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We investigate the single-point probability density function of the velocityin three-dimensional stationary and decaying homogeneous isotropic turbulence.To this end we apply the statistical framework of the Lundgren-Monin-Novikovhierarchy combined with conditional averaging, identifying the quantities thatdetermine the shape of the probability density function. In this framework theconditional averages of the rate of energy dissipation, the velocity diffusionand the pressure gradient with respect to velocity play a key role. Directnumerical simulations of the Navier-Stokes equation are used to complement thetheoretical results and assess deviations from Gaussianity.

On the velocity distribution in homogeneous isotropic turbulence: correlations and deviations from Gaussianity