Topological Features in Time-Dependent Advection-Diffusion Flow

Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)

Abstract

Concepts from vector field topology have been successfully applied to a wide range of phenomena so far—typically to problems involving the transport of a quantity, such as in flow fields, or to problems concerning the instantaneous structure, such as in the case of electric fields. However, transport of quantities in time-dependent flows has so far been topologically analyzed in terms of advection only, restricting the approach to quantities that are solely governed by advection. Nevertheless, the majority of quantities transported in flows undergoes simultaneous diffusion, leading to advection-diffusion problems. By extending topology-based concepts with diffusion, we provide an approach for visualizing the mechanisms in advection-diffusion flow. This helps answering many typical questions in science and engineering that have so far not been amenable to adequate visualization. We exemplify the utility of our technique by applying it to simulation data of advection-diffusion problems from different fields.

Notes

Acknowledgements

This work was supported by the Cluster of Excellence in Simulation Technology (EXC 310/1) and the Collaborative Research Center SFB-TRR 75 at University of Stuttgart.

References

  1. 1.
    D. Bürkle, T. Preußer, M. Rumpf, Transport and anisotropic diffusion in time-dependent flow visualization, in Proceedings of the IEEE Visualization, San Diego, 2001, pp. 61–67Google Scholar
  2. 2.
    D. Eberly, Ridges in Image and Data Analysis. Computational Imaging and Vision (Kluwer, Boston, 1996)CrossRefMATHGoogle Scholar
  3. 3.
    R. Fuchs, J. Kemmler, B. Schindler, F. Sadlo, H. Hauser, R. Peikert, Toward a Lagrangian vector field topology. Comput. Graph. Forum 29(3), 1163–1172 (2010)CrossRefGoogle Scholar
  4. 4.
    C. Garth, G.-S. Li, X. Tricoche, C.D. Hansen, H. Hagen, Visualization of coherent structures in transient 2d flows, in Topology-Based Methods in Visualization II, ed. by H.-C. Hege, K. Polthier, G. Scheuermann (Springer, Berlin, 2009), pp. 1–13Google Scholar
  5. 5.
    A. Globus, C. Levit, T. Lasinski, A tool for visualizing the topology of three-dimensional vector fields, in Proceedings of the IEEE Visualization, San Diego, 1991, pp. 33–40, 408Google Scholar
  6. 6.
    G. Haller, Finding finite-time invariant manifolds in two-dimensional velocity fields. Chaos 10(1), 99–108 (2000)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    G. Haller, Distinguished material surfaces and coherent structures in three-dimensional fluid flows. Physica D 149(4), 248–277 (2001)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    G. Haller, A variational theory of hyperbolic Lagrangian coherent structures. Phys. D: Nonlinear Phenom. 240(7), 574–598 (2011)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    J. Helman, L. Hesselink, Representation and display of vector field topology in fluid flow data sets. IEEE Comput. 22(8), 27–36 (1989)CrossRefGoogle Scholar
  10. 10.
    J. Helman, L. Hesselink, Visualizing vector field topology in fluid flows. IEEE Comput. Graph. Appl. 11(3), 36–46 (1991)CrossRefGoogle Scholar
  11. 11.
    M. Hlawatsch, J. Vollrath, F. Sadlo, D. Weiskopf, Coherent structures of characteristic curves in symmetric second order tensor fields. IEEE Trans. Vis. Comput. Graph. 17(6), 781–794 (2011)CrossRefGoogle Scholar
  12. 12.
    F. Hussain, Coherent structures and turbulence. J. Fluid Mech. 173, 303–356 (1986)CrossRefGoogle Scholar
  13. 13.
    K. Ide, D. Small, S. Wiggins, Distinguished hyperbolic trajectories in time-dependent fluid flows: analytical and computational approach for velocity fields defined as data sets. Nonlinear Process. Geophys. 9(3/4), 237–263 (2002)CrossRefGoogle Scholar
  14. 14.
    G.K. Karch, F. Sadlo, D. Weiskopf, C.-D. Munz, T. Ertl, Visualization of advection-diffusion in unsteady fluid flow. Comput. Graph. Forum 31(3), 1105–1114 (2012)CrossRefGoogle Scholar
  15. 15.
    J. Kasten, I. Hotz, B. Noack, H.-C. Hege, On the extraction of long-living features in unsteady fluid flows, in Topological Methods in Data Analysis and Visualization. Theory, Algorithms, and Applications, ed. by V. Pascucci, X. Tricoche, H. Hagen, J. Tierny (Springer, Berlin/Heidelberg, 2010), pp. 115–126Google Scholar
  16. 16.
    A. Kuhn, T. Senst, I. Keller, T. Sikora, H. Theisel, A Lagrangian framework for video analytics, in Proceedings of the IEEE Workshop on Multimedia Signal Processing, Banff, 2012Google Scholar
  17. 17.
    Y. Levy, D. Degani, A. Seginer, Graphical visualization of vortical flows by means of helicity. AIAA 28(8), 1347–1352 (1990)CrossRefGoogle Scholar
  18. 18.
    H. Löffelmann, H. Doleisch, E. Gröller, Visualizing dynamical systems near critical points, in Proceedings of the Spring Conference on Computer Graphics and Its Applications, Budmerice, 1998, pp. 175–184Google Scholar
  19. 19.
    E.N. Lorenz, A study of the predictability if a 28-variable atmospheric model. Tellus 17, 321–333 (1965)CrossRefGoogle Scholar
  20. 20.
    M. Otto, T. Germer, H. Theisel, Uncertain topology of 3d vector fields, in Proceedings of the IEEE Pacific Visualization Symposium, Hong Kong, 2011, pp. 67–74Google Scholar
  21. 21.
    R. Peikert, M. Roth, The “parallel vectors” operator – a vector field visualization primitive, in Proceedings of the IEEE Visualization, San Francisco, 1999, pp. 263–270Google Scholar
  22. 22.
    R. Peikert, F. Sadlo, Topology-guided visualization of constrained vector fields, in Topology-Based Methods in Visualization, ed. by H. Hauser, H. Hagen, H. Theisel (Springer, Berlin/New York, 2007), pp. 21–34CrossRefGoogle Scholar
  23. 23.
    R. Peikert, F. Sadlo, Visualization methods for vortex rings and vortex breakdown bubbles, in Proceedings of the Joint Eurographics/IEEE VGTC Conference on Visualization, Norrköping, 2007, pp. 211–218Google Scholar
  24. 24.
    R. Peikert, F. Sadlo, Flow topology beyond skeletons: visualization of features in recirculating flow, in Topology-Based Methods in Visualization II, ed. by H.-C. Hege, K. Polthier, G. Scheuermann (Springer, Berlin, 2009), pp. 145–160Google Scholar
  25. 25.
    A.E. Perry, M.S. Chong, A description of eddying motions and flow patterns using critical-point concepts. Annu. Rev. Fluid Mech. 19, 125–155 (1987)CrossRefGoogle Scholar
  26. 26.
    A. Pobitzer, R. Peikert, R. Fuchs, B. Schindler, A. Kuhn, H. Theisel, K. Matković, H. Hauser, On the way towards topology-based visualization of unsteady flow – the state of the art, in Eurographics 2010 State of the Art Reports, Norrköping, 2010, pp. 137–154Google Scholar
  27. 27.
    S.K. Robinson, Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23, 601–639 (1991)CrossRefGoogle Scholar
  28. 28.
    F. Sadlo, R. Peikert, Efficient visualization of Lagrangian coherent structures by filtered amr ridge extraction. IEEE Trans. Vis. Comput. Graph. 13(6), 1456–1463 (2007)CrossRefGoogle Scholar
  29. 29.
    F. Sadlo, R. Peikert, Visualizing Lagrangian coherent structures and comparison to vector field topology, in Topology-Based Methods in Visualization II, ed. by H.-C. Hege, K. Polthier, G. Scheuermann (Springer, Berlin, 2009)Google Scholar
  30. 30.
    F. Sadlo, M. Üffinger, T. Ertl, D. Weiskopf, On the finite-time scope for computing Lagrangian coherent structures from Lyapunov exponents, in Topological Methods in Data Analysis and Visualization II, ed. by R. Peikert et al. (Springer, Heidelberg/New York, 2012), pp. 269–281CrossRefGoogle Scholar
  31. 31.
    F. Sadlo, D. Weiskopf, Time-dependent 2-d vector field topology: an approach inspired by Lagrangian coherent structures. Comput. Graph. Forum 29(1), 88–100 (2010)CrossRefGoogle Scholar
  32. 32.
    A. Sanderson, G. Chen, X. Tricoche, D. Pugmire, S. Kruger, J. Breslau, Analysis of recurrent patterns in toroidal magnetic fields. IEEE Trans. Vis. Comput. Graph. 16(6), 1431–1440 (2010)CrossRefGoogle Scholar
  33. 33.
    A. Sanderson, C.R. Johnson, R.M. Kirby, Display of vector fields using a reaction-diffusion model, in Proceedings of the IEEE Visualization, Austin, 2004, pp. 115–122Google Scholar
  34. 34.
    B. Schindler, R. Fuchs, S. Barp, J. Waser, A. Pobitzer, R. Carnecky, K. Matkovic, R. Peikert, Lagrangian coherent structures for design analysis of revolving doors. IEEE Trans. Vis. Comput. Graph. 18(12), 2159–2168 (2012)CrossRefGoogle Scholar
  35. 35.
    D. Schneider, J. Fuhrmann, W. Reich, G. Scheuermann, A variance based ftle-like method for unsteady uncertain vector fields, in Topological Methods in Data Analysis and Visualization II, ed. by R. Peikert et al. (Springer, Heidelberg/New York, 2012), pp. 255–268CrossRefGoogle Scholar
  36. 36.
    S.C. Shadden, F. Lekien, J.E. Marsden, Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Phys. D: Nonlinear Phenom. 212, 271–304 (2005)CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    K. Shi, H. Theisel, H. Hauser, T. Weinkauf, K. Matkovic, H.-C. Hege, H.-P. Seidel, Path line attributes – an information visualization approach to analyzing the dynamic behavior of 3D time-dependent flow fields, in Topology-Based Methods in Visualization II, ed. by H.-C. Hege, K. Polthier, G. Scheuermann (Springer, Berlin, 2009), pp. 75–88CrossRefGoogle Scholar
  38. 38.
    D. Sujudi, R. Haimes, Identification of swirling flow in 3d vector fields, in Proceedings of the 12th AIAA Computational Fluid Dynamics Conference, 1995, pp. 95–1715Google Scholar
  39. 39.
    H. Theisel, J. Sahner, T. Weinkauf, H.-C. Hege, H.-P. Seidel, Extraction of parallel vector surfaces in 3D time-dependent fields and application to vortex core line tracking, in Proceedings of the IEEE Visualization, Minneapolis, 2005, pp. 631–638Google Scholar
  40. 40.
    H. Theisel, T. Weinkauf, H.-C. Hege, H.-P. Seidel, Saddle connectors – an approach to visualizing the topological skeleton of complex 3d vector fields, in Proceedings of the IEEE Visualization, Seattle, 2003, pp. 225–232Google Scholar
  41. 41.
    H. Theisel, T. Weinkauf, H.-C. Hege, H.-P. Seidel, Stream line and path line oriented topology for 2d time-dependent vector fields, in Proceedings of the IEEE Visualization, Austin, 2004, pp. 321–328Google Scholar
  42. 42.
    X. Tricoche, C. Garth, A. Sanderson, K. Joy, Visualizing invariant manifolds in area-preserving maps, in Topological Methods in Data Analysis and Visualization II, ed. by R. Peikert et al. (Springer, Heidelberg/New York, 2012), pp. 109–124Google Scholar
  43. 43.
    X. Tricoche, M. Hlawitschka, S. Barakat, C. Garth, Beyond topology: a Lagrangian metaphor to visualize the structure of 3d tensor fields, in New Developments in the Visualization and Processing of Tensor Fields, ed. by D. Laidlaw, A. Vilanova (Springer, Berlin/New York, 2012)Google Scholar
  44. 44.
    M. Üffinger, F. Sadlo, T. Ertl, A time-dependent vector field topology based on streak surfaces. IEEE Trans. Vis. Comput. Graph. 19(3), 379–392 (2013)CrossRefGoogle Scholar
  45. 45.
    T. Weinkauf, H. Theisel, H.-C. Hege, H.-P. Seidel, Boundary switch connectors for topological visualization of complex 3D vector fields, in Proceedings of the VisSym, Konstanz, 2004, pp. 183–192Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.University of StuttgartStuttgartGermany

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