On the consistency of Reynolds stress turbulence closures with hydrodynamic stability theory

The consistency of second-order closure models with results from hydrodynamic stability theory is analyzed for the simplified case of homogeneous turbulence. In a recent study, Speziale, Gatski, and Mac Giolla Mhuiris [Phys. Fluids A 2, 1678 (1990)] showed that second-order closures are capable of yielding results that are consistent with linear stability theory for the case of homogeneous shear flow in a rotating frame. It is demonstrated in this paper that this success is due to the fact that the stability boundaries for rotating homogeneous shear flow are not dependent on the details of the spatial structure of the disturbances. For those instances where they are—such as in the case of elliptical flows where the instability mechanism is more subtle—the results are not so favorable. The origins and extent of this modeling problem are examined in detail along with a possible resolution based on Rapid Distortion Theory (RDT) and its implications for turbulence modeling.