Homogenization Theory for a Replenishing Passive Scalar Field

Homogenization theory provides a rigorous framework for calculating the effective diffusivity of a decaying passive scalar field in a turbulent or complex flow. We extend this framework to the case where the passive scalar fluctuations are continuously replenished by a source (and/or sink). The basic structure of the homogenized equations carries over, but in some cases the homogenized source can involve a non-trivial coupling of the velocity field and the source. We derive expressions for the homogenized source term for various multiscale source structures and interpret them physically.