Numerical Validation of Lyapunov-Liouville Theory and Non-Diffusive Closures in Decaying Isotropic Fluid and Scalar Turbulence

Abstract

This work validates the Lyapunov--Liouville framework and its non-diffusive closures in decaying homogeneous isotropic turbulence (HIT) by numerically integrating the closed von Kármán--Howarth and Corrsin equations via an adaptive solver. Three initial states (Saffman--Birkhoff, Loitsiansky, Gaussian) are analyzed across Prandtl numbers from $Pr=10^{-3}$ to $1000$. The model reproduces the distinct decay paths, with Saffman--Birkhoff yielding velocity and thermal exponents $m\simeq n\simeq -1.25$, while Loitsiansky condition accelerates mechanical decay ($m\simeq -1.51$) and increases thermal persistence ($n\simeq -0.89$). The Gaussian profile induces rapid decay ($m\simeq -2.7$), approaching a critical threshold at $t\simeq 33$ initial Lyapunov times. At $Pr=1000$, the thermal microscale drops below the Kolmogorov scale, with the Batchelor constant settling around $C_B\simeq 3.5$. Calculated Kolmogorov ($C_K\simeq 1.72-1.75$) and Obukhov--Corrsin ($C_{OC}\simeq 1.8$) constants align with benchmarks. Finally, velocity and temperature increment PDFs successfully capture multi-scale intermittency and non-Gaussian statistics, matching DNS and experimental data.

v2

References in corpus (4)

• Liouville Spectral Gap and Bifurcation Driven Lagrangian Eulerian Decoupling with Nondiffusive Turbulence Closures
• From Bifurcations to State-Variable Statistics in Isotropic Turbulence: Internal Structure, Intermittency, and Kolmogorov Scaling via Non-Observable Quasi-PDFs
• Quantitative Evaluation of Forward and Backward Scattering in Isotropic Turbulence via Hänggi--Klimontovich and Itô Stochastic Processes
• The Asymptotic State of Decaying Turbulence