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.     

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.     

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.     

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.     

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.     

Lifting wings of optimum shape in a viscous hypersonic stream

The variational problem of the shape of a low-aspect-ratio wing with maximum lift-to-drag ratio in a viscous hypersonic stream is formulated with allowance for the flow structure in the thin compressed layer and the state of the boundary layer, and a numerical-analytic solution of the problem is given. The characteristic shapes of optimum wings are obtained together with the corresponding pressure distributions. The bifurcation of the optimum regime with variation of the wing span is found to exist. It is shown that viscosity, when included in the optimization procedure, can result in a change in the optimized wing shape and reduce the maximum lift-to-drag ratio; however, the gain in lift-to-drag ratio, as compared with the limiting Newtonian value, is still quite appreciable.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 154–164, November–December, 1995.     

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.     

Forces exerted by a weak shock wave on a wing of complicated shape in plan at supersonic velocities

A method is presented for calculating the supersonic distributed and total aerodynamic characteristics of a wing of complicated shape in plan, including a wing having the shape of an aircraft in plan which encounters a weak shock wave of arbitrary orientation. The problem is solved in the linear formulation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 121–127, July–August, 1982.     

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.     

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.     

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.     

On the propulsive efficiency of vibrating bodies in a subsonic gas flow

In the cited work [1], a variational problem of the minimum power required to obtain a given thrust for a wing with a surface able to vibrate in a supersonic flow is considered. In the present work the theoretical maximum efficiency in a subsonic flow of bodies whose shape can be simulated by unsteady sources is found. For axisymmetric bodies a shape is found giving an efficiency as close as required to the theoretical limit. The results obtained make it possible to evaluate possible improvements not only of artificial devices but also of living creatures, since they move at subsonic speeds.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 128–132, July–August, 1984.     

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.     

Supercritical regimes of hypersonic flow past a planar yawed delta wing

Hypersonic viscous perfect gas flow past a planar delta wing in the viscous-inviscid interaction regime is considered. The effect of the yaw angle on the parameters of the laminar boundary layer on the cold wing and the formation of subcritical and supercritical flow regions is studied.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 151–158, July–August, 1995.     

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.     

Dynamics of a nonuniformly conducting gas flow in a magnetic field

The dynamics of a nonuniformly conducting gas flow in the channel of a MHD generator are investigated on the basis of numerical modeling. The initial shape of the plasmoids periodically entering the MHD channel qualitatively correspond to that noted in [6, 7]. The alkali metal seed is uniformly distributed over the entire flow. The mechanism of dynamic interaction of the plasmoids and the low-conductivity gas flowing over them, the variation of the shape of the plasmoids and the evolution of the gas dynamic structure of the flow are studied.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 135–145, September–October, 1989.     

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.     

Method of calculating subsonic gas flow with separation past wings

The steady nonlinear problem of subsonic compressible gas flow past a wing of arbitrary shape in plan is considered. A numerical method was devized for solving the problem; this is a further development of the method of discrete vortices. The surface of the body and the vortex wake behind it are simulated by systems of discrete vortex sections, but, in contrast to the case of an incompressible medium, it is necessary in this case for the sources to be distributed outside the wing. The circulations of the attached vortices, the strengths of the sources, and the shape of the wake are determined by iterations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 140–147, July–August, 1984.     

Design of airfoil with flap simulated by a point vortex

The method of quasisolutions of inverse boundary-value problems (see [4]) is used to solve the problem of designing an airfoil with a flap, replaced by a fixed vortex, from given velocity distribution along the contour of the wing main part. Profiles are constructed and the effect of the flap (vortex) on the shape and aerodynamic properties of the mechanized wing is examined.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 3–9, January–February, 1992.     

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.