A nem approach to characterizing the equilibrium states of the Reynolds stress anisotropy in homogeneous turbulence

Using representation theory, a system of three scalar invariant relationships for homogeneous turbulence is developed which is equivalent to the general algebraic relationship for the Reynolds stress anisotropy tensor at equilibrium. From these three relations, which account for any additional anisotropy effect, a single scalar, algebraic relationship between three state variables, the turbulent production-to-dissipation rate ratio, a scaled turbulent time scale, and a ratio of mean rotation rate and strain rate invariants, is derived. This provides a unique characterization for all planar homogeneous turbulent fleas. With this result, equilibrium can be predicted based solely on the form of the pressure-strain rate closure model, including both linear and nonlinear models. Consequences of this formulation to the description of two-dimensional inhomogeneous flows and their prediction are also deduced.