Dynamics of probability density functions for decaying passive scalars in periodic velocity fields

The probability density function (PDF) for a decaying passive scalar advected by a deterministic, periodic, incompressible fluid flow is numerically studied using a variety of random and coherent initial scalar fields. We establish the dynamic emergence at large Péclet numbers of a broad-tailed PDF for the scalar initialized with a Gaussian random measure, and further explore a rich parameter space involving scales of the initial scalar field and the geometry of the flow. We document that the dynamic transition of the PDF to a broad-tailed distribution is similar for shear flows and time-varying nonsheared flows with positive Lyapunov exponent, thereby showing that chaos in the particle trajectories is not essential to observe intermittent scalar signals. The role of the initial scalar field is carefully explored. The long-time PDF is sensitive to the scale of the initial data. For shear flows we show that heavy-tailed PDFs appear only when the initial field has sufficiently small-scale variation. We also connect geometric features of the scalar field with the shape of the PDFs. We document that the PDF is constructed by a subtle balance between spatial regions of strong and weak shear in conjunction with the presence of small-scale scalar variation within the weak shear regions. For cellular flows we document a lack of self-similarity in the PDFs when periodic time dependence is present, in contrast to the self-similar decay for time independent flow. Finally, we analyze the behavior of the PDFs for coherent initial fields and the parametric dependence of the variance decay rate on the Péclet number and the initial wavenumber of the scalar field

Dynamics of probability density functions for decaying passive scalars in periodic velocity fields

The probability density function (PDF) for a decaying passive scalar advected by a deterministic, periodic, incompressible fluid flow is numerically studied using a variety of random and coherent initial scalar fields. We establish the dynamic emergence at large Péclet num-bers of a broad-tailed PDF for the scalar initialized with a Gaussian random measure, and further explore a rich parameter space involv-ing scales of the initial scalar field and the geometry of the flow. We document that the dynamic transition of the PDF to a broad tailed distribution is similar for shear flows and time-varying non-sheared flows with positive Lyapunov exponent, thereby showing that chaos in the particle trajectories is not essential to observe intermittent scalar signals. The role of the initial scalar field is carefully explored. The long time PDF is sensitive to the scale of the initial data. For shear flows we show that heavy-tailed PDFs appear only when the initial field has sufficiently small-scale variation. We also connect geometric features of the scalar field with the shape of the PDFs. We docu-ment that the PDF is constructed by a subtle balance between spatial regions of strong and weak shear in conjunction with the presence of 1 To appear: Physics of Fluids small-scale scalar variation within the weak shear regions. For cellular flows we document a lack of self-similarity in the PDFs when periodic time dependence is present, in contrast to the self-similar decay for time independent flow. Finally we analyze the behavior of the PDFs for coherent initial fields and the parametric dependence of the vari-ance decay rate on the Péclet number and the initial wavenumber of the scalar field. 2