Comparison of Interface Capturing Methods using OpenFOAM 4th OpenFOAM 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.

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

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

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

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

Numerical Approach: Volume of Fluid OpenFOAM uses MULES-VOF Advantages 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

Numerical Approach: interFoam with Level-Set 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) Time Step Species Mass Conservation & Level-set transport equation: g Eqn. Momentum Predictor: UEqn Pressure Poisson Eqn.: pEqn Momentum Corrector Reinitilization Procedure

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

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)

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 Reinitilization LS-4: (Kinzel 2008) Notes: Neglecting smeared mass e2 is amount neglected

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

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

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

Test Cases: Dam Break

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

Test Cases: 2-D Water Drop in Oil Mass conservation Mixed conditions 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

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

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

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

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

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

Test Cases: Submerged Hydrofoil Free surface flows Sharp interface Level-Set sharpening Signed Distance Boundary Conditions

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

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. 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

Summary Conclusions Clearly problem dependent Future 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

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.