Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-18T21:39:27.195Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear potential theory of steady internal supersonic flow with quasi-cylindrical geometry. Part 1. Flow in ducts

Published online by Cambridge University Press:  26 April 2006

Andreas Dillmann
Andreas Dillmann
Affiliation:
Deutsche Forschungsanstalt für Luft- und Raumfahrt, Bunsenstraße 10, D-37073 Göttingen, Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

Based on linear potential theory, the general three-dimensional problem of steady supersonic flow inside quasi-cylindrical ducts is formulated as an initial-boundary-value problem for the wave equation, whose general solution arises as an infinite double series of the Fourier–Bessel type. For a broad class of solutions including the general axisymmetric case, it is shown that the presence of a discontinuity in wall slope leads to a periodic singularity pattern associated with non-uniform convergence of the corresponding series solutions, which thus are unsuitable for direct numerical computation. This practical difficulty is overcome by extending a classical analytical method, viz. Kummer's series transformation. A variety of elementary flow fields is presented, whose complex cellular structure can be qualitatively explained by asymptotic laws governing the propagation of small perturbations on characteristic surfaces.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 281 , 25 December 1994 , pp. 159 - 191
Copyright
© 1994 Cambridge University Press

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

Abramowitz, M. & Stegun, I. A. 1972 Handbook of Mathematical Functions. Dover.
Bleistein, N. & Handelsman, R. A. 1986 Asymptotic Expansions of Integrals. Dover.
Courant, R. & Hilbert, D. 1931 Methoden der Mathematischen Physik. Springer.
Dillmann, A. & Grabitz, G. 1994 On a method to evaluate Fourier–Bessel series with poor convergence properties and its application to linearized supersonic free jet flow. Q. Appl. Maths (in press).Google Scholar
Friedlander, F. G. 1958 Sound Pulses. Cambridge University Press.
Kármán, Th. Von 1907 Über stationäre Wellen in Gasstrahlen. Physik. Z. 8, 209–211.Google Scholar
Karmán, Th. Von & Moore, N. B. 1932 Resistance of slender bodies moving with supersonic velocities. Trans. ASME 54, 303–310.Google Scholar
Kolodner, I. 1950 On the linearized theory of supersonic flows through axially symmetrical ducts. Commun. Pure Appl. Maths 3, 133–152.Google Scholar
Knopp, K. 1928 Theory and Application of Infinite Series. Blackie & Son.
Lamb, H. 1975 Hydrodynamics. Cambridge University Press.
Lighthill, M. J. 1970 Fourier Analysis and Generalized Functions. Cambridge University Press.
Ludloff, H. F. & Reiche, F. 1949 Linearized treatment of supersonic flow through ducts. J. Aero. Sci. 16, 5–21.Google Scholar
Mack, Ch. 1947 Linearized treatment of supersonic flow through and around ducted bodies of narrow cross-section. PhD thesis, New York University.
Oswatitsch, K. 1952 Theoretische Gasdynamik. Springer.
Powell, E. O. 1952 A table of the generalized Riemann zeta function in a particular case. Q. J. Mech. Appl. Maths 5, 116–123.Google Scholar
Prandtl, L. 1904 Über die stationären Wellen in einem Gasstrahl. Physik. Z. 5, 559–601.Google Scholar
Tolstov, G. P. 1976 Fourier Series. Dover.
Ward, G. N. 1945 A note on compressible flow in a tube of slightly varying cross-section. Aero. Res. Counc. R. & M. 2183.
Ward, G. N. 1948 The approximate external and internal flow past a quasi–cylindrical tube moving at supersonic speeds. Q. J. Mech. Appl. Maths 1, 225–245.Google Scholar
Ward, G. N. 1955 Linearized Theory of Steady High-Speed Flow. Cambridge University Press.
Watson, G. N. 1944 A Treatise on the Theory of Bessel Functions. Cambridge University Press.
Whittaker, E. T. & Watson, G. N. 1927 A Course of Modern Analysis. Cambridge University Press.