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Vortex sheet roll-up revisited

Published online by Cambridge University Press:  18 September 2018

A. C. DeVoria Open the ORCID record for A. C. DeVoria [Opens in a new window]  and
K. Mohseni Open the ORCID record for K. Mohseni [Opens in a new window]
A. C. DeVoria
Affiliation:
Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611, USA
K. Mohseni*
Affiliation:
Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611, USA Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611, USA
*
†Email address for correspondence: mohseni@ufl.edu
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Abstract

The classical problem of roll-up of a two-dimensional free inviscid vortex sheet is reconsidered. The singular governing equation brings with it considerable difficulty in terms of actual calculation of the sheet dynamics. Here, the sheet is discretized into segments that maintain it as a continuous object with curvature. A model for the self-induced velocity of a finite segment is derived based on the physical consideration that the velocity remain bounded. This allows direct integration through the singularity of the Birkhoff–Rott equation. The self-induced velocity of the segments represents the explicit inclusion of stretching of the sheet and thus vorticity transport. The method is applied to two benchmark cases. The first is a finite vortex sheet with an elliptical circulation distribution. It is found that the self-induced velocity is most relevant in regions where the curvature and the sheet strength or its gradient are large. The second is the Kelvin–Helmholtz instability of an infinite vortex sheet. The critical time at which the sheet forms a singularity in curvature is accurately predicted. As observed by others, the vortex sheet strength forms a finite-valued cusp at this time. Here, it is shown that the cusp value rapidly increases after the critical time and is the impetus that initiates the roll-up process.

JFM classification

Vortex Flows: Vortex dynamics Vortex Flows: Vortex Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 855 , 25 November 2018 , pp. 299 - 321
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Copyright
© 2018 Cambridge University Press 

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References

Baker, G. R. 1980 A test of the method of Fink and Soh for following vortex-sheet motion. J. Fluid Mech. 100 (1), 209–220.Google Scholar
Baker, G. R. 1983 Generalized vortex methods for free surface flows. In Waves on Fluid Interfaces, pp. 53–81. Academic Press.Google Scholar
Baker, G. R. & Pham, L. D. 2006 A comparison of blob methods for vortex sheet roll-up. J. Fluid Mech. 547, 297–316.Google Scholar
Chorin, A. J. & Bernard, P. S. 1973 Discretization of a vortex sheet with an example of roll-up. J. Comput. Phys. 13, 423–429.Google Scholar
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
Higdon, J. J. L. & Pozrikidis, C. 1985 The self-induced motino of vortex sheets. J. Fluid Mech. 150, 203–231.Google Scholar
Hoeijmakers, H. W. M. & Vaatstra, W. 1983 A higher-order panel method applied to vortex sheet roll-up. AIAA J. 21 (4), 516–523.Google Scholar
Kaden, H. 1931 Aufwicklung einer unstabilen Unstetigkeitsflache. Ing.-Arch. 79, 93–112.Google Scholar
Krasny, R. 1986a Desingularization of periodic vortex sheet roll-up. J. Comput. Phys. 65 (2), 292–313.Google Scholar
Krasny, R. 1986b A study of singularity formation in a vortex sheet by the point-vortex approximation. J. Fluid Mech. 167, 65–93.Google Scholar
Krasny, R. 1987 Computation of vortex sheet roll-up in the Trefftz plane. J. Fluid Mech. 184, 123–155.Google Scholar
Krasny, R. & Nitsche, M. 2002 The onset of chaos in vortex sheet flow. J. Fluid Mech. 454, 47–69.Google Scholar
Lindsay, K. & Krasny, R. 2001 A particle method and adaptive treecode for vortex sheet motion in three-dimensional flow. J. Comput. Phys. 172, 879–907.Google Scholar
Meiron, D. I., Baker, G. R. & Orszag, S. A. 1982 Analytic structure of vortex sheet dynamics. Part I. Kelvin–Helmholtz instability. J. Fluid Mech. 123, 477–501.Google Scholar
Moore, D. W.1971 The discrete vortex approximation of a vortex sheet. Report 1084-69. Airforce Office of Scientific Research.Google Scholar
Moore, D. W. 1974 A numerical study of the roll-up of a finite vortex sheet. J. Fluid Mech. 63, 225–235.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
Nitsche, M. & Krasny, R. 1994 A numerical study of vortex ring formation at the edge of a circular tube. J. Fluid Mech. 276, 139–161.Google Scholar
Pozrikidis, C. 2000 Theoretical and computation aspects of the self-induced motion of three-dimensional vortex sheets. J. Fluid Mech. 425, 335–366.Google Scholar
Pullin, D. I. 1978 The large-scale structure of unsteady self-similar rolled-up vortex sheets. J. Fluid Mech. 88 (3), 401–430.Google Scholar
Pullin, D. I. & Phillips, W. R. C. 1981 On a generalization of Kaden’s problem. J. Fluid Mech. 104, 45–53.Google Scholar
Rosenhead, L. 1931 The formation of vortices from a surface of discontinuity. Proc. R. Soc. A 134 (823), 170–192.Google Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Saffman, P. G. & Baker, G. R. 1979 Vortex interactions. Annu. Rev. Fluid Mech. 11, 95–122.Google Scholar
Smith, J. H. B. 1968 Improved calculations of leading-edge separation from slender, thin, delta wings. Proc. R. Soc. Lond. A 306, 67–90.Google Scholar
Sohn, S.-I., Yoon, D. & Hwang, W. 2010 Long-time simulations of the Kelvin–Helmholtz instability using an adaptive vortex method. Phys. Rev. E 82 (4), 046711.Google Scholar
Sugioka, I. & Widnall, S. E.1985 A panel method study of vortex sheets with special emphasis on sheets of axisymmetric geometry. Tech. Rep. NASA-CR-177365. Massachusetts Inst. of Tech.; Fluid Dynamics Research Lab.; Cambridge, MA, NASA Contract Report.Google Scholar
Westwater, F. L.1935 The rolling up of a surface of discontinuity behind an aerofoil of finite span. Report R&M 1692. Aeronautical Research Council.Google Scholar