Isotropy and homogeneity of the Taylor–Green vortex at late stages using different initial conditions

The Taylor–Green vortex (TGV) serves as a canonical benchmark for studying the transition from laminar to turbulent flow in the absence of solid boundaries. Despite its widespread use in turbulence model validation, the degree to which the TGV exhibits true isotropy and homogeneity, particularly at late stages of decay, remains insufficiently examined. This study employs high-order numerical simulations to investigate these properties for both the standard and isotropic variants of the TGV. Statistical measures, including Reynolds stress anisotropy, coherent structure functions, homogeneity indices and integral length scales, are used to assess flow behaviour over time. Results show that the standard TGV remains anisotropic and inhomogeneous even during late decay stages, with unequal longitudinal length scales and directionally dependent homogeneity indices. The isotropic TGV maintains isotropy by design but still deviates from the characteristics of ideal homogeneous isotropic turbulence, exhibiting larger transverse than longitudinal length scales. Both configurations reveal persistent spatial inhomogeneities manifested as fixed peaks in turbulent kinetic energy and the coherent structure function of plane-averaged statistics. The findings highlight that while the isotropic TGV provides a more balanced and symmetric configuration, neither flow achieves fully homogeneous isotropic turbulence.