The Lundgren-Monin-Novikov Hierarchy: Kinetic Equations for Turbulence

TL;DR

This paper reviews the use of the Lundgren-Monin-Novikov hierarchy for describing turbulence statistically via PDFs, highlighting analytical techniques and applications to various turbulent systems using DNS data.

Contribution

It provides a comprehensive overview of the LMN hierarchy's application to turbulence, including new insights from DNS data for unclosed terms.

Findings

Application to 2D vorticity statistics

Analysis of 3D velocity and vorticity PDFs

Insights into Rayleigh-Bénard convection and Burgers turbulence

Abstract

We present an overview of recent works on the statistical description of turbulent flows in terms of probability density functions (PDFs) in the framework of the Lundgren-Monin-Novikov (LMN) hierarchy. Within this framework, evolution equations for the PDFs are derived from the basic equations of fluid motion. The closure problem arises either in terms of a coupling to multi-point PDFs or in terms of conditional averages entering the evolution equations as unknown functions. We mainly focus on the latter case and use data from direct numerical simulations (DNS) to specify the unclosed terms. Apart from giving an introduction into the basic analytical techniques, applications to two-dimensional vorticity statistics, to the single-point velocity and vorticity statistics of three-dimensional turbulence, to the temperature statistics of Rayleigh-B\'enard convection and to Burgers turbulence are discussed.