Speed-direction description of turbulent flows

Abstract

In this note we introduce speed and direction variables to describe the motion of incompressible viscous flows. Fluid velocity $\mathbf{u}$ is decomposed into $\mathbf{u}=u\mathbf{r}$, with $u=|\mathbf{u}|$ and $\mathbf{r}=\mathbf{u}/|\mathbf{u}|$. We consider a directional split of the Navier-Stokes equations into a coupled system of equations for $u$ and for $\mathbf{r}$. 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 $u$-$\mathbf{r}$ 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.