Fluid turbulence and the renormalization group: A preliminary calculation of the eddy viscosity

A new method of averaging the Navier-Stokes equations has recently led to an equation for the mean velocity in which the Reynolds stresses are systematically replaced by a renormalized eddy viscosity. Here we give the results for an iterative calculation of the eddy viscosity ${\nu }_n\mathrm{}\left(k\right)\mathrm{}$ on the assumption that the energy spectrum is given by $E\left(k\right)=\alpha {\epsilon }^{\frac{2}{3}}{k}^{-\frac{5}{3}}$. A renormalization-group analysis shows that ${\nu }_n\mathrm{}\left(k\right)\mathrm{}$ reaches the fixed functional form $\nu \mathrm{}\left(k\right)=2.5\mathrm{}{\alpha }^{\frac{1}{2}}{\epsilon }^{\frac{1}{3}}{k}^{-\frac{4}{3}}$.