Identification and Calculation of the Universal Asymptote for Drag Reduction by Polymers in Wall Bounded Turbulence

Drag reduction by polymers in wall turbulence is bounded from above by a universal maximal drag reduction (MDR) velocity profile that is a log law, estimated experimentally by Virk as $V^{+}(y^{+})\approx 11.7\log y^{+}-17$. Here $V^{+}(y)$ and $y^{+}$ are the mean streamwise velocity and the distance from the wall in “wall” units. In this Letter we propose that this MDR profile is an edge solution of the Navier-Stokes equations (with an effective viscosity profile) beyond which no turbulent solutions exist. This insight rationalizes the universality of the MDR and provides a maximum principle which allows an ab initio calculation of the parameters in this law without any viscoelastic experimental input.

Figure 1

Mean normalized velocity profiles as a function of the normalized distance from the wall. The data points from numerical simulations (green circles) [8] and the experimental points (open circles) refer to Newtonian flows. The red data points (squares) [1] represent the maximum drag reduction (MDR) asymptote. The dashed red curve represents the log law (4). The blue filled triangles [16] and green open triangles represent the crossover, for intermediate concentrations of the polymer, from the MDR asymptote to the Newtonian plug.

Figure 2

The solution for $10\sqrt{W^{+}}$ (dashed line) and $y^{+}{S}^{+}$ (solid line) in the asymptotic region $y^{+}\gg {\delta }^{+}$, as a function of $\alpha$. The vertical solid line $\alpha =1/2{\delta }^{+}=1/12$ which is the edge of turbulent solutions; since $\sqrt{W^{+}}$ changes sign here, to the right of this line there are only laminar states. The horizontal solid line indicates the highest attainable value of the slope of the MDR logarithmic law $1/{\kappa }_{\mathrm{v}}=12$.