Skip to main content

The ‘Hurst phenomenon’ in grid turbulence

  • K. N. Helland (a1) and C. W. Van Atta (a1)

Measurements of the statistical property called the ‘rescaled range’ in grid-generated turbulence exhibit a Hurst coefficient H = 0·5 for 43 < UT/M < 1850, where M/U is a characteristic time scale associated with the grid size M and mean velocity U. Theory predicts that H = 0·5 for independence of two observations separated by a time interval T, and the deviation from H = 0·5 is referred to as the ‘Hurst phenomenon’. The rescaled range obtained for grid turbulence contains an initial region UT/M < 43 of large H, approaching 1·0, corresponding approximately to the usual region of a finite non-zero autocorrelation of turbulent velocity fluctuations. For UT/M > 1850 the rescaled range breaks from H = 0·5 and rises at a significantly faster rate, H = 0·7-0·8, implying a long-term dependence or possibly non-stationarity at long times. The measured autocorrelations remain indistinguishable from zero for UT/M > 20. The break in the trend H = 0·5 is probably caused by motions on scales comparable to characteristic time scales of the wind-tunnel circulation. Rescaled-range analysis is a powerful statistical tool for determining the time scale separating the grid turbulence from the background wind-tunnel motions.

Hide All
Anis, A. A. & Lloyd, E. H. 1953 On the range of partial sums of a finite number of independent normal variates. Biometrika 40, 35.
Batchelor, G. K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.
Comte-Bellot, G. & Corrsin, S. 1971 Simple Eulerian time correlation of full- and narrowband velocity signals in grid-generated, ‘isotropic’ turbulence. J. Fluid Mech. 48, 273.
Feller, W. 1951 The asymptotic distribution of the range of sums of independent random variables. Ann. Math. Statist. 22, 427.
Fischer, H. B. 1973 A possible explanation for Nordin, McQuivey and Mejia's observation of the Hurst phenomenon in turbulence. Water Resources Res. 9, 492.
Frenkiel, F. N. & Klebanoff, P. S. 1967 Higher-order correlations in a turbulent field. Phys. Fluids, 10, 507.
Hurst, H. E. 1951 Long-term storage capacity of reservoirs. Trans. A.S.C.E. 116, 770.
KLEMEšS, V. 1974 The Hurst phenomenon: a puzzle?. Water Resources Res. 10, 675.
Lloyd, E. H. 1967 Stochastic reservoir theory. Adv. in Hydrosci. 4, 281.
Mandelbrot, B. B. 1965 Une classe de procèssus stochastiques homothétiques à sui; application à la loi climatologique de H. E. Hurst. C. R. Acad. Sci. Paris 260, 3274.
Mandelbrot, B. B. 1975 Limit theorems on the self-normalized range for weakly and strongly dependent processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete 31, 271.
Mandelbrot, B. B. & Van ness, J. W. 1968 Fractional Brownian motions, fractional noises and applications. S.I.A.M. Rev. 4, 422.
Mandelbrot, B. B. & Wallis, J. R. 1969a Computer experiments with fractional Gaussian noises. 1, Averages and variances; 2, Rescaled ranges and spectra; 3, Mathematical appendix. Water Resources Res. 5, 228.
Mandelbrot, B. B. & Wallis, J. R. 1969b Some long-run properties of geophysical records. Water Resources Res. 5, 321.
Nordin, C. F., Mcquivey, R. S. & Mejia, J. M. 1972 Hurst phenomenon in turbulence. Water Resources Res. 8, 1480.
Nordin, C. F., Mcquivey, R. S. & Mejia, J. M. 1973 Reply. Water Resources Res. 9, 494.
Van Atta, C. W. & Chen, W. Y. 1968 Correlation measurements in grid turbulence using digital harmonic analysis. J. Fluid Mech. 34, 497.
Van Atta, C. W. & Chen, W. Y. 1969 Correlation measurements in turbulence using digital Fourier analysis. Phys. Fluids Suppl. 12, II 264.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 13 *
Loading metrics...

Abstract views

Total abstract views: 50 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 20th November 2017. This data will be updated every 24 hours.