Investigation of supersonic three-dimensional flow past pointed axisymmetric bodies

The flow pattern near bodies of revolution with very long cylindrical and pointed nose sections is studied in the framework of an ideal gas model by means of a numerical method based on MacCormack's difference scheme. The existence of internal shock waves, oriented in both the longitudinal and the transverse directions, in the shock layer is established. The variation of the aerodynamic coefficients of the configuration with its length, angle of attack, and free stream Mach number is investigated. The calculated and experimental data are compared, and the connection between the flow parameters on the body surface and the position of the separation line of the boundary layer on its lateral face is established. A method of calculating the influence of the boundary layer on the values of the aerodynamic coefficients of bodies of revolution of large aspect ratio at small angles of attack is proposed. Axisymmetric flow near blunt bodies has been studied in detail in [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 127–133, September–October, 1986.The author expresses his gratitude to A. N. Pokrovskii for his help in calculating the boundary layer parameters on the surfaces of the considered configurations.     

Asymptotic theory of the flow near the rear end of a slender axisymmetric body

Steady high-Reynolds-number flow of a viscous incompressible fluid past a slender axisymmetric body is considered. The structure of the near wake and the boundary layer in the vicinity of the rear end of the body is studied. The relationship between the maximum relative body thickness and the rearend shape giving a local boundary layer — potential flow interaction zone in a small neighborhood of the rear end is found. The boundary value problem for this region is solved numerically.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 68–77, September–October, 1993.     

Characteristics of laminar-turbulent boundary layer transition on a cone

The effect of the unit Reynolds number and the Mach number on the transition on a slender circular cone is experimentally investigated. The perturbation spectra in the boundary layer on the cone are measured. It is shown that the location of the transition is determined by the perturbation level at the frequencies causing transition.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 23–27, November–December, 1985.     

Universal equations for the laminar boundary layer on a solid of revolution in an oblique gas flow

We consider the problem of laminar gas motion in the boundary layer on a solid of revolution oriented at an angle of attack. The parametric method of L. G. Loitsyanskii is used for the solution. The effect of the external current and the form of the body are considered by introduction of three series of parameters. A corresponding system of universal equations is obtained, which is then numerically integrated over a wide range of parameters and their combinations. The results permit evaluation of the general principles of flow in a boundary layer on a solid of revolution in an oblique gas flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 32–41, July–August, 1973.     

Onset of separation in hydrodynamic impact on a floating body

The vertical impact problem is considered for a body of revolution immersed in an ideal incompressible fluid bounded from below by a bottom in the shape of a surface of revolution. For a certain class of bodies it is proved that separation begins on the intersection between the wetted surface of the body and the meridional plane in which the shock impulse is located. As shown by the examples of spindle-shaped surfaces of revolution and an ellipsoid of revolution, separation can take place at one of two points on the boundary of the wetted surface of the body, one farther from and the other nearer to the point of application of the impulse.Rostov-on-Don. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 99–104, November–December, 1996.     

Influence of second-order boundary layer effects in hypersonic flow over slender blunt cones

The aim of this study is to determine the influence of second-order effects in the aggregate on supersonic axisymmetric flow over slender blunt cones and also to determine the region of applicability of approximate methods of taking into account the strongest of these second-order effects — entropy layer absorption. A system of complete viscous shock layer equations containing all the terms of the gas dynamic Euler equations and all the second-order terms of asymptotic boundary layer theory is chosen as the gas-dynamic model. Within the calculation domain the problem is solved in a unified manner.Translated from Izvestiya Rossiiskoi Akademii Nauk, Meknanika Zhidkosti i Gaza, No.4, pp. 129–134, July–August, 1992.     

Nonuniqueness of the solution of the problem of viscous interaction on an axisymmetric body

The article discusses solutions of the equations of the hypersonic boundary layer on an axisymmetric offset slender body (with a power exponent equal to 3/4), taking account of interactions with a nonviscous flow. It is shown that, in this case, the equations of the boundary layer have solutions differing from the self-similar solution corresponding to flow around a semi-infinite body. The solutions obtained are analogous to solutions for a strong interaction on a plate with slipping and triangular vanes [1–4], but are obtained over a wide range of values of the parameter of viscous interaction. An asymptotic solution is given to the problem with the approach to zero of the interaction parameter.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 41–47, September–October, 1973.The authors thank V. V. Mikhailova for discussion of the work and useful advice.     

Supersonic drag of three-dimensional bodies with a star-shaped cross section and its calculation

The effect on the aerodynamic drag of the real properties of the gas in the shock layer around pyramidal star-shaped bodies (the viscosity, the displacement thickness of the boundary layer, its separation under the influence of the inner shocks) is considered. It is shown that the models for calculating the total drag of star-shaped bodies which do not take into account the displacement thickness of the boundary layer are applicable only at low supersonic free-stream velocities (M < 3). A model of the boundary layer displacement thickness is proposed and tested over a broad range of variation of the parameters that determine the geometry of the pyramidal bodies for high supersonic or hypersonic speeds. A comparison with the experimental data shows that the calculation procedure adequately reflects the results of experiments on the aerodynamic drag of star-shaped bodies in cases in which the inner shocks in the shock layer do not lead to boundary layer separation and can be used in optimization problems.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 57–69, January–February, 1993.     

Method of mean-mass parameters for a three-dimensional boundary layer in a flow with vorticity

When a gas flows with hypersonic velocity over a slender blunt body, the bow shock induces large entropy gradients and vorticity near the wall in the disturbed flow region (in the high-entropy layer) [1]. The boundary layer on the body develops in an essentially inhomogeneous inviscid flow, so that it is necessary to take into account the difference between the values of the gas parameters on the outer edge of the boundary layer and their values on the wall in the inviscid flow. This vortex interaction is usually accompanied by a growth in the frictional stress and heat flux at the wall [2, 3]. In three-dimensional flows in which the spreading of the gas on the windward sections of the body causes the high-entropy layer to become narrower, the vortex interaction can be expected to be particularly important. The first investigations in this direction [4–6] studied the attachment lines of a three-dimensional boundary layer. The method proposed in the present paper for calculating the heat transfer generalizes the approach realized in [5] for the attachment lines and makes it possible to take into account this effect on the complete surface of a blunt body for three-dimensional laminar, transition, or turbulent flow regime in the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 80–87, January–February, 1981.     

Flow near the rear end of a slender axisymmetric body

The steady flow of a viscous incompressible fluid at high Reynolds numbers near a body of revolution of finite length whose radius coincides in order of magnitude with the thickness of the boundary layer is considered. The structure of the boundary layer in the neighborhood of the rear end of the body is investigated on the assumption that it has a power-law shape with values of the exponent n 1/2. A solution is also obtained for the near wake.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 10–18, September–October, 1990.     

Singular solution of boundary layer equations which can be extended continuously through the point of zero surface friction

It is shown that the Prandtl equations for an incompressible boundary layer admit a solution which can be extended continuously through the point of zero friction on the surface and is singular at this point. A solution of this type is realized, in particular, at the leading edge of a slender profile at an angle of attack to the oncoming flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 42–52, October–December, 1931.I thank V. V. Sychev and Vik. V. Sychev for discussing the work and helpful comments.     

Character of the boundary layer near its origin on a body rotating in a fluid at rest

The evolution of the boundary layer on bodies of revolution rotating about the symmetry axis in a fluid at rest is largely determined by the position of its origin with respect to the axis of rotation. If the origin of the boundary layer coincides with a pole of the rotating body, then under fairly general assumptions as to the shape of the body the boundary layer has a nonzero thickness in the vicinity of the pole, and the flow in it is described by a particular self-similar solution of the boundary-layer equations [1, 2]. The applicability of existing approximate methods for calculation of the boundary layer [2, 3] is restricted to this case. The results of the present article refer to the case in which the boundary originates at the leading edge at a finite distance from the rotation axis. The behavior of the solution of the boundary-layer equations near the edge is determined. A transformation of variables that reduces the system of boundary-layer equations to a form suitable for analysis and solution is derived. This transformation is used to obtain universal equations determining the local behavior of the boundary layer in the vicinity of its origin in both of the cases indicated above.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 137–140, July–August, 1976.     

Simulation of the aerodynamic drag of three-dimensional star-shaped bodies at hypersonic velocities

The domain of the parameters in which the aerodynamic drag of hypersonic pyramidal bodies, whose wave component is calculated within the framework of conical flows with the boundary layer displacement thickness taken into account, agrees satisfactorily with the experimental data is found. The calculation model is also applicable in the region of minimum aerodynamic drag of star-shaped bodies in the class of conical bodies equivalent in length and mid-sectional area.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 69–79, September–October, 1996.     

Simulation of turbulent heat-mass transfer on ablating surfaces

One of the world's biggest gasdynamic tubes intended for studying heat-mass transfer in the turbulent boundary layer on an ablating surface is described. It is shown that the facility can be used to simulate the heat-mass transfer and ablated shapes of meteorites and other bodies travelling in the atmosphere.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 133–142, January–February, 1996.     

Influence of the viscous boundary layer on the deposition of particles from a gas suspension flowing past a sphere

The flow of incompressible gas containing particles past bodies of simple shapes at moderate and high velocities is investigated in [1–5], in which the flow of the carrier medium is assumed to be irrotational. The estimates made in [3] for the neighborhood of the stagnation point show that it is necessary to take into account the viscous boundary layer in the case of fine particles. In the present paper, the viscous flow of a gas suspension over the front surface of a sphere at Reynolds numbers R = 10³–10⁷ is considered. It is assumed that the carrier gas is incompressible and the particle concentra ion negligibly small. The influence of the boundary layer on the particle trajectories and the deposition of the disperse phase on the surface of the sphere is investigated. It is shown that there is a wide range of flow parameters for the gas suspension in which the influence of the boundary layer is important. The limits of this range are established.Translated from Izvestiya Akademli Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 59–66, January–February, 1982.I thank Yu. P. Savel'ev for a helpful discussion of the work.     

Computation of the two-dimensional viscous incompressible fluid flow with separation at the nlet edge around a cascade

A complex flow consisting of an outer inviscid stream, a dead-water separation domain, and a boundary layer, which interact strongly, is formed in viscous fluid flows with separation at the streamlined profile with high Re numbers. Different jet and vortex models of separation flow are known for an inviscid fluid; numerical, asymptotic, and integral methods [1–3] are used for a viscous fluid. The plane, stationary, turbulent flow through a turbine cascade by a constant-density fluid without and with separation from the inlet edge of the profile and subsequent attachment of the stream to the profile (a short, slender separation domain) is considered in this paper.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 34–44, May–June, 1978.     

Calculation of the viscosity effect on the aerodynamic characteristics of slender bodies at hypersonic flow velocities

Approximating dependences of the local coefficients of friction, heat transfer, and pressure induced by a boundary layer on the generalized similarity parameters, including the inviscid flow characteristics, are obtained on the basis of the results of a numerical calculation of hypersonic flow past a number of plane and axisymmetric bodies. If the inviscid flow characteristics are known, these relations can be used to take the viscosity approximately into account under conditions of interaction between the laminar boundary layer and the hypersonic inviscid stream [1].Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 142–150, July–August, 1995.     

Friction and heat transfer in laminar and turbulent boundary layers on axisymmetric bodies in nonuniform supersonic flows

Generalized expressions are obtained for calculating the heat fluxes and frictional stresses of the laminar and turbulent flow regimes in a boundary layer in the case of uniform and nonuniform flow past bodies.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 65–72, March–April, 1984.     

Influence of viscosity on the flow over the sharp edge of bodies consisting of a spherical segment joined to an inverted cone

The results are given of an experimental investigation of the supersonic axisymmetric flow over a body consisting of a spherical segment joined to an inverted cone in the neighborhood of the point of inflection of the profile (Fig. 1a). For the limiting case of a cylinder with a flat end and M = 3, a study was made of the influence of the Reynolds number and the state of the boundary layer on the parameters of the local separation region formed near the inflection (Fig. 1b). It was found that there is an appreciable decrease in the length of the separation region and the pressure in it when the Reynolds number increases in the range Re = 10⁵– 10⁷ in the case of a laminar boundary layer on the flat end near the inflection point. A low level of the pressure on the surface of the body was achieved — of the order of thousandths of the pressure behind a normal shock. There was found to be a sharp increase in the pressure in the separation region when the boundary layer on the end becomes turbulent with transition to a flow regime that is self-similar with respect to the Reynolds number. Under conditions of a turbulent boundary layer, systematic experimental data on the pressure on the inverted cone near the point of inflection of such bodies were obtained and generalized.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 154–157, January–February, 1981.     

Three-dimensional laminar boundary layer on a permeable surface in the neighborhood of a symmetry plane

Analytical and numerical methods are used to investigate a three-dimensional laminar boundary layer near symmetry planes of blunt bodies in supersonic gas flows. In the first approximation of an integral method of successive approximation an analytic solution to the problem is obtained that is valid for an impermeable surface, for small values of the blowing parameter, and arbitrary values of the suction parameter. An asymptotic solution is obtained for large values of the blowing or suction parameters in the case when the velocity vector of the blown gas makes an acute angle with the velocity vector of the external flow on the surface of the body. Some results are given of the numerical solution of the problem for bodies of different shapes and a wide range of angles of attack and blowing and suction parameters. The analytic and numerical solutions are compared and the region of applicability of the analytic expressions is estimated. On the basis of the solutions obtained in the present work and that of other authors, a formula is proposed for calculating the heat fluxes to a perfectly catalytic surface at a symmetry plane of blunt bodies in a supersonic flow of dissociated and ionized air at different angles of attack. Flow near symmetry planes on an impermeable surface or for weak blowing was considered earlier in the framework of the theory of a laminar boundary layer in [1–4]. An asymptotic solution to the equations of a three-dimensional boundary layer in the case of strong normal blowing or suction is given in [5, 6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 37–48, September–October, 1980.