Inviscid Supersonic Flow Research Articles

Surface pressures on a wing moving with supersonic speed

Estimates for pressures on the surface of a given delta wing at zero incidence in a steady uniform stream of air are obtained by numerically integrating two semi-characteristic forms of equations which govern the inviscid supersonic flow of an ideal gas with constant specific heats. In one form of the equations coordinate surfaces are fixed in space so that the surface of the wing, which has round sonic leading edges, is a coordinate surface. In the other, two families of coordinates are chosen to be stream-surfaces. For each form of the equations, a finite difference method has been used to compute the supersonic flow around the wing. Convergence of the numerical results, as the mesh is refined, is slow near the leading edge of the wing and an extrapolation procedure is used to predict limiting values for the pressures on the surface of the wing at two stations where theoretical and experimental results have been given earlier by another worker. At one station differences between the results given here and the results given earlier are significant. The two methods used here produce consistent values for the pressures on the surface of the wing and, on the basis of this numerical evidence together with other cited numerical results, it is concluded that the pressures given here are close to the true theoretical values.

Velocity Distribution Functions Near the Leading Edge of a Flat Plate

References 1 Jaslow, H., Relationships Inherent in Impact Theory, AIAA Journal, Vol. 6, No. 4, April 1968, pp. 608-612. 2 Pike, J., Lift and Drag of Axisymmetric Bodies in Flow, AIAA Journal, Vol. 7, No. 1, Jan. 1969, pp. 185-186. 3 Pike, J., Newtonian Lift and Drag of Blunt Cone-Cylinder Bodies, AIAA Journal, Vol. 10, No. 2, Feb. 1972, pp. 176-180. 4 Pike, J., Newtonian Aerodynamic Forces from Poisson's Equation, AIAA Journal, Vol. 11, No. 4, April 1973, pp. 499-504. 3 Hobson, E. W., The Theory oj Spherical and Ellipsoidal Harmonics, Cambridge University Press, Cambridge, England, 1931. 6 Jones, D. J., Tables of Inviscid Supersonic Flow about Circular Cones at Incidence y = 1.4, AGARD-AG-137-71, Dec. 1971. 7 Babenko, K. I., Voskresenskiy, G. P., Lyubimov, A. N., and Rusanov, U. V., Three-Dimensional Flow of Ideal Gas past Smooth Bodies, TT F-380, April 1966, NASA. 8 Beecham, L. J. and Titchener, I. M., Some Notes on an Approximate Solution for the Free Oscillation Characteristics of NonLinear Systems typified by x + F(x, x) = 0, R & M 3651, 1971, Aeronautical Research Council, London, England. Moseley, W. C, Jr., Moore, R. H., Jr., and Hughes, J. E., Stability Characteristics of the Apollo Command Module, TN D-3890, March 1967, NASA.

An analytic characteristic method for steady three-dimensional isentropic flow

The analytical characteristic method is an effective method for computing non-linear effects in inviscid supersonic flow problems. Although only linear equations have to be solved, the results are essentially nonlinear, in the sense that the functional relations between physical state variables and space co-ordinates are nonlinear in the small perturbation parameter introduced, like the thickness ratio or incidence of a wing. This holds even for the first-order approximation of the method.In the case of two-dimensional (plane or axisymmetric) flow the independent variables are characteristic co-ordinates, i.e. they are chosen so as to be constant along corresponding characteristic lines. The space co-ordinates are considered as dependent variables. In three dimensions there is no unique definition of a characteristic co-ordinate system, because the manifold of characteristic surfaces or bi-characteristics is larger than is necessary for defining a co-ordinate system. The success of a three-dimensional analytical characteristic method, however, depends on the proper choice of the co-ordinate system.The present analytical Characteristic method for three-dimensional flow is based on the fact that three-dimensional flow behaves locally like axisymmetric flow if it is considered in the osculating plane. The corresponding ‘distance from the axis’ is a function of space depending on the flow field. No change of pressure occurs normal to the osculating plane and in isentropic flow no change of speed either. Therefore no co-ordinate perturbation is performed in this normal direction. In the osculating plane the analytical characteristic methodis applied locally as in axisymmetric flow. In the large the space co-ordinates are obtained by integration along the main bi-characteristics.As an example the flow field on the suction side of a flat delta wing with sub-sonic leading edges is computed. As a main result one obtains shock waves in the neighbourhood of the leading edges following the expansion.

A computerized analysis of supersonic nonuniform flows over sharp and spherically blunted cones at angle of attack

A system of computer programs has been developed to predict supersonic inviscid and viscous nonuniform flow fields over sharp and spherically blunted cones at angle of attack. For blunt cones the flow fields considered were axisymmetric wake flows positioned such that the flow in the subsonic nose region remained axisymmetric. For sharp cones, both axisymmetric wake flows and two-dimensional shear flows were considered. The programs used in solving inviscid flow fields incorporate a modified inverse method for solving subsonic flow regions and modified axiymmetric and three-dimensional method of characteristics procedures for solving the supersonic flow regions. Body properties predicted by the inviscid solutions were used as edge data for solution of the corresponding laminar boundary-layers over the bodies. The viscous flow solutions were obtained using axisymmetric and full three-dimensional boundary-layer programs. Typical results from inviscid calculations have shown the development of strong adverse pressure gradients over both sharp and blunt cones in wake flows. In addition a thin entropy layer was found near the surface of both bodies; however, the normal pressure gradient was found to be negligible for the nonuniform flows considered. For the sharp cone in shear flow, property variation along the body was found to be almost linear. In all cases the aerodynamic coefficients were found to be significantly affected by the free-stream nonuniformity. Typical viscous flow field results have shown that relative to uniform flow values the skin friction and heat transfer increase along the windward streamline of both blunt and sharp cones in the nonuniform flows considered. Decreasing the width of the wake in wake flow increases the heat transfer and skin friction.

Computation of Three Dimentional Flows about aircraft configurations

The object of this paper is to describe an accurate and efficient numerical procedure to compute the supersonic inviscid flow field about complicated three-dimentional aircraft configurations and to demonstrate how the technique works in a practical case. A two level second order marching scheme is used to integrate Euler's equations in regions where the flow is continuous. Shocks are, however, treated explicitly as descontinuities satisfying the Renkine-Hugoniot equations. Use is made of conformal mappings in order to provide proper resolution in critical areas. Results are presented for several three-dimentional flow field computations including a complete aircraft and the results are compared with experiments.

Side Forces on Ogive-Cylinder Bodies at High Angles of Attack in Transonic Flow

Flow characteristics about slender axisymmetric bodies of revolution at high angles of attack in the high subsonic and transonic speed range are poorly understood. This may be ascribed to several extremely complex interactions involving inviscid mixed subsonic and supersonic flows, viscous attached and separated boundary-layer flow, and the resulting vortex flow. It was reported in Ref. 1 that at high subsonic and transonic flows and at angles of attack above 20°, with zero side-slip, ogive-cylinder bodies of revolution exhibited rather large steady side forces. The direction and magnitude of these side forces were unpredictable. Pressure distribution measurements and schlieren photographs in Ref. 2 indicated that an unsymmetric vortex buildup was causing these forces. In spite of reported failures in various research programs due to unexpected side forces, no systematic data exist which deal with physical causes or effects. Consequently, the objective of the present investigation was to obtain systematic information about the effects of Mach number, nose geometry, angle of attack, and boundary layer conditions on the measured side forces. The experiments were conducted in the NSRDC 18-in. indraft tunnel using a slotted test section. Unit Reynolds numbers varied between 2.7 x 10/ft and 4.3 x 10/ft for the speed range of M = 0.5-M=1.1 (0.5, 0.6, 0.7, 0.8, 0.9, and 1.1). Tangent ogive noses of fineness ratios (F.R.) 2, 3, and 4 were tested with bluntness ratios (B.R.) of 0 to 50% (100 nose tip radius/base radius) using a cylindrical afterbody of 1.1 in. base diameter and 8.688 in. length. Schematic diagram of the tested configurations are shown in Fig. 1. Aerodynamic forces were obtained by a six component internal balance. Schlieren photography and oil flow techniques provided both external and surface flow visualization. At several speeds, roll angles were changed from 0-90° and 180° to check the flow sensitivity to small geometric changes. At M = 0.5, 0.7, and 0.9, both laminar and turbulent boundary layers were examined. Turbulent boundary layers were generated by a i in. wide strip of No. 54 Grit placed from the nose to the base of the models along the windward meridian. The flow model which can be reconstructed from the schlieren photographs, oil flow experiments and aerodynamic

Boundary-Layer Separation on Slender Cones at Angle of Attack

6 McClure, J. D., On Perturbed Boundary Layer Flows, Fluid Dynamic Research Lab. Rept. 62-2, July 1962, MIT, Cambridge, Mass. 7 Lighthill, M. J., Reflection at a Laminar Boundary Layer of a Weak Steady Disturbance to a Supersonic Stream, Neglecting Velocity and Heat Conduction, Quarterly Journal of Mechanics and Applied Mathematics, Vol. 3, No. 3, March 1950, pp. 302-325. 8 Lighthill, M. J., On Boundary Layers and Upstream Influence II Supersonic Flows Without Separation, Proceedings of the Royal Society, Vol. 217, 1953, pp. 478-507, 9 Benjamin, T. B., Shearing Flow Over a Wavy Boundary, Journal of Fluid Mechanics, Vol. 6, No. 2, 1959, pp. 161-205. 10 Lees, L. and Reshotko, E., Stability of the Compressible Laminar Boundary Layer, Journal of Fluid Mechanics, Vol. 12, No. 4, 1962, pp. 555-590. 11 Inger, G. R., Discontinuous Supersonic Flow Past an Ablating Wavy Wall, AIAA Journal, Vol. 7, No. 4, April 1969, pp. 762-764. 12 Inger, G. R., Rotational Inviscid Supersonic Flow Past a Wavy Wall, Bulletin of the American Physical Society, Vol. 13, No. 11, Nov. 1968, p. 1579. 13 Martellucci, H., Rie, H., and J. F. Somtowski, J. F., Evaluation of Several Eddy Viscosity Models through Comparison with Measurements in Hypersonic Flows, AIAA Paper 69-688, San Francisco, Calif., 1969. 14 Brown, W. B., Stability of Compressible Boundary Layers, AIAA Journal, Vol. 5, No. 10, Oct. 1967, pp. 1753-1759. 15 Lew, H. G. and Li, H., The Role of the Turbulent Viscous Sublayer in the Formation of Surface Patterns, Rept. R68SD12, June 1968, General Electric Missle and Space Div., Valley Forge, Pa. 16 Klein, E. J., Liquid Crystals in Aerodynamic Testing, Astronautics and Aeronautics, Vol. 6, No. 7, July 1968, pp. 20-25. 17 Williams, E. P. and Inger, G. R., Investigations of Ablation Surface Cross Hatching, SAMSO TR-70-246, June 1970, McDonnell Douglas Astronautics, Huhtington Beach, Calif.

Supersonic wing-body interference

We consider here the supersonic inviscid flow past a wing-body combination consisting of a semi-infinite plane wing symmetrically placed about an infinite convex cylinder. When the cylinder is circular in cross-section a formal solution for the Laplace transform of the velocity potential exists (Neilson(l), and Stewartson, 1951, unpublished). This solution is given in the form of a series which is only slowly convergent near the boundary surface separating the disturbed and the undisturbed regions. However, this slow convergence has been overcome by Waechter(4) using the Poisson Summation formula on the series solution. This enables the solution to be expressed in terms of integrals which can be evaluated as a convergent residue series, taking on different forms in the different regions of the interaction.

Large-amplitude slow oscillation of wedges in inviscid hypersonic and supersonic flows

Large-amplitude slow oscillation of wedges in inviscid hypersonic and supersonic flows

Theory of laminar viscous-inviscid interactions in supersonic flow

This investigation is concerned with those fluid-mechanical problems in which the pressure distribution is determined by the interaction between an external, supersonic inviscid flow and an inner, laminar viscous layer. The boundary-layer approximations are assumed to remain valid throughout the viscous region, and the integral or moment method of Lees and Reeves, extended to include flows with heat transfer; is used in the analysis. The general features of interacting flows are established, including the important distinctions between subcritical and supercritical viscous layers. The eigensolution representing self-induced boundary-layer flow along a semi-infinite flat plate is determined, and a consistent set of departure conditions is derived for determining solutions to interactions caused by external disturbances. Complete viscous-inviscid interactions are discussed in detail, with emphasis on methods of solution for both subcritical and supercritical flows. The method is also shown to be capable of predicting the laminar flow field in the near wake of blunt bodies. Results of the present theory are shown to be in good agreement with the measurements of Lewis for boundary-layer separation in adiabatic and non-adiabatic compression corners, and with the near-wake experiments of Dewey and McCarthy for adiabatic flow over a circular cylinder. Extensions of the method to flows with mass injection at the surface and to subsonic interactions are indicated.

Limit case for supersonic inviscid flow in the corner of intersecting wedges.

Limit case for supersonic inviscid flow in the corner of intersecting wedges.

Solutions for supersonic rotational flow around a corner using a new co-ordinate system

A co-ordinate system consisting of the left-running characteristics (α = const.) and the streamlines (ϕ = const.) is used. The governing equations are derived in terms of α and ϕ for a two-dimensional steady supersonic rotational inviscid flow of a perfect gas. The equations are applied to the problem of an initially parallel supersonic rotational flow which expands around a convex corner. The velocity of the incoming flow at the wall is considered to be either supersonic (case 1) or sonic (case 2). For each case, solutions uniformly valid in the region near the leading characteristic and in the region near the corner, are found for the Mach angle and flow deflexion angle in terms of their values on the leading characteristic and at the corner. In case 2, a transonic similarity solution is found and composite solutions are constructed for each region. Comparisons are made with existing exact numerical results.

Vortical singularities in conical flow.

In this paper, we investigate the behavior of inviscid supersonic conical flow fields near crossflow stagnation points by constructing coordinate expansions of the exact equations of conical flow. The local cross-flow streamline pattern is shown to be determined primarily by the curvature of the surface pressure distribution (P and by the direction of the surface velocity near the stagnation point. The local streamline pattern at a maximum of the surface pressure is shown to be either a saddle point or a nodal point with cross-flow streamlines normal to the body. The solution corresponding to the node changes to a saddle point at a critical value, (P == — 8, which thus signals the liftoff of the vortical singularity first suggested by Ferri. The streamline pattern at a minimum of the pressure distribution is a node with streamlines tangent to the body if (P is less than a critical value, (P = 1, and does not exist if (P > 1. The local solutions are used to infer the general characteristics of the cross-flow streamline pattern generated by yawed circular cones and flat plates. They are also used to analyze the nonuniformity of the small yaw and thin-shock-layer expansions at vortical singularities.

Supersonic interference flow along the corner of intersecting wedges.

Abstract : An experimental investigation of the inviscid supersonic flow field in a corner formed by two intersecting wedges is reported. The study includes measurements of the distribution of the surface pressure and the wave structure of the flow field in the corner region both for a symmetrical corner configuration (e. e., equal wedge angles) in the Mach-number range between 2.5 and 4 and for an asymmetrical model at one Mach number as a function of the incidence angle of one wedge. A clearly identifiable shock structure is discovered that divides the interference field into four zones. The total extent of the disturbance near the surfaces of the model is approximately twice the extent indicated by available small-perturbation analyses. The highest pressure rise occurs in the inner zone near the corner; the magnitude of this rise is related successfully to the second-order linearized theory. The inviscid flow field is shown to involve strong cross-stream pressure and entropy gradients both at the corner proper and away from it, which cannot be disregarded in analyses of the viscous boundary layer. (Author)

On wing–body interference in supersonic flow

Although the theory of supersonic inviscid flow past thin wings and slender bodies has received an enormous amount of attention in recent years, comparatively little attention has been paid to the interaction between wings and finite bodies. The principal reason stems from the extensive numerical work that has to be done in order to form a detailed picture of the flow properties—even with the present generation of highspeed computers the student of this problem is faced with a formidable task. The main effort in this interaction problem has been put into the special case when the body is a circular cylinder symmetrically disposed towards the oncoming flow, and here Nielsen (1951) and Stewartson (1951, unpublished) have shown that a formal solution can be obtained in a simple way in terms of Laplace transforms. Interpretation of these formulae however was still a formidable task, but has been greatly facilitated by the publication of an extensive set of tables of the basic functions (Nielsen 1957). Using these tables a number of workers (Randall 1965, Chan and Sheppard 1965, Treadgold, unpublished, and others) have successfully computed pressure distributions on the wing and the body, except for the neighbourhoods of the lines separating the interaction regions from the remainder of the wing and body. The aim of this paper is to provide formulae for the pressure distributions in these neighbourhoods in order to enable the solutions already obtained to be completed.

A direct method for calculation of the flow about an axisymmetric blunt body at angle of attack.

Direct numerical method for calculation of supersonic inviscid flow about axisymmetric blunt body at large angle of attack

Variational problems of gas dynamics of nonequilibrium and equilibrium flows

The problem of shape determination is examined for two-dimensional and axisymmetric bodies with minimum drag and for nozzles with maximum thrust under conditions of steady supersonic flow of inviscid and thermally nonconducting gas in the presence and in the absence of irreversible processes of the type of chemical reactions proceeding at finite rates. It assumed that the region of influence of the part of the contour which is to be determined is bounded by characteristics and does not contain shock waves. The boundaries with respect to the body contour are arbitrary; the dimensions of the body, the area of the surface, the volume etc. can be prescribed. In the present study parameters on the surface of the body determined by a system of non-linear equations in partial differentials, appear as functional s of a form which is unknown In advance. In problems solved up to recent time [1 to 9] this difficulty is overcome by a transformation to a control contour as suggested by Nikol'skii [10]. However, this transformation is applicable when only the dimensions of the body-are prescribed and when irreversible processes are not present. A method for solution of problems which do not permit such a transformation was proposed recently by Guderley and Armitage [11] and independently by Sirazetdinov [12]. Application of this method to problems of the present study permits to obtain the necessary conditions of an extremum which serves as a basis for the construction of optimum contours. Furthermore, it is demonstrated that in a number of cases it is necessary to permit discontinuities in Lagrange's multipliers for continuous parameters of flow. It is shown that these discontinuities can occur along characteristics and streamlines. Relationships for discontinuities are obtained.

Linearized pressure distributions with strong supersonic entropy layers

The pressure distribution which results from the disturbance of initially plane, parallel, and constant-pressure inviscid supersonic flow in which there is a strong entropy variation near a surface is investigated. The disturbance considered may arise from a small deflection of the surface or may be the result of a simple wave impinging on the entropy layer. This linear problem has been treated previously by a somewhat different technique for transonic entropy layers by Howarth (1948), Tsien & Finston (1949), and Lighthill (1950, 1953). The results obtained in this investigation are useful principally in that they permit the general behaviour of such pressure distributions to be deduced without resort to the more general method of characteristics for rotational flows.

An Approximate Method of Determining Axisymmetric Inviscid Supersonic Flow Over a Solid Body and Its Wake

An Approximate Method of Determining Axisymmetric Inviscid Supersonic Flow Over a Solid Body and Its Wake

An Introduction to Second-Order Wing Theory

In this paper the particular integral for the second-order iteration equation of inviscid supersonic flow, discovered by Fell and the author, is applied to wing theory. This enables us to calculate second-order derivatives for wings of finite aspect ratio, and thence to determine how far the second-order strip theory of Busemann is in error for such wings. Remarking that the most interesting effect in second-order wing theory is the influence of thickness on lifting derivatives, and that corrections to strip theory will be largest on a wing with subsonic edges, we choose as our example the lift coefficient of a rectangular wing with a simple thickness distribution. Some details are given of the calculation, which is fairly complex; use of the f method introduced by Perry and the author ' is found to reduce the labor involved very considerabty.