Do fluid particles separate exponentially in the dissipation range?

Abstract

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, the recent paper by Afik \& Steinberg (Nat. Commun. \textbf{8}: 468, 2017) argues 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 the particle separation over very long periods of time to see if we ever see 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, for the first time, convincing evidence that at sufficiently long times the moments do grow exponentially. However, this exponential growth is only observed after extremely long times $\gtrsim 200τ_η$, where $τ_η$ is the Kolmogorov timescale. 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 particle 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..