A model is proposed for the new stress term in the momentum equation of a modified generalised Newtonian fluid that is used to mimic viscoelastic effects of fluids exhibiting drag reduction in turbulent pipe flow. The new stress quantifies the cross-correlation between the fluctuating viscosity and the fluctuating rate of strain and had been neglected in the k-epsilon low Reynolds number model originally developed by Cruz and Pinho [J. Non-Newt. Fluid Mechanics 114 (2003) 109]. With the inclusion of this new stress, the predictions of turbulent kinetic energy (k+) in drag reducing pipe flow are significantly improved at the cost of a slight deterioration in the prediction of other quantities for some of the fluids. The value of the coefficient C in the original model of Cruz and Pinho was also modified to correct a mistake and this was shown to improve the predictions for all fluids except for the solution of 0.125% PAA used to calibrate the model. Comparison of predictions with DNS results for a FENE-P model showed large overprediction of drag reduction and emphasize the need for improvements in extensional rheometry. (C) 2004 Elsevier B.V. All rights reserved.