Effect of viscous-convective subrange on passive scalar statistics at high Reynolds number

An assumption common to many approaches to understanding and modeling turbulent mixing is that the statistics of a passive scalar at small length scales will approach the analogous statistics of the velocity field as the scale separation increases between the flow-specific outer scales and the small scales of interest. This assumption follows from the hypotheses of Kolmogorov, Oboukhov, Corrsin, and Batchelor, although it is well recognized that differences in the scalar and velocity dynamics prevent the overall statistics of the two being the same. Considerable research on the effect of Reynolds number and Schmidt number on the scalar statistics has shown that the assumption is not very good at either high Reynolds number or high Schmidt number individually. Because of limitations in laboratory measurements, direct numerical simulations (DNSs), and in situ measurements of the ocean and atmosphere, it is difficult to obtain data in which the Reynolds number is high enough for an unequivocal inertial-convective subrange, and the Schmidt number is simultaneously high enough for a clear viscous-convective subrange. Our hypothesis is that, when both subranges exist, then there is sufficient scale separation in the velocity field for its statistics to be approximately universal, and also sufficient additional scale separation for the scalar to relax so that its small-scale statistics approach those of the velocity. We explore this hypothesis with DNS resolved on up to $14256\times 14256\times 14256$ grid points, Taylor Reynolds number ${\mathrm{Re}}_{\lambda }=633$, and Schmidt number $\mathrm{Sc}=[0.1,0.7,1.0,7.0]$, with the aim of studying the validity of this hypothesis for mixing in air and water. When the Schmidt number is not greater than unity at high Reynolds number, a viscous-convective subrange does not exist and we find that small-scale isotropy, the intermittency exponent, and the probability density function of the scalar dissipation rate are all much different from the analogous velocity statistics, as reported widely in literature. However, when the Schmidt number is greater than unity at high Reynolds number, inertial-convective and viscous-convective subranges both exist, and the velocity and scalar statistics are similar. Since this similarity is a consequence of the presence of both inertial convective and viscous convective subranges, it is achieved only at both high Reynolds and Schmidt numbers, and the statistics show a sharp change when the Schmidt number is changed from from 1 to 7. This suggests that at high Reynolds numbers, the modeling assumption of similarity between velocity and scalar statistics is valid for mixing in water, but not in air.

Figure 1

Sketches of scaled spectra. Solid portions of the curves are the inertial, inertial-convective, and viscous-convective spectral subranges defined by (3) with commonly used values for $C_k$, $C_{\text{oc}}$, and $C_b$. The gray shadings indicate nonexhaustive ranges of these constants collected from the literature and tabulated in Table 1.

Figure 2

Scalar contours for each simulation case on a region of the $xz$ plane centered at $y=0$; the region is approximately 1/50th of the size of the full plane in each direction. The scalar values are normalized by the rms value for the region shown, and there are eight contours spaced evenly between the minimum and maximum values. The data plotted are at a lower resolution than the actual simulations and are intended purely for visualization.

Figure 3

Scaled velocity and scalar spectra $f\left(\kappa {\eta }_k\right)$ and $g\left(\kappa {\eta }_k\right)$ for our data.

Figure 4

Solid and pattern curves are $g\left(\kappa {\eta }_k\right)$ for ${\varphi }_a$, ${\varphi }_b$, ${\varphi }_c$, and ${\varphi }_d$. The small-dashed curves in same color are Hill's model fit to the DNS data as discussed in Sec. 3c. The horizontal dash-dotted line is is the value for $C_{\text{oc}}$ for the dashed lines. The locations of $\kappa {\eta }_c$ for case ${\varphi }_d$ and $\kappa {\eta }_b$ for case ${\varphi }_a$ are indicated on the horizontal axis to indicate graphically the resolution of the simulations.

Figure 5

Scaled scalar spectrum $h\left(\kappa {\eta }_k\right)$.

Figure 6

Trace of scalars in $z$ direction at $x\approx 0$ and $y=0$.

Figure 7

Skewness of the parallel scalar derivative as a function of the Schmidt number. The dotted line denotes a slope of ${\text{Sc}}^{-1/2}$, as predicted by the model in (10) for high Schmidt number. Our results are consistent with this prediction.

Figure 8

The scale dependence of the second-order moments of the dissipation rates. The dashed vertical line is drawn at the value of the Taylor microscale. The labels ${\chi }_a$, etc., are shorthand for ${\chi }_r$ for the case indicated by the subscript. ${\epsilon }_r$ and ${\chi }_r$ are averages over spheres of radius $r$ and computed exactly in Fourier space as described by Almalkie and de Bruyn Kops [52].

Figure 9

PDF of the natural logarithm $log\left({\chi }_0\right)$ and $log\left({\epsilon }_0\right)$ for scalars ${\varphi }_a$, ${\varphi }_b$, ${\varphi }_c$, and ${\varphi }_d$. Each curve is normalized by its mean and variance per the definition of the standard Gaussian, and the Gaussian is marked with filled circles. The labels ${\chi }_a$, etc., are shorthand for ${\chi }_0$ for the case indicated by the subscript. The error bars are described in the text and are too short to see except near the tails of the distribution. The curves for ${\chi }_b$ and ${\chi }_c$ are indistinguishable at the resolution of the figure.