Abstract
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.
Highlights
The statistical nature of turbulence has been much studied regarding correlations of flow velocity or quantities relevant with them at one point or many points so far
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 of the Monin-Lundgren hierarchy to see its substantial effect on the dynamics of homogeneous isotropic turbulence, the pressure term in which is proved to vanish, as argued in “One-point velocity statistics in decaying homogeneous isotropic turbulence,” Phys
Summary
The statistical nature of turbulence has been much studied regarding correlations of flow velocity or quantities relevant with them at one point or many points so far. We propose to consider it in terms of characteristic function of velocity distribution, in homogeneous isotropic turbulence. It is meaningful to know a form of two-point velocity characteristic function because it must involve several-order velocity correlations at two points in turbulence; in particular it is expected to be the simplest when turbulence is homogeneous and isotropic. By consideration of homogeneity and isotropy of turbulence, the joint PDF must locationally depend on a scalar distance s alone, so that longitudinal and vertical velocity correlations are naturally written as, respectively, ρ1 (s) and ρ2 (s). It is interesting to see whether this form yield a new knowledge to the first equation of the Monin-Lundgren hierarchy which involves the pressure term expressed by the two-point velocity PDF. The closed second equation of the Monin-Lundgren hierarchy obtained can be a starting point to study homogeneous, isotropic turbulence
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