A theory of turbulence based on scale relativity

Abstract: The internal interactions of fluids occur at all scales therefore the resulting force fields have no reason to be smooth and differentiable. The release of the differentiability hypothesis has important mathematical consequences, like scale dependence and the use of a higher algebra. The law of mechanics transfers directly these properties to the velocity of fluid particles whose trajectories in velocity space become fractal and non-deterministic. The principle of relativity is used to find the form of the equation governing velocity in scale space. The solution of this equation contains a fractal and a non-fractal term. The fractal part is shown to be equivalent to the Lagrangian version of the Kolmogorov law of fully-developed and isotropic turbulence. It is therefore associated with turbulence, whereas the non-fractal deterministic term is associated with a laminar behavior. These terms are found to be balanced when the typical velocity reaches a level at which the Reynolds number is equal to one, in agreement with the empirical observations. The rate of energy dissipated by turbulence in a flow passing an obstacle that was only known from experiments can be derived theoretically from the equation's solution. Finally, a quantum-like equation in velocity space is proposed in order to find the probability of having a given velocity at a given location. It may eventually explain the presence of large scale coherent structures in geophysical turbulent flows, like jet-streams.