Spectral analysis of mixing in chaotic flows via the mapping matrix formalism: Inclusion of molecular diffusion and quantitative eigenvalue estimate in the purely convective limit

Abstract

This paper extends the mapping matrix formalism to include the effects of molecular diffusion in the analysis of mixing processes in chaotic flows. The approach followed is Lagrangian, by considering the stochastic formulation of advection-diffusion processes via the Langevin equation for passive fluid particle motion. In addition, the inclusion of diffusional effects in the mapping matrix formalism permits to frame the spectral properties of mapping matrices in the purely convective limit in a quantitative way. Specifically, the effects of coarse graining can be quantified by means of an effective Péclet number that scales as the second power of the linear lattice size. This simple result is sufficient to predict the scaling exponents characterizing the behavior of the eigenvalue spectrum of the advection-diffusion operator in chaotic flows as a function of the Péclet number, exclusively from purely kinematic data, by varying the grid resolution. Simple but representative model systems and realistic physically realizable flows are considered under a wealth of different kinematic conditions-from the presence of large quasi-periodic islands intertwined by chaotic regions, to almost globally chaotic conditions, to flows possessing "sticky islands"-providing a fairly comprehensive characterization of the different numerical phenomenologies that may occur in the quantitative analysis of mapping matricesultimately of chaotic mixing processes. © 2012 American Institute of Physics

Analysis of Advective–Diffusive Transport in Complex Mixing Devices by the Diffusive Mapping Method

“…Numerical diffusion—and thus —can be estimated in several ways . Spectral analysis performed in terms of dominant decay exponents , where are dominant eigenvalues of the mapping matrix, shows that exponents computed from the diffusive mapping matrix reliably approximate exponents of the associated continuous advection‐diffusion operator . This is demonstrated in Figure for typical situations in the mixers under investigation here.…”

“…The diffusive mapping method serves as a reliable coarse‐grained approximation of the advection–diffusion operator on a finite grid . The method is applicable to both time‐periodic and spatially‐periodic flow systems and essentially leans on the accurate tracking of a large set of passive fluid particles.…”

“…Gorodetskyi et al have shown that the coarse‐graining of the mapping method acts as a superimposed additional diffusive contribution to the mixing process (numerical diffusion). The effect of numerical diffusion admits quantification by the effective Péclet number associated with the purely advective mapping matrix . Thus, numerical diffusion can be exploited for simulation of advective–diffusive fluid mixing at Péclet numbers .…”

“…The conventional mapping method is restricted to purely advective transport. However, a recent extension of the method enables incorporation of diffusion as well . This expands the area of application tremendously.…”

Designing a Finite-Time Mixer: Optimizing Stirring for Two-Dimensional Maps

“…This can be estimated-for a fixed map f -by a one step iteration of a large number of trial points in each cell. Such a discretization introduces an effective diffusion that will decrease the variance (12) [KGA + 01], and explicit diffusion can also be included in the process [GGA12]. However, this method is not practical for our purposes, since the stochastic matrix would need to be recomputed for each new map in the sequence (1).…”