Chaotic and turbulent mixing of passive scalar

Spatio-temporal deterministic chaos at small Taylor-Reynolds numbers $R{e}_{\lambda }\lesssim 40$ 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 $8\le R{e}_{\lambda }<700$ and of a reacting turbulent mixing layer. It is shown that the deterministic chaos in the passive scalar fluctuations at the small $R{e}_{\lambda }$ is characterized by exponential spatial (wavenumber) spectrum: $E(k) \propto \exp-(k/k_c)$,whereasthedistributedchaosatturbulent$Re_{\lambda}$ 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 $k_c\mathrm{/}{k}_{bat}$ and $k_{\beta }\mathrm{/}{k}_{bat}$ exhibit universality.