Slender bodies of revolution with minimum wave drag in nonequilibrium supersonic flow

We examine the problem of finding the generatrix shape of a body of revolution which travels at supersonic speed and has minimum wave drag. We assume that any number of nonequilibrium processes can take place in the flow. The pressure distribution over the body surface is taken in the linear approximation [1, 2]. A survey of studies using linear theory to find bodies of revolution of optimal form in supersonic perfect gas flow can be found in [3]. The solution of the problem of finding the form of two-dimensional slender bodies of minimum wave drag in nonequilibrium supersonic flow was obtained in [4]. In the following we examine the optimization of only those bodies of revolution for which the leading point lies on the axis of symmetry.The author wishes to thank A. N. Kraiko for his helpful comments.     

Stress singularities in an anisotropic body of revolution

To fill the gap in the literature on the application of three-dimensional elasticity theory to geometrically induced stress singularities, this work develops asymptotic solutions for Williams-type stress singularities in bodies of revolution that are made of rectilinearly anisotropic materials. The Cartesian coordinate system used to describe the material properties differs from the coordinate system used to describe the geometry of a body of revolution, so the problems under consideration are very complicated. The eigenfunction expansion approach is combined with a power series solution technique to find the asymptotic solutions by directly solving the three-dimensional equilibrium equations in terms of the displacement components. The correctness of the proposed solution is verified by convergence studies and by comparisons with results obtained using closed-form characteristic equations for an isotropic body of revolution and using the commercial finite element program ABAQUS for orthotropic bodies of revolution. Thereafter, the solution is employed to comprehensively examine the singularities of bodies of revolution with different geometries, made of a single material or bi-materials, under different boundary conditions.     

Stabilization law for transonic flows around bodies of revolution

A mathematical foundation is presented for the law of gas-parameter stabilization during near-sonic flow of a stream around bodies, which is known in experimental aerodynamics. The quantitative formulation of this law relies on a simple asymptotic analysis of the solutions of the Karman equations. The numerical computation performed for the velocity field around a body of revolution whose meridian section is a Chaplygin profile confirmed the deductions of asymptotic theory to high accuracy.     

Conical wing of maximum lift-drag ratio in a supersonic gas flow

The calculation of supersonic flow past three-dimensional bodies and wings presents an extremely complicated problem, whose solution is made still more difficult in the case of a search for optimum aerodynamic shapes. These difficulties made it necessary to simplify the variational problems and to use the simplest dependences, such as, for example, the Newton formula [1–3]. But even in such a formulation it is only possible to obtain an analytic solution if there are stringent constraints on the thickness of the body, and this reduces the three-dimensional problem for the shape of a wing to a two-dimensional problem for the shape of a longitudinal profile. The use of more complicated flow models requires the restriction of the class of considered configurations. In particular, paper [4] shows that at hypersonic flight velocities a wing whose windward surface is concave can have the maximum lift-drag ratio. The problem of a V-shaped wing of maximum lift-drag ratio is also of interest in the supersonic velocity range, where the results of the linear theory of [5] or the approximate dependences of the type of [6] can be used.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 128–133, May–June, 1986.We note in conclusion that this analysis is valid for those flow regimes for which there are no internal shock waves in the shock layer near the windward side of the wing.     

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.     

Problems of incompressible flow past thin blades and correction of their shape

Solutions to the problem of the flow of an ideal fluid past a blade or cascade of blades with low blocking factor are found in the framework of the first approximation of the theory of perturbations of the flow past infinitely thin arcs. Problems of correction of the shape of the blades are also considered. Problems associated with the application of perturbation theory in problems of flow past bodies are discussed in Van Dyke's monograph [1]. The present paper includes an example of realization of this theory for the thin curved blades that are widely used in compressor construction. Searches for effective methods for calculating the shape of blades to ensure necessary gas-dynamic properties, for example, a given distribution on the blades of the velocity of separationless flow, led to the appearance of algorithms based on the theory of small perturbations for a thin wing of finite span [2] and a single airfoil in a gas flow [3]. In such an approach, the problem of constructing the required profile can be formulated as a sequence of corrections of the boundary of the flow region with respect to small variations of the boundary values of the flow velocity. The paper contains a general formulation of the linear problem of the correction of the flow boundary, an algorithm for its solution in the case of thin blades in an incompressible flow, and analysis of the obtained solutions. Examples of calculations are presented.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 130–137, January–February, 1992.We thank S.A. Smirnov for assistance in the calculations.     

Asymptotic solution of the two-dimensional equations for a thin viscous rarefied shock layer on a cold surface

Hypersonic rarefied flow past blunt bodies is studied in the continuum-free-molecular transition regime. On the basis of an asymptotic analysis three rarefied gas flow patterns are established depending on the relation between the relevant parameters of the problem. In the first regime corresponding to a cold surface asymptotic solutions of the equations of a thin viscous shock layer are derived at low Reynolds numbers in the axisymmetric and plane cases. Simple analytical expressions for the pressure and the heat transfer and friction coefficients are obtained as functions of the freestream parameters and the body geometry. With decrease in the Reynolds number the coefficients approach the values corresponding to free-molecular flow. In this regime a similarity parameter for the hypersonic rarefied flow past bodies is determined. The asymptotic solutions are compared with numerical solutions and the results of direct statistical simulation by the Monte Carlo method.     

Base pressure of bodies of revolution when gas is blown through their surface into a supersonic flow

One of the most important problems of high-velocity aerodynamics is the investigation of the base pressure and the wake flow behind bodies of revolution under the conditions of surface mass transfer. The present paper reports careful experimental investigations of the flow past bodies of revolution in the presence on their surface of a transverse mass flux, especially at a low blowing intensity. These investigations have yielded new data on the dependence of the base pressure on the main determining parameters, extending and making more accurate the modern picture of the mechanism by which surface mass transfer influences the base pressure and its connection with the state of the boundary layer.     

On turbulent separated flow past a truncated body of revolution

Results of testing a series of truncated bodies of revolution with convergent afterbodies in a hydrodynamic tunnel are presented. It is shown that the base pressure can be substantially raised and hence the total drag reduced by varying the shape and convergence of the afterbodies. This effect is caused by intense reverse jets formed as a result of the collision of flow particles moving toward the axis of symmetry.The turbulent flow past the bodies is calculated using the method of viscous-inviscid interaction. The formulas derived for the base pressure and drag coefficients agree satisfactorily with the experimental data.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 50–55, November–December, 1996.     

The pattern and restructuring of supersonic flow past a pair of bodies

The pattern of symmetric and asymmetric supersonic flow past a pair of isolated/connected bodies is analyzed. Semiempirical dependences of the critical parameterl ^*, determining direct and reverse flow restructuring, on the geometry and permeability of the bodies and the Mach and Reynolds numbers are derived using experimental data. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 158–165, January–February, 1994.     

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.     

Hypersonic flow past ducted blunt bodies of revolution

Chernyi [1, 2] has examined the problem of hypersonic flow past a ducted cone with sharp leading edge. In the following we present an analysis of the characteristic features of this problem in the case of a blunt leading edge. We use hypersonic theory for flow past slender bodies with nose blunting of relatively small dimensions [1, 3, 4], based on replacing the nose by a concentrated force and use of the nonsteady analogy. It has been shown in [4, 5] that within the framework of this theory the effect of the violation of the law of plane sections and also the effect of the chemical and physical transformations of the gas in the high-entropy layer is qualitatively equivalent to a change in the drag coefficient of the nose. This approach makes it possible to establish useful similarity laws. The development of these ideas in the direction of the study of the flow structure behind the bow shock wave and analysis of the parameters defining this structure is given in [6–8] in which, in particular, the role of the entropy distribution with respect to the streamlines in the transitional section between the nose and the side surface was clarified and the important practical empirical result was established that this distribution is universal for noses of any form for given flow conditions. In the following these results are extended to blunt bodies of revolution with a duct in the nose. We examine the flow region which is external to the duct under the assumption that the external flow regime corresponds to maximum flow rate through the duct. A characteristic feature of the problem is associated with the additional characteristic linear dimension r₀, which determines the gas mass lost through the duct.     

Certain three-dimensional flows with Mach interaction of shock waves

In [1, 2] it is shown that there is a range of values of the Mach number and geometric parameters for which flow past conical bodies is realized with plane attached shocks and regular intersection of the shocks in space. In the present article we determine the class of solutions corresponding to flow past conical star-shaped bodies with a configuration of the shocks in space of the Mach type.     

Construction of Equivalent Axisymmetric Bodies for Calculating Heat Transfer and Viscous Stress on Three-Dimensional Bodies at Incidence

An effective approximate technique for calculating heat transfer, viscous stress, and species components on the windward side of three-dimensional bodies at incidence in hypersonic flow is developed. Using the similarity method, the solution of the three-dimensional problem is reduced to the solution of an axisymmetric problem. For determining the heat flux on a real body, modified two-dimensional equations are solved for equivalent axisymmetric bodies, specially constructed for meridional planes of the original body. For an arbitrary three-dimensional geometry and angle of attack formulas are derived and a conversion program is developed. These make it possible to determine all the parameters of the equivalent body corresponding to a given meridional plane of the original body; then these parameters are used as input data for calculating the viscous flow past the body. The solutions of the two-dimensional equations for the equivalent bodies are in good agreement with more exact solutions of the three-dimensional equations.     

Flow past a thin axisymmetric body with a given velocity distribution along part of its surface

Longitudinal flow past a thin body of revolution, part of whose surface is not known a priori and is to be determined from the tangential velocity specified there (free-flow boundary), is considered. The flow is assumed to be vortex-free, and the fluid to be ideal and incompressible. An integral equation for the form of the free surface is derived and is solved by the method of successive approximations. Conditions for the existence and uniqueness of the solution are given. A constant velocity flow along the free boundary (cavitation flow) is considered as a particular example of the general theory.     

Drag of Bodies of Revolution in Supersonic Flow with Heat and Mass Addition to the Near Wake

It is shown experimentally that the base drag of bodies of revolution in Mach 1.15 to 3.0 flow can be completely eliminated using special techniques for injecting hydrogen and the products of incomplete combustion of pyrotechnic compounds into the near wake. The experimental data obtained are generalized.     

Far field scattering from bodies of revolution

Summary By use of approximations based on physical reasoning radar cross-section results for bodies of revolution are found. In the Rayleigh region (wavelength large with respect to object dimensions) approximate solutions are found. Examples given include a finite cone, a lens, an elliptic ogive, a spindle and a finite cylinder. In the physical optics region (wavelength very small with respect to all radii of curvature) Kirchhoff theory and also geometric optics can be used. When the body dimensions are only moderately large with respect to the wavelength, Fock or Franz theory can be applied, and examples of the circular and elliptic cylinder are presented. In the region where some dimensions of the body are large with respect to the wavelength and other dimensions are small with respect to the wavelength, special techniques are used. One example, the finite cone, is solved by appropriate use of the wedgelike fields locally at the base. Another example is the use of traveling wave theory for obtaining approximate solutions for the prolate spheroid and the ogive. Other results are obtained for cones the base perimeter of which is of the order of a wavelength by using known results for rings of the same perimeter.This paper with minor revision is as the author presented it at the URSI XIIth General Assembly in Boulder, Colorado August 22–September 5, 1957.     

Bodies of revolution with minimal drag coefficient and low heat transfer at high supersonic speeds

On the basis of modified Newtonian theory and the theory of selfsimilar hypersonic flows we study the form of the optimal contour of a body of revolution with minimal drag coefficient at hypersonic speeds. It is shown that bodies of optimal form also have a small heat transfer coefficient, much smaller than for a conical body. It is established experimentally that the optimal properties of these bodies of revolution are also retained for moderate supersonic flight speeds.In concluslion the author wishes to thank V. V. Sychev for valuable discussions of this problem.     

Solution of a problem of supersonic gas flow past solids of revolution in the theory of small perturbations

A small-parameter method is widely used today for solving many problems of aerodynamics. It has made it possible to obtain results which are interesting from the practical point of view. In particular, it has helped to solve a problem dealing with a substantially three-dimensional field of flow around a wing with finite wingspread past which a gas is flowing at supersonic speed [1]. However, in the problem of flow past solids of revolution, certain difficulties have been encountered in the application of the method. In the present paper, within the framework of the theory of small perturbations, we consider a method for obtaining a solution in which these difficulties can be avoided. We obtain simple analytic expressions for the aerodynamic characteristics of solids of revolution in a supersonic stream of gas. We give a comparison of the results with experimental data and with calculations carried out by the method of characteristics.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 99–106, November–December, 1977.In conclusion, the authors wish to thank V. V. Sychev for his comments on the results of the study.     

A comparative assessment of fluid flow and heat transfer parameters in laminar flow around a circular cylinder and a sphere

Theoretical studies have been carried out for a comparative assessment of hydrodynamic boundary layer thickness, displacement thickness and shear stress at the wall for laminar flow around a circular cylinder and a sphere with the help of the approximate method due to Karman and Pohlhausen for two dimensional flow and the method as applied to bodies of revolution based on the work of F. W. Scholkemeier, respectively. Thermal boundary layer thickness and Nusselt number have been evaluated around the surface of the solids. Comparison is made with available solutions. The graphical presentation of the results depicts a concise and relative assessment of fluid flow and heat-transfer parameters for flow around cylinder and sphere.