柯尔莫哥洛夫微尺度

柯尔莫哥洛夫微尺度是湍流中最小的尺度。在柯尔莫哥洛夫尺度上，粘度占主导地位，湍流动能消散为热量。它们由^[1]定义

柯尔莫哥洛夫长度尺度	$\eta =\left({\frac {\nu ^{3}}{\varepsilon }}\right)^{1/4}$
柯尔莫哥洛夫时间尺度	$\tau _{\eta }=\left({\frac {\nu }{\varepsilon }}\right)^{1/2}$
柯尔莫哥洛夫速度标度	$u_{\eta }=\left(\nu \varepsilon \right)^{1/4}$

其中 $\varepsilon$ 是每单位质量的湍流动能的平均耗散率，和 $\nu$ 是流体的运动粘度。柯尔莫哥洛夫长度尺度的典型值，对于大气运动，其中对于千米级的大涡流，尺度范围大约从 0.1 到 10 毫米；对于较小的流，例如在实验室系统中的流， $\eta$ 可能要小得多。 ^[2]

柯尔莫哥洛夫在他1941年发表的理论中称，最小尺度的湍流是普遍的（对于每一个湍流都相似），并且它们只依赖于 $\varepsilon$ 和 $\nu$ 。柯尔莫哥洛夫微尺度的定义可以通过这种理论和量纲分析得到。由于运动粘度的维度是长度² /时间，单位质量的能量耗散率的维度是长度² /时间³ ，所以唯一具有时间维度的组合是 $\tau _{\eta }=(\nu /\varepsilon )^{1/2}$ 这是柯尔莫罗戈夫时间尺度。同样，柯尔莫哥洛夫长度尺度是唯一的组合 $\varepsilon$ 和 $\nu$ 具有长度尺寸。

或者，可以从均方应变率张量的倒数中获得柯尔莫哥洛夫时间尺度的定义， $\tau _{\eta }=(2\langle E_{ij}E_{ij}\rangle )^{-1/2}$ 这也给出了 $\tau _{\eta }=(\nu /\varepsilon )^{1/2}$ 使用单位质量的能量耗散率的定义 $\varepsilon =2\nu \langle E_{ij}E_{ij}\rangle$ .那么柯尔莫哥洛夫长度尺度可以得到雷诺数等于1的尺度， ${\mathit {Re}}=UL/\nu =(\eta /\tau _{\eta })\eta /\nu =1$ .

柯尔莫哥洛夫 1941 理论是一种平均场理论，因为它假设相关的动态参数是平均能量耗散率。在流体湍流中，能量耗散率随空间和时间波动，因此可以将微尺度视为也在空间和时间上变化的量。但是，标准做法是使用平均场值，因为它们代表给定流量中最小尺度的典型值。

另见

参考文献

^ M. T. Landahl; E. Mollo-Christensen. Turbulence and Random Processes in Fluid Mechanics 2nd. Cambridge University Press. 1992: 10. ISBN 978-0521422130.
^ George, William K. "Lectures in Turbulence for the 21st Century." Department of Thermo and Fluid Engineering, Chalmers University of Technology, Göteborg, Sweden (2005).p 64 [online] http://www.turbulence-online.com/Publications/Lecture_Notes/Turbulence_Lille/TB_16January2013.pdf （页面存档备份，存于互联网档案馆）

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