Bounding the scalar dissipation scale for mixing flows in the presence of sources

We investigate the mixing properties of scalars stirred by spatially smooth, divergence-free flows and maintained by a steady source-sink distribution. We focus on the spatial variation of the scalar field, described by the dissipation wavenumber, k , that we define as a function of the mean variance of the scalar and its gradient. We derive a set of upper bounds that for large Péclet number (Pe > 1) yield four distinct regimes for the scaling behaviour of k , one of which corresponds to the Batchelor regime. The transition between these regimes is controlled by the value of Pe and the ratio p = ℓ /ℓ , where ℓ and ℓ are, respectively, the characteristic length scales of the velocity and source fields. A fifth regime is revealed by homogenization theory. These regimes reflect the balance between different processes: scalar injection, molecular diffusion, stirring and bulk transport from the sources to the sinks. We verify the relevance of these bounds by numerical simulations for a two-dimensional, chaotically mixing example flow and discuss their relation to previous bounds. Finally, we note some implications for three-dimensional turbulent flows.