On modeling the Reynolds stresses in turbulent shear flows

The issue of modeling various turbulent shear flows by virtue of Reynolds stress and related models is addressed on the basis of three interrelated methodologies. First, the idea of algebraic Reynolds stress models is utilized for wall-bounded flows. Second, a solution to the modeled Reynolds stress transport equation is presented for three-dimensional flows yielding a more complex model also valid for non-equilibrium flows. Third, the rapid pressure-strain correlation model as one of the crucial terms in full Reynolds stress closures is modeled for homogeneous and non-homogeneous flows. A nonlinear stress-strain model derived as an equilibrium solution to the modeled Reynolds stress transport equation is modified to account for the near-wall effects in wall-bounded turbulent shear flows. The results based on the new model are compared with numerical and experimental data for channel flows and boundary layers. To include non-equilibrium effects, a higher order model only neglecting turbulent transport effects is developed for three-dimensional flows utilizing the Cayley-Hamilton theorem. The solution is cast in terms of five tensors involving the strain and vorticity field and is valid for the whole range of turbulent time scales. The five coefficients multiplying the tensors are determined by a set of nonlinear first order differential equations. Numerical solutions for various homogeneous flow fields are compared with existing nonlinear stress-strain models. Furthermore, tensor representation theory is utilized for the quadratic expansion of the two-point velocity correlation tensor in terms of the Reynolds stress tensor and a separation vector. This model allows the analytical integration of the Poisson equation for the fluctuating pressure and leads to a model for the rapid part of the pressure-strain correlation. The new methodology is developed for homogeneous flow situations yielding pressure-strain coefficients solely based on numerical two-point correlation data. To accommodate inhomogeneity effects a similarity hypothesis for the two-point correlation tensor is proposed. The new non-homogeneous formalism yields a dependence of the model closure on the gradient of the turbulent length and velocity scale and involves the second derivative of the mean velocity. The new model is compared with experimental and numerical data for the pressure-strain correlation tensor.