Analysis of turbulence by statistics based on generalized entropies. (English) Zbl 0978.76034
Summary: We derive non-perturbatively an analytical formula for scaling exponents of velocity structure function for fully developed turbulence, by assuming that its underlying statistics is based on generalized measures of entropy, the Renyi entropy or the Havrda-Charvat-Tsallis (HCT) entropy. It is revealed by a self-consistent analysis for the observed value \(\mu=0.220\;(\pm 1\%)\) that the formula explains experimental data very well for the value \(q=0.343\) of an index which appears in the measures of Renyi entropy or of the HCT entropy. We also obtain the probability density functions of velocity fluctuation and of velocity gradient.
MSC:
| 76F55 | Statistical turbulence modeling |
Keywords:
velocity structure function; probability density functions; velocity fluctuation; scaling exponent; Renyi entropy; Havrda-Charvat-Tsallis entropyReferences:
| [1] | Kolmogorov, A. N., C. R. Acad. Sci. USSR, 30, 301; 538 (1941) |
| [2] | Heisenberg, W., Z. Phys., 124, 628 (1948) · Zbl 0034.26903 |
| [3] | Kraichnan, R. H., J. Fluid Mech., 5, 497 (1959) · Zbl 0093.41202 |
| [4] | T. Nakano, F. Tanaka, Prog. Theoret. Phys. 65 (1981) 120 and the references therein.; T. Nakano, F. Tanaka, Prog. Theoret. Phys. 65 (1981) 120 and the references therein. |
| [5] | Oboukhov, A. M., J. Fluid Mech., 13, 77 (1962) · Zbl 0112.42002 |
| [6] | Kolmogorov, A. N., J. Fluid Mech., 13, 82 (1962) · Zbl 0112.42003 |
| [7] | Yaglom, A. M., Sov. Phys. Dokl., 11, 26 (1966) |
| [8] | Frisch, U.; Sulem, P.-L.; Nelkin, M., J. Fluid Mech., 87, 719 (1978) · Zbl 0395.76051 |
| [9] | Parisi, G.; Frisch, U., (Ghil, M.; Benzi, R.; Parisi, G., Turbulence and Predictability in Geophysical Fluid Dynamics (1985), North-Holland: North-Holland Amsterdam), 84 |
| [10] | Benzi, R.; Paladin, G.; Parisi, G.; Vulpiani, A., J. Phys. A, 17, 3521 (1984) |
| [11] | Meneveau, C.; Sreenivasan, K. R., Phys. Rev. Lett., 59, 1424 (1987) |
| [12] | Meneveau, C.; Sreenivasan, K. R., Nucl. Phys. B (Proc. Suppl.), 2, 49 (1987) |
| [13] | Hosokawa, I., Phys. Rev. Lett., 66, 1054 (1991) |
| [14] | She, Z.-S.; Leveque, E., Phys. Rev. Lett., 72, 336 (1994) |
| [15] | She, Z.-S.; Waymire, E., Phys. Rev. Lett., 74, 262 (1995) |
| [16] | Arimitsu, T.; Arimitsu, N., Phys. Rev. E, 61, 3237 (2000) |
| [17] | Renyi, A., Proc. 4th Berkley Symp. Maths. Stat. Prob., 1, 547 (1961) · Zbl 0106.33001 |
| [18] | Havrda, J. H.; Charvat, F., Kybernatica, 3, 30 (1967) · Zbl 0178.22401 |
| [19] | Tsallis, C., J. Stat. Phys., 52, 479 (1988) · Zbl 1082.82501 |
| [20] | C. Tsallis, Braz. J. Phys. 29 (1999) 1 and the references therein.; C. Tsallis, Braz. J. Phys. 29 (1999) 1 and the references therein. |
| [21] | Lyra, M. L.; Tsallis, C., Phys. Rev. Lett., 80, 53 (1998) |
| [22] | Costa, U. M.S.; Lyra, M. L.; Plastino, A. R.; Tsallis, C., Phys. Rev. E, 56, 245 (1997) |
| [23] | Arimitsu, T.; Arimitsu, N., J. Phys. A, 33, L235 (2000) |
| [24] | T. Arimitsu, N. Arimitsu, Chaos, Solitons and Fractals (2000), in press.; T. Arimitsu, N. Arimitsu, Chaos, Solitons and Fractals (2000), in press. · Zbl 1051.82002 |
| [25] | T. Arimitsu, N. Arimitsu, Analysis of fully developed turbulence by a generalized statistics, 2000, submitted for publication.; T. Arimitsu, N. Arimitsu, Analysis of fully developed turbulence by a generalized statistics, 2000, submitted for publication. · Zbl 1005.76038 |
| [26] | Anselmet, F.; Cagne, Y.; Hopfinger, E. J.; Antonia, R. A., J. Fluid Mech., 140, 63 (1984) |
| [27] | Meneveau, C.; Sreenivasan, K. R., J. Fluid Mech., 224, 429 (1991) · Zbl 0717.76061 |
| [28] | Benzi, R.; Ciliberto, S.; Baudet, C.; Chavarria, C. R., Physica D, 80, 385 (1995) · Zbl 0899.76196 |
| [29] | Chhabra, A. B.; Sreenivasan, K. R., Phys. Rev. A, 43, 1114 (1991) |
| [30] | Benzi, R.; Biferale, L.; Paladin, G.; Vulpiani, A.; Vergassola, M., Phys. Rev. Lett., 67, 2299 (1991) |
| [31] | Beck, C., Physica A, 277, 115 (2000) |
| [32] | C. Beck, G.S. Lewis, H.L. Swinney, preprint, 2000.; C. Beck, G.S. Lewis, H.L. Swinney, preprint, 2000. |
| [33] | F.M. Ramos, C. Rodrigues Neto, R.R. Rosa, preprint, 2000.; F.M. Ramos, C. Rodrigues Neto, R.R. Rosa, preprint, 2000. |
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