One equation model for turbulence pipe flow with second order viscoelastic corrections

Abstract

A modified second order viscoelastic constitutive equation is used to derive a k–l type turbulence closure to qualitatively assess the effects of elastic stresses on fully-developed channel flow. Specifically, the second order correction to the Newtonian constitutive equation gives rise to a new term in the momentum equation involving the time-averaged elastic shear stress and in the turbulent kinetic energy transport equation quantifying the interaction between the fluctuating elastic stress and rate of strain tensors, denoted by Pw, for which a closure is developed and tested. This closure is based on arguments of isotropic turbulence and equilibrium in boundary layer flows and a priori Pw could be either positive or negative. When Pw is positive, it acts to reduce the production of turbulent kinetic energy and the turbulence model predictions qualitatively agree with direct numerical simulation (DNS) results obtained for more realistic viscoelastic fluid models with memory which exhibit drag reduction. In contrast, Pw 0 but has also a small region of negative Pw in the buffer layer, was also successfully tested. This second model for Pw leads to predictions of drag reduction, in spite of the enhancement of turbulence production very close to the wall, but the equilibrium conditions in the inertial sub-layer were not strictly maintained.