An efficient approach for eigenmode analysis of transient distributive mixing by the mapping method

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The mapping method is an efficient tool to investigate distributive mixing induced by periodic flows. Computed only once, the mapping matrix can be applied a number of times to determine the distribution of concentration inside the flow domain. Spectral analysis of the mapping matrix reveals detailed properties of the distributive mixing as all relevant information is stored in its eigenmodes. Any vector that describes a distribution of concentration can be expanded in the complete system of linearly independent eigenvectors of the mapping matrix. The rapid decay of the contribution of each mode in the eigenmode decomposition allows for a truncation of the eigenmode expansion from the whole spectrum to only the dominant eigenmodes (characterized by a decay rate significantly lower than the duration of the mixing process). This truncated decomposition adequately represents the distribution of concentration inside the flow domain already after a low number of periods, because contributions of all non-dominant eigenmodes rapidly become insignificant. The truncation is determined independently of the initial distribution of concentration and based on the decay rates of the eigenmodes, which are inversely proportional to the corresponding eigenvalues. Only modes with eigenvalues above a certain threshold are retained. The key advantage of the proposed compact eigenmode representation of the mapping method is that it includes practically relevant transient states and not just the asymptotic one. As such the method enables an eigenmode analysis of realistic problems yet with a substantial reduction in computational effort compared to the conventional approach.
Original languageEnglish
Article number053602
Pages (from-to)053602-1/16
Number of pages16
JournalPhysics of Fluids
Issue number5
Publication statusPublished - 2012


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