Asymptotic results for backwards two-particle dispersion in a turbulent flow

We derive an exact equation governing two-particle backwards mean-squared dispersion for both deterministic and stochastic tracer particles in turbulent flows. For the deterministic trajectories, we probe the consequences of our formula for short times and arrive at approximate expressions for the mean-squared dispersion which involve second order structure functions of the velocity and acceleration fields. For the stochastic trajectories, we analytically compute an exact $t^3$ contribution to the squared separation of stochastic paths. We argue that this contribution appears also for deterministic paths at long times and present direct numerical simulation results for incompressible Navier-Stokes flows to support this claim. We also numerically compute the probability distribution of particle separations for the deterministic paths and the stochastic paths and show their strong self-similar nature.

Figure 1

Plots of the different terms of Eqs. (9) and (10). The dash-dotted line is proportional to ${\tau }^5$, and the dashed lines represent $\frac{4}{3}{\langle \varepsilon \rangle }_{\mathbf{x}}{\tau }^3$. The inset is a close-up view in semilogarithmic scale to compare the dispersion to $\mathcal{E}\left(\tau \right)$.

Figure 2

Particle dispersion $\langle |\mathbf{\rho }(\tau ;{\mathbf{r}}_f){{|}^2\rangle }_{{\mathbf{x}}_f}$ for eight different initial separations in pure DNS data compared to the dispersion for the stochastic advection model. The curves are compensated by $\frac{4}{3}{\langle \varepsilon \rangle }_{\mathbf{x}}{\tau }^3$. The inset shows the same pure DNS curves (without the stochastic case) rescaled by $\frac{{\tau }^4}{4}{S}_2^{\mathbf{a}}\left({\mathbf{r}}_f\right)$. The dashed line represents $\frac{{\tau }^4}{4}{S}_2^{\mathbf{a}}\left({\mathbf{r}}_f\right)$.

Figure 3

Pair separation PDF for eight different initial separations in pure DNS data compared to the dispersion for the stochastic advection model with similarity scaling. The straight dashed line is the Infinite Reynolds self-similar Richardson PDF.

Figure 4

Pair separation PDF for seven different times for the stochastic advection case with similarity scaling. The straight dashed line is the infinite Reynolds self-similar PDF.