Reynolds stress under a change of frame of reference

In this paper, we study the characteristics of the Reynolds stress under a change of frame, as defined by the Euclidean group of transformation. We show that being subject to the dynamical processes induced from the mean Navier-Stokes equations, the invariance group of the fluctuating velocity and the Reynolds stress is no longer the Euclidean group of transformation, which is merely a kinematical aspect, but reduces to the extended Galilean group of transformation. As a consequence, in contrast to developing the constitutive equations for the Cauchy stress in continuum mechanics, wherein the principle of material frame-indifference is a guiding principle, the frame-dependent kinematical quantities, e.g., the mean spin tensor, may be allowed to play an effective role as the constitutive variable in turbulence modeling.