The Effects of Polymer Concentration on Turbulent Drag Reduction

In 1948 the phenomenon of turbulent drag reduction by polymers was discovered [B. Toms, Proc. Int'l Rheological Congress 2, 135 (1948)]. It was found that the addition of a small amount of polymers can significantly reduce the frictional drag in wall-bounded turbulent flows. The reduction in drag is characterized by an increase in the fluid throughput under fixed applied pressure gradient, or equivalently, a smaller pressure gradient is required to drive the flow at the same flow rate. This phenomenon has been studied extensively in the past decades but a theory that can give predictions was proposed only recently [I. Procaccia, V.L. L'ov, and R. Benzi, Rev. Mod. Phys. 80, 225 (2008)]. This theory is based on the balance of momentum and energy, and shows that the effect of the polymers can be represented by a position-dependent effective viscosity. 

In this thesis, we work out further predictions of this theory that can be tested by experiments. In particular, we have worked out the various effects of polymer concentration for both flexible and rodlike polymers. We have focused on the pipe geometry for easy comparison with experiments. We have calculated the relation of the friction factor with the Reynolds number and studied how this relation changes with the zero shear viscosity of the polymer solution. The zero-shear viscosity of the polymer solution depends on the concentration of the polymers in the solvent. Using additional experimental knowledge of the concentration dependence of viscosity, we have found how the friction factor relation changes with concentration for the rodlike polymer xanthan gum. We have also calculated how the Reynolds stress profile varies with the concentration of the polymer. Our results reveal a linear dependence of the maximum Reynolds stress on the position of the maximum. This linear dependence is a new theoretical prediction, which can be tested experimentally. We have found that the viscosity dependence of the percentage drag reduction at different Reynolds numbers can be collapsed into a universal curve with different functional forms for flexible and rodlike polymers. We have also compared our results with experimental observations.