Finite Scale Lyapunov Analysis of Temperature Fluctuations in Homogeneous Isotropic Turbulence

TL;DR

This paper uses finite scale Lyapunov analysis to explain temperature fluctuations and energy cascades in homogeneous isotropic turbulence, deriving scaling laws and statistical properties consistent with existing theories and experiments.

Contribution

It introduces a Lyapunov-based approach to analyze temperature fluctuations, providing closure for the Corrsin equation and detailed statistical descriptions.

Findings

Temperature spectrum follows $\kappa^n$ scaling laws with different exponents.

PDF of temperature derivative is non-Gaussian with increasing intermittency.

Results align with Obukhov--Corrsin and Batchelor theories, supported by simulations and experiments.

Abstract

This study analyzes the temperature fluctuations in incompressible homogeneous isotropic turbulence through the finite scale Lyapunov analysis of the relative motion between two fluid particles. The analysis provides an explanation of the mechanism of the thermal energy cascade, leads to the closure of the Corrsin equation, and describes the statistics of the longitudinal temperature derivative through the Lyapunov theory of the local deformation and the thermal energy equation. The results here obtained show that, in the case of self-similarity, the temperature spectrum exhibits the scaling laws $\kappa^n$, with $n \approx{-5/3}$, ${-1}$ and ${-17/3} \div -11/3$ depending upon the flow regime. These results are in agreement with the theoretical arguments of Obukhov--Corrsin and Batchelor and with the numerical simulations and experiments known from the literature. The PDF of the longitudinal temperature derivative is found to be a non--gaussian distribution function with null skewness, whose intermittency rises with the Taylor scale P\'eclet number. This study applies also to any passive scalar which exhibits diffusivity.