Mixing-scale dependent dispersion for transport in heterogeneous flows

Dispersion quantifies the impact of subscale velocity fluctuations on the effective movement of particles and the evolution of scalar distributions in heterogeneous flows. Which fluctuation scales are represented by dispersion, and the very meaning of dispersion, depends on the definition of the subscale, or the corresponding coarse-graining scale. We study here the dispersion effect due to velocity fluctuations that are sampled on the homogenization scale of the scalar distribution. This homogenization scale is identified with the mixing scale, the characteristic length below which the scalar is well mixed. It evolves in time as a result of local-scale dispersion and the deformation of material fluid elements in the heterogeneous flow. The fluctuation scales below the mixing scale are equally accessible to all scalar particles, and thus contribute to enhanced scalar dispersion and mixing. We focus here on transport in steady spatially heterogeneous flow fields such as porous media flows. The dispersion effect is measured by mixing-scale dependent dispersion coefficients, which are defined through a filtering operation based on the evolving mixing scale. This renders the coarse-grained velocity as a function of time, which evolves as velocity fluctuation scales are assimilated by the expanding scalar. We study the behaviour of the mixing-scale dependent dispersion coefficients for transport in a random shear flow and in heterogeneous porous media. Using a stochastic modelling framework, we derive explicit expressions for their time behaviour. The dispersion coefficients evolve as the mixing scale scans through the pertinent velocity fluctuation scales, which reflects the fundamental role of the interaction of scalar and velocity fluctuation scales in solute mixing and dispersion.