A number of authors [94, 81, 22, 20, 59] have addressed the problem of passive scalar diffusion under simple flow conditions, and non-trivial time scales have been identified and explained in different cases. The present study we believe contributes a more complete global understanding of the various scalings that such problems can exhibit. In particular, with the formulation of the analysis as an eigenvalue problem we have identified and calculated the explicit long-lived slow modes, and further sorted these modes into two different categories. The first is connected to the Taylor regime, governed by the homogenized evolution equation. The second category is connected to the intermediate- and short-time anomalous evolution. While first computed for idealized periodic flows, we have also shown how such modes provide insight for more physically relevant shears, such as the example of the Poiseuille channel flow. Explicit analysis of a sawtooth shear flow provides the detailed structure and decay properties of the tracer’s boundary layer near flat walls, while in the interior shear-free regions (near locations where u′(y) = 0) more general WKBJ asymptotic analysis provides the
longest lived anomalous modes, which persists well beyond the wall boundary layer modes in the limit ǫ→0.
The analysis characterizes different stages of evolution, each one carrying the sig- nature of a different spectral band. From the spectral point of view of the advection- diffusion problem, the Taylor regime should be regarded as the limiting state in which any component of the spectrum has decayed and become negligible with respect to the
n = 0 modes in the k ≪ Pe−1 range. The modes inside the range k ≫ Pe−1 charac- terize the structure of the solution in the super-Gaussian anomalous diffusive regime described, for instance, in the work by Latini & Bernoff. Considering the point source distributions discussed by these authors, the cross-section-averaged distribution is ini- tialized from a flat Fourier spectrum, which necessarily excites all three classes of modes
and exhibits three regimes along the evolution. We observe how a simulation in aLx×Ly periodic domain would require a fundamental x wavenumber kx0 = 2π/Lx ≪ 1/Pe in order to observe the Taylor regime. A smaller domain, with lower resolution in the wavenumber domain, leads to a cut-off of the Taylor modes, limiting the possible ob- servable regimes up to the “anomalous diffusion” stage. By adjusting the initial relative energy in the bands we demonstrated how the WKBJ may in principle be extended well beyond the classical cross-stream-diffusion timescale r2/D (or τ
D in non-dimensional units). Additionally, compared to previous studies, the present investigation provides deeper insight into the geometrical spatial structures arising during time-evolution.
It is interesting to consider the implications of our analysis for the case of several superimposed passive scalars with different diffusion coefficients. For example, one can consider a setup consisting of two different chemical species that are injected in the same point within a shear flow. If the molecular diffusivities are different, our analysis indicates that the velocity field acts as a separator for the two scalars, owing to their different interplay with advection. If the scalars are reactive, one could in turn expect the separation to affect reaction, possibly suppressing it, when the time scales of advection-diffusion are comparable to the time scale of reaction. A detailed investigation in this direction is interesting and will be considered for further studies.
Future studies will also include the extension of the concepts presented in this work to the more realistic setups of flows in both two and three dimensions, with open and closed streamlines and physical boundary conditions, where similar phenomena including long-lived modes have been observed. In particular, the axially symmetric geometry naturally merits study for its relevance to pipe flows. Further extensions of the methods presented here should also be directed to addressing time-dependent flows possessing multiple scales and even randomness.
Part III
NUMERICAL SIMULATIONS OF DENSE-CORE
Chapter 9
DENSE-CORE VORTEX RINGS IN A SHARPLY STRATIFIED ENVIRONMENT
In this last part of my dissertation it is reported my contribution to the project “Vortex rings dynamics in presence of stratification” currently being pursued at the UNC Fluid Lab. Aim of this project is to isolate essential elements of mixing, trap- ping, and escape through stratified fluid by focusing on the specific case of dense-core vortex rings settling in sharply stratified miscible ambient fluids for near two-layer con- figurations. The core of the vortex rings is made of fluid which has density higher than both the top and bottom layers of the ambient fluid, and is fully miscible in both layers. This setup ensures a rich phenomenology including, in particular, a critical (bifurca- tion) phenomenon which distinguishes long-time behavior of the falling vortex ring in either being fully trapped in the ambient density layer, or continuing through the layer in its downward motion.
My work consists in performing 3D direct simulations supporting the lab experi- ments conducted by R. Camassa, S. Khatri, R. McLaughlin and K. Mertens, assisted by the undergraduate students D. Nenon and C. Smith. The numerical package VARDEN [6] is used for the simulations. As will be illustrated next, this task presents serious challenges, not limited to the considerable computational size of the problem. In fact, the experimental set up is not reproducible in its entirety, and several efforts go into
surface and the following formation process, see discussion in§9.4.2) and yet is able to capture the relevant dynamics. Several significant results have been obtained, which will be presented in this report. However, the study is still open to improvement under several technical aspects and can be completed by collecting further data.