RAL SEMINAR: Energy Dissipation Rate Estimation from a Statistical Signal Processing Perspective

Calculation of energy dissipation rates (ε) is ubiquitous in turbulence theory and applications. One of the reasons for this is due to a similarity hypothesis of A. Kolmogorov, which states that for high Reynolds number isotropic flow, there is a range of spatial scales – defining the so-called inertial subrange - for which certain statistical properties of the flow are solely a function of ε. Furthermore, the hypothesis leads to a relatively simple form for the second-order structure functions and power spectral densities of components of the velocity field and passive scalar fields (e.g., temperature) in the inertial subrange. Using Taylor’s hypothesis, which relates spatial and temporal statistics, time series measurements from sensors (e.g., anemometers or aircraft) can then be used to estimate ε (or as is common, 𝜀2/3), using the inertial subrange expressions. These expressions are theoretical, in that they assume that the velocity or scalar processes are stationary continuous-time ones and that they are measured over an infinitely long temporal window. Accommodating the real-world characteristics of discrete sampling and finite window lengths are relatively commonplace in the statistical signal processing literature when calculating correlation functions (and hence structure functions) as well as power spectra. Most practical estimates of 𝜀2/3 are based on the theoretical expressions – as given by the Kolmogorov relations - and it is the exception rather than the rule that the estimation methods take into account the real-world characteristics.

In this talk, we will discuss practical calculations of 𝜀2/3 with a focus on accommodating the real world discrete sampling and finite window aspects – especially for scenarios where the window lengths are not large. Furthermore, consideration will be given to how the statistical properties of the structure functions and power spectra, specifically their covariance properties, tells us something about the statistical performance of the estimators, and can be used to improve their performance. Standard estimation methods, based on maximum likelihood as well as unweighted and weighted least squares approaches will be discussed. The pros and cons of the standard techniques will be delineated, and a discussion as to their performance based on a simulation analysis will be presented. For example, it will be shown that the covariance properties of structure functions (i.e., high correlation between lags) make their utility more problematic than spectral methods – unless those properties are taken into account. We end the talk by presenting a novel and numerically efficient 𝜀2/3 estimation algorithm that accentuates the positive aspects and mitigates many of the negative aspects of the other methods.