Euler-Lagrange point-particle simulation has emerged as a premier methodology for studying dispersed particle-laden flows. This method’s popularity stems from its ability to resolve fine-scale fluid structures while also tracking individual particles at reduced cost using an appropriate particle acceleration model. However, the point-particle model has known convergence issues in that refinement of the fluid grid can lead to changes in the predicted statistics. The reasons for non-convergence are two-fold: the point-particle two-way coupling force in the Navier-Stokes equations requires a numerical regularization and without careful implementation, yields a singular force on the fluid with grid refinement. The second factor that yields grid dependent statistics is that the point-particle force model in general depends on the undisturbed fluid velocity. When the undisturbed fluid velocity is not robustly modelled in a grid insensitive way, the calculated force for both particles and fluid will be grid dependent, contaminating their respective statistics. While the first issue regarding regularizing the point-particle source term has received attention in the literature, the consequences of robustly modelling the undisturbed velocity in the context of grid refinement of turbulence has received little attention. In this work, we will consider decaying homogeneous isotropic turbulence laden with particles at different Stokes numbers. For a given Stokes number, we systematically refine the grid and demonstrate that explicitly modelling the undisturbed fluid velocity yields relative grid insensitivity for the energy of the particle and fluid phases, as well as acceleration of the particles. We also demonstrate that an appropriately defined dissipation rate is also grid insensitive when an undisturbed fluid velocity correction is used. In contrast, when the undisturbed fluid velocity is modelled using the conventional approach of interpolating the local fluid velocity to the particle location, we show this procedure yields divergent statistics with grid refinement. In particular, we show that higher-order interpolation of the fluid velocity in two-way coupled problems is worse than lower-order interpolation, in the absence of a correction procedure to estimate the undisturbed fluid velocity. We also examine velocity derivative statistics of the fluid phase and demonstrate that these statistics are not in general convergent even when the undisturbed fluid velocity is explicitly modelled. Collectively, the observations in this work are used to present a philosophy on the types of questions which are answerable with point-particle methods.

中文翻译：

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双向耦合的颗粒-湍流相互作用：数值和分辨率对流体和颗粒统计的影响

欧拉-拉格朗日点粒子模拟已经成为研究分散的载有粒子流的主要方法。该方法之所以流行，是因为它具有解析精细流体结构的能力，同时还可以使用适当的粒子加速模型以较低的成本跟踪单个粒子。但是，点粒子模型存在已知的收敛问题，因为流体网格的细化可能导致预测统计量的变化。不收敛的原因有两个：Navier-Stokes方程中的点-粒子双向耦合力需要数值正则化，而没有仔细实现，则随着网格的细化会在流体上产生奇异力。产生依赖于网格的统计数据的第二个因素是，点粒子力模型通常取决于不受干扰的流体速度。如果未以网格不敏感的方式对未扰动的流体速度进行鲁棒性建模，则粒子和流体的计算力将取决于网格，从而污染它们各自的统计数据。尽管关于点粒子源项正则化的第一个问题在文献中已受到关注，但在湍流网格细化的情况下稳健建模不受干扰速度的结果却很少受到关注。在这项工作中，我们将考虑充满不同斯托克斯数的均质各向同性湍流的衰减。对于给定的斯托克斯数，我们系统地优化了网格，并证明了对不受干扰的流体速度进行显式建模会产生相对于网格对粒子和流体相能量以及粒子加速度的不敏感性。我们还证明，使用不受干扰的流体速度校正时，适当定义的耗散率对网格也不敏感。相反，当使用将局部流体速度插值到粒子位置的常规方法对未扰动流体速度进行建模时，我们显示此过程会产生网格精化的发散统计量。尤其是，我们表明，在缺少用于估计未扰动流体速度的校正程序的情况下，双向耦合问题中流体速度的高阶插值比低阶插值差。我们还检查了流体相的速度导数统计数据，并证明即使显式建模了未扰动的流体速度，这些统计数据通常也不收敛。集体地，在这项工作中的观察被用来提出关于点粒子方法可以回答的问题类型的哲学。