Determination of the vorticity on a wing of small aspect ratio in a hypersonic stream

The theory of a thin shock layer [1–3] is used to obtain a formula for calculating the component of the vorticity in the direction of the flow on a wing of small aspect ratio in a hypersonic gas stream. It is shown that for definite shapes of the wing and flow regimes zones may occur with large local values of the vorticity, which, as is well known, have a significant influence on the structure of the flow field.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 175–178, September–October, 1980.     

Hypersonic flow past a wing at large angles of attack with detached shock wave

The thin shock layer method [1–3] has been used to solve the problem of hypersonic flow past the windward surface of a delta wing at large angles of attack, when the shock wave is detached from the leading edge (but attached to the apex of the wing) and the velocity of the gas in the shock layer is of the same order as the speed of sound. A classification of the regimes of flow past a delta wing at large angles of attack has been made. A general solution has been obtained for the problem of three-dimensional hypersonic flow past the wing allowing for nonequilibrium physicochemical processes of thermal radiation of the gas at high temperatures.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 149–157, May–June, 1985.     

On the theory of a wing with small aspect ratio in a hypersonic flow

A study is made of flow over three-dimensional wings of small aspect ratio with shape close to that of a flat delta-shaped wing. The obtained results make it possible to estimate the influence of the plan shape of the leading edge and the curvature of the wing on the pattern of the flow over its windward surface and on the corresponding gas-dynamic functions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 112–117, July–August, 1980.     

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.     

Hypersonic viscous gas flow over a delta wing in the intermediate interaction regime with allowance for wake flow

The results of calculating the hypersonic flow over a plane delta wing of finite length with allowance for wake flow in the intermediate interaction regime are presented. A comparison is made with the data for flow over a delta wing with given pressure at the trailing edge.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 142–149, September–October, 1993.     

Mathematical simulation of unsteady flow past a wing with jets

The various approximate approaches to the investigation of the unsteady aerodynamic characteristics of an airfoil with jet flap [1–3] are applicable only for an airfoil, low jet intensity, and low oscillation frequencies. In the present paper, the method of discrete vortices [4] is generalized to the case of unsteady flow past a wing with jets and arbitrary shape in plan. The problem is solved in the linear formulation; the conditions used are standard: no flow through the wing and jet, finite velocities at the trailing edges where there is no jet, and also a dynamical condition on the jet. The wing and jet are assumed to be thin and the medium inviscid and incompressible.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 139–144, May–June, 1982.     

Visual study of the three-dimensional flow diagram around a delta wing in a subsonic stream

Experimental methods, particularly visualization methods, permit a sufficiently detailed representation of the flow around bodies of complex shape, whose analysis meets with a considerable number of difficulties. The flow around a delta wing in the 1–90-m/sec free-stream velocity range is studied in this paper by using three-dimensional visual methods. Since stream separation and vortex-system formation play the main role in the flow formation over a wing surface, the main purpose of the experiment was to trace the physical process of dynamic development of the flow resulting in separation and vortex formation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 190–194, March–April, 1976.     

Skin effect in two-phase flow over a wing

The author proposes a mathematical model of the skin effect — the flow in the thin film formed on the surface of a wing in a two-phase stream and consisting of the particle component [1–6]. The possible regimes are classified and the influence of the skin effect on the overall aerodynamic characteristics of a wing moving through heavy rain is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 49–55, January–February, 1990.The author is grateful to A. N. Kraiko for discussing the topic and for his valuable comments.     

Strong viscous interaction on triangular and slipping wings

The flow past triangular slipping wings is considered in strong viscous interaction conditions. It is shown that the solution of each of the problems discussed is not unique. The condition is obtained which has to be satisfied by the solution on the axis of symmetry of a triangular wing of infinite dimensions. The class of self-similar flows in the boundary layer of a thin triangular wing is found.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 94–99, November–December, 1970.     

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.     

Calculation of the flow of an ideal fluid past a wing of finite thickness

The problem of irrotational flow past a wing of finite thickness and finite span can be reduced by Green's formula to the solution of a system of Fredholm equations of the second kind on the surface of the wing [1]. The wake vortex sheet is represented by a free vortex surface. Besides panel methods (see, for example, [2]) there are also methods of approximate solution of this problem based on a preliminary discretization of the solution along the span of the wing in which the two-dimensional integral equations are reduced to a system of one-dimensional integral equations [1], for which numerical methods of solution have already been developed [3–6]. At the same time, a discretization is also realized for the wake vortex sheet along the span of the wing. In the present paper, this idea of numerical solution of the problem of irrotational flow past a wing of finite span is realized on the basis of an approximation of the unknown functions which is piecewise linear along the span. The wake vortex sheet is represented by vortex filaments [7] in the nonlinear problem. In the linear problem, the sheet is represented both by vortex filaments and by a vortex surface. Examples are given of an aerodynamic calculation for sweptback wings of finite thickness with a constriction, and the results of the calculation are also compared with experimental results.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 124–131, October–December, 1981.     

Asymptotic theory of flow separation from wings of low aspect ratio

Numerous methods have been developed to calculate the aerodynamic characteristics of wings of low aspect ratio in the case when there is flow separation from the wing edges. Among the methods based on direct solution of the three-dimensional Euler equations there are the method of discrete vortices [1, 2] and the panel method [3]. In addition, numerical and asymptotic methods [4, 5] based on the theory of slender bodies [6] are used. One of the most important shortcomings of this theory is the dependence of the flow pattern at a given section of the wing on only the upstream flow. The obtained solutions therefore contain no information about the influence of the trailing edge of the wing, on which, as is well known, the Chaplygin-Zhukovskii condition is satisfied. The aim of the present paper is to construct an asymptotic theory of higher approximation and a corresponding numerical method for calculating flow separation from wings of low aspect ratio in which this shortcoming is absent.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 141–147, July–August, 1982.     

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.     

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.     

Hypersonic flow of gas over a slender wing of variable shape

In a formulation analogous to [1–3], a study is made of the flow of a uniform homogeneous hypersonic ideal gas over the windward side of a slender wing whose surface profile depends on the time. The problem is solved by the thin shock layer method [4]. The bow shock is assumed to be attached to the leading edge of the wing at at least one point. The corrections of the first approximation to the main Newtonian flow are found. For wings of finite aspect ratio, when the bow shock is attached along the whole of the leading edge of the wing, computational formulas are obtained for determining the parameters of the gas in the shock layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 94–101, July–August, 1979.     

Hypersonic flow past an airfoil with a mach system of shock waves

A method based on the use of the two-approximation theory developed in [1, 2] is proposed for the computation of hypersonic flow past a conical wing with a Mach-type shock configuration.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 104–116, May–June, 1972.     

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.     

Boundary layer transition on the surface of a delta wing in supersonic flow

Laminar-turbulent transition on the surface of a delta wing has been experimentally investigated in a supersonic wind tunnel at Mach numbers M_t8=3–5. It is shown that when M,=3, Re_L=6.5·10⁶, and =–5.5° much of the upper surface of the wing in the neighborhood of the line of symmetry is occupied by a wedge-shaped region of turbulent flow. In this region the heat fluxes reach the same values as at the heat transfer maxima induced here by separated flows and may exceed the turbulent heat flux level on the windward surface of the wing. Changing the shape of the under surface of the wing from plane to pyramidal leads to acceleration of the boundary layer transition on the under surface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 87–92, May–June, 1989.     

Influence of transverse slits on the hydrodynamic coefficients of a wing of finite span in stationary and nonstationary motion near a wall

The method of matched asymptotic expansions is used to construct an approximate solution to the problem of the influence of narrow transverse slits on the hydrodynamic coefficients of a thin rectangular wing moving near a wall. The flow in the neighborhood of a slit is described by a local asymptotic solution satisfying the condition of continuity of the pressure on the leading edge of the slit and matched to the main solution. Results of the calculations illustrate the influence of the slits on the hydrodynamic characteristics of the wing at different Strouhal numbers and aspect ratios.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 122–128, November–December, 1980.     

Discontinuous solutions of an interacting three-dimensional hypersonic boundary layer

The paper is a mathematical study of the three-dimensional flow of viscous gas in a hypersonic boundary layer that develops along a flat wing whose leading edge has a step shape. The flow interacts with a flap on the wing set at a small angle. A linear solution to the problem is constructed under the assumption that the deflection angle of the flap is small and the difference between the length of the plates is of order unity. It is shown that an important part in the formation of the flow near and behind the flap may be played by the change in the pressure along the span of the wing due to the step shape of the leading edge. It is significant that although the pressure and displacement thickness are continuous functions of the transverse coordinate, the longitudinal and transverse components of the friction force have discontinuities.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 19–26, March–April, 1991.I thank V. V. Sychev and A. I. Ruban for suggesting the problem, for valuable advice, and assistance.