Scaling of polymer drag reduction with polymer and flow parameters in turbulent channel flow

Abstract

The scaling of polymer drag reduction with polymer and flow parameters has been investigated using results from direct numerical simulations (DNS) of dilute, homogeneous polymer solutions in turbulent channel flow. Simulations were performed using a mixed Eulerian-Lagrangian scheme with a FENE-P dumbbell model of the polymer. The full range of drag reduction from onset to maximum drag reduction (MDR) is reproduced in DNS with realistic polymer parameters, giving results in good agreement with available experimental data. Onset of drag reduction is found to be a function of both the polymer concentration and the Weissenberg number, in agreement with the predictions of de Gennes[1]. The magnitude of drag reduction increases monotonically with decreasing viscosity ratio, β, for 1.0 > β > 0.98, saturates to a plateau for 0.98 > β > 0.9, and slowly decays for 0.9 > β when the solution ceases to be dilute. The magnitude of drag reduction at saturation is a strong function of the Weissenberg number. A \( We_\tau \sim O\left( {{\mathop{\rm Re}\nolimits} _\tau /3} \right) \) is needed to achieve MDR. The presence of the polymer results in attenuation of the small scales along with enhancement of the large scales in the spectra of the streamwise turbulent velocity fluctuations, and attenuation of all scales in the spectra of cross-stream turbulent velocity fluctuations. The degree of attenuation and the range of affected scales increase with the Weissenberg number and with the polymer concentration up to the saturation concentration. At saturation concentration, the cross-stream size of the largest attenuated eddies in the streamwise spectra conform to the predictions of Lumley's theory[2, 3], while at concentrations below the saturation, they conform to a modified version of de Gennes's theory[1]. The net effect of the polymer can be represented by an effective viscosity with a peak magnitude of \( v_{eff} \sim O\left( {0.1\lambda u_\tau ^2 } \right) \) at saturation, in agreement with the predictions of Lumley's theory[2, 3].