The viscous-convective subrange in nonstationary turbulence

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Abstract
The similarity form of the scalar-variance spectrum at high Schmidt numbers is investigated for nonstationary turbulence. Theoretical arguments show that Batchelor scaling may apply only at high Reynolds numbers. At low Reynolds numbers, Batchlor scaling is not possible unless the turbulence is stationary or the enstrophy decays asymptotically as t−2. When this latter condition is satisfied, it is shown from an analysis using both the Batchelor and Kraichnan models for the scalar-variance transfer spectrum that the k−1 power law in the viscous-convective subrange is modified. Results of direct numerical simulations of high Schmidt number passive scalar transport in stationary and decaying two-dimensional turbulence are compared to the theoretical analysis. For stationary turbulence, Batchelor scaling is shown to collapse the spectra at different Schmidt numbers and a k−1 viscous-convective subrange is observed. The Kraichnan model is shown to accurately predict the simulation spectrum. For nonstationary turbulence decaying at constant Reynolds number for which the enstrophy decays as t−2, scalar fields for different Schmidt numbers are simulated in situations with and without a uniform mean scalar gradient. The Kraichnan model is again shown to predict the spectra in these cases with different anomalous exponents in the viscous-convective subrange.