Turbulent stress invariant analysis: Clarification of existing terminology

As a means of studying the structure of the turbulence, the time mean invariants, defined according to the theory of Lumley “Computational modelling turbulent flows,” Adv. Appl. Mech. 18, 123 (1978) has proven to be a useful and popular tool. According to the theory there is a domain, known as the anisotropy invariant map, within which all realizable Reynolds’s stress invariants must lie. The borders of this domain describe different states of the turbulent stress tensor. It has been found that there is some confusion in the terminology used when describing these states. The confusion is related to whether the notation is used to describe the shape of the stress tensor or the eddies of the turbulence. Choi and Lumley “The return of isotropy of homogeneous turbulence,” J. Fluid Mech. 436, 59 (2001) noted the same controversy in terminology, but since the confusion is still found to exist, it was thought timely to describe the fundamental relationships which relate the shape of the stress tensor and its invariants.