Realizable algebraic Reynolds stress closure

The normalized Reynolds (NR-) stress is a symmetric, non-negative, dyadic-valued operator. An analysis of the hydrodynamic equation governing velocity fluctuations of a constant property Newtonian fluid shows that this operator is related to a prestress operator that is also symmetric and non-negative. The prestress operator accounts for local spatial changes in the fluctuating pressure and in the fluctuating instantaneous Reynolds stress. The Cayley–Hamilton theorem from linear algebra is used to complete the closure with a non-negative mapping of the normalized Reynolds stress into the prestress. The non-negative mapping between the prestress operator and the Reynolds stress depends on a scalar-valued turbulent transport time related to the relaxation of a Green's function associated with a convective–viscous parabolic differential operator and the relaxation of a two-point, space–time correlation related to turbulent velocity fluctuations. The preclosure equation also depends on a kinematic operator related to the average velocity gradient and a rotational operator related to the angular velocity of the frame. The resulting universal realizable anisotropic prestress (URAPS-) closure is realizable for all non-rotating and rotating turbulent flows, provided the complementary transport equations for the turbulent kinetic energy and the turbulent dissipation are formulated to yield non-negative solutions. Experimental data and DNS results previously reported in the literature for non-rotating homogeneous simple shear and for non-rotating and rotating homogeneous decay are used to determine the closure constants. For rotating homogeneous simple shear, the URAPS-closure predicts the existence of self-similar states for finite positive and negative rotation numbers. The URAPS-closure for the NR-stress predicts anisotropic states consistent with expected behavior.