School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA.
Department of Mechanical and Aerospace Engineering, Department of Physics, and Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA.
Phys Rev E. 2019 Feb;99(2-1):023102. doi: 10.1103/PhysRevE.99.023102.
Small-scale characteristics of turbulence such as velocity gradients and vorticity fluctuate rapidly in magnitude and oscillate in sign. Much work exists on the characterization of magnitude variations, but far less on sign oscillations. While in homogeneous turbulence averages performed on large scales tend to zero because of the oscillatory character, those performed on increasingly smaller scales will vary with the averaging scale in some characteristic way. This characteristic variation at high Reynolds numbers is captured by the so-called cancellation exponent, which measures how local averages tend to cancel out as the averaging scale increases, in space or time. Past experimental work suggests that the exponents in turbulence depend on whether one considers quantities in full three-dimensional (3D) space or uses their one- or two-dimensional cuts. We compute cancellation exponents of vorticity and longitudinal as well as transverse velocity gradients in isotropic turbulence at Taylor-scale Reynolds numbers up to 1300 on 8192^{3} grids. The 2D cuts yield the same exponents as those for full 3D, while the 1D cuts yield smaller numbers, suggesting that the results in higher dimensions are more reliable. We make the case that the presence of vortical filaments in isotropic turbulence leads to this conclusion. This effect is particularly conspicuous in magnetohydrodynamic turbulence, where an increased degree of spatial coherence develops along the direction of an imposed magnetic field.
小尺度湍流的特征，如速度梯度和涡度的大小迅速波动，并且符号也在不断振荡。关于大小变化的特征已经有很多研究，但关于符号振荡的研究则少得多。在均匀湍流中，由于振荡特性，大尺度上的平均趋于零，但随着平均尺度的减小，越来越小的尺度上的平均会以某种特征方式变化。在高雷诺数下，这种特征变化由所谓的抵消指数来捕捉，它衡量的是随着平均尺度的增加，局部平均如何趋于抵消，无论是在空间还是时间上。过去的实验工作表明，湍流中的指数取决于人们是考虑全三维（3D）空间中的量，还是使用它们的一维或二维切片。我们在泰勒尺度雷诺数高达 1300 的各向同性湍流中计算了涡度和纵向以及横向速度梯度的抵消指数，使用的网格尺寸为 8192^{3}。二维切片得到的指数与全三维的指数相同，而一维切片得到的指数较小，这表明更高维的结果更可靠。我们认为各向同性湍流中涡丝的存在导致了这一结论。在磁流体力学湍流中，这种效应尤为明显，其中沿外加磁场方向会出现更大程度的空间相干性。
