Time Evolution of Modeled Reynolds Stresses in Planar Homogeneous Flows

The analytic expression of the time evolution of the Reynolds stress anisotropy tensor in all planar homogeneous ows is obtained by exact integration of the modeled di erential Reynolds stress equations. The procedure is based on results of tensor representation theory, is applicable for general pressure-strain correlation tensors, and can account for any additional turbulence anisotropy e ects included in the closure. An explicit solution of the resulting system of scalar ordinary di erential equations is obtained for the case of a linear pressure-strain correlation tensor. The properties of this solution are discussed, and the dynamic behavior of the Reynolds stresses is studied, including limit cycles and sensitivity to initial anisotropies.