Non-Kolmogorov scaling for two-particle relative velocity in two-dimensional inverse energy-cascade turbulence

Herein we numerically examine the relative dispersion of Lagrangian particle pairs in two-dimensional inverse energy-cascade turbulence. Behind the Richardson-Obukhov $t^3$ law of relative separation, we discover that the second-order moment of the relative velocity have a temporal scaling exponent different from the prediction based on the Kolmogorov's phenomenology. The results also indicate that time evolution of the probability distribution function of the relative velocity is self-similar. The findings are obtained by enforcing the Richardson-Obukhov law either by considering a special initial separation or by conditional sampling. In particular, we demonstrate that the conditional sampling removes the initial-separation dependence of the statistics of the separation and relative velocity. Furthermore, we demonstrate that the conditional statistics are robust with respect to the change in the parameters involved, and that the number of the removed pairs from the sampling decreases when the Reynolds number increases. We also discuss the insights gained as a result of conditional sampling.

Figure 1

(a) Time-averaged energy flux $\mathrm{\Pi }\left(k\right)$ for ${\text{Re}}_{\alpha }=40$ (red line), ${\text{Re}}_{\alpha }=80$ (blue line), and ${\text{Re}}_{\alpha }=160$ (green line). Ranges between the two gray dotted lines correspond to the inertial ranges as determined by the plateau regions of the energy flux in the Fourier space. The inset shows the time-averaged energy spectrum $E\left(k\right)$ for ${\text{Re}}_{\alpha }=40$ (red line), ${\text{Re}}_{\alpha }=80$ (blue line), and ${\text{Re}}_{\alpha }=160$ (green line). (b) Normalized PDFs of the longitudinal velocity increments for ${\text{Re}}_{\alpha }=40$ (red) and ${\text{Re}}_{\alpha }=80$ (blue) at various separations, $r/l_f=1.1,1.5,2.2,3.2,4.8$, and 7.1. Here $l_f=2\pi /k_f$ is the forcing scale and longitudinal velocity increment, $\delta u_l$, normalized by the second-order moment and is denoted by $s$: $s=\delta u_l/\sqrt{\langle \delta u_l^2\rangle }$. The dotted line denotes the Gaussian distribution with zero mean and unit variance. The inset shows the second-order longitudinal structure function, $S_2\left(r\right)$ for ${\text{Re}}_{\alpha }=40$ (red) and ${\text{Re}}_{\alpha }=80$ (blue). The dashed line represents the K41 scaling, $r^{2/3}$, for $S_2\left(r\right)$. (c) Same as (a) except for the normal viscous case ($h=1$). (d) Same as (b) except for the normal viscous case ($h=1$).

Figure 2

(a) Mean-squared separation for the hyperviscous case with ${\text{Re}}_{\alpha }=40$ as a function of time for various initial separations. Here $t_f={(l_f^2/\varepsilon )}^{1/3}$ denotes the forcing timescale. The ranges between the two horizontal lines correspond to the inertial range determined by the region of the mean exit time that is proportional to $r^{2/3}$, which is discussed later. Inset: the logarithmic local slope for the special initial separation, $r_0=0.088l_f\sim r_0^{\left(p\right)}$, where $\varepsilon$ denotes the mean energy flux in the inertial range. The gray dashed line corresponds to the Richardson scaling exponent, 3. (b) Mean-squared relative velocity for various initial separations. Inset: the logarithmic local slope for the special initial separation, $r_0=0.088l_f$. The gray dashed line corresponds to the Kolmogorov scaling exponent, 1. (c) Same as (a) except for the normal viscous case. (d) Same as (b) except for the normal viscous case.

Figure 3

(a) Mean exit time for the hyperviscous cases with ${\text{Re}}_{\alpha }=40$ (dash-dotted line) and ${\text{Re}}_{\alpha }=80$ (solid line). Black solid line denotes $r^{2/3}$ power law. The ranges between the two gray dashed lines denote the inertial range estimated with the exit time at ${\text{Re}}_{\alpha }=40$ and ${\text{Re}}_{\alpha }=80$. (b) The PDFs of exit time normalized by the mean at ${\text{Re}}_{\alpha }=40$ (red) and 80 (blue) for $j=0$ to 6. Black dashed line and solid line denote longtime asymptotic forms for the PDF of exit times, $P\left(T_E\right)\langle T_E\rangle \sim C{\sum }_{i=1}^N{j}_{2,i}{J}_2^{\prime }\left(j_{2,i}\right)exp\left[-\frac{j_{2,i}^2}{12}\frac{{\rho }^{2/3}-1}{{\rho }^{2/3}}\frac{t}{\langle T_E\rangle }\right]$ [13], at $N=1$ and $N=3$, respectively. Here $C$ denotes the normalized factor, $J_2$ denotes the Bessel function, and $j_{2,i}$ denotes the $i\mathrm{th}$ zero point of $J_2$ and $\rho =1.1$.

Figure 4

Conditionally sampled second-order moments of relative separations. The initial separations exceed the proper initial separation $r_0^{\left(p\right)}$. (a) Conditional sampling for the hyperviscous ${\text{Re}}_{\alpha }=40$ case starting from various initial separations. (b) Conditional sampling for the hyperviscous ${\text{Re}}_{\alpha }=80$ starting from various initial separations. (c) Conditional sampling for the viscous ${\text{Re}}_{\alpha }=39$ case starting from various initial separations. Inset shows the logarithmic local slopes of the conditionally sampled $\langle r^2\left(t\right)\rangle$. Black solid curve shows the second-order moment of the relative separation without any conditional sampling starting from the proper initial separation, $r_0=r_0^{\left(p\right)}$. The gray dashed line denotes $t^3$ power law. The range between the two horizontal gray solid lines denotes the inertial range estimated with the exit time.

Figure 5

(a) Mean-squared separations for the hyperviscous case with ${\text{Re}}_{\alpha }=40$ (dashed line) and ${\text{Re}}_{\alpha }=80$ (solid line), although the initial separation is subtracted for various initial separations. This is compensated by the scaling $\varepsilon t^3$. Inset shows the same plots as the main panel although the horizontal axis is compensated by the Bitane timescale, $t_0=S_2\left(r_0\right)/2\varepsilon$. Gray line denotes the scaling law, $\langle \mathit{r}\left(t\right)-{\mathit{r}}_0\rangle =g\varepsilon t^3(1+C{t}_0/t)$ as suggested by Bitane et al. [18]. Here $C=0.6$. (b) Cubed-local slopes of $\langle r^2\left(t\right)\rangle$ for various initial separations at the hyperviscous ${\text{Re}}_{\alpha }=40$ (dashed line) and ${\text{Re}}_{\alpha }=80$ (solid line). In panels (a) and (b), the time is normalized with the Batchelor time, $t_{\mathrm{B}}=r_0^{2/3}{\langle \varepsilon \rangle }^{-1/3}$. In the inset of panel (a), the time is normalized with Bitane time, $t_0=S_2\left(r_0\right)/2\varepsilon$.

Figure 6

Second-order (left panels) and first-order (right panels) moments of the longitudinal relative velocity for conditionally sampled data starting from various initial separations and for the unconditioned data starting from the proper initial separation, $r_0^{\left(p\right)}$. Inset: Logarithmic local slope. (a), (b) ${\text{Re}}_{\alpha }=40$ with hyperviscosity, (c), (d) ${\text{Re}}_{\alpha }=80$ with hyperviscosity, (e), (f) ${\text{Re}}_{\alpha }=39$ with normal viscosity. Dashed line denotes $t^{1.23}$ and $t^{0.7}$ scalings for second- and first-order moments, respectively.

Figure 7

${\text{Re}}_{\alpha }$ dependence of the logarithmic local slope of (a) $\langle v_l^2\left(t\right)\rangle$ and (b) $\langle v_l\left(t\right)\rangle$ for unconditioned data at the proper initial separation, $r_0^{\left(p\right)}$ at ${\text{Re}}_{\alpha }=40,80$, and 160. Gray dashed lines correspond to (a) 1.23 and (b) 0.70, respectively.

Figure 8

(a) PDFs of the rescaled longitudinal relative velocity, $\zeta \left(t\right)=v_l\left(t\right)/t^{0.7}$ for unconditioned data at different instances $t/t_f=1.0,1.5,2.0,2.5,3.0$ for ${\text{Re}}_{\alpha }=80$ with hyperviscosity. Here the initial separation corresponds to $r_0=0.48l_f$, which is different from the proper initial separation. It should be noted that $\zeta$ is nondimensionalized as it is divided by $l_f/t_f^{1.7}$. Inset: PDFs without rescaling of the longitudinal relative velocity for the unconditioned data with the initial separation $r_0=0.48l_f$. (b) Same as (a) except for the conditionally sampled data. (c) Same as (a) except for ${\text{Re}}_{\alpha }=39$ with normal viscosity at different instances $t/t_f=0.9,1.4,1.8,2.3,2.7$. Here the initial separation corresponds to $r_0=0.39l_f$. (d) Same as (c) albeit for conditionally sampled data.

Figure 9

Evolution of PDFs of the rescaled longitudinal relative velocity, $\zeta \left(t\right)=v_l\left(t\right)/t^{0.7}$, of the unconditioned pairs starting from the proper initial separation, $r_0^{\left(p\right)}$. The PDFs are for (a) ${\text{Re}}_{\alpha }=40$, (b) ${\text{Re}}_{\alpha }=80$, and (c) ${\text{Re}}_{\alpha }=160$.

Figure 10

(a) Mean separation rate, ${\langle r\left(t\right)v_l\left(t\right)\rangle }_c$ for conditioned data at various initial separations and for unconditioned data at the proper initial separation, $r_0^{\left(p\right)}$ for ${\text{Re}}_{\alpha }=40$ with hyperviscosity. Dashed line denotes $t^2$ scaling. Inset: The logarithmic local slopes of the data shown in the outset. Dashed line denotes $t^2$ scaling. (b) Same as (a) but for ${\text{Re}}_{\alpha }=80$ with hyperviscosity.

Figure 11

(a) Second-order moments of the relative separations, $\langle r^2\left(t\right)\rangle$, starting from the proper initial separations in the quasi-steady simulations with resolution $N=1024,2048$, and 4096. Dashed lines denote $t^3$ scaling. Inset:the logarithmic local slope of $\langle r^2\left(t\right)\rangle$. (b) Same as (a) except for the second-order moments of the longitudinal relative velocity, $\langle v_l^2\left(t\right)\rangle$. Dashed line denotes $t^{1.2}$ scaling. Inset: the logarithmic local slope of $\langle v_l^2\left(t\right)\rangle$.