Advertisement

An objective-adaptive refinement criterion based on modified ridge extraction method for finite-time Lyapunov exponent (FTLE) calculation

  • Haotian Hang
  • Bin Yu
  • Yang Xiang
  • Bin ZhangEmail author
  • Hong Liu
Regular Paper
  • 16 Downloads

Abstract

Visualizing finite-time Lyapunov exponent (FTLE) efficiently and accurately has long been a research objective in identifying the coherent structures of turbulence or vortex flows. In this field, adaptive mesh refinement shows its effectiveness. The proposed objective-adaptive refinement (OAR) criterion can refine adaptive particles in the vicinity of FTLE ridges by a modified gradient climbing method. While error-based refinement methods suffer from ineffective refinement when the initial velocity field contains error, and refinement methods based on FTLE magnitude have issues with undulate ridges, our objective OAR criterion always steers the refinement toward the vicinity of FTLE ridges. Testing cases include Bickley jet, mild FTLE ridge and experimental single vortex, three-dimensional ABC flow. The results demonstrate that the proposed OAR criterion can give the right refinement region, and thus enhance computation efficiency, by means of accurate extraction of FTLE ridges.

Graphic abstract

Keywords

Flow structures visualization Objective-adaptive refinement (OAR) Lagrangian coherent structures (LCS) Finite-time Lyapunov exponent (FTLE) 

Notes

Acknowledgements

The authors would like to thank the Center for High-Performance Computing of SJTU for providing the super computer \(\pi \) to support this research. This work is supported by the National Natural Science Foundation of China (NSFC-91741113) and National Science Foundation for Young Scientists of China (Grant No. 51676203). Furthermore, Geng Liang is appreciated in assistance in three-dimensional cases and refinement criterion. Also, Chunhui Tang, Haiyan Lin and Linying Li are appreciated in assisting the completion of the paper.

References

  1. Barakat SS, Tricoche X (2013) Adaptive refinement of the flow map using sparse samples. IEEE Trans Vis Comput Graph 19(12):2753–2762Google Scholar
  2. Berger MJ, Colella PJ (1989) Local adaptive mesh refinement for shock hydrodynamics. J Comput Phys 82(1):64–84zbMATHGoogle Scholar
  3. Beron Vera FJ, Olascoaga MJ, Haller G, Farazmand M, Trianes J, Wang Y (2015) Dissipative inertial transport patterns near coherent lagrangian eddies in the ocean. Chaos 25(8):087,412MathSciNetGoogle Scholar
  4. Blazevski D, Haller G (2014) Hyperbolic and elliptic transport barriers in three-dimensional unsteady flows. Phys D Nonlinear Phenom 273–274(2):46–62MathSciNetzbMATHGoogle Scholar
  5. Chong MS, Perry AE, Cantwell BJ (1990) A general classification of three-dimensional flow fields. Phys Fluids A 2(5):765–777MathSciNetGoogle Scholar
  6. Duc LH, Siegmund S (2008) Hyperbolicity and invariant manifolds for planar nonautonomous systems on finite time intervals. Int J Bifurc Chaos 18(03):0802,056MathSciNetzbMATHGoogle Scholar
  7. Fortin A, Briffard T, Garon A, Briffard T, Garon A (2015) A more efficient anisotropic mesh adaptation for the computation of lagrangian coherent structures. J Comput Phys 285(C):100–110zbMATHGoogle Scholar
  8. Froyland G, Padberg-Gehle K (2014) Almost-invariant and finite-time coherent sets: directionality, duration, and diffusion. Springer, New York, pp 171–216zbMATHGoogle Scholar
  9. Garth C, Gerhardt F, Tricoche X, Hagen H (2007) Efficient computation and visualization of coherent structures in fluid flow applications. IEEE Trans Vis Comput Graph 13(6):1464–1471Google Scholar
  10. Green MA, Rowley CW, Haller G (2007) Detection of lagrangian coherent structures in three-dimensional turbulence. J Fluid Mech 572(572):111–120MathSciNetzbMATHGoogle Scholar
  11. Hadjighasem A, Haller G (2014) Geodesic transport barriers in jupiter’s atmosphere: a video-based analysis. IEEE Trans Commun 58(1):536–551MathSciNetzbMATHGoogle Scholar
  12. Haller G (2001) Distinguished material surfaces and coherent structures in three-dimensional fluid flows. Phys D Nonlinear Phenom 149(4):248–277MathSciNetzbMATHGoogle Scholar
  13. Haller G (2005) An objective definition of a vortex. J Fluid Mech 525(525):1–26MathSciNetzbMATHGoogle Scholar
  14. Haller G (2011) A variational theory of hyperbolic lagrangian coherent structures. Phys D Nonlinear Phenom 240(7):574–598MathSciNetzbMATHGoogle Scholar
  15. Haller G (2015) Lagrangian coherent structures. Annu Rev Fluid Mech 47(1):137–162MathSciNetGoogle Scholar
  16. Haller G, Yuan G (2000) Lagrangian coherent structures and mixing in two-dimensional turbulence. Phys D Nonlinear Phenom 147(3C4):352–370MathSciNetzbMATHGoogle Scholar
  17. Huang W, Russell RD (2011) Adaptive moving mesh methods. Springer, New YorkzbMATHGoogle Scholar
  18. Hunt JCR (1988) Eddies, streams, convergence zones in turbulent flows. In: Studying turbulence using numerical simulation databases, pp 193–208Google Scholar
  19. Jeong J, Hussain F (1995) On the identification of a vortex. J Fluid Mech 285(285):69–94MathSciNetzbMATHGoogle Scholar
  20. Jimbo T, Tanahashi T (2003) Numerical simulations of unsteady shock waves around complex bodies. J Vis 6(3):212–212Google Scholar
  21. Karch GK, Sadlo F, Weiskopf D, Ertl T (2016) Visualization of 2d unsteady flow using streamline-based concepts in space-time. J Vis 19(1):115–128Google Scholar
  22. Karrasch D, Huhn F, Haller G (2015) Automated detection of coherent lagrangian vortices in two-dimensional unsteady flows. Proc R Soc A Math Phys Eng Sci 471(2173):20140639MathSciNetzbMATHGoogle Scholar
  23. Keith W, Demetri T (1991) Modeling and animating faces using scanned data. J Vis Comput Anim 2(4):123–128Google Scholar
  24. Liang G, Yu B, Zhang B, Xu H, Liu H (2019) Hidden flow structures in compressible mixing layer and a quantitative analysis of entrainment based on lagrangian method. J Hydrodyn.  https://doi.org/10.1007/s42241-019-0027-z Google Scholar
  25. Lipinski D, Mohseni K (2010) A ridge tracking algorithm and error estimate for efficient computation of lagrangian coherent structures. Chaos 20(1):017,504MathSciNetzbMATHGoogle Scholar
  26. Mathur M, Haller G, Peacock T, Ruppert-Felsot JE, Swinney HL (2007) Uncovering the lagrangian skeleton of turbulence. Phys Rev Lett 98(14):144,502Google Scholar
  27. Miron P, Vtel J, Garon A, Delfour M, Hassan ME (2012) Anisotropic mesh adaptation on lagrangian coherent structures. J Comput Phys 231(19):6419–6437MathSciNetzbMATHGoogle Scholar
  28. Ng KW, Wong YP (2007) Adaptive model simplification in real-time rendering for visualization. J Vis 10(1):111–121Google Scholar
  29. O’Farrell C, Dabiri JO (2010) A lagrangian approach to identifying vortex pinch-off. Chaos 20(1):261–300Google Scholar
  30. Olcay AB, Krueger PS (2008) Measurement of ambient fluid entrainment during laminar vortex ring formation. Exp Fluids 44(2):235–247Google Scholar
  31. Onu K, Huhn F, Haller G (2014) Lcs tool: a computational platform for lagrangian coherent structures. J Comput Sci 7:26–36Google Scholar
  32. Plewa T, Linde T, Weirs VG, Numerik (2005) Adaptive mesh refinement—theory and applications. Springer, BerlinzbMATHGoogle Scholar
  33. Qin S, Liu H, Xiang Y (2017) Lagrangian flow visualization of multiple co-axial co-rotating vortex rings. J Vis 31:1–9Google Scholar
  34. Qin S, Liu H, Xiang Y (2018) On the formation modes in vortex interaction for multiple co-axial co-rotating vortex rings. Phys Fluids 30(1):011,901Google Scholar
  35. Sadlo F, Peikert R (2007) Efficient visualization of lagrangian coherent structures by filtered amr ridge extraction. IEEE Trans Vis Comput Graph 13(13):1456–1463Google Scholar
  36. Shadden SC, Dabiri JO, Marsden JE (2006) Lagrangian analysis of fluid transport in empirical vortex ring flows. Phys Fluids 18(4):047,105MathSciNetzbMATHGoogle Scholar
  37. Shadden SC, Katija K, Rosenfeld M, Marsden JE, Dabiri JO (2007) Transport and stirring induced by vortex formation. J Fluid Mech 593(593):315–331zbMATHGoogle Scholar
  38. Sulman MHM, Huntley HS, Lipphardt BL Jr, Kirwan AD Jr (2013) Leaving flatland: diagnostics for lagrangian coherent structures in three-dimensional flows. Phys D Nonlinear Phenom 258(5):77–92MathSciNetzbMATHGoogle Scholar
  39. Tallapragada P, Ross SD, Schmale III DG (2011) Lagrangian coherent structures are associated with fluctuations in airborne microbial populations. Chaos 21(3):033,122Google Scholar
  40. Tang W, Chan PW, Haller G (2010) Accurate extraction of lagrangian coherent structures over finite domains with application to flight data analysis over hong kong international airport. Chaos 20(1):017,502MathSciNetzbMATHGoogle Scholar
  41. Tian S, Gao Y, Dong X, Liu C (2018) Definitions of vortex vector and vortex. J Fluid Mech 849:312C339.  https://doi.org/10.1017/jfm.2018.406 MathSciNetzbMATHGoogle Scholar
  42. Wang H, Ai Z, Cao Y, Xiao L (2016) A parallel preintegration volume rendering algorithm based on adaptive sampling. J Vis 19(3):437–446Google Scholar
  43. Zhang Y, Liu K, Xian H, Du X (2017) A review of methods for vortex identification in hydroturbines. Renew Sustain Energy Rev 81:1269–1285Google Scholar

Copyright information

© The Visualization Society of Japan 2019

Authors and Affiliations

  • Haotian Hang
    • 1
  • Bin Yu
    • 1
  • Yang Xiang
    • 1
  • Bin Zhang
    • 1
    Email author
  • Hong Liu
    • 1
  1. 1.School of Aeronautics and AstronauticsShanghai Jiao Tong UniversityShanghaiChina

Personalised recommendations