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The purpose of the present paper is to derive a partial differential equation(PDE) for the single-time single-point probability density function (PDF) ofthe velocity field of a turbulent flow. The PDF PDE is a highly non-linearparabolic-transport equation, which depends on two conditional statisticalnumerics of important physical significance. The PDF PDE is a general form ofthe classical Reynolds mean flow equation, and is a precise formulation of thePDF transport equation. The PDF PDE provides us with a new method for modellingturbulence. An explicit example is constructed, though the example is seeminglyartificial, but it demonstrates the PDF method based on the new PDF PDE.

On distributions of velocity random fields in turbulent flows