Three-dimensional turbulent relative dispersion by the Gledzer-Ohkitani-Yamada shell model

We study pair dispersion in a three-dimensional incompressible high Reynolds number turbulent flow generated by Fourier transforming the dynamics of the Gledzer-Ohkitani-Yamada (GOY) shell model into real space. We show that GOY shell model can successfully reproduce both the Batchelor and the Richardson-Obukhov regimes of turbulent relative dispersion. We also study how the crossover time scales with the initial separations of a particle pair and compare it to the prediction by Batchelor.

Figure 1

(Color online) Two particles being advected in a random force field generated by the GOY shell model.

Figure 2

(Color online) The initial separation $r_0$ is ${10}^{-4}$ and the data are averaged over $10000$ different simulations. One can distinctly note the presence of both the R-O law and the Batchelor’s scaling law. The crossover time $t_0$ is somewhere in the third decade on the $x$ axis. The deviation of the blue curve [marked $⟨{\left|\mathbf{r}\left(t\right)-{\mathbf{r}}_0\right|}^2⟩/t^3$] from R-O scaling at higher times is due to the fact that the pair separations are no longer in the inertial regime. All the units are in GOY-shell-model units.

Figure 3

(Color online) The crossover times $t_0$ plotted the initial pair separations $r_0$. The dashed line has the theoretical slope predicted by Batchelor. All the units are in GOY-shell-model units.