Dissipation-scale fluctuations and mixing transition in turbulent flows

Abstract

A small separation between reactants, not exceeding 10−8 − 10−7 cm, is the necessary condition for various chemical reactions. It is shown that random advection and stretching by turbulence leads to the formation of scalar-enriched sheets of strongly fluctuating thickness ηc. The molecular-level mixing is achieved by diffusion across these sheets (interfaces) separating the reactigants. Since the diffusion time scale is $\tau_{d}\,{\propto}\,\eta_{c}^{2}$, knowledge of the probability density Q(ηc, Re) is crucial for evaluation of mixing times and chemical reaction rates. According to Kolmogorov–Batchelor phenomenology, the stretching time τeddy ≈ L/urms = O(1) is independent of large-scale Reynolds number Re = urmsL/ν and the diffusion time $\tau_{d}\,{\approx}\,\tau_{\it eddy}/\sqrt{{\it Re}}\,{\ll}\, \tau_{\it eddy}$ is very small. Therefore, in previous studies, molecular diffusion was frequently neglected as being too fast to contribute substantially to the reaction rates. In this paper, taking into account strong intermittent fluctuations of the scalar dissipation scales, this conclusion is re-examined. We derive the probability density Q(ηc, Re, Sc), calculate the mean scalar dissipation scale and predict transition in the reaction rate behaviour from ${\cal R}\,{\propto}\,\sqrt{Re}$ ($Re\,{\leq}\, 10^{3}-10^{4})$ to the high-Re asymptotics ${\cal R}\,{\propto}\, {\it Re}^{0}$. These conclusions are compared with known experimental and numerical data.