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

TL;DR

This paper develops a theoretical framework for understanding the velocity probability density function in fully developed turbulence, linking deviations from Gaussianity to correlations of dynamical quantities, and validates it with numerical simulations.

Contribution

It introduces a joint analytical and numerical approach to quantify velocity PDF deviations in turbulence, providing a stationary solution and validation with DNS data.

Findings

Deviations from Gaussianity are linked to correlations of pressure gradient, forcing, and dissipation.

Stationary PDF solutions are derived analytically.

Numerical simulations confirm sub-Gaussian tails in the velocity PDF.

Abstract

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.