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

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## Abstract

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 (*R*_{λ} = 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 *V*_{η} ≡ *η*/τ_{η} in which *η* and τ_{η} are the Kolmogorov space and time scales, respectively, scales with the Reynolds number *R*_{λ} 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

O357.5## 2010 Mathematics Subject Classification

76F05 82C40## Preview

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## References

- [1]DIMOTAKIS, P. E. Turbulent mixing.
*Annual Review of Fluid Mechanics*,**37**, 329–356 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - [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]SAWFORD, B. Turbulent relative dispersion.
*Annual Review of Fluid Mechanics*,**33**, 289–317 (2001)CrossRefzbMATHGoogle Scholar - [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]TOSCHI, F. and BODENSCHATZ, E. Lagrangian properties of particles in turbulence.
*Annual Review of Fluid Mechanics*,**41**, 375–404 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - [6]TAYLOR, G. Diffusion by continuous movements.
*Proceedings of the London Mathematical Society*,**20**, 196–212 (1922)MathSciNetCrossRefzbMATHGoogle Scholar - [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]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]RICHARDSON, L. F. Atmospheric diffusion shown on a distance-neighbour graph.
*Proceedings of the Royal Society A*,**110**, 709–737 (1926)CrossRefGoogle Scholar - [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]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]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]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]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]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]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]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]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]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]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]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]POPE, S. B.
*Turbulent Flows*, Cambridge University Press, Cambridge (2000)CrossRefzbMATHGoogle Scholar - [23]MONIN, A. S. and YAGLOM, A. M.
*Statistical Fluid Mechanics: Mechanics of Turbulence*, MIT Press, Cambridge (1975)Google Scholar - [24]BIFERALE, L. Lagrangian structure functions in turbulence: experimental and numerical results.
*Physics of Fluids*,**20**, 065103 (2008)CrossRefzbMATHGoogle Scholar