Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T00:52:18.420Z Has data issue: false hasContentIssue false
  • English
  • Français

Elements of an improved linear perturbation theory of steady supersonic flow with axial symmetry

Published online by Cambridge University Press:  24 October 2008

C. W. Jones
C. W. Jones
Affiliation:
Department of Applied MathematicsUniversity of Liverpool
Get access Rights & Permissions [Opens in a new window]

Abstract

The usual linearized theory neglects small terms, of second and higher orders of the velocity perturbation, in the coefficients of the equation for the potential. In this paper these terms are approximated by constant mean values and the familiar formulae of the first approximation are then generalized. The characteristic lines of this improved linear theory thereby differ from the Mach lines of the undisturbed flow, though still being straight and parallel. The new formulae thus give a better calculation of the velocity field near bodies not slender enough for the ordinary theory to be accurate. A numerical example, for a 10° cone in a stream of Mach number about 3, illustrates this fact, and shows that a shock of constant strength may be fitted to the improved solution.

Although the improved theory as it stands must fail at large distances from the axis of symmetry, just as the first approximation fails, the tentative suggestion is made that a technique due to G. B. Whitham may be applied to the former, as to the latter, perhaps leading to an improved description of the whole field of flow.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 52 , Issue 2 , April 1956 , pp. 336 - 343
Copyright
Copyright © Cambridge Philosophical Society 1956

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

(1)KÁrmÁn, T. and Moore, N. B.Trans. Amer. Soc. meoh. Engrs, 54 (1932), 303–10.Google Scholar
(2)Ward, G. N. Chapter VIII of Modern developments in fluid dynamics: high speed flow (ed. Howarth, ; Oxford, 1953).Google Scholar
(3)Ward, G. N.Linearized theory of steady high speed flow (Cambridge, 1955).Google Scholar
(4)Whitham, G. B.Proc. roy. Soc. A, 201 (1950), 89–109.Google Scholar
(5)Whitham, G. B.Commun. pure appl. Math. 5 (1952), 301–48.CrossRefGoogle Scholar
(6)Whittaker, E. T. and Watson, G. N.A course of modern analysis, 4th ed. (Cambridge, 1935), p. 293.Google Scholar