A stochastic analysis of a nonlinear flow response

In this article a stochastic analysis of velocity fluctuations from a nonlinear flow response is presented. The statistical analysis is based on probability averages of the flow quantities evaluated over several realizations of the examined turbulent flow. In order to define the realizations, a long-time record of a turbulent velocity signal is cut up into pieces of length TT, where TT is much longer than the characteristic velocity fluctuation correlation time occurring in the flow. These pieces are then treated as observations of different responses in an ensemble of similarly simulated flows. The ergodic assumption is investigated, and it is shown that in some regions of the flow the standard statistical approach of time average fails to capture the nonlinear response of the flow. The ergodicity deviations based on samples from three-dimensional numerical simulations are compared with theoretical predictions given by scaling arguments and also with data from experimental observations. A very good agreement is observed. Calculations of the normalized auto-correlation functions and power spectra are also performed.