Further assessments of a “return to equilibrium” anisotropy model for non-equilibrium Reynolds stress closures

The “return to equilibrium” closure to the Reynolds-averaged Navier–Stokes equations is a model for the Reynolds stress anisotropy bij, proposed to enable more accurate predictions of flows with temporal and spatial unsteadiness. It is based on the premise that the response of turbulence to imposed rates of fluid deformation can be bounded by the extremes of equilibrium behavior when dbij/dt=0 and rapid distortion behavior when |dbij/dt|≫0, with a model bij transport equation to blend the response between these extremes. It was shown to describe quite accurately the behavior of the primary Reynolds stresses in homogeneous turbulent flows undergoing time-dependent shear and plane strains [G. J. Brereton, Phys. Fluids 36, 035121 (2024)]. In this paper, the model bij transport equation is derived for both primary and secondary Reynolds stresses, for homogeneous turbulence in any two-dimensional, time-dependent mean flow, in a stationary or rotating reference frame. Model predictions for the Reynolds stresses were in good agreement with simulation data for homogeneous turbulence undergoing a step change in shear in a reference frame rotating at different rates, and oscillatory shear in an inertial frame.