Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-05T10:54:36.211Z Has data issue: false hasContentIssue false
  • English
  • Français

The linearised theory of conical fields in supersonic flow, with applications to plane aerofoils

Published online by Cambridge University Press:  07 June 2016

S. Goldstein  and
G. N. Ward
S. Goldstein
Affiliation:
Department of Mathematics, The University, Manchester
G. N. Ward
Affiliation:
Department of Mathematics, The University, Manchester
Get access

Summary

In many important problems of supersonic flow, either for the whole field of flow or a part of it, the velocity components are constant on straight lines through a fixed point. Such velocity fields are called conical fields. In the usual theory of the linearised perturbations of a steady supersonic flow, the velocity is assumed to differ only slightly from a uniform undisturbed velocity, and in the defining equations and boundary conditions all non-linear terms in the components of the perturbation velocity (and their space derivatives) are neglected. In this paper the equations of linearised supersonic conical fields, and their general solution, are set out both for the region inside and for the region outside the Mach cone of the origin in the conical field. The results are applied to flow past plane triangular aerofoils with straight edges downstream of the vertex (of which there are six cases), to flow past those plane aerofoils of more extended shape and finite span for which the solution may be obtained by combining a finite number of conical fields, and to the problem of plane triangular vanes in semi-infinite free jets. In the applications, the velocity fields are determined in considerable detail, but a main purpose in setting them out was to exhibit the mathematical methods used and the physical considerations that enter in determining the mathematical solutions.

Type
Research Article
Information
Aeronautical Quarterly , Volume 2 , Issue 1 , May 1950 , pp. 39 - 84
Copyright
Copyright © Royal Aeronautical Society. 1950

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

References

A. Conical Fields

1. Browne, S. H., Friedman, L., and Hodes, I. (1948). A Wing-Body Problem in a Supersonic Conical Flow. Journal of the Aeronautical Sciences, 15, 1948, 443–452.Google Scholar
2. Beschkine, L. (1947). Ecoulements Coniques Supersoniques. Les Cahiers d’AeYo- dynamique, Groupes d’Etudes Aéronautiques de la Faculté des Sciences, Université de Paris, No. 6, 1947, 24–38; Forces Ae’rodynamiques Agissant sur les Surfaces Portantes aux Vitesses Supersoniques, ibid., 39–50.Google Scholar
3. Busemann, A. (1935). Aerodynamischer Auftrieb bei Uberschallgeschwindigkeit. Atti dei Convegni 5, Reale Accademia d’ltalia (V Volta Congress), Rome, 1936, pp. 328–360; Luftfahrtforschung, 12, 1935, 210–219.Google Scholar
4. Busemann, A. (1943). Infinitesimale kegelige Uberschallstromung. Schriften der Deutschen Akademie der Luftfahrtforschung, 7B, 1943, 105–122.Google Scholar
5. Germain, P. (1948). Applications of the Theory of Conical Motion to Supersonic Dynamics. VII International Congress of Applied Mechanics, 1948. (Unpublished).Google Scholar
6. Hayes, W. D. (1946). Linearized Conical Supersonic Flow. VI International Congress of Applied Mechanics, 1946. (Unpublished).Google Scholar
7. Hayes, W. D., Browne, S. H., and Lew, R. J. (1947). Linearized Theory of Conical Supersonic Flow, with Application to Triangular Wings. North American Aviation, Inc., Report No. NA-46-818, 1947. (Unpublished).Google Scholar
8. Hayes, W. D. (1947). Linearized Supersonic Flow. Ph.D. Thesis, California Institute of Technology, 1947; North American Aviation, Inc., Report No. AL-222. (Unpublished).Google Scholar
9. Lagerstrom, P. A. (1948). Linearized Supersonic Theory of Conical Wings. N.A.C.A. Technical Note 1685, 1948.Google Scholar
10. Lagerstrom, P. A., and Graham, M. E. Downwash and Sidewash induced by Three- dimensional Lifting Wings in Supersonic Flow. Douglas Aircraft Company Inc., Report No. SM-13007. (Unpublished).Google Scholar
11. Lagerstrom, P. A., and Graham, M. E. (1948). Linearized Theory of Supersonic Control Surfaces. Douglas Aircraft Company Inc. Report No. SM-13060. (Unpublished). For abbreviated version see Journal of the Aeronautical Sciences, 16 (1949), 31–34.Google Scholar
12. Lagerstrom, P. A., and Graham, M. E. LOW Aspect Ratio Rectangular Wings in Supersonic Flow. Douglas Aircraft Company Inc., Report No. SM-13110. (Unpublished).Google Scholar
13. Laporte, O., and Bartels, R. C. F. (1948). An Investigation of the Exact Solutions of the Linearized Equations for the Flow past Conical Bodies. Bumblebee Series Report No. 75, Engineering Research Institute, University of Michigan, 1948. (Unpublished; abstract in Aero. Engineering Review, February 1949, p. 48).Google Scholar
14. Laporte, O., and Bartels, R. C. F. (1948). Supersonic Flow past a Delta Wing at Angles of Attack and Yaw. Engineering Research Institute, University of Michigan, 1948. (Unpublished).Google Scholar
15. Poritsky, H. (1949). Linearized Compressible Flow. Quarterly of Applied Mathematics, 6, 1949, 389–403.Google Scholar
16. Roberts, R. C. (1948). Note on the Lift of a Triangular Wing at Supersonic Speeds. Journal of Mathematics and Physics, 27, 1948, 49–55.Google Scholar
17. Roper, G. M. (1948). The Flat Delta Wing at Incidence, at Supersonic Speeds, when the Leading Edges lie outside the Mach Cone of the Vertex. Quarterly Journal of Mechanics and Applied Mathematics, 1, 1948, 327–343.Google Scholar
18. Snow, R. M. (1948). Aerodynamics of Thin Quadrilateral Wings at Supersonic Speeds. Quarterly of Applied Mathematics, 5, 1948, 417–428.Google Scholar
19. Stewart, H. I. (1946). The Lift of a Delta Wing at Supersonic Speeds. Quarterly of Applied Mathematics, 4, 1946, 246–254.Google Scholar
20. Ward, G. N. (1946). The Pressure Distribution on some Flat Laminar Aerofoils at Incidence at Supersonic Speeds. R. & M. 2206, 1946.Google Scholar
21. Gunn, J. C. (1947). Linearized Supersonic Aerofoil Theory. Philosophical Transactions of the Royal Society, 240, 1947, 327–373.Google Scholar
22. Hayes, W. D. (1947). Linearized Supersonic Flow. Ph.D. Thesis, California Institute of Technology, 1947; North American Aviation, Inc., Report No. AL-222. (Unpublished).Google Scholar
23. Lighthill, M.'j. (1944). The Supersonic Theory of Wings of Finite Span. R. & M. 2001, 1944.Google Scholar
24. Robinson, A. (1946). Aerofoil Theory of a Flat Delta Wing at Supersonic Speeds. A.R.C. 10222, 1946. (Unpublished).Google Scholar
25. Schlichting, H. (1936). Tragflügeltheorie bei Uberschallgeschwindigkeit. Luftfahrtforschung, 13 (1936), 320–335.Google Scholar
26. Taunt, D. R., and Wajrd, G. N. (1946). Wings of Finite Aspect Ratio at Supersonic Velocities. R. & M. 2421 (1946).Google Scholar
27. Ward, G. N. (1949). Supersonic Flow past Thin Wings, 1: General Theory. Quarterly Journal of Mechanics and Applied Mathematics, Part II, Vol. II, June 1949, p. 136.CrossRefGoogle Scholar