The formation of caustics by inertial particles is distinctive of dispersed flows. Their pressureless nature allows crossing trajectories resulting in singularities that cannot be captured accurately by standard Lagrangian approaches due to their fine spatial scale. A promising method for the investigation of caustics is the Osiptsov method or fully Lagrangian approach (FLA). The FLA has the advantage of identifying caustics, but its applicability is hindered by the occurrence of singularities. We present an original robust framework based on the FLA that provides an explicit expression of the dispersed phase structure that does not degenerate in the vicinity of caustics, using a single representative particle. The FLA is extended to account for the Hessian of the dispersed continuum (DC). It demonstrates the integrability of the FLA number density and allows for the calculation of the number density on a given length scale, retaining the functionality of the FLA. Number density models based on the second-order representation of the DC and on the one-dimensional structure of the particle distribution, that account for the anisotropy of the DC on caustics, are derived and applied for analytical flows. The number density is linked to a finite length scale, needed for the introduction of the FLA to spatially filtered flow fields. Finally, the method is used for the calculation of the interparticle separation on caustics. The identification of the structure of caustics presented in this work paves the way to a robust understanding of the mechanisms of particle accumulation.

中文翻译：

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用于研究分散流中苛性层二阶结构的模型

由惯性粒子形成的焦散与分散流不同。它们的无压力性质允许交叉轨迹，导致由于其精细的空间尺度而无法通过标准拉格朗日方法准确捕获的奇点。Osiptsov 方法或完全拉格朗日方法 (FLA) 是研究焦散的一种很有前景的方法。FLA 具有识别焦散的优势，但它的适用性受到奇点的出现的阻碍。我们提出了一个基于 FLA 的原始稳健框架，该框架使用单个代表性粒子提供了在焦散附近不会退化的分散相结构的明确表达。FLA 被扩展以解释分散连续体 (DC) 的 Hessian。它展示了 FLA 数密度的可积分性，并允许在给定长度尺度上计算数密度，同时保留 FLA 的功能。导出基于 DC 的二阶表示和粒子分布的一维结构的数密度模型，这些模型解释了 DC 在焦散上的各向异性，并将其应用于分析流。数密度与有限长度尺度相关联，这是将 FLA 引入空间过滤流场所需的。最后，该方法用于计算焦散粒子间的分离。这项工作中提出的焦散结构的识别为深入了解粒子积累机制铺平了道路。