Eigenmode analysis of advective-diffusive transport by the compact mapping method

The present study concerns an efficient spectral analysis of advective-diffusive transport in periodic flows by the way of a compact version of the diffusive mapping method. Key to the compact approach is the representation of the scalar evolution by only a small subset of the eigenmodes of the mapping matrix, and capturing the relevant features of the transient towards the homogeneous state. This has been demonstrated for purely advective transport in an earlier study by Gorodetskyi et al. (2012). Here this ansatz is extended to advective-diffusive transport and more complex 3D flow fields, motivated primarily by the importance of molecular diffusion in many mixing processes. The study exposed an even greater potential for such transport problems due to the progressive widening of the spectral gaps in the eigenvalue spectrum of the mapping matrix with increasing diffusion. This facilitates substantially larger reductions of the eigenmode basis compared to the purely advective limit for a given approximation tolerance. The compact diffusive mapping method is demonstrated for a representative three-dimensional prototype micro-mixer. This revealed a reliable prediction of (transient) scalar evolutions and mixing patterns with reductions of the eigenmode basis by up to a factor 2000. The accurate estimation of the truncation error from the eigenvalue spectrum enables the systematic determination of the spectral cut-off for a desired degree of approximation. The validity and universality of the presumed correlation between spectral cut-off and truncation error have been established. This has the important practical consequence that the cut-off can a priori be chosen such that the truncation error remains within a preset tolerance. This offers a way to systematically (and reliably) employ the compact mapping method for an in-depth analysis of advective-diffusive transport.