Self-similar two-particle separation model

We present a new stochastic model for relative two-particle separation in turbulence. Inspired by material line stretching, we suggest that a similar process also occurs beyond the viscous range, with time scaling according to the longitudinal second-order structure function S2(r), e.g.; in the inertial range as epsilon−1/3r2/3. Particle separation is modeled as a Gaussian process without invoking information of Eulerian acceleration statistics or of precise shapes of Eulerian velocity distribution functions. The time scale is a function of S2(r) and thus of the Lagrangian evolving separation. The model predictions agree with numerical and experimental results for various initial particle separations. We present model results for fixed time and fixed scale statistics. We find that for the Richardson-Obukhov law, i.e., =gepsilont3, to hold and to also be observed in experiments, high Reynolds numbers are necessary, i.e., Relambda>[script O](1000), and the integral scale needs to be large compared to initial separation, i.e., [script L]/r0>30 and d/[script L]>3 need to be fulfilled, where d is the size of the field of view. Removing the constraint of finite inertial range, the model is used to explore separation dynamics in the asymptotic regime. As Relambda-->[infinity], the distance neighbor function takes on a constant shape, almost as predicted by the Richardson diffusion equation. For the Richardson constant we obtain that g-->0.95 as Relambda-->[infinity]. This asymptotic limit is reached at Relambda>1000. For the Richardson constant g, the model predicts a ratio of gb/gf[approximate]1.9 between backwards and forwards dispersion. ©2007 American Institute of Physics