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A moment model for vortex interactions of the two-dimensional Euler equations. Part 1. Computational validation of a Hamiltonian elliptical representation

Published online by Cambridge University Press:  21 April 2006

M. V. Melander ,
N. J. Zabusky  and
A. S. Styczek
M. V. Melander
Affiliation:
Institute for Computational Mathematics and Applications, Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
N. J. Zabusky
Affiliation:
Institute for Computational Mathematics and Applications, Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
A. S. Styczek
Affiliation:
Institute for Computational Mathematics and Applications, Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260 Permanent address: Institut Techniki Cieplnej, Politechnika Warszawska, Warsaw, Poland.
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Abstract

We consider the evolution of finite uniform-vorticity regions in an unbounded in viscid fluid. We perform a perturbation analysis based on the assumption that the regions are remote from each other and nearly circular. Thereby we obtain a self-consistent infinite system of ordinary differential equations governing the physical-space moments of the individual regions. Truncation yields an Nth-order moment model. Special attention is given to the second-order model where each region is assumed elliptical. The equations of motion conserve local area, global centroid, total angular impulse and global excess energy, and the system can be written in canonical Hamiltonian form. Computational comparison with solutions to the contour-dynamical representation of the Euler equations shows that the model is useful and accurate. Because of the internal degrees of freedom, namely aspect ratio and orientation, two like-signed vorticity regions collapse if they are near each other. Although the model becomes invalid during a collapse, we find a striking similarity with the merger process of the Euler equations.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 167 , June 1986 , pp. 95 - 115
Copyright
© 1986 Cambridge University Press

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References

Aref, H. 1983 Integrable, chaotic, and turbulent vortex motion in two-dimensional flows. Ann. Rev. Fluid Mech. 15, 345–389.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Mechanics. Cambridge University Press.
Beale, J. T. & Majda, A. 1984 Higher order accurate vortex methods with explicit velocity kernels. J. Comp. Phys. 58, 188–208.Google Scholar
Dritschel, D. 1985 The stability and energetics of corotating uniform vortices. J. Fluid Mech. 157, 95–134.Google Scholar
Hernan, M. A. & Jimenez, J. 1982 Computer analysis of a high-speed film of the plane turbulent mixing layer. J. Fluid Mech. 119, 323–345.Google Scholar
Howard, L. N. 1957/58 Arch. Rat. Mech. Anal. 1, 113–123.
Joseph, D. D. & Nield, D. A. 1975 Stability of bifurcating time-periodic and steady state solutions of arbitrary amplitude. Arch. Rat. Mech. Anal. 58, 369.Google Scholar
Kida, S. 1981 Motion of an elliptic vortex in a uniform shear flow. J. Phys. Soc. Japan 50, 3517–3520.Google Scholar
Leonard, A. 1980 Vortex methods for flow simulation. J. Comp. Phys. 37, 289–355.Google Scholar
Mc Williams, J. C. 1984 The emergence of isolated vortices in turbulent flow. J. Fluid Mech. 146, 21–43.Google Scholar
Melander, M. V., McWillams, J. C. & Zabusky, N. J. 1986 A model for symmetric vortex-merger. Trans. 3rd Army Conf. on Applied Mathematics and Computing; ARO Rep. 867.
Melander, M. V., Styczek, A. S. & Zabusky, N. J. 1984 Elliptically desingularized vortex model for the two-dimensional Euler equations. Phys. Rev. Lett. 53, 1222–1225.Google Scholar
Nakamura, Y., Leonard, A. & Spalart, P. 1982 Vortex simulation of an inviscid shear layer. A1AA/ASME 3rd Joint Thermophysics, Fluids, Plasma and Heat Transfer Conf.; Paper AIAA-82-0948.
Neu, J. C. 1984 The dynamics of a columnar vortex in an imposed strain. Phys. Fluids 27, 2397–2402.Google Scholar
Overman, E. A. & Zabusky, N. J. 1982 Evolution and merger of isolated vortex structures. Phys. Fluids 25, 1297–1305.Google Scholar
Poincaré, H. 1893 Theorie des Tourbillons (ed. G. Carre), chap. IV. Deslis Freres.
Saffman, P. G. & Szeto, R. 1980 Equilibrium shapes of a pair of equal uniform vortices. Phys. Fluids 23, 2339–2342.Google Scholar
Winant, C. D. & Browand, F. K. 1974 Vortex pairing: a mechanism of turbulent mixing layer growth at moderate Reynolds number. J. Fluid Mech. 63, 237–255.Google Scholar
Wu, H. M., Overman, E. A. & Zabusky, N. J. 1984 Steady state solutions of the Euler equations in two dimensions. Rotating and translating V-states with limiting cases. I. Numerical results. J. Comp. Phys. 53, 42–71.Google Scholar
Zabusky, N. J. 1984 Contour dynamics: a method for inviscid and nearly inviscid two-dimensional flows. In Proc. IUTAM Symp. on Turbulence and Chaotic Phenomena, pp. 251–257. North-Holland.
Zabusky, N. J., Hughes, M. H. & Roberts, K. V. 1979 Contour dynamics for the Euler equations in two dimensions. J. Comp. Phys. 30, 96–106.Google Scholar