On modelling the Reynolds stress in the context of continuum mechanics

We explore an approach to the constitutive equations for the Reynolds stress, assuming that it is determined by the history of the mean deformation of turbulence but has a fading memory. By considering the first and the subsequent higher order approximations to the mean deformation of turbulence, we further assume that the Reynolds stress depends on the history of the mean relative deformation gradient of turbulence and its higher gradients up to some finite order. To exemplify this approach, a simple but nonlinear constitutive equation for the Reynolds stress is proposed with emphasis on some history effects of turbulence; its approximate form is shown to include as special cases a few previously proposed models, which however were developed based on different approaches and reasoning. ? 2003 Elsevier B.V. All rights reserved.EI05543-559