Stationary flow past the lower surface of a piecewise planar delta wing with supersonic leading edges

The considered wing has any finite number of inflections in its plane with lines of inflection intersecting at the point of inflection of the leading edge. In the present paper, this generalizes the author's earlier work [1] on flow past the undersurface of a flat wing at unite angle of attack with finite angle of slip and supersonic leading edges. In [1], calculations were not given. The special case of flow without slip in the same situation was considered later in [2], However, this paper contains errors, indicated at the end of the present paper. The calculations given in [2] are not correct. In the quoted papers, the gas flow is assumed to be a perturbation of a homogeneous flow behind a plane oblique shock wave. Such flows are treated systematically in [3]. Here and in [1], we use and generalize the representation of the linearized conservation laws across the shock front as the conditions of a boundary-value problem for an analytic function of a complex variable as obtained in [4, 5]. Calculations are given of the pressure distribution over the span for a number of different flow regimes and the pressure coefficients in the middle of the wing are compared with a numerical solution presented partly in [6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 80–90, September–October, 1979.I am very grateful to V. I. Lapygin for making available a large number of variants of his numerical solution, and to L. E. Pekurovskii for assistance in the calculations.     

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.     

Nonlinear aeroelastic analysis of a two-dimensional wing with control surface in supersonic flow

Based on the piston theory of supersonic flow and the energy method, a two dimensional wing with a control surface in supersonic flow is theoretically modeled, in which the cubic stiffness in the torsional direction of the control surface is considered. An approximate method of the cha- otic response analysis of the nonlinear aeroelastic system is studied, the main idea of which is that under the condi- tion of stable limit cycle flutter of the aeroelastic system, the vibrations in the plunging and pitching of the wing can approximately be considered to be simple harmonic excita- tion to the control surface. The motion of the control surface can approximately be modeled by a nonlinear oscillation of one-degree-of-freedom. The range of the chaotic response of the aeroelastic system is approximately determined by means of the chaotic response of the nonlinear oscillator. The rich dynamic behaviors of the control surface are represented as bifurcation diagrams, phase-plane portraits and PS diagrams. The theoretical analysis is verified by the numerical results.     

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.     

Influence of intense surface cooling on hypersonic viscous flow past a delta wing

The influence of intense surface cooling on the parameters of a laminar boundary layer flow on a thin delta wing in a hypersonic viscous perfect-gas stream is studied for the strong viscous-inviscid interaction regime. The effect of the power-law shape of the wing cross-section and the wing thickness to boundary-layer displacement thickness ratio on the local and total aerodynamic characteristics is numerically investigated. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 57–64, July–August, 1998.     

Viscous shock layer on the surface of a blunt body in a diverging supersonic flow

The results are given of calculations of supersonic diverging flow past the spherical front of a body obtained using the Navier-Stokes equations. Flows at moderate Reynolds numbers are considered. A study is made of the influence of the nonuniformity of the oncoming flow on the flow field in the shock layer and on the distribution of the pressure, the friction coefficient, and the heat flux over the surface of the sphere.     

Numerical analysis of the flow pattern and vortex breakdown over a pitching delta wing at supersonic speeds

A supersonic compressible flow over a 60° swept delta wing with a sharp leading edge undergoing pitching oscillations is computationally studied. Numerical simulations are performed by the finite volume method with the use of the k?ω turbulence model for various Mach numbers and angles of attack. Variations of flow patterns in a crossflow plane, hysteresis loops associated with the vortex core location, and vortex breakdown positions during a pitching cycle are investigated. Trends for various Mach numbers, mean angles of attack, pitching amplitudes, and pitching frequencies are illustrated.     

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.     

Cone in a supersonic flow near a surface with a turbulent boundary layer

The published information about the interaction of incident shocks and a turbulent boundary layer relate to cases of a thin boundary layer ( 1–3 mm) on a flat surface. The present study relates to supersonic flow with Mach number M = 3 and stagnation pressure p₀=1.2 MPa past cones near a surface with a thick boundary layer formed on a plate abutting the lower edge of a plane nozzle.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 177–180, July–August, 1991.     

Evolution of disturbances in a laminarized supersonic boundary layer on a swept wing

Experimental data on stability of a three-dimensional supersonic boundary layer on a swept wing are presented. The experiments are performed on a swept wing model with a lenticular profile with a 40° sweep angle of the leading edge at a zero angle of attack. The supersonic boundary layer on the swept wing was laminarized with the use of distributed roughness. A pioneering study of interaction of traveling and stationary disturbances is performed. Some specific features of this interaction are identified. The main reason for turbulence emergence in a supersonic boundary layer on a swept wing is demonstrated to be secondary crossflow instability. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 40–46, March–April, 2008.     

Vortex control by the spanwise suction flow on the upper surface of delta wing

The numerical investigation has been performed to explore the feasibility of vortex control by leading edge sucking excitation on a delta wing. The results reveal that the flow on the upper surface of the delta wing changes significantly in a wide range of the angle of attack. For the vortical flow at moderate angle of attack, the secondary and tertiary vortices are weakened or suppressed, and the total lift is almost unchanged. For the stalled flow at high angle of attack, the leading edge concentrated vortex is recovered, and the lift is enhanced with increasing suction rate. For the bluff-body flow at even high angles of attack, the lift can still be improved. The concentrated vortex disappears on the upper surface, and the load increment is nearly unchanged along the chordwise direction. The project supported by the National Natural Science Foundation of China (19802018).     

Boundary conditions on a shock wave in a supersonic flow

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 49–56, January–February, 1989.     

Linear evolution of controlled disturbances in the supersonic boundary layer on a swept wing

The linear development of controlled disturbances in the three-dimensional supersonic boundary layer on a swept model wing with a sharp leading edge is experimentally investigated at the Mach number 2. The spatial-temporal and spectral-wave characteristics of the wave train of unstable disturbances are obtained. The asymmetry of these characteristics, due to the secondary flow in the three-dimensional boundary layer, is confirmed.     

Normal force of a flat triangular wing in a supersonic flow

On the basis of an analysis of the results of numerical calculations for flow conditions with a forward shock wave attached to the leading edges, the dependence of the coefficient of the normal force of a flat triangular wing on the parameter sin is obtained in a wide range of change in the angle of attack, the angle of sweepback, and the Mach numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 162–164, May–June, 1977.     

Boundary layer on a vaporizing surface

The article discusses the laminar boundary layer on the vaporizing surface of liquid hydrogen over which there is a flow of molecular hydrogen. The pressure p in the boundary layer corresponds to the saturation temperature of the oxygen T₀, which is lower than the temperature of the oncoming flow T, but higher than the temperature of the surface of the liquid hydrogen. Under such conditions, within the boundary layer there is condensation of oxygen in the volume, which leads to the formation of drops of liquid oxygen of different sizes. It is proposed in the article that, with the condensation of gas in the volume, drops of one size are formed. Drops of a chosen mean size are regarded as molecules of a heavy gas. The gas drop is a third component present in the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 48–53, September–October, 1973.The author thanks G. I. Petrov for his interest in and direction of the work.     

Aerodynamic characteristics of a delta wing in hypersonic flow

Novosibirsk. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 2, pp. 66–69, March–April, 1994.