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Small-scale statistics in high-resolution direct numerical simulation of turbulence: Reynolds number dependence of one-point velocity gradient statistics

  • T. ISHIHARA (a1), Y. KANEDA (a1), M. YOKOKAWA (a2), K. ITAKURA (a3) and A. UNO (a2)...

Abstract

One-point statistics of velocity gradients and Eulerian and Lagrangian accelerations are studied by analysing the data from high-resolution direct numerical simulations (DNS) of turbulence in a periodic box, with up to 40963 grid points. The DNS consist of two series of runs; one is with kmaxη ~ 1 (Series 1) and the other is with kmaxη ~ 2 (Series 2), where kmax is the maximum wavenumber and η the Kolmogorov length scale. The maximum Taylor-microscale Reynolds number Rλ in Series 1 is about 1130, and it is about 675 in Series 2. Particular attention is paid to the possible Reynolds number (Re) dependence of the statistics. The visualization of the intense vorticity regions shows that the turbulence field at high Re consists of clusters of small intense vorticity regions, and their structure is to be distinguished from those of small eddies. The possible dependence on Re of the probability distribution functions of velocity gradients is analysed through the dependence on Rλ of the skewness and flatness factors (S and F). The DNS data suggest that the Rλ dependence of S and F of the longitudinal velocity gradients fit well with a simple power law: S ~ −0.32Rλ0.11 and F ~ 1.14Rλ0.34, in fairly good agreement with previous experimental data. They also suggest that all the fourth-order moments of velocity gradients scale with Rλ similarly to each other at Rλ > 100, in contrast to Rλ < 100. Regarding the statistics of time derivatives, the second-order time derivatives of turbulent velocities are more intermittent than the first-order ones for both the Eulerian and Lagrangian velocities, and the Lagrangian time derivatives of turbulent velocities are more intermittent than the Eulerian time derivatives, as would be expected. The flatness factor of the Lagrangian acceleration is as large as 90 at Rλ ≈ 430. The flatness factors of the Eulerian and Lagrangian accelerations increase with Rλ approximately proportional to RλαE and RλαL, respectively, where αE ≈ 0.5 and αL ≈ 1.0, while those of the second-order time derivatives of the Eulerian and Lagrangian velocities increases approximately proportional to RλβE and RλβL, respectively, where βE ≈ 1.5 and βL ≈ 3.0.

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Corresponding author

Author to whom correspondence should be addressed: ishihara@cse.nagoya-u.ac.jp.

References

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Antonia, R., Chambers, A. & Satyaprakash, B. 1981 Reynolds number dependence of high-order moments of the streamwise turbulent velocity derivative. Boundary Layer Met. 21, 159171.
Belin, F., Maurer, J., Tabeling, P. & Willaime, H. 1997 Velocity gradient distribution in fully developed turbulence: An experimental study. Phys. Fluids 9, 38433850.
Biferale, L., Boffetta, G., Celani, A., Devenish, B. J., Lanotte, A. & Toschi, F. 2004 Multifractal statistics of Lagrangian velocity and acceleration in turbulence. Phys. Rev. Lett. 93, 064502.
Biferale, L., Boffetta, G., Celani, A., Lanotte, A. & Toschi, F. 2005 Particle trapping in three-dimensional fully developed turbulence. Phys. Fluids 17, 021701.
Champagne, F. H. 1978 The fine-scale structure of the turbulent velocity field. J. Fluid Mech. 86, 67108.
Gotoh, T. & Fukayama, D. 2001 Pressure spectrum in homogeneous turbulence. Phys. Rev. Lett. 86, 37753779.
Gotoh, T., Fukayama, D. & Nakano, T. 2002 Velocity field statistics in homogeneous steady turbulence obtained using a high-resolution direct numerical simulation. Phys. Fluids 14, 10651081.
Gotoh, T. & Rogallo, R. S. 1999 Intermittency and scaling of pressure at small scales in forced isotropic turbulence. J. Fluid Mech. 396, 257285.
Gylfason, A., Ayyalasomayajula, S. & Warhaft, Z. 2004 Intermittency, pressure and acceleration statistics from hot-wire measurements in wind-tunnel turbulence. J. Fluid Mech. 501, 213229.
Hierro, J. & Dopazo, C. 2003 Fourth-order statistical moments of the velocity gradient tensor in homogeneous, isotropic turbulence. Phys. Fluids 15, 34343442.
Hill, R. J. 2002 a Scaling of acceleration in locally isotropic turbulence. J. Fluid Mech. 452, 361370.
Hill, R. J. 2002 b Possible alternative to R λ-scaling of small-scale turbulence statistics. J. Fluid Mech. 463, 403412.
Ishihara, T. & Kaneda, Y. 2002 High resolution DNS of incompressible homogeneous forced turbulence – Time dependence of the statistics. In Proc. Intl Workshop on Statistical Theories and Computational Approaches to Turbulence (ed. Kaneda, Y. & Gotoh, T.), pp. 179188. Springer.
Ishihara, T., Kaneda, Y., Yokokawa, M., Itakura, K. & Uno, A. 2003 Spectra of energy dissipation, enstrophy and pressure by high-resolution direct numerical simulations of turbulence in a periodic box. J. Phys. Soc. Japan 72 983986.
Ishihara, T., Kaneda, Y., Yokokawa, M., Itakura, K. & Uno, A. 2005 Energy spectrum in the near dissipation range of high resolution direct numerical simulation of turbulence. J. Phys. Soc. Japan 74 14641471.
Jiménez, J., Wray, A. A., Saffman, P. G. & Rogallo, R. S. 1993 The structure of intense vorticity in isotropic turbulence. J. Fluid Mech. 255, 6590.
Kaneda, Y. & Ishihara, T. 2006 High-resolution direct numerical simulation of turbulence. J. Turbulence 7, 20.
Kaneda, Y., Ishihara, T. & Gotoh, K. 1999 Taylor expansions in powers of time of Lagrangian and Eulerian two-point two-time velocity correlations in turbulence. Phys. Fluids 11, 21542166.
Kaneda, Y., Ishihara, T., Yokokawa, M., Itakura, K. & Uno, A. 2003 Energy dissipation rate and energy spectrum in high resolution direct numerical simulations of turbulence in a periodic box. Phys. Fluids 15, L21L24.
Kerr, R. M. 1985 Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence. J. Fluid Mech. 153, 3158.
Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds number. C. R. Acad. Sci. URSS 30, 299303.
Kraichnan, R. H. 1964 Kolmogorov hypotheses and Eulerian turbulence theory, Phys. Fluids. 7, 17231734.
La Porta, A., Voth, G. A., Crawford, A. M., Alexander, J. & Bodenschatz, E. 2001 Fluid particle accelerations in fully developed turbulence. Nature 409, 10171019.
Sawford, B. L., Yeung, P. K., Borgas, M. S., LaPorta, A. Porta, A., Crawford, A. M. & Bodenschatz, E. 2003 Conditional and unconditional acceleration statistics in turbulence. Phys. Fluids 15, 34783489.
Siggia, E. D. 1981a Numerical study of small-scale intermittency in three-dimensional turbulence. J. Fluid Mech. 107, 375406.
Siggia, E. D. 1981 b Invariants for the one-point vorticity and strain rate correlation functions Phys. Fluids 24, 19341936.
Sreenivasan, K. R. & Antonia, R. A. 1997 The phenomenology of small-scale turbulence. Annu. Rev. Fluid Mech. 29, 435472.
Tabeling, P., Zocchi, G., Belin, F., Maurer, J. & Willaime, H. 1996 Probability density functions, skewness, and flatness in large Reynolds number turbulence. Phys. Rev. E 53, 16131621.
Tsinober, A., Vedula, P. & Yeung, P. K. 2001 Random Taylor hypothesis and the behavior of local and convective accelerations in isotropic turbulence. Phys. Fluids 13, 19741984.
Vedula, P. & Yeung, P. K. 1999 Similarity scaling of acceleration and pressure statistics in numerical simulations of isotropic turbulence. Phys. Fluids 11, 12081220.
Vincent, A. & Meneguzzi, M. 1991 The spatial structure and statistical properties of homogeneous turbulence. J. Fluid Mech. 225, 120.
Wang, L.-P., Chen, S., Brasseur, J. G. & Wyngaard, J. C. 1996 Examination of hypotheses in the Kolmogorov refined turbulence theory through high-resolution simulations. Part 1. Velocity field. J. Fluid Mech. 309, 113156.
Yakhot, V. 2003 Pressure-velocity correlations and scaling exponents in turbulence. J. Fluid Mech. 495, 135143.
Yakhot, V. & Sreenivasan, K. R. 2005 Anomalous scaling of structure functions and dynamic constraints on turbulence simulations. J. Statist Phys. 121, 823841.
Yamazaki, Y., Ishihara, T. & Kaneda, Y. 2002 Effects of wavenumber truncation on high-resolution direct numerical simulation of turbulence J. Phys. Soc. Japan 71 (3), 771781.
Yeung, P. K. 2002 Lagrangian investigations of turbulence. Annu. Rev. Fluid Mech. 34, 115142.
Yeung, P. K., Pope, S. B., Lamorgese, A. G. & Donzis, D. A. 2006 Acceleration and dissipation statistics of numerically simulated isotropic turbulence. Phys. Fluids 18, 065103-1-14.
Yokokawa, M., Itakura, K., Uno, A., Ishihara, T. & Kaneda, Y. 2002 16.4-Tflops direct numerical simulation of turbulence by a Fourier spectral method on the Earth Simulator Proceeding of the IEEE/ACM SC2002 Conference (CD-ROM), Baltimore. (http://www.sc-2002.org/paperp.d.f.s/pap.pap273.pdf).
Zhou, T. & Antonia, R. A. 2000 Reynolds number dependence of the small-scale structure of grid turbulence J. Fluid Mech. 406, 81107.
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Small-scale statistics in high-resolution direct numerical simulation of turbulence: Reynolds number dependence of one-point velocity gradient statistics

  • T. ISHIHARA (a1), Y. KANEDA (a1), M. YOKOKAWA (a2), K. ITAKURA (a3) and A. UNO (a2)...

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