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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)
8 e 9 de Junho de 20061/25
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
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
8 e 9 de Junho de 2006
= 0⋅∇ uConservation of linear momentum
D pDt
ρ = − + ⋅ τ∇ ∇u
3/25
η γ•
τ =
Newtonian non-Newtonian
( )η γ γ• •
τ =
3 - EQUATIONS3 - EQUATIONS
Conservation of mass
8 e 9 de Junho de 2006
= 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
8 e 9 de Junho de 2006
( )1
0 1
na a
η η η η λ γ
−
•
∞ ∞
⎡ ⎤⎛ ⎞= + − +⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦
4/25
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−
5/25
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
6/25
3
1
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.
24/25
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
25/25
University of Beira Interior (Portugal)