Scale covariance and scale-ratio covariance in turbulent front propagation
Résumé
We address the implications of scale symmetries in out-of-equilibrium systems within the paradigm of turbulent front propagation. Usually, in this kind of issues, the attention is laid on the scale properties of objects, i.e. of their geometry, as for instance in fractal analysis. In contrast, the main objective here will be to infer the scale properties of laws, i.e. of the relationships between physical variables. This leads us to introduce the concept of scale covariance of laws which may include, but goes beyond, the scale similarity of objects. In particular, emphasizing the fact that turbulent front propagation refers not to isolated scales but to scale ranges, we put forward a scale symmetry different than the usual scale invariance: the scale-ratio invariance, based on the similarity with respect to changes of scale-ratios. Implementing the resulting constraints on turbulent front propagation provides the selection of a relationship that gives the effective front velocity. Both the form of this relation and it scale properties are validated from experiment. Interestingly, this scale symmetric law enables us to recover the construction of fronts in scale space from the euclidean regime to the fractal regime. This provides an explanation of the variation of the fractal dimensions of fronts with turbulence intensity and unifies in a single framework the euclidean and the fractal regimes which were previously thought to be intrinsically different.