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

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.Comment: Accepted for publication in C. R. Acad. Sc

Theory for the single-point velocity statistics of fully developed turbulence

We investigate the single-point velocity probability density function (PDF) in three-dimensional fully developed homogeneous isotropic turbulence within the framework of PDF equations focussing on deviations from Gaussianity. A joint analytical and numerical analysis shows that these deviations may be quantified studying correlations of dynamical quantities like pressure gradient, external forcing and energy dissipation with the velocity. A stationary solution for the PDF equation in terms of these quantities is presented, and the theory is validated with the help of direct numerical simulations indicating sub-Gaussian tails of the PDF.Comment: 6 pages, 4 figures, corrected typo in eq. (4

Vorticity statistics in fully developed turbulence

We study the statistical properties of fully developed hydrodynamical turbulence. To this end, a theoretical framework is established relating basic dynamical features of fluid flows to the shape and evolution of probability density functions of, for example, the vorticity. Starting from the basic equations of motion, the theory involves terms, which presently cannot be calculated from first principles. This missing information is taken from direct numerical simulations. A parallel pseudospectral code is used to to obtain well-resolved numerical simulations of homogeneous isotropic turbulence. The results yield a consistent description of the vorticity statistics and provide insights into the structure and dynamics of fully developed turbulence

On the velocity distribution in homogeneous isotropic turbulence: correlations and deviations from Gaussianity

We investigate the single-point probability density function of the velocity in three-dimensional stationary and decaying homogeneous isotropic turbulence. To this end, we apply the statistical framework of the Lundgren-Monin-Novikov hierarchy combined with conditional averaging, identifying the quantities that determine the shape of the probability density function. In this framework, the conditional averages of the rate of energy dissipation, the velocity diffusion and the pressure gradient with respect to velocity play a key role. Direct numerical simulations of the Navier-Stokes equation are used to complement the theoretical results and assess deviations from Gaussianity