arl penn state computational mechanics comparison of interface capturing methods using openfoam 4 th...
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ARLPenn StateCOMPUTATIONAL MECHANICS
Comparison of Interface Capturing Methods using OpenFOAM
4th OpenFOAM Workshop4 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.
ARLPenn StateCOMPUTATIONAL MECHANICS
• Background– Motivation– Interface Capturing
• Numerical Approach– Volume of Fluid– Level Set Methods
• Test Cases• Summary
Outline
ARLPenn StateCOMPUTATIONAL MECHANICS
• Supercavitating vehicle simulation – Drag reduction– Performance predictions– Vehicle dynamics– Ventilation gas required
• Methods of cavity formation – Ventilation
• Air ventilated
– Vaporous• Water boils
Background: Motivation
ARLPenn StateCOMPUTATIONAL MECHANICS
• 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
Background: Interface Capturing
ARLPenn StateCOMPUTATIONAL MECHANICS
• Background– Motivation– Interface Capturing
• Numerical Approach– Volume of Fluid– Level Set Methods
• Test Cases• Summary
Outline
ARLPenn StateCOMPUTATIONAL MECHANICS
• 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
Numerical Approach: Volume of Fluid
0,1
0D
Dt
ARLPenn StateCOMPUTATIONAL MECHANICS
• 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)
Numerical Approach: interFoam with Level-Set
Time Step
Species Mass Conservation & Level-set transport equation:
g Eqn.
Momentum Predictor: UEqn
Pressure Poisson Eqn.: pEqn
Reinitilization Procedure
Momentum Corrector
ARLPenn StateCOMPUTATIONAL MECHANICS
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: Advantages/Disadvantages
ARLPenn StateCOMPUTATIONAL MECHANICS
• 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
Numerical Approach: Reinitilization Approaches
Reinitilization LS-2: (Olsson et al. 2007)
ˆ ˆ ˆ1 n n n
Reinitilization LS-1: (Sussman et al. 1994)
Transform g→ :f
Reinitialize :f
Transform f → :g
1tanh 2 1k
11 tanh
2 k
1sign
Notes: • Approximating Heaviside as:
• e is 0.5 interface thickness• Consistency with original H is given when k ~ 0.379
11 tanh
2H
k
ARLPenn StateCOMPUTATIONAL MECHANICS
• Signed-Distance Function– Without variable transformations (Kinzel 2008)
• Realizable Scaled (Kinzel 2008, Kinzel et al. 2009)
– Algebraic sharpening. No solution to PDE!
Numerical Approach: Reinitilization Approaches
Reinitilization LS-4: (Kinzel 2008)
Notes: • Approximating Heaviside as:
• e is 0.5 interface thickness• Consistency with original H is given when k ~ 0.379
11 tanh
2H
k
Reinitilization LS-3: (Kinzel 2008)
where:
12 1
2 2
2 11
2 1 1 2 1
k
1 21
2
1 0.5min max 1 ,0 ,1
2 0.5
nn
Notes: • Neglecting smeared mass • e2 is amount neglected
ARLPenn StateCOMPUTATIONAL MECHANICS
• 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)
Numerical Approach: Reinitilization Approaches
1 * 1/2 *m m m mrf
Notes: • m*: after gEqn• m+1/2: after reinitialization
ARLPenn StateCOMPUTATIONAL MECHANICS
• Background– Motivation– Interface Capturing
• Numerical Approach– Volume of Fluid– Level Set Methods
• Test Cases• Summary
Outline
ARLPenn StateCOMPUTATIONAL MECHANICS
• Mass conservation• Wave propagation• Sub-grid mixing
Test Cases: Dam Break
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
ARLPenn StateCOMPUTATIONAL MECHANICS
Test Cases: Dam Break
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5
Wat
er in
Dom
ain,
Vl(t
) /V
l,t=0
Time (s)
VOF
Olssen
Realizable-Scale
Sussman W/ VOF Transport
Kinzel Sharpening
• Initial Wave• Captured with all
methods
• Subsequent events • Level-set -> mass
loss
• Scheme/parameter dependent
• VOF ->Mass conserved
ARLPenn StateCOMPUTATIONAL MECHANICS
• 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
Test Cases: 2-D Water Drop in Oil
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
ARLPenn StateCOMPUTATIONAL MECHANICS
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
ARLPenn StateCOMPUTATIONAL MECHANICS
Test Cases: 2-D Water Drop in Oil
• Mass-Conserving (LS-2)– ε increases
conservation
– Damping and periodic reinitialization lowered conservation
ARLPenn StateCOMPUTATIONAL MECHANICS
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
ARLPenn StateCOMPUTATIONAL MECHANICS
Test Cases: 2-D Water Drop in Oil
• Realizable-Scaled (LS-4)– Higher ε clips
more, conserves less
– Damping and periodic reinitialization help
ARLPenn StateCOMPUTATIONAL MECHANICS
Test Cases: Submerged Hydrofoil
• Free surface flows• Sharp interface• Level-Set sharpening• Signed Distance
Boundary Conditions
ARLPenn StateCOMPUTATIONAL MECHANICS
• Background– Motivation– Interface Capturing
• Numerical Approach– Volume of Fluid– Level Set Methods
• Test Cases• Summary
Outline
ARLPenn StateCOMPUTATIONAL MECHANICS
• 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
Summary
ARLPenn StateCOMPUTATIONAL MECHANICS
• 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
Summary
ARLPenn StateCOMPUTATIONAL MECHANICS
1. 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
2. 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
3. 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
4. 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.
5. 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.
References