Drag reduction by polymers in turbulent channel flows: Energy redistribution between invariant empirical modes

We address the phenomenon of drag reduction by a dilute polymeric additive to turbulent flows, using direct numerical simulations (DNS) of the FENE-P model of viscoelastic flows. It had been amply demonstrated that these model equations reproduce the phenomenon, but the results of DNS were not analyzed so far with the goal of interpreting the phenomenon. In order to construct a useful framework for the understanding of drag reduction we initiate in this paper an investigation of the most important modes that are sustained in the viscoelastic and Newtonian turbulent flows, respectively. The modes are obtained empirically using the Karhunen-Loéve decomposition, allowing us to compare the most energetic modes in the viscoelastic and Newtonian flows. The main finding of the present study is that the spatial profile of the most energetic modes is hardly changed between the two flows. What changes is the energy associated with these modes, and their relative ordering in the decreasing order from the most energetic to the least. Modes that are highly excited in one flow can be strongly suppressed in the other, and vice versa. This dramatic energy redistribution is an important clue to the mechanism of drag reduction as is proposed in this paper. In particular, there is an enhancement of the energy containing modes in the viscoelastic flow compared to the Newtonian one; drag reduction is seen in the energy containing modes rather than the dissipative modes, as proposed in some previous theories.