Figure 1

Measured second order spatial velocity structure function $D_2\left(r\right)$ for Reynolds numbers $R_{\lambda }=372$, 740, and 1115. For intermediate distances the desired $r^{2/3}$ scaling is observed. For smaller distances the structure function does not show the behavior given in Eq. (4) because our model is not able to correlate the velocities sufficiently at small distances. The insets show the higher order velocity structure functions rescaled with the large distance limits from Eq. (6) for $n=2,\cdots ,8$ (top to bottom) for $R_{\lambda }=740$. The increase of the exponents of the power laws is clearly visible.Reuse & Permissions

Figure 2

Measured spatial velocity correlation functions at $R_{\lambda }=740$. The velocity correlation $R_{\parallel }(r/\eta )$ is too small at small distances. First, this is due to the fact that our model is not able to perfectly correlate the directions of the fluid particle velocities for small $r$ as shown by $R_{\mathrm{d}}(r/\eta )$. Second, the Hamiltonian (7) does not correlate the magnitudes of the fluid velocities, which leads to $R_{\mathrm{a}}(r/\eta )$ being almost independent of $r$.Reuse & Permissions

Figure 3

Numerically measured exponents of the spatial velocity structure functions up to eighth order. For $n>3$ the measured exponents are smaller than the prediction of Kolmogorov's K41 theory. This result is confirmed by experiments [9, 20].Reuse & Permissions

Figure 4

Measured mean square displacement of fluid particles at $R_{\lambda }=372$, 740, and 1115 rescaled by the short time behavior $3{\sigma }_{\mathrm{u}}^2{t}^2$. Simulations without correlations (solid lines) show a clear ballistic ($\sim$$t^2$) regime at short times and a transition to a normal diffusion ($\sim$$t$) regime for long times. The latter is indicated by the solid black line. For higher Reynolds numbers this transition happens later since $T_{\mathrm{L}}$ is larger for higher $R_{\lambda }$. For the cases with correlations (dashed lines) there is no clear ballistic range visible for $R_{\lambda }=372$ and 740, and the transition to a diffusion regime happens much earlier than in the case without correlations. For $R_{\lambda }=1115$ the results are closer to the uncorrelated ones.Reuse & Permissions

Figure 5

Measured dispersion of pairs of fluid particles rescaled by $t^2$. The expected Richardson $t^3$ could not be observed. The vertical line indicates the large eddy lifetime $T_{\mathrm{E}}$, the arrow shows the range of $t_0$ in Eq. (21), and the transverse lines indicate normal diffusion with slope $-1$. We observe a large dependence on the initial separation between fluid particles ${\Delta }_0$ for higher $R_{\lambda }$. (a) Results for $R_{\lambda }=372$. Since $T_{\mathrm{E}}\sim t_0$, no Richardson scaling is observed. (b) Results for $R_{\lambda }=740$. For short $t$ a Batchelor $t^2$ scaling is observed. For $t$ in the range of $t_0$ the increase of the pair separation is slower than $t^2$. For the smallest initial separation ${\Delta }_0$ a quadratic scaling can be observed again for times larger than $T_{\mathrm{E}}$. (c) Results for $R_{\lambda }=1115$. Here for the smallest initial separation ${\Delta }_0$ an increase of the pair separation faster than $t^2$ can be found for $T_{\mathrm{E}}<t<1000t_{\eta }$, which may be an indication of Richardson scaling in this region.Reuse & Permissions

Figure 6

Numerically measured mean square displacement for real heavy particles at two different Reynolds numbers compared to the MSD of fluid particles. The main effect of the inertia of the real particles is a reduction of the mean particle velocity and a prolongation of the ballistic range at small $t$. (a) $R_{\lambda }=740$ and (b) $R_{\lambda }=1115$. For $r_{\mathrm{p}}={10}^{-5}\mathrm{m}$ the MSD of fluid and real particles are almost the same. For $r_{\mathrm{p}}={10}^{-4}\mathrm{m}$ an increase of the ballistic range at short times is visible. For long times the MSDs are again almost the same. For the largest particles at $R_{\lambda }=1115$ the ballistic range is increased further and the transition to normal diffusion happens later than for the fluid particles.Reuse & Permissions

Figure 7

Numerically measured MSS for different particle radii and Reynolds numbers. (a) $R_{\lambda }=740$, $r_{\mathrm{p}}={10}^{-5}$ m, (b) $R_{\lambda }=1115$, $r_{\mathrm{p}}={10}^{-5}$ m, (c) $R_{\lambda }=740$, $r_{\mathrm{p}}={10}^{-3}$ m, and (d) $R_{\lambda }=1115$, $r_{\mathrm{p}}={10}^{-3}$ m. For the smallest particle radius the curves are almost the same as for fluid particles. For larger particles the $t^2$ regime at short times is enlarged, and the mean particle velocities are reduced. For the largest particles also the long time behavior of the MSS is influenced.Reuse & Permissions