Three basic approaches:

1
FEM and Free Surface Flow

• Three basic approaches
• 1. Fixed mesh and free boundary is tracked
• 2. Deformed spatial mesh using a 3-stage
  iterative cycle
• Stage 1 Assume shape of free boundary
• Stage 2 BVP solved after discarding 1 BC on
  free boundary
• Stage 3 Shape of boundary is updated using
  previously neglected BC
• Cylce repeated until convergence is acheived
• 3. Deformed mesh and define nodes on free
  boundary
• Nodes give extra degrees of freedom
• Field variables and boundary position solved
  simultaneously using a Newton-Rapson interative
  procedure

2
FEM and Free Surfaces New Approach

• Combination of FEM and VOF technique
• FEM solves for the field variables on a deforming
  boundary
• VOF used to advect the boundary interface
• Advantages
• Simulate Large Surface Deformations (i.e.
  Mergering and Breaking)
• Accurate implementation of Boundary Conditions
• Increases computational efficiency

3
FEM-VOF Governing Equations

• Governing Equations (non-dimensional form)
• Continuity
• Navier-Stokes (Momentum)

4
FEM-VOF Boundary Conditions

• BCs are given by
• Surface Traction is related to Radii of
  Curvatures
• Radius of Curvature is defined as

5
FEM-VOF Formulation

• Two restictions
• 1. Solution in terms of primitive variables based
  on linear quadrilateral elements
• 2. Model must handle pressure, velocity,
  velcoity gradient and stress boundary conditions
  directly
• Penalty function
• Apply Galerkin Method to Momentum equations

6
FEM-VOF Mesh Generation

• Master element Isoparametric linear
  quadrilateral element
• 9 possible cases regarding intersection points

7
FEM-VOF Surface Advection

• Once velocities obtained interface is
  advected using FLAIR
• Velocities at nodes NOT adequate for advection
  technique
• Calculate mean velocity fom two node velocities
  
• Axisymmetric r-z plane mapped to master element
  in ??? plane

8
FEM-VOF Moving Nodes

• Governing Equations for Moving Nodes
• Motion only in R-direction
• Extra terms will modify finite element formulation

9
FEM-VOF Solution Procedure

• 1 Specify inital surface geometry and velocities
• 2 Determine inital f-field based on geometry
• 3 Using FLAIR reconstruct surface interface
• 4 Mesh domain
• 5 Solve for nodal velocities using the
  Navier-Stokes Eqs.
• 6 Transform nodal velocities to cell face
  velocities
• 7 Determine new f-field by advecting old f-field
  using FLAIR
• 8 Reconstruct new surface interface
• 9 Increment time and repeat 4-8 until done

10
FEM-VOF Algorithm Steps
11
Conclusion

• Volume of Fluid (VOF) Methods
• Reconstructs interface surfaces
• Able to handle large surface deformation
• Easy implementation
• Many forms exist
• Hybrid FEM-VOF technique for Free Surface Flows
• Combination eliminates short-comings of each
  method
• Handles BCs accurately
• Handles Large Surface Deformation (i.e. Merging
  Breakup)
• Accurate and versatile