Turbulence With Pressure

TL;DR

This paper derives exact results for the Navier-Stokes equations with pressure, using advanced methods to analyze the velocity field and probability distribution functions, including perturbative solutions for small pressure.

Contribution

It extends the operator product expansion framework to include pressure effects and provides a perturbative approach to solve the Navier-Stokes equations in the mean field approximation.

Findings

Derived exact N-point generating functions for velocity fields.

Generalized operator product expansion to include pressure effects.

Obtained the first correction to the probability distribution function.

Abstract

We investigate the exact results of the Navier-Stokes equations using the methods developed by Polyakov. It is shown that when the velocity field and the density are not independent, the Burgers equation is obtained leading to exact N-point generating functions of velocity field. Our results show that, the operator product expansion has to be generalized both in the absence and the presence of pressure. We find a method to determine the extra terms in the operator product expansion and derive its coefficients and find the first correction to probablity distribuation function. In the general case and for small pressure, we solve the problem perturbatively and find the probablity distribuation function for the Navier-Stokes equation in the mean field approximation.