Irreversibility inversions in two-dimensional turbulence

In this paper, we consider a recent theoretical prediction [Bragg et al., Phys. Fluids 28, 013305 (2016)] that for inertial particles in two-dimensional (2D) turbulence, the nature of the irreversibility of the particle-pair dispersion inverts when the particle inertia exceeds a certain value. In particular, when the particle Stokes number, $\mathrm{St}$, is below a certain value, the forward-in-time (FIT) dispersion should be faster than the backward-in-time (BIT) dispersion, but for $\mathrm{St}$ above this value, this should invert so that BIT becomes faster than FIT dispersion. This nontrivial behavior arises because of the competition between two physically distinct irreversibility mechanisms that operate in different regimes of $\mathrm{St}$. In three-dimensional (3D) turbulence, both mechanisms act to produce faster BIT than FIT dispersion, but in 2D turbulence, the two mechanisms have opposite effects because of the flux of energy from the small to the large scales. We supplement the qualitative argument given by Bragg et al. [Phys. Fluids 28, 013305 (2016)] by deriving quantitative predictions of this effect in the short time limit. We confirm the theoretical predictions using results of inertial particle dispersion in a direct numerical simulation of 2D turbulence. A more general finding of this analysis is that in turbulent flows with an inverse energy flux, inertial particles may yet exhibit a net downscale flux of kinetic energy because of their nonlocal-in-time dynamics.

Figure 1

DNS results for $\mathcal{I}(\xi ,t)$ for various $\mathrm{St}$ and (a) $\xi /{\ell }_f=2$, (b) $\xi /{\ell }_f=5$, (c) $\xi /{\ell }_f=8$, and (d) $\xi /{\ell }_f=10$. Solid line denotes $\mathrm{St}=0$ results, and the horizontal line indicates $\mathcal{I}=0$.

Figure 2

DNS results for (a) ${\xi }^{-2}|\langle \parallel {\mathit{r}}^p{\left(t\right)\parallel }^2-\parallel {\mathit{r}}^p{(-t)\parallel }^2{\rangle }_{\xi }|$ and (b) ${\xi }^{-2}\langle \parallel {\mathit{r}}^p{\left(t\right)\parallel }^2-\parallel {\mathit{r}}^p(-t){{\parallel }^2\rangle }_{\xi }$ for various $\text{St}$ and $\xi /{\ell }_f=2$. Whereas plot (a) emphasizes the $t^3$ scaling, plot (b) reveals the change in sign of the short-time behavior of $\langle \parallel {\mathit{r}}^p{\left(t\right)\parallel }^2-\parallel {\mathit{r}}^p(-t){{\parallel }^2\rangle }_{\xi }$ as $\mathrm{St}$ is increased, signifying the irreversibility inversion.

Figure 3

DNS results for (a) $\mathcal{J}(\xi ,t)$ and (b) $\mathcal{J}(\xi ,-t)$ for various $\mathrm{St}$ and $\xi /{\ell }_f=2$.

Figure 4

DNS results for ${\mathcal{S}}_{\parallel }^p\left(r\right)$ for various $\mathrm{St}$. The continuous line is the skewness of the fluid velocity (i.e., $\mathrm{St}=0$), given for reference. Note the change in sign of ${\mathcal{S}}_{\parallel }^p\left(r\right)$ as $\mathrm{St}$ is increased.