Second-order velocity structure functions in direct numerical simulations of turbulence with $R_{\lambda }$ up to 2250

We report results of a series of high-resolution direct numerical simulations (DNSs) of forced incompressible isotropic turbulence with the number of grid points and the Taylor scale Reynolds number $R_{\lambda }$ up to $12{288}^3$ and $\sim 2250$, respectively. The DNSs show that there exists a scaling range (approximately $100<r/\eta <400$), at which the second-order two-point velocity structure functions $S_2\left(r\right)$ fit well with a simple power-law, $S_2\left(r\right)/{\left(〈\epsilon 〉r\right)}^{2/3}=C_2{(r/L_0)}^{\zeta }$, where $r$ is the distance between the two points, $\eta$ is the Kolmogorov length scale, $〈\epsilon 〉$ is the mean rate of energy dissipation per unit mass, and $L_0$ is the integral length scale. The exponent $\zeta$ is constant independent from $R_{\lambda }$. However, the coefficient $C_2$ is dependent of $R_{\lambda }$ or the viscosity. This implies that the power-law scaling range of $100<r/\eta <400$ for $R_{\lambda }$ up to $\sim 2250$ is not the so-called “inertial subrange” in the sense that the statistics in the range are independent from the viscosity, as assumed in various turbulence theories. This suggests that the constancy of the scaling exponent of a structure function within a certain range does not necessarily mean that the exponent is the scaling exponent in “the inertial subrange.”

Figure 1

$S_2^{\text{LC}}\left(r\right)$ and $S_3^{\text{LC}}\left(r\right)$ at $R_{\lambda }\sim 730$ in Run2048-1. The solid lines and shadow regions show the average over several snapshots, and the standard deviations, respectively. The broken lines show several instances of snapshot data. (a) Linear-log plot and (b) log-log plot.

Figure 2

(a) $S_2^{\text{LC}}\left(r\right)$ and $S_3^{\text{LC}}\left(r\right)$ as a function of $r/\eta$, and (b) the same as (a) but as a function of $r/L_0$. For “T-range” see the text.

Figure 3

The same as described in the caption of Fig. 2 but the vertical axis is in log-scale.

Figure 4

(a) Overhead view of $S_2^{\text{LC}}\left(r\right)$ as a function of $r/L_0$ and $R_{\lambda }$. (b) The same as described in the caption of Fig. 3 but as a function of $X=r/\left(L_0{R}_{\lambda }^{-1.09}\right)$.

Figure 5

K-DNS data of $S_2^{\text{LC}}\left(r\right)$ and $S_2^{\text{TC}}\left(r\right)$ by $E_{\mathrm{DNS}}\left(k\right)$ ($\circ$), and by taking the average over the directions of $\mathbf{e}$ (solid lines). The broken lines indicate model Eq. (10). (a) Linear-log plot and (b) log-log plot.

Figure 6

${\zeta }_2^L$ and ${\zeta }_2^T$ in Run12288-1, and by model Eq. (10). The dots show the experimental data of ${\zeta }_2^L$ by Tsuji [24].

Figure 7

DNS values of “$a$” given by Eq. (27) in Run 12288-1 as a function of $r/\eta$, with $4/3-2z=2/3$ or $4/3-2z={\zeta }_2^L=0.065$. The vertical arrow shows the position of $r$ at $r=L_0$.