An algebraic preclosure theory for the Reynolds stress

Abstract

An algebraic preclosure theory for the Reynolds stress 〈u′u′〉 is developed based on a smoothing approximation which compares the space–time relaxation of a convective-diffusive Green’s function with the space–time relaxation of turbulent correlations. The formal preclosure theory relates the Reynolds stress to three distinct statistical properties of the flow: (1) a relaxation time τR associated with the temporal structure of the turbulence; (2) the spatial gradient of the mean field; and, (3) a prestress correlation related to fluctuations in the instantaneous Reynolds stress and the pressure field. Closure occurs by using an isotropic model for the prestress. For simple shear flows, the theory predicts the existence of a nonzero primary normal stress difference and an eddy viscosity coefficient which depends on the temporal relaxation of the turbulent structure and a characteristic time scale associated with the mean field. The asymptotic state of homogeneously sheared turbulence shows that τRS∼1, where S represents the mean shear rate. The Reynolds stress model and a set of recalibrated k−ε transport equations predict that the relaxation of homogeneously sheared turbulence to an asymptotic state requires development distances larger than 20 ×〈uz〉(0)/S, a theoretical result consistent with experimental observations.