Frame-invariance of turbulence constitutive relations

The applicability of the principle of material frame-indifference to turbulence and its consequences for turbulence modelling are considered. By introducing a distinction between arbitrary time-dependent orthogonal transformations of (i) the fluid motion, and (ii) the observer, we find that frame-invariant tensorial measures of the absolute angular velocity and acceleration can be defined. This follows from consideration of a four-dimensional formulation of classical mechanics which formally possesses the property of invariance under arbitrary changes of frame of the observer. These tensors allow the construction of constitutive equations which depend, for example, on the absolute angular velocity of the fluid, and which also satisfy the principle of material frame-indifference. The results of this analysis have application outside the field of turbulence modelling. The invariance properties of some previously published Reynolds stress transport turbulence models are examined and modifications are suggested.