Generalization of Taylor's formula to particles of arbitrary inertia

One of the cornerstones of turbulent dispersion is the celebrated Taylor's formula. This formula expresses the rate of transport (i.e., the eddy diffusivity) of a tracer as a time integral of the fluid velocity autocorrelation function evaluated along the fluid particle trajectories. Here, we review the hypotheses which permit us to extend Taylor's formula to particles of any inertia. The hypotheses are independent of the details of the inertial particle model. We also show by explicit calculation that the hypotheses encompass cases when memory terms such as Basset's and Faxén's corrections are taken into account in the modeling of inertial particle dynamics.

Figure 1

Sketch of a shear flow along the direction ${\mathit{e}}_1$ in a reference frame with angular velocity $\mathbf{\Omega }$ along ${\mathit{e}}_3$.