[论文解读] Exact regularized point particle method for multi-phase flows in the two-way coupling regime

P. Gualtieri, Francesco Picano+2 more|arXiv (Cornell University)|May 27, 2014

Particle Dynamics in Fluid Flows参考文献 20被引用 10

一句话总结

本文提出了一种用于模拟双向耦合多相流的精确正则化点粒子方法，其中通过小刚性球体周围非定常Stokes流的闭式解计算粒子引起的流体扰动。该方法严格正则化点粒子带来的奇异性，通过利用局部化、时间精确的动量交换模型，实现了在低计算成本下对数万个粒子的高效、高保真模拟。

ABSTRACT

Particulate flows have been largely studied under the simplifying assumptions of one-way coupling regime where the disperse phase do not react-back on the carrier fluid. In the context of turbulent flows, many non trivial phenomena such as small scales particles clustering or preferential spatial accumulation have been explained and understood. A more complete view of multiphase flows can be gained calling into play two-way coupling effects, i.e. by accounting for the inter-phase momentum exchange between the carrier and the suspended phase, certainly relevant at increasing mass loading. In such regime, partially investigated in the past by the so-called Particle In Cell (PIC) method, much is still to be learned about the dynamics of the disperse phase and the ensuing alteration of the carrier flow. In this paper we present a new methodology rigorously designed to capture the inter-phase momentum exchange for particles smaller than the smallest hydrodynamical scale, e.g. the Kolmogorov scale in a turbulent flow. In fact, the momentum coupling mechanism exploits the unsteady Stokes flow around a small rigid sphere where the transient disturbance produced by each particle is evaluated in a closed form. The particles are described as lumped, point masses which would lead to the appearance of singularities. A rigorous regularization procedure is conceived to extract the physically relevant interactions between particles and fluid which avoids any "ah hoc" assumption. The approach is suited for high efficiency implementation on massively parallel machines since the transient disturbance produced by the particles is strongly localized in space around the actual particle position. As will be shown, hundred thousands particles can therefore be handled at an affordable computational cost as demonstrated by a preliminary application to a particle laden turbulent shear flow.

研究动机与目标

• 开发一种在数学上严格的方法，用于建模双向耦合状态下流体与粒子之间的动量交换，特别是在粒子尺寸小于最小湍流尺度（如Kolmogorov尺度）时。
• 通过点粒子周围瞬态Stokes流的闭式解，推导出物理解释一致的相互作用，消除正则化中的任意假设。
• 通过确保扰动场在空间上强局域化，实现对粒子载流湍流的大规模模拟，适用于大规模并行架构。
• 通过基于热核和正则化格林函数的时间精确半拉格朗日方法，精确捕捉每个粒子的自感应扰动流。

提出的方法

• 该方法将每个粒子建模为带有正则化力分布的质点，用高斯平滑函数替代奇异的狄拉克δ函数，以避免数学奇异性。
• 利用从热核和正则化格林函数推导出的非定常Stokes流闭式解，计算流体扰动场，确保动量交换的精确性。
• 通过三步时间积分计算自感应扰动流：施加力、扩散（利用热方程的半群性质），以及投影以满足不可压缩性。
• 扰动场以涉及粒子距离和时变扩散尺度 σ = √(2ν(εR + Δt)) 的无量纲变量表达，包含误差函数项以处理非球对称性。
• 通过限制积分域和使用修正的角限界，处理部分粒子-滤波器交叠问题，防止奇异性行为。
• 最终速度场以包含误差函数及其导数的径向和切向分量的闭式组合形式计算，确保流场无散度。

实验结果

研究问题

• RQ1 当粒子尺寸小于最小湍流尺度时，如何在双向耦合状态下精确建模小粒子与载流体之间的动量交换？
• RQ2 点粒子模型的正确正则化程序是什么，能够避免任意假设并产生物理解释一致的流体-粒子相互作用？
• RQ3 能否以闭式形式计算粒子运动引起的瞬态扰动场，以实现对大规模粒子系统的精确、高效且稳定的模拟？
• RQ4 该方法如何在大规模并行机上高效实现，同时保持对粒子载流湍流的高精度？
• RQ5 当粒子与计算滤波器边界相交时，正则化扰动场的行为如何？如何避免奇异性？

主要发现

• 该方法提供了基于热核和正则化格林函数的点粒子周围非定常Stokes流扰动的闭式解，确保了精确的动量交换。
• 自感应扰动场通过误差函数和径向分量以闭式形式计算，即使在粒子间距很小时解仍保持有限。
• 扰动场在空间上强局域化，支持高效计算，使该方法适用于包含数十万粒子的大规模模拟。
• 通过严格的正则化程序，用高斯平滑函数替代点力，消除了奇异性，避免了任意假设。
• 通过限制积分域和调整角限界，该方法在粒子-滤波器边界交叠时仍保持精度。
• 对粒子载流剪切湍流的初步模拟验证了该方法在大规模计算中的高效性和数值稳定性。