Fluid particles only separate exponentially in the dissipation range of turbulence after extremely long times

In this paper, we consider how the statistical moments of the separation between two fluid particles grow with time when their separation lies in the dissipation range of turbulence. In this range, the fluid velocity field varies smoothly and the relative velocity of two fluid particles depends linearly upon their separation. While this may suggest that the rate at which fluid particles separate is exponential in time, this is not guaranteed because the strain rate governing their separation is a strongly fluctuating quantity in turbulence. Indeed, Afik and Steinberg [Nat. Commun. 8, 468 (2017)] argue that there is no convincing evidence that the moments of the separation between fluid particles grow exponentially with time in the dissipation range of turbulence. Motivated by this, we use direct numerical simulations (DNS) to compute the moments of particle separation over very long periods of time in a statistically stationary, isotropic turbulent flow to see if we ever observe evidence for exponential separation. Our results show that if the initial separation between the particles is infinitesimal, the moments of the particle separation first grow as power laws in time, but we then observe convincing evidence that at sufficiently long times the moments do grow exponentially. However, this exponential growth is only observed after extremely long times $\gtrsim 200{\tau }_{\eta }$, where ${\tau }_{\eta }$ is the Kolmogorov time scale. This is due to fluctuations in the strain rate about its mean value measured along the particle trajectories, the effect of which on the moments of the particle separation persists for very long times. We also consider the backward-in-time (BIT) moments of the article separation, and observe that they too grow exponentially in the long-time regime. However, a dramatic consequence of the exponential separation is that at long times the difference between the rate of the particle separation forward in time (FIT) and BIT grows exponentially in time, leading to incredibly strong irreversibility in the dispersion. This is in striking contrast to the irreversibility of their relative dispersion in the inertial range, where the difference between FIT and BIT is constant in time according to Richardson's phenomenology.

Figure 1

DNS data for ${\langle r^2\left(t\right)\rangle }_{\xi }-{\xi }^2$ using two different kinds of initial conditions, focusing on the short-time behavior.

Figure 2

DNS data for (a) ${\mathcal{Q}}_2\left(t\right)$, (b) ${\mathcal{Q}}_4\left(t\right)$, (c) ${\mathcal{Q}}_6\left(t\right)$, and (d) ${\mathcal{Q}}_8\left(t\right)$, where ${\mathcal{Q}}_N\left(t\right)=({\tau }_{\eta }/t)ln\left[{\xi }^{-N}{\langle r^N\left(t\right)\rangle }_{\xi }\right]$. The BIT results correspond to ${\mathcal{Q}}_N(-t)=({\tau }_{\eta }/t)ln\left[{\xi }^{-N}{\langle r^N(-t)\rangle }_{\xi }\right]$.

Figure 3

DNS data for (a) ${\mathcal{Q}}_4\left(t\right)/{\mathcal{Q}}_2\left(t\right)$, (b) ${\mathcal{Q}}_6\left(t\right)/{\mathcal{Q}}_2\left(t\right)$, and (c) ${\mathcal{Q}}_8\left(t\right)/{\mathcal{Q}}_6\left(t\right)$, where ${\mathcal{Q}}_N\left(t\right)=({\tau }_{\eta }/t)ln\left[{\xi }^{-N}{\langle r^N\left(t\right)\rangle }_{\xi }\right]$. The BIT results correspond to ${\mathcal{Q}}_N(-t)=({\tau }_{\eta }/t)ln\left[{\xi }^{-N}{\langle r^N(-t)\rangle }_{\xi }\right]$.

Figure 4

DNS data for ${\langle r^2(-t)\rangle }_{\xi }/{\langle r^2\left(t\right)\rangle }_{\xi }$.

Figure 5

DNS data for ${\tau }_{\eta }{\langle \mathcal{S}\left(t\right)\rangle }_{\xi }$, with plot (a) emphasizing the behavior at short times. For the BIT data, we plot $-{\tau }_{\eta }{\langle \mathcal{S}(-t)\rangle }_{\xi }$ since ${\langle \mathcal{S}(-t)\rangle }_{\xi }\le 0$.

Figure 6

DNS data for $\sigma /{\mathcal{M}}_1$, where $\sigma \equiv \sqrt{({\mathcal{M}}_2-{\mathcal{M}}_1^2)}$. The BIT results are $-\sigma /{\mathcal{M}}_1$ since ${\mathcal{M}}_1<1$ BIT.

Figure 7

DNS data for the PDFs (a) $\mathcal{P}(\gamma ,\xi ,t)$ and (b) $\mathcal{P}(\gamma ,\xi ,-t)$, plotted in standard form.