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

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.