Skip to main content

The dynamics of rolled vortex sheets tightly winded around slender vortex filaments in inviscid incompressible flow

Dynamique du noyau des nappes tourbillonnaires enroulées autour de filaments tourbillonnaires élancés

Part of the Lecture Notes in Mathematics book series (LNM,volume 594)

Abstract

The purpose of this paper is to develop an asymptotic theory for the core of a tightly winded rolled vortex sheet, when winding arises around a line. The physical picture replaces the core by a region of distributed vorticity, the so-called vortex filament. The (pseudo) mathematical picture relies on an algorithm which allows to relate (in a quite convincing but non rigorous way) a flow with vorticity spread over a region to a corresponding irrotational flow with vorticity concentrated on a rolled sheet. Of course, the correspondence holds only in the asymptotic limit that the turns of the sheet are infinitely close to each other. The technique for that, which was developed by Guiraud & Zeytounian, 1977, relies on a multiple scale expansion using a closeness parameter which is the ratio of the distance from one turn of the sheet to the next one, to a length characteristic of the flow. The leading approximation is, then, a rotational flow. In the present paper, an application of the theory is given when this leading approximation has the structure of a slender vortex filament and is itself given by an inner and outer expansion technique. Inner and outer approximations to the dynamics of vortex filaments is reviewed, the results of Guiraud & Zeytounian are presented and, finally, the specific application is worked out.

Keywords

  • Vortex Ring
  • Axial Flow
  • Vortex Sheet
  • Delta Wing
  • Lead Edge Vortex

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   44.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   59.00
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • G.K. BATCHELOR 1967-An introduction to fluid mechanics. Cambridge Univ. Press.

    Google Scholar 

  • L.E. FRAENKEL 1970-On steady vortex rings of small cross section in an ideal fluid. Proc. Roy. Soc. A 316, p. 29.

    CrossRef  MATH  Google Scholar 

  • L.E. FRAENKEL 1972-Examples of steady vortex rings of small cross section in an ideal fluid. JFM 51, 1, p. 119–135.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • J.P. GUIRAUD, R. Kh. ZEYTOUNIAN 1977-A double scale investigation of the asymptotic structure of rolled up vortex sheets. To appear in JFM.

    Google Scholar 

  • J.P. GUIRAUD, R. Kh. ZEYTOUNIAN 1977a-Une théorie pour le noyau de nappes tourbillonnaires à enroulement serré. To appear in “La Recherche Aérospatiale”.

    Google Scholar 

  • H. LAMB 1932-Hydrodynamics. Sixth edition, Cambridge University Press.

    Google Scholar 

  • D.W. MOORE & P.G. SAFFMAN 1972-The motion of a vortex filament with axial flow. Phil. Trans. Roy. Soc., London, A 76, p. 403–429, no 1226.

    CrossRef  MATH  Google Scholar 

  • S.H. OLSEN, A. GOLDBURG, M. ROGERS 1971-Aircraft wake turbulence and its protection. Proc. of Symposium held in Seattle, September 1970. Plenum Press.

    Google Scholar 

  • L. PRANDTL 1922-In L. Prandtl: Gesammelte Abhandlungen, Berlin (1961) 2, p. 697–713. Uber die Entstehung von wirbeln in der idealen Flüssigkeit mit Anwendung auf die Tragflugel Theorie und andere Aufgabe.

    Google Scholar 

  • L. TING 1971-Studies in the motion and decay of vortices. In Olsen & al 1971 p. 11–39.

    Google Scholar 

  • L. TING, C. TUNG 1965-Motion and decay of a vortex in a non uniform stream. Phys. Fluids 8, p. 1039–1051.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • C. TUNG, L. TING 1967-Motion and decay of vortex rings. Phys. Fluids 10, p. 901–910.

    CrossRef  Google Scholar 

  • S.E. WIDNALL, D. BLISS, A. ZALAY 1971-Theoretical and experimental study of the stability of a vortex pair. In Olsen & al. 1971, p. 305–338.

    Google Scholar 

  • S.E. WIDNALL, J.P. SULLIVAN 1973-On the stability of vortex rings. Proc. Roy. Soc. A 332, p. 335–353.

    CrossRef  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1977 Springer-Verlag

About this paper

Cite this paper

Guiraud, JP. (1977). The dynamics of rolled vortex sheets tightly winded around slender vortex filaments in inviscid incompressible flow. In: Brauner, CM., Gay, B., Mathieu, J. (eds) Singular Perturbations and Boundary Layer Theory. Lecture Notes in Mathematics, vol 594. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086090

Download citation

  • DOI: https://doi.org/10.1007/BFb0086090

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08258-3

  • Online ISBN: 978-3-540-37340-7

  • eBook Packages: Springer Book Archive