Dissipation Scale Fluctuations and Chemical Reaction Rates in Turbulent Flows

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 formation of scalar-enriched sheets of {\it strongly fluctuating thickness} $\eta_{c}$. The molecular-level mixing is achieved by diffusion across these sheets (interfaces) separating the reactants. Since diffusion time scale is $\tau_{d}\propto \eta_{c}^{2}$, the knowledge of probability density $Q(\eta_{c},Re)$ is crucial for evaluation of chemical reaction rates. In this paper we derive the probability density $Q(\eta_{c},Re,Sc)$ and predict a transition in the reaction rate behavior from ${\cal R}\propto \sqrt{Re}$ ($Re\leq 10^{4}$) to the high-Re asymptotics ${\cal R}\propto Re^{0}$. The theory leads to an approximate universality of transitional Reynolds number $Re_{tr}\approx 10^{4}$. It is also shown that if chemical reaction involves short-lived reactants, very strong anomalous fluctuations of the length-scale $\eta_{c}$ may lead to non-negligibly small reaction rates

Probability Densities in Strong Turbulence

According to modern developments in turbulence theory, the "dissipation" scales (u.v. cut-offs) $\eta$ form a random field related to velocity increments $\delta_{\eta}u$. In this work we, using Mellin's transform combined with the Gaussain large -scale boundary condition, calculate probability densities (PDFs) of velocity increments $P(\delta_{r}u,r)$ and the PDF of the dissipation scales $Q(\eta, Re)$, where $Re$ is the large-scale Reynolds number. The resulting expressions strongly deviate from the Log-normal PDF $P_{L}(\delta_{r}u,r)$ often quoted in the literature. It is shown that the probability density of the small-scale velocity fluctuations includes information about the large (integral) scale dynamics which is responsible for deviation of $P(\delta_{r}u,r)$ from $P_{L}(\delta_{r}u,r)$. A framework for evaluation of the PDFs of various turbulence characteristics involving spatial derivatives is developed. The exact relation, free of spurious Logarithms recently discussed in Frisch et al (J. Fluid Mech. {\bf 542}, 97 (2005)), for the multifractal probability density of velocity increments, not based on the steepest descent evaluation of the integrals is obtained and the calculated function $D(h)$ is close to experimental data. A novel derivation (Polyakov, 2005), of a well-known result of the multi-fractal theory [Frisch, "Turbulence. {\it Legacy of A.N.Kolmogorov}", Cambridge University Press, 1995)), based on the concepts described in this paper, is also presented.Comment: 25 pages and 9 figure