Towards a Simplified Dynamic Wake Model using POD Analysis

We apply the proper orthogonal decomposition (POD) to large eddy simulation data of a wind turbine wake in a turbulent atmospheric boundary layer. The turbine is modeled as an actuator disk. Our analyis mainly focuses on the question whether POD could be a useful tool to develop a simplified dynamic wake model. The extracted POD modes are used to obtain approximate descriptions of the velocity field. To assess the quality of these POD reconstructions, we define simple measures which are believed to be relevant for a sequential turbine in the wake such as the energy flux through a disk in the wake. It is shown that only a few modes are necessary to capture basic dynamical aspects of these measures even though only a small part of the turbulent kinetic energy is restored. Furthermore, we show that the importance of the individual modes depends on the measure chosen. Therefore, the optimal choice of modes for a possible model could in principle depend on the application of interest. We additionally present a possible interpretation of the POD modes relating them to specific properties of the wake. For example the first mode is related to the horizontal large scale movement. Besides yielding a deeper understanding, this also enables us to view our results in comparison to existing dynamic wake models

The Langevin Approach: An R Package for Modeling Markov Processes

We describe an R package developed by the research group Turbulence, Wind energy and Stochastics (TWiSt) at the Carl von Ossietzky University of Oldenburg, which extracts the (stochastic) evolution equation underlying a set of data or measurements. The method can be directly applied to data sets with one or two stochastic variables. Examples for the one-dimensional and two-dimensional cases are provided. This framework is valid under a small set of conditions which are explicitly presented and which imply simple preliminary test procedures to the data. For Markovian processes involving Gaussian white noise, a stochastic differential equation is derived straightforwardly from the time series and captures the full dynamical properties of the underlying process. Still, even in the case such conditions are not fulfilled, there are alternative versions of this method which we discuss briefly and provide the user with the necessary bibliography

An open source MATLAB package to perform basic statistical analysis of turbulence data and other complex systems along with its application to Fokker-Planck equation and Integral fluctuation theorem

We present a user-friendly open-source \proglang{MATLAB\textsuperscript{\textregistered}} package developed by the research group Turbulence, Wind energy and Stochastics (TWiSt) at the Carl von Ossietzky University of Oldenburg. This package enables to perform a standard analysis of given turbulent data and extracts the stochastic equations describing the scale-dependent cascade process in turbulent flows through Fokker-Planck equations. As a precondition, Markovian properties of the process in scale are tested. Such a stochastic scale-dependent cascade process allows a comprehensive statistical description in terms of the complexity of the data. Cascade trajectories can be defined as single events, for each of which a total entropy production can be determined. For such entropy fluctuations a rigorous law of non-equilibrium stochastic thermodynamics, namely the integral fluctuation theorem, will be verified. As the analysis of the scale-dependent cascade process through a hierarchy of spatial and temporal scales in turbulent flows is an integral part of turbulence theory, this interdisciplinary treatment of the turbulent cascade process has the potential for a new way to link the statistical description of turbulence (via common two-point increment statistics), non-equilibrium stochastic thermodynamics and local turbulent flow structures. The presented package can be used also for the analysis of other data with turbulent like complexity.Comment: This open source MATLAB package can be downloaded with all the supplementary material (data, source code and standalone applications (64-bit) for Windows, macOS and Linux) to replicate all the results presented in this paper from the repository on GitHub https://github.com/andre-fuchs-uni-oldenburg/OPEN_FPE_IF