A unified analysis of planar homogeneous turbulence using single-point closure equations

A unified approach for assessing and characterizing both the non-equilibrium and equilibrium states of planar homogeneous flows is analysed within the framework of single-point turbulence closure equations. The underlying methodology is based on the replacement of the modelled evolution equation for the Reynolds stress anisotropy tensor by an equivalent set of three equations for characteristic scalar invariants or state variables. For stress anisotropy evolution equations which use modelled pressure-strain rate correlations that are quasi-linear, this equivalence then leads to an analytic solution for the time evolution of the Reynolds stress anisotropy. With this analysis, the transient system characteristics can be studied, including the dependence on initial states, the occurrence of limit-cycle behaviour, and the system global stability. In the fixed-point asymptotic limit, these results are consistent with and unify previous equilibrium studies, and provide additional information allowing the resolution of some questions that could not be answered in the framework of previous developments. A new result on constraints applicable to the development of realizable pressure-strain rate models is obtained from a re-examination of the stress anisotropy invariant map. With the analytic solution for the transient behaviour, some recent non-equilibrium models, which incorporate relaxation effects, are evaluated in a variety of homogeneous flows in inertial and non-inertial frames.