Intermittency in the isotropic component of helical and nonhelical turbulent flows

We analyze the isotropic component of turbulent flows spanning a broad range or Reynolds numbers. The aim is to identify scaling laws and their Reynolds number dependence in flows under different mechanical forcings. To this end, we applied an SO(3) decomposition to data stemming from direct numerical simulations with spatial resolutions ranging from ${64}^3$ to ${1024}^3$ grid points, and studied the scaling of high order moments of the velocity field. The study was carried out for two different flows obtained forcing the system with a Taylor-Green vortex or the Arn’old-Beltrami-Childress flow. Our results indicate that helicity has no significant impact on the scaling exponents as obtained from the generalized structure functions. Intermittency effects increase with the Reynolds number in the range of parameters studied, and in some cases are larger than what can be expected from several models of intermittency in the literature. The observed dependence of intermittency with the Reynolds number decreases if extended self-similarity is used to estimate the exponents.

Figure 1

Third order structure functions $G_3\left(\mathbf{l}\right)$ and isotropic component $G_3^{00}\left(l\right)$ as a function of $l$ for the T3 (nonhelical) run. Gray dots correspond to the 73 different directions while the thick solid line is the $G_3^{00}\left(l\right)$ average. The straight lines indicate $\sim l^3$ and $\sim l$ scaling.

Figure 2

Third order structure functions $-S_3\left(\mathbf{l}\right)$ compensated by $ϵl$ for the same run as in Fig. 1. The average $-S_3^{00}/ϵl$ is indicated by the thick solid line. The horizontal line indicates 4/5.

Figure 3

Isotropic component of the second order structure function $G_2^{00}\left(=S_2^{00}\right)$ for the nonhelical runs T1, T2, and T3 with increasing Reynolds number. The slopes of 2/3 and 2 are shown as references. The inset shows the three structure functions compensated by 2/3.

Figure 4

Above: $G_p^{00}\left(l\right)$ structure functions as function of the increment $l$ for $p$ from 1 to 8 for the nonhelical T3 run. $G_3^{00}\left(l\right)$ is indicated by the thick curve. Only a range of scales near the inertial range is shown. Below: same for the isotropic generalized structure functions for the helical run A3. In both figures, the error bars correspond to the error $e$.

Figure 5

Above: scaling exponents ${\zeta }_p$ in the nonhelical runs with increasing Reynolds number T1 (diamonds), T2 (crosses), and T3 (triangles). The K41 prediction is given as reference. Below: same exponents for the helical runs A1 (diamonds), A2 (crosses), and A3 (triangles).

Figure 6

Second order (above) and fourth order (below) scaling exponents as a function of the Reynolds number for all runs (triangles are for helical forcing and crosses for nonhelical forcing). Error bars correspond to the error $e$. The dotted line shows the same exponents using ESS.

Figure 7

Intermittency exponent as a function of the Reynolds number for the four runs at larger Reynolds number (labels as in Fig. 6). As in Fig. 6, the dotted line show the intermittency exponent when ESS is used.

Figure 8

Comparison between the ${\zeta }_p$ exponents from the nonhelical T3 (crosses) and helical A3 runs (triangles). Shown as a reference are the K41 (solid straight line), log-normal (dash), mean field (dotted), She-Leveque (dash-dotted), and Arimitsu and Arimitsu (solid) predictions. The mean-field model and Arimitsu and Arimitsu model are indistinguishable. The inset shows a zoom for the highest order moments, with error bars based on the error $e$.