Lyapunov exponent and Wasserstein metric as validation tools for assessing short-time dynamics and quantitative model evaluation of large-eddy simulation

Abstract

and velocity. This data is obtained from various diagnostics techniques, including nonintrusive methods such as laser spectroscopy and particle image velocimetry, or intrusive techniques such as exhaustgas sampling or thermocouple measurements [7,8,9,10]. Instead of directly measuring physical quantities, they are typically inferred from measured signals, introducing several correction factors and uncertainties [11]. Depending on the experimental technique, these measurements are generated from single-point data, line measurements, line-of-sight absorption, or multidimensional imaging at acquisition rates ranging from single-shot to high-repetition rate measurements to resolve turbulent dynamics [12]. This data is commonly processed in the form of statistical results from Favre and Reynolds averaging, conditional data, probability density functions, and scatter data. However, the diversity of data, it makes it difficult to fully utilize this data for a complete model assessment.3. Dynamic description: LES is inherently unsteady and a dynamic measure is desirable to further characterize the LES quality in representing the dynamic content of a simulation. This is particularly relevant for flows that are inherently transient. The key observation is that turbulence is a deterministic chaotic phenomenon, which is characterized by an aperiodic long-term behavior exhibiting high sensitivity to the initial conditions. Different approaches exist to measure and characterize a chaotic solution [13,14,15,16,17]. On one hand, geometric approaches estimate the fractal dimension of the chaotic attractor, which, in turn, gives an estimate of the active degrees of freedom of the chaotic dynamical system. An accurate measure is the Hausdorff dimension [14], which is often approximated by box counting, based on phase-space partitioning and correlation dimension based on time series analysis [16]. On the other hand, dynamical approaches estimate the entropy content of the solution, namely the frequency with which a solution visits different regions of an attractor, for example by the Kolmogorov-Sinai entropy.The objective of this contribution is to introduce methods that enable the quantitative analysis of LES.The first metric introduces the Lyapunov exponent [18] as a metric for the assessment of the dynamic content of LES-calculations. Specifically, the Lyapunov exponent provides a measure of chaos in turbulent flows, and represents the rate of separation, and its reciprocal is closely related to the predictability horizon of a chaotic solution. In particular, the Lyapunov exponent, λ, is amenable to a simple physical interpretation:If a system is chaotic, given an infinitesimal initial perturbation to the solution, two trajectories of the system separate in time exponentially until nonlinear saturation. The average exponential separation is the Lyapunov exponent. A solution is typically regarded as being chaotic if there exists at least one positive Lyapunov exponent. The Lyapunov exponent is (i) a robust indicator of chaos, (ii)...

Towards the Suitability of Information Entropy as an LES Quality Indicator

“…The Lyapunov exponent is a dynamical measure, which can be used to characterize the dynamical processes in a turbulent system and can lead to statements about the predictability time of a turbulent system. A further analysis can be found in Nastac et al (2017); Wu et al (2018).…”

“…of occurance of a certain solution or the degree of freedom of the solution-state Ruelle (1979); Eckmann and Ruelle (1985); Wu et al (2018).…”

“…While the perturbated and unperturbated solution separate until non-linear saturation in a chaotic system, the exponent quantifies the average separation of the solutions. Hence, the Lyapunovexponent solely remains on the turbulent dynamics in an LES calculation Wu et al (2018); Nastac et al (2017).…”

“…A single state is not meaningful for this set of data and should be replaced by a distribution instead. The Wasserstein metric can be seen as a measure for "dissimilarity" between the data of two different states Johnson et al (2017); Wu et al (2018).…”

Requirements Towards Predictive Simulations of Turbulent Combustion

“…To this end, the Lyapunov exponent has been introduced as a metric for assessing the chaotic dynamics of LES calculations. 33,34 The Lyapunov exponent, λ, is amenable to a simple physical interpretation: If a system is chaotic, given an infinitesimal initial perturbation to the solution, two trajectories of the system separate in time exponentially until nonlinear saturation. The average exponential separation is the Lyapunov exponent.…”