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 K\'arm\'an--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.