1. 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 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 AERC 2005 22 nd to 24 th April 2005 Grenoble, France

2. 3D Viscoelastic flow in smooth contractions European Congress on Rheology AERC 2005 Flow geometry Experiments of Poole et al (2004) with solutions of PAA Area ratio R = d/D = 0.7 Upstream spanwise velocity profiles (x-z plane) at x/h=-8.33 and 0 d = 28mm, h = 6mm, D = 40mm, w = 80mm Inlet duct: 120 D H long (area ratio > 2/3  double backward-facing step ) Aspect ratios A 1 = w/h = 13.3 A 2 = w/d= 2.86

3. 3D Viscoelastic flow in smooth contractions European Congress on Rheology AERC 2005 Experimental and numerical findings 0.1% PAA Re  120 Cat’s ears Spanwise variation at y/D=0.5 GNF PTT (N 2 =0)

4. 3D Viscoelastic flow in smooth contractions European Congress on Rheology AERC 2005 Experimental and numerical findings 3 0.1% PAA Re  120 Downstream

5. 3D Viscoelastic flow in smooth contractions European Congress on Rheology AERC 2005 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. 3D Viscoelastic flow in smooth contractions European Congress on Rheology AERC 2005 Governing equations 1) Mass 2) Momentum 3) Constitutive equation Full PTT (linear stress coefficient) Newtonian solvent

7. 3D Viscoelastic flow in smooth contractions European Congress on Rheology AERC 2005 Numerical method: brief description 2) Structured, colocated and non-orthogonal meshes 3) Momentum (u i ) solvent polymer 1) Finite volume method ( Oliveira et al,1998; Oliveira & Pinho, 1999) 4) Discretization (formally 2 nd 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. 3D Viscoelastic flow in smooth contractions European Congress on Rheology AERC 2005 Computational domain and mesh 5 m (62 D H ) 120 h 20 cells 30 cells 102 000 total cells 1 020 000 DF

9. 3D Viscoelastic flow in smooth contractions European Congress on Rheology AERC 2005 Inlet flow x/h=-16

10. 3D Viscoelastic flow in smooth contractions European Congress on Rheology AERC 2005 Non-dimensional numbers Reynolds number Deborah number with and Extensional parameter Slip parameter Bulk velocity at contraction exit

11. 3D Viscoelastic flow in smooth contractions European Congress on Rheology AERC 2005 Effect of  : 1 kitten’s ears Absence of kitten’s ears Several values of 

12. 3D Viscoelastic flow in smooth contractions European Congress on Rheology AERC 2005 Effect of  : 2 kitten’s ears:high De, high , low 

13. 3D Viscoelastic flow in smooth contractions European Congress on Rheology AERC 2005 Effect of  : 3 kitten’s ears x/h=-8 x/h=-4 x/h=-2.06 x/h=-0.1 Effect of inertia

14. 3D Viscoelastic flow in smooth contractions European Congress on Rheology AERC 2005 Effect of  Closed symbols: kitten’s ears x/h=-8 x/h=-4 x/h=-2.06 x/h=-0.1 (a) (b) Effect of De (next slide)

15. 3D Viscoelastic flow in smooth contractions European Congress on Rheology AERC 2005 Effect of De De

16. 3D Viscoelastic flow in smooth contractions European Congress on Rheology AERC 2005 Effect of Re’: 1 Re=0.6 Re’=0.48 Re=0.6 Re’=0.43

17. 3D Viscoelastic flow in smooth contractions European Congress on Rheology AERC 2005 Effect of Re’: 2 Re=1.7 Re’=1.4 Re=1.7 Re’=1.3

18. 3D Viscoelastic flow in smooth contractions European Congress on Rheology AERC 2005 Effect of Re’: 3 Re=3.4 Re’=2.8 Re=3.4 Re’=2.6

19. 3D Viscoelastic flow in smooth contractions European Congress on Rheology AERC 2005 Effect of Re’: 4 Re=6.3 Re’=5.2 Re=6.3 Re’=4.7

20. 3D Viscoelastic flow in smooth contractions European Congress on Rheology AERC 2005 Conclusions Cat’s ears are qualitatively predicted by PTT (kitten’s ears) N 2 ≠ 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 ???