Corrections for higher-order spectral estimates of Gaussian processes

A common assumption utilized in turbulence theory is the tendency towards Gaussian statistics for streamwise velocity fluctuations. The analytical expressions associated with the probability distribution functions and statistics that yield from a Gaussian process allow for simplifications in turbulence theories, especially with respect to higher order statistics of velocity fluctuations. In the case of the Random Sweeping Decorrelation Hypothesis, the assumption of Gaussian statistics allows an estimate of the higher-order spectral content, and thus higher-order moments, of the streamwise velocity fluctuations solely from the first-order spectrum. However, most real flows do not follow a normal distribution across all scales, and therefore will depart from these idealized behaviors. Deviations in the spectral content from the ideal Gaussian behavior are investigated with phase randomized turbulence with varying levels of skewness and kurtosis. It is found that combinations of non-zero skewness and sub-gaussian kurtosis combine to compensate for each other, resulting in Gaussian approximations of higher-order moments nearly equating to actual moment calculations. The resulting implications on the Random Sweeping Decorrelation Hypothesis, including both spectral estimates and the logarithmic scaling of the moments, are discussed.