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Linear potential theory of steady internal supersonic flow with quasi-cylindrical geometry. Part 2. Free jet flow

Published online by Cambridge University Press:  26 April 2006

Andreas Dillmann
Deutsche Forschungsanstalt für Luft- und Raumfahrt, Bunsenstraße 10, D-37073 Göttingen, Germany


By extending the methods of Part 1, the general problem of steady cylindrical supersonic free jet flow is treated in a similar manner to the flow in quasi-cylindrical ducts. It is shown that the presence of a finite pressure jump at the nozzle lip gives rise to a periodic singularity pattern in the flow field. Basic examples of free jet flows are discussed, and for the case of a nearly ideally expanded axisymmetric jet, theoretical Mach—Zehnder interferograms are calculated by analytical integration of the density field. Excellent agreement with experiment proves the validity of linear theory even close to the singularities and far downstream of the nozzle orifice. Furthermore, it is shown that Pack's formula for the wavelength of the shock cell structure is inconsistent; the correct formula is derived and excellent agreement with Emden's empirical fit is found.

Research Article
© 1995 Cambridge University Press

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Abramowitz, M. & Stegun, I. A. 1972 Handbook of Mathematical Functions. Dover.
Bartels-Lehnhoff, H.-H., Baumann, P. H., Bretthauer, B. & Meier, G. E. A. 1993 Computer aided evaluation of interferograms. Exps. Fluids 16, 4653.Google Scholar
Bleistein, N. & Handelsman, R. A. 1986 Asymptotic Expansions of Integrals. Dover.
Carpenter, P. W. 1978 A small perturbation theory for supersonic jets. AIAA Paper 78-1151.
Dillmann, A. 1994a Linear potential theory of steady internal supersonic flow with quasi-cylindrical geometry. Part 1. Flow in ducts. J. Fluid Mech. 281, 159191.Google Scholar
Dillmann, A. 1994b On the calculation of theoretical Mach–Zehnder interferograms from the given velocity potential of a cylindrical supersonic free jet. Acta Mechanica 104, 143157.Google Scholar
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
Emden, R. 1899 Über die Ausströmungserscheinungen permanenter Gase. Ann. Phys. Chem. 69, 264289 and 426453.Google Scholar
Goldstein, R. J. 1983 Fluid Mechanics Measurements. Hemisphere.
Grabitz, G. 1975 Analytische Lösung für den stationären rotationssymmetrischen Überschallfreistrahl in linearer Näherung. Z. Angew. Math. Mech. 55, T127T130.Google Scholar
Grabitz, G., Hiller, W. J. & Meier, G. E. A. 1979 Die Dichteverteilung in einem stationären, rotationssymmetrischen Überschallfreistrahl, der gegen schwachen Unter- oder Überdruck austritt. In Recent Developments in Theoretical and Experimental Fluid Mechanics (ed. U. Müller, K. G. Roesner & B. Schmidt), pp. 144156. Springer.
Oswatitsch, K. 1952 Theoretische Gasdynamik. Springer.
Pack, D. C. 1950 A note on Prandtl's formula for the wavelength of a supersonic gas jet. Qt. J. Mech. Appl. Maths 3, 173181.Google Scholar
Powell, E. O. 1952 A table of the generalized Riemann zeta function in a particular case. Qt. J. Mech. Appl. Maths 5, 116123.Google Scholar
Powell, A. 1992 Physics of standing waves in circular underexpanded jets. J. Acoust. Soc. Am. 91, 2354.Google Scholar
Prandtl, L. 1904 Über die stationären Wellen in einem Gasstrahl. Physik. Z. 5, 599601.Google Scholar
Rayleigh, Lord 1916 On the discharge of gases under high pressures. Phil. Mag. (6) 32, 177187.Google Scholar
Tolstov, G. P. 1976 Fourier Series. Dover.
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.