Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



On a method to evaluate Fourier-Bessel series with poor convergence properties and its application to linearized supersonic free jet flow

Authors: A. Dillmann and G. Grabitz
Journal: Quart. Appl. Math. 53 (1995), 335-352
MSC: Primary 76M25; Secondary 65T20, 76J20
DOI: https://doi.org/10.1090/qam/1330656
MathSciNet review: MR1330656
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: An analytical method based on Kummer's series transformation is presented, which allows for the evaluation of Fourier-Bessel series with poor convergence properties and, in addition, yields the singularities of the series in closed form. This method is applied to the Fourier-Bessel series which arise as solutions of the linearized gas dynamic potential equation for the cylindrical flow field of a supersonic free jet. As an illustrative example, the presented method is applied to D. C. Pack's classical solution for the axisymmetric free jet with initial homogeneous pressure perturbation, which in its original form cannot be evaluated directly. It is shown that in contrast to the plane jet, the flow field of the axisymmetric jet does not exhibit a strictly periodic behaviour, whereas its singularities are distributed periodically.

References [Enhancements On Off] (What's this?)

  • [1] G. N. Ward, Linearized Theory of Steady High-Speed Flow, Cambridge Univ. Press, Cambridge, 1955
  • [2] K. Knopp, Theory and Application of Infinite Series, Blackie & Son, London, 1928
  • [3] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972
  • [4] D. C. Pack, A note on Prandtl's formula for the wave-length of a supersonic gas jet, Quart. J. Mech. and Appl. Math. 3, 173 (1950)
  • [5] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, Cambridge, 1922
  • [6] G. P Tolstow, Fourierreihen, VEB Deutseher Verlag der Wissenschaften, Berlin, 1950
  • [7] G. Grabitz, Analytische Lösung für den stationären rotationssymmetrischen Überschallfreistrahl in linearer Näherung, Z. Angew. Math. Mech. 55, T127 (1975)
  • [8] G. Grabitz, Über die Singularitäten in der linearisierten Lösung für den rotationssymmetrischen Überschallfreistrahl, Z. Angew. Math. Mech. 60, T187 (1980)
  • [9] G. Grabitz and H. W. Burmann, Die Singularitäten der Lösung des schwach gestörten rotationssymmetrischen Überschallfreistrahls, Report 6/1980, Max-Planck-Institut für Strömungsforschung, Göttingen, 1980
  • [10] A. Dillmann, Analytische Theorie zylindrischer dreidimensionaler Überschallfreistrahlen, Report IB 222-92 A24, Deutsche Forschungsanstalt für Luft- und Raumfahrt, Göttingen, 1992
  • [11] A. Erdelyi (ed.). Higher Transcendental Functions, McGraw-Hill, New York, 1953
  • [12] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, Academic Press, New York, 1965
  • [13] B. Baule, Die Mathematik des Naturforschers und Ingenieurs, Hirzel, Leipzig, 1950
  • [14] G. Grabitz, W. Hiller, und G. E. A. Meier, Zur Wellenstruktur eines schwach über- oder unterexpandierten, stationären, rotationssymmetrischen Überschallfreistrahles, Z. Angew. Math. Mech. 59, T231 (1979)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 76M25, 65T20, 76J20

Retrieve articles in all journals with MSC: 76M25, 65T20, 76J20

Additional Information

DOI: https://doi.org/10.1090/qam/1330656
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society