Turbulent drag reduction by flexible and rodlike polymers: Crossover effects at small concentrations

Drag reduction by polymers is bounded between two universal asymptotes, the von Kármán log law of the law and the maximum drag reduction (MDR) asymptote. It is theoretically understood why the MDR asymptote is universal, independent of whether the polymers are flexible or rodlike. The crossover behavior from the Newtonian von Kármán log law to the MDR is, however, not universal, showing different characteristics for flexible and rodlike polymers. In this paper we provide a theory for this crossover phenomenology.

Figure 1

Typical velocity profiles taken from Ref. 4. The dashed line shows the von Kármán law (2), while the MDR (3) is shown as the continuous black line. In all cases the mean velocity follows the same viscous behavior for $y^{+}<10$. After that the scenario is different for flexible and rodlike polymers. The typical behavior for the former is presented by the open triangles, which follow the MDR up to a crossover point that depends on the concentration of the polymer and on the value of Re. The rodlike behavior is exemplified by the solid triangles and the open squares; the mean velocity profiles appear to interpolate smoothly between the two asymptotes as a function of the concentration of the rodlike polymer.

Figure 2

The mean velocity profiles for flexible polymer additives with $\tilde{\nu }=1$, 5, 10, 20, 50, 100, and 500 from below to above. Note that the profile follows the MDR until it crosses over back to the Newtonian plug.

Figure 3

The mean velocity profile for rodlike polymer additives with $\tilde{\nu }=1$, 5, 10, 20, 50, 100, 500, 1000, 5000, and 10 000 from below to above. Note the typical behavior expected for rodlike polymers, i.e., that the profile diverges from the von Kármán log law, reaching the MDR only asymptotically.

Figure 4

The turbulent kinetic energy profile $K^{+}\left(y^{+}\right)$ for the flexible (solid line) and rodlike (dashed line) cases, respectively. We note the quadratic and linear dependence respectively, as anticipated in the text.