Intermittency of turbulent velocity and scalar fields using 3D local averaging

The DNS data examined in this work are obtained by solving the incompressible Navier-Stokes equations:

where ${\mathbf{u}}$ is the divergence free velocity field ($\nabla\cdot{\mathbf{u}}=0$), $P$ is the kinematic pressure and $\mathbf{f}$ is the large-scale forcing term to maintain statistical stationarity. The DNS corresponds to the canonical setup of isotropic turbulence in a periodic domain Ishihara et al. (2009), allowing the use of highly accurate Fourier pseudo-spectral methods, with aliasing errors controlled using a combination of grid-shifting and truncation Rogallo (1981). The database for the present work corresponds to recent works Buaria and Sreenivasan (2020); Buaria et al. (2020a, 2022); Buaria and Sreenivasan (2022), with the Taylor-scale Reynolds number ${R_{\lambda}}$ in the range $140-1300$. Convergence with respect to resolution and statistical sampling has also been thoroughly established in all these previous works.

The passive scalar is obtained by simultaneously solving the advection-diffusion equation in the presence of mean uniform gradient:

The uniform mean gradient is set as $\nabla\Theta=(G,0,0)$ along the first Cartesian directions, and provides the forcing needed to achieve statistical stationarity for the scalar Overholt and Pope (1996). The database for scalars utilized here is the same as in our recent papers Buaria et al. (2021a, b), and corresponds to ${R_{\lambda}}$ in the range $140-650$, and $Sc$ in the range $1-512$. As noted in Buaria et al. (2021a), the data were generated using conventional Fourier pseudo-spectral methods for $Sc=1$, and a hybrid approach for higher $Sc$ Clay et al. (2017a, 2018, b); this approach consisted of solving the velocity field pseudo-spectrally while resolving the Kolmogorov length scale $\eta_{K}$, while resolving the scalar field using compact finite differences on a finer grid, so as to resolve the Batchelor scale $\eta_{B}=\eta_{K}Sc^{-1/2}$ Batchelor (1959).

To implement the 3D local averaging efficiently, Eq. (1) is rewritten as:

where $G({\mathbf{x}}^{\prime})=3/4\pi r^{3}$, satisfying $\int_{{\mathbf{x}}^{\prime}}G({\mathbf{x}}^{\prime})d{\mathbf{x}}^{\prime}=1$, represents an isotropic box filter Pope (2000). Following recent work in Ref. Buaria et al. (2020b); Buaria and Pumir (2021), this filtering operation can be easily evaluated in the Fourier-space for a chosen value of $r$, as:

Here, $\hat{(\cdot)}$ denotes the Fourier transform, $\mathbf{k}$ is the wave-vector, with $|\mathbf{k}|=k$, and $f(kr)$ is the transfer function corresponding to $G({\mathbf{r}})$. Unlike in previous works Iyer et al. (2015); Yeung and Ravikumar (2020), this local averaging is evaluated exactly in an isotropic spherical volume. It can also be easily shown that this isotropic box filter also satisfies the consistency condition: $\langle A_{r}\rangle=\langle A\rangle$.

We first consider the intermittency of dissipation and enstrophy, a topic that has received considerable attention in the literature Siggia (1981); Chen et al. (1997); Grossmann et al. (1997); Buaria et al. (2019); Yeung and Ravikumar (2020). Figure 1a shows the second moments of locally-averaged dissipation and enstrophy at ${R_{\lambda}}=1300$ (the highest in the current work). As anticipated, for small $r$, the moments of local averages simply tend to those of instantaneous quantities, those for enstrophy being larger (as is known); whereas at large $r$, the moments tend to the same value of unity (as they should). In the inertial range, dissipation exhibits the power-law $r^{-0.23}$ (with an error bar of $0.02$) for a reasonable range of $r$, implying $\mu\approx 0.23\pm 0.02$, in very good agreement with previous works Sreenivasan and Kailasnath (1993). At the same time, within the same range of $r$, the enstrophy curve suggests a slightly larger intermittency exponent, consistent with previous works Chen et al. (1997); Yeung and Ravikumar (2020); see later for more details.

An alternative means of extracting the intermittency exponent is to consider the variance of $\epsilon_{r}$ (and $\Omega_{r}$), computed after subtracting the respective means for each $r$; for instance,

note that $\langle\epsilon_{r}\rangle=\langle\epsilon\rangle$. The expectation is that $\sigma(\epsilon_{r})\sim r^{-\mu}$ in the inertial range Sreenivasan and Kailasnath (1993). Figure 1b shows the variance of $\epsilon_{r}$ and $\Omega_{r}$, and indeed the nature of their scaling follow expectations — although the scaling ranges are somewhat different from Fig. 1a.

To be stringent about the power law exponents, it is helpful to take the log-log derivatives (or the local slope) of the curves in Fig. 1a. Figure 2a shows the local slope of second moments of $\epsilon_{r}$ and $\Omega_{r}$ for various ${R_{\lambda}}$. It can be seen that the quality of results in the inertial range depend on ${R_{\lambda}}$ and, in fact, a constant local slope (corresponding to a true power-law) is convincing only at the highest ${R_{\lambda}}$ ($=1300$). However, guided by this feature, one can look for signs of approximate power laws at lower ${R_{\lambda}}$, and find slightly large exponent values.

It is worth noting that in Fig. 2a, the local slope of enstrophy (in the inertial range) is always larger than that of dissipation for every ${R_{\lambda}}$. To better document this behavior, Fig. 2b shows the ratio of the local slope of enstrophy with respect to that of dissipation (in the spirit of extended self-similarity). Remarkably, the curves are always above unity in the inertial range. Further, there is a weak but clear tendency for this ratio to approach unity with increasing ${R_{\lambda}}$, but the rate of approach is so slow that the difference remains in place at all finite Reynolds numbers of interest. That is, enstrophy in the inertial range is slightly more intermittent than dissipation.

We first consider locally averaged scalar dissipation rate for $Sc=1$. Figure 3a shows the second moments of locally averaged energy dissipation, enstrophy and scalar dissipation (at ${R_{\lambda}}=650$), with the corresponding local slopes shown in Fig. 3b. The intermittency exponent of the scalar $\mu_{\theta}$ is larger than those of both dissipation and enstrophy. While a clear plateau for dissipation and enstrophy is not achieved at ${R_{\lambda}}=650$ (in Fig. 3b) the curve for scalar dissipation shows a distinct plateau, giving $\mu_{\theta}\approx 0.35$. This result is in excellent agreement with earlier experimental results of Sreenivasan et al. (1997); Prasad et al. (1988).

The effect of increasing $Sc$ is considered next. Since increasing $Sc$ imposes a stricter constraint on small-scale resolution, we consider data at slightly lower ${R_{\lambda}}=390$, but still large enough to display inertial range characteristics. Figure 4a shows the local slope of second moment of scalar dissipation at ${R_{\lambda}}=390$ and $Sc=1-8$. The curve corresponding to ${R_{\lambda}}=650$ and $Sc=1$ is also shown for comparison. The scalar intermittency exponent monotonically decreases with increasing $Sc$ (also displaying a weak ${R_{\lambda}}$ dependence).

Due to this possible ${R_{\lambda}}$-dependence, it is difficult to extract precisely the intermittency exponents at higher $Sc$. Instead, if we were to compare the ratio of $\mu_{\theta}$ to $\mu$, a trend can be established; this should shed light on the asymptotic limit of $Sc\to\infty$. To this end, Fig. 4b shows the local slope of scalar dissipation with respect to that of energy dissipation. The ${R_{\lambda}}$-dependence is somewhat more prominent than in Fig. 4a, likely because $\mu_{\theta}$ seemingly has a stronger ${R_{\lambda}}$-dependence than $\mu$. Nevertheless, it is evident that the ratio $\mu_{\theta}/\mu$ monotonically decreases with $Sc$.

For a definitive answer on the high $Sc$ limit, one obviously needs to obtain data at substantially higher $Sc$ at ${R_{\lambda}}=390$ (and also at higher ${R_{\lambda}}$). But these are unlikely to be attained anytime soon. Instead, we analyze data at lower ${R_{\lambda}}=140$, for which inertial range characteristics just begin to manifest Ishihara et al. (2009); Iyer et al. (2015). Figure 5a shows the local slope of second moment of scalar dissipation with respect to that of energy dissipation up to $Sc$ as high as $512$. The curve corresponding to ${R_{\lambda}}=390$ and $Sc=8$ is also shown for comparison.

Two main conclusions can be drawn from this figure. First, the effect of ${R_{\lambda}}$ is weaker at higher $Sc$ (in comparison to that at $Sc=1$, as evident from Fig. 4b). Second, the ratio $\mu_{\theta}/\mu$ keeps decreasing further, possibly suggesting that $\mu_{\theta}\to 0$ as $Sc\to\infty$. We extract the ratio $\mu_{\theta}/\mu$ from Fig. 5a and plot its inverse as a function of $Sc$ in Fig. 5b. The data show a weak power-law dependence (with an exponent of about $0.16$), though a $\log Sc$-behavior is equally appropriate. The $\log Sc$-behavior of $\mu/\mu_{\theta}$ is loosely based on how the mean scalar dissipation rate scales with $Sc$ Buaria et al. (2021b).