A new approach to decaying homogeneous isotropic turbulence. Reynolds-number dependence of the Kolmogorov constant (1941), skewness of velocity difference and the characteristic function of two-point velocity probability density

The non-dimensional prefactor of the Kolmogorov 2/3 law in the inertial range of homogeneous isotropic turbulence (HIT) has been believed to be a universal constant for many years, and then people have continued intensive search for evaluation of the constant by theory, experiment, and direct numerical simulation (DNS). Here a simple consideration is presented on this problem. First, a piecewise analytical form of longitudinal velocity autocorrelation ρ₁ is contrived such as to satisfy various physical conditions, so far known from theory and experiment. On this basis, it turns out that the constant must vary depending on Reynolds number. Next, by exactly solving the Karman-Howarth equation (for the ρ₁ given), the longitudinal third-order velocity correlation σ₁ is formulated for a decaying HIT. There it is verified that the Kolmogorov −4/5 law for third-order velocity structure function approximately holds near the origin for high Reynolds numbers inside the inertial range. Furthermore, the characteristic function of two-point velocity probability density function (PDF) in the Monin-Lundgren hierarchy, related to the already known one-point velocity PDF, is sketched towards the search for the fourth-order cumulants which must completely govern the intermittency in HIT. (author)