Department of Mechanical Engineering and Center for Environmental and Applied Fluid Mechanics, The Johns Hopkins University, 3400 N. Charles Street, Baltimore, Maryland 21218, USA.
Phys Rev E. 2016 Mar;93(3):033118. doi: 10.1103/PhysRevE.93.033118. Epub 2016 Mar 21.
A key feature of three-dimensional fluid turbulence is the stretching and realignment of vorticity by the action of the strain rate. It is shown in this paper, using the cumulant-generating function, that the cumulative vorticity stretching along a Lagrangian path in isotropic turbulence obeys a large deviation principle. As a result, the relevant statistics can be described by the vorticity stretching Cramér function. This function is computed from a direct numerical simulation data set at a Taylor-scale Reynolds number of Re(λ)=433 and compared to those of the finite-time Lyapunov exponents (FTLE) for material deformation. As expected, the mean cumulative vorticity stretching is slightly less than that of the most-stretched material line (largest FTLE), due to the vorticity's preferential alignment with the second-largest eigenvalue of strain rate and the material line's preferential alignment with the largest eigenvalue. However, the vorticity stretching tends to be significantly larger than the second-largest FTLE, and the Cramér functions reveal that the statistics of vorticity stretching fluctuations are more similar to those of the largest FTLE. In an attempt to relate the vorticity stretching statistics to the vorticity magnitude probability density function in statistically stationary conditions, a model Kramers-Moyal equation is constructed using the statistics encoded in the Cramér function. The model predicts a stretched-exponential tail for the vorticity magnitude probability density function, with good agreement for the exponent but significant difference (35%) in the prefactor.
三维流体湍流的一个重要特征是涡度通过应变率的作用而被拉伸和重新排列。本文利用累积生成函数表明，各向同性湍流中拉格朗日路径上的累积涡度拉伸服从大偏差原理。因此，相关的统计数据可以用涡度拉伸的克拉默函数来描述。该函数是根据瑞利数为 Re(λ)=433 的直接数值模拟数据集计算得到的，并与物质变形的有限时间李雅普诺夫指数（FTLE）进行了比较。正如预期的那样，由于涡度优先与应变率的第二大特征值对齐，而物质线优先与最大特征值对齐，因此平均累积涡度拉伸略小于最拉伸的物质线（最大 FTLE）。然而，涡度拉伸往往明显大于第二大 FTLE，并且克拉默函数揭示了涡度拉伸波动的统计数据与最大 FTLE 的统计数据更为相似。为了将涡度拉伸统计数据与统计稳态条件下的涡度大小概率密度函数联系起来，使用克拉默函数中编码的统计数据构建了一个 Kramers-Moyal 模型方程。该模型预测了涡度大小概率密度函数的拉伸指数分布尾部，其指数与模型预测值吻合较好，但在预因子方面存在显著差异（35%）。
