Investigating turbulent particulate flows with the aid of invariant solutions to the Navier-Stokes equations

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Many natural and man-made fluidic systems feature a disperse particulate phase, and the majority of these flows is turbulent. Examples are hydro-meteors (rain, snow or hail) in the earth's atmosphere, sediment particles in surface water bodies, or the solid matter transported through an industrial pipeline system. Despite much past research effort our understanding of turbulent particulate flows is still rather limited at this point due to two factors. First, obtaining high-fidelity data in such multi-phase flow systems still poses a formidable challenge to modern experimental and numerical techniques. Second, high-quality data-sets feature an enormous number of degrees of freedom, and they are therefore extremely tedious to analyze. The situation is even worse in the case of particles with a size larger than the smallest relevant flow scales, since they are fully coupled to the fluid motion and no simplified description for their dynamics is available. Here we propose to reduce the complexity of the systems by replacing fully developed turbulence in the carrier phase with one of the relevant invariant solutions (i.e. travelling waves or periodic orbits) pertaining to the specific flow configuration. Our strategy follows the spirit of the dynamical systems approach to turbulence, which considers that invariant solutions contain most of the essential information and thereby form the skeleton of turbulence. Analyzing the dynamics of solid particles in these simpler flows is then much less resource-intensive (thereby allowing for extensive parameter sweeps), while the results can still be directly related to fully-developed turbulence. Our working hypothesis is that carrying out this program will allow us to shed new light on long-standing open questions such as the influence of turbulence upon the settling velocity and the clustering of finite-size particles.