Lagrangian multi-particle statistics

  • Beat Lüthi
  • Jacob Berg
  • Søren Ott
  • Jakob Mann
Conference paper
Part of the ERCOFTAC Series book series (ERCO, volume 11)


Combined measurements of the Lagrangian evolution of particle constellations and the coarse grained velocity derivative tensor \( \partial \tilde u_i /\partial x_j \) are presented. The data is obtained from three dimensional particle tracking measurements in a quasi isotropic turbulent flow at intermediate Reynolds number. Particle constellations are followed for as long as one integral time and for several Batchelor times. We suggest a method to obtain quantitatively accurate \( \partial \tilde u_i /\partial x_j \) from velocity measurements at discrete points. We obtain good scaling with \( t_* = \sqrt {2r^2 /15S_r (r)} \) for filtered strain and vorticity and present filtered R-Q invariant maps with the typical ‘tear drop’ shape that is known from velocity gradients at viscous scales. Lagrangian result are given for the growth of particle pairs, triangles and tetrahedra. We find that their principal axes are preferentially oriented with the eigenframe of coarse grained strain, just like constellations with infinitesimal separations are known to do. The compensated separation rate is found to be close to its viscous counterpart as \( 1/2{\text{ }}(dr^2 /dt)/r^2 \cdot t_* /\sqrt 2 \approx 0.11 - 0.14 \). It appears that the contribution from the coarse grained strain field, \( r_i r_j \tilde s_{ij} \) filtered at scale Δ = r, is responsible only for roughly 50% of the separation rate. The rest stems from contributions with scales Δ < r.


Particle Image Velocimetry Inertial Range Particle Track Velocimetry Particle Pair Linear Polynomial 
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.


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  1. [1]
    Brian Sawford. Turbulent relative dispersion. Ann. Rev. Fluid Mech., 33:289–317, 2001.CrossRefGoogle Scholar
  2. [2]
    Jacob Berg, Beat Lüthi, Jakob Mann, and Søren Ott. An experimental investigation: backwards and forwards relative dispersion in turbulent flow. Phys. Rev. E, 74(1):016304, 2006.CrossRefGoogle Scholar
  3. [3]
    Beat Lüthi, Jacob Berg, Søren Ott, and Jakob Mann. Self similar two particle separation model. 1st revision, Physics of Fluids, 2006.Google Scholar
  4. [4]
    Laurent Mydlarski, Alain Pumir, Boris I. Shraiman, Eric D. Siggia, and Zellman Warhaft. Structures and multipoint correlators for turbulent advection: Predictions and experiments. Phys. Rev. Lett., 81(20):4373–4376, 1998.CrossRefGoogle Scholar
  5. [5]
    Misha Chertkov, Alain Pumir, and Boris I. Shraiman. Lagrangian tetrad dynamics and the phenomenology of turbulence. Physics of Fluids, 11(8):2394–2410, August 1999.CrossRefGoogle Scholar
  6. [6]
    Alain Pumir, Boris I. Shraiman, and Misha Chertkov. Geometry of Lagrangian dispersion in turbulence. Phys. Rev. Lett., 85(25):5324–5327, December 2000.CrossRefGoogle Scholar
  7. [7]
    A. Naso and A. Pumir. Scale dependence of the coarse-grained velocity derivative tensor structure in turbulence. Phys. Rev. E, 72:056318, 2005.CrossRefGoogle Scholar
  8. [8]
    L. Biferale, G. Boffetta, A. Celani, B. J. Devenish, A. Lanotte, and F. Toschi. Multiparticle dispersion in fully developed turbulence. Physics of Fluids, 17(11):111701, 1–4, 2005.CrossRefGoogle Scholar
  9. [9]
    A. Naso and A. Pumir. Scale dependence of the coarse-grained velocity derivative tensor: Influence of large-scale shear on small-scale turbulence. J. of Turbulence, 7(41):1–11, 2006.CrossRefGoogle Scholar
  10. [10]
    Vadim Borue and Steven A. Orzag. Local energy flux and subgrid-scale statistics in three-dimensional turbulence. J. Fluid Mech., 366:1–31, 1998.CrossRefGoogle Scholar
  11. [11]
    Fedderik van der Bos, Bo Tao, Charles Meneveau, and Joseph Katz. Effects of small-scale turbulent motions on the filtered velocity gradient tensor as deduced from holographic particle image velocimetry measurements. Physics of Fluids, 14(7):2456–2474, 2002.CrossRefGoogle Scholar
  12. [12]
    G. K. Batchelor. Diffusion in a field of homogeneous turbulence. II The relative motion of particles. Proc. Cambridge Phil. Soc., 48:345–363, 1952.CrossRefGoogle Scholar
  13. [13]
    Alain Pumir, Boris I. Shraiman, and Misha Chertkov. The Lagrangian view of energy transfer in turbulent flow. EuroPhys. Lett., 56(3):379–385, 2001.CrossRefGoogle Scholar
  14. [14]
    Beat Lüthi, Arkady Tsinober, and Wolfgang Kinzelbach. Lagrangian measurement of vorticity dynamics in turbulent flow. J. Fluid Mech., 528:87–118, 2005.CrossRefGoogle Scholar
  15. [15]
    T. Chang and G. Taterson. Application of image processing to the analysis of three-dimensional flow fields. Opt. Engng, 23:283–287, 1983.Google Scholar
  16. [16]
    R. Racca and J. Dewey. A method for automatic particle tracking in a three-dimensional flow field. Experiments in Fluids, 6:25–32, 1988.CrossRefGoogle Scholar
  17. [17]
    Marko Virant and Themistocles Dracos. 3D PTV and its application on Lagrangian motion. Meas. Sci. Technol., 8:1529–1552, 1997.CrossRefGoogle Scholar
  18. [18]
    Søren Ott and Jakob Mann. An experimental investigation of the relative diffusion of particle pairs in three dimensional turbulent flow. J. Fluid Mech., 422:207–223, 2000.CrossRefGoogle Scholar
  19. [19]
    A. La Porta, Greg A. Voth, Alice M. Crawford, Jim Alexander, and Eberhard Bodenschatz. Fluid particle accelerations in fully developed turbulence. Nature, 409:1016–1017, Feb. 2001.CrossRefGoogle Scholar
  20. [20]
    Mickaël Bourgoin, Nicholas T. Ouellette, Haitao Xu, Jacob Berg, and Eberhard Bodenschatz. Pair Dispersion in Turbulence. Science, 311:835–838, 2006.CrossRefGoogle Scholar
  21. [21]
    Michael S. Borgas and P.K. Yeung. Relative dispersion in isotropic turbulence. Part 2. A new stochastic model with Reynolds-number dependence. J. Fluid Mech., 503:125–160, 2004.CrossRefGoogle Scholar
  22. [22]
    A. Tsinober. An Informal Introduction to Turbulence. KluwerAcademic Publishers, 2001.Google Scholar
  23. [23]
    M. Kholmyansky, A. Tsinober, and S. Yorish. Velocity derivatives in the atmospheric surface layer at Re-lambda=104. Physics of Fluids, 13(1):311–314, 2001.CrossRefGoogle Scholar
  24. [24]
    B.J. Cantwell. Exact solution of a restricted euler equation for the velocity-gradient tensor. Physics of Fluids, 4(4):782–793, 1992.CrossRefGoogle Scholar
  25. [25]
    S. S. Girimaji and S. B. Pope. Material-element deformation in isotropic turbulence. J. Fluid Mech., 220:427–458, 1990.CrossRefGoogle Scholar
  26. [26]
    Michele Guala, Beat Lüthi, Alex Liberzon, Arkady Tsinober, and Wolfgang Kinzelbach. On the evolution of material lines and vorticity in homogeneous turbulence. J. Fluid Mech., 533:339–359, 2005.CrossRefGoogle Scholar

Copyright information

© Springer 2007

Authors and Affiliations

  • Beat Lüthi
    • 1
  • Jacob Berg
    • 1
  • Søren Ott
    • 1
  • Jakob Mann
    • 1
  1. 1.Risø National Laboratory, Wind Energy DepartmentRoskildeDenmark

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