Onset and universality of drag reduction in the turbulent Kolmogorov flow

We investigate the phenomenon of drag reduction in a viscoelastic fluid model of dilute polymer solutions. By means of direct numerical simulations of the three-dimensional turbulent Kolmogorov flow we show that drag reduction takes place above a critical Reynolds number $Re_c$. An explicit expression for the dependence of $Re_c$ on polymer elasticity and diffusivity is derived. The values of the drag coefficient obtained for different fluid parameters collapse onto a universal curve when plotted as a function of the rescaled Reynolds number $Re/Re_c$. The analysis of the momentum budget allows to gain some insight on the physics of drag reduction, and suggests the existence of a maximum drag reduction asymptote.