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Contour dynamics in complex domains

Published online by Cambridge University Press:  23 November 2007

DARREN CROWDY  and
AMIT SURANA
DARREN CROWDY
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology 2-384, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
AMIT SURANA
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology 3-335, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
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Abstract

This paper demonstrates that there is a contour dynamics formulation for the evolution of uniform vortex patches in any finitely connected planar domain bounded by impenetrable walls. A general numerical scheme is presented based on this formulation. The algorithm makes use of conformal mappings and follows the evolution of a conformal pre-image of a given vortex patch in a canonical multiply connected circular pre-image region. The evolution of vortex patches can be computed given just the conformal map from this pre-image region to the physical fluid region. The efficacy of the scheme is demonstrated by illustrative examples.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 593 , 25 December 2007 , pp. 235 - 254
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Ablowitz, M. & Fokas, A. S. 1995 Complex Variables. Cambridge University Press.Google Scholar
Backhaus, E. Y., Fajans, J. & Wurtele, J. S. 1998 Application of contour dynamics to systems with cylindrical boundaries. J. Comput. Phys. 145, 462–468.CrossRefGoogle Scholar
Coppa, G. G. M., Peano, F. & Peinetti, F. 2002 Image-charge method for contour dynamics in systems with cylindrical boundaries. J. Comput. Phys. 182, 392–417.CrossRefGoogle Scholar
Crowdy, D. G. 2006 Point vortex motion of the surface of a sphere with impenetrable boundaries. Phys. Fluids 18, 036602.CrossRefGoogle Scholar
Crowdy, D. G. & Marshall, J. S. 2005 a Analytical formulae for the Kirchoff–Routh path function in multiply connected domains. Proc. R. Soc. 461, 2477–2501.CrossRefGoogle Scholar
Crowdy, D. G. & Marshall, J. S. 2005 b The motion of a point vortex around multiple circular islands. Phys. Fluids 17, 056602.CrossRefGoogle Scholar
Crowdy, D. G. & Marshall, J. S. 2006 The motion of a point vortex through gaps in walls. J. Fluid Mech. 551, 31–48.CrossRefGoogle Scholar
Crowdy, D. G. & Marshall, J. S. 2007 Computing the Schottky–Klein prime function on the Schottky double of planar domains. Comput. Meth. Func. Theory 7 (1), 293–308.CrossRefGoogle Scholar
Driscoll, T. & Trefethen, L. N. 2002 Schwarz–Christoffel Mappings. Cambridge University Press.CrossRefGoogle Scholar
Dritschel, D. G. 1988 a Contour surgery: a topological reconnection scheme for extended integrations using contour dynamics. J. Comput. Phys. 77, 240–266.CrossRefGoogle Scholar
Dritschel, D. G. 1988 b Contour dynamics/surgery on the sphere. J. Comput. Phys. 79, 477.CrossRefGoogle Scholar
Flucher, M. 1999 Variational Problems with Concentration. Birkhauser.CrossRefGoogle Scholar
Flucher, M. & Gustafsson, B. 1997 Vortex motion in two dimensional hydrodynamics. TRITA-MAT-1997-MA 02, Stockholm, Royal Institute of Technology.Google Scholar
Goluzin, G. M. 1969 Geometric Theory of Functions of a Complex Variable. Translations of Mathematical Monographs, 26, American Mathematical Society.CrossRefGoogle Scholar
Johnson, E. R. & McDonald, N. R. 2005 a The motion of a vortex near two circular cylinders. Proc. R. Soc. Lond. A 460, 939–954.CrossRefGoogle Scholar
Johnson, E. R. & McDonald, N. R. 2005 b The motion of a vortex near a gap in a wall. Phys. Fluids 16 (2), 462.CrossRefGoogle Scholar
Johnson, E. R. & McDonald, N. R. 2005 c Vortices near barriers with multiple gaps. J. Fluid Mech. 531, 335–358.CrossRefGoogle Scholar
Jomaa, Z. & Macaskill, C. 2005 The embedded finite-difference method for the Poisson equation in a domain with an irregular boundary and Dirichlet boundary conditions. J. Comput. Phys. 202, 488–506.CrossRefGoogle Scholar
Kidambi, R. & Newton, P. K. 2000 Point vortex motion on a sphere with solid boundaries. Phys. Fluids 12, 581.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Lin, C. C. 1941 a On the motion of vortices in two dimensions. I: Existence of the Kirchhoff–Routh function. Proc. Natl Acad. Sci. 27, 570–575.CrossRefGoogle ScholarPubMed
Lin, C. C. 1941 b On the motion of vortices in two dimensions. II: Some further investigations of the Kirchhoff–Routh function. Proc. Natl Acad. Sci. 27, 575–577.CrossRefGoogle ScholarPubMed
Love, A. E. H. 1983 On the stability of certain vortex motions. Proc. Lond/ Math. Soc. 25, 18–42.Google Scholar
Macaskill, C. Padden, W. E. P. & Dritschel, D. G. 2003 The CASL algorithm for quasi-geostrophic flow in a cylinder. J. Comput. Phys. 188, 232–251.CrossRefGoogle Scholar
Mitchell, T. B. Wang, X. J. & Rossi, L. F. 2006 Stability of elliptical electron vortices. In Non-neutral Plasma Physics VI, Workshop on Non-neutral Plasmas (ed M. Drewsen, U. Uggerhoj & H. Knudsen).Google Scholar
Nehari, Z. 1982 Conformal Mapping. Dover.Google Scholar
Newton, P. K. 2001 The N-Vortex Problem. Springer.CrossRefGoogle Scholar
Pozrikidis, C. 1997 Introduction to Theoretical and Computational Fluid Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Pullin, D. I. 1981 The nonlinear behaviour of a constant vorticity layer at a wall. J. Fluid Mech, 108, 401.CrossRefGoogle Scholar
Pullin, D. I. 1992 Contour dynamics methods Annu. Rev. Fluid Mech. 24, 89–115.CrossRefGoogle Scholar
Routh, E. J. 1881 Some applications of conjugate functions. Proc. Lond. Math. Soc. 12, 73–89.Google Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Zabusky, N. J., Hughes, M. H. & Roberts, K. V. 1979 Contour dynamics for the Euler equations in two dimensions. J. Comput. Phys. 30, 96.CrossRefGoogle Scholar