Numerical Study of non-Newtonian Inelastic Fluid Flow in a 2D...

Helder M. Matos , Paulo J. Oliveira

Universidade da Beira InteriorDep.to Eng. Electromecânica

Numerical Study of non-Newtonian Inelastic Fluid Flow in a 2D Bifurcation at

Ninety Degrees

1 - T-JUNCTION FLOWS1 - T-JUNCTION FLOWS

The human circulatory system (many bifurcations)

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2 - OBJECTIVES2 - OBJECTIVES

Evaluate the effect of inertia and flow rate ratio in flow through a T-junction (dividing flow arrangement) for Newtonian and non-Newtonian inelastic fluids (fixed n).

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1

0

Re u Hρη

=

50≤ Re ≤1000

3

1

QQ

β =

0.1≤ β ≤0.9

2 - OBJECTIVES2 - OBJECTIVES

Evaluate the effect of inertia and flow rate ratio in flow through a T-junction (dividing flow arrangement) for Newtonian and non-Newtonian inelastic fluids (fixed n).

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Evaluate the effect of power law exponent nvariation in the Carreau-Yasuda model for this kind of flow.

Conservation of mass

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= 0⋅∇ uConservation of linear momentum

D pDt

ρ = − + ⋅ τ∇ ∇u

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η γ•

τ =

Newtonian non-Newtonian

( )η γ γ• •

τ =

3 - EQUATIONS3 - EQUATIONS

Conservation of mass

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= 0⋅∇ uConservation of linear momentum

D pDt

ρ = − + ⋅ τ∇ ∇u

3/25

non-Newtonian

( )η γ γ• •

τ =

GNF fluid

Carreau-Yasudamodel

3 - EQUATIONS3 - EQUATIONS

3 - EQUATIONS3 - EQUATIONS

Carreau-Yasuda model

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( )1

0 1

na a

η η η η λ γ

−

•

∞ ∞

⎡ ⎤⎛ ⎞= + − +⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

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Blood parameters (Banerjee et al. 1997)

– Power law exponent, n=0.3568 (for results with fixed n);– Carreau parameter, a=2;– Zero shear rate viscosity, η0=0.056 Pa.s; – Infinite shear rate viscosity, η∞ =0.00345 Pa.s; – Time constant λ=3.313 s.

4 - NUMERICAL METHOD4 - NUMERICAL METHOD

Finite-volume method for discretization of equations.

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Nonstaggered mesh arrangement.– Pressure-velocity coupling: Rhie e Chow, 1983.– Stress-velocity coupling: Oliveira et al., 1999.

Convective terms: CUBISTA scheme (Alves et al. 2003).

Pressure-correction SIMPLEC algorithm with time-marching.

Convergence tolerance for iterative process when norm of residuals less than .810−

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5 - FLOW GEOMETRY5 - FLOW GEOMETRY

x

y

H

X R

Q 1 Q 2

Q 3

Y tot = 21 H

X tot = 26 H

H

X LX S

Y L

Y R

Y S

H

Fully developed upstream flow

Inlet

0 0orx y∂ ∂≡ ≡

∂ ∂

Vanishing axial variation for all

quantities ( )

except pressure (linear extrapolation)

Outlet

Walls No slip condition u = v = 0

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3

1

QQ

β =

Flow rate

6 - MESHES6 - MESHES

Orthogonal but non uniform meshes

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– MESH M1, 12800 VC (Xtot=26H Ytot=21H)2

min min 2.5 10x y −Δ = Δ = ×

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– MESH M2, 22400 VC (Xtot=66.5H Ytot=60.5H)2

min min 2.5 10x y −Δ = Δ = ×

6 - MESHES6 - MESHES

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X-4 -2 0 2 4

0

2

4

Y

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MESH M1 (12800 VC)2

min min 2.5 10x y −Δ = Δ = ×

Streamlines for increasing Reynolds number

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7 - RESULTS7 - RESULTS

7 - RESULTS7 - RESULTS

Vertical recirculation length

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X

X

X

X

XX X X X X X X

X

Re0 250 500 750 10000

2

4

6

8

10

YL

GNF

X

XX

XX

XX

X X X X X X X X X X X X X

Re0 250 500 750 10000

2

4

6

8

10β=0.1β=0.2β=0.3β=0.4β=0.5β=0.6β=0.7β=0.8β=0.9

X

YL

Newtonian

Streamlines for increasing Reynolds number

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7 - RESULTS7 - RESULTS

7 - RESULTS7 - RESULTS

Vortex strength of vertical recirculation

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X

X

XX

X X X X XX

XX

X

Re0 250 500 750 10000

0.02

0.04

0.06

0.08

0.1

β=0.1β=0.2β=0.3β=0.4β=0.5β=0.6β=0.7β=0.8β=0.9

X

ψV

GNF

X

X

XX

X X X X X X X X X X X X X X X X

Re0 250 500 750 10000

0.02

0.04

0.06

0.08

0.1

ψV

Newtonian

7 - RESULTS7 - RESULTS

Vortex strength of vertical recirculation

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Re0 250 500 750 1000

0

0.02

0.04

0.06

0.08

0.1

NewtonianGNF

ψV

β=0.2

Re0 250 500 750 1000

0

0.02

0.04

0.06

0.08

0.1

NewtonianGNF

ψV

β=0.8

Streamlines (secondary branch)

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7 - RESULTS7 - RESULTS

7 - RESULTS7 - RESULTS

Horizontal recirculation length

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XX

XX

XX

XX

X X X XX

Re0 250 500 750 1000

0

5

10

15

20

XL

GNF

XX X X X X X X X X X X X X X X X X X X

Re0 250 500 750 1000

0

5

10

15

20β=0.1β=0.2β=0.3β=0.4β=0.5β=0.6β=0.7β=0.8β=0.9

XXL

Newtonian

7 - RESULTS7 - RESULTS

Vortex strength of horizontal recirculation

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X

X

XX X X X X X X X X X X X X X X X X

Re0 250 500 750 1000

0

0.01

0.02

0.03

0.04

ψH

Newtonian

X

X

XX X X X X X X X X X

Re0 250 500 750 1000

0

0.01

0.02

0.03

0.04β=0.1β=0.2β=0.3β=0.4β=0.5β=0.6β=0.7β=0.8β=0.9

X

ψH

GNF

7 - RESULTS7 - RESULTS

Shear stress field: variation with Re

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7 - RESULTS7 - RESULTS

Maximum values of shear stress

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X

XX

XX

X X X X X X X X X X X X X X X

Re0 250 500 750 1000

-10

-5

0

τXYmin

Newtonian

X X XX

X X X X X X X X X XX

XX

XX

X

Re0 250 500 750 1000

1

2

3 β=0.1 (Newt)β=0.2 (Newt)β=0.3 (Newt)β=0.4 (Newt)β=0.5 (Newt)β=0.6 (Newt)β=0.7 (Newt)β=0.8 (Newt)β=0.9 (Newt)

X

τXYmax

Newtonian

7 - RESULTS (variation with n)7 - RESULTS (variation with n)

Streamlines for increasing power law index

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7 - RESULTS (variation with n)7 - RESULTS (variation with n)

Recirculation length (fixed β)

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Re0 200 400 6000

2

4

6

8

10

12n=0.10n=0.20n=0.30n=0.40n=0.50n=0.60

YL

β=0.5

Re0 200 400 600

0

2

4

6

8

10

12n=0.10n=0.20n=0.30n=0.40n=0.50n=0.60

XL

β=0.5

Vertical Horizontal

7 - RESULTS (variation with n)7 - RESULTS (variation with n)

Recirculation length (fixed Re)

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β0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6n=0.10n=0.20n=0.30n=0.40n=0.50n=0.60

YL

Re =102

β0 0.2 0.4 0.6 0.8 1-1

0

1

2

3

4n=0.10n=0.20n=0.30n=0.40n=0.50n=0.60

XL

Re =102

Vertical Horizontal

7 - RESULTS (variation with n)7 - RESULTS (variation with n)

Vortex strength of recirculations (fixed β)

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Re0 200 400 600

0

0.02

0.04

0.06

n=0.10n=0.20n=0.30n=0.40n=0.50n=0.60

ψV

β=0.5

Re0 200 400 600

0

0.005

0.01

0.015 n=0.10n=0.20n=0.30n=0.40n=0.50n=0.60

ψH

β=0.5

Vertical Horizontal

7 - RESULTS (variation with n)7 - RESULTS (variation with n)

Shear stress field: variation with n

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8 - CONCLUSIONS8 - CONCLUSIONS

The length and intensity of the recirculations increase with Re for both fluids.

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The recirculation lengths YL and XL increase with βfor β≤0.6 and decrease for β>0.6, for both fluids.

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Non-Newtonian fluids show larger recirculation lengths.

Reduction of growth rates of XL and YL with Re after a new recirculation is formed at the opposite wall (distal and top walls).

Sudden increase of ψV with Re due to the division of the vertical recirculation into two vortices.

8 - CONCLUSIONS8 - CONCLUSIONS

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Decrease of ψV with β, for low Re numbers and vice-versa. Increase of ψH with β.

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Stress increase (in modulus) with Re and β .

Larger stress maxima with Newtonian fluid.

Low stresses inside recirculating zones and high stresses in the re-entrant corners of the bifurcation.

Length and intensity of recirculations decrease with increase of power law exponent (n).

Stress increase (in modulus) with increase of power law exponent (n).

9 - ACKNOWLEDGEMENTS9 - ACKNOWLEDGEMENTS

European Union - FEDER

PROJECT: BD/18062/2004PROJECT: POCI/EME/58657/2004

ECCOMAS Scholarship

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University of Beira Interior (Portugal)