Turbulence and scale relativity

We develop a new formalism for the study of turbulence using the scale relativity framework (applied in v-space, following de Montera’s proposal). We first review some of the various ingredients which are at the heart of the scale relativity approach (scale dependence and fractality, chaotic paths, irreversibility) and recall that they indeed characterize fully developed turbulent flows. Then, we show that, in this framework, the time derivative of the Navier-Stokes equation can be transformed into a macroscopic Schrodinger-like equation. The local velocity Probability Distribution Function (PDF), Pv(v), is given by the squared modulus of a solution of this equation. This implies the presence of null minima Pv(vi) ≈ 0 in this PDF. We also predict a new acceleration component, Aq(v)=±Dv ∂v⁡lnPv, which is divergent in these minima. Then, we check these theoretical predictions by data analysis of available turbulence experiments: (1) Empty zones are in effect detected in observed Lagrangian velocity PDFs. (2) A direct proof of the existence of the new acceleration component is obtained by identifying it in the data of a laboratory turbulence experiment. (3) It precisely accounts for the intermittent bursts of the acceleration observed in experiments, separated by calm zones which correspond to Aq ≈ 0 and are shown to remain perfectly Gaussian. (4) Moreover, the shape of the acceleration PDF can be analytically predicted from Aq, and this theoretical PDF precisely fits the experimental data, including the large tails. (5) Finally, numerical simulations of this new process allow us to recover the observed autocorrelation functions of acceleration magnitude and the exponents of structure functions.We develop a new formalism for the study of turbulence using the scale relativity framework (applied in v-space, following de Montera’s proposal). We first review some of the various ingredients which are at the heart of the scale relativity approach (scale dependence and fractality, chaotic paths, irreversibility) and recall that they indeed characterize fully developed turbulent flows. Then, we show that, in this framework, the time derivative of the Navier-Stokes equation can be transformed into a macroscopic Schrodinger-like equation. The local velocity Probability Distribution Function (PDF), Pv(v), is given by the squared modulus of a solution of this equation. This implies the presence of null minima Pv(vi) ≈ 0 in this PDF. We also predict a new acceleration component, Aq(v)=±Dv ∂v⁡lnPv, which is divergent in these minima. Then, we check these theoretical predictions by data analysis of available turbulence experiments: (1) Empty zones are in effect detected in observed Lagrangian velocity PDFs. (2...