Experiments in Fluids

, 59:19 | Cite as

A Lagrangian perspective towards studying entrainment

  • Giuseppe A. RosiEmail author
  • David E. Rival
Research Article


This study presents a method for characterizing entrainment using Lagrangian data. Specifically, the method exploits the temporal evolution of enstrophy along pathlines to determine if pathlines undergo entrainment. The method only recently became feasible with the development of four-dimensional particle tracking velocimetry [4D-PTV; see Schanz et al. (Exp Fluids 57(5):7, 2016)], which produces pathlines of temporal length and spatial density that was previously unattainable. The proposed entrainment method was tested on experimentally acquired 4D-PTV data of a turbulent starting vortex forming behind a linearly accelerating circular plate. The method is shown to be insensitive to its control parameters. The first control parameter is an enstrophy threshold used to identify pathlines that always exhibit low enstrophy. The second control parameter is the size of an “enstrophy-source region”, which is placed within the measurement domain to identify pathlines that gain enstrophy through interactions with an enstrophy source. In the case of the starting vortex, the method reveals topological features significant to the entrainment process, such as the roll-up of irrotational fluid between the shear layer and vortex core, and pockets of entrainment that exist within undulations on the shear layer’s outboard side. Whereas the method proposed here measures the entrainment ratio directly, the use of Lagrangian coherent structures (LCSs) to quantify entrainment is shown to be infeasible for turbulent flows due to the presence of complex material manifolds for such flows. Furthermore, unlike results from Rosi and Rival (J Fluid Mech 811:3750, 2017), which analyzed the spreading of enstrophy isocontours to characterize entrainment into a starting vortex, the proposed method produces an entrainment rate that is insensitive to the enstrophy threshold, and clearly reveals mechanisms of entrainment. Using the starting vortex data, the proposed technique measured an entrainment ratio range similar to that for laminar vortex rings. The result suggests that, for this particular class of flows, turbulence does not play a significant role with respect to entrainment. The proposed method is seen as an effective tool, given its ability to quantify the entrainment, reveal its salient topological features, and potentially be easily adapted to other classes of flows.



The authors wish to thank the Natural Sciences and Engineering Council of Canada (NSERC) for their financial backing. The authors also wish to acknowledge contributions made by John N. Fernando and Jaime Wong to the development and characterization of the experiment.

Supplementary material

348_2017_2465_MOESM1_ESM.mp4 (23.1 mb)
Supplementary material 1 (MP4 23,684 KB)


  1. Brown GL, Roshko A (1974) On density effects and large structure in turbulent mixing layers. J Fluid Mech 64(4):775–816CrossRefGoogle Scholar
  2. Brunton SL, Rowley CW (2010) Fast computation of FTLE fields for unsteady flows: a comparison of methods. Chaos 20:1–12CrossRefGoogle Scholar
  3. Caflisch R (1989) Mathematical aspects of vortex dynamics. In: Proceedings in applied mathematics series, Society for Industrial and Applied MathematicsGoogle Scholar
  4. Chauhan K, Philip J, de Silva CM, Hutchins N, Marusic I (2014) The turbulent/non-turbulent interface and entrainment in a boundary layer. J Fluid Mech 742:119–151CrossRefGoogle Scholar
  5. Corrsin S, Kistler AL (1955) Free-stream boundaries of turbulent flows. Tech. Rep, NACAGoogle Scholar
  6. Dabiri JO, Gharib M (2004) Fluid entrainment in isolated vortex rings. J Fluid Mech 511:311–331CrossRefzbMATHGoogle Scholar
  7. Didden N (1977) Untersuchung laminarer, instabiler ringwirbel mittels laser doppler anemometrie. Technical Report 67Google Scholar
  8. Dimotakis PE, Brown GL (1976) The mixing layer at high reynolds number: large-structure dynamics and entrainment. J Fluid Mech 78:535–560CrossRefGoogle Scholar
  9. Fernando JN, Rival DE (2016a) On vortex evolution in the wake of axisymmetric and non-axisymmetric low-aspect-ratio accelerating plates. Phys Fluids 28(1):017,102CrossRefGoogle Scholar
  10. Fernando JN, Rival DE (2016b) Reynolds-number scaling of vortex pinch-off on low-aspect-ratio propulsors. J Fluid Mech 799Google Scholar
  11. Green MA, Rowley CW, Haller G (2007) Detection of Lagrangian coherent structures in three-dimensional turbulence. J Fluid Mech 572:111–120MathSciNetCrossRefzbMATHGoogle Scholar
  12. Head MR, Bandyopadhay PR (1981) New aspects of turbulent boundary-layer structure. J Fluid Mech 107:297–338CrossRefGoogle Scholar
  13. Holzner M, Lüthi B (2011) Laminar superlayer at the turbulence boundary. Phys Rev Lett 106(134):503Google Scholar
  14. Huang Z, Kawall JG, Keffer JF, Ferre JA (1995) On the entrainment process in plane turbulent wakes. Phys Fluids 7(5):1130–1141CrossRefGoogle Scholar
  15. Knisely CW (1990) Strouhal numbers of rectangular cylinders at incidence: a review and new data. J Fluids Struct 4(4):371–393CrossRefGoogle Scholar
  16. Krug D, Holzner M, Lüthi B, Wolf M, Kinzelbach W, Tsinober A (2013) Experimental study of entrainment and interface dynamics in a gravity current. Exp Fluids 54(1530):1–13zbMATHGoogle Scholar
  17. Lüthi B, Tsinobar A, Kinzelbach W (2005) Lagrangian measurement of vorticity dynamics in turbulent flow. J Fluid Mech 528:87118CrossRefGoogle Scholar
  18. Mathew J, Basu AJ (2002) Some characteristics of entrainment at a cylindrical turbulence boundary. Phys Fluids 14(7):2065–2072MathSciNetCrossRefzbMATHGoogle Scholar
  19. Mistry D, Philip J, Dawson JR, Marusic I (2016) Entrainment at multi-scales across the turbulent/non-turbulent interface in an axisymmetric jet. J Fluid Mech 802:690–725MathSciNetCrossRefGoogle Scholar
  20. Mohebi M (2016) Influence of thickness and angle of attack on the dynamics of rectangular cylinder wakeGoogle Scholar
  21. Nakamura Y, Hirata K (1989) Critical geometry of oscillating bluff bodies. J Fluid Mech 208:375393CrossRefGoogle Scholar
  22. Olcay AB, Krueger PS (2008) Measurement of ambient fluid entrainment during laminar vortex ring formation. Exp Fluids 44(2):235–247CrossRefGoogle Scholar
  23. Olcay AB, Krueger PS (2010) Momentum evolution of ejected and entrained fluid during laminar vortex ring formation. Theoret Comput Fluid Dyn 24(5):465–482CrossRefzbMATHGoogle Scholar
  24. Phillip J, Marusic I (2012) Large-scale eddies and their roles in entrainment in turbulent jets and wakes. Phys Fluids 48(055108)Google Scholar
  25. Raben SG, Ross SD, Vlachos PP (2014) Computation of finite time Lyapunov exponents from time resolved particle image velocimetry data. Exp FluidsGoogle Scholar
  26. Raffel M, Willert CE, Wereley ST, Kompenhans J (2007) Particle image velocimetry: a practical guide, 2nd edn. Springer, BerlinGoogle Scholar
  27. Rival DE, Kriegseis J, Schaub P, Widmann A, Tropea C (2014) Characteristic length scales for vortex detachment on plunging profiles with varying leading-edge geometry. Exp Fluids 55(1):1660CrossRefGoogle Scholar
  28. Rosi GA, Rival DE (2017) Entrainment and topology of accelerating shear layers. J Fluid Mech 811:3750. MathSciNetCrossRefGoogle Scholar
  29. Rosi GA, Walker AM, Rival DE (2015) Lagrangian coherent structure identification using a Voronoi tesselation-based networking algorithm. Exp Fluids 56(10):1–14CrossRefGoogle Scholar
  30. Schanz D, Gesemann S, Schröder A (2016) Shake-the-box: Lagrangian particle tracking at high particle image densities. Exp Fluids 57(5):7CrossRefGoogle Scholar
  31. Shadden SC, Lekien F, Marsden JE (2005) Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Phys D 212:271–304MathSciNetCrossRefzbMATHGoogle Scholar
  32. Shadden SC, Dabiri JO, Marsden JE (2006) Lagrangian analysis of fluid transport in empirical vortex ring flows. Phys Fluids 18(047105):1–11MathSciNetzbMATHGoogle Scholar
  33. Shadden SC, Katija K, Rosenfeld M, Marsden JE, Dabiri JO (2007) Transport and stirring induced by vortex formation. J Fluid Mech 593:315–331CrossRefzbMATHGoogle Scholar
  34. Wang M, Tao J, Wang C (2013) Flowvisual: Design and evaluation of a visualization tool for teaching 2D flow field concepts. 120th ASEE annual conference and expositionGoogle Scholar
  35. Westerweel J, Hofmann T, C F, J H (2009) Momentum and scalar transport at the turbulent/non-turbulent interface of a jet. J Fluid Mech 631:199–230CrossRefzbMATHGoogle Scholar
  36. Wolf M, Lüthi B, Holzner M, Krug D, Kinzelbach W, Tsinober A (2012) Investigations on the local entrainment velocity in a turbulent jet. Phys Fluids 24(10):105,110CrossRefGoogle Scholar
  37. Wolf M, Holzner M, Krug D, Lüthi B, Kinzelbach W, Tsinober A (2013) Effects of mean shear on the local turbulent entrainment process. J Fluid Mech 731:95–116CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Mechanical and Materials EngineeringQueen’s UniversityKingstonCanada

Personalised recommendations