Transcript Re - Multi-Scale Modeling and Simulation Laboratory

Pressure-driven Flow in a Channel
with Porous Walls
Qianlong Liu1 & Andrea Prosperetti 1,2
1
2
Department of Mechanical Engineering
Johns Hopkins University, USA
Department of Applied Science
University of Twente, The Netherlands
Funded by NSF CBET-0754344
Results :
 Detailed flow structure
 Hydrodynamic force/torque
 Dependence on Re
 Lift Force on spheres
 Slip Condition vs. Beavers-Joseph model
(See JFM paper
submitted)
Numerical Method: PHYSALIS, combination of spectral and
immersed boundary method
• Spectrally accurate near particle
• No-slip condition satisfied exactly
• No integration needed for force and torque
Flow Field
1  H  a 3G
Re    2
12  a  

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
Flow Field
1  H  a 3G
Re    2
12  a  

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
Flow Field
1  H  a 3G
Re    2
12  a  

Re = 833

y/a=0.8,0.5,0.3,0

More evident features

Three-dimensional
separation
Pressure Distribution
1  H  a 3G
Re    2
12  a  

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
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
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
Hydrodynamic Force




F*  Ga 2 H
2
Re p 
aH  G
2
Normalized 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
Hydrodynamic Torque





T*  Ga3 H
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
Slip Condition
Beavers-Joseph model
dU

U i  U D 

dz

modified model
dU

U i  U D 

dz


Using Beavers-Joseph model, different
results for shear- and pressure-driven flows

Modified with another parameter

Good fit of experimental results
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
Thank you!
Rotation Axis  Wall: Force

F
2
a

force directed toward
the plane
low pressure between
the sphere and the
wall
Re 
Re
F

2
a
Re
small Re
const.
large Re
a
2

Rotation Axis  Wall: Couple

L
1
3
8
a

low Re: torque
increases by wallinduced viscous
dissipation
high Re: velocity
smaller on wall side:
dissipation smaller
Re
Re
a
2

Rotation Axis
Wall: Streamsurfaces
Re=50
Re=1
Rotation Axis
Wall: Streamsurfaces
Re=50
Force Normal to Wall



Re
force in wall direction:
sign change
low Re: viscous
repulsive force
pushes particle away
from the wall
high Re: attractive
force from Bernoullitype effect
Pressure distribution on wall
axis
Force Parallel to Wall



Re
force in z direction:
complex, sign change
low Re: negative,
viscous effect
dominates
high Re: positive to
negative
Approximate Force Scaling


force in x and z
directions
Scaling of gap:
collapse

F
d  
d

1

f

Re
1





2



a
a  
a



Particle in a Box
Unbounded Flow: couple

Hydrodynamic couple
for rotating sphere in
unbounded flow

Accurate results

Zero force
Unbounded Flow: maximum w



Poleward flow exert
equal and opposite
forces
Wall: destroy the
symmetry
Continuity equation:
∂w
≃
∂z
Thus,
Re
w
/
=
−1 ∂
a
ru ≃
r ∂r
a
w
− 1/2
≃ Re
a
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