Skip to main content Accessibility help

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

  • Andreas Dillmann (a1)


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.



Hide All
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).
Friedlander, F. G. 1958 Sound Pulses. Cambridge University Press.
Kármán, Th. Von 1907 Über stationäre Wellen in Gasstrahlen. Physik. Z. 8, 209211.
Karmán, Th. Von & Moore, N. B. 1932 Resistance of slender bodies moving with supersonic velocities. Trans. ASME 54, 303310.
Kolodner, I. 1950 On the linearized theory of supersonic flows through axially symmetrical ducts. Commun. Pure Appl. Maths 3, 133152.
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, 521.
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, 116123.
Prandtl, L. 1904 Über die stationären Wellen in einem Gasstrahl. Physik. Z. 5, 559601.
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, 225245.
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.
MathJax is a JavaScript display engine for mathematics. For more information see

Related content

Powered by UNSILO

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

  • Andreas Dillmann (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.