Applied Mathematics and Mechanics

, Volume 39, Issue 11, pp 1605–1616 | Cite as

Effects of the Reynolds number on a scale-similarity model of Lagrangian velocity correlations in isotropic turbulent flows

  • Zhaoyu Shi
  • Jincai Chen
  • Guodong JinEmail author


A scale-similarity model of a two-point two-time Lagrangian velocity correlation (LVC) was originally developed for the relative dispersion of tracer particles in isotropic turbulent flows (HE, G. W., JIN, G. D., and ZHAO, X. Scale-similarity model for Lagrangian velocity correlations in isotropic and stationary turbulence. Physical Review E, 80, 066313 (2009)). The model can be expressed as a two-point Eulerian space correlation and the dispersion velocity V. The dispersion velocity denotes the rate at which one moving particle departs from another fixed particle. This paper numerically validates the robustness of the scale-similarity model at high Taylor micro-scale Reynolds numbers up to 373, which are much higher than the original values (Rλ = 66, 102). The effect of the Reynolds number on the dispersion velocity in the scale-similarity model is carefully investigated. The results show that the scale-similarity model is more accurate at higher Reynolds numbers because the two-point Lagrangian velocity correlations with different initial spatial separations collapse into a universal form compared with a combination of the initial separation and the temporal separation via the dispersion velocity. Moreover, the dispersion velocity V normalized by the Kolmogorov velocity Vηηη in which η and τη are the Kolmogorov space and time scales, respectively, scales with the Reynolds number Rλ as \(V/V_\eta\propto{R_\lambda^{1.39}}\) obtained from the numerical data.

Key words

turbulent mixing relative dispersion Lagrangian velocity correlation scale-similarity model dispersion velocity Reynolds number effect 

Chinese Library Classification


2010 Mathematics Subject Classification

76F05 82C40 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    DIMOTAKIS, P. E. Turbulent mixing. Annual Review of Fluid Mechanics, 37, 329–356 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    BOURGOIN, M., OUELLETTE, N. T., XU, H. T., BERG, J., and BODENSCHATZ, E. The role of pair dispersion in turbulent flow. Science, 311, 835–838 (2005)CrossRefGoogle Scholar
  3. [3]
    SAWFORD, B. Turbulent relative dispersion. Annual Review of Fluid Mechanics, 33, 289–317 (2001)CrossRefzbMATHGoogle Scholar
  4. [4]
    SALAZAR, J. P. L. C. and COLLINS, L. R. Two-particle dispersion in isotropic turbulent flows. Annual Review of Fluid Mechanics, 41, 405–432 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    TOSCHI, F. and BODENSCHATZ, E. Lagrangian properties of particles in turbulence. Annual Review of Fluid Mechanics, 41, 375–404 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    TAYLOR, G. Diffusion by continuous movements. Proceedings of the London Mathematical Society, 20, 196–212 (1922)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    BATCHELOR, G. K. The application of the similarity theory of turbulence to atmospheric diffusion. Quarterly Journal of the Royal Meteorological Society, 76, 133–146 (1950)CrossRefGoogle Scholar
  8. [8]
    BATCHELOR, G. K. Diffusion in a field of homogeneous turbulence: II. The relative motion of particles. Mathematical Proceedings of the Cambridge Philosophical Society, 48, 345–362 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    RICHARDSON, L. F. Atmospheric diffusion shown on a distance-neighbour graph. Proceedings of the Royal Society A, 110, 709–737 (1926)CrossRefGoogle Scholar
  10. [10]
    DHARIWAL, R. and BRAGG, A. Tracer particles only separate exponentially in the dissipation range of turbulence after extremely long times. Physical Review Fluids, 3, 034604 (2018)CrossRefGoogle Scholar
  11. [11]
    SMITH, F. and HAY, J. The expansion of clusters of particles in the atmosphere. Quarterly Journal of the Royal Meteorological Society, 87, 82–101 (1961)CrossRefGoogle Scholar
  12. [12]
    HE, G. W., JIN, G. D., and ZHAO, X. Scale-similarity model for Lagrangian velocity correlations in isotropic and stationary turbulence. Physical Review E, 80, 066313 (2009)CrossRefGoogle Scholar
  13. [13]
    JIN, G. D., HE, G. W., and WANG, L. P. Large-eddy simulation of turbulent collision of heavy particles in isotropic turbulence. Physics of Fluids, 22, 055106 (2010)CrossRefzbMATHGoogle Scholar
  14. [14]
    JIN, G. D. and HE, G. W. A nonlinear model for the subgrid timescale experienced by heavy particles in large eddy simulation of isotropic turbulence with a stochastic differential equation. New Journal of Physics, 15, 035011 (2013)CrossRefGoogle Scholar
  15. [15]
    HE, G. W., WANG, M., and LELE, S. K. On the computation of space-time correlations by large-eddy simulation. Physics of Fluids, 16, 3859–3867 (2004)CrossRefzbMATHGoogle Scholar
  16. [16]
    HE, G. W., RUBINSTEIN, R., and WANG, L. P. Effects of subgrid-scale modeling on time correlations in large eddy simulation. Physics of Fluids, 14, 2186–2193 (2002)CrossRefzbMATHGoogle Scholar
  17. [17]
    YANG, Y., HE, G.W., and WANG, L. P. Effects of subgrid-scale modeling on Lagrangian statistics in large-eddy simulation. Journal of Turbulence, 9, 1–24 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    POJE, A. C., OZGÖKMEN, T. M., LIPPHARDT, B. L., JR, HAUS, B. K., RYAN, E. H., HAZA, A. C., JACOBS, G. A., RENIERS, A. J., OLASCOAGA, M. J., NOVELLI, G., GRIFFA, A., BERON-VERA, F. J., CHEN, S. S., COELHO, E., HOGAN, P. J., KIRWAN, A. D., Jr, HUNTLEY, H. S., and MARIANO, A. J. Submesoscale dispersion in the vicinity of the deepwater horizon spill. Proceedings of the National Academy of Sciences of the United States of America, 111, 12693–12698 (2014)CrossRefGoogle Scholar
  19. [19]
    ESWARAN, V. and POPE, S. B. An examination of forcing in direct numerical simulations of turbulence. Computers and Fluidse, 16, 257–278 (1988)CrossRefzbMATHGoogle Scholar
  20. [20]
    YEUNG, P. K. and POPE, S. B. Lagrangian statistics from direct numerical simulations of isotropic turbulence. Journal of Fluid Mechanics, 207, 531–586 (1989)MathSciNetCrossRefGoogle Scholar
  21. [21]
    PRESS, W. H., TEUKOLSKY, S. A., VETTERLING, W. T., and FLANNERY, B. Numerical Recipes in Fortran: the Art of Scientific Computing, Cambridge University Press, New York (1993)zbMATHGoogle Scholar
  22. [22]
    POPE, S. B. Turbulent Flows, Cambridge University Press, Cambridge (2000)CrossRefzbMATHGoogle Scholar
  23. [23]
    MONIN, A. S. and YAGLOM, A. M. Statistical Fluid Mechanics: Mechanics of Turbulence, MIT Press, Cambridge (1975)Google Scholar
  24. [24]
    BIFERALE, L. Lagrangian structure functions in turbulence: experimental and numerical results. Physics of Fluids, 20, 065103 (2008)CrossRefzbMATHGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.State Key Laboratory of Nonlinear Mechanics (LNM), Institute of MechanicsChinese Academy of SciencesBeijingChina
  2. 2.School of Engineering ScienceUniversity of Chinese Academy of SciencesBeijingChina
  3. 3.School of EngineeringSun Yat-sen UniversityGuangzhouChina

Personalised recommendations