NSF/GOALI: Validation of a New Algebraic Reynolds Stress Model for Statistically Stationary Flows

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AbstractCTS-0083229C. Petty, Michigan State UniversityThe ability to predict the low-order statistical properties of turbulent flows is an important aspect of modern engineering design. The exact equations governing the instantaneous pressure and velocity fields of constant density, Newtonian fluids are known. However, the Reynolds averaged Navier-Stokes (RANS-) equation for the mean velocity is statistically unclosed. This closure problem has been an intense area of research for more that fifty years.Current commercial CFD codes offer closure models of varying complexity. Although the Boussinesq approximation for the Reynolds stress limits the utility of the k-e theory, engineers attempting to understand the flow behavior within complex spatial domains nevertheless apply this theory. Unfortunately, for many practical flows with streamline curvature, the use of a linear gradient type model for the Reynolds stress may produce qualitatively results. Significant efforts has been expended to address the short comings of the k-e theory by developing closure models for statistical correlation's that appear in the unclosed second-order moment equation for the Reynolds stress. The approach is less appealing for design in astute as six additional non-linear partial differential equations must be solved in addition to the vector-valued RANS equation, the scalar-valued e-equation, and the continuity equation. LES and DNS methods are limited to benchmark flows for model calibration. Unfortunately, computational demands associated with LES and DNS prohibit the use of these methods for routine engineering design calculations in complex geometries.With this research a spatial smoothing approximation is used to derive an algebraic preclosure representation for the Reynolds stress in terns of a prestress correlation. This novel theoretical approach stems directly from the exact dynamical equation governing velocity fluctuations. The prestress, which is a self-correlation of an effective fluctuating force per unit mass of fluid, stems from the divergence of fluctuations in the instantaneous Reynolds stress and pressure fluctuations. The working hypothesis for the research is that the prestress correlation, which depends explicitly on fourth order statistical quantities, is more amenable to a universal phenomenological closure that the Reynolds stresses itself.The resulting new algebraic Reynolds stress model will open up new opportunities for engineering design. An isotropic prestress (IPS-) model and algebraic prestress (AAPS-) model will be validated for flows with strong streamline curvature and integrated into a commercial software code, Fluent. The new closure model will be tested and optimized for a class of statistically stationary benchmark flows with streamline curvature. The IPS- and AAPS- models will be used to describe the internal flow structures encountered in hydrocyclone separators, industrial scale mixers, and swirl combustors.