The boundary layer in the scale-relativity theory of turbulence

Abstract

We apply the scale-relativity theory of turbulence to the turbulent boundary layer problem. On the basis of Kolmogorov's scaling, the time derivative of the Navier–Stokes equations can be integrated under the form of a macroscopic Schrödinger equation acting in velocity-space. In this equation, the potential coming from pressure gradients takes the form of a quantum harmonic oscillator (QHO) in a universal way. From the properties of QHOs, we can then derive the possible values of the ratio of turbulent intensities in the shear flow, R=σu/σv=1.35±0.05. We show that the Karman constant is theoretically predicted to be κ=1/R3, in good agreement with its typical value κ≈0.4 and its observed possible variations. Then, we find a generic solution of our equations for the normal Reynolds stress pure profile, which closely fits the data from laboratory and numerical experiments. Its amplitude, μB, is the solution of an implicit equation that we solve numerically and analytically through power series, yielding to lowest order μB−1.35≈−2(R−1.35), plus smaller contributions from other parameters. Consequently, the correlation coefficient of velocities is given by ρ≈1/R μB2≈1/R3≈0.4 and is therefore equal to the Karman constant to lowest order, in agreement with its universally measured value ≈0.4 for all shear flows. We also find a general similarity between turbulent round jets and boundary layers in their outer region. These results therefore apply to a wide set of turbulent flows, including jets, plane boundary layers, and, to some extent, channels and pipes.