Speed-direction description of turbulent flows

TL;DR

This paper introduces a novel approach to describing turbulent flows using speed and direction variables, simplifying the Navier-Stokes equations and providing new insights into turbulence statistics without relying on the Reynolds stress tensor.

Contribution

It proposes a new variable decomposition for incompressible flows, leading to simplified equations and a novel way to analyze turbulence statistics.

Findings

Derived a coupled system for speed and direction variables in Navier-Stokes equations.

Showed the approach simplifies energy balance analysis.

Applied the method to describe mean statistics of shear turbulence.

Abstract

In this note we introduce speed and direction variables to describe the motion of incompressible viscous flows. Fluid velocity ${\bf u}$ is decomposed into ${\bf u}=u{\bf r}$, with $u=|{\bf u}|$ and ${\bf r}={\bf u}/|{\bf u}|$. We consider a directional split of the Navier-Stokes equations into a coupled system of equations for $u$ and for ${\bf r}$. Equation for $u$ is particularly simple but solely maintains the energy balance of the system. Under the assumption of a weak correlation between fluctuations in speed and direction in a developed turbulent flow, we further illustrate the application of $u$-${\bf r}$ variables to describe mean statistics of a shear turbulence. The standard (full) Reynolds stress tensor does not appear in a resulting equation for the mean flow profile.