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Plane sudden expansion flows of viscoelastic liquids: effect of expansion ratio Robert J Poole Department of Engineering, University of Liverpool, UK Manuel.

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Presentation on theme: "Plane sudden expansion flows of viscoelastic liquids: effect of expansion ratio Robert J Poole Department of Engineering, University of Liverpool, UK Manuel."— Presentation transcript:

1 Plane sudden expansion flows of viscoelastic liquids: effect of expansion ratio Robert J Poole Department of Engineering, University of Liverpool, UK Manuel A Alves CEFT, Faculdade de Engenharia, Universidade do Porto, Portugal Fernando T Pinho a CEFT, Faculdade de Engenharia, Universidade do Porto, Portugal b Universidade do Minho, Portugal Paulo J Oliveira Departamento de Engenharia Electromecânica, Universidade da Beira Interior, Portugal AERC 2007 4th Annual European Rheology Conference April 12-14, Napoli - Italy

2 Outline Introduction Governing equations Numerical method / grid dependency issues Newtonian results UCM simulations: “High” ER followed by “Low” ER Conclusions AERC 2007 4th Annual European Rheology Conference April 12-14, Napoli - Italy

3 Introduction Prevailing view….vortex suppressed by elasticity and totally eliminated at “high” Deborah AERC 2007 4th Annual European Rheology Conference April 12-14, Napoli - Italy Not the whole story (AERC 2006 Poole et al, JNNFM 2007 to appear) UCM/Oldroyd-B (β = 1/9) simulations, 1:3 expansion ratio, creeping flow Maximum obtainable De ≈ 1 Effect of elasticity is to reduce but not eliminate recirculation Enhanced pressure drop observed Why investigate expansion flows of viscoelastic liquids?

4 Governing equations 1) Mass 2) Momentum (creeping flow) 3) Constitutive equationUpper Convected Maxwell model (UCM) AERC 2007 4th Annual European Rheology Conference April 12-14, Napoli - Italy Essentially phenomenological model “Simplest” viscoelastic differential model Capable of capturing qualitative features of many highly-elastic flows

5 Numerical method 1) Finite-volume method (Oliveira et al (1998), Oliveira & Pinho (1999)) 2) Structured, collocated and non-orthogonal meshes 3) Discretization (formally second order) Diffusive terms: central differences (CDS) Convective terms: CUBISTA (Alves et al (2003)) 4) Special formulations for cell-face velocities and stresses AERC 2007 4th Annual European Rheology Conference April 12-14, Napoli - Italy

6 Computational domain and meshes Y X ER=D/d L 2 = 100d L 1 = 20d h d D UBUB symmetry axis AERC 2007 4th Annual European Rheology Conference April 12-14, Napoli - Italy Expansion ratios (ER) 1:1.5 1:2 1:3 1:4 1:8 1:16 1:32 Fully-developed inlet velocity and stress profiles Neumann b.c.s at exit Low ER High ER

7 Representative mesh details ER = 4NCDOF (  x MIN )/d M115 00090 0000.01 M260 000360 0000.005 AERC 2007 4th Annual European Rheology Conference April 12-14, Napoli - Italy ER = 16NCDOF (  x MIN )/d M121 500129 0000.01 M286 000516 0000.005 ER = 1.5NCDOF (  x MIN )/d M114 50087 0000.005 M258 000348 0000.0025

8 Representative grid dependency and numerical accuracy ER and fluidX R (= x R / d)XR#XR# % error M1M2 Newtonian ER =1.50.33000.3298 0.02% Newtonian ER = 20.59150.59140.59130.01% Newtonian ER =41.49771.49941.49990.04% Newtonian ER = 166.56036.55736.55620.02% De = 1.0 ER =1.50.33660.34260.34470.59% De = 1.0 ER =20.55280.55010.54920.16% De = 1.0 ER =41.23391.23031.22910.12% De = 1.0 ER =166.25456.24906.24710.03% # denotes extrapolated value using Richardson’s technique AERC 2007 4th Annual European Rheology Conference April 12-14, Napoli - Italy

9 Newtonian simulations: X R variation with ER AERC 2007 4th Annual European Rheology Conference April 12-14, Napoli - Italy d Linear fit to data for ER  4 (R 2 =1)

10 Newtonian simulations: X R variation with ER AERC 2007 4th Annual European Rheology Conference April 12-14, Napoli - Italy Deviations from linear fit as ER  1

11 Newtonian simulations: X R variation with ER AERC 2007 4th Annual European Rheology Conference April 12-14, Napoli - Italy D H

12 “High” ER viscoelastic : X R variation with De and ER AERC 2007 4th Annual European Rheology Conference April 12-14, Napoli - Italy Δ M1 X M2  Extrapolated

13 1:4 expansion ratio AERC 2007 4th Annual European Rheology Conference April 12-14, Napoli - Italy

14 De = 0.0De = 0.2De = 0.4De = 0.6De = 0.8De = 1.0 1:4 expansion ratio (M2) AERC 2007 4th Annual European Rheology Conference April 12-14, Napoli - Italy

15 “High” ER viscoelastic : scaling of X R AERC 2007 4th Annual European Rheology Conference April 12-14, Napoli - Italy ER =4 ER =8 ER =16 ER =32

16 “Low” ER viscoelastic : X R variation with De and ER AERC 2007 4th Annual European Rheology Conference April 12-14, Napoli - Italy

17 1:1.5 expansion ratio1:2 expansion ratio AERC 2007 4th Annual European Rheology Conference April 12-14, Napoli - Italy

18 1:1.5 expansion ratio AERC 2007 4th Annual European Rheology Conference April 12-14, Napoli - Italy De = 0.0De = 0.1De = 0.2De = 0.3De = 0.4De = 0.6De = 0.8De = 1.0 De = 0.0De = 0.1De = 0.2De = 0.3De = 0.4De = 0.6De = 0.8De = 1.0

19 “Low” ER viscoelastic : scaling of X R AERC 2007 4th Annual European Rheology Conference April 12-14, Napoli - Italy

20 Maximum De  1.0? AERC 2007 4th Annual European Rheology Conference April 12-14, Napoli - Italy McKinley et al scaling criterion for onset of purely elastic instabilities: independent of ER Streamlines at De = 1 for ER = 4, 8 and 16

21 Maximum De  1.0? AERC 2007 4th Annual European Rheology Conference April 12-14, Napoli - Italy McKinley scaling criterion for onset of purely elastic instabilities: Streamlines at De = 1 for ER = 4, 8 and 16

22 Conclusions AERC 2007 4th Annual European Rheology Conference April 12-14, Napoli - Italy For large expansion ratios (  8) Recirculation length normalised with downstream duct height scales with a Deborah number based on bulk velocity at inlet and downstream duct height (De/ER) For small expansion ratios (  2) X R initially decreases before increasing at a given level of elasticity (De/ER ~ 0.4) In range of De for which steady solutions could be obtained X R decreases with elasticity Maximum obtainable De is approximately 1.0: independent of ER

23 Enhanced pressure drop AERC 2007 4th Annual European Rheology Conference April 12-14, Napoli - Italy

24 0.15% polyacrylamide solutionNewtonian ‘2D’ 1: 13.3 Planar Expansion Townsend and Walters (1993) Re < 10 De O(1)? AERC 2007 4th Annual European Rheology Conference April 12-14, Napoli - Italy

25 Stress variation around sharp corner Hinch (1993) JnNFM Stresses around sharp corner go to infinity as: r AERC 2007 4th Annual European Rheology Conference April 12-14, Napoli - Italy

26 Normal stresses (ER = 3) AERC 2007 4th Annual European Rheology Conference April 12-14, Napoli - Italy

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