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

许多天然的和人造的流体系统具有分散的颗粒相，并且这些流动中的大多数是湍流的。例如地球大气中的水流星（雨、雪或冰雹），地表水体中的沉积物颗粒，或通过工业管道系统输送的固体物质。尽管过去的研究工作很多，但由于两个因素，我们对湍流颗粒流动的理解在这一点上仍然相当有限。首先，在这种多相流系统中获得高保真的数据对现代实验和数值技术仍然是一个巨大的挑战。其次，高质量的数据集具有大量的自由度，因此分析起来非常繁琐。对于尺寸大于最小相关流动尺度的颗粒，情况甚至更糟，因为它们与流体运动完全耦合，并且无法对其动力学进行简化描述。在这里，我们建议用一种与特定流动构型相关的不变量解（即行波或周期轨道）取代载流子相中完全发展的湍流来降低系统的复杂性。我们的策略遵循湍流的动力系统方法的精神，它认为不变解包含了大多数基本信息，从而形成了湍流的骨架。在这些简单的流动中分析固体颗粒的动力学，资源消耗要少得多（从而允许广泛的参数扫描），而结果仍然可以直接与完全发展的湍流相关。我们的工作假设是，执行这个程序将使我们能够对长期悬而未决的问题，如湍流对沉降速度的影响和有限大小颗粒的聚集有新的认识。

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.