arl penn state computational mechanics comparison of interface capturing methods using openfoam 4 th...

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.


• Background– Motivation– Interface Capturing

• Numerical Approach– Volume of Fluid– Level Set Methods

• Test Cases• Summary

Outline


• Supercavitating vehicle simulation – Drag reduction– Performance predictions– Vehicle dynamics– Ventilation gas required

• Methods of cavity formation – Ventilation

• Air ventilated

– Vaporous• Water boils

Background: Motivation


• 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





Outline


• 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


• 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


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


• 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


• Signed-Distance Function– Without variable transformations (Kinzel 2008)

• Realizable Scaled (Kinzel 2008, Kinzel et al. 2009)

– Algebraic sharpening. No solution to PDE!


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


• 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)


1 * 1/2 *m m m mrf

Notes: • m*: after gEqn• m+1/2: after reinitialization





Outline


• 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



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


• 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



LS-2 LS-4 LS-1 LS-3



• SDF Sharpening w/ VOF transport (LS-1)– ε has effect when

fls=1 and fr=1

– Damping and periodic reinitialization help



• Mass-Conserving (LS-2)– ε increases

conservation

– Damping and periodic reinitialization lowered conservation



• Transformed SDF Sharpening w/ VOF transport (LS-3)– ε has effect

when fls=1




• Realizable-Scaled (LS-4)– Higher ε clips

more, conserves less



Test Cases: Submerged Hydrofoil

• Free surface flows• Sharp interface• Level-Set sharpening• Signed Distance

Boundary Conditions





Outline


• 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


• 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


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

http://dx.doi.org/10.1016/j.jcp.2005.04.007

arl penn state computational mechanics comparison of interface capturing methods using openfoam 4 th...

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