AN ALGEBRAIC PRECLOSURE THEORY FOR THE REYNOLDS STRESS

An algebraic preclosure theory for the Reynolds stress [u'u'] is devel oped based on a smoothing approximation which compares the space-time relaxation of a convective-diffusive Green's function with the space-t ime relaxation of turbulent correlations. The formal preclosure theory relates the Reynolds stress to three distinct statistical properties of the flow: (1) a relaxation time tau(R) associated with the temporal structure of the turbulence; (2) the spatial gradient of the mean fie ld; and, (3) a prestress correlation related to fluctuations in the in stantaneous Reynolds stress and the pressure field. Closure occurs by using an isotropic model for the prestress. For simple shear flows, th e theory predicts the existence of a nonzero primary normal stress dif ference and an eddy viscosity coefficient which depends on the tempora l relaxation of the turbulent structure and a characteristic time scal e associated with the mean field. The asymptotic state of homogeneousl y sheared turbulence shows that tau(R)S similar to 1, where S represen ts the mean shear rate. The Reynolds stress model and a set of recalib rated k-epsilon transport equations predict that the relaxation of hom ogeneously sheared turbulence to an asymptotic state requires developm ent distances larger than 20 x[u(z)](O)/S, a theoretical result consis tent with experimental observations. (C) 1998 American Institute of Ph ysics.