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Asymptotic vorticity structure and numerical simulation of slender vortex filaments

Published online by Cambridge University Press:  26 April 2006

Rupert Klein  and
Omar M. Knio
Rupert Klein
Affiliation:
Institut für Technische Mechanik, RWTH, Templergraben 64, 52056 Aachen, Germany
Omar M. Knio
Affiliation:
Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA
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Abstract

A new asymptotic analysis of slender vortices in three dimensions, based solely on the vorticity transport equation and the non-local vorticity–velocity relation gives new insight into the structure of slender vortex filaments. The approach is quite different from earlier analyses using matched asymptotic solutions for the velocity field and it yields additional information. This insight is used to derive three different modifications of the thin-tube version of a numerical vortex element method. Our modifications remove an O(1) error from the node velocities of the standard thin-tube model and allow us to properly account for any prescribed physical vortex core structure independent of the numerical vorticity smoothing function. We demonstrate the performance of the improved models by comparison with asymptotic solutions for slender vortex rings and for perturbed slender vortex filaments in the Klein–Majda regime, in which the filament geometry is characterized by small-amplitude–short-wavelength displacements from a straight line. These comparisons represent a stringent mutual test for both the proposed modified thin-tube schemes and for the Klein–Majda theory. Importantly, we find a convincing agreement of numerical and asymptotic predictions for values of the Klein–Majda expansion parameter ε as large as ½. Thus, our results support their findings in earlier publications for realistic physical vortex core sizes.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 284 , 10 February 1995 , pp. 275 - 321
Copyright
© 1995 Cambridge University Press

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References

Arms, R. J. & Hama, F. R. 1965 Phys, Fluids 8, 533.
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Beale, J. T. 1986 Math. Comput. 46, 401.
Beale, J. T. & Majda, A. 1982a Math. Comput. 39, 1.
Beale, J. T. & Majda, A. 1982b Math. Comput. 39, 29.
Beale, J. T. & Majda, A. 1985 J. Comput. Phys. 58, 188.
Callegari, A. J. & Ting, L. 1978 SIAM J. Appl. Maths. 35, 148.
Chorin, A. J. 1980 SIAM Sci. Statist. Comput. 1, 1.
Chorin, A. J. 1982 Commun. Math. Phys. 83, 517.
Chorin, A. J. & Akao, J. 1991 Physica D 52, 403.
Crow, S. C. 1970 AIAA J. 8, 2172.
Fraenkel, L. E. 1972 J. Fluid Mech. 51, 119.
Ghoniem, A. F., Heidarinejad, G. & Krishnan, A. 1988 J. Comput. Phys. 79, 135.
Greengard, C. 1986 Math. Comput. 47, 387.
Hasimoto, H. 1972 J. Fluid Mech. 51, 477.
Klein, R. 1994 Zur Dynamik schlanker Wirbel. einger. zur Habilitation. RWTH Aachen.
Klein, R. & Majdà, A. 1991c Physica D 49, 323.
Klein, R. & Majda, A. 1991b Physica D 53, 267.
Klein, R. & Majda, A. 1993 Phys. Fluids A 5, 369.
Klein, R., Majda, A. & Mclaughlin, R. M. 1992 Phys. Fluids A 4, 2271.
Klein, R. & Ting, L. 1992 Appl. Math. Lett. 5, 99–103.
Klein, R., Ting, L. & Knio, O. M. 1994 Accurate numerical computation of stretched, high-Reynolds-number, slender vortices in three space dimensions. In preparation for 3rd. Intl Congress on Industrial and Applied Maths, Hamburg, Germany, June 1995.
Knio, O. M. & Ghoniem, A. F. 1990 J. Comput. Phys. 86, 75.
Knio, O. M. & Ghoniem, A. F. 1991 J. Comput. Phys. 97, 172.
Knio, O. M. & Ghoniem, A. F. 1992 J. Fluid Mech. 243, 353.
Krasny, R. 1983 J. Fluid Mech. 167, 65.
Leonard, A. 1985 Ann. Rev. Fluid Mech. 17, 525.
Liu, C. H., Tavantsis, J. & Ting, L. 1986 AIAA J. 24, 1290.
Pumir, A. & Siggia, E. 1990 Phys. Fluids A 2, 220.
Rosenhead, L. 1930 Proc. R. Soc. Lond. A 127, 590.
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.
Thomson, J. J. 1983 A Treatise on the Motion of Vortex Rings. Macmillan.
Ting, L. 1971 In Aircraft Wake Turbulence and its Detection (ed. J. H. Olsen, A. Goldburg & M. Rogers), pp. 11–39. Plenum.
Ting, L. & Klein, R. 1991 Viscous Vortical Flows, Lecture Notes in Physics, vol. 374. Springer.
Vitting, T. 1991 Struktur von Flugzeugrandwirbeln und Massnahmen zur Wirbelabschwaechung. VDI Verlag Duesseldorf, Germany.
Widnall, S. 1975 Ann. Rev. Fluid Mech. 7, 141–165.