Review about scale relativity applied to turbulence

This review summarizes the application of Laurent Nottale’s scale relativity (SRT) theory to hydrodynamic turbulence, a framework he has developed over four decades by rethinking physical laws under the principle of scale relativity. Initially aimed to derive quantum mechanics in fractal position space–time, SRT has more recently been extended to turbulent flows, with equations written in velocity space. This innovative approach enables SR to address long-standing issues in turbulence, such as non-Gaussian velocity distributions and intermittency, through macroscopic analogues of quantum mechanics. Specifically, SR has already provided theoretical insights into: (1) homogeneous, isotropic turbulence, where it predicts deviations consistent with empirical observations for accelerations; (2) rotating turbulence, where it accounts for rotation-induced patterns relevant to planetary and stellar flows; and (3) shear flows, such as turbulent jets, offering accurate predictions of key flow parameters, including turbulent intensity profiles and Reynolds stress distributions. Overall, SR opens new avenues for understanding turbulence across various contexts by providing a unified, non-classical framework in order to fulfill this purpose.