Symmetries and the closure problem of turbulence

When applying Lie-group symmetry analysis systematically to turbulent flows, any arbitrary invariant scaling law for the mean velocity profiles can be generated consistent to any higher order in the velocity correlations. This problem of arbitrariness in invariant scaling persists even if we would formally consider the infinite statistical hierarchy of all multi-point correlation equations. The closure problem of turbulence simply cannot be circumvented by just employing the method of Lie-group symmetry analysis alone: as the statistical equations are unclosed, so are their symmetries! Hence, an a priori prediction as how turbulence scales is thus not possible. Only a posteriori by anticipating what to expect from numerical or experimental data the adequate invariant scaling law can be generated through an iterative trial-and-error process.