Asymptotic theory of a wing moving near a solid wall

In this study we use the method of matched asymptotic expansions to obtain an approximate solution of the problem of the nonstationary motion of a lifting surface near a solid wall. The region of flow is provisionally subdivided into characteristic zones, in which, using the appropriate coordinates, we construct asymptotic expansions for the velocity potential, which thereafter coalesce in the regions of common validity. In the first approximation (extremely small heights of flight) the problem reduces to the solution of a Poisson equation in a plane region bounded by the contour of the wing in the horizontal plane with boundary conditions established from the coalescence.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 115–124, November–December, 1977.     

Development of a two-dimensional turbulent jet in an infinite space

The second and third terms in the asymptotic expansion of the stream function in the nonsimilar problem of the development of a two-dimensional turbulent jet in an unbounded space are found in final form. Results of experimental investigations of free turbulent jets are cited, and the effect of the initial velocity profile on the aerodynamic characteristics of the jet is considered. The problem of the development of a two-dimensional turbulent jet in an unbounded space has been considered in [1–3]. The existing solution is similar, and is valid only at a sufficiently large distance from the slit. Allowance for the finite dimensions of the slit leads to a nonsimilar problem. The papers [4–6] are devoted to the experimental investigation of the free two-dimensional turbulent jet.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 137–142, July–August, 1971.     

On the possibility of enhancing the efficiency of tangential blowing of a slit jet from an airfoil surface

Experimental results are presented for a teardrop-shaped wing section with tangential slit jet blowing in the forward part of the section. It is shown that such airfoils can provide a lift significantly greater than the lift which can be achieved by blowing a tangential slit jet from a wing of ordinary shape with a flap.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 182–186, July–August, 1996.     

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.     

Asymptotic nonlinear solution of the problem of the penetration of a compressible fluid by a slender body

A combined solution of the problem of the penetration of a compressible fluid by a slender wedge and a cone is found by the method of matched asymptotic expansions. The new solution is based on taking into account the nonlinear terms in the Cauchy-Lagrange integral and is uniformly applicable in the neighborhood of the nose.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 49–57, May–June, 1993.     

Asymptotic method for solving problems with radial laminar jets of immiscible liquids

Two problems involving radial laminar jets of immiscible liquids are considered: a free radial slit jet and a jet on a rotating disk. An asymptotic method of solution is proposed that makes it possible to determine the flow parameters far from the source. The difference between these flows and those of homogeneous liquids is demonstrated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 192–28, May–June, 1980.     

Asymptotic solution to the problem of viscous flow in the neighborhood of the axis of a vortex sheet

An asymptotic solution is constructed to the problem of the flow of a viscous incompressible fluid in the neighborhood of the axis of a vortex sheet generated by flow separation from sharp edges of a delta wing of small aspect ratio at large values of the Reynolds number and small angles of attack.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 57–65, January–February, 1984.     

Convective flow with uniform vorticity in a vertical layer

The free convective circulation of liquid in plane vertical slits of circular and square cross section with a longitudinal horizontal temperature gradient at the boundaries was investigated experimentally. It was found that under such heating conditions there is a uniform-vorticity flow with a region of quasirigid rotation, which has the shape of a disk in a circular slit and the shape of a cross in a square slit; in each longitudinal section of this zone the liquid moves along concentric trajectories with constant angular velocity. Dimensionless numbers for the problem were established by tests with various liquids and cavities of different dimensions. In dimensionless form, the angular velocity of the vortex and the temperature gradient in it depend linearly on the temperature difference at the boundaries of the layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 160–165, May–June, 1984.     

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.     

Contact Problem for a Narrow Annular Punch. Unknown Region of Contact

The nonlinear problem of determining the contact stresses and the contact zone under the base of a narrow annular punch is studied. An asymptotic model of one–sided contact along the line is constructed by the method of matched asymptotic expansions. Explicit asymptotic formulas for the line–pressure density are obtained. The asymptotic representation of the contact arc is given.     

Optimization of suction-blowing in a viscous fluid

The problem of drag minimization in a viscous fluid by means of controlled suction (blowing) is considered. In the low Reynolds number approximation matched asymptotic expansions are used to construct the second approximation and analytic solutions of the optimization problem are found for a sphere and a circular cylinder. Transition from unseparated to separated flow is accompanied by a qualitative restructuring of the optimal solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 27–32, May–June, 1989.     

Stress state of an elastic plane weakened by an infinite series of longitudinal-transverse cracks

Use of the fact that a singular operator transforms a polynomial again into a polynomial permitted obtaining substantially new results in [1], devoted to wing theory. This property of singular operators is used to solve the plane problem of elasticity theory for a plane weakened by cracks. The criterion for the beginning of crack growth is related in the linear theory of fracture to the stress-intensity factor at its end. An investigation of the influence of the mutual arrangement of cracks on the intensity factor is of considerable interest. The intensity factor is zero in the stretching of a plane weakened by a longitudinal slit, but this factor grows in the presence of a transverse slit and may even exceed the intensity factor at the end of the transverse slit. In this case stratification of the material, the development of cracks located along the loading line, starts. Fractures of this kind have been observed in experiments. To solve the problem of determining the stress-intensity factor at the end of a longitudinal crack in the presence of a transverse crack, the consideration of a periodic system of longitudinal-transverse cracks turns out to be effective. Introduction of symmetry simplifies the construction of the solution of the problem, on the one hand, and is a good approximation to the problem of the mutual influence of two cracks for a sufficient mutual removal of the slits, on the other.     

Influence of moisture transfer in the aeration region on the thermal regime of soil

One-dimensional heat and moisture transfer in the aeration region of reclaimed land is considered for two limiting cases. In the first, the heat transfer is mainly due to the thermal conductivity of the soil; in the second, to the motion of moisture. The influence of the water-table depth on the thermal regime is investigated. A solution to the problem of unsteady heat transfer in soil is found analytically by the method of matched asymptotic expansions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 63–71, September–October, 1981.We thank V. S. Berman for discussions and valuable comments.     

Asymptotic form of the solution to the problem of multicomponent mixture flow through a porous medium with a boundary layer

In problems of two-phase mixture flow through a porous medium in a subterranean stratum a boundary layer phenomenon arises caused by the fact that relative phase motion exists in the system, and so having no analogy with the single-phase case. The physical nature of boundary layer phenomena is explained, and an asymptotic solution is constructed for the self-similar problem with an arbitrary number of components in the system, by using the method of matched asymptotic forms. The conditions are established for the motions of a multicomponent and a binary mixture to be equivalent, and a study is made of the role of convective factors in the formation of averaged working indices for the stratum.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 94–100, July–August, 1985.     

Nonlinear analysis of the flow initiated by the sudden motion of a wedge

Certain self-similar problems involving the sudden motion of a wedge which were treated in the linear approximation in [1–3] are studied by the method of matched asymptotic expansions. The nature of the wave boundary of the perturbed region is determined. Second-approximation solutions are constructed which describe flows behind weak shock fronts propagating in a stationary gas and behind fronts of weak discontinuity lines propagating by known uniform flows. A boundary-value problem is formulated whose solution describes, in first approximation, flows in the neighborhoods of points of interaction of the fronts. The existence of similarity rules of flows in these nieghborhoods is estimated. An approximate solution of the problems is given.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 37–47, May–June, 1976.     

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.     

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.     

Supersonic flow over a slender wing with subsonic edges

The method of integral equations is generalized to calculate steady flow past wings with an arbitrary shape in plan with subsonic leading and trailing edges. The determination of the velocity potential in the leading part of the wing, where there is no influence of the vortex sheet, is reduced to the solution of a two-dimensional integral equation of the second kind. The trailing part, which is subject to the influence of the vortex sheet, is divided into a number of subregions, in which the calculation of the acceleration potential reduces to the solution of one-dimensional equations of the type of Fredholm equations of the second kind and to quadrature. The unique solvability of the obtained integral equations is investigated; it is shown that they can be solved by successive approximation. As an example, the solution to the problem of flow past a flat delta-shaped wing is found and compared with the exact solution to the problem found by the method of conic flows [4, 6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 119–127, September–October, 1981.I thank G. Yu. Stepanov for discussing the paper.     

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.     

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.