A Lagrangian stochastic model for the trajectories of particle pairs and its application to the prediction of concentration variance within plant canopies

Abstract

A simple Lagrangian stochastic model for the trajectories of particle pairs in high Reynolds-number turbulent flows is presented. In this model, the velocities of particle pairs are initially correlated but subsequently each particle moves independently. The independent single-particle trajectories are simulated using Thomson's model [J. Fluid Mech. 180, 529–556, 1987]. This two-particle model exactly satisfies the well-mixed condition for Gaussian turbulence when length scales, characterizing the two-point Eulerian velocity correlation function, vanish. Temperature variances, due to heat released as a passive scalar from an elevated plane source, within a model plant canopy (Coppin et al. Boundary Layer Meteorol. 35, 167–191, 1986) are shown to be well predicted by the model. It is suggested that for strongly inhomogeneous flows, the two-point Eulerian velocity function is of secondary importance in determining the simulated trajectories of particle pairs compared to the importance of ensuring satisfaction of the two-to-one constraint (Borgas and Sawford. J. Fluid Mech. 279, 69–99, 1994); i.e ensuring that one-particle statistics obtained from the two-particle model are the same as those obtained from the corresponding one-particle model. Limitations of this modelling approach are discussed.