| | This Article | |
| |
| |
| | Share | |
| |
| |
| | Bibliographic References | |
| |
| |
| | Add to: | |
| |
 Digg
 Furl
 Spurl
 Blink
 Simpy
 Google
 Del.icio.us
 Y!MyWeb
| |
| | Search | |
| |
| |
| | |
Understanding the Structure of the Turbulent Mixing Layer in Hydrodynamic Instabilities
September-October 2006 (vol. 12 no. 5)
pp. 1053-1060
When a heavy fluid is placed above a light fluid, tiny vertical perturbations in the interface create a characteristic structure of rising bubbles and falling spikes known as Rayleigh-Taylor instability. Rayleigh-Taylor instabilities have received much attention over the past half-century because of their importance in understanding many natural and man-made phenomena, ranging from the rate of formation of heavy elements in supernovae to the design of capsules for Inertial Confinement Fusion. We present a new approach to analyze Rayleigh-Taylor instabilities in which we extract a hierarchical segmentation of the mixing envelope surface to identify bubbles and analyze analogous segmentations of fields on the original interface plane. We compute meaningful statistical information that reveals the evolution of topological features and corroborates the observations made by scientists. We also use geometric tracking to follow the evolution of single bubbles and highlight merge/split events leading to the formation of the large and complex structures characteristic of the later stages. In particular we (i) Provide a formal definition of a bubble; (ii) Segment the envelope surface to identify bubbles; (iii) Provide a multi-scale analysis technique to produce statistical measures of bubble growth; (iv) Correlate bubble measurements with analysis of fields on the interface plane; (v) Track the evolution of individual bubbles over time. Our approach is based on the rigorous mathematical foundations of Morse theory and can be applied to a more general class of applications.
[1] 1053 U. Alon and D. Shvarts, Two-phase flow model for rayleigh-taylor and richtmeyer-meshkov mixing. In Proc. Fifth Int. Workshop on Compressible Tur Mixing. World Scientific, 1996.[2] C. Bajaj, B. Sohn, Time-varying contour topology. IEEE Trans, on Vis. and Comp. Graph., 12 (1): 14–25, 2005.[3] P.-T. Bremer, H. Edelsbrunner, B. Hamann, and V. Pascucci, A topological hierarchy for functions on triangulated surfaces. IEEE Trans, on Vis. and Comp. Graph., 10 (4): 385–396, 2004.[4] P.-T. Bremer and V. Pascucci, A practical approach to two-dimensional scalar topology. H. Hauser, H. Theisel, and H. Hagen, editors, To appear., 2006.[5] H. Carr, J. Snoeyink, and M. van de Panne, Simplifying flexible isosurfaces using local geometric measures. In IEEE Vis., 2004.[6] T.T. Clark, A numerical study of the statistics of a two dimensional Rayleigh-Taylor mixing layer. Phys. Fluids, 15: 2413–2423.[7] A.W. Cook, W. Cabot, and P.L. Miller, The mixing transition in Rayleigh-Taylor instability. J. Fluid Mech., to appear, 2004.[8] A.W. Cook and P. E. Dimotakis, Transition stages of Rayleigh-Taylor instability between miscible fluids. J. Fluid Mech., 443: 66–99.[9] D. Kartoon, D. Oron, L. Arazi, and D. Shvarts, Three-dimensional multimode rayleigh-taylor and richtmeyer instabilities at all density ratios. Laser and Particle Beams, 21: 327–334.[10] W. de Leeuw and R. van Liere, Collapsing flow topology using area metrics. In Proc. IEEE Vis., pages 349–354. 1999.[11] H. Edelsbrunner and J. Harer, Jacobi sets of multiple Morse functions. In Foundations of Computational Mathematics, Minneapolis 2002, pages 37–57. Cambridge Univ. Press, England.[12] H. Edelsbrunner, J. Harer, A. Mascarenhas, and V. Pascucci, Time-varying Reeb graphs for continuous space-time data. In 20th Symp. on Comp. Geom., pages 366–372, New York, NY, USA, 2004.[13] H. Edelsbrunner, J. Harer, V. Natarajan, and V. Pascucci, Morse-Smale complexes for piecewise linear 3-manifolds. In Proc. 19th Sympos. Comput. Geom., pages 361–370, 2003.[14] H. Edelsbrunner, J. Harer, and A. Zomorodian, Hierarchical Morse-Smale complexes for piecewise linear 2-manifolds. Discrete Comput. Geom., 30: 87–107, 2003.[15] C. Garth, X. Tricoche, and G. Scheuermann, Tracking of vector field singularities in unstructured 3D time-dependent datasets. In Proc. IEEE Visualization, pages 329–336. 2004.[16] A. Gyulassy, V. Natarajan, V. Pascucci, P.-T. Bremer, and B. Hamann, Topology-based simplification for feature extraction from 3D scalar fields. IEEE Trans, on Vis. and Comp. Graph., 2006. to appear.[17] J. L. Helman and L. Hesselink, Visualizing vector field topology in fluid flows. IEEE Comp. Graph. and Appl., 11 (3): 36–46. 1991.[18] M. Isenburg, P. Lindstrom, S. Gumhold, and J. Snoeyink, Large mesh simplification using processing sequences. In Proc. IEEE Vis., pages 465–472. 2003.[19] G. Ji and H.-W. Shen, Efficient isosurface tracking using precomputed correspondence table. In Proc. Symp. on Vis., pages 283–292. 2004.[20] G. Ji, H.-W. Shen, and R. Wegner, Volume tracking using higher dimensional isocontouring. In Proc. IEEE Vis., pages 209–216. 2003.[21] J. Milnor, Morse Theory. Princeton Univ. Press, New Jersey, 1963.[22] F. Reinders, F.H. Post, and H. J. W. Spoelder, Visualization of time-dependent data with feature tracking and event detection. The Visual Computer, 17 (1): 55–71, 2001.[23] R. Samtaney, D. Silver, N. Zabusky, and J. Cao, Visualizing features and tracking their evolution. IEEE Computer, 27 (7): 20–27, 1994.[24] G. Scheuermann and X. Tricoche, Topological methods in flow visualization. In Visualization Handbook, pages 341–356. Elsevier, 2004.[25] D. Silver and X. Wang, Tracking and visualizing turbulent 3D features. IEEE Trans, on Vis. and Comp. Graph., 3 (2): 129–141, 1997.[26] D. Silver and X. Wang, Tracking scalar features in unstructured datasets. In Proc. IEEE Vis., pages 79–86. 1998.[27] A. Szymczak, Subdomain-aware contour trees and contour tree evolution in time-dependent scalar fields. In Proc. Shape Modeling International (SMI) '05, pages 136–144. IEEE Computer Society, 2005.[28] H. Theisel, T. Weinkauf, H.-C. Hege, and H.-P. Seidel, Saddle connectors - An approach to visualizing the topological skeleton of complex 3D vector fields. In Proc. IEEE Vis., pages 225–232, 2003.[29] H. Theisel, T. Weinkauf, H.-C. Hege, and H.-P. Seidel, Topological methods for 2d time-dependent vector fields based on stream lines and path lines. IEEE Trans, on Vis. and Comp. Graph., 2005.[30] X. Tricoche, G. Scheuermann, and H. Hagen, A topology simplification method for 2d vector fields. In IEEE Vis., pages 359–366. 2000.[31] X. Tricoche, T. Wischgoll, G. Scheuermann, and H. Hagen, Topology tracking for the visualization of time-dependent two-dimensional flows. Computer & Graphics, 26 (2): 249–257, 2002.[32] G. Weber, G. Scheuermann, H. Hagen, and B. Hamann, Exploring scalar fields using critical isovalues. In M. Gross, K. I. Joy, and R. J. Moorhead, editors, Proc. IEEE Vis., pages 171–178. 2002.[33] T. Weinkauf, H. Theisel, K. Shi, H.-C. Hege, and H.-P. Seidel, Extracting higher order critical points and topological simplification of 3D vector fields. In Proc. IEEE Vis., pages 559–566. 2005.
Index Terms:
topology, multi-resolution, Morse theory
Citation:
D. Laney, P.-T. Bremer, A. Mascarenhas, P. Miller, V. Pascucci, "Understanding the Structure of the Turbulent Mixing Layer in Hydrodynamic Instabilities," IEEE Transactions on Visualization and Computer Graphics, vol. 12, no. 5, pp. 1053-1060, Sept. 2006, doi:10.1109/TVCG.2006.186
