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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 #N , with Dr. Kam Ng as contract monitor.
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Outline Background Numerical Approach Test Cases Summary Motivation
Interface Capturing Numerical Approach Volume of Fluid Level Set Methods Test Cases Summary
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Background: Motivation
Supercavitating vehicle simulation Drag reduction Performance predictions Vehicle dynamics Ventilation gas required Methods of cavity formation Ventilation Air ventilated Vaporous Water boils
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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
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Outline Background Numerical Approach Test Cases Summary Motivation
Interface Capturing Numerical Approach Volume of Fluid Level Set Methods Test Cases Summary
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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
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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, & 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
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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
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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)
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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
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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
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Outline Background Numerical Approach Test Cases Summary Motivation
Interface Capturing Numerical Approach Volume of Fluid Level Set Methods Test Cases Summary
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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
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Test Cases: Dam Break
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Test Cases: Dam Break Initial Wave Subsequent events
Captured with all methods Subsequent events Level-set -> mass loss Scheme/parameter dependent VOF ->Mass conserved
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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 = kg/(m-s) ρoil = 850 kg/m3 μoil = 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
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Test Cases: 2-D Water Drop in Oil LS-2 LS-4 LS-1 LS-3
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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
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Test Cases: 2-D Water Drop in Oil Mass-Conserving (LS-2)
ε increases conservation Damping and periodic reinitialization lowered conservation
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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
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Test Cases: 2-D Water Drop in Oil Realizable-Scaled (LS-4)
Higher ε clips more, conserves less Damping and periodic reinitialization help
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Test Cases: Submerged Hydrofoil
Free surface flows Sharp interface Level-Set sharpening Signed Distance Boundary Conditions
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Outline Background Numerical Approach Test Cases Summary Motivation
Interface Capturing Numerical Approach Volume of Fluid Level Set Methods Test Cases Summary
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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
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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
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References Sussman, M., Smereka, P., and Osher, S A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114, 1 (Sep. 1994), DOI= Olsson, E., Kreiss, G., and Zahedi, S A conservative level set method for two phase flow II. J. Comput. Phys. 225, 1 (Jul. 2007), DOI= Olsson, E. and Kreiss, G A conservative level set method for two phase flow. J. Comput. Phys. 210, 1 (Nov. 2005), DOI= 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.
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