Drag reduction

The phenomenon of drag reduction, reported for the first time by the British chemist Toms in 1949, is probably the effect produced by polymer addiction in fluids which has attracted the most attention, because of its relevance for applications. While performing experiments on the degradation of polymers, Toms observed that the addition of few parts per million of long chain polymers in turbulent flow produces a dramatic reduction of the friction drag.

The adimensional quantity which is normally used to measure the friction drag in a pipe flow is the Fanning friction factor $f$ defined as:

$$f = {\Delta p \over \rho V^2}{R \over L}$$ (3.44)

where $R$ is the radius of the pipe, $\Delta p$ is the pressure drop measured across a distance $L$ in the pipe, $\rho$ is the density of the fluid and $V$ is the mean velocity over the section.

The physical meaning of the friction factor is the ratio between the input of energy provided by an external pressure difference and the kinetic energy of the resulting mean flow in the pipe, and essentially it gives a measure of the force that is required to sustain a certain mean flow. Rephrasing the drag reduction in this terms, it means that the force necessary to pump a fluid through a pipe can be reduced of a factor $80 \%$ with the simple addition of few ppm of polymers. The relevance for practical applications is thus enormous.

In Newtonian fluids the friction drag is a function of the Reynolds number, which for the pipe flow reads $Re = 2R V/\nu$ where $\nu$ is the kinematic viscosity. The dependence of the friction drag on the $Re$ number is conventionally shown in the so-called Prandtl-Karman coordinates: $1 /\sqrt{f}$ versus $\log (Re \sqrt{f})$ (see Figure (3.3)).

Figure 3.3: A schematic illustrating the onset of drag reduction and the MDR asymptote in the Prandtl-Karman coordinates. The Prandtl-Karman law correspond to the turbulent behavior of Newtonian fluids. The dotted line represent qualitatively the friction reduction in the viscoelastic case.

In the laminar regime the friction drag decrease as $Re^{-1}$ until the critical $Re$ number is reached. Transition to turbulence causes a sudden increase of the friction drag which for fully developed turbulence reaches an almost constant value with only a weak logarithmic dependence on $Re$ described by the Prandtl-Karman (P-K) law (a straight line in P-K coordinates).

Dilute polymer solutions deviate from the P-K law: while for $Re$ number smaller than a critical threshold their behavior is similar to the Newtonian fluid, for larger $Re$ numbers the friction drag is drastically reduced with respect to the Newtonian case and it finally reaches a universal asymptote which is independent on the kind of polymers or the concentration of the solution, and is known in literature as the Maximum Drag Reduction asymptote (MDR). A theory based on the elastic behavior of polymers was proposed by Tabor & de Gennes in 1986[58] to explain both the onset of the drag reduction and the presence of the universal upper bound, but its validity is still controversial. An overview of this theory can be found in Sreenivasan and White[59].

Recent works have provided new insights on the matter. A shell-model based on the Fene-P model has been proposed by Benzi et al. [64], which provides a simple and usefull tool for understanding the phenomenon of drag reduction. Numerical simulations of Oldroyd-B and Fene-P models are able to reproduce, at least qualitatively, the phenomenology of the problem [60,62], and and have proved that drag reduction can occurs also in absence of boundaries [55]. Moreover, some experiments seems to indicate a peel off from the MDR asymptote at high $Re$ numbers, opening new questions.

The interest for drag reduction is clearly amplified by its possible applications. Indeed nowadays it is widely applied in oil and water pipelines and specific polymers have been developed to reduce hydraulic friction for industrial and petrochemical applications.