Direct numerical simulations of the instantaneous Navier-Stokes equation for constant property fluids as well as direct experimental evidence clearly show that the Reynolds stress cannot be represented as an algebraic dyadic-valued function of the mean strain rate, as suggested by Boussinesq more than 100 years ago. However, the governing equation for turbulent fluctuations does show that under certain conditions the normalized Reynolds stress can be related to a non-negative prestress operator by an algebraic preclosure equation (Parks et al., 1998).

A self-consistent argument supports the idea that the normalized prestress can be expressed as an explicit algebraic function of the normalized Reynolds stress. This hypothesis together with the preclosure equation yields a non-linear algebraic equation for the Reynolds stress, which can be solved by successive substitution.  In this presentation, the efficacy of the new closure for the Reynolds stress will be illustrated for a class of benchmark flows in inertial and non-inertial frames. The results will be compared with a realizable algebraic closure previously developed by Shih et al., 1994.  

Parks, S.M., K. Weispfennig, and C.A. Petty, 1998, “An Algebraic Preclosure Theory for the Reynolds Stress”, Phys. Fluids, 10(3), 645-653.

Shih T.H., J. Zhu and J.L. Lumley, 1994, “A New Reynolds Stress Algebraic Equation Model”, NASA TM-106644.