On the Lundgren hierarchy of helically symmetric turbulence

Abstract This paper analyzes the reduction of the infinite Lundgren–Monin–Novikov (LMN) hierarchy of probability density functions (PDFs) in the statistical theory of helically symmetric turbulence. Lundgren’s hierarchy is considered a complete model, i.e. fully describes the joint multi-point statistic of turbulence though at the expense of dealing with an infinite set of integro-differential equations. The LMN hierarchy and its respective side-conditions are transformed to helical coordinates and thus are dimesionally reduced. In the course of development, a number of key questions were solved, namely in particular the transformation of PDFs and sample space velocities into orthonormal coordinate systems. In a validity check it is shown, that the mean momentum equations derived from the helical LMN hierarchy via statistical moment integration are identical to the mean momentum equations derived by direct ensemble averaging the Navier–Stokes equation, in helically symmetric form. Finally, we derive the equation for the characteristic function equivalent to the PDF equation in a helically symmetric frame, which allows to generate arbitrary <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi>t</mml:mi> <mml:mi>h</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> -order statistical moments by simple differentiation.