A Theory of Fluid Mixing

A recently developed theory of uid mixing is summarized here. It describes linear advection of a passive scalar by a random velocity eld. Asymptotic scaling laws are given, which are consistently derived from leading order results of primitive and renormalized perturbation theory in both the Eulerian and Lagrangian pictures. The theory has been veri ed by numerical simulation, for moderately large perturbation parameters, and by comparison to an exactly solvable case. The theory is suitable for the description of the passive transport of pollutants in ground water. In this context, the time independent velocity v is a random eld, of nonzero mean v, with a uctuation v = v v which is used as the perturbation expansion parameter. For the purposes of deriving analytic asymptotics, v is assumed to be a Gaussian random eld. For the simulation studies of this theory, v is derived from Darcy's law, with heterogeneous geology speci ed by a log{normal random permeability eld. The perturbation theory distinguishes three qualitatively distinct regimes. The infra{red nite case corresponds to a Fickean, or classical, di usion process. The super{renormalizable case corresponds to anomalous di usion where the dominant divergences in lowest order are of di usion type. Field and experimental data appear to fall into this case. The infra{red nonrenormalizable case results in in nite scaling exponents, or non unique scaling behavior, which depends essentially on an infra{red regularization (cuto scale). Practical applications of this theory to Kriging and conditional simulation are proposed.