Abstract
The Shannon entropy is a rigorous measure to evaluate the complexity in dynamical systems. Shannon entropy can be directly calculated from any set of experimental or numerical data and yields the uncertainty of a given dataset. Originating from information theory, the concept can be generalized from assessing the uncertainty in a message to any dynamical system. Following the concept of ergodicity, turbulence forms another class of dynamical systems, which is generally assessed using statistical measures. The quantification of resolution quality is a crucial aspect in assessing turbulent-flow simulations. While a vast variety of statistical measures for the evaluation of resolution is available, measures closer representing the dynamics of a turbulent systems, such as the Wasserstein metric or the Ljapunov exponent become popular. This study investigates how the Shannon entropy can lead to useful insights in the quality of turbulent-flow simulations. The Shannon entropy is calculated based on distributions, which enables the direct evaluation from unsteady flow simulations or by post-processing. A turbulent channel flow and a planar turbulent jet are used as validation tests. The Shannon entropy is calculated for turbulent velocity- and scalar-fields and correlations with physical quantities, such as turbulent kinetic energy and passive scalars, are investigated. It is shown that the spatial structure of the Shannon entropy can be related to flow phenomena. This is illustrated by the investigation of the entropy of the velocity fluctuations, passive scalars and turbulent kinetic energy. Grid studies reveal the Shannon entropy as a converging measure. It is demonstrated, that classical turbulent-kinetic-energy-based quality measures struggle with the identification of insufficient resolution, while the Shannon entropy has demonstrated potential to form a solid basis for LES quality assessment.
Highlights
Turbulence theory has evolved and many statistical measures are available for describing turbulence through statistical analysis tools
While measures based on the turbulent kinetic energy are often criticized for generating misleading statements about the simulation quality Davidson (2009); Nguyen and Kempf (2017)— as emphasized in Figs. 6/10—an entropy-measure based on the total fluctuation value may be considered, as it obtains its information about the flow physics rather from the statistical structure of the flow field than from absolute values
The utility of the Shannon entropy has been studied for different flow variables in different canonical flows in large-eddy simulation (LES)
Summary
Turbulence theory has evolved and many statistical measures are available for describing turbulence through statistical analysis tools. A large number of statistical quantities can be considered to describe a turbulent flow—including spectral information, spatial correlations, length-scales, structure functions and others—which often require joint measurements or sampling at two different points in space. Filtering, the physical model for the subgrid scales, the choice of numerical algorithm, grid resolution and filter size determine the quality of an LES Nastac et al (2017). There are plenty of forms of entropy in the context of dynamical systems, they are all related together as they assess the complexity of a given set This can be divided into the study of the properties unpredictability, incompressibility, asymmetry and delayed recurrenceHillman (1998); Wyner (1994)
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