Spectral decay of a passive scalar in chaotic mixing

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Abstract
In this paper we take a closer look at the decay phase of a passive, diffusing, scalar field undergoing steady, three-dimensional chaotic advection. The energy spectrum of the scalar is obtained by numerical simulation of the advection–diffusion equation at high Péclet number. At large times, the spectral decay is found to be exponential and self-similar. It is emphasized that the asymptotic decay-time is an important measure of mixing efficiency, alongside the time required for diffusion to first become effective. The large-wavenumber spectral form, representing the distribution of scalar energy over small scales, is analyzed. Power-law behavior is found at scales intermediate between the large ones, comparable in size with the entire flow volume, and the smallest ones, at which diffusion is effective and the spectrum falls off exponentially with increasing wavenumber. Fitting of the numerical results allows the exponent of the power-law to be estimated. It is observed to vary with the parameters of the flow, taking negative values which can be either less than or greater than −1. This implies that the dominant spectral energy at high P may be either at small, large or intermediate scales, depending on the flow. In consequence, the qualitative nature of the scalar field during the decay phase varies from flow to flow, resulting in differing behavior of the predicted decay times in the large P limit obtained by asymptotic analysis. The case in which the spectral exponent exceeds −1 is shown to produce more rapid mixing and the corresponding asymptotic expression for the decay time, independent of P and involving two spectral parameters, is suggested as a quantitative means for optimizing the flow.

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