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FEM and Free Surface Flow

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

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FEM and Free Surface Flow

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  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. 0 0 0 0 0 0 0 0 0 0 .98 .86 .59 .15 0 1 1 1 .91 0 .91 1 1 1 1 0 0 0 0 0 .15 0 0 0 .25 1 .89 .67 .16 0 1 1 1 .95 .19 .55 1 1 1 1 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

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