Chaotic and turbulent mixing of passive scalar

Spatio-temporal deterministic chaos at small Taylor-Reynolds numbers $Re_{\lambda} \lesssim 40$ and distributed chaos at turbulent $Re_{\lambda} \gtrsim 40$ in passive scalar dynamics have been studied using results of direct numerical simulations of homogeneous incompressible flows (with and without mean gradient of the passive scalar) for $8 \leq Re_{\lambda} < 700$ and of a reacting turbulent mixing layer. It is shown that the deterministic chaos in the passive scalar fluctuations at the small $Re_{\lambda}$ is characterized by exponential spatial (wavenumber) spectrum: $E(k) \propto \exp-(k/k_c)$, whereas the distributed chaos at turbulent $Re_{\lambda}$ is characterized by stretched exponential spectrum $E(k) \propto \exp-(k/k_{\beta})^{3/4}$. The Birkhoff-Saffman invariant related to the momentum conservation and, due to the Noether theorem, to the spatial homogeneity has been used as a theoretical basis for this stretched exponential spectrum. Although the $k_c$ and $k_{\beta}$ represent the large-scale structures a relevance of the Batchelor scale $k_{bat}$ has been established as well: the normalized values $k_c/k_{bat}$ and $k_{\beta}/k_{bat}$ exhibit universality.