On the modelling of the return to isotropy of homogeneous non-isotropic turbulence

This paper is concerned with the modelling of the return to isotropy part of the pressure strain term in homogeneous anisotropic turbulent flows. Analytical solutions of the transport equations of the invariants of the anisotropy tensor as well as that of turbulent kinetic energy as a function of the natural time of decay are provided and discussed. Principal components of the Reynolds stresses are obtained from the solution of a cubic equation which involves the invariants. It is shown that current models based on Rotta's hypothesis are subject to a constraint which is only satisfied by axisymmetric homogeneous turbulence, and can be eliminated by non-linear modelling. A physical picture of energy transfer among the Reynolds stress components which takes into account the influence of the third invariant on the process of return to isotropy is presented.