  We use high resolution direct numerical simulations to study the anisotropic
contents of a turbulent, statistically homogeneous flow with random transitions
among multiple energy containing states. We decompose the velocity correlation
functions on different sectors of the three dimensional group of rotations,
SO(3), using a high-precision quadrature. Scaling properties of anisotropic
components of longitudinal and transverse velocity fluctuations are accurately
measured at changing Reynolds numbers. We show that independently of the
anisotropic content of the energy containing eddies, small-scale turbulent
fluctuations recover isotropy and universality faster than previously reported
in experimental and numerical studies. The discrepancies are ascribed to the
presence of highly anisotropic contributions that have either been neglected or
measured with less accuracy in the foregoing works. Furthermore, the anomalous
anisotropic scaling exponents are devoid of any sign of saturation with
increasing order. Our study paves the way to systematically assess persistence
of anisotropy in high Reynolds number flows.
