Characteristics of backward and forward two-particle relative dispersion in turbulence at different Reynolds numbers

A new algorithm based on post-processing of saved trajectories has been developed and applied to obtain well-sampled backward and forward relative dispersion statistics in stationary isotropic turbulence, over a range of initial separations ranging from Kolmogorov to energy-containing scales. Detailed results are obtained over a range of Taylor-scale Reynolds numbers, up to 1000, which is higher than in recent work in the literature. Backward dispersion is faster, especially at intermediate times after the ballistic range and before long-time diffusive behavior is reached. Richardson scaling has been demonstrated for the mean-squared separation, and forward and backward Richardson constants estimated to be gf = 0.55 and gb = 1.5, which are close to or comparable to other estimates. However, because of persistent dissipation sub-range effects no corresponding scaling was observed for higher order moments of the separation. Analysis of the separation probability density function showed only transitory agreement with the well-known Richardson prediction. The strong exponential growth of the separation on dissipation sub-range scales was analyzed in terms of a central limit theory approximation. The resulting predictions for the ratio of the growth rates of the third- and fourth-order moments are reasonably consistent with the theory. The backward growth rates, corresponding to the ratio of the magnitude of the smallest to largest Lyapunov exponents, are about 50% greater than the forward growth rates, somewhat higher than other estimates. The predicted asymmetry between backward and forward relative displacements at early times, manifested in a t3 variation of the difference in the backward and forward mean-square relative displacement, was confirmed numerically and explicitly traced to Eulerian properties at the small scales. However, this t3 growth is not simply connected to the t3 growth in the Richardson regime and the asymmetry manifested there by the difference in the backward and forward Richardson constants. Asymmetry in time for higher order moments was also explained using a Taylor-series analysis at early times.