Numerical investigation on the partial return to isotropy of freely decaying homogeneous axisymmetric turbulence

Abstract

The return to isotropy of freely decaying homogeneous axisymmetric turbulence is numerically studied using the Eddy-Damped Quasi-Normal Markovian closure. The model, whose classical formulation has been extended to moderately anisotropic flows by Cambon, Jeandel, and Mathieu [“Spectral modelling of homogeneous non-isotropic turbulence,” J. Fluid Mech. 104, 247–262 (1981)], allows for an accurate description of the turbulence decay for very long times. More specifically, the observed results encompass both the high and low Reynolds number asymptotic regimes. Such an analysis escapes both wind tunnel experiments and direct numerical simulations possibilities at the present time. Anisotropy generation mechanisms considered in the present paper do not affect the nature of nonlinear interactions and the related energy cascade mechanisms. Their influence on the decay regime is quantified by the investigation of the features of the initial three dimensional kinetic energy spectrum. The initial anisotropy level is always chosen in agreement with experimental grid turbulence observations. The present results show that the relaxation towards an isotropic state is observed in the inertial range of the energy spectrum E(k, t) during the initial high Reynolds regime, while the large scales conserve the imposed anisotropy level if the permanence of large eddies hypothesis is verified. A direct consequence is that the ratio between axial and transverse kinetic energies γ increases when the low Reynolds asymptotic regime is reached. Saturation effects, i.e., effects related to the boundedness of the physical domain under consideration, are also investigated. Isotropic saturation (same cut off scale in every direction) leads to the relaxation towards a fully isotropic state in all cases, whereas anisotropic saturation (unbounded domain in the axial direction) leads to an amplification of the initial anisotropy.