Analysis of passive scalar advection in parallel shear flows: Sorting of modes at intermediate time scales

Abstract

The time evolution of a passive scalar advected by parallel shear flows is studied for a class of rapidly varying initial data. Such situations are of practical importance in a wide range of applications from microfluidics to geophysics. In these contexts, it is well-known that the long-time evolution of the tracer concentration is governed by Taylor’s asymptotic theory of dispersion. In contrast, we focus here on the evolution of the tracer at intermediate time scales. We show how intermediate regimes can be identified before Taylor’s, and in particular, how the Taylor regime can be delayed indefinitely by properly manufactured initial data. A complete characterization of the sorting of these time scales and their associated spatial structures is presented. These analytical predictions are compared with highly resolved numerical simulations. Specifically, this comparison is carried out for the case of periodic variations in the streamwise direction on the short scale with envelope modulations on the long scales, and show how this structure can lead to “anomalously” diffusive transients in the evolution of the scalar onto the ultimate regime governed by Taylor dispersion. Mathematically, the occurrence of these transients can be viewed as a competition in the asymptotic dominance between large Péclet (Pe) numbers and the long/short scale aspect ratios (LVel/LTracer≡k), two independent nondimensional parameters of the problem. We provide analytical predictions of the associated time scales by a modal analysis of the eigenvalue problem arising in the separation of variables of the governing advection-diffusion equation. The anomalous time scale in the asymptotic limit of large k Pe is derived for the short scale periodic structure of the scalar’s initial data, for both exactly solvable cases and in general with WKBJ analysis. In particular, the exactly solvable sawtooth flow is especially important in that it provides a short cut to the exact solution to the eigenvalue problem for the physically relevant vanishing Neumann boundary conditions in linear-shear channel flow. We show that the life of the corresponding modes at large Pe for this case is shorter than the ones arising from shear free zones in the fluid’s interior. A WKBJ study of the latter modes provides a longer intermediate time evolution. This part of the analysis is technical, as the corresponding spectrum is dominated by asymptotically coalescing turning points in the limit of large Pe numbers. When large scale initial data components are present, the transient regime of the WKBJ (anomalous) modes evolves into one governed by Taylor dispersion. This is studied by a regular perturbation expansion of the spectrum in the small wavenumber regimes.