1. Pressure-driven Flow in a Channel with Porous Walls Funded by NSF CBET-0754344 Qianlong Liu & Andrea Prosperetti 11,2 Department of Mechanical Engineering Johns Hopkins University, USA 1 2 Department of Applied Science University of Twente, The Netherlands

2. Numerical Method: PHYSALIS, combination of spectral and immersed boundary method Results : Detailed flow structure Hydrodynamic force/torque Dependence on Re Lift Force on spheres Slip 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

3. 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

4. Flow Field 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

5. Flow Field Re = 833 y/a=0.8,0.5,0.3,0 More evident features Three-dimensional separation

6. Pressure Distribution 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

7. 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

8. 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

9. Hydrodynamic Force 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

10. Hydrodynamic Torque 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

11. Slip Condition Using Beavers-Joseph model, different results for shear- and pressure-driven flows Modified with another parameter Good fit of experimental results Beavers-Joseph model modified model

12. 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

13. Thank you!

14. Rotation Axis Wall: Force force directed toward the plane low pressure between the sphere and the wall const. small Re Re large Re Re

15. Rotation Axis Wall: Couple low Re: torque increases by wall- induced viscous dissipation high Re: velocity smaller on wall side: dissipation smaller Re

16. Re=50 Re=1 Rotation Axis Wall: Streamsurfaces

17. Re=50 Rotation Axis Wall: Streamsurfaces

18. 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 effect Re

19. Pressure distribution on wall axis

20. Force Parallel to Wall force in z direction: complex, sign change low Re: negative, viscous effect dominates high Re: positive to negative Re

21. Approximate Force Scaling force in x and z directions Scaling of gap: collapse

22. Particle in a Box

23. Unbounded Flow: couple Hydrodynamic couple for rotating sphere in unbounded flow Accurate results Zero force

24. Unbounded Flow: maximum w Poleward flow exert equal and opposite forces Wall: destroy the symmetry Continuity equation: Thus, Re

25. 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