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

摘要

A simple model accounting for the ejection of heavy particles from thevortical structures of a turbulent flow is introduced. This model involves aspace and time discretization of the dynamics and depends on only twoparameters: the fraction of space-time occupied by rotating structures of thecarrier flow and the rate at which particles are ejected from them. The lattercan be heuristically related to the response time of the particles and hencemeasure their inertia. It is shown that such a model reproduces qualitativelymost aspects of the spatial distribution of heavy particles transported byrealistic flows. In particular the probability density function of the mass $m$in a cell displays an power-law behavior at small values and decreases fasterthan exponentially at large values. The dependence of the exponent of the firsttail 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, thedistribution of mass averaged over several cells is shown to obey rescalingproperties as a function of the coarse-grain size and of the ejection rate ofthe particles. Contrarily to what has been observed in direct numericalsimulations 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 theparticles and do not involve any scaling properties in the spatial structure ofthe carrier flow.