On distributions of velocity random fields in turbulent flows

The present paper aims to derive a partial differential equation (PDE) for the single-time single-point probability density function (PDF) of the velocity field of a turbulent flow. The PDF PDE is a highly non-linear parabolic-transport equation, which depends on two conditional statistical numerics of important physical significance. The PDF PDE is a general form of the classical Reynolds mean flow equation (O. Reynolds, Philos. Trans. R. Soc. Lond. Ser. A 186 (1895), 123–164. http://doi.org/10.1098/rsta.1895.0004) and is a precise formulation of the PDF transport equation (S. B. Pope, Turbulent flows, Cambridge University Press, Cambridge, 2000. https://doi.org/10.1017/CBO9780511840531.). The PDF PDE provides us with a new method for modelling turbulence. An explicit example is constructed. Though the example is seemingly artificial, it demonstrates the PDF method based on the new PDF PDE.