A simplified Reynolds-stress model and verification and validation in comprehensive two-dimensional and three-dimensional cases

Reynolds-stress model (RSM) is considered a more advanced turbulence closure due to its superiority in predicting the anisotropic distributions of turbulent kinetic energy and their evolutionary history, compared to eddy viscosity models (EVMs). However, the large number of transport equations and their strong coupling relationships in RSMs result in relatively low computational efficiency and numerical robustness, which limit the widespread application of RSMs. This paper develops a simplified RSM (semi-differential RSM, Semi-RSM) based on the moving equilibrium relation. The simplification is implemented by preserving three transport equations of Reynolds normal stress and an ω-scale equation originating from the hybrid Speziale–Sarkar–Gatski/Launder–Reece–Rodi (SSG/LRR) model and solving the three components of the shear stress tensor through algebraic relations. To comprehensively evaluate the characteristics of the Semi-RSM, systematic verification and validation are conducted. The verification process involves a two-dimensional turbulent flat plate and a three-dimensional (3D) bump in the channel case. The relevant results confirm that Semi-RSM exhibits favorable grid convergence characteristics. The validation is conducted on periodic hill, wall-mounted hump, curvature boundary layer, 3D supersonic square duct, and common research model wing–body configuration from the National Aeronautics and Space Administration, all of which are challenging for traditional EVMs. The Semi-RSM delivers results comparable to those of the SSG/LRR-ω RSM at a lower computational cost. Especially in streamwise separation cases, it effectively eliminates the unphysical streamline backbending problem near reattachment points observed in traditional RSMs.