Fractal iso-level sets in high-Reynolds-number scalar turbulence

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 DF can be obtained for iso-levels below about three 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 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.