WWhittle Maximum Likelihood Estimate of spectral properties of Rayleigh-Taylor interfacial mixing using hot-wire anemometry experimental data

Abstract: Investigating the power density spectrum of fluctuations in Rayleigh-Taylor (RT) interfacial mixing is a means of studying characteristic length- and time-scales, anisotropies and anomalous processes. Guided by group theory, analysing the invariance-based properties of the fluctuations, our work examines raw time series from hot-wire anemometry measurements in the experiment by Akula et al., JFM 816, 619-660 (2017). The results suggest that the power density spectrum can be modelled as a compound function presented as the product of a power law and an exponential. The data analysis is based on Whittle's approximation of the power density spectrum for independent zero-mean near-Gaussian signals to construct a Maximum likelihood Estimator (MLE) of the parameters. Those that maximise the log-likelihood are computed numerically through Newton-Raphson iteration. The Hessian of the log-likelihood is used to evaluate the Fisher information matrix and provide an estimate of the statistical error on the obtained parameters. The Kolmogorov-Smirnov test is applied to analyse the goodness-of-fit, by verifying the hypothesis that the ratio between the observed periodogram and the estimated power density spectrum follows a chi-squared probability distribution. The dependence of the parameters of the compound function is investigated on the range of mode numbers over which the fit is performed. In the domain where the relative errors of the power law exponent and the exponential decay rate are small and the goodness-of-fit is excellent, the parameters of the compound function are clearly defined, in agreement with the theory. The study of the power-law spectra in RT mixing data suggests that rigorous physics-based statistical methods can help researchers to see beyond visual inspection.