The ‘Hurst phenomenon’ in grid turbulence

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Abstract
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.