Prandtl number and the specific heats were constant, and that the dynamic viscosity was proportional to temperature. Heat transfer has been examined [4] near a cylinder exposed to a flow of dissociated air.Here we give results from numerical solution of a system of ordinary differential equations for the flow of a compressible gas in a laminar boundary layer on streamlines in conical external flow, with or without influx or withdrawal of a homogeneous gas. It is assumed that the gas is perfect and that the dynamic viscosity has a power-law temperature dependence.     

Transonic flow of a gas in axisymmetric Laval nozzles with steep walls

A large number of articles have recently appeared [1–5] in which various numerical methods have been proposed for solution of both the direct and inverse problem of calculating the flow of an ideal gas in a Laval nozzle. An analysis of the results presented in these articles shows that, in spite of the two-dimensional character of the flow fields obtained, the distribution of the pressure at the wall and along the central line of the flow differs only slightly from the values calculated using the hydraulic theory, in which, as is well known, the transverse distribution of the parameters is neglected. In what follows, an analysis is presented of the numerical results of a calculation of the gas flow in the transitional region of round Laval nozzles with very steep walls, where the flow parameters vary considerably in a transverse direction and where their values differ strongly from those obtained in a hydraulic approximation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 55–58, November–December, 1970.     

Laminar boundary layer on swept-back wings of infinite span at an angle of attack

A study is made of the flow of a compressible gas in a laminar boundary layer on swept-back wings of infinite span in a supersonic gas flow at different angles of attack. The surface is assumed to be either impermeable or that gas is blown or sucked through it. For this flow and an axisymmetric flow an analytic solution to the problem is obtained in the first approximation of an integral method of successive approximation. For large values of the blowing or suction parameters, asymptotic solutions are found for the boundary layer equations. Some results of numerical solution of the problem obtained by the finite-difference method are given for wings of various shapes in a wide range of angles characterizing the amount by which the wings are swept back and also the blowing or suction parameters. A numerical solution is obtained for the equations of the three-dimensional mixing layer formed in the case of strong blowing of gas from the surface of the body. The analytic and numerical solutions are compared and the regions of applicability of the analytic expressions are estimated. On the basis of the solutions obtained in the present paper and studies of other authors a formula is proposed for the calculation of the heat fluxes to a perfectly catalytic surface of swept-back wings in a supersonic flow of dissociated and ionized air at different angles of attack. Flow over swept-back wings at zero angle of attack has been considered earlier (see, for example, [1–4]) in the theory of a laminar boundary layer. In [5], a study was made of flow over swept-back wings at nonzero angle of attack at small and moderate Reynolds numbers in the framework of the theory of a hypersonic viscous shock layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 27–39, May–June, 1980.We thank G. A. Tirskii for a helpful discussion of the results.     

Weakly rarefied gas flows between nonuniformly heated parallel plates

The aim of this research is to establish the validity of the predictions of the theory of slow nonisothermal flows, to study the limits of applicability (with respect to the Knudsen number) of the conclusions reached and to determine the effect of the Knudsen layers on these flows on the basis of a numerical investigation of slow nonisothermal weakly rarefied gas flow in a plane infinite channel with weakly nonequilibrium heating of the walls and a finite wall temperature difference. The gas flow is described by a relaxation transport equation. The results obtained show how quickly, as the Knudsen number decreases, the solutions of the transport equation outside the Knudsen layers tend to the solution of the equations of gas dynamics of slow nonisothermal flows (and not to the solution of the Navier-Stokes equations).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 115–121, January–February, 1988.     

Airfoil design using quasisolutions of inverse boundary-value problems

A mathematically rigorous approach to the solution of inverse boundary-value problems of aerohydrodynamics — problems in which the airfoil contour is found from the given distribution of the velocity as a function of the arc abscissa – is proposed. The approach is based on the notion of a quasisolution in the theory of improperly posed problems and involves a controlled modification of the initial data for the purpose of obtaining a unique solution in a given class of contours. The approach is explained with reference to the problem of designing an isolated airfoil in a plane steady ideal incompressible flow and is extended to the cases of subsonic gas flow and viscous incompressible flow. Some numerical results illustrating the possibilities of the method are also presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 5–13, May–June, 1988.The authors wish to thank G. Yu. Stepanov for frequent useful discussions and valuable advice.     

Solution of Navier-Stokes equations near the trailing edge of a plate

The appearance of a local singularity in the solutions for the neighborhood of the trailing edge of a plate in a sub- or supersonic flow makes it necessary to consider the flow in the local region which is described in the first approximation by the Navier-Stokes equations for an incompressible gas. In this paper numerical solutions are obtained for such a region for both a thin plate and a plate with thickness. The streamline patterns and the distributions of the flow functions over the surface of the plate and in the wake behind it are presented. For the plate with finite thickness, a numerical solution is obtained in a wide range of variation of the local Reynolds number.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 173–176, July–August, 1985.     

Two-phase couette flow

A study is made of plane laminar Couette flow, in which foreign particles are injected through the upper boundary. The effect of the particles on friction and heat transfer is analyzed on the basis of the equations of two-fluid theory. A two-phase boundary layer on a plate has been considered in [1, 2] with the effect of the particles on the gas flow field neglected. A solution has been obtained in [3] for a laminar boundary layer on a plate with allowance for the dynamic and thermal effects of the particles on the gas parameters. There are also solutions for the case of the impulsive motion of a plate in a two-phase medium [4–6], and local rotation of the particles is taken into account in [5, 6]. The simplest model accounting for the effect of the particles on friction and heat transfer for the general case, when the particles are not in equilibrium with the gas at the outer edge of the boundary layer, is Couette flow. This type of flow with particle injection and a fixed surface has been considered in [7] under the assumptions of constant gas viscosity and the simplest drag and heat-transfer law. A solution for an accelerated Couette flow without particle injection and with a wall has been obtained in [6]. In the present paper fairly general assumptions are used to obtain a numerical solution of the problem of two-phase Couette flow with particle injection, and simple formulas useful for estimating the effect of the particles on friction and heat transfer are also obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 42–46, May–June, 1976.     

Formula for the Flow Resistance Factor in a Pipe with a Sudden Expansion at Small Reynolds Numbers

A formula for the flow resistance factors in a pipe with a sudden expansion of the cross section at Reynolds numbers of 0.2 to 10 is obtained by numerical solution of the complete Navier–Stokes equations for incompressible fluids. The flow resistance factors obtained using the derived formula are compared to those found by numerical solution of the Navier–Stokes equations.     

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.     

Unsteady flows of an erosion plasma in irradiation of solid targets by annular beams

In recent times high-pressure physics has made ever wider application of constructions which use convergent shock waves [1–8]. The study of gas dynamic flows with convergent shock waves imposes the need for more careful calculation of the motions of a gas in regions whose dimensions are much less than the characteristic dimensions of the flow. In the present study the numerical method is used to study the gas dynamic phenomena accompanying the irradiation of solid obstacles by annular beams of monochromatic radiation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 179–182, November–December, 1988.In conclusion we note that at very short durations t t_k the solution to the problem is similar to the flow during separation of a gaseous toroid [19].     

Transonic flows around an airfoil in wind tunnels with porous walls

A study is made of two-dimensional transonic flows of gas around an airfoil in the working part of a wind tunnel with porous walls. The values of the flow parameters are determined by the numerical solution of a boundary-value problem for the equation of the velocity potential; this problem simulates the gas flow around the profile in the tunnel with porous walls. The obtained results are then used to construct an asymptotic theory of the influence of the wind-tunnel height and the Mach number M of the flow in it on the characteristics of the flow around the airfoil.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 99–107, September–October, 1980.     

Calculation of the swirling flow of an ideal gas in a laval nozzle

The numerical solution of the problem of the motion of a swirling flow of an ideal gas in a Laval nozzle in axisymmetric formulation is obtained by the method of stabilization. As a result, a number of effects appear that are essentially not one-dimensional, in particular, the drawing-in of the sonic line into the nozzle, an effect that leads to a decrease in the nozzle's expansion coefficient. The dependence of this coefficient on the intensity of the swirling is obtained. A number of problems connected with the control of the expansion of a gas through a Laval nozzle and with variation of the thrust of a nozzle can be solved successfully in cases where a rotary motion is imparted to the flow of gas exhausted from the nozzle. Investigation of such a swirling flow in [1, 2] and a number of other papers are based on a one-dimensional model of gas flow, which makes it possible in principle to obtain integrated characteristics of the flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 72–76, September–October, 1971.     

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.     

Numerical investigation of viscous gas secondary flows in a rotating cylinder with sources and sinks

A source-sink model of secondary flow excitation in a rotating cylinder, which describes the interaction between a circulator and a rotating gas, is proposed for a nonlinear system of Navier-Stokes equations, and the results of a numerical calculation of the resulting circulating flows are presented. The modified Newton's method employed in the numerical solution makes use of regularizing perturbations to ensure its stability and convergence at low Ekman numbers and high rates of rotation of the cylinder. The combined effect of mechanical and thermal means of flow excitation and the influence of viscous energy dissipation are considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 39–44, July–August, 1989.     

Analysis of flow in an annular duct with large injection at the walls

Viscous heat-conducting compressible fluid flow in an annular duct formed by two coaxial cylinders with large injection at the walls is investigated. An asymptotic solution exhibiting the influence of the axial symmetry of the duct is obtained in the vicinity of the y axis and is compared with the results of exact numerical calculations. Asymptotic solutions of the Navier-Stokes equations have been obtained earlier for flows in a plane channel with various rates of wall injection in the case of an incompressible gas [1, 2] and a compressible gas [3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 135–139, May–June, 1976.