Extending the weak-equilibrium condition for algebraic Reynolds stress models to rotating and curved flows

Weis & Hutter (J. Fluid Mech. vol. 476, 2003, p. 63) obtained an implicit algebraic Reynolds stress model from a differential Reynolds stress transport equation valid in an arbitrarily rotating time-dependent coordinate frame (relative to an inertial system). Although the form of this implicit algebraic equation differed from previous implicit forms, its correctness was argued based on the objective tensor form of the implicit algebraic equation. It is shown here that such conclusions based on simple coordinate frame transformations are incomplete, and that additional considerations taking into account flow rotation and curvature, for example, are necessary. By properly accounting for both the arbitrary motions of the observer coordinate frame as well as the motion of the flow itself, it is shown that previous formulations and application of the weak-equilibrium condition are correct in contrast to the results of Weis Hutter.