Area rule for integral quantities

In hydrodynamics and aerodynamics there is an area rule for nearly axisymmetric bodies. It states that the drag [1–3], the coefficient of heat transfer and the ablation [4], and also the wake parameters [5] of a three-dimensional body are equal to the analogous quantities for an axisymmetric body which has the same distribution of the cross sectional area along the axis. In some cases, the area rule holds for bodies which depart strongly from axial symmetry [6]. It is shown in the present paper that equality also holds for other integral quantities and not only in hydrodynamic problems.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 157–159, July–August, 1981.We thank Yu. B. Lifshits for helpful comments.     

Distribution of radiant heat fluxes over the surface of three-dimensional and axisymmetric bodies in a supersonic ideal-gas flow

Numerous studies have been devoted to the calculation of supersonic radiating flow past bodies (see the bibliography in [1, 2]). In almost all these studies the gas flows investigated are plane or axisymmetric. Three-dimensional flows, however, have received little attention. The flow of air past three-dimensional bodies was considered in [3], where the chief object was to investigate the accuracy with which a real radiating volume can be simulated by the widely used plane-layer approximation. In [4] the flow of a hydrogen-helium mixture past three-dimensional bodies was investigated in the hypersonic approximation in the neighborhood of the stagnation point and in two planes of symmetry with allowance for the screening of the radiation of vaporized material.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 92–97, January–February, 1986.The authors wish to thank V. V. Lunev for his useful observations and advice.     

Radiative and convective heat transfer during blowing into a compressed hypersonic layer of two-phase products from the ablation of a covering

The study examines the screening of the radiative heat flux in conditions of hypersonic flow around blunt bodies with ablated carbon-based coverings. In contrast to the studies already known [1–3], allowance is made for the presence of condensed microscopic particles in the products of ablation. In [4] the problem of radiative transfer is considered in a layer of two-phase ablation products with parametrically prescribed dimensions, particle temperature, and layer thickness. The present study uses a closed system of equations which describes the processes of heat and mass transfer. This gives rise to considerable differences in the numerical results, according to the degree of screening.Translated fron Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 161–166, November–December 1985.deceased     

Analytic investigation of friction and heat transfer in the neighborhood of a three-dimensional stagnation point at low and moderate Reynolds numbers

A study is made of hypersonic three-dimensional flow of a viscous gas past blunt bodies at low and moderate Reynolds numbers with allowance for the effects of slip and a jump of the temperature across the surface. The equations of the three-dimensional viscous shock layer are solved by an integral method of successive approximation and a finite-difference method in the neighborhood of the stagnation point. In the first approximation of the method an analytic solution to the problem is found. Analysis of the obtained solution leads to the proposal of a simple formula by means of which the calculation of the heat flux to a three-dimensional stagnation point is reduced to the calculation of the heat flux to an axisymmetric stagnation point. A formula for the relative heat flux obtained by generalizing Cheng's well-known formula [1] is given. The accuracy and range of applicability of the obtained expressions are estimated by comparing the analytic and numerical solutions. Three-dimensional problems of the theory of a supersonic viscous shock layer at small Reynolds numbers were considered earlier in [2–5] in a similar formulation but without allowance for the effects of slip.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 143–150, March–April, 1988.     

Axisymmetric analogy for three-dimensional viscous flow problems

A method of solving three-dimensional flow problems with the aid of two-dimensional solutions, which can be used for any Reynolds numbers, is proposed. The method is based on the use of similarity relations obtained in the theoretical analysis of the approximate analytic solution of the equations of a three-dimensional viscous shock layer. These relations express the heat flux and the friction stress on the lateral surface of a three-dimensional body in terms of the values on the surface of an axisymmetric body. The accuracy is estimated by comparing the results with those of a numerical finite-difference calculation of the flow past bodies of various shapes. Similar similarity relations were previously obtained in [1] for the plane of symmetry of a blunt body.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 111–118, November–December, 1991.The authors are grateful to G. A. Tirskii for his interest in their work.     

Investigation of hypersonic three-dimensional flow past blunt bodies within the framework of the parabolized Navier-Stokes equations

The flow of a homogeneous gas in a three-dimensional hypersonic viscous shock layer, which includes the shock wave structure, is examined within the framework of the parabolic approximation of the Navier-Stokes equations. The Navier-Stokes equations are simplified on the basis of the asymptotic analysis carried out in [1], are written in variables of the Dorodnitsyn type [2] and are solved by the method proposed in [3, 4] extended to the case of three-dimensional flows. The flow at zero angle of attack past elliptic paraboloids, two-sheeted hyperboloids and triaxial ellipsoids is calculated. Some results of investigating the flow past such bodies are presented. Flow past a sphere in the analogous approximation was considered in [5], where a comparison was also made with the solution of the complete Navier-Stokes equations [6, 7].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 134–142, July–August, 1987.In conclusion, the authors wish to express their warm thanks to V. V. Lunev and G. A. Tirskii for useful discussion and valuable comments.     

Investigation of the shape of the transverse contour of a conical three-dimensional body of minimal drag moving in rarefied gas

The shape of bodies of minimal drag moving in rarefied gas in the entire range of flight heights is investigated at present on the basis of the use of local interaction models [1]. Corresponding theoretical investigations have been published in [2, 3] for the case of bodies of rotation and in [4] for three-dimensional winged bodies, and detailed numerical investigations have been carried out in [2, 5–7]. In the present paper, an analytic investigation is carried out for the purpose of determining the optimal shape of three-dimensional bodies with minimal drag in an intermediate region of flight heights in rarefied gas.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 112–117, September–October, 1986.     

Steady-state shapes of bodies melted by aerodynamic heating

The variation of the shape of bodies exposed to aerodynamic ablation has been the subject of a considerable amount of research. One particular aspect of the general problem, namely, the problem of the steady-state shapes, i.e., those that do not vary as a result of ablation, was solved in [1, 2] for convective heat transfer on the assumption that the effective enthalpy of the material is constant. In this case a distinctive feature of the solution is the presence of corner points (breaks) on the steady-state shapes. Here, the problem is solved without assuming a constant effective enthalpy — the ablation rate is determined on the basis of a numerical solution of the equations describing the ablation of glassy materials, the flow of the molten film over the surface being calculated on the basis of the complete system of boundary-layer equations for an incompressible fluid [3]. It is shown that for a uniform heat transfer regime (laminar or turbulent) the steady-state shapes are smooth bodies without corner points. In the mixed heat transfer regime, in the general case the problem has no solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 122–127, July–August, 1987.     

Aerodynamic coefficients of non-conical bodies with a star-shaped transverse cross section

It is well known that, in a supersonic flow, the wave resistance of a body of non-round transverse cross section can be less than the resistance of an equivalent body of revolution with the same length and volume. Starting from 1959, when an exact solution was obtained to the problem of supersonic flow around conical bodies with a pyramidal system of flat discontinuities [1], a number of publications have appeared [2–5] developing this direction. Article [3] pointed out the possibility of achieving a flow with reflected shock waves, normal to the faces of a pyramidal body, by selection of the form of the leading edge. In [6, 7], using the Newton resistance law, bodies were constructed with a transverse cross section of a star-shaped form, having a wave resistance several times less than for an equivalent body of revolution. Just such forms, with certain limitations, have the least wave resistance and retain optimality with respect to the total resistance, taking approximate account of friction forces. Still two more exact solutions were then found, corresponding to flow around star-shaped bodies with regular and Mach interaction between shock waves [8, 9]. At a seminar of the Institute of Mechanics of Moscow State University, G. G. Chernyi advanced the postulation of the existence of certain classes of three-dimensional bodies not having the property of similitude and retaining optimality with respect to determined characteristics, for example, the resistance, the aerodynamic quality, or the torque, and stated partial problems of finding various forms of optimal bodies. Classes of bodies, optimal with respect to the resistance, were obtained within the framework of the Newton theory; the bodies consisted of helical surfaces, as well as of sections of planes and conical surfaces, formed by straight lines connecting the leading edges with a round contour. As a result of calculations using the Newton theory and experimental investigations it was established that bodies with a wedge-shaped nose part, with determined geometric parameters, have greater values of the lifting and of the aerodynamic quality than round cones [10]. The possibility of lowering the resistance and increasing the aerodynamic quality of aircraft by giving them shapes of the transverse cross section in the form of a star [11–14] leads to new investigations of three-dimensional bodies which retain optimality with respect to their aerodynamic characteristics, and are used in conjunction with bodies of revolution. This latter factor is of decisive importance with the use of such configurations as the nose part of the aircraft, or of a multi-step diffusor. The present article gives the results of an experimental investigation of flow around two classes of such bodies: multi-wedge and helical.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 127–132, November–December, 1974.     

Three-dimensional chemically nonequilibrium viscous shock layer on a catalytic surface with allowance for associated heat transfer

A three-dimensional flow of dissociating air past blunt bodies is investigated in the framework of the thin viscous shock layer theory. Multicomponent diffusion and homogeneous chemical reactions, including dissociation, recombination, and exchange reactions, are taken into account. The generalized Rankine-Kugoniot conditions are specified on the shock wave and the conditions which take into account the heterogeneous catalytic reactions, on the surface of the body. The viscous shock layer equations are solved together with the heat equations inside the coating, which is carbon with a deposited thin film of SiO₂, or quartz. The case of a thermally insulated surface is also considered. The problem for the case of the motion of a body along the re-entry trajectory into Earth's atmosphere is investigated numerically. The temperature of the surface and the heat flux toward it are given as a dependence on the height (tine) of the flight for different cases of the specification of the catalytic reactions. It is shown that the difference between the heat fluxes towards the thermally insulated surface and the fluxes toward the heat-conducting surface in the neighborhood of the stagnation point is of the order of 6–12% for all the cases considered. This makes it possible to decouple the solution of the problem of heat conduction in the body.Translated fron Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 140–146, November–December, 1985.deceased     

On the center of pressure of conical bodies

The profiles of conical bodies for which the position of the center of pressure in a supersonic flow with symmetry plane does not depend on the flow parameters are considered. The theoretical investigation of the aerodynamic characteristics of circular cones [1] has shown that their center of pressure does not depend on the angle of attack when the shock wave is attached to the apex of the cone. It was established experimentally in [2, 3] for star-shaped bodies that the position of the center of pressure for such bodies hardly changes in a wide range of Mach numbers and angles of attack.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 99–104, January–February, 1980.I thank G. G. Chernyi for discussing the results.     

Spatial Thermoviscoplastic Problems

Methods and results of studies of the three-dimensional viscoplastic stress–strain state of engineering structures under thermomechanical loading are presented. The following classes of thermoviscoplastic problems are considered: axisymmetric problems, nonaxisymmetric problems for bodies of revolution, three-dimensional problems for bodies of arbitrary shapes, and three-dimensional problems for anisotropic bodies of revolution     

Arbitrary body,close to a wedge,in a supersonic flow

The problem of supersonic flow around bodies close to a wedge was first discussed in the two-dimensional case in [1]. The shock wave was assumed to be attached, and the flow behind it to be supersonic; taking this into account, the angle of the wedge was assumed to be arbitrary. The surface of the body was also arbitrary, provided that it was close to the surface of the wedge. In solution of the three-dimensional problem, there was first considered flow around two supporting surfaces with only slightly different angles of attack [2], and then around a delta wing [3, 4]. In all these articles, the Lighthill method was used to solve the Hilbert boundary-value problem [5, 6]. A whole class of surfaces of bodies with arbitrary edges, under the assumption that the surface of the body was cylindrical, with generatrices directed along the flow lines of the unperturbed flow behind an oblique shock wave, was discussed in [7]. In the present work, the problem is regarded for a broad class of surfaces of bodies, using a new method which generalizes the results of [8].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 109–117, July–August, 1974.The author thanks G. G. Chernyi for his direction of the work.     

Similarity of three-dimensional and axisymmetric chemically-nonequilibrium flows in the neighborhood of a plane of symmetry

Three-dimensional chemically-nonequilibrium flow past blunt bodies in the neighborhood of the plane of symmetry is investigated within the framework of viscous shock layer theory. The similarity of three-dimensional and axisymmetric flows, previously established in [1] for a uniform gas, is extended to chemically-nonequilibrium gas flows. It is shown that the problem of determining the heat fluxes and friction stress in the neighborhood of the line of flow divergence can be reduced to the problem of determining these quantities for the axisymmetric body. The validity of the axisymmetric analogy is verified by carrying out numerical calculations for bodies of different shapes re-entering the earth's atmosphere along a gliding trajectory. Various models of surface catalytic activity are considered. The use of similarity relations makes it possible to apply existing programs for calculating axisymmetric flows to the solution of three-dimensional problems.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 115–120, March–April, 1990.     

Two-dimensional problem for internal waves generated by moving singular sources

When bodies move in a liquid with inhomogeneous density in a gravitational field waves are excited even at low velocities and in the absence of boundaries. They are the so-called internal waves (buoyancy waves), which play an important part in geophysical processes in the ocean and the atmosphere [1–4]. A method based on the replacement of the bodies by systems of point sources is now commonly used to calculate the fields of internal waves generated by moving bodies. However, even so the problems of the generation of waves by a point source and dipole are usually solved approximately or numerically [5–11]. In the present paper, we obtain exact results on the spectral distribution of the emitted waves and the total radiation energy per unit time for some of the simplest sources in the two-dimensional case for an incompressible fluid with exponential density stratification. The wave resistance is obtained simply by dividing the energy loss per unit time by the velocity of the source. In the final section, some results for the three-dimensional case are briefly formulated for comparison.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 77–83, March–April, 1981.     

Averaged temperature fields in asymmetrical turbulent streams over localized heat sources

A modification of the shadow method which can be used to measure the three-dimensional averaged fields of the refractive index in turbulent flows is described. The method is applied to the measurement of a temperature field in a turbulent convective flow. A stream of heated fluid flowing slowly out of a circular opening is used as the heat source. The measurement results are compared with asymptotic dependences for buoyant convective fluids [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 199–203, March–April, 1977.     

Investigation of flow past blunt bodies with allowance for radiation by the large-particle method

Much recent work has been done on developing methods of solving gas-dynamic problems in which radiation plays a part (see, for example, [1–7]). This is because the temperature in the shock layer associated with flight in the atmosphere at hypersonic velocities can reach values exceeding 10⁴ °K. In such a case, heat transfer by radiation can make an important contribution to the total heat transfer. With increasing flight velocities, the importance of radiation in heat transfer increases and then becomes predominant. In the present paper, the large-particle method as developed by Belotserkovskii and Davydov [8] is developed to calculate flows with radiation around blunt bodies, including the case when there is distributed blowing from the surface of the bodies into the shock layer, which simulates ablation of a heat-shielding covering under the influence of strong heating by radiation. The results are given of systematic calculations of flow past blunt bodies of various shapes by a stream of radiating air, and the results are compared with the data of other methods.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 106–112, July–August, 1982.     

Three-dimensional stability theory of deformable bodies. Stability of construction elements

Conclusion In [8, 9] and in the present paper we analyzed the possibilities of using the approximate approach [15, 18] in the three-dimensional stability theory of deformable bodies as applied to effects of internal and surface instability and to stability of thinwalled structural elements. The analysis mentioned has been performed by comparing for standard problems the results obtained by the approximate approach [15, 18] with the results for the similar problems, obtained within the three-dimensional linearized stability theory of deformable bodies (for example [2–5, 7, 10, 19]), constructed with the accuracy usually adopted in mechanics. The following conclusions are drawn as a result of the analysis.Applied to effects of internal and surface instability, the approximate approach leads to result in disagreement with the corresponding results of the three-dimensional linearized stability theory of deformable bodies.As applied to the study of stability of thin-walled structural elements, the use of the approximate approach is justified if we restrict ourselves to a calculational accuracy of critical loads corresponding to that of the Kirchhoff-Love hypothesis.In connection with the discussion above, numerous publications carried out on the basis of the approximate approach require further study to clarify the validity limits of the results obtained.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 22, No. 2, pp. 3–17, February, 1986.     

Flow past a sphere of two co-axial supersonic streams

We investigate the flow past a sphere of a parallel supersonic stream which is nonuniform in magnitude; such a flow can be considered as two co-axial streams of an ideal gas. The problem is solved numerically by the method of establishment [1]. Supersonic flow of nonuniform magnitude and direction past blunt bodies and a plane wall was investigated in [2–5],Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 89–94, September–October, 1970.The author wishes to thank G. F. Telenin for his attention to the paper.     

Method of determining the aerodynamic coefficients of asymmetric bodies with allowance for nonlinear body shape influence factors

The numerical method proposed makes it possible to determine the aerodynamic coefficients of asymmetric bodies of fairly arbitrary shape (including those with discontinuities of the transverse contour) at small solid angles of attack. The method allows an aerodynamically sound transition from the three-dimensional system of equations of gas dynamics to a two-dimensional system, which considerably simplifies the problem and reduces by an order the machine time required. The method takes into account the nonlinear body shape influence factors, which substantially improves the accuracy of the calculations. The efficiency and accuracy of the method are demonstrated by comparing the results of the calculations with the results of a numerical solution of the three-dimensional problem.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 121–128, March–April, 1992.