Scalings and Relative Scalings in the Navier-Stokes Turbulence

High-resolution direct numerical simulations of 3D Navier-Stokes turbulence with normal viscosity and hyperviscosity are carried out. It is found that the inertial-range statistics, both the scalings and the probability density functions, are independent of the dissipation mechanism, while the near-dissipation-range fluctuations show significant structural differences. Nevertheless, the relative scalings expressing the dependence of the moments at different orders are universal, and show unambiguous departure from the Kolmogorov 1941 description, including the $2/3$ law for the kinetic energy. Implications for numerical modeling of turbulence are discussed.