Perrin Vincent E, Jonker Harm J J
Delft University of Technology, Delft, The Netherlands.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Oct;92(4):043022. doi: 10.1103/PhysRevE.92.043022. Epub 2015 Oct 30.
The distribution of relative velocities between particles provides invaluable information on the rates and characteristics of particle collisions. We show that the theoretical model of Gustavsson and Mehlig [K. Gustavsson and B. Mehlig, J. Turbul. 15, 34 (2014)], within its anticipated limits of validity, can predict the joint probability density function of relative velocities and separations of identical inertial particles in isotropic turbulent flows with remarkable accuracy. We also quantify the validity range of the model. The model matches two limits (or two types) of relative motion between particles: one where pair diffusion dominates (i.e., large coherence between particle motion) and one where caustics dominate (i.e., large velocity differences between particles at small separations). By using direct numerical simulation combined with Lagrangian particle tracking, we assess the model prediction in homogeneous and isotropic turbulence. We demonstrate that, when sufficient caustics are present at a given separation and the particle response time is significantly smaller than the integral time scales of the flow, the distribution exhibits the same universal power-law form dictated by the correlation dimension as predicted by the model of Gustavsson and Mehlig. In agreement with the model, no strong dependency on the Taylor-based Reynolds number is observed.
粒子间相对速度的分布为粒子碰撞的速率和特性提供了宝贵信息。我们表明，古斯塔夫松和梅利格[K. 古斯塔夫松和B. 梅利格，《湍流杂志》15, 34 (2014)]的理论模型在其预期的有效范围内，能够以极高的精度预测各向同性湍流中相同惯性粒子相对速度和间距的联合概率密度函数。我们还对该模型的有效范围进行了量化。该模型匹配了粒子间相对运动的两种极限情况（或两种类型）：一种是对扩散占主导（即粒子运动间具有较大的相关性），另一种是焦散占主导（即小间距时粒子间具有较大的速度差异）。通过结合直接数值模拟和拉格朗日粒子追踪，我们评估了该模型在均匀各向同性湍流中的预测。我们证明，当在给定间距处存在足够的焦散且粒子响应时间远小于流动的积分时间尺度时，分布呈现出与古斯塔夫松和梅利格模型所预测相同的由关联维数决定的通用幂律形式。与模型一致，未观察到对基于泰勒的雷诺数有强烈依赖性。
