Steep Cliffs and Saturated Exponents in Three-Dimensional Scalar Turbulence

The intermittency of a passive scalar advected by three-dimensional Navier-Stokes turbulence at a Taylor-scale Reynolds number of 650 is studied using direct numerical simulations on a $4096^3$ grid; the Schmidt number is unity. By measuring scalar increment moments of high orders, while ensuring statistical convergence, we provide unambiguous evidence that the scaling exponents saturate to 1.2 for moment orders beyond about 12, indicating that scalar intermittency is dominated by the most singular shocklike cliffs in the scalar field. We show that the fractal dimension of the spatial support of steep cliffs is about 1.8, whose sum with the saturation exponent value of 1.2 adds up to the space dimension of 3, thus demonstrating a deep connection between the geometry and statistics in turbulent scalar mixing. The anomaly for the fourth and sixth order moments is comparable to that in the Kraichnan model for the roughness exponent of $4/3$.

Figure 1

Ramp-cliff structures in a scalar field, $\mathrm{\Theta }\equiv \theta +Gx$, here $\theta$ is the scalar fluctuation and $(G,0,0)$ is the mean gradient, at $R_{\lambda }=650$ and the Schmidt number $\text{Sc}\equiv \nu /D=1$, where $\nu$ is the kinematic viscosity of the fluid and $D$ is the scalar diffusivity. $L$ is the size of the computational cube in one direction. The main figure to the left plots four 1D profiles of $\mathrm{\Theta }$ in the $x$ direction, along which the mean gradient is imposed. Examples for ramps and cliffs are indicated by arrows as is the mean scalar concentration profile by dashed lines. Profiles are shifted in steps of 6 units with respect to each other for clarity. The vertical solid lines indicate the spatial positions for the magnifications of the scalar fluctuation profiles plotted to the right. Grid resolution and Kolmogorov length ${\eta }_K$ are indicated.

Figure 2

Scalar increment exponent ${\zeta }_p^{\theta }$ versus moment order $p$. (a) Present DNS: ${\text{Pe}}_{\lambda }=650$, dashed line at saturation exponent, ${\zeta }_{\infty }^{\theta }=1.2$. Error bars indicate 95% confidence interval. (b) Comparison of present DNS (shaded region) with previous results: (▽) ${\text{Pe}}_{\lambda }=220$ [52], (◯) ${\text{Pe}}_{\lambda }=280$ [29], (△) ${\text{Pe}}_{\lambda }=396$ [32], ($◇$) ${\text{Pe}}_{\lambda }=580$ [34]. Dash-dotted line shows normal scaling ${\zeta }_p^{\theta }=p/3$ and dashed line is the model of Ref. [28]. A more extensive data comparison can be found in [43].

Figure 3

Integrands of scalar increment moments [$\mathcal{P}(·)$ denotes PDF of $(·)$] as functions of scalar increments, for orders 6 and 16 at $r/{\eta }_K=55$ (lower end of the inertial range), on lin-log scales; moments of orders up to 20 converge as well and confirm the saturation of exponents but are not shown here. The integrands are normalized by respective moments such that the area under each curve is unity.

Figure 4

PDF of scalar increments across the inertial range, multiplied by $r^{-{\zeta }_{\infty }^{\theta }}$, where ${\zeta }_{\infty }^{\theta }$ is the saturation exponent (Fig. 2). The PDF tails collapse, confirming saturation of exponents.

Figure 5

Log-log plot of the number $N(r)$ of cubes of side $r$ containing the steepest fronts versus size $r$. The ordinate is compensated by $r^{D_F}$, where $D_F=1.8$ is the fractal codimension of the fronts. The plateau region (filled circle) corresponds to the scaling $r^{-D_F}$, indicated by the horizontal dashed line. At the smallest and largest $r$, the dimension of the fronts is 2 (corresponding to flat fronts) and 3 (which is the Euclidean dimension of the flow), respectively. Inset shows the PDF of the normalized gradient $z$, $|z|>0.2$ (dotted lines) is used to calculate $N(r)$.

Figure 6

Flatness anomaly $({\psi }_4^{\theta }=2{\zeta }_2^{\theta }-{\zeta }_4^{\theta })$ (open symbols) and hyperflatness anomaly $({\psi }_6^{\theta }=3{\zeta }_2^{\theta }-{\zeta }_6^{\theta })$ (closed symbols) for the scalar versus flow roughness $\xi$. Circles correspond to 3D Kraichnan model, (◯) [55] and (filled circle) [56], while triangles are for 3D NS flow at ${\text{Pe}}_{\lambda }=650$, with the roughness parameter $\xi =4/3$.