Schmidt-number influence on passive-scalar spectral exponent

Determine whether variation of the Schmidt number Sc changes the power-law decay index of the passive-scalar energy spectrum E_T(k, t) in homogeneous isotropic decaying turbulence, as computed from the dual-theory spectrum g(k, t) (equations (res1)–(res2)), or whether Sc only reshapes the spectrum without altering the exponent; quantitatively characterize the dependence of E_T(k, t) on Sc across the inertial–diffusive range.

Background

The paper develops an exact analytic framework for passive scalar mixing in decaying turbulence using a dual formulation based on Euler ensembles and loop functionals, deriving a universal spectral form g(k, t) and predicting a leading decay index close to the classical Kolmogorov–Obukhov–Corrsin law. The analysis shows that while the asymptotic decay appears universal, the detailed spectral shape depends on the Schmidt number Sc through the Mellin-transformed function f(ω) and related universal functions, suggesting possible regime changes.

In discussing empirical comparisons and DNS/experimental data, the authors note that the observed spectra vary with Sc but that a key uncertainty remains: whether Sc modifies the power-law exponent itself or only the overall spectral shape. Resolving this would validate the universality claims of the dual theory and clarify how diffusivity–viscosity ratios impact passive-scalar decay in turbulent flows.