Experiments in Fluids

, 55:1638 | Cite as

Computation of finite-time Lyapunov exponents from time-resolved particle image velocimetry data

  • Samuel G. Raben
  • Shane D. Ross
  • Pavlos P. Vlachos
Research Article


This work presents two new methods for computing finite-time Lyapunov exponents (FTLEs) from noisy spatiotemporally resolved experimentally measured image data of the type used for particle image velocimetry (PIV) or particle tracking velocimetry (PTV). These new approaches are based on the simple insight that the particle images recorded during PIV experiments represent Lagrangian flow tracers whose trajectories lend themselves to the direct computation of flow maps, and related quantities such as flow map gradients and FTLEs. We show that using this idea we can improve the reliability and accuracy of FTLE calculation through the use of either direct pathline flow map (PFM) calculation, where individual particle pathlines over a fixed period of time are used to determine the flow map, or particle tracking flow map compilation (FMC), where instantaneous tracking results are used to estimate small snapshots of the flow map which are then compiled to describe the complete flow map. Comparisons of the traditional velocity field integration (VFI) method for computing FTLE fields with these new methods show that FMC produces significantly more accurate estimates of the FTLE field for both synthetic data and experimental data especially in cases where the particle number density is low. This is because the VFI estimates particle motion while PTV directly measures particle motion and therefore generates a more accurate flow map. Overall, our results suggest that VFI is not always a reliable approach when applied to noisy experimental PIV data. For cases where particle loss between frames is minimal, the PFM can also produce better results, but the final field is susceptible to error due to the unstructured nature of the raw flow maps. When comparing the ability to match the true separatrix of a flow, FMC is shown to be a far superior method. The separatrix from FMC has an 80 % overlap with the true solution as compared to approximately 25 % for the PFM and only 1 % for the VFI method. FMC shows a significant advantage when the particle seeding is low, which is particularly relevant for applications to environmental or biological flows where adding seed particles is not always practical, and investigation of Lagrangian flow structures must rely on naturally occurring flow tracers.


Particle Image Velocimetry Vortex Ring Seeding Density Particle Tracking Velocimetry Lagrangian Coherent Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The authors would like to thank Roderick La Foy, Amir Bozorg Magham, Shibabrat Naik, and John Charonko for their helpful conversations and for commenting on the manuscript as well as Kelley Stewart and Cassie Niebel for providing the experimental data. SDR gratefully acknowledges partial support by the National Science Foundation under Grant No. CMMI-1150456. PPV gratefully acknowledges the partial support of NIH Grant 1R21HL106276-01A1.


  1. Adrian RJ (1991) Particle-imaging techniques for experimental fluid-mechanics. Annu Rev Fluid Mech 23(1):261–304CrossRefGoogle Scholar
  2. Adrian RJ (2005) Twenty years of particle image velocimetry. Exp Fluids 39:159–169CrossRefGoogle Scholar
  3. Adrian R, Westerweel J (2011) Particle image velocimetry. Cambridge University Press, CambridgeGoogle Scholar
  4. BozorgMagham AE, Ross SD et al (2013) Real-time prediction of atmospheric Lagrangian coherent structures based on forecast data: an application and error analysis. Phys D 258(1):47–60CrossRefMathSciNetGoogle Scholar
  5. Brady MR, Raben SG et al (2009) Methods for digital particle image sizing (DPIS): comparisons and improvements. Flow Meas Instrum 20(6):207–219CrossRefGoogle Scholar
  6. Brunton SL, Rowley CW (2010) Fast computation of finite-time Lyapunov exponent fields for unsteady flows. Chaos 20:1–12CrossRefMathSciNetGoogle Scholar
  7. Cardwell ND, Vlachos PP et al (2010) A multi-parametric particle pairing algorithm for particle tracking in single and multiphase flows. Meas Sci Technol 22:105406 Google Scholar
  8. Charonko J, Kumar R et al (2013) Vortices formed on the mitral valve tips aid normal left ventricular filling. Ann Biomed Eng 41(5):1049–1061Google Scholar
  9. Du Toit PC (2010) Transport and separatrices in time dependent flows. Ph.D., California Institute of TechnologyGoogle Scholar
  10. Dubuisson MP, Jain AK (1994) A modified Hausdorff distance for object matching. Pattern Recognit 1:566–568Google Scholar
  11. Duncan J, Dabiri D et al (2010) Universal outlier detection for particle image velocimetry (PIV) and particle tracking velocimetry (PTV) data. Meas Sci Technol 21(5):057002CrossRefGoogle Scholar
  12. Eckstein A, Vlachos PP (2009a) Assessment of advanced windowing techniques for digital particle image velocimetry (DPIV). Meas Sci Technol 20(7):075402CrossRefGoogle Scholar
  13. Eckstein A, Vlachos PP (2009b) Digital particle image velocimetry (DPIV) robust phase correlation. Meas Sci Technol 20(5):055401Google Scholar
  14. Etebari A, Vlachos PP (2005) Improvements on the accuracy of derivative estimation from DPIV velocity measurements. Exp Fluids 39(6):1040–1050CrossRefGoogle Scholar
  15. Green MA, Rowley CW et al (2011) The unsteady three-dimensional wake produced by a trapezoidal pitching panel. J Fluid Mech 685:117–145CrossRefzbMATHGoogle Scholar
  16. Haller G (2001) Distinguished material surfaces and coherent structures in three-dimensional fluid flows. Phys D 149:248–277CrossRefzbMATHMathSciNetGoogle Scholar
  17. Haller G (2002) Lagrangian coherent structures from approximate velocity data. Phys Fluids 14(6):1851–1861CrossRefMathSciNetGoogle Scholar
  18. Haller G (2011) A variational theory of hyperbolic Lagrangian coherent structures. Phys D 240(7):574–598CrossRefzbMATHMathSciNetGoogle Scholar
  19. Haller G, Yuan G (2000) Lagrangian coherent structures and mixing in two-dimensional turbulence. Phys D 147:352–370CrossRefzbMATHMathSciNetGoogle Scholar
  20. Holmes P, Lumley JL et al (1996) Turbulence, coherent structures, dynamical systems and symmetry. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  21. Kahler CJ, Scharnowski S et al (2012) On the uncertainty of digital PIV and PTV near walls. Exp Fluids 52(6):1641–1656CrossRefGoogle Scholar
  22. Karri S, Charonko J et al (2009) Robust wall gradient estimation using radial basis functions and proper orthogonal decomposition (POD) for particle image velocimetry (PIV) measured fields. Meas Sci Technol 20(4):045401Google Scholar
  23. Lekien F, Ross SD (2010) The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds. Chaos 20(1):017505CrossRefMathSciNetGoogle Scholar
  24. Lipinski D, Mohseni K (2010) A ridge tracking algorithm and error estimate for efficient computation of Lagrangian coherent structures. Chaos 20(1):017504Google Scholar
  25. Mathur M, Haller G et al (2007) Uncovering the Lagrangian skeleton of turbulence. Phys Rev Lett 98:1–4Google Scholar
  26. Mikheev AV, Zubtsov VM (2008) Enhanced particle-tracking velocimetry (EPTV) with a combined two-component pair-matching algorithm. Meas Sci Technol 19(8):085401CrossRefGoogle Scholar
  27. Ohmi K, Li H-Y (2000) Particle-tracking velocimetry with new algorithms. Meas Sci Technol 11:603–616CrossRefGoogle Scholar
  28. Ohmi K, Panday SP (2009) Particle tracking velocimetry using the genetic algorithm. J Vis 12(3):217–232CrossRefGoogle Scholar
  29. Olcay AB, Pottebaum TS et al (2010) Sensitivity of Lagrangian coherent structure identification to flow field resolution and random errors. Chaos 20(1):017506Google Scholar
  30. Peng J, Dabiri JO (2009) Transport of inertial particles by Lagrangian coherent structures: application to predator-prey interaction in jelleyfish feeding. J Fluid Mech 623:75–84CrossRefzbMATHGoogle Scholar
  31. Raffel M, Willert CE et al (1998) Particle image velocimetry. Springer, BerlinCrossRefGoogle Scholar
  32. Ruiz T, Boree J et al (2010) Finite time Lagrangian analysis of an unsteady separation induced y a near wall wake. Phys Fluids 22(7):075193CrossRefGoogle Scholar
  33. Scarano F (2002) Iterative image deformation methods in PIV. Meas Sci Technol 13:R1–R19Google Scholar
  34. Scarano F, Riethmuller ML (1999) Iterative multigrid approach in PIV image processing with discrete window offset. Exp Fluids 26(6):513–523CrossRefGoogle Scholar
  35. Senatore C, Ross SD (2011) Detection and characterization of transport barriers in complex flows via ridge extraction of the finite time Lyapunov exponent field. Int J Numer Methods Eng 86:116301174CrossRefGoogle Scholar
  36. Shadden SC (2011) Lagrangian coherent structures. In: Grigoriev R (ed) Transport and mixing in laminar flows: from microfluidics to oceanic currents. Wiley-VCH Verlag GmbH & Co. KGaA, WeinheimGoogle Scholar
  37. Shadden SC, Lekien F et al (2005) Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Phys D 212:271–304CrossRefzbMATHMathSciNetGoogle Scholar
  38. Shadden SC, Dabiri JO et al (2006) Lagrangian analysis of fluid transport in empirical vortex ring flows. Phys Fluids 18:1–11CrossRefMathSciNetGoogle Scholar
  39. Shadden SC, Katija K et al (2007) Transport and stirring induced by vortex formation. J Fluid Mech 593:315–331CrossRefzbMATHGoogle Scholar
  40. Shinneeb A, Balachandar R et al (2006). Quantitative investigation of coherent structures in a free jet using PIV and POD. ASME Joint US—European Fluids Engineering Summer Meeting. Miami, Fl, pp 1–8Google Scholar
  41. Sirovich L (1987) Turbulence and the dynamics of coherent structures I–III. Q Appl Math 45(567–571):573–590MathSciNetGoogle Scholar
  42. Solomon TH, Gollub JP (1988a) Chaotic particle-transport in time-dependent Rayleigh-Benard convection. Phys Rev A 38(12):6280–6286CrossRefGoogle Scholar
  43. Solomon TH, Gollub JP (1988b) Passive transport in steady Rayleigh-Benard convection. Phys Fluids 31(6):1372–1379CrossRefGoogle Scholar
  44. Stewart K, Niebel C et al (2012) The decay of confined vortex rings. Exp Fluids 53(1):163–171Google Scholar
  45. Tallapragada P, Ross SD (2008) Particle segregation by Stokes number for small neutrally buoyant spheres in a fluid. Phys Rev E 78:1–9CrossRefGoogle Scholar
  46. Tallapragada P, Ross SD (2013) A set oriented definition of finite-time Lyapunov exponents and coherent sets. Commun Nonlinear Sci Numer Simul 18(5):1106–1126CrossRefzbMATHMathSciNetGoogle Scholar
  47. Voth GA, Haller G et al (2002) Experimental measurements of stretching fields in fluid mixing. Phys Rev Lett 88(25)Google Scholar
  48. Westerweel J, Scarano F (2005) Universal outlier detection for PIV data. Exp Fluids 39(6):1096–1100CrossRefGoogle Scholar
  49. Wilson MM, Peng J et al (2009) Lagrangian coherent structures in low Reynolds number swimming. J Phys Condens Matter 21:204105Google Scholar
  50. Zhao C, Shi W et al (2005) A new Hausdorff distance for image matching. Pattern Recognit Lett 26(5):581–586Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Samuel G. Raben
    • 1
  • Shane D. Ross
    • 2
  • Pavlos P. Vlachos
    • 1
  1. 1.Department of Mechanical EngineeringVirginia TechBlacksburgUSA
  2. 2.Department of Engineering Science and MechanicsVirginia TechBlacksburgUSA

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