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 th Annual European Rheology Conference April 12-14, Napoli - Italy

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

Introduction Prevailing view….vortex suppressed by elasticity and totally eliminated at “high” Deborah AERC th 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?

Governing equations 1) Mass 2) Momentum (creeping flow) 3) Constitutive equationUpper Convected Maxwell model (UCM) AERC th 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

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 th Annual European Rheology Conference April 12-14, Napoli - Italy

Computational domain and meshes Y X ER=D/d L 2 = 100d L 1 = 20d h d D UBUB symmetry axis AERC th 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

Representative mesh details ER = 4NCDOF (  x MIN )/d M M AERC th Annual European Rheology Conference April 12-14, Napoli - Italy ER = 16NCDOF (  x MIN )/d M M ER = 1.5NCDOF (  x MIN )/d M M

Representative grid dependency and numerical accuracy ER and fluidX R (= x R / d)XR#XR# % error M1M2 Newtonian ER = % Newtonian ER = % Newtonian ER = % Newtonian ER = % De = 1.0 ER = % De = 1.0 ER = % De = 1.0 ER = % De = 1.0 ER = % # denotes extrapolated value using Richardson’s technique AERC th Annual European Rheology Conference April 12-14, Napoli - Italy

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

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

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

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

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

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

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

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

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

1:1.5 expansion ratio AERC th 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

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

Maximum De  1.0? AERC th 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

Maximum De  1.0? AERC th 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

Conclusions AERC th 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

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

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

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

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