Multiscale mixing efficiencies for steady sources

Multiscale mixing efficiencies for passive scalar advection are defined in terms of the suppression of variance weighted at various length scales. We consider scalars maintained by temporally steady but spatially inhomogeneous sources, stirred by statistically homogeneous and isotropic incompressible flows including fully developed turbulence. The mixing efficiencies are rigorously bounded in terms of the Péclet number and specific quantitative features of the source. Scaling exponents for the bounds at high Péclet number depend on the spectrum of length scales in the source, indicating that molecular diffusion plays a more important quantitative role than that implied by classical eddy diffusion theories.

Figure 1

Upper bound for the mixing efficiency ${\mathcal{E}}_0$ as a function of Péclet number for a small source with $\mathcal{l}={10}^{-8}L$ stirred by a three-dimensional statistically homogeneous and isotropic flow [computed from Eq. (10b)]. The intermediate ${\mathrm{Pe}}^{1∕2}$ scaling for $1⪡\mathrm{Pe}⪡(L∕\mathcal{l})$ is evident.