We study the fractal scaling of iso-level sets of a passive scalar mixed by three-dimensional homogeneous and isotropic turbulence at high Reynolds numbers. The scalar field is maintained by a linear mean scalar gradient and the Schmidt number is unity. A fractal box-counting dimension $D_F$ can be obtained for iso-levels below about $3$ standard deviations of the scalar fluctuation on either side of its mean value. The dimension varies systematically with the iso-level, with a maximum of about $8/3$ for the iso-level at the mean scalar value; this maximum dimension also follows as an upper bound from the geometric measure theory. We interpret this result to mean that mixing in turbulence is always incomplete. A unique box-counting dimension for all iso-levels results when we consider the spatial support of the steep cliffs of the scalar conditioned on local strain-rate; that unique dimension, independent of the iso-level set, is about $4/3$.

中文翻译：

---

高雷诺数标量湍流中的分形等值集

我们研究了在高雷诺数下由三维均质和各向同性湍流混合的无源标量等值集的分形标度。标量场由线性平均标量梯度保持，并且施密特数为1。分形盒计数维$d_F$ 可以获得低于大约 $3$标量波动的平均值两侧的标准偏差。尺寸随等值线系统变化，最大值约为$8/3$等量于平均标量值；这个最大尺寸也是几何测量理论的上限。我们将此结果解释为意味着湍流中的混合总是不完整的。当我们考虑以局部应变率为条件的标量陡峭悬崖的空间支撑时，对于所有等水平线，都会得到唯一的计数盒维数。独立于等水平线集的唯一维大约是$4/3$。