Two-time Lagrangian velocity correlation function for particle pairs in two-dimensional inverse energy-cascade turbulence

We numerically investigate a two-time Lagrangian velocity correlation function (TTLVCF) for particle pairs in two-dimensional energy inverse-cascade turbulence. We consider self-similarity of the correlation function by means of incomplete similarity. In this framework, we propose a self-similar form of the correlation function, whose scaling exponents cannot be determined by only using the dimensional analysis based on the Kolmogorov's phenomenology. As a result, the scaling laws of the correlation function can depend on the initial separation. This initial-separation dependency is frequently observed in laboratory experiments and direct numerical simulations of the relative dispersion, which is directly related to the correlation function, at moderate Reynolds numbers. We numerically verify the self-similar form by direct numerical simulations of two-dimensional energy inverse-cascade turbulence. The involved scaling exponents and the dependencies on finite Reynolds-number effects are determined empirically. Then, we consider implication of the scaling laws of the correlation function on the relative dispersion, i.e., the Richardson-Obukhov $t^3$ law. Our results suggest a possibility not to recover the Richardson-Obukhov $t^3$ law at infinite Reynolds number.

Figure 1

Color map of the TTLVCF, Eq. (5), with $T_B=3.4T_{\eta }$ for ${\mathrm{Re}}_{\alpha }=160$, where $T_B$ and $T_{\eta }$ are the Batchelor time and the Kolmogorov dissipation timescale, respectively. The infrared Reynolds number ${\mathrm{Re}}_{\alpha }$ is defined in the main text. See Sec. 3 for details.

Figure 2

Color maps of time evolution of the TTLVCF $C_{ii}^L(r_0,t_1,t_2)$ defined in Eq. (5) with $T_B={(r_0^2/\varepsilon )}^{1/3}=3.5T_{\eta }$. (a) ${\mathrm{Re}}_{\alpha }=40$, (b) ${\mathrm{Re}}_{\alpha }=80$, (c) ${\mathrm{Re}}_{\alpha }=160$. The white lines indicate $t_i=T_B$ ($i=1,2$). Here the time axes, $t_i/T_{\eta }$ ($i=1,2$), span from 0 to $T_L/T_{\eta }$.

Figure 3

(a) Time evolution of $C_d^L(T,T_B)$ normalized by ${(r_0/T_B)}^2$ for various $T_B$'s at ${\mathrm{Re}}_{\alpha }=40$ (dashed dotted), 80 (dashed), and 160 (solid). (b) Logarithmic local slope (LLS) of $C_d^L(T,T_B)$ at ${\mathrm{Re}}_{\alpha }=40$ (dashed) and 160 (solid). The filled circle on each line indicates the position of the maximum. (c) Value of the exponent $\gamma$ suggested by the maximum value of the LLS plotted as a function of $(T_B-T_{\eta })/T_L$ at ${\mathrm{Re}}_{\alpha }=40$ (red) and 160 (green). The blue solid line corresponds to $1.34{[(T_B-T_{\eta })/T_L]}^{0.5}$, which is determined by the least square fit in the range $0.039\le (T_B-T_{\eta })/T_L\le 0.16$.

Figure 4

Normalized correlation function, $C_p^L(T,\tau ,T_B)$, defined in Eq. (32) as a function of the relative time $\tau$ with $T_B=3.5T_{\eta }$ for various $T$'s. The average time varies in $T_{\eta }<T<0.54T_L$ and the corresponding curves are colored from black to yellow. The three panels correspond to (a) ${\mathrm{Re}}_{\alpha }=40$, (b) ${\mathrm{Re}}_{\alpha }=80$, and (c) ${\mathrm{Re}}_{\alpha }=160$. Two vertical dashed lines in each panel show $\tau /T_B=\pm T_{\eta }/T_B$, respectively. The horizontal axis spans in $-T_L/T_B\le \tau /T_B\le T_L/T_B$. The insets show the same plots as the panels but in the lin-log coordinates.

Figure 5

$n\mathrm{th}$ decay timescale, ${\tau }_{1/n}\left(T\right)$, as a function of the average time $T$ for (a) $n=2$, (b) $n=8$, (c) $n=32$ at ${\mathrm{Re}}_{\alpha }=40$ (dashed dotted), ${\mathrm{Re}}_{\alpha }=80$ (dashed), and ${\mathrm{Re}}_{\alpha }=160$ (solid). The insets show the LLS of ${\tau }_{1/n}\left(T\right)$ shown in the panels.

Figure 6

Compensated plots of ${\tau }_{1/n}\left(T\right)$ by $T^{1-{\beta }_{1/n}}$ for (a) $n=2$, (b) $n=8$, (c) $n=32$ at ${\mathrm{Re}}_{\alpha }=80$ (dashed) and ${\mathrm{Re}}_{\alpha }=160$ (solid). The values of ${\beta }_{1/n}$ are determined in such a way that each of the compensated graphs has the widest flat region.

Figure 7

Scaling exponents, ${\beta }_{1/n}$, evaluated from the compensated plots of Fig. 5 for (a) $n=2$, (b) $n=8$, and (c) $n=32$ at ${\mathrm{Re}}_{\alpha }=80$ (red) and ${\mathrm{Re}}_{\alpha }=160$ (green). The inset of panel (a) shows the same plots in the outset, but the horizontal axis is changed to $T_B/T_L$ from $(T_B-T_{\eta })/T_L$. The orange solid line shows ${[T_B/T_L]}^{0.4}$. The blue solid line shows ${[T_B/T_L+0.01]}^{0.4}$. For panels (b) and (c), the orange solid line shows ${[(T_B-T_{\eta })/T_L]}^{0.4}$. The blue solid line shows ${[(T_B-T_{\eta })/T_L+0.01]}^{0.4}$.

Figure 8

Normalized correlation function, $C_p^L(T,\tau ,T_B)$ for $T_B=2.2T_{\eta }$ rescaled as a function of (a) $\tau /\left[T_B^{{\beta }_{1/2}}{T}^{1-{\beta }_{1/2}}\right]$ and (b) $\tau /\left[T_B^{{\beta }_{1/32}}{T}^{1-{\beta }_{1/32}}\right]$, where ${\beta }_{1/2}=0.64$ and ${\beta }_{1/32}=0.76$, for $T_B<T<0.54T_L$ at ${\mathrm{Re}}_{\alpha }=160$. The colors of the curves change from black to yellow as the average time $T$ increases, which is similar to Fig. 4. Black dashed lines show (a) $exp(-2.3|\tau |/\left[T_B^{{\beta }_{1/2}}{T}^{1-{\beta }_{1/2}}\right])$ and (b) $exp(-3.0|\tau |/\left[T_B^{{\beta }_{1/32}}{T}^{1-{\beta }_{1/32}}\right])$. (c) Same as panel (a) but for $T_B=3.5T_{\eta }$, where ${\beta }_{1/2}=0.42$. (d) Same as panel (b) but for $T_B=3.5T_{\eta }$, where ${\beta }_{1/32}=0.30$.

Figure 9

(a) Time evolution of $C_d^L(T,T_B)$ for various $T_B$'s at ${\mathrm{Re}}_{\alpha }=40$ (dashed dotted), 80 (dashed), and 160 (solid). (b) LLSs of $C_d^L(T,T_B)$ at ${\mathrm{Re}}_{\alpha }=40$ (dashed) and 160 (solid). Filled circles on the curves show positions of their maximum value. (c) Values of the exponent $\check{\gamma }$ as a function of $T_B/T_{\eta }$, which are measured by the maximum value of the LLSs at ${\mathrm{Re}}_{\alpha }=40$ (red) and 160 (green). The blue solid line shows $ln(T_B/T_{\eta })-0.25$.

Figure 10

Time evolution of the $n\mathrm{th}$ decay timescale, ${\tau }_{1/n}\left(T\right)$ for (a) $n=2$, (b) $n=8$, and (c) $n=32$ at ${\mathrm{Re}}_{\alpha }=160$. The insets show the same plots as the panels but at ${\mathrm{Re}}_{\alpha }=80$.

Figure 11

Compensated graphs of ${\tau }_{1/n}\left(T\right)$ by ${\check{\beta }}_{1/n}$ for (a) $n=2$, (b) $n=8$, and (c) $n=32$ at ${\mathrm{Re}}_{\alpha }=80$ (dashed) and ${\mathrm{Re}}_{\alpha }=160$ (solid). The insets show ${\beta }_{1/n}$ as a function of $T_B/T_{\eta }$ at ${\mathrm{Re}}_{\alpha }=80$ (red) and ${\mathrm{Re}}_{\alpha }=160$ (green).

Figure 12

Normalized correlation function, $C_p^L(T,\tau ,\varepsilon )$, with $T_B=0.47T_{\eta }$ as a function of rescaled variable (a) $\tau /\left[T_B^{{\check{\beta }}_{1/2}}{T}^{1-{\check{\beta }}_{1/2}}\right]$ and (b) $\tau /\left[T_B^{{\check{\beta }}_{1/32}}{T}^{1-{\check{\beta }}_{1/32}}\right]$ under the small $T_B$ condition. Here the exponents are ${\check{\beta }}_{1/2}=0.82$ and ${\check{\beta }}_{1/32}=0.96$. Different curves have different $T\mathrm{s}$ in $T_B<T<0.54T_L$ at ${\mathrm{Re}}_{\alpha }=160$. The black dashed lines correspond to (a) $exp(-2.7|\tau |/\left[T_B^{{\check{\beta }}_{1/2}}{T}^{1-{\beta }_{1/2}}\right])$ and (b) $exp(-4.8|\tau |/\left[T_B^{{\check{\beta }}_{1/32}}{T}^{1-{\beta }_{1/32}}\right])$.

Figure 13

Mean-squared relative separation of particle pairs, $\langle r^2\left(t\right)\rangle$, for various initial separations at ${\mathrm{Re}}_{\alpha }=160$ (colored lines) and the scaling law (50) (black lines). Two vertical dashed lines show the dissipation timescale $T_{\eta }$ (left) and the integral scale $T_L$ (right).