On relating Eulerian and Lagrangian velocity statistics: single particles in homogeneous flows

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Abstract
Various theories seeking to relate the velocity statistics of Lagrangian particles to the statistics of the Eulerian flow in which they are embedded are examined. Mean particle drift, mean-square particle velocity and the frequency spectrum of velocity are examined for stationary, homogeneous and joint-normally distributed Eulerian fields. Predictions based on a third-order weak-interaction expansion, the successive approximation procedure of Phythian (1975), the quasi-normal approximation of Saffman (1969), the parametrized model of Saffman (1962), and a new procedure based on a statistical estimator of the kinematic equation are compared with simulations of particle motion in one-dimensional flow. Only the statistical estimator produces both acceptable mean-drift and frequency-spectrum predictions.