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.     

Stationary flow past the lower surface of a piecewise planar delta wing with supersonic leading edges

The considered wing has any finite number of inflections in its plane with lines of inflection intersecting at the point of inflection of the leading edge. In the present paper, this generalizes the author's earlier work [1] on flow past the undersurface of a flat wing at unite angle of attack with finite angle of slip and supersonic leading edges. In [1], calculations were not given. The special case of flow without slip in the same situation was considered later in [2], However, this paper contains errors, indicated at the end of the present paper. The calculations given in [2] are not correct. In the quoted papers, the gas flow is assumed to be a perturbation of a homogeneous flow behind a plane oblique shock wave. Such flows are treated systematically in [3]. Here and in [1], we use and generalize the representation of the linearized conservation laws across the shock front as the conditions of a boundary-value problem for an analytic function of a complex variable as obtained in [4, 5]. Calculations are given of the pressure distribution over the span for a number of different flow regimes and the pressure coefficients in the middle of the wing are compared with a numerical solution presented partly in [6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 80–90, September–October, 1979.I am very grateful to V. I. Lapygin for making available a large number of variants of his numerical solution, and to L. E. Pekurovskii for assistance in the calculations.     

Passage of a supersonic flow over intersecting surfaces

Using the linear formulation, the problem of passage of a supersonic flow over slightly curved intersecting surfaces whose tangent planes form small dihedral angles with the incident flow velocity at every point is considered. Conditions on the surfaces are referred to planes parallel to the incident flow forming angles 0≤γ≤2π at their intersection [1]. The problem reduces to finding the solution of the wave equation for the velocity potential with boundary conditions set on the surfaces flowed over and the leading characteristic surface. The Volterra method is used to find the solution [2]. This method has been applied to the problem of flow over a nonplanar wing [3] and flow around intersecting nonplanar wings forming an angle γ=π/n (n=1, 2, 3, ...) with consideration of the end effect on the wings forming the angle [4]. In [5] the end effect was considered for nonplanar wings with dihedral angle γ=m/nπ. In the general case of an arbitrary angle 0≤γ≤2π the problem of finding the velocity potential reduces to solution of Volterra type integrodifferential equations whose integrands contain singularities [1]. It was shown in [6] that the integrodifferential equations may be solved by the method of successive approximation, and approximate solutions were found differing slightly from the exact solution over the entire range of interaction with the surface and coinciding with the exact solution on the characteristic lines (the boundary of the interaction region, the edge of the dihedral angle). The solution of the problem of flow over intersecting plane wings (the conic case) for an arbitrary angle γ was obtained in terms of elementary functions in [7], which also considered the effect of boundary conditions set on a portion of the leading wave diffraction. In [8, 9] the nonstationary problem of wave diffraction at a plane angle π≤γ≤2π was considered. On the basis of the wave equation solution found in [8], this present study will derive a solution which permits solving the problem of supersonic flow over nonplanar wings forming an arbitrary angle π≤γ≤2π in quadratures. The solutions for flow over nonplanar intersecting surfaces for the cases 0≤γ≤π [6] and π≤γ≤2π, found in the present study, permit calculation of gasdynamic parameters near a wing with a prismatic appendage (fuselage or air intake). The study presents a method for construction of solutions in various zones of wing-air intake interaction.     

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.     

Velocity field generated by wing vibrations propagating along the surface

The velocity field generated by wing vibrations propagating along an elastic wing surface with finite velocity is studied.The gasdynamic problem is reduced to a mixed boundary-value problem with a moving boundary for the three-dimensional wave equation. The solution is obtained in closed form when the wing travels at supersonic velocity following an arbitrary law, the vibration propagation front is an arbitrary curve displacing along the wing surface, and the wing edges are supersonic.     

A variational problem of one-dimensional nonequilibrium gasdynamics

In [1] the problem of optimal profiling of the supersonic portion of a plane or axisymmetric nozzle for nonequilibrium flow is reduced to a boundary-value problem for a hyperbolic system of equations which includes the flow equations and the equations for the Lagrange multipliers. In view of the complexity of the solution of that system, the present paper presents an analogous study based on the one-dimensional formulation. The solution is illustrated by examples. It is noted that a similar solution undertaken in [2] is in error.     

Velocity field near a wing—Fuselage combination at an angle of attack in a supersonic stream

To investigate interference between the wing and fuselage at supersonic flight velocities, one can, besides numerical methods based on the exact equations of motion, make effective use of the theory of small perturbations [1]. This is the direction adopted, in particular, in [2–4], in which the problem is solved in the framework of linear theory. In [5], the results obtained in the first approximations are corrected by taking into account the following term in the expansion of the potential function in a series in a small parameter. The present paper considers the velocity field near an arbitrarily profiled wing with supersonic edges and the features due to the presence of the fuselage. A general expression is found for the singular term of the asymptotic expansion of the solution of the linear equation in the neighborhood of the Mach cone with apex at the point of intersection of the leading edge of the wing with the surface of the fuselage. A uniformly exact solution for the linear differential equation for the additional velocity potential is constructed. The position and intensity of the shock wave on the upper surface of the wing are determined. Analytic dependences and quantitiative estimates are obtained for the local downwashes below the wing in the region of the flow where the linear theory leads to the largest errors. The obtained results are important for the correct determination of the aerodynamic characteristics of aircraft in the three-dimensional velocity field produced by the wing-fuselage combination.Translated from Izvestlya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 136–148, November–December, 1980.I am grateful to M. F. Pritulo for discussing the results of the work.     

Direct and inverse problems of flow over a wing of finite span in the linear formulation

A correspondence between the solutions of the direct and the inverse problem for wing theory is established for a wing of finite span in the framework of linear theory on the basis of solution of a wave equation in Volterra form for supersonic flow and solution of the Laplace equation in the form of Green's formula for subsonic flow. For the direct problem in the case of supersonic flow an expression is derived for finding the load on the wing with maximal allowance for the wing geometry. In the inverse problem for supersonic and subsonic flows, expressions are derived for finding the wing geometry from given values of the load on the wing and the variation of the load along the span of the wing. The solution of the inverse problem is presented in the form of integrals that converge for interior points of the wing surface in the sense of the Cauchy principal value, the wing surface being represented as a vortex surface of mutually orthogonal vortex lines. The conditions of finiteness of the velocities on the edges are discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 114–125, September–October, 1979.     

Thin flat bodies of minimum wave drag in nonequilibrium supersonic flow

In [1] the problem of optimal profiling of the contours of plane and axisymmetric bodies in supersonic nonequilibrium flow without the formation of a shock wave (these bodies include, in particular, the contours of base sections and nozzles) is reduced to the boundary value problem for a hyperbolic system of equations, which includes the flow equations and the equations for the Lagrange multipliers (there is an error in Eq. (4.5) of [1]; there should be a minus sign in front of the third term in the braces). In view of the solution complexity, in [2] the construction of the optimum nozzle contour is based on the one-dimensional approximation. Although this approach does permit establishing the order of the possible gain, the conclusions concerning the contour shape which result from this approach are basically qualitative. In the following the construction of thin plane bodies of minimal wave drag in a nonequilibrium supersonic flow is carried out in the linear approximation, which leads to a more complete picture of the form of the optimum contours. Numerous examples of the use of linear theory for optimizing body shape in supersonic perfect gas flow are given in [3].The authors wish to thank L. E. Sternin for continued support.     

Plane nonstationary gas flow with a strong discontinuity

The problem of plane, nonstationary gas motion under the effect of a piston in the shape of a dihedral angle moving at constant velocity in the gas is considered. In contrast to one-dimensional motion under the effect of a flat piston, a curvilinear shockwave originates here, and the flow becomes nonisentropic and vortical. This problem is examined herein in a linear formulation when the angle of the piston breakpoint is assumed small. The linear problem reduces to an inhomogeneous Riemann—Hilbert problem whose solution is found explicitly. The problem under consideration adjoins a circle of problems associated with shockwave diffraction and reflection studied by Lighthill [1], Smyrl [2], Ter-Minassiants [3], etc.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 45–50, May–June, 1971.The author is grateful to L. V. Ovsyannikov for interest in the research and useful comments.     

Reflection of disturbances caused by a weak energy-release source from a flat shock wave

The process of reflection of linear disturbances from a plane shock wave is considered in the case when these disturbances are caused by a weak energy-release source in a uniform supersonic flow of an inviscid non-heat-conducting gas. It is shown that, in the basic range of constitutive parameters, this interaction proceeds in such a way that the quantity characterizing the disturbance which provides a force load on the lateral surface of a body substantially changes when the reflection from the shock wave occurs.     

Supersonic Flow Around a V-Shaped Wing at Incidence and Yaw

A solution of the problem of the flow around a V-wing with supersonic leading edges at low angles of attack and yaw is obtained within the framework of the linear theory. Possible patterns of nonsymmetric flow around the wing are analyzed as functions of the wing geometry and the freestream velocity direction, and the ranges of angles of attack and yaw on which these patterns are realized are established. Some previously undescribed shock wave configurations are found to exist in the wing-induced conical flows.     

Numerical solution of the problem of the flow of a supersonic gas stream over the upper surface of a delta wing in the expansion region

A numerical method is described for the calculation of supersonic flow over the arbitrary upper surface of a delta wing in the expansion region. The shock wave must be attached everywhere to the leading edge of this wing from the side of the lower surface. The stream flowing over the wing is assumed to be nonviscous. A problem with initial conditions at some plane and with boundary conditions at the wing surface and the characteristic surface is set up for the nonlinear system of equations of gas dynamics. The difference system of equations, which approximates the original system of differential equations on a grid, has a second order of accuracy and is solved by the iteration system proposed in [1]. The initial conditions are determined by the method of establishment of self-similar flow. A number of examples are considered. Comparison is made with the solutions of other authors and with experiment.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 76–81, November–December, 1973.The author thanks A. S. II'ina who conducted the calculations and V. S. Tatarenchik for advice.     

Theory of diffraction of weak shock waves

A numerical solution is considered to the universal nonlinear boundary-value diffraction problem which occurs in various problems of weak interaction [1, 2] in the asymptotic analysis of the flow in a region with large gradients of the parameters near the point of intersection of the incident, diffracted, and reflected waves. The analytical solutions to this type of problem usually approximately satisfy the conditions on the diffracted front, the position of which is not known beforehand, but is found along with the solution. In the present paper, the problem is solved by the numerical method of [3], which reduces the initial boundary-value problem for the system of short-wave equations with an unknown boundary to the solution of a series of boundary-value problems with a fixed boundary. The problem of the diffraction of a weak shock wave on a wedge with a finite apex angle is considered as an application of the solution. The data calculated by the asymptotic theory agree significantly better with the experimental data [5] than the theoretical data of [4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6. pp. 176–178, November–December, 1984.     

Thin weakly curved wings with maximum aerodynamic quality

Lifting wings that only slightly disturb the supersonic gas flow are considered. The plan shape and thickness distribution of the wing and the free-stream parameters are given. The flow problem is solved within the framework of the Prandtl model. The outer potential flow is determined in accordance with the linear theory. The turbulent boundary layer is found by the method of plane sections with allowance for the three-dimensional inviscid flow pattern. A numerical model of the flow is constructed in the class of piecewise-constant functions on characteristic calculation grids [1]. The variational problem of finding the weakly curved middle surface of the wing giving maximum aerodynamic quality is reduced, by analogy with [2], to a problem of nonlinear programming and is solved by the gradient projection method.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 165–168, July–August, 1991.     

Combined method of calculating supersonic ideal-gas flow over a wing with a supersonic blunt leading edge

A combined numerical method, based on the successive calculation of the flow regions near the blunt leading edge and center of a wing, is proposed on the assumption that the angle of attack and the relative thickness and bluntness radius of the leading edge are small. The flow in the neighborhood of the leading edge of the wing is assumed to be identical to that on the windward surface of a slender body coinciding in shape with the surface of the blunt nose of the wing and is numerically determined in accordance with [1]. The flow parameters near the center of the wing are calculated within the framework of the law of plane sections [2]. In both regions the equations of motion of the gas are integrated by the Godunov method. The flow fields around elliptic cones are obtained within the framework of the combined method and the method of [3], A comparative analysis of the results of the calculations makes it possible to estimate the region of applicability of the method proposed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 159–164, January–February, 1989.The authors wish to express their gratitude to A. A. Gladkov for discussing their work, and to G. P. Voskresenskii, O. V. Ivanov, and V. A. Stebunov for making available a program for calculating supersonic flow over a wing with a detached shock.     

Numerical simulation of shock wave intersections

A large number of papers, generalized and classified in [1, 2], have been devoted to unsteady gas flows arising in shock wave interaction. Experimental results [3–5] and theoretical analysis [6–9] indicate that the most interesting and least studied types of interaction arise in cases when there are several shock waves. At the same time, nonlinear effects, which depend largely on the nature of the shock wave intersections, become appreciable. Regions of existence of different types, of plane shock wave intersections have been analyzed in [10–13]. It has been shown that in a number of cases the simultaneous existence of different types of intersections is possible. The aim of the present paper is to study unsteady shock wave intersections in the framework of a numerical solution of the axisymmetric boundary-value problem that arises in the diffraction of a plane shock wave on a cone in a supersonic gas flow. Flow regimes that augment the experimental data of [3–5] and the theoretical analysis of [9] are considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 134–140, September–October, 1986.     

Study of flow in the vicinity of V-shaped wings formed by the stream surfaces behind a plane shock

In calculating the flow about bodies with plane surfaces and sharp edges it is assumed that in the flow regimes with attached shock the latter may be defined in a section normal to the edge from the corresponding relations for the wedge [1, 2], The solution is taken corresponding to a weak shock on a wedge with supersonic velocity behind it. While in the plane case (wedge) this solution will be the only physically realizable solution, in the case of three-dimensional bodies, when there is a slip velocity along the leading edge, the realization of a second wedge solution with a strong shock is conceivable in the section normal to the leading edge if the total velocity behind the shock (with account for the slip velocity along the edge) is supersonic [3].Relative to the undisturbed stream velocity both of these solutions correspond to a weak shock. We present an example when the solution with a strong shock in the section normal to the edge is possible.     

Interaction between an external compression shock and a blunt body in a hypersonic stream

A complex shock configuration with two triple points can occur during the interaction between an external oblique compression shock and the detached shock ahead of a blunt body (for instance, ahead of a wing or stabilizer edge). This results in the formation of a high-pressure, low-entropy supersonic gas jet [1–6]. Here two flow modes are possible [1], which differ substantially in the intensity of the thermal and dynamic effects of the stream on the blunt body: mode I corresponds to the impact of a supersonic jet [2–6], while the supersonic jet in mode II does not reach the body surface in the domain of shock interaction because of curvature under the effect of a pressure drop. Conditions for the realization of the above-mentioned flow modes are investigated experimentally and theoretically, and an approximate method is proposed to determine the magnitude of the compression shock standoff in the interaction domain. Blunt bodies with plane and cylindrical leading edges are examined. The results of a computation agree satisfactorily with experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 97–103, January–February, 1976.The author is grateful to V. V. Lunev for discussing the research and for useful remarks.     

Supersonic motion of bodies in a gas with shock waves

The authors consider the problem of supersonic unsteady flow of an inviscid stream containing shock waves round blunt shaped bodies. Various approaches are possible for solving this problem. The parameters in the shock layer on the axis of symmetry have been determined in [1, 2] by using one-dimensional theory. The authors of [3, 4] studied shock wave diffraction on a moving end plane and wedge, respectively, by the through calculation method. This method for studying flow around a wedge with attached shock was also used in [5]. But that study, unlike [4], used self-similar variables, and so was able to obtain a clearer picture of the interaction. The present study gives results of research into the diffraction of a plane shock wave on a body in supersonic motion with the separation of a bow shock. The solution to the problem was based on the grid characteristic method [6], which has been used successfully to solve steady and unsteady problems [7–10]. However a modification of the method was developed in order to improve the calculation of flows with internal discontinuities; this consisted of adopting the velocity of sound and entropy in place of enthalpy and pressure as the unknown thermodynamic parameters. Numerical calculations have shown how effective this procedure is in solving the present problem. The results are given for flow round bodies with spherical and flat (end plane) ends for various different values of the velocities of the bodies and the shock waves intersected by them. The collision and overtaking interactions are considered, and there is a comparison with the experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 141–147, September–October, 1984.