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(d), 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 Peclet number (Pe >> 1) yield four distinct regimes for the scaling behaviour of kd, one of which corresponds to the Batchelor regime. The transition between these regimes is controlled by the value of Pe and the ratio rho = l(u)/l(s), where l(e) and l(s) 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.