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

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

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

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

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

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

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 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

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

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

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

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

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

Re=50

Re=1

Rotation Axis Wall: Streamsurfaces

Re=50

Rotation Axis Wall: Streamsurfaces

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

Pressure distribution on wall

axis

Force Parallel to Wall

force in z direction: complex, sign change

low Re: negative, viscous effect dominates

high Re: positive to negative

Re

Approximate Force Scaling

force in x and z directions

Scaling of gap: collapse

1Re1

2 a

df

a

d

a

F

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:

Thus,

∂w∂ z

≃ w

/=−1r

∂∂ r

r u ≃ aa

w a

≃Re−1/2

Re

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