Colloquium: Theory of drag reduction by polymers in wall-bounded turbulence

The flow of fluids in channels, pipes, or ducts, as in any other wall-bounded flow (like water along the hulls of ships or air on airplanes) is hindered by a drag, which increases manyfold when the fluid flow turns from laminar to turbulent. A major technological problem is how to reduce this drag in order to minimize the expense of transporting fluids like oil in pipelines, or to move ships in the ocean. It was discovered that minute concentrations of polymers can reduce the drag in turbulent flows by up to 80%. While experimental knowledge had accumulated over the years, the fundamental theory of drag reduction by polymers remained elusive for a long time, with arguments raging whether this is a “skin” or a “bulk” effect. In this Colloquium the phenomenology of drag reduction by polymers is summarized, stressing both its universal and nonuniversal aspects, and a recent theory is reviewed that provides a quantitative explanation of all the known phenomenology. Both flexible and rodlike polymers are treated, explaining the existence of universal properties like the maximum drag reduction asymptote, as well as nonuniversal crossover phenomena that depend on the Reynolds number, on the nature of the polymer and on its concentration. Finally other agents for drag reduction are discussed with a stress on the important example of bubbles.

Figure 1

The channel geometry.

Figure 2

(Color online) Mean normalized velocity profiles as a function of the normalized distance from the wall. The data points (solid circles) from numerical simulations of 16 and the experimental points (open circles) (62) refer to Newtonian flows. The solid line is a theoretical formula developed by 41. The red data points (squares) (58) represent the maximum drag reduction (MDR) asymptote. The dashed red curve represents the log law (1.5) which was derived from first principles by 5. The blue filled triangles (50) and green open triangles (51) represent the crossover, for intermediate concentrations of the polymer, from the MDR asymptote to the Newtonian plug.

Figure 3

%DR in a counter-rotating disks experiment with $\lambda$-DNA as the drag reducing polymer. Note that the %DR is proportional to $c_p$. When the length $N_p$ reduces by a factor of 2 and, simultaneously, $c_p$ increases by factor of 2, the %DR reduces by a factor of 4.

Figure 4

Typical velocity profiles taken from 24. Dashed line notes the von Kármán law (1.3), while the MDR (1.6) is shown as the continuous 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 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 5

The drag in Prandtl-Kármán coordinates for a low concentration solution of NaCl in water mixed with the rodlike polymer PAMH B1120 in a pipe flow, see 60 for details. The symbols represent the concentrations in wppm (weight parts per million) as given.

Figure 6

Second-order structure function $S_2\left(r\right)$ for homogeneous and isotropic turbulence. Two cases are represented: with polymers (line with symbols) and without polymer (dashed line). The Lumley scale is indicated by an arrow. The numerical simulations with polymers are performed using a ${128}^3$ resolution with periodic boundary conditions. The energy content for scales below the Lumley scale is reduced indicating a significant energy transfer from the velocity field to the polymer elastic energy.

Figure 7

Mean velocity profiles for the Newtonian and for the viscoelastic simulations with $\mathrm{Re}=125$ (6). Solid line, Newtonian. Dashed line, viscoelastic. Straight lines represent a log law with the classical von Kármán slope. Notice that in this simulation the modest Reynolds number results in an elastic layer in the region $y^{+}⩽20$.

Figure 8

(Color online) Comparison of the DNS data (54) for mean profiles of $R_{xx}$ and $10R_{yy}$, the components of the dimensionless conformation tensor, with analytical predictions. In our notation ${\Pi }_0^{ij}\equiv {\nu }_{0p}{R}_{\max }^2{R}_{ij}∕{\tau }_p{R}_{\mathrm{eq}}^2$. Squares, DNS data for the streamwise diagonal component $R_{xx}$, that according to our theory has to decrease as $1∕y$ with the distance to the wall. Solid line, the function $1∕y^{+}$. Open circles, DNS data for the wall-normal component $10R_{yy}$, for which we predicted a linear increase with $y^{+}$ in the log-law turbulent region. Dashed line: linear dependence, $\propto y^{+}$.

Figure 9

The Newtonian viscosity profile and four examples of close to linear viscosity profiles employed in the numerical simulations. Solid line, run $N$; – –, run $R$; ⋯, run $S$; –∙–, $T$; – ∙ ∙ –, run $U$.

Figure 10

The reduced mean velocity as a function of the reduced distance from the wall. The line types correspond to those used in Fig. 9.

Figure 11

The solution for $10\sqrt{W^{+}}$ (dashed line) and $y^{+}{S}^{+}$ (solid line) in the asymptotic region $y^{+}⪢{\delta }^{+}$, 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 $1∕{\kappa }_{\mathrm{V}}=12$.

Figure 12

(Color online) Schematic mean velocity profiles. Region 1, $y^{+}<y_{\mathrm{N}}^{+}$, viscous sublayer. Region 2, $y^{+}>y_{\mathrm{N}}^{+}$, logarithmic layer for the turbulent Newtonian flow. Region 3, $y_{\mathrm{N}}^{+}<y^{+}<y_{\mathrm{V}}^{+}=(1+Q)y_{\mathrm{N}}^{+}$, MDR asymptotic profile in the viscoelastic flow. Region 4, $y^{+}>y_{\mathrm{V}}^{+}$, Newtonian plug in the viscoelastic flow.

Figure 13

(Color online) Drag reduction efficiency %DR as function of the drag reducing parameter $Q$ for different centerline Reynolds numbers Re: $1.2\times {10}^5$, $1.2\times {10}^6$, and $1.2\times {10}^7$ (from top to bottom).

Figure 14

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 15

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 polyemrs, i.e., the profile diverges from the von Kármán log law, reaching the MDR only asymptotically.

Figure 16

$f^{-1∕2}$ as a function of ${\log }_{10}\left(\mathrm{Re}{f}^{1∕2}\right)$ with $\lambda =1$ with various values of ${\nu }_p$.

Figure 17

(Color online) Percentage drag reduction as a function of Re for two values of the concentration of the rodlike polymer. Symbols are results of experiments and solid lines are theory, cf. 1.

Figure 18

Predicted values of drag reduction with $\alpha =2$ and different values of $A$. Dashed line reproduces the predictions of Legner’s theory which suffer from an unphysical drag enhancement at $C=0$. For $A=0$ (rigid spheres) we find only increasing drag enhancement as a function of $C$. For small values of $A$ we have first a slight drag enhancement, and then modest drag reduction. For large values of $A$, associated with strong bubble oscillations, we find significant values of drag reduction.