Evidence for scale similarity of internal intermittency in turbulent flows at large Reynolds numbers

Abstract

Some new measurements and a reassessment of previous data on statistical properties of the breakdown coefficients qr,l in high Reynolds number turbulence show the existence of a range of scale similarity for scales larger than those in the viscous range (l [ges ] 36η). The rate of variation of the probability density p(qr,l) with changing outer scale l/η decreases as l/η increases, becoming fairly insignificant for the largest values of l/η. Measurements of characteristic functions of the probability densities show a substantial degree of statistical independence for sequential adjoint values of qr,l, consistent with the small values of the correlation coefficients for these variables. The data for the moments of qr,l exhibit a behaviour very close to that predicted by the scale-similarity theory when only data for r [ges ] 36η are considered, i.e. data for smaller inner length scales are excluded. The moments and corresponding values of the parameters μp are in good agreement with our previous results and with some earlier data of Kholmyansky, but some rather large unresolved differences in the probability densities of qr,l are found on comparing the present data with those of Kholmyansky. The present measurements of breakdown coefficients for ζ1 = [uscr ]− 1∂u/∂t = ∂(In [uscr ])/∂tand ζ2 = U−1∂u/∂t the time derivatives of the streamwise velocity and its logarithm measured in the atmospheric boundary layer, resolve some previous questions concerning the sensitivity of the results obtained to the choice of positive variable, varying sampling rates and the values of external parameters.For low sampling rates, a systematic change in the shape of the probability densities p(qr,l) with varying digital sampling rate is found using either ζ1 or ζ2. For sufficiently high sampling rates, the probability densities are independent of the sampling rate; and invariant results are obtained when the sampling rate is at least one-quarter of the Kolmogorov frequency associated with the viscous length scale based on the turbulent dissipation rate. The probability densities p(qr,l) measured using either ζ1 or ζ2 are very similar to the corresponding spectra of ζ1 or ζ2 respectively. Comparison of the mean-square values of ζ1 and ζ2 with an extended form of Taylor's hypothesis shows that the variable ζ1 is not a good approximation to the true spatial derivative ∂u/∂x, and the use of such an approximation can lead to results that are both qualitatively and quantitatively incorrect.