Toward a phenomenological approach to the clustering of heavy particles in turbulent flows

A simple model accounting for the ejection of heavy particles from the vortical structures of a turbulent flow is introduced. This model involves a space and time discretization of the dynamics and depends on only two parameters: the fraction of space-time occupied by rotating structures of the carrier flow and the rate at which particles are ejected from them. The latter can be heuristically related to the response time of the particles and hence measure their inertia. It is shown that such a model reproduces qualitatively most aspects of the spatial distribution of heavy particles transported by realistic flows. In particular the probability density function of the mass $m$ in a cell displays an power-law behavior at small values and decreases faster than exponentially at large values. The dependence of the exponent of the first tail upon the parameters of the dynamics is explicitly derived for the model. The right tail is shown to decrease as $\exp (-C m \log m)$. Finally, the distribution of mass averaged over several cells is shown to obey rescaling properties as a function of the coarse-grain size and of the ejection rate of the particles. Contrarily to what has been observed in direct numerical simulations of turbulent flows (Bec et al., http://arxiv.org/nlin.CD/0608045), such rescaling properties are only due in the model to the mass dynamics of the particles and do not involve any scaling properties in the spatial structure of the carrier flow.