Electronic supplementary material on the total length of vortex axes plotted in figures 2, 4(b). from Self-similar hierarchy of coherent tubular vortices in turbulence

Energy transfers from larger to smaller scales in turbulence. This energy cascade is a process of the creation of smaller-scale coherent vortices by larger ones. In our recent study (Yoneda, Goto and Tsuruhashi 2021), we reformulated the energy cascade in terms of this stretching process and derived the -5/3 law of the energy spectrum under physically reasonable assumptions. In the present study, we provide a quantitative verification of these assumptions by using direct numerical simulations. We decompose developed turbulence in a periodic cube into scales by using the band-pass filter and identify the axes of coherent tubular vortices by the low-pressure method. Even when the turbulent kinetic energy and its dissipation rate temporally fluctuate about their temporal means, the total length of the vortices at each scale varies little with time. This result is consistent with our assumption of the temporal stationarity on the vorticity decomposition. The present numerical analysis also shows that the hierarchy of vortex axes is self-similar in a wide range of scales, i.e. in the inertial range and a lower part of the dissipation range and that the volume fraction occupied by the tubular vortices at each scale is independent of the scale.This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.

湍流（turbulence）中能量从大尺度向小尺度传递。这一能量级串（energy cascade）是大尺度涡旋生成小尺度相干涡旋（coherent vortices）的过程。在我们近期的研究（Yoneda、Goto与Tsuruhashi，2021）中，我们基于这一拉伸过程重新构建了能量级串的理论框架，并在合理的物理假设下推导出了能谱的-5/3幂律。在本研究中，我们通过直接数值模拟（direct numerical simulations）对上述假设开展定量验证。我们利用带通滤波器（band-pass filter）将周期立方体中的充分发展湍流按尺度分解，并通过低压法（low-pressure method）识别管状相干涡旋的轴线。即便湍流动能（turbulent kinetic energy）及其耗散率（dissipation rate）围绕各自的时间均值随时间波动，各尺度下涡旋的总长度随时间的变化仍极小。这一结果与我们关于涡量分解（vorticity decomposition）的时间平稳性假设相一致。本次数值分析还表明，涡旋轴线的层级结构在广泛的尺度范围内具有自相似性——覆盖惯性区（inertial range）与耗散区（dissipation range）的下半部分——且各尺度下管状涡旋所占的体积分数（volume fraction）与尺度无关。本文属于主题刊《物理流体动力学中的数学问题（第二部分）》（Mathematical problems in physical fluid dynamics (part 2)）的一部分。