Footprints of Lagrangian flow structures in Eulerian concentration distributions in periodic mixing flows

Transport of passive tracers may be described through the spatio-temporal evolution of Eulerian concentration distributions or via the geometrical composition of the Lagrangian flow structure. The present study seeks to deepen insight into the connection between the Eulerian and Lagrangian perspectives by investigating the role of Lagrangian coherent structures (LCSs) in the Eulerian concentration distributions in time-periodic and spatially-periodic mixing flows. Representation of the Eulerian transport by the mapping method, describing realistic transport problems by distribution matrices, admits a generic analysis based on matrix and graph theory. This reveals that LCSs–and the transport barriers that separate them–leave a distinct “footprint” in the eigenmode spectrum of the distribution matrix and, by proxy, of the underlying Eulerian transport operator. Transport barriers impart a block-diagonal structure upon the mapping matrix, where each block matrix A corresponds with a given LCS. Its kind is reflected in the spectrum of A; higher-order periodicity yields a distinct permutation within A. The composition of the distribution matrix versus the Lagrangian flow structure thus predicted is demonstrated by way of examples. These findings increase fundamental understanding of transport phenomena and have great practical potential for e.g. flow and mixing control.