J. Fluid Mech. (2011), vol. 677, pp. 272–304. c  Cambridge University Press 2011 doi:10.1017/jfm.2011.84 Dynamics of high-Deborah-number entry flows: a numerical study A. M. AFONSO 1 , P. J. OLIVEIRA 2 , F. T. PINHO 3 AND M. A. ALVES 1 † 1 Departamento de Engenharia Qu´ ımica, Centro de Estudos de Fen´ omenos de Transporte, Faculdade de Engenharia da Universidade do Porto, Rua Doutor Roberto Frias, 4200-465 Porto, Portugal 2 Departamento de Engenharia Electromecˆ anica, Unidade de Materiais Texteis e Papeleiros, Universidade da Beira Interior, 6201-001 Covilh˜ a, Portugal 3 Departamento de Engenharia Mecˆ anica, Centro de Estudos de Fen´ omenos de Transporte, Faculdade de Engenharia da Universidade do Porto, Rua Doutor Roberto Frias, 4200-465 Porto, Portugal (Received 21 May 2010; revised 3 January 2011; accepted 11 February 2011; first published online 13 April 2011) 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 higher De were possible only with the log-conformation technique and showed that the periodic unsteadiness grows with De leading 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 high De. 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 high De were attained (De ≈ 20 for PTT with ε =0.02 and De ≈ 10 000 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 high De, as observed experimentally by Sousa et al. (J. Non-Newtonian Fluid Mech., vol. 160, 2009, p. 122). Key words: complex fluids, low-Reynolds-number flows, viscoelasticity 1. Introduction Viscoelastic entry flows, and in devices with geometric singularities such as flows through contractions or contraction/expansions, are important in polymer processing † Email address for correspondence: mmalves@fe.up.pt