On Reynolds averaging and the zero integral scale constraint

Classically, the integral scale is the integral of the autocorrelation over all lags. It is often used to evaluate whether events in a turbulence time series are independent for the purpose of judging the accuracy of an average or, alternatively, for deciding how long to average to achieve some arbitrary measurement accuracy (Lumley and Panofsky 1964, p. 35ff.). Beginning with Navier-Stokes we derive using Reynolds averaging a surface-layer budget equation for the autocovariance as a function of lag rather than time. We then impose on this differential equation the zero integral scale constraint reported by several investigators (Comte-Bellot and Corrsin 1971; Kaimal and Finnigan 1994, p. 276; Sreenivasan et al. 1978; de Waele et al. 2002) and deduce that the variance must accordingly also be zero. This result is unacceptable. We propose that the cause of this anomaly is that the principles of Reynolds averaging are flawed. Specifically, the autocovariance is determined by averaging over the same T used to estimate the mean. This conflicts with time series theory (cf. Papoulis 1965, p. 330) which dictates that defining the requisite interval for the autocovariance (and thus the variance) requires knowledge of fourth-order moments. This requirement is consistent with the fact that mean and variance represent different degrees of freedom and should therefore each be determined by averaging over different time scales. It is also consistent with the fact that autocovariances determined by ensemble averaging do not necessarily have zero integral scale. Therefore, both standard practice and physical reasoning suggest that it is unwise to apply the same averaging time to mean fields and to turbulence statistics--they exhibit different temporal behaviors. Finally, we suggest that the Time Dependent Memory Method (Treviño and Andreas 2000) offers one avenue for circumventing these problems.

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