Form Invariance and Frame Indifference of Closure Relations in LES

In this paper the principle of Turbulent Frame Indifference is revised. It is demonstrated that not all the turbulent closure relations in LES must fulfil the principle of Turbulent Frame Indifference: in the turbulent closure relations, the modelled expressions of an unknown objective tensor must be formulated in terms of objective tensors (allowing the closure relations to fulfil the requirement of form invariance) and must retain the same dependence (on the angular velocity of the frame) of the unknown tensor. It is demonstrated that, since the generalized SGS turbulent stress tensor is objective and frame independent, closure relations for this tensor must fulfil the principle of Turbulent Frame Indifference. A new closure relation for the generalized SGS turbulent stress tensor is proposed. The proposed closure relation complies with the principle of Turbulent Frame Indifference. In the proposed model the generalized SGS turbulent stress tensor is related exclusively to the generalized SGS turbulent kinetic energy (which is calculated by means of its balance equation) and the modified Leonard tensor. The viscous dissipation  of the generalized SGS turbulent kinetic energy is calculated by solving the balance equation of . It is demonstrated that the balance equation of the viscous dissipation is form-invariant but frame-dependent under Euclidean transformations of the frame; the closure relations proposed in this paper allow the modeled balance equation of  to respect the properties of form-invariance and frame-dependence of the exact equation.