Geometric evolution of the Reynolds stress tensor in three-dimensional turbulence

The dynamics of the Reynolds stress tensor is given by an evolution equation coupling geometrical effects and turbulent source terms. The effects of the mean flow geometry are shown up when the source terms are neglected. In this case, the Reynolds stress tensor is expressed as the sum of three tensor products of vector fields only associated with the mean flow. The vector fields are governed by differential equations similar to a distorted gyroscopic equation. Along the trajectories of mean flow, the fluctuations of velocity are determined by a differential equation whose coefficients only depend on the eigenvalues of the mean rate of deformation tensor.