Power Law Decay Estimation for Turbulent Spectral Densities

Turbulent flows commonly feature power law decay in one or more field quantities, such as the -5/3 inertial subrange power law for velocity spectra. Assuming sufficient time series data are collected, the problem of estimating the power law decay rate of a turbulent spectral density relies on two factors: the correct choice of data window in statistical signal processing, and an objective procedure to estimate the power law decay rate. In this context the single most important factor for a data window is the side-lobe decay rate. Ensuring the side-lobe decay rate exceeds that of measured data avoids the subtle error of spectral leakage. An objective procedure to estimate the power law decay rate is based on a maximum likelihood estimator. Under the assumption that the Fourier transform of turbulence time series is a circularly-symmetric complex normal random variable a likelihood function for the power spectral density is based on a Gamma distribution. Maximizing the log-likelihood, with a spectral model that parameterizes the Gamma distributions, leads to a robust estimator for the power law decay rate. These concepts are illustrated through synthetic realizations of colored noise, acoustic measurements of a supersonic turbulent jet, and atmospheric surface-layer turbulence.