A simple nonlinear model for the return to isotropy in turbulence

A quadratic nonlinear generalization of the linear Rotta model for the slow pressure-strain correlation of turbulence is developed for high Reynolds number flows. The model is shown to satisfy realizability and to give rise to no stable nonzero equilibrium solutions for the anisotropy tensor in the case of vanishing mean velocity gradients. In order for any model to predict a return to isotropy for all relaxational flows, it is necessary to ensure that there is no nonzero stable fixed point that attracts realizable initial conditions. Both the phase space dynamics and the temporal behavior of the model are examined and compared against experimental data for the return to isotropy problem. It is demonstrated that the quadratic model successfully captures the experimental trends which clearly exhibit nonlinear behavior. Comparisons are also made with the predictions of the linear Rotta model, the quasilinear Lumley model, and the nonlinear model of Shih, Mansour, and Moin. The simple quadratic model proposed in this study does better than the Rotta model as anticipated, and also compares quite favorably with the other more complicated nonlinear models.