Reynolds number scaling of velocity increments in isotropic turbulence

Using the largest database of isotropic turbulence available to date, generated by the direct numerical simulation (DNS) of the Navier-Stokes equations on an ${8192}^3$ periodic box, we show that the longitudinal and transverse velocity increments scale identically in the inertial range. By examining the DNS data at several Reynolds numbers, we infer that the contradictory results of the past on the inertial-range universality are artifacts of low Reynolds number and residual anisotropy. We further show that both longitudinal and transverse velocity increments scale on locally averaged dissipation rate, just as postulated by Kolmogorov's refined similarity hypothesis, and that, in isotropic turbulence, a single independent scaling adequately describes fluid turbulence in the inertial range.

Figure 1

Isotropy tests at $R_{\lambda }=140$ (dotted line) and $R_{\lambda }=1300$ (solid line), as functions of scale separation $r$ in a cubic box with edge length $2L_0$. Panels (a) and (b) show $g\left(r\right)$, the right-hand side of Eq. (1) divided by the left-hand side of the same, against $r/\eta$ and $r/L_0$, respectively. Horizontal line at unity is the isotropic value, $g\left(r\right)=1$, for reference. Panels (c) and (d) show the skewness of the transverse structure function against $r/\eta$ and $r/L_0$, respectively [Eq. (2)]. The reference line at zero shows the exact isotropic value.

Figure 2

Normalized third-order longitudinal velocity structure function at ${8192}^3,R_{\lambda }=1300$. The dashed horizontal line corresponds to the plateau at $4/5$. Vertical lines demarcate inertial range $50<r/\eta <400$, considered as the range within $5\%$ deviation from $4/5$. Error bars indicate $95\%$ confidence intervals.

Figure 3

Inertial range local slopes at $R_{\lambda }=1300$ as a function of spatial separation. Dashed and solid lines correspond to $S_L^p\left(r\right)$ and $S_T^p\left(r\right)$, respectively. Colors red, blue, and green correspond to orders $p=4,6,\text{and}8$, respectively. Corresponding shaded regions denote $95\%$ confidence intervals.

Figure 4

Relative scaling of transverse structure function against longitudinal structure function at two different Reynolds numbers. Plotted against $\langle {\left(\delta u_r\right)}^6\rangle$ is $\langle {\left(\delta v_r\right)}^6\rangle$ at (a) $R_{\lambda }=240$ and (b) $R_{\lambda }=1300$. The $\left(•\right)$ points are for $r$ in the inertial range, while $\left(\odot \right)$ correspond to $r$ outside this range. The dashed line is for $\langle {\left(\delta v_r\right)}^6\rangle \sim \langle {\left(\delta u_r\right)}^6\rangle$. The insets show similar results for the eighth-order structure functions, with the inertial range data highlighted by $\left(\blacksquare \right)$.

Figure 5

Ratio of inertial range exponents of $S_T^p\left(r\right)$ and $S_L^p\left(r\right)$ as a function of the Taylor-scale Reynolds number. Symbols $\left(○\right)$, $\left(▵\right)$, and $\left(\square \right)$ correspond to orders $p=4$, 6, and 8, respectively. The horizontal line at unity is for reference. Solid vertical lines indicate $95\%$ confidence intervals. The corresponding result of Ref. [8] for $p=8$ $\left(\blacksquare \right)$ with error bars, shown by a dashed vertical line, is given for comparison.

Figure 6

Normalized PDF of $V_L$ and $V_T$ at $R_{\lambda }=1300$ for various inertial range separations. Curves with filled symbols correspond to $V_L$ for different values of $r$: $\left(♦\right)$ $r/\eta =99$, $\left(\blacksquare \right)$ $r/\eta =118$, $\left(▴\right)$ $r/\eta =143$, $\left(•\right)$ $r/\eta =170$, $\left(▾\right)$ $r/\eta =206$. Corresponding curves for $V_T$ given by open symbols. The dashed line shows the standard Gaussian for reference.

Figure 7

Probability distribution of the dimensionless relative velocity $\mathbf{V}\left(\mathbf{r}\right)$ at inertial range separation $r/\eta =100,R_{\lambda }=1300$. (a) Logarithm of the JPDF of $V_L$ and $V_T$. Slivers of width ${\delta }_L$ and ${\delta }_T$ of the JPDF along $V_L$ and $V_T$ (marked by vertical lines) are shown in (b) and (c) to reveal the asymmetric and symmetric parts, respectively.

Figure 8

Verification of the RSH scaling relations [Eqs. (6) and (7)] at $R_{\lambda }=1300$. Symbols $\left(\square \right)$ and $\left(○\right)$ correspond to the longitudinal and transverse cases, respectively. If RSH is correct, the measured differences between ${\zeta }_L^p$ and ${\tau }_{p/3}$ and that between ${\zeta }_T^p$ and ${\tau }_{p/3}$ for various values of $p$ must equal $p/3$, which is the dashed line. The correspondence is very good. Vertical bars indicate the extent of $95\%$ confidence intervals (which are significant mainly at higher orders).