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.     

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.     

The problem of supersonic flow over the lower surface of a triangular wing

In formulating the problem we make no assumption of smallness of the angle of attack; the attached three-dimensional compression shock which arises under the lower surface of the wing may be of arbitrary intensity, and in form is assumed to differ little from a plane shock; a finite yaw angle is allowed. We consider linear supersonic conical flow which is realized, with the exception of a characteristic linear dimension, in the portion of space bounded by the shock, the plane of the wing, and the surface of a disturbance cone with vertex at the discontinuity of the supersonic leading edge and which is a disturbance of the uniform flow behind the plane shock wave.The problem studied reduces to the homogeneous Hilbert boundary-value problem for an analytic function of a complex variable, whose real and imaginary parts are the partial derivatives of the unknown pressure disturbance with respect to the similarity coordinates.In the solution of the boundary-value problem, the effective method of Lighthill, developed with application to diffraction problems [1, 2], is generalized to the problem of an asymmetric region.The particular case of hypersonic flow about an unyawed triangular wing has been studied by Malmuth [3]; the author obtains the problem considered by Lighthill in [2] and writes out the solution contained in that work.     

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.     

A wave propagation problem for non-uniform screw dislocation motion in a viscoelastic half-plane

The wave propagation problem for a largely arbitrary anti-plane displacement discontinuity imposed along a line perpendicular to the surface of a stress-free linearly viscoelastic half-plane is considered. The general Laplace transform solution is obtained and then inverted for the case of a screw dislocation moving at an arbitrary speed in a Maxwell material. It is shown that the material viscoelasticity alters the coefficient of the dislocation edge stress singularity and damps the surface displacements from the elastic values. The surface damping increases with time, distance from the dislocation path and dislocation speed, whether sub- or supersonic.     

An adjoint method for the calculation of remote sensitivities in supersonic flow

This paper presents an adjoint method for the calculation of remote sensitivities in supersonic flow. The goal is to develop a set of discrete adjoint equations and their corresponding boundary conditions in order to quantify the influence of geometry modifications on the pressure distribution at an arbitrary location within the domain of interest. First, this paper presents the complete formulation and discretization of the discrete adjoint equations. The special treatment of the adjoint boundary condition to obtain remote sensitivities or sensitivities of pressure distributions at points remotely located from the wing surface are discussed. Secondly, we present results that demonstrate the application of the theory to a three-dimensional remote inverse design problem using a low sweep biconvex wing and a highly swept blunt leading edge wing. Lastly, we present results that establish the added benefit of using an objective function that contains the sum of the remote inverse and drag minimization cost functions.     

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.     

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.     

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.     

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.     

Existence and Stability of Multidimensional Transonic Flows through an Infinite Nozzle of Arbitrary Cross-Sections

We establish the existence and stability of multidimensional steady transonic flows with transonic shocks through an infinite nozzle of arbitrary cross-sections, including a slowly varying de Laval nozzle. The transonic flow is governed by the inviscid potential flow equation with supersonic upstream flow at the entrance, uniform subsonic downstream flow at the exit at infinity, and the slip boundary condition on the nozzle boundary. Our results indicate that, if the supersonic upstream flow at the entrance is sufficiently close to a uniform flow, there exists a solution that consists of a C ^1,α subsonic flow in the unbounded downstream region, converging to a uniform velocity state at infinity, and a C ^1,α multidimensional transonic shock separating the subsonic flow from the supersonic upstream flow; the uniform velocity state at the exit at infinity in the downstream direction is uniquely determined by the supersonic upstream flow; and the shock is orthogonal to the nozzle boundary at every point of their intersection. In order to construct such a transonic flow, we reformulate the multidimensional transonic nozzle problem into a free boundary problem for the subsonic phase, in which the equation is elliptic and the free boundary is a transonic shock. The free boundary conditions are determined by the Rankine–Hugoniot conditions along the shock. We further develop a nonlinear iteration approach and employ its advantages to deal with such a free boundary problem in the unbounded domain. We also prove that the transonic flow with a transonic shock is unique and stable with respect to the nozzle boundary and the smooth supersonic upstream flow at the entrance.     

Wing of simple planform in a sharp-edged gust

The entry of a wing into the zone of a sharp-edged gust is considered in the linear formulation. The case is studied when the wing velocity is supersonic and its edges satisfy the supersonic flow condition. The gust intensity is considered to be variable, and its edge may move into the undisturbed medium. Equations in finite form are obtained for the forces and moments for a rectangular wing of infinite span, and also for triangular wings with positive and negative sweep, for the case when the gust intensity varies linearly. Sudden envelopment of the wing and penetration of the wing into a gust whose edge is fixed relative to the undisturbed medium are considered.     

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.     

Inverse problem of wing aerodynamics in a supersonic flow

The inverse problem of wing aerodynamics—the determination of the lifting surface shape from a specified load—is solved within the framework of linear theory. Volterra's solution of the wave equation is used. Solutions are found in the class of bounded functions if certain conditions imposed on the governing parameters of the problem are satisfied. Solutions of inverse problems of supersonic flow are presented for an infinite-span wing, a triangular wing with completely subsonic edges, and a rectangular wing. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 3, pp. 86–91, May–June, 1998.     

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.     

Flutter of an elastic strip in a longitudinal supersonic flow

An elastic strip in a longitudinal supersonic gas flow is considered. An excess pressure formula is obtained on the basis of the linearized theory of supersonic potential flow. It is shown that, in the case of high-speed supersonic flow, the critical flutter velocity is equal to the phase velocity of perturbation waves propagated along the strip. In the framework of the classical piston theory, the flutter problem is solved for the case of a rigidly fixed strip in a longitudinal flow.     

Three-dimensional laminar boundary layer on a permeable surface in the neighborhood of a symmetry plane

Analytical and numerical methods are used to investigate a three-dimensional laminar boundary layer near symmetry planes of blunt bodies in supersonic gas flows. In the first approximation of an integral method of successive approximation an analytic solution to the problem is obtained that is valid for an impermeable surface, for small values of the blowing parameter, and arbitrary values of the suction parameter. An asymptotic solution is obtained for large values of the blowing or suction parameters in the case when the velocity vector of the blown gas makes an acute angle with the velocity vector of the external flow on the surface of the body. Some results are given of the numerical solution of the problem for bodies of different shapes and a wide range of angles of attack and blowing and suction parameters. The analytic and numerical solutions are compared and the region of applicability of the analytic expressions is estimated. On the basis of the solutions obtained in the present work and that of other authors, a formula is proposed for calculating the heat fluxes to a perfectly catalytic surface at a symmetry plane of blunt bodies in a supersonic flow of dissociated and ionized air at different angles of attack. Flow near symmetry planes on an impermeable surface or for weak blowing was considered earlier in the framework of the theory of a laminar boundary layer in [1–4]. An asymptotic solution to the equations of a three-dimensional boundary layer in the case of strong normal blowing or suction is given in [5, 6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 37–48, September–October, 1980.     

Calculating the effect of gusts on an arbitrary thin wing

A numerical method of calculating the unsteady flow about a thin wing moving in an ideal incompressible medium is developed on the basis of the lifiting surface scheme. The variation of the boundary conditions on the wing surface with time and coordinates may be arbitrary. Therefore, the method makes it possible to examine the aperiodic motion of a wing as a rigid body, consider any wing deformations, analyze the wing entry into a gust, study the effect of a weak shock wave on the wing, etc. In addition, practically no limitation is imposed on the shape of the thin lifting surface: the method is applicable to monoplane wings of any planform, to annular wings, to systems of similar wings, etc.Studies in which the effect of a gust on a wing is analyzed have been reviewed in [1, 2]. Without dwelling on this review, we note that at subsonic speeds an effective solution of the problem has been obtained only for a profile.The author wishes to thank E. P. Kapustina for working the examples.     

混凝土动态破坏过程超声波测试研究

混凝土在荷载作用下会产生损伤或破坏,用超声波的方法可测量损伤或破坏的程度。但在荷载作用下混凝土的破坏是一个动态过程,由于测试面发生变形,使常规超声波测试无法完成。本文研制了一个混凝土材料破坏过程超声波测试辅助装置,结合原有的非金属材料超声波检测仪,使测试混凝土材料动态破坏过程成为可能。同时通过辅助装置还可消除试件在破坏过程中由于变形带来的测试误差,可方便、快捷、准确地测试混凝土在荷载作用下的动态破坏过程。采用常规混凝土和冻融混凝土为试件,在材料万能试验机上进行了破坏实验,得到了不同荷载作用下超声波的波速。实验结果表明,混凝土的超声波波速随混凝土的破坏程度不断增大而减小,由此表示可采用超声波测试的方法确定混凝土的动态破坏过程。     

On the role of a balancing load in the problem of aerodynamic drag minimization in the supersonic flight regime

The problem of reducing the aerodynamic losses through balancing in the supersonic flight regime is considered. An analysis and a comparison of the drag components due to the aerodynamic surface deformation and the balancing weigh distribution is made with reference to the examples of a zero-thickness airfoil and a three-dimensional configuration of an aircraft with a wing of complicated planform. It is shown that minimum values of the aerodynamic drag are achieved as a result of complex optimization including the dead load mass as a varied parameter.