Mixing and spreading in stratified flow

G. I. Taylor [Proc. R. Soc. London, Ser. A 219, 186 (1953)] quantified enhanced solute mixing in the flow through a tube at asymptotically long times by the constant Taylor dispersion coefficient, which provides a good representation of both the asymptotic dispersion dynamics and evolution of the solute concentration. At preasymptotic times, however, the use of the constant Taylor dispersion coefficient does not facilitate a faithful representation of either the actual mixing or spreading, which are controlling factors for chemical reaction rates. Transport in spatially varying flow fields often displays non-Fickian or anomalous behavior, which is reflected by the fact that effective dispersion evolves in time. Here we study and quantify the mechanisms leading to enhanced solute mixing in spatially nonhomogeneous flow fields using local spatial moments, i.e., moments of the transport Green function. On the basis of such a local moment formulation, we define effective dispersion coefficients to characterize effective preasymptotic solute spreading and mixing. We apply these concepts to the characterization of effective transport in general stratified flows, and illustrate them for the particular case of transport in a two-dimensional channel. We study effective mixing and spreading at preasymptotic times in terms of explicit analytical expressions for the effective dispersion coefficients as well as by numerical random walk simulations. We find that the vertically averaged concentration profiles contain little information on the physical mixing and spreading processes occurring at preasymptotic times. This leads us to define an alternative average, effective concentration, whose evolution is characterized by the effective dispersion coefficient and which reflects mainly the average effective solute mixing rather than purely advective spreading of the initial solute distribution.