Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-22T07:21:16.671Z Has data issue: false hasContentIssue false
  • English
  • Français

Equation of motion of a diffusing vortex sheet

Published online by Cambridge University Press:  26 April 2006

M. R. Dhanak
M. R. Dhanak
Affiliation:
Department of Ocean Engineering, Florida Atlantic University, Boca Raton, FL 33431, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Moore's (1978) equation for following the evolution of a thin layer of uniform vorticity in two dimensions is extended to the case of a non-uniform, instantaneously known, vorticity distribution, using the method of matched asymptotic expansions. In general, the vorticity distribution satisfies a boundary-layer equation. This has a similarity solution in the case of a vortex layer of small thickness in a viscous fluid. Using this solution, an equation of motion of a diffusing vortex sheet is obtained. The equation retains the simplicity of Birkhoff's integro-differential equation for a vortex sheet, while incorporating the effect of viscous diffusion approximately. The equation is used to study the growth of long waves on a Rayleigh layer.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 269 , 25 June 1994 , pp. 265 - 281
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

Birkoff, G. 1962 Proc. Symp. Appl. Math. Am. Math. Soc. 13, 55.
Baker, G. R. & Shelley, M. J. 1990 J. Fluid Mech. 215, 161–194.
Dhanak, M. R. 1980 The motion of vortex layers and vortex filaments. PhD dissertation. Imperial College, London.
Dhanak, M. R. 1994 Stud. Appl. Maths. In press.
Drazin, P. G. & Howard, L. N. 1962 J. Fluid Mech. 14, 257.
Goldstein, S. 1938 Modern Developments in Fluid Mechanics, vol. I. Clarendon.
Moore, D. W. 1978 Stud. Appl. Maths 58, 119–140.
Moore, D. W. 1981 SIAM J. Sci. Stat. Comput. 2, 65–84.Google Scholar
Rayleigh, Lord 1894 Theory of Sound, 2nd edn, chap. XXI. Macmillan.
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.
Tryggvason, G., Dahm, W. J. A. & Skeih, K. 1991 Trans. ASME I: J. Fluids Engng 113, 31–36.