Typical form of characteristic function of two-point velocity distribution in homogeneous isotropic turbulence and its extension to the three-point version

A necessary and sufficient form of two-point velocity characteristic function to embody two-point velocity distribution in turbulence is constructed on the mathematical basis of homogeneity and isotropy. This is applied in the first equation (for one-point velocity probability density) of the Monin-Lundgren hierarchy to see its substantial effect on the dynamics of homogeneous isotropic turbulence, the pressure term in which then is proved to vanish, as argued in “One-point velocity statistics in decaying homogeneous isotropic turbulence,” Phys. Rev. E 78, 066312 (2008). Furthermore, an approximate form of three-point velocity characteristic function is searched on this basis, so that we obtain a simple closed hierarchy at the second equation stage. Thereby a certain closure method for the hierarchy in homogeneous, isotropic turbulence is illuminated from a new point of view