Rotation Of Trajectories In Lagrangian Stochastic Models Of Turbulent Dispersion

Abstract

We present a new measure for the rotation of Lagrangian trajectories in turbulence that simplifies and generalises that suggested by Wilson and Flesch ( Boundary-Layer Meteorol. 84, 411–426). The new measure is the cross product of the velocity and acceleration and is directly related to the area, rather than the angle, swept out by the velocity vector. It makes it possible to derive a simple but exact kinematic expression for the mean rotation <d s> of the velocity vector and to partition this expression into terms <d S> that are closed in terms of Eulerian velocity moments up to second order and unclosed terms. The unclosed terms <d s'> arise from the interaction of the fluctuating part of the velocity and the rate of change of the fluctuating velocity.

We examine the mean rotation of a class of Lagrangian stochastic models that are quadratic in velocity for Gaussian inhomogeneous turbulence. For some of these models, including that of Thomson ( J. Fluid Mech. 180, 113–153), the unclosed part of the mean rotation <d s'> vanishes identically, while for other models it is non-zero. Thus the mean rotation criterion clearly separates the class of models into two sets, but still does not provide a criterion for choosing a single model.

We also show that models for which <d s'> = 0 are independent of whether the model is derived on the assumption that total Lagrangian velocity is Markovian or whether the fluctuating part is Markovian.