Effects of the mean velocity field on the renormalized turbulent viscosity and correlation function

We perform renormalization group analysis of the Navier Stokes equation in the Eulerian framework in the presence of mean velocity field $U_0$, and observe that that the renormalized viscosity $\nu(k)$ is independent of $U_0$, where $k$ is the wavenumber. Thus we show that $\nu(k)$ in the Eulerian field theory is Galilean invariant. We also compute $\nu(k)$ using numerical simulations and verify the above theoretical prediction. The velocity-velocity correlation function however exhibits damped oscillations whose time period of oscillation and damping time scales are given by $1/k U_0$ and $1/( \nu(k) k^2)$ respectively. In a modified form of Kraichnan's direct interaction approximation (DIA), the ``random mean velocity field'' of the large eddies sweeps the small-scale fluctuations. The DIA calculations also reveal that in the weak turbulence limit, the energy spectrum $E(k) \sim k^{-3/2}$, but for the strong turbulence limit, the random velocity field of the large-scale eddies is scale-dependent that leads to Kolmogorov's energy spectrum.\footnote{M. K. Verma and A. Kumar, arXiv:1411.2693 (2015).}