1. FLAIR:General Idea Reconstructs interface by: New “f” calculated by:
sloped line
fitted at the boundary of every TWO neighboring cells
New “f” calculated by:
integrating the “advected” area underneath the interface line segment
Original Geometry
Reconstructed

2. FLAIR: Fluid Cases Possible fluid combinations for TWO cells 1 2 3 4 56 7 8 9

3. FLAIR: Subcases Determine Interface Line All cases reduce to Case 9
Line Segment: y = ax + b a,b = f(fa, fb) fb fa fb fa 9a 9a 9b fb fa fb fa 9c 9d

4. FLAIR:Flux Definition
Determine fluid flux moving from one cell to another.
Define: s = ut/h
Fluxes:
f+ = s[a + b* - as/2]
f- = s[a + b* + as/2]
Variables a and b geometry dependent

5. FLAIR: Flux Diagram y x y y x x* x x s j + 1/2 j - 1/2 i - 1/2 i + 1/2

6. FLAIR: Case Distinction Diagram
Criteria to determine which Subcase is required

7. FLAIR: Single Interface Cell
Full cell moving into Empty cell
4 possible cases 1 2     a b c d

8. FLAIR: Configuration Distinction
Criteria to determine which line-segment will be in a single interfacial cell.

9. General Solution Procedure using FLAIR
1: Specify intital surface geometry and velocities
2: Move surface based on velocities and FLAIR
3: Find Surface Curvature using f-field
4: Calculate Surface pressure using local curvatures
5: Calculate remaining pressures using Poisson’s Eq.
6: Solve for velocity field everywhere in domain using Momentum Eqs.
7: Interate between 5 & 6 until convergence
8: Increment time and repeat 1-8 until done

10. FLAIR: Sample Application
Problem: Capillary-Driven Drop Motion
2-D, Imcompressible, constant properties
Governing Equations:
Continuity
Navier-Stokes Equations
Boundary Conditions:
No shear forces at fluid surface
Effect of velocity gradients on surface pressure are neglected

11. Staggered Grid: Poisson Equation
vi, j+3/2
ui-1/2, j+1
ui+1/2, j+1
Pi, j+1
zj+1
vi-1, j+1/2
vi+1, j+1/2
vi, j+1/2
Pi-1, j
ui+1/2, j
Pi+1, j
ui+3/2, j
Pi, j
zj
ui+1/2, j
ui-1/2, j
vi-1, j+1/2
vi+1, j-1/2
vi, j-1/2
Pi,j-1
ui-1/2, j-1
ui+1/2, j-1
zj-1
vi, j-3/2
ri
ri+1
ri-1

12. Cell Variable Definitions
vi, j-1/2
j+1/2
Pi, j j
ui-1/2, j
Fi, j
ui+1/2, j
vi, j-1/2
j-1/2
i-1/2 i
i+1/2

13. FLAIR: Surface Pressure
Pi, j Ps Pf dc
dfs

14. A-FLAIR: General Idea Axisymmetric version of FLAIR
Need Advection Criterion:
Satisfies Continuity Equation
Axial Direction: u = Constant
Radial Direction: rv = Constant

15. A-FLAIR: Interface Reconstruction
Similar to FLAIR but....
Movement along the “z” and “r” directions do NOT obey same relations
Different relations must be used depending on “+” or “-” movement
Line: ab
No closed form solution for a & b

16. A-FLAIR: Flux Diagramsz
i - 1/2
i + 1/2 *
i + 3/2 r
j - 1/2
j + 1/2  u
i+1/2, j  z  z r z
i - 1/2
i + 1/2 *
i + 3/2 r
j - 1/2
j + 1/2  u
i+1/2, j  z  z

17. A-FLAIR: Z-direction Interface orientation? 9 cases
Need to identify
9 cases
All reduce to case 9, 4 subcases
Fluxes: f= f(r, a, b, u, t)
Can’t solve for a,b directly
Solve a,b for EACH subcase and check limits

18. A-FLAIR: R-direction Most general case case 9, 4 subcases
Fluxes: solve cubic or Newton-Rapson
Cases: 1,2,7 & 8: single interface cell
16 possible cases: use f and  to identify case
Fluxes: case specific -  ,  and V()  
i.e. or