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The self-induced motion of vortex sheets

Published online by Cambridge University Press:  20 April 2006

J. J. L. Higdon  and
C. Pozrikidis
J. J. L. Higdon
Affiliation:
Department of Chemical Engineering, University of Illinois, Urbana. IL 61801
C. Pozrikidis
Affiliation:
Department of Chemical Engineering, University of Illinois, Urbana. IL 61801
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Abstract

A method is presented for following the self-induced motion of vortex sheets. In this method, we use a piecewise analytic representation of the sheet consisting of circular arcs with trigonometric polynomials for the circulation. The procedure is used to study the evolution of the motion in two special cases: a circular vortex sheet with sinusoidal circulation distribution and an infinite plane vortex sheet subject to periodic disturbances. The first problem was studied by Baker (1980) as a test of the method of Fink & Soh (1978), while the second has been studied by a number of authors, notably Meiron, Baker & Orszag (1982). In each case, we follow the motion of the sheet up to the appearance of a singularity at a finite time. The singularity takes the form of an exponential spiral with the simultaneous development of singularities in the curvature and in the circulation distribution. In the final stages of the calculations, up to 155 marker points are used to specify the position of the sheet. If it were possible to execute a stable calculation with equally spaced point vortices, approximately 106 points would be required to achieve the same resolution. Problems with instabilities have been reduced, but not entirely eliminated, and prevent a rigorous verification of the results obtained.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 150 , January 1985 , pp. 203 - 231
Copyright
© 1985 Cambridge University Press

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References

Baker G. R.1980 A test of the method of Fink & Soh for following vortex-sheet motion. J. Fluid Mech. 100, 209–220.Google Scholar
Batchelor G. K.1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Birkhoff G.1962 Helmholtz and Taylor instability. Proc. Symp. Appl. Maths Am. Math. Soc. 13, 55–76.Google Scholar
Bofah K. K.1975 Ph.D. thesis, California Institute of Technology.
Fink, P. T. & Soh W. K.1974 Calculation of vortex sheets in unsteady flow and applications in ship hydrodynamics. Proc. 10th Symp. Naval Hydrodynamics, Cambridge, Mass., pp. 463–488.
Fink, P. T. & Soh W. K.1978 A new approach to roll-up calculations of vortex sheets Proc. R. Soc. Lond. A 362, 195–209.Google Scholar
Gaunt, D. S. & Guttmann A. J.1974 Asymptotic analysis of coefficients. In Phase Transitions and Critical Phenomena, vol. 3. Academic.
Helmholtz H. Von1868 Über discontinuirliche Flüssigkeitsbewegungen. Monatsber. Akad. Wiss. Berlin 1868, pp. 215–228.Google Scholar
Longuet-Higgins, M. S. & Cokelet E. L.1976 The deformation of steep surface waves on water. I. A numerical method of computation Proc. R. Soc. Lond. A 350, 1–26.Google Scholar
Mangler, K. W. & Smith J. H. B.1959 A theory of the flow past a slender delta wing with leading edge separation Proc. R. Soc. Lond. A 251, 200–217.Google Scholar
Meiron D. I., Baker, G. R. & Orsag S. A.1982 Analytical structure of vortex-sheet dynamics. Part 1. Kelvin-Helmholtz instability. J. Fluid Mech. 114, 283–298.Google Scholar
Moore D. W.1979 The spontaneous appearance of a singularity in the shape of an evolving vortex sheet Proc. R. Soc. Lond. A 365, 105–119.Google Scholar
Moore D. W.1981 On the point vortex method. SIAM J. Sci. Stat. Comp. 2, 65–84.Google Scholar
Pozrikidis C.1983 M.S. thesis, University of Illinois.
Pullin D. I.1982 Numerical studies of surface tension effects in nonlinear Kelvin—Helmholtz and Rayleigh—Taylor instability. J. Fluid Mech. 119, 57–532.Google Scholar
Rosenhead L.1931 The formation of vortices from a surface of discontinuity Proc. R. Soc. Lond. A 134, 170–192.Google Scholar
Saffman, P. G. & Baker G. R.1979 Vortex interactions. Ann. Rev. Fluid Mech. 11, 95–122.Google Scholar
Schwartz L. W.1981 A semi-analytical approach to the self induced motion of vortex sheets. J. Fluid Mech. 111, 475–490.Google Scholar
Van De Vooren A. I.1980 A numerical investigation of the rolling up of vortex sheets Proc. R. Soc. London A 373, 67–91.Google Scholar
Van Dyke M.1974 Analysis and improvement of perturbation series. Q. J. Mech. Appl. Maths 27, 423–450.Google Scholar
Westwater F. L.1935 Rolling up of the surface of discontinuity behind an aerofoil of finite span. Aero. Res. Comm. R. & M. 1692.Google Scholar