Generalized Langevin equation for relative turbulent dispersion

The relative velocity of two fluid particles in homogeneous and stationary turbulence is considered. Looking for reduced dynamics of turbulent dispersion, we apply the nonlinear Mori-Zwanzig projector method to the Navier-Stokes equations. The projector method decomposes the Lagrangian acceleration into a conditionally averaged part and a random force. The result is an exact generalized Langevin equation for the Lagrangian velocity differences accounting for the exact equation of the Eulerian probability density. From the generalized Langevin equation, we obtain a stochastic model of relative dispersion by stochastic estimation of conditional averages and by assuming the random force to be Gaussian white noise. This new approach to dispersion modelling generalizes and unifies stochastic models based on the well-mixed condition and the moments approximation. Furthermore, we incorporate viscous effects in a systematic way. At a moderate Reynolds number, the model agrees qualitatively with direct numerical simulations showing highly non-Gaussian separation and velocity statistics for particle pairs initially close together. Ar very large Reynolds numbers, the mean-square separation obeys a Richardson law with coefficient of the order of 0.1.