On the realizability of reynolds stress turbulence closures

The realizability of Reynolds stress models in homogeneous turbulence is critically assessed from a theoretical standpoint. It is proven that a well known second-order closure model formulated using the strong realizability constraints of Schumann (1977) and Lumley (1978) is, in fact, not a realizable model. The problem arises from the failure to properly satisfy the necessary positive second time derivative constraint when a principal Reynolds stress vanishes-a flaw that becomes apparent when the nonanalytic terms in the model are made single-valued as required on physical grounds. More importantly, arguments are advanced which suggest that it is impossible to identically satisfy the strong from of realizability in any version of the present generation of second-order closures. On the other hand, models properly formulated to satisfy the weak form of realizability—wherein states of one or two component turbulence are made inaccessible in finite time via the imposition of a positive first derivative condition—are found to be realizable. However, unlike the simpler and more commonly used second-order closures, these models can be ill-behaved near the extreme limits of realizable turbulence due to the way that higher-degree nonlinearities are often unnecessarily introduced to satisfy realizability. Illustrative computations of homogeneous shear flow are presented to demonstrate these points which can have important implications for turbulence modeling.