R. J. Poole and M. P. Escudier Dept. Engineering, Mechanical Engineering, University of Liverpool

1. 3D FLOW OF VISCOELASTIC FLUIDS OVER A BACKWARD-FACING STEP PRECEDED BY A GRADUAL CONTRACTION A. Afonso Centro de Estudos de Fenómenos de Transporte, DEMEGI Faculdade de Engenharia, Universidade do Porto, Portugal, aafonso@fe.up.pt F. T. Pinho Centro de Estudos de Fenómenos de Transporte, Dep. Eng. Mecânica Escola de Engenharia, Universidade do Minho, Portugal, fpinho@dem.uminho.pt R. J. Poole and M. P. Escudier Dept. Engineering, Mechanical Engineering, University of Liverpool Liverpool L69 3GH, UK, robpoole@liv.ac.uk,escudier@liv.ac.uk AERC 2005 22nd to 24th April 2005 Grenoble, France

2. Flow geometry Experiments of Poole et al (2004) with solutions of PAA Upstream spanwise velocity profiles (x-z plane) at x/h=-8.33 and 0 Aspect ratios A1 = w/h = 13.3 A2 = w/d= 2.86 d = 28mm, h = 6mm, D = 40mm, w = 80mm Inlet duct: 120 DH long Area ratio R = d/D = 0.7 (area ratio > 2/3  double backward-facing step )

3. Experimental and numerical findings Spanwise variation at y/D=0.5 GNF 0.1% PAA Re  120 PTT (N2=0) Cat’s ears

4. Experimental and numerical findings 3 0.1% PAA Re  120 Downstream

5. Objective Cat’s ears: Why? Shear-thinning:No Elasticity - : No Qualitative calculation with PTT: parametric investigation Effect of Effect of De Effect of Effect of Re Individual and combined effects

6. Governing equations 1) Mass 2) Momentum 3) Constitutive equation Newtonian solvent Full PTT (linear stress coefficient)

7. Numerical method: brief description 1) Finite volume method (Oliveira et al,1998; Oliveira & Pinho, 1999) 2) Structured, colocated and non-orthogonal meshes 3) Momentum (ui) polymer solvent 4) Discretization (formally 2nd order) Diffusive terms: central differences (CDS) Advective terms: CUBISTA (deferred correction) (Alves et al, 2000, 2003) 5) Special formulations for cell-face velocities and stresses

8. Computational domain and mesh 102 000 total cells 1 020 000 DF 5 m (62 DH) 120 h 20 cells 30 cells

9. Inlet flow x/h=-16

10. Non-dimensional numbers Reynolds number and with Bulk velocity at contraction exit Deborah number Extensional parameter Slip parameter

11. Effect of : 1 Several values of  kitten’s ears Absence of kitten’s ears

12. Effect of : 2 kitten’s ears:high De, high , low 

13. Effect of : 3 Effect of inertia kitten’s ears x/h=-0.1 x/h=-2.06 x/h=-4 x/h=-8

14. Effect of  x/h=-0.1 Closed symbols: kitten’s ears (b) (a) x/h=-2.06 x/h=-4 x/h=-8 Effect of De (next slide)

15. Effect of De De De

16. Effect of Re’: 1 Re=0.6 Re’=0.43 Re=0.6 Re’=0.48

17. Effect of Re’: 2 Re=1.7 Re’=1.3 Re=1.7 Re’=1.4

18. Effect of Re’: 3 Re=3.4 Re’=2.6 Re=3.4 Re’=2.8

19. Effect of Re’: 4 Re=6.3 Re’=4.7 Re=6.3 Re’=5.2

20. Conclusions • Cat’s ears are qualitatively predicted by PTT (kitten’s ears) • N2≠ 0 (essential)— high  • Low  • High De • Intermediate Re • Sometimes enhanced peaks observed at corners • Low Re: very slim profiles at contraction exit, no peaks • High Re: flat profiles at contracton exit, no peaks • Accurate predictions: different transient properties ???