Dynamics of high-Deborah-number entry flows: a numerical study

Abstract

High-elasticity simulations of flows through a two-dimensional (2D) 4 : 1 abrupt contraction and a 4 : 1 three-dimensional square–square abrupt contraction were performed with a finite-volume method implementing the log-conformation formulation, proposed by Fattal & Kupferman (J. Non-Newtonian Fluid Mech., vol. 123, 2004, p. 281) to alleviate the high-Weissenberg-number problem. For the 2D simulations of Boger fluids, modelled by the Oldroyd-B constitutive equation, local flow unsteadiness appears at a relatively low Deborah number (De) of 2.5. Predictions at higherDewere possible only with the log-conformation technique and showed that the periodic unsteadiness grows withDeleading to an asymmetric flow with alternate back-shedding of vorticity from pulsating upstream recirculating eddies. This is accompanied by a frequency doubling mechanism deteriorating to a chaotic regime at highDe. The log-conformation technique provides solutions of accuracy similar to the thoroughly tested standard finite-volume method under steady flow conditions and the onset of a time-dependent solution occurred approximately at the same Deborah number for both formulations. Nevertheless, for Deborah numbers higher than the critical Deborah number, and for which the standard iterative technique diverges, the log-conformation technique continues to provide stable solutions up to quite (impressively) high Deborah numbers, demonstrating its advantages relative to the standard methodology. For the 3D contraction, calculations were restricted to steady flows of Oldroyd-B and Phan-Thien–Tanner (PTT) fluids and very highDewere attained (De ≈ 20 for PTT with ϵ = 0.02 andDe≈ 10000 for PTT with ϵ = 0.25), with prediction of strong vortex enhancement. For the Boger fluid calculations, there was inversion of the secondary flow at highDe, as observed experimentally by Sousaet al. (J. Non-Newtonian Fluid Mech., vol. 160, 2009, p. 122).