Titre/Title : Homogeneous Isotropic Turbulence: Is Kolmogorov’s theory necessary?

Contact : Sedat Tardu (équipe EDT)

Résumé/Abstract : The problem of homogeneous isotropic turbulence (HIT) is revisited within the analytical framework of the Navier-Stokes equations, with the view to assessing rigorously the consequences of the scale-invariance (an exact property of the Naviers-Stokes equations) for any Reynolds number. The analytical development, which is independent of the 1941 (K41) and 1962 (K62) theories of Kolmogorov for HIT for inﬁnitely large Reynolds number, is applied to the transport equations for the second and third order moments of the longitudinal velocity increment, (δu). Once the normalised equations and the constraints required for complying with the scale-invariance property of the equations are presented, results derived from these equations and constraints are discussed and compared with measurements. It is found that the ﬂuid viscosity, ν, and the mean kinetic energy dissipation rate, e, are the only scaling parameters that make the equations scale-invariant. The analysis further leads to expressions for the distributions of the skewness and the ﬂatness factor of (δu) and shows that these distributions, must exhibit plateaus (of diﬀerent magnitudes) in the dissipative and inertial ranges, as the Taylor microscale Reynolds number Reλ increases indeﬁnitely. Also, the skewness and ﬂatness factor of the longitudinal velocity derivative become constant as Reλ increases; this is supported by experimental data. Further, the analysis, backed up by experimental evidence, shows that, beyond the dissipative range, the behaviour of (δu)n with n = 2, 3 and 4 cannot be represented by a power law.