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

Abstract

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 consequences of our formula for short time 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 (DNS) results for incompressible NavierStokes 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.The dispersion of fluid particles backwards-in-time is critical for understanding the mixing properties of turbulent flows [1]. Mixing is a process by which scalar or vector quantities such as dye concentration, temperature or magnetic fields, are transported due to the action of an underlying flow [2,3]. These fields are stretched and contorted, often violently, and if they have intrinsic diffusive properties (as most naturally occurring systems do), they are dissipated. If the substance is non-diffusive, its value at a point is the initial value at the starting point of the backwards deterministic trajectory ending there. The separation of these trajectories ending at nearby points is the principle subject of study for backward deterministic particle dispersion. For diffusive substances, initially separate bits of material can be brought together by the action of the turbulent flow and molecular effects. Here an averaging process over backwards stochastic trajectories recovers the value at the field at any point. The reversal of the process described above -coalescence of forward perturbed trajectories -is known as backwards stochastic particle dispersion. Backwards particle dispersion is a key feature to understanding many physical problems ranging from hydrodynamic turbulence [4], passive scalar transport and dissipative anomalies [5][6][7] It has been argued that in a turbulent flow, a pair of tracer particles separate proportionally to t 3 on average [19]. This prediction is now known as the RichardsonObukhov scaling law [20] and it remains a subject of controversy and inquiry. Experimental and numerical studies of two-particle dispersion are challenging as they require setting wide range separation between the dissipation scale and the integral scale of the flow [1,21]. Backward dispersion is particularly difficult because it involves integrating the Navier-Stokes equation forward in time and tracking the tracers backward in time. Most of the work on deterministic particle dispersion as therefore been in the forward-in-time setting. There the t 3 scaling has been numerically and experimentally thoroughly investig...

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