Non-equilibrium statistical mechanics models can be used to construct reduced order models from the time-dynamics data such as numerical or physical fluid mechanics experiments. One of the well-established statistical projection methods is the Kramers-Moyal expansion (KM) method. The first two terms of the KM expansion result can be used to construct a non-linear Langevin equation, which can serve as the statistically-trained reduced-order model. This non-linear Langevin equation can be approximated to the Fokker-Planck equation, which is similar to Advection-Diffusion equation, thereby preserving some characteristics of fluctuations associated with fluid mechanics. The KM method captures continuous-time dynamics, however, any data obtained through measurement is discrete. In order to accurately capture the time dynamics of the discrete data, the method for calculating the KM coefficients must be carefully chosen and implemented. To better represent the solution from discrete data, the drift and diffusion coefficients can be calculated at multiple time scales and then extrapolated to a time scale of zero, assuming a linear correlation. One challenge in using this method is that the calculated KM coefficients are only accurate for time scales greater than the Taylor microscale. This means that the extrapolation must use only the KM coefficients calculated for time scales greater than the Taylor microscale, however, this value is not always provided from the data nor simple to calculate. This work presents a method of approximating the Taylor microscale from the data through the relationship between the Markov property and the Taylor microscale and implementing this method to find the extrapolated KM coefficients. The KM method implementing the Taylor microscale estimation was applied to existing DNS turbulent channel flow data to model a time series. This generated time series was then compared to the DNS data using a statistical analysis including probability density function, autocorrelation, and power spectral density.

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