Reynolds number dependence of velocity gradient skewness in a renormalized scheme of fluid turbulence

We obtain the behavior of skewness of the longitudinal velocity gradient with respect to the Reynolds number of a turbulent fluid from the incompressible Navier–Stokes dynamics. This is achieved in a renormalized scheme incorporating a von Kàrmàn interpolation function between the production and inertial ranges together with an interpolation between the inertial and dissipation ranges. The corresponding second- and third-order renormalized moments of the longitudinal velocity gradient and the resulting skewness values are found to depend on the Reynolds number in nontrivial fashions. For intermediate and moderately high Reynolds numbers, the skewness value does not exhibit any scaling behavior with respect to the Reynolds number. Instead, the skewness value remains bounded for any Reynolds number and it approaches a constant value for infinitely high Reynolds numbers. In the same scheme, we also incorporate another interpolation function between the production and inertial ranges coming from energy balance equation and inputs from renormalization-group analysis. We find exact agreement between the asymptotic values of skewness at high Reynolds numbers while the skewness values are in the same neighborhood in the finite Reynolds number (non-universal) regime. (paper)