2.0 Drag reduction in curved pipes

Virk [28] published an extensive review on drag reduction in straight pipes. The paper highlighted some important aspects of drag reduction such as mechanism of polymer drag reduction, turbulence structure and velocity profile. The work also pioneered the concepts of maximum drag reduction (MDR) and drag reduction envelop. The Virk’s envelop for polymer drag reduction in straight pipes is shown in Fig. 1 on the Prandtl-Karman coordinates. Eqs. (6) – (8) give the equations for laminar flow, turbulent flow and maximum drag reduction asymptote. The maximum drag reduction law holds irrespective of polymer specie used, its concentration or molecular weight [29].
\(\frac{1}{\sqrt[f]{}}=\frac{N_{\text{Re}}\sqrt[f]{}}{16}\) (6)
\(\frac{1}{\sqrt[f]{}}=4N_{\text{Re}}\sqrt[f]{}\ -0.4\) (7)
\(\frac{1}{\sqrt[f]{}}=19N_{\text{Re}}\sqrt[f]{}\ -3.24\) (8)