Analysis of Turbulence Intermittency Using Variable Interval Reynolds Time Averaging

In the traditional formulation, a turbulent flow is decomposed into MEAN and RANDOM components using time averages taken over a sufficiently long time interval. Here, "sufficiently long" is defined to be consistent with the presumption that the turbulence is in steady state over the interval. Many intermittent features of turbulent behavior, however, are lost or masked by such averages. We propose an averaging process (VIRTA) which uses variable intervals much like that proposed by Gupta et al. (1971), Blackwelder and Kaplan (1976), and Gupta (1996), except that VIRTA is also applicable to real-time analysis of turbulence, as opposed to strictly post-processing. We also investigate the sensitivity of VIRTA to different specifications of the sensor accuracy.

Specifically, VIRTA begins with the turbulence time series S which is sampled for the period [t-T,t], where t is the present time and T can be on the order of hours. S is first time-averaged over [t-M,t], where M. Such an M also defines the instantaneous random part according to s(x)=S(x)-, where x is a parameter denoting time between t-M and t. From the random part, VIRTA then defines the time scale (R) for analyzing second-order correlations of s(x).

The interesting physics of VIRTA is that M is not the same as R. Both are dynamic (as opposed to strictly kinematic) and vary randomly with time. This is not the case in traditional formulations. Specifically, M is larger than R and is obtained from instantaneous properties of the random part and the accuracy of the measurements. R, on the other hand, is determined from a property called the MEMORY, which is much like an integral scale. The inequality between M and R is compatible with the fact that mean and random components represent different degrees-of-freedom.

Researchers recognize that the proper choice of scales is important in the search for Reynolds-number-independent features. Yet they typically use the same scale for Reynolds averaging all correlations being examined. The fact that classical Reynolds averaging has been used to investigate the structure of higher-order correlations and has not yet yielded an appreciable understanding of the turbulence problem might be due to an inappropriate choice of averaging times. Flows encountered in nature are always nonstationary, while Reynolds averaging requires they be stationary. An averaging procedure that adapts to the natural scales of the turbulence has the advantage that it eliminates user-specific biases. An averaging procedure that can analyze turbulence in real time avoids the corruption caused by an averaging scheme which assumes that the future impacts the present.

We apply VIRTA to a time series of longitudinal wind speed measured with a three-axis sonic anemometer positioned 4 m above the ground in horizontally homogeneous terrain and digitized at 10 Hz. According to the manufacturer, this anemometer has an accuracy that is ±1% of the mean wind. This accuracy is the key to determining M. Our results clearly quantify the intermittency.