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 $D_F$ 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 $4/3$.

Figure 1

A two-dimensional slice of the passive scalar fluctuation field $\theta$ in the homogeneous isotropic turbulent flow. In (a) the contours of the cut are shown, as is the direction of the mean scalar gradient $G$; (b) shows the magnification of the contours within the black square in panel (a). Two segments of a scalar front are highlighted by A and B. The color bar below (a) holds for both (a) and (b) in units of $G{L}_0$, which is the maximum available difference of the mean scalar in the box. (c) The corresponding zoom of the scalar dissipation field, ${\epsilon }_{\theta }=D{\left(\nabla \theta \right)}^2$ in units of $u^{\prime }{G}^2{L}_0$, probes the magnitude of the scalar gradient. The contour levels are chosen in units of the logarithm to base 10 of ${\epsilon }_{\theta }$ and correspond to the color bar below (c).

Figure 2

Box-counting results for scalar iso-levels for $\tilde{\theta }=\theta /{\theta }^{\prime }$ of 0 (red circles), 1.5 (blue squares), and 3 (purple diamonds). These symbols have the same meaning in panels (a), (c), and (d). The results are averaged for equal iso-levels of positive and negative $\theta$. (a) The number of boxes $N\left(r\right)$ required to cover different iso-levels $\theta /{\theta }^{\prime }$ versus box size $r$. The box count is evaluated as $N\left(r\right)=\langle N\left(r\right)\rangle {(L_0/r)}^3$, where $\langle N\left(r\right)\rangle$ is the average box number from a gliding-box algorithm [29]. The box count obtained by considering nonoverlapping boxes is in good agreement with that from the gliding-box algorithm. Dashed horizontal line at 8 corresponds to $N\left(r\right)$ at $r=L_0/2$, which yields the box-counting dimension $D_F=3$ for $r\sim L_0$, shown by the dotted line. Also shown is the fractal dimension $D_F=2$ for diffusive scales by the dash-dot line. The power laws in the intermediate scale range are marked by solid lines. (b) The box-counting dimension $D_F$ as a function of the iso-level in the region $|\tilde{\theta }|\le 3$. (c) Double-logarithmic plot of $N\left(r\right)$ compensated by ${(r/\eta )}^{D_F}$ versus $r/\eta$, where $D_F$ is obtained from (a) using the method of least-squares. (d) Logarithmic local slope $D_F\left(r\right)=-d[logN\left(r\right)]/d[logr]$ of three different iso-levels $\tilde{\theta }$. Solid horizontal lines drawn for ordinate values 2 and 3 denote the small- and large-scale dimension limits, respectively.

Figure 3

Local scaling dimension of the Hausdorff volume $D_g\left(\tilde{r}\right)$ versus $r/\eta$. The quantity is obtained for a passive scalar graph over a three-dimensional ball $B_r$ of varying radius $r$ for $R_{\lambda }=650$ (squares) and for an additional DNS run at $R_{\lambda }=240$ (inverted triangles). In the inertial range (demarcated by the vertical lines), $D_g=11/3$, indicated by the horizontal dashed line, and is consistent with the fractal dimension of 2.67 for the zero iso-level shown in Fig. 2.

Figure 4

Scalar cliff region identification. (a) The line trace taken diagonally across the data shown in Fig. 1 (a) from the lower left to the upper right corner. Red points indicate the positions at which the derivative magnitude $|\partial \theta /\partial x|$ exceeds the threshold given in the legend. The criterion is taken from Ref. [10]. The mean scalar gradient is added to show the ramp-cliff structure more clearly. (b) The corresponding derivative trace $\partial \theta /\partial x$ in units of ${\theta }^{\prime }/\eta$ along the same diagonal as (a), indicating that the highest amplitudes are captured by this criterion.

Figure 5

Scaling of scalar iso-level conditioned on the high strain rate of the cliffs. (a) The compensated log-log plot of $N\left(r\right)$ versus size $r$, for the strain-dominated regions of the flow. The plateau region marked by the solid line corresponds to the $4/3$ scaling and is exhibited by all iso-levels sets shown. The small- and large-scale regimes exhibit a $-1/3$ (dashed line) and $-3$ (dash-dot line) scaling dependency, respectively. (b) Relative volume fraction of strain-dominated (filled triangles) and rotation-dominated regions (open triangles) for the chosen scalar iso-level set $\theta /{\theta }^{\prime }$. This analysis relates to the strain-dominated cliff regions only with ($|\partial \theta /\partial x|>0.2{\theta }^{\prime }/\eta$).