1. Driven Lid Partially Filled Porous Cavity - Single Domain Approach
G P Raja Sekhar
Credit: Sourav Dutta (Masters Project)
Currently at Texas A&M as a PhD student

2. The problem statement Rectangular cavity partially filled with
a porous medium(porosity and
Permeability being known)
Lid moving at constant velocity , u
To solve for flow inside the cavity

3. Approach Solve: Single Domain:
Incompressible Navier-Stokes in fluid domain and Brinkman extended Darcy equation in porous domain.
Single Domain:
No matching of material dependant parameters at the interface. They need to be determined empirically. Hence this method is favored.

4. Single Domain: Mathematical Formulation
After non-dimensionalization:

5. Vorticity-Streamfunction form
Goal: To remove pressure
Combining (3) &(4) and using in (1) &(2) we get :
Eq.(6) is the poisson equation for Ψ variable.

6. Grid & Discretization
We use the non staggered grid ie. placing ζ & Ψ at the same point as shown.
Several schemes used are: forward for time
And central for all space variables.

7. Vorticity-Streamfunction Algorithm
Initial values are set as ζ=0 & Ψ=0

8. Boundary Conditions
Simple first order expressions for ζ derivatives used at the walls. Ψ = 0 is set at all the walls. No-slip condition yields:

9. Results
Effect of Re, Da, porosity on flow characteristics has been studied. Viscosity factor is 1. All simulations are done in MATLAB with 50X50 grid.

10. Effect of Reynolds number: Fully porous, Da = 0.01, porosity = 0.5
As Re increases, volume flow entering porous region decreases.

11. Effect of porosity: Re = 10, Da = 0.01, fully porous
As porosity decreases, less liquid penetrates into the medium, though not too much evident from the plots.

12. Effect of Darcy Number: fully porous, Re = 10, porosity = 0.5
Darcy number is proportional to permeability. As Da decreases less liquid penetrates the medium.

13. Conclusion
The streamfunction vorticity approach is unable to provide results for high Re number calculations.
Not efficient enough.
Streamfunction velocity formulation adopted.
[Gupta and Kalita]

14. Future Work
Use Streamfunction-velocity formulation to solve the hydrodynamic problem and continue the analysis for :
High Re values
Effect of porous fraction
Effect of aspect ratio
Mass transport problem

15. Velocity Streamfunction formulation

17. Method Ψ=0 at the walls and (u=1,v=0) at top wall.
Everywhere else (u=0,v=0).
Iteration:
Solve by BiCGSTAB algorithm
Solve tri-diagonal system (12) & (13) using Thomas Algorithm.
Continue till convergence to the order of 10^-6

18. References

20. Thank you