Oscillations Modulating Power Law Exponents in Isotropic Turbulence: Comparison of Experiments with Simulations

Inertial-range features of turbulence are investigated using data from experimental measurements of grid turbulence and direct numerical simulations of isotropic turbulence simulated in a periodic box, both at the Taylor-scale Reynolds number $R_{\lambda }\sim 1000$. In particular, oscillations modulating the power-law scaling in the inertial range are examined for structure functions up to sixth-order moments. The oscillations in exponent ratios decrease with increasing sample size in simulations, although in experiments they survive at a low value of 4 parts in 1000 even after massive averaging. The two datasets are consistent in their intermittent character but differ in small but observable respects. Neither the scaling exponents themselves nor all the viscous effects are consistently reproduced by existing models of intermittency.

Figure 1

Replot of data from [12] of the ratio of logarithmic local slopes of fourth-order longitudinal velocity structure function ($S_4$) to the corresponding quantity for the second order ($S_2$) vs spatial separation $r$ normalized by Kolmogorov scale ($\eta$); see Eq. (2). The two thinner curves were computed from datasets about 10 times shorter than the others, as explained in [12]. The legend shows the microscale Reynolds numbers $R_{\lambda }$.

Figure 2

Ratio of logarithmic local slopes of $S_4$ and $S_2$, ${\zeta }_{4,2}(r)\equiv d[\log S_4(r)]/d[\log S_2(r)]$, vs spatial separation $r$ normalized by Kolmogorov scale $\eta$. (a) Data from EXP at $R_{\lambda }=1030$ (open circles) are from [12]; the isotropic sector from DNS at $R_{\lambda }=1300$ (solid line) [16] are compared with the $p$ model (dash-dot line) of Meneveau and Sreenivasan [22], She-Leveque model (dashed line) [23], and that by Yakhot (dotted line) [24]. Horizontal line at ${\zeta }_{4,2}=2$ corresponds to nonintermittent scaling [5]. Error bars indicate the standard error obtained from temporal fluctuations of local slopes. (b) Isotropic DNS data at different Reynolds numbers: $R_{\lambda }=240$ (dotted line), 650 (dash-dot line), and 1300 (solid line) are plotted to show that the viscous bottleneck around $r/\eta =80$ decreases in amplitude with increasing $R_{\lambda }$ in the direction shown. (c) Data from EXP (open circles), compared with the DNS data for the isotropic sector (solid line) and the one-dimensional cuts in DNS along the three Cartesian directions: $\widehat{\mathbf{r}}=(1,0,0)$ (dotted line), $\widehat{\mathbf{r}}=(0,1,0)$ (dashed line), and $\widehat{\mathbf{r}}=(0,0,1)$ (dash-dot line). The latter three illustrate that the inertial range value of ${\zeta }_{4,2}(r)$ can be affected by nonuniversal large-scale effects when not projecting onto the isotropic sector.

Figure 3

Log-log plot of the standard deviation of the oscillation amplitude in $d[\log (S_4)]/d[\log (S_2)]$ from its mean value in the range $r/\eta \in (100,1000)$ in Fig. 2 as function of sample size $n$ for (a) the experiments and (b) the isotropic sector from DNS. Error bars indicate the variation obtained by changing the interval $r/\eta \in [100,1000]$ by 10% on either side. We include an analysis of a second EXP dataset (green squares) acquired at $R_{\lambda }=961$ to illustrate typical variation in large $n$ behavior. The standard deviation in the experiment is approximately constant for about two orders of magnitude for $n>{10}^8$, or equivalently for more than ${10}^5$ turnover times, while that in DNS shows a $-0.31\pm 0.04$ power-law decay for $n>3\times {10}^{12}$, as indicated by the solid line. The sample size in DNS cannot be easily translated to turnover timescales. The dashed line in (a) corresponds to the $-1/2$ scaling of random noise.

Figure 4

Ratio of logarithmic local slopes of $S_6$ and $S_2$, ${\zeta }_{6,2}(r)\equiv d[\log S_6(r)]/d[\log S_2(r)]$ vs spatial separation $r$ normalized by Kolmogorov scale $\eta$. Error bars indicate standard error obtained from temporal fluctuations of local slopes. Data from experiment (open circles) and the isotropic sector of DNS (solid line) are compared with the $p$ model (dash-dot line) of Meneveau and Sreenivasan [22], She-Leveque model (dashed line) [23], and that by Yakhot (dotted line) [24]. Horizontal line at ${\zeta }_{6,2}=3$ corresponds to self-similar nonintermittent scaling [5]. Inset shows a blowup of the inertial range and to either side of it to highlight differences between EXP and DNS.