Summary
A pattern often found in regions of recirculating flow is the vortex ring. Smoke rings and vortex breakdown bubbles are two familiar instances of this pattern. A vortex ring requires at least two critical points, and in fact this minimum number is observed in many synthetic or real-world examples. Based on this observation, we propose a visualization technique utilizing a Poincaré section that contains the pair of critical points. The Poincaré section by itself can be taken as a visualization of the vortex ring, especially if streamlines are seeded on the stable and unstable manifolds of the critical points. The resulting image reveals the extent of the structure, and more interestingly, regions of chaos and islands of stability. As a next step, we describe for the case of incompressible flow an algorithm for finding invariant tori in an island of stability. The basic idea is to find invariant closed curves in the Poincaré plane, which are then taken as seed curves for stream surfaces. For visualization the two extremes of the set of nested tori are computed. This is on the inner side the periodic orbit toward which the tori converge, and on the outer side, a torus which marks the boundary between ordered and chaotic flow, a distinction which is of importance for the mixing properties of the flow. For the purpose of testing, we developed a simple analytical model of a perturbed vortex ring based on Hill's spherical vortex. Finally, we applied the proposed visualization methods to this synthetic vector field and to two hydromechanical simulation results.
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References
Abraham R. H., Shaw C. D.: Dynamics, the Geometry of Behavior. 2nd ed. Addison-Wesley, 1992.
Brackbill J., Barnes D.: The effect of nonzero ▿ · B on the numerical solution of the magnetohydrodynamic equations. J. Comput. Phys. 35 (1980), 426–430.
Guckenheimer J., Holmes P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Applied Mathematical Sciences, Vol. 42. Springer, New York, Berlin, Heidelberg, Tokyo, 1983.
Globus A., Levit C., Lasinski T.: A tool for visualizing the topology of three- dimensional vector fields. In Proc. IEEE Visualization '91 (1991), pp. 33–40.
Garth C., Tricoche X., Salzbrunn T., Bobach T., Scheuermann G.: Surface techniques for vortex visualization. In VisSym (2004), pp. 155–164, 346.
Haller G.: Lagrangian structures and the rate of strain in a partition of two- dimensional turbulence. Phys. Fluids 13 (2001), 3365–3385.
Helman J., Hesselink L.: Representation and display of vector field topology in fluid flow data set. IEEE Computer (August 1989), 27–36.
Krasny R., Fritsche M.: The onset of chaos in vortex sheet flow. J. Fluid Mech. 454 (2002), 47–69.
Löffelmann H., Doleisch H., Gröller E.: Visualizing dynamical systems near critical points. In 14th Spring Conference on Computer Graphics (April 1998), Kalos L. S., (Ed.), pp. 175–184.
Löffelmann H., Gröller E.: Enhancing the visualization of characteristic structures in dynamical systems. In Visualization in Scientific Computing '98 (1998), Bartz D., (Ed.), Springer, pp. 59–68.
Löffelmann H., Kucera T., Gröller E.: Visualizing poincaré maps together with the underlying flow. In Mathematical Visualization. Proceedings of the International Workshop on Visualization and Mathematics '97 (1997), Hege H.-C., Polthier K., (Eds.), Springer, pp. 315–328.
MacKay R. S., Meiss J. D., Percival I. C.: Transport in hamiltonian systems. Physica D 13D (1984), 55–81.
Norbury J.: A family of steady vortex rings. J. Fluid Mech. 57, Pt. 3 (1973), 417–431.
Saffman P. G.: Vortex Dynamics. Cambridge Univ. Press, Cambridge, UK, 1992.
Sil'nikov L. P.: A case of the existence of a denumerable set of periodic motions. Sov. Math. Dokl. 6 (1965), 163–166.
Spohn A., Mory M., Hopfinger E.: Experiments on vortex breakdown in a confined flow generated by a rotating disc. Journal of Fluid Mechanics 370 (1998), 73–99.
Sadlo F., Peikert R.: Topology-guided visualization of constrained vector fields. In Proceedings of the 2005 Workshop on Topology-Based Methods in Visualization, Budmerice, Slovakia (2007), p. (to appear).
Sotiropoulos F., Ventikos Y., Lackey T. C.: Chaotic advection in three- dimensional stationary vortex-breakdown bubbles: Sil'nikov's chaos and the devil's staircase. J. Fluid Mech. 444 (2001), 257–297.
Stalling D., Zöckler M., Hege H.-C.: Fast display of illuminated field lines. IEEE Transactions on Visualization and Computer Graphics 3, 2 (1997), 118–128.
Tricoche X., Garth C., Kindlmann G., Deines E., Scheuermann G., Ruetten M., Hansen C.: Visualization of intricate flow structures for vortex breakdown analysis. In Proc. IEEE Visualization 2004 (October 2004), IEEE Computer Society, pp. 187–194.
Thompson M. C., Hourigan K.: The sensitivity of steady vortex breakdown bubbles in confined cylinder flows to rotating lid misalignment. Journal of Fluid Mechanics 496 (Dec. 2003), 129–138.
Toth G.: The div b=0 constraint in shock-capturing magnetohydrodynamics codes. Journal of Computational Physics 161 (2000), 605–652.
Theisel H., Weinkauf T., Hege H.-C., Seidel H.-P.: Saddle connectors - an approach to visualizing the topological skeleton of complex 3d vector fields. In Proc. IEEE Visualization 2003 (Oct. 2003), pp. 225–232.
Weinkauf T., Hege H.-C., Noack B., Schlegel M., Dillmann A.: Coherent structures in a transitional flow around a backward-facing step. Physics of Fluids 15, 9 (September 2003), S3.
Wischgoll T., Scheuermann G.: Detection and visualization of closed streamlines in planar flows. IEEE Transactions on Visualization and Computer Graphics 7, 2 (2001), 165–172.
Ye X., Kao D., Pang A.: Strategy for scalable seeding of 3d streamlines. In Proc. IEEE Visualization '05 (2005), pp. 471–478.
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Peikert, R., Sadlo, F. (2009). Flow Topology Beyond Skeletons: Visualization of Features in Recirculating Flow. In: Hege, HC., Polthier, K., Scheuermann, G. (eds) Topology-Based Methods in Visualization II. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88606-8_11
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DOI: https://doi.org/10.1007/978-3-540-88606-8_11
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