Euclidean invariance and weak-equilibrium condition for the algebraic Reynolds stress model

Abstract

Taking into account the frame-invariance of a model expression under arbitrarily rotating transformations, Weis & Hutter (J. Fluid Mech. vol. 476, 2003, p. 63) proposed a Euclidean-objective weak-equilibrium condition for the algebraic Reynolds stress model (ARSM). However, Gatski & Wallin (J. Fluid Mech. vol. 518, 2004, p. 147) pointed out that the weak-equilibrium condition proposed is not correct in actual rotating flows such as a rotating channel flow and showed that a non-objective weak-equilibrium condition extended to curved and rotating flows should be assumed. The frame-invariance is an important issue not only for the ARSM but also for general nonlinear eddy-viscosity models. By introducing the corotational derivative of the Reynolds stress, the transport equation for the Reynolds stress can be written to be frame-invariant. It is shown that a frame-invariant expression is desirable as a general model by comparing the error of model expressions in different rotating frames. The extended weak-equilibrium condition of Gatski & Wallin is examined to show that it is in reality objective and it does not contradict a frame-invariant model expression for the Reynolds stress.