×

zbMATH — the first resource for mathematics

Adaptive panel representation for 3D vortex ring motion and instability. (English) Zbl 1233.76083
Summary: We develop a hierarchical panel method for representing vortex sheet surface motion in 3D flows. Unlike previously employed filament methods, each panel is a leaf of the tree, so it can be subdivided locally, which allows for an efficient adaptive point insertion. In addition, we develop curvature-based insertion criteria which allow to localize point insertion to the most complicated curved regions of the sheet. The particles representing the sheet are advected by a regularized Biot-Savart integral with Rosenhead-Moore kernel. The particle velocities are evaluated by an adaptive tree-code algorithm based on Taylor expansions in Cartesian coordinates due to K. Lindsay and R. Krasny [J. Comput. Phys. 172, No. 2, 879–907 (2001; Zbl 1002.76093)]. The method allows to consider much later stages of a vortex ring instability, which may shed light on this complicated flow phase directly leading to the turbulent flow.
MSC:
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76B47 Vortex flows for incompressible inviscid fluids
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] K. Marten, K. Shariff, S. Psarakos, and D. J. White, “Ring bubbles of dolphins,” Scientific American, vol. 275, no. 2, pp. 82-87, 1996.
[2] W. Thomson, “On vortex atoms,” Philosophical Magazine, vol. 34, pp. 15-24, 1867. · JFM 15.0767.01
[3] A. Roshko, “Structure of a turbulent shear flows: a new look,” American Institute of Aeronautics and Astronautics Journal, vol. 14, no. 10, pp. 1349-1357, 1976.
[4] F. K. Browand and P. D. Weidman, “Large scales in the developing mixing layer,” Journal of Fluid Mechanics, vol. 76, pp. 127-144, 1976.
[5] G. W. Rayfield and F. Reif, “Evidence for the creation and motion of quantized vortex rings in superfluid helium,” Physical Review Letters, vol. 11, no. 7, pp. 305-308, 1963.
[6] P. H. Roberts and R. J. Donnelly, “Superfluid mechanics,” Annual Review of Fluid Mechanics, vol. 6, pp. 179-225, 1974. · Zbl 0298.76007
[7] T. Minota, T. Murakami, and T. Kambe, “Acoustic emission from interaction of a vortex ring with a sphere,” Fluid Dynamics Research, vol. 3, no. 1-4, pp. 357-362, 1988.
[8] D. G. Akhmetov, B. A. Lugovtsov, and V. F. Tarasov, “Extinguishing gas and oil well fires by means of vortex rings,” Combustion, Explosion, and Shock Waves, vol. 16, no. 5, pp. 490-494, 1980.
[9] G. L. Chahine and P. F. Genoux, “Collapse of a cavitating vortex ring,” Journal of Fluids Engineering, vol. 105, no. 4, pp. 400-405, 1983. · Zbl 0564.76022
[10] G. S. Settles, H. C. Ferree, M. D. Tronosky, Z. M. Moyer, and W. J. McGann, “Natural aerodynamic portal sampling of trace explosives from the human body,” in Proceedings of the 3rd International FAA Symposium on Explosives Detection and Aviation Security, Atlantic City, NJ, USA, November 2001.
[11] H. A. Gowadia and G. S. Settles, “The natural sampling of airborne trace signals from explosives concealed upon the human body,” Journal of Forensic Sciences, vol. 46, no. 6, pp. 1324-1331, 2001.
[12] A. Leonard, “Computing three-dimensional incompressible flows with vortex elements,” Annual Review of Fluid Mechanics, vol. 17, pp. 523-559, 1985. · Zbl 0596.76026
[13] E. G. Pucket, “Vortex methods: an introduction and survey of selected research topics,” in Incompressible Computational Fluid Dynamics Trends and Advances, p. 335, Cambridge University Press, Cambridge, UK, 1993.
[14] E. Meiburg, “Three dimensional vortex dynamics simulations,” in Fluid Vortices, p. 651, Kluwer Academic, Dordrecht, The Netherlands, 1995.
[15] G.-H. Cottet and P. D. Koumoutsakos, “Vortex Methods: Theory and Practice,” Cambridge University Press, Cambridge, UK, 2000. · Zbl 1140.76002
[16] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, vol. 27 of Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, UK, 2002. · Zbl 0983.76001
[17] K. Lindsay and R. Krasny, “A particle method and adaptive treecode for vortex sheet motion in three-dimensional flow,” Journal of Computational Physics, vol. 172, no. 2, pp. 879-907, 2001. · Zbl 1002.76093
[18] T. Maxworthy, “Turbulent vortex rings,” Journal of Fluid Mechanics, vol. 64, pp. 227-240, 1974. · Zbl 0279.76029
[19] T. Maxworthy, “Some experimental studies of vortex rings,” Journal of Fluid Mechanics, vol. 81, pp. 465-495, 1977.
[20] T. Maxworthy, “Waves on vortex cores,” Fluid Dynamics Research, vol. 3, no. 1-4, pp. 52-62, 1988.
[21] Y. Kaneda, “A representation of the motion of a vortex sheet in a three-dimensional flow,” Physics of Fluids A, vol. 2, no. 3, pp. 458-461, 1990. · Zbl 0704.76011
[22] R. E. Caflisch, “Mathematical analysis of vortex dynamics,” in Mathematical Aspects of Vortex Dynamics (Leesburg, VA, 1988), pp. 1-24, SIAM, Philadelphia, Pa, USA, 1989. · Zbl 0674.76012
[23] K. Lindsay, A three-dimensional cartesian tree code and applications to vortex sheet roll-up, Ph.D. thesis, The University of Michigan, Ann Arbor, Mich, USA, 1997.
[24] L. Rosenhead, “The spread of vorticity in the wake behind a cylinder,” Proceedings of the Royal Society of London. Series A, vol. 127, no. 806, pp. 590-612, 1930. · JFM 56.1253.03
[25] D. W. Moore, “Finite amplitude waves on aircraft trailing vortices,” Aeronautical Quarterly, vol. 23, pp. 307-314, 1972.
[26] H. Samet, “The quadtree and related hierarchical data structures,” Computing Surveys, vol. 16, no. 2, pp. 187-260, 1984.
[27] H. Samet, Applications of Spatial Data Structures, Addison Wesley, Reading, Mass, USA, 1990. · Zbl 1070.35129
[28] R. Krasny, “A computation of vortex sheet roll-up in the Trefftz plane,” Journal of Fluid Mechanics, vol. 184, pp. 123-155, 1987.
[29] M. Brady, A. Leonard, and D. I. Pullin, “Regularized vortex sheet evolution in three dimensions,” Journal of Computational Physics, vol. 146, no. 2, pp. 520-545, 1998. · Zbl 0928.76079
[30] F. Losasso, F. Gibou, and R. Fedkiw, “Simulating water and smoke with an octree data structure,” ACM Transactions on Graphics, vol. 23, no. 3, pp. 457-462, 2004. · Zbl 05457455
[31] A. J. Line and R. E. Brown, “Efficient high resolution wake modelling using vorticity transport equation,” in Proceedings of the 60th Annual Forum of the American Helicopter Society, Baltimore, Md, USA, June 2004.
[32] O. Klaas and M. S. Shephard, “Automatic generation of octree based three dimensional discretizations for partition of unity methods,” Computational Mechanics, vol. 25, no. 2-3, pp. 296-304, 2000. · Zbl 0956.65113
[33] V. Cristini, J. Blawzdziewicz, and M. Loewenberg, “An adaptive mesh algorithm for evolving surfaces: simulations of drop breakup and coalescence,” Journal of Computational Physics, vol. 168, no. 2, pp. 445-463, 2001. · Zbl 1153.76382
[34] T. W. Sederberg, J. Zheng, A. Bakenov, and A. Nasri, “T-splines and T-NURCCs,” ACM Transactions on Graphics, vol. 22, no. 3, pp. 477-484, 2003. · Zbl 05457575
[35] T. W. Sederberg, D. L. Cardon, G. T. Finnigan, N. S. North, J. Zheng, and T. Lyche, “T-spline simplification and local refinement,” ACM Transactions on Graphics, vol. 23, no. 3, pp. 276-283, 2004. · Zbl 05457574
[36] M. Pauly, M. Gross, and L. P. Kobbelt, “Efficient high resolution wake modelling using vorticity transport equation,” in Proceedings of IEEE Conference on Visualization (VIS /02), pp. 163-170, Boston, Mass, USA, October 2002.
[37] W. Gropp, E. Lusk, and A. Skjellum, Using MPI, The MIT Press, Cambridge, Mass, USA, 1999. · Zbl 0808.65124
[38] Y. M. Marzouk and A. F. Ghoniem, “K-means clustering for optimal partitioning and dynamic load balancing of parallel hierarchical N-body simulations,” Journal of Computational Physics, vol. 207, no. 2, pp. 493-528, 2005. · Zbl 1176.70005
[39] O. M. Knio and A. F. Ghoniem, “Numerical study of a three-dimensional vortex method,” Journal of Computational Physics, vol. 86, no. 1, pp. 75-106, 1990. · Zbl 0681.76025
[40] C. H. Krutzsch, “Uber eine experimentel bebachtete erscheinung a wirbelringendbel ihrer translatorischen bewengugng in wirklichen flussignketen,” Annals of Physics, vol. 35, no. 5, pp. 497-523, 1939.
[41] S. E. Widnall, J. P. Sullivan, and S. Ezekiel, “A study of vortex rings using a laser Doppler velocimeter,” American Institute of Aeronautics and Astronautics Journal, vol. 11, no. 10, pp. 1384-1389, 1973.
[42] T. Maxworthy, “The structure and stability of vortex rings,” Journal of Fluid Mechanics, vol. 51, pp. 15-32, 1972.
[43] C. Liess and N. Didden, “Formation of vortex rings,” Zeitschrift für Angewandte Mathematik und Physik, vol. 56, p. 206, 1976.
[44] N. Didden, “On the formation of vortex rings: rolling up and production of circulation,” Zeitschrift für Angewandte Mathematik und Physik, vol. 30, no. 1, pp. 101-116, 1979.
[45] A. Glezer, “The formation of vortex rings,” Physics of Fluids, vol. 31, no. 12, pp. 3532-3542, 1988. · Zbl 1062.81110
[46] A. Dazin, P. Dupont, and M. Stanislas, “Experimental observation of the straining field responsible for vortex ring instability,” Comptes Rendus. Mécanique, vol. 332, no. 3, pp. 231-236, 2004.
[47] A. Dazin, P. Dupont, and M. Stanislas, “Experimental study of instability of vortex rings,” in preparation.
[48] A. Dazin, P. Dupont, and M. Stanislas, “Experimental study of nonlinear phase of vortex ring instability,” in preparation.
[49] S. E. Widnall, D. H. Bliss, and A. Zelay, “Theoretical and experimental study of the stability of the vortex,” in Aircraft Wake Turbulence and Its Detection, p. 305, Plenum Press, New York, NY, USA, 1971.
[50] S. E. Widnall and J. P. Sullivan, “On the stability of vortex rings,” Proceedings of the Royal Society of London. Series A, vol. 332, no. 1590, pp. 335-353, 1973. · Zbl 0259.76022
[51] S. E. Widnall, D. B. Bliss, and C. Y. Tsai, “The instability of short waves on a vortex ring,” Journal of Fluid Mechanics, vol. 66, no. 1, pp. 35-47, 1974. · Zbl 0289.76027
[52] C.-Y. Tsai and S. E. Widnall, “The stability of short waves on a straight vortex filament in a weak externally imposed strain field,” Journal of Fluid Mechanics, vol. 73, no. 4, pp. 721-733, 1976. · Zbl 0326.76045
[53] S. E. Widnall and C. Y. Tsai, “The instability of the thin vortex ring of constant vorticity,” Philosophical Transactions of the Royal Society of London. Series A, vol. 287, no. 1344, pp. 273-305, 1977. · Zbl 0371.76022
[54] D. W. Moore and P. G. Saffman, “The instability of a straight vortex filament in a strain field,” Proceedings of the Royal Society of London. Series A, vol. 346, no. 1646, pp. 413-425, 1975. · Zbl 0326.76046
[55] P. G. Saffman, “The number of waves on unstable vortex rings,” Journal of Fluid Mechanics, vol. 84, no. 4, pp. 625-639, 1978.
[56] K. Shariff and A. Leonard, “Vortex rings,” in Annual Review of Fluid Mechanics, Vol. 24, pp. 235-279, Annual Reviews, Palo Alto, Calif, USA, 1992. · Zbl 0743.76024
[57] T. T. Lim and T. B. Nickels, “Vortex rings,” in Fluid Vortices, pp. 95-153, Kluwer Academic, Dordrecht, The Netherlands, 1995.
[58] A. Lifschitz, W. H. Suters, and J. T. Beale, “The onset of instability in exact vortex rings with swirl,” Journal of Computational Physics, vol. 129, no. 1, pp. 8-29, 1996. · Zbl 0883.76066
[59] E. Meiburg, J. E. Martin, and J. C. Lasheras, “Experimental and numerical analysis of the three-dimensional evolution of an axisymmetric jet,” Tech. Rep., Stanford University, Stanford, Calif, USA, 1991. Proceedings of the 7th Symposium on Turbulent Shear Flows, 1989.
[60] A. J. Chorin, “Hairpin removal in vortex interactions,” Journal of Computational Physics, vol. 91, no. 1, pp. 1-21, 1990. · Zbl 0711.76047
[61] A. J. Chorin, “Hairpin removal in vortex interactions. II,” Journal of Computational Physics, vol. 107, no. 1, pp. 1-9, 1993. · Zbl 0778.76071
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.