Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-06-16T21:48:19.715Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite rotating and translating vortex sheets

Published online by Cambridge University Press:  29 July 2021

Bartosz Protas Open the ORCID record for Bartosz Protas [Opens in a new window] ,
Stefan G. Llewellyn Smith Open the ORCID record for Stefan G. Llewellyn Smith [Opens in a new window]  and
Takashi Sakajo Open the ORCID record for Takashi Sakajo [Opens in a new window]
Bartosz Protas*
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada
Stefan G. Llewellyn Smith
Affiliation:
Department of Mechanical and Aerospace Engineering, Jacobs School of Engineering, UCSD, La Jolla, CA 92093-0411, USA Scripps Institution of Oceanography, UCSD, La Jolla, CA 92093-0209, USA
Takashi Sakajo
Affiliation:
Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan
*
†Email address for correspondence: bprotas@mcmaster.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the rotating and translating equilibria of open finite vortex sheets with endpoints in two-dimensional potential flows. New results are obtained concerning the stability of these equilibrium configurations which complement analogous results known for unbounded, periodic and circular vortex sheets. First, we show that the rotating and translating equilibria of finite vortex sheets are linearly unstable. However, while in the first case unstable perturbations grow exponentially fast in time, the growth of such perturbations in the second case is algebraic. In both cases the growth rates are increasing functions of the wavenumbers of the perturbations. Remarkably, these stability results are obtained entirely with analytical computations. Second, we obtain and analyse equations describing the time evolution of a straight vortex sheet in linear external fields. Third, it is demonstrated that the results concerning the linear stability analysis of the rotating sheet are consistent with the infinite aspect ratio limit of the stability results known for Kirchhoff's ellipse (Love, Proc. Lond. Math. Soc., vol. s1–25, 1893, pp. 18–43; Mitchell & Rossi, Phys. Fluids, vol. 20, 2008, p. 054103) and that the solutions we obtained accounting for the presence of external fields are also consistent with the infinite aspect ratio limits of the analogous solutions known for vortex patches.

JFM classification

Vortex Flows: Vortex instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 923 , 25 September 2021 , A23
Check for updates
Copyright
© The Author(s), 2021. Published by 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

REFERENCES

Alben, S. 2009 Simulating the dynamics of flexible bodies and vortex sheets. J. Comput. Phys. 228, 2587–2603.CrossRefGoogle Scholar
Alben, S. 2015 Flag flutter in inviscid channel flow. Phys. Fluids 27, 033603.CrossRefGoogle Scholar
Baker, G.R. & Shelley, M.J. 1990 On the connection between thin vortex layers and vortex sheets. J. Fluid Mech. 215, 161–194.CrossRefGoogle Scholar
Batchelor, G.K. 1988 An Introduction to Fluid Dynamics, 7th edn. Cambridge University Press.Google Scholar
Benedetto, D. & Pulvirenti, M. 1992 From vortex layers to vortex sheets. SIAM J. Appl. Maths 52, 1041–1056.CrossRefGoogle Scholar
Boyd, J.P. 2001 Chebyshev and Fourier Spectral Methods. Dover.Google Scholar
Brady, M., Leonard, A. & Pullin, D.I. 1998 Regularized vortex sheet evolution in three dimensions. J. Comput. Phys. 146, 520–545.CrossRefGoogle Scholar
Caflisch, R.E. (Ed.) 1989 Mathematical Aspects of Vortex Dynamics. SIAM.Google Scholar
DeVoria, A.C. & Mohseni, K. 2018 Vortex sheet roll-up revisited. J. Fluid Mech. 855, 299–321.CrossRefGoogle Scholar
DLMF 2020 NIST Digital Library of Mathematical Functions. Available at: http://dlmf.nist.gov/, release 1.1.0 of 2020-12-15 (ed. F.W.J. Olver, A.B. Olde Daalhuis, D.W. Lozier, B.I. Schneider, R.F. Boisvert, C.W. Clark, B.R. Miller, B.V. Saunders, H.S. Cohl & M.A. McClain).Google Scholar
Elling, V. & Gnann, M.V. 2019 Variety of unsymmetric multibranched logarithmic vortex spirals. Eur. J. Appl. Maths 30, 23–38.CrossRefGoogle Scholar
Estrada, R. & Kanwal, R.P. 2012 Singular Integral Equations. Birkhäuser.Google Scholar
Jones, M.A. 2003 The separated flow of an inviscid fluid around a moving flat plate. J. Fluid Mech. 496, 405–441.CrossRefGoogle Scholar
Jones, M.A. & Shelley, M.J. 2005 Falling cards. J. Fluid Mech. 540, 393–425.CrossRefGoogle Scholar
Kida, S. 1981 Motion of an elliptic vortex in a uniform shear flow. J. Phys. Soc. Japan 50, 3517–3520.CrossRefGoogle Scholar
Krasny, R. 1986 a Desingularization of periodic vortex sheet roll-up. J. Comput. Phys. 65, 292–313.CrossRefGoogle Scholar
Krasny, R. 1986 b A study of singularity formation in a vortex sheet by the point vortex approximation. J. Fluid Mech. 167, 65–93.CrossRefGoogle Scholar
Krasny, R. 1987 Computation of vortex sheet roll-up in the Trefftz plane. J. Fluid Mech. 184, 123–155.CrossRefGoogle Scholar
Krasny, R. & Nitsche, M. 2002 The onset of chaos in vortex sheet flow. J. Fluid Mech. 454, 47–69.CrossRefGoogle Scholar
Lopes Filho, M.C., Nussenzveig Lopes, H.J. & Schochet, S. 2007 A criterion for the equivalence of the Birkhoff-Rott and Euler descriptions of vortex sheet evolution. Trans. Am. Math. Soc. 359, 4125–4142.CrossRefGoogle Scholar
Lopes Filho, M.C., Nussenzeig Lopes, H.J. & Souza, M.O. 2003 On the equation satisfied by a steady Prandtl-Munk vortex sheet. Commun. Math. Sci. 1, 68–73.CrossRefGoogle Scholar
Love, A.E.H. 1893 On the stability of certain vortex motions. Proc. Lond. Math. Soc. s1–25, 18–43.CrossRefGoogle Scholar
Majda, A.J. & Bertozzi, A.L. 2002 Vorticity and Incompressible Flow. Cambridge University Press.Google Scholar
Marchioro, C. & Pulvirenti, M. 1993 Mathematical Theory of Incompressible Nonviscous Fluids. Springer.Google Scholar
Michalke, A. & Timme, A. 1967 On the inviscid instability of certain two-dimensional vortex-type flows. J. Fluid Mech. 29, 647–666.CrossRefGoogle Scholar
Milne-Thomson, L.M. 1973 Theoretical Aerodynamics. Dover Publications.Google Scholar
Mitchell, T.B. & Rossi, L.F. 2008 The evolution of Kirchhoff elliptic vortices. Phys. Fluids 20, 054103.CrossRefGoogle Scholar
Moore, D.W. 1979 The spontaneous appearance of a singularity in the shape of an evolving vortex sheet. Proc. R. Soc. A 365, 105–119.Google Scholar
Moore, D.W. & Saffman, P.G. 1971 Structure of a line vortex in an imposed strain. In Aircraft Wake Turbulence (ed. G. Olsen & E. Rogers), pp. 339–354. Plenum.CrossRefGoogle Scholar
Munk, M.M. 1919 Isoperimetrische Aufgaben aus der Theorie des Fluges. PhD thesis, University of Göttingen.Google Scholar
Muskhelishvili, N.I. 2008 Singular Integral Equations. Boundary Problems of Function Theory and Their Application to Mathematical Physics, 2nd edn. Dover.Google Scholar
O'Neil, K.A. 2018 a Dipole and multipole flows with point vortices and vortex sheets. Regular Chaotic Dyn. 23, 519–529.CrossRefGoogle Scholar
O'Neil, K.A. 2018 b Relative equilibria of point vortices and linear vortex sheets. Phys. Fluids 30, 107101.CrossRefGoogle Scholar
Perko, L. 2008 Differential Equations and Dynamical Systems. Texts in Applied Mathematics 7. Springer.Google Scholar
Protas, B. & Sakajo, T. 2020 Rotating equilibria of vortex sheets. Physica D: Nonlinear Phenom. 403, 132286.CrossRefGoogle Scholar
Pullin, D.I. & Sader, J.E. 2021 On the starting vortex generated by a translating and rotating flat plate. J. Fluid Mech. 906, A9.CrossRefGoogle Scholar
Saffman, P.G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Sakajo, T. 2001 Numerical computation of a three-dimensional vortex sheet with swirl flow. Fluid Dyn. Res. 28 (6), 423–448.CrossRefGoogle Scholar
Sakajo, T. & Okamoto, H. 1996 Numerical computation of vortex sheet roll-up in the background shear flow. Fluid Dyn. Res. 17, 195–212.CrossRefGoogle Scholar