Lower-dimensional manifold (algebraic) representation of Reynolds stress closure equations

For complex turbulent flows, Reynolds stress closure modeling (RSCM) is the lowest level at which models can be developed with some fidelity to the governing Navier-Stokes equations. Citing computational burden, researchers have long sought to reduce the seven-equation RSCM to the so-called algebraic Reynolds stress model which involves solving only two evolution equations for turbulent kinetic energy and dissipation. In the past, reduction has been accomplished successfully in the weak-equilibrium limit of turbulence. In non-equilibrium turbulence, attempts at reduction have lacked mathematical rigor and have been based on ad hoc hypotheses resulting in less than adequate models. In this work we undertake a formal (numerical) examination of the dynamical system of equations that constitute the Reynolds stress closure model to investigate the following questions. (i) When does the RSCM equation system formally permit reduced representation? (ii) What is the dimensionality (number of independent variables) of the permitted reduced system? (iii) How can one derive the reduced system (algebraic Reynolds stress model) from the full RSCM equations? Our analysis reveals that a lower-dimensional representation of the RSCM equations is possible not only in the equilibrium limit, but also in the slow-manifold stage of non-equilibrium turbulence. The degree of reduction depends on the type of mean-flow deformation and state of turbulence. We further develop two novel methods for deriving algebraic Reynolds stress models from RSCM equations in non-equilibrium turbulence. The present work is expected to play an important role in bringing much of the sophistication of the RSCM into the realm of two-equation algebraic Reynolds stress models. Another objective of this work is to place the other algebraic stress modeling efforts in the lower-dimensional modeling context.