The turbulent jet in the scale-relativity framework

Abstract

The turbulent jet plays an important role in fluid mechanics since it is a prototype of free turbulent flow which presents many unsolved problems, among which the absence of closure of the governing equations. Despite the chaotic nature of turbulence, it is characterized by the existence of many universal structures, in particular dimensionless invariants such as its opening angle or the correlation coefficient of velocities. After having given new exact solutions of the Reynolds Averaged Navier–Stokes equations (RANS), we suggest a solution to this puzzle in the scale-relativity framework. In this theory, the time derivative of the Navier–Stokes equations is integrated in terms of a macroscopic Schrödinger equation acting in velocity-space. This equation involves a constant ℏv which can be identified with the energy dissipation rate, while the pressure gradient manifests itself as a quantum harmonic oscillator (QHO) potential. The squared modulus of its solutions yields the probability density function (PDF) of velocities. The Reynolds stresses can then be derived from this PDF, so that the closure problem is solved in this case. This allows us to obtain a theoretical prediction for the turbulent intensity radial profile (and therefore for the pressure) which agrees with the experimental data. The ratio of axial over radial velocity fluctuations is found to be R=1.3–1.4 from QHO properties, in good agreement with its experimental values; we theoretically predict a jet opening angle α=1/2R3 accounting for its universal value ≈1/5; the mean ratio of turbulent intensity amplitudes over jet centerline axial velocity is predicted to be X=19/80, in good agreement with its universally measured value ≈1/4; from these parameters we derive possible values of the radial turbulent intensity amplitude μ=0.19–0.22 consistent with experiments; finally, we find a correlation coefficient of velocities ρ=1/R3=2 α, suggesting an explanation for the universal value ≈0.4 observed for the turbulent jet and for all free shear flows.