Pressure-driven Flow in a Channel with Porous Walls
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Pressure-driven Flow in a Channel with Porous Walls
Funded by NSF CBET-0754344
Qianlong Liu & Andrea Prosperetti1 1,2
Department of Mechanical EngineeringJohns Hopkins University, USA
1
2 Department of Applied ScienceUniversity of Twente, The Netherlands
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Numerical Method: PHYSALIS, combination of spectral and immersed boundary method
Results :Detailed flow structureHydrodynamic force/torqueDependence on ReLift Force on spheresSlip Condition vs. Beavers-Joseph model (See JFM paper
submitted)
• Spectrally accurate near particle • No-slip condition satisfied exactly• No integration needed for force and torque
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Flow Field
Re = 0.833
y/a=0.8,0.5,0.3,0
Streamlines on the symmetry midplane and neighbor similar to 2D case
At outermost cut, open loop similar to 2D results at small volume fraction
2D features
2
3
12
1Re
Ga
a
H
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Flow Field2
3
12
1Re
Ga
a
H
Re = 83.3
y/a=0.8,0.5,0.3,0
Marked upstream and downstream
Clear streamline separation from the upstream sphere and reattachment to the downstream one
Different from 2D features
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Flow Field2
3
12
1Re
Ga
a
H
Re = 833
y/a=0.8,0.5,0.3,0
More evident features
Three-dimensional separation
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Pressure Distribution
2
3
12
1Re
Ga
a
H
Pressure on plane of symmetry for Re=0.833, 83.3, 833
High and low pressures near points of reattachment and separation
Maximum pressure smaller than minimum pressure
Point of Maximum pressure lower than that of minimum pressure
Combination of these two features contributes to a lift force
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Horizontally Averaged Velocity
In the porous media for Re=0.833, 83.3, 833
Two layers of spheres
Below the center of the top sphere, virtually identical averaged velocity
Consistent to experimental results of the depth of penetration
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Horizontally Averaged Velocity
In the channel for Re=0.833, 83.3, 833
Circles: numerical results
Solid lines: parabolic fit allowing for slip at the plane tangent to spheres
A parabolic-like fit reproduces very well mean velocity profile
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Hydrodynamic ForceNormalized lift force as a
function of the particle Reynolds number
Total force, pressure and viscous components
Dependence of channel height and porosity is weak, implying scales adequately capture the main flow phenomena
Slope 1: Low Re
Constant: High Re
HGaF 2* 2
2Re
GaH
p
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Hydrodynamic Torque
HGaT 3* 2
Normalized Torque as a function of the particle Reynolds number
Decease with increasing Re_p in response to the increasing importance of flow separation
Weak dependence on channel height H/a=10, 12
Dependence on volume fraction, although weak
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Slip Condition
Using Beavers-Joseph model, different results for shear- and pressure-driven flows
Modified with another parameter
Good fit of experimental results
Di UUdz
dU
Di UUdz
dU
Beavers-Joseph model
modified model
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Conclusions
Finite-Reynolds-number three-dimensional flow in a channel bounded by one and two parallel porous walls studied numerically
Detailed results on flow structure
Hydrodynamic force and torque
Dependence on Reynolds number
Lift force on spheres
Modification of slip condition
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Thank you!
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Rotation Axis Wall: Force
force directed toward the plane
low pressure between the sphere and the wall
2a
F
const.
small ReRe
large Re
Re
2a
F
2
Rea
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Rotation Axis Wall: Couple
low Re: torque increases by wall-induced viscous dissipation
high Re: velocity smaller on wall side:
dissipation smaller
18 3
aL
Re
2
Rea
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Re=50
Re=1
Rotation Axis Wall: Streamsurfaces
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Re=50
Rotation Axis Wall: Streamsurfaces
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Force Normal to Wall
force in wall direction: sign change
low Re: viscous repulsive force pushes particle away from the wall
high Re: attractive force from Bernoulli-type effectRe
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Pressure distribution on wall
axis
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Force Parallel to Wall
force in z direction: complex, sign change
low Re: negative, viscous effect dominates
high Re: positive to negative
Re
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Approximate Force Scaling
force in x and z directions
Scaling of gap: collapse
1Re1
2 a
df
a
d
a
F
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Particle in a Box
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Unbounded Flow: couple
Hydrodynamic couple for rotating sphere in unbounded flow
Accurate results
Zero force
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Unbounded Flow: maximum w
Poleward flow exert equal and opposite forces
Wall: destroy the symmetry
Continuity equation:
Thus,
∂w∂ z
≃ w
/=−1r
∂∂ r
r u ≃ aa
w a
≃Re−1/2
Re
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Perpendicular Wall: Pathline
Start near the wall, spirals up and outward toward the rotating sphere, and spirals back toward the wall
Resides on a toroidal surface