Isotropic prestress theory for fully developed channel flows

The anisotropic distribution of turbulent kinetic energy in fully developed channel flows is examined by using an algebraic preclosure which relates the Reynolds stress to the mean field gradient and to a prestress correlation, (I̳+τ R ∇〈u_〉) T ⋅〈u_ ′ u_ ′ 〉⋅(I̳+τ R ∇〈u_〉)=τ R 2 〈f_ ′ f_ ′ 〉. Local fluctuations in the pressure field and in the instantaneous Reynolds stress are responsible for the prestress correlation τ R 2 〈f_ ′ f_ ′ 〉. Closure requires a phenomenological model for the anisotropic prestress 2kH̳, defined by 2kH̳≡τ R 2 〈f_ ′ f_ ′ 〉−2αI̳/3. The prestress coefficient α(=τ R 2 〈f_ ′ ⋅f_ ′ 〉/2) depends algebraically on the components of the Reynolds stress, the mean velocity gradient, the relaxation time τ R , and the turbulent kinetic energy k. Previously reported direct numerical simulations (DNS) results for fully developed channel flows (δ + =395) are used to evaluate the behavior of the Reynolds stress for an isotropic prestress (IPS) correlation (i.e., H̳=O̳ ). The IPS theory predicts the existence of a nonzero primary normal stress difference and shows that a significant transfer of kinetic energy occurs from the transverse and normal components of the Reynolds stress to the longitudinal component for τ R ‖∇〈u_〉‖≫1. The spatial distributions of the two nontrivial invariants of the anisotropic stress predicted by the IPS theory are consistent with DNS results for 10⩽y + ⩽395. The practical utility of the isotropic prestress theory is further demonstrated by predicting the low-order statistical properties of the turbulence in the outer region of fully developed channel flows. Transport equations for the turbulent kinetic energy and the turbulent dissipation are used to estimate the spatial distributions of the turbulent time scales k/ε and τ R .