2. Theoretical Principles and Galilean Invariance

This chapter focuses on those theoretical principles which are required to formulate physically correct mathematical closure equations for modelling turbulent flows. The importance of the Galilean invariance in the Newtonian physics is to ensure that the conservation laws of turbulent flow motions remain the same in any two reference frames. Therefore, we devote a particular attention to the Galilean transformation and the derivation of the Galilean invariance of the Reynolds momentum equation (1.​43), the Reynolds stress tensor (1.​54), the rate-of-strain tensor (1.​114) and the generalised Boussinesq hypothesis on the Reynolds stresses (1.​113). The principle of Galilean invariance for the Reynolds stress tensor will also be taken into account in the proposal to the new hypothesis on the anisotropic Reynolds stress tensor in Chap. 5. In addition to the Galilean invariance, the consistency of physical dimensions, the coordinate system independence of physical laws and the realisability condition have also been considered as relevant criteria in the mathematical description of the Reynonds stress tensor. The derivations included in the present chapter make an attempt to bring closer a theoretically demanding advanced subject to a wider audience.