Comparison of Interface Capturing Methods using OpenFOAM 4 th OpenFOAM Workshop 4 June 2009

1. Comparison of Interface Capturing Methods using OpenFOAM 4thOpenFOAM Workshop 4 June 2009 Montreal, Canada Sean M. McIntyre, Michael P. Kinzel, Jules W. Lindau Applied Research Laboratory, Penn State University This work was supported by the Office of Naval Research, contract #N00014-07-1-0134, with Dr. Kam Ng as contract monitor.

2. Outline • Background • Motivation • Interface Capturing • Numerical Approach • Volume of Fluid • Level Set Methods • Test Cases • Summary

3. Background: Motivation • Supercavitating vehicle simulation • Drag reduction • Performance predictions • Vehicle dynamics • Ventilation gas required • Methods of cavity formation • Ventilation • Air ventilated • Vaporous • Water boils

4. Background: Interface Capturing • Interface tracking • Conforming mesh • Issues • Breaking waves • Sub-grid mixing • Interface capturing • Scalar variable • Identify species: volume fraction, mass fraction, concentration, signed distance functions • Improvements • Breaking interfaces • Sub-grid mixing

5. Outline • Background • Motivation • Interface Capturing • Numerical Approach • Volume of Fluid • Level Set Methods • Test Cases • Summary

6. Numerical Approach: Volume of Fluid • OpenFOAM uses MULES-VOF • Phase fraction: • Limited/conservative solution to: • Advantages • Conserves species mass • Single scalar equation • Allows sub-grid mixing • Disadvantages • Interface smearing (for sharp interface problems) • Homogeneous mixing

7. Numerical Approach: interFoam with Level-Set Time Step • Simple extension from VOF • g-equation level-set transport (Olsson & Kreiss 2005, Olsson et al. 2007) • Φ-analytically equivalent for incompressible flows(Kinzel, 2008 & Kinzel et al. 2009) • Various reinitialization schemes explored • Volume fraction field: g-based (Olsson et al. 2007, Kinzel 2008) • Signed Distance Function: f-based (Sussman et al. 1994, Kinzel 2008) Species Mass Conservation & Level-set transport equation: g Eqn. Momentum Predictor: UEqn Pressure Poisson Eqn.: pEqn Momentum Corrector Reinitilization Procedure

8. Numerical Approach: Advantages/Disadvantages VOF-based level-set methods • Advantages • Easy extension from VOF code • Conservative variable basis • Extensions to other flows (Kinzel 2008, Kinzel et al. 2009) • Cavitation/Boiling • Compressible-multiphase flows • Mass-conservation issues obvious • Alleviation (Olsson & Kreiss 2005, Olsson et al. 2007) • Arbitrary number of species • Straightforward boundary conditions • Disadvantages • Numerical accuracy: Only relevant at the interface

9. Numerical Approach: Reinitilization Approaches • Signed-Distance Function (Sussman et al. 1994) • Using variable transformations (Kinzel 2008) • Mass conserving (Olsson et al. 2007) • Only need to reinitialize the gamma field Reinitilization LS-1: (Sussman et al. 1994) Transform g→f: Reinitialize f: Transform f →g: • Notes: • Approximating Heaviside as: • e is 0.5 interface thickness • Consistency with original • H is given when k ~ 0.379 Reinitilization LS-2: (Olsson et al. 2007)

10. Numerical Approach: Reinitilization Approaches • Signed-Distance Function • Without variable transformations (Kinzel 2008) • Realizable Scaled (Kinzel 2008, Kinzel et al. 2009) • Algebraic sharpening. No solution to PDE! Reinitilization LS-3: (Kinzel 2008) where: • Notes: • Approximating Heaviside as: • e is 0.5 interface thickness • Consistency with original • H is given when k ~ 0.379 • Notes: • Neglecting smeared mass • e2 is amount neglected Reinitilization LS-4: (Kinzel 2008)

11. Numerical Approach: Reinitilization Approaches • Numerical solution to reinitilization • Pseudo time reinitialization • 4 Stage Runge-Kutta method • OpenFOAMfvc constructs used – adopts parallel capability • Stable solution highly dependent on fvScheme • Periodic reinitialization • Initialized every 1/flstimesteps • Improves stability and mass conservation • Relaxing reinitilization (Kinzel et al. 2009) • Notes: • m*: after gEqn • m+1/2: after reinitialization

12. Outline • Background • Motivation • Interface Capturing • Numerical Approach • Volume of Fluid • Level Set Methods • Test Cases • Summary

13. Test Cases: Dam Break • Mass conservation • Wave propagation • Sub-grid mixing Black: Sussman (SDF Level-Set) Gray: Sussman w/ VOF (LS-1) Pink: Olssen (LS-2) Yellow: Transformed (LS-3) Green: Realizable (LS-4) Background: VOF

14. Test Cases: Dam Break

15. Test Cases: Dam Break • Initial Wave • Captured with all methods • Subsequent events • Level-set -> mass loss • Scheme/parameter dependent • VOF ->Mass conserved

16. Test Cases: 2-D Water Drop in Oil • Mass conservation • Effect of level set parameters • Mixed conditions • Sharp interface • Sub-grid mixing • Parameters: • 1 x 3 meter domain • 50 x 150 cells • Water drop radius = 0.25 m • ρwater= 1000 kg/m3 • μwater = 0.001 kg/(m-s) • ρoil= 850 kg/m3 • μoil = 0.0272 kg/(m-s) • g = 9.81 m/s2 • Surface tension = 0 Black: Sussman (SDF Level-Set) Gray: Sussman w/ VOF (LS-1) Pink: Olssen (LS-2) Yellow: Transformed (LS-3) Green: Realizable (LS-4) Background: VOF

17. Test Cases: 2-D Water Drop in Oil LS-2 LS-4 LS-1 LS-3

18. Test Cases: 2-D Water Drop in Oil • SDF Sharpening w/ VOF transport (LS-1) • ε has effect when fls=1 and fr=1 • Damping and periodic reinitialization help

19. Test Cases: 2-D Water Drop in Oil • Mass-Conserving (LS-2) • ε increases conservation • Damping and periodic reinitialization lowered conservation

20. Test Cases: 2-D Water Drop in Oil • Transformed SDF Sharpening w/ VOF transport (LS-3) • ε has effect when fls=1 • Damping and periodic reinitialization help

21. Test Cases: 2-D Water Drop in Oil • Realizable-Scaled (LS-4) • Higher ε clips more, conserves less • Damping and periodic reinitialization help

22. Test Cases: Submerged Hydrofoil • Free surface flows • Sharp interface • Level-Set sharpening • Signed DistanceBoundary Conditions

23. Outline • Background • Motivation • Interface Capturing • Numerical Approach • Volume of Fluid • Level Set Methods • Test Cases • Summary

24. Summary • Compared Interface Capturing Methods • Using simple test cases • Volume of Fluid Vs. Level Set Methods • Test Cases • Dam Break: • Level-set methods: nice initial wave, mass conservation issues. Olssen method best of level set schemes. • VOF: Performs well • Water drop in Oil: • Level-set methods: good until breakup, mass conservation issues. Olssen method best of level set schemes. • VOF: Performs well • Duncan submerged hydrofoil: • Level-set methods: Good results. BC difficulties. Olssen method best of level set schemes. • VOF Performs well, more diffuse and less experimental agreement than Olssen

25. Summary • Conclusions • Clearly problem dependent • VOF all around best approach • Olssen conserves mass well, best of level-set methods. • Realizable scaling is cheaper, and performs similar to SDF methods • Future • Level-set parameter space • Performance on unstructured meshes • Reinitialization: performance/mass conservation

26. References • Sussman, M., Smereka, P., and Osher, S. 1994. A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114, 1 (Sep. 1994), 146-159. DOI= http://dx.doi.org/10.1006/jcph.1994.1155 • Olsson, E., Kreiss, G., and Zahedi, S. 2007. A conservative level set method for two phase flow II. J. Comput. Phys. 225, 1 (Jul. 2007), 785-807. DOI= http://dx.doi.org/10.1016/j.jcp.2006.12.027 • Olsson, E. and Kreiss, G. 2005. A conservative level set method for two phase flow. J. Comput. Phys. 210, 1 (Nov. 2005), 225-246. DOI= http://dx.doi.org/10.1016/j.jcp.2005.04.007 • Kinzel, M. P. Computational Techniques and Analysis of Cavitating-Fluid Flows. Dissertation in Aerospace Engineering, University Park, PA, USA : The Pennsylvania State University, May 2008. • Kinzel, M. P. Lindau, J.W., and Kunz, R.F.,”A Level-Set Approach for Compressible, Multiphase Fluid Flows with Mass Transfer,” AIAA CFD Conference, San Antonio TC, USA, June 2009.