Exit-time statistics and the inference of Richardson scaling in numerical simulations of turbulent dispersion.

Unambiguous observation of Richardson inertial-range behavior for particle-pair dispersion and the associated scaling constant ($g$) in turbulence is often difficult in both experiment and computation, because of limitations in Reynolds number, effects of initial separation, and other factors. The concept of exit-time (time taken for the distance $l$ between two fluid particles to increase by a prescribed factor) has attracted recent interest in dispersion statistics viewed as functions of instantaneous length scale instead of time of travel. We consider both mean-squared dispersion and mean exit time, obtained from direct numerical simulations of forced isotropic turbulence up to Taylor-scale Reynolds numbers of about 650 on a $2048^3$ grid. Moments of exit time at fixed thresholds of particle pair separation are computed, and estimates of $g$ are inferred by assuming that the probability density function of $l$ follow self-similar forms such as that predicted by Richardson (1926). Subject to uncertainties due to temporal variability of space-averaged dissipation rate in the simulations, the present analyses suggest a trend towards $g$ in the range 0.4--0.6. However, high-Reynolds-number simulations longer than recently reported in the literature are needed.