Calculation of the flow around wings of arbitrary planform over a wide range of angles of attack

A numerical method is described for the calculation of the distributed and total aerodynamic characteristics of a thin wing of any planform. We use only the generally accepted hypotheses-smoothness of flow around the wing and the Chaplygin-Zhukovskii condition of finite velocity at the trailing edges. The medium is considered ideal and incompressible.The development of a nonlinear theory for the wing of small aspect ratio in a compressible medium is one of the most important and difficult problems of wing theory. It has long attracted the attention of the aerodynamiscists. Chaplygin touched on this question in his 1913 report On the vortex theory of the finite span wing, presented to the Moscow Mathematical Society. Several interesting ideas and schemes were proposed by Golubev (see, for example, [2]). The first adequately correct and effective attempt to determine theoretically the nonlinear variation of wing normal force with angle of attack was that of Bollay [3]. In this work he studied rectangular wings of very small aspect ratio. The circulation variation law along the span was taken to be constant, and along the chord it was taken the same as for a flat plate of infinite span. It was also assumed that the centerlines of the free vortices trailing from the wing tips are straight lines and form the same angle with the plane of the wing. The magnitude of this angle was calculated from the average value of the relative velocity. The boundary condition at the wing was satisfied at a single point.In several later studies [4–8] attempts have been made to extend this approach somewhat. In [7] the circulation variation law along the wing chord is calculated, and the boundary conditions are satisfied more exactly. However, attempts to convert to the study of wings of more complex planform, when the circulation can no longer be considered constant along the span, are hydrodynamically incorrect [5, 6, 8]. In these studies schemes are used in which with smooth flow around the wing the free vortices stand off from the wing surface. The angles which the vortex centerlines form with the wing surface are assumed or are calculated on the basis of very arbitrary hypotheses.In the present paper the vortex layer which simulates the wing surface, just as in the linear theory [9, 10], is replaced by a system of discrete vortices. The free vortices away from the wing then are discrete curvilinear vortex filaments. Each of them is replaced by a series of rectilinear vortex segments. The number of bound and free discrete vortices may be increased without limit. The position of the free vortex segments is determined in the computation process, which is carried out sequentially for a series of angles of attack , beginning with 0 when the linear theory scheme holds. We note that the question of accounting for the effect of the leading-edge free vortex sheet is not considered here, although the method described may also be used to obtain results for this problem.The proposed method turned out to be very general, flexible and convenient for the digital computer. It permits studying the practical convergence of the solution, and also permits obtaining not only the total and distributed characteristics of the wing of arbitrary planform, but also studying such delicate questions as the rollup of the vortex sheet behind the wing.The author wishes to thank O. N. Sokolov and T. M. Muzychenko for the example calculations.     

Form of a free surface during steady flow of a capillary fluid in a rectangular channel

The two-dimensional problem of the form of a free surface of an ideal incompressible fluid during steady flow from a rectangular channel through a thin slot with simultaneous uniform delivery of fluid through the side walls is examined. Forces of gravity and surface tension are taken into account. The nonlinear problem of the simultaneous determination of the free surface and velocity field of the fluid is solved by the iteration method. Convergence of the iterations to the solution of the problem for small values of the parameters is investigated. The solution of the linearized problem is obtained in a closed form for a small depth of the discharge and small width of the channel, which is compared with the solution of the problem in a complete formulation. Graphs of the free surface of the fluid for different values of the parameters, obtained as a result of numerical solution of the nonlinear problem, are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 67–75, January–February, 1977.     

Investigation of diffuser characteristics in an aerodynamic shock tube

An investigation has been made in an aerodynamic shock tube, with M=8, of the diffuser starting with variable Reynolds numbers, and its throttle characteristics have been recorded. The results obtained enable conclusions to be made regarding the possibility of investigating diffusers in such types of tubes. In [1] the possibility of, in principle, determining the throat starting of a diffuser in an aerodynamic shock tube, and the time to establish flow in the diffuser channel, which was 300 sec, was measured. This paper is devoted to the further investigation of diffuser operation with variable Reynolds numbers, and to determining the throttle characteristics, in particular, the total pressure reduction coefficient.Moscow. Translated from Izvestiya Akademii Nauk SSSR, mekhanika Zhidkosti i Gaza, No. 1, pp. 156–161, January–February, 1972.     

Flow around a sphere with material cross flow at low reynolds numbers

An analytical solution is carried out for the problem of the flow around a sphere with material cross flow at Reynolds numbers less than 1 and a blowing velocity less than the free stream velocity. The method of asymptotic expansions of Pearson and Proudman is used for the solution. Expressions are obtained for the distribution of the current and velocity component functions as well as for the aerodynamic drag coefficient of the sphere. It is shown that blowing diminishes the sphere drag, where its influence will increase as the Reynolds number grows.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 103–109, May–June, 1972.     

Effect of proximity of barrier on lifting force produced by vertical solid jets

The effect of proximity to the ground on the lifting force generated by a vertical solid jet is studied in connection with development of vertical takeoff and landing devices and of air cushion devices. Such a study was made in [1 ] for planar flow by an incompressible ideal fluid. There a generalization of the results obtained on a compressible fluid was made by the approximation method. In the present work the planar problem of streamline flow past a dihedral barrier of a gas jet emerging from a channel with parallel walls was solved by the Chaplygin-Fal'kovich method [2, 3], The results of [1, 4–9] follow as a particular case from the solution obtained. Calculations were carried out clarifying the effect of the proximity of a barrier and the lifting effect of a fluid on flow characteristics at subsonic speeds.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 123–131, September–October, 1971.     

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.     

Motion of a viscoplastic liquid in a porous inhomogeneous medium

Equations of motion are derived for a viscoplastic liquid in a nonuniform medium of type 2 (piecewise uniform) or type 3 (with a variable filtration coefficient) [1] on the assumption that the motion is of steady-state type. Solutions are presented for a parallel flow and a flow with axial symmetry.