coupling of vof with lpt to improve cavitation modeling · introduction multi-scale modelbubbles...

Introduction Multi-scale model Bubbles identification Results Conclusions

Coupling of VOF with LPTto improve cavitation modeling

Aurelia Vallier 1

Johan Revstedt 1

Hakan Nilsson 2

1 Fluid Mechanics/Energy Sciences, LTH Lund University, Sweden2 Applied Mechanics/Fluid Dynamics, Chalmers University of Technology, Sweden

2011-06-15

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 1 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

Experiments

Numerical simulations

ImprovementModel the small bubbles which are present after thebreakup of the attached cavity.

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 2 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

Experiments

Numerical simulations

ImprovementModel the small bubbles which are present after thebreakup of the attached cavity.

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 2 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

Experiments

Numerical simulations

ImprovementModel the small bubbles which are present after thebreakup of the attached cavity.

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 2 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

Mass transfer cavitation modelbased on VOF (Volume Of Fluid)

Bubbles larger than the grid size

High vapor volume fraction

Irregular structures: need todescribe the interface

Bubble cavitation model based onLPT (Lagrangian Particle Tracking)

Bubbles smaller than the grid size

Low vapor volume fraction

Shape can be considered spherical

1 Coupling of Eulerian and Lagrangian models : multi-scale model

2 Identify the small bubbles suitable for LPT

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 3 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

Mass transfer cavitation modelbased on VOF (Volume Of Fluid)

Bubbles larger than the grid size

High vapor volume fraction

Irregular structures: need todescribe the interface

Bubble cavitation model based onLPT (Lagrangian Particle Tracking)

Bubbles smaller than the grid size

Low vapor volume fraction

Shape can be considered spherical

1 Coupling of Eulerian and Lagrangian models : multi-scale model

2 Identify the small bubbles suitable for LPT

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 3 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

Mass transfer cavitation modelbased on VOF (Volume Of Fluid)

Bubbles larger than the grid size

High vapor volume fraction

Irregular structures: need todescribe the interface

Bubble cavitation model based onLPT (Lagrangian Particle Tracking)

Bubbles smaller than the grid size

Low vapor volume fraction

Shape can be considered spherical

1 Coupling of Eulerian and Lagrangian models : multi-scale model

2 Identify the small bubbles suitable for LPT

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 3 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

Mass transfer cavitation modelbased on VOF (Volume Of Fluid)

Bubbles larger than the grid size

High vapor volume fraction

Irregular structures: need todescribe the interface

Bubble cavitation model based onLPT (Lagrangian Particle Tracking)

Bubbles smaller than the grid size

Low vapor volume fraction

Shape can be considered spherical

1 Coupling of Eulerian and Lagrangian models : multi-scale model

2 Identify the small bubbles suitable for LPT

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 3 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

VOF for mass transfer cavitation model

Liquid volume fraction α ∈ [0, 1].

ρ = αρg + (1− α)ρl,

µ = αµg + (1− α)µl,

Transport equation for the vapor volume fraction

∂α

∂t+∇ · (αU) +∇ · (α(1− α)Ur) = Sα

Sα = − mρv

Mass and momentum equations for the mixture

∇ ·U = 0,∂ρU∂t

+∇ · (ρU⊗U) = −∇p+ µ∇2U + ρg − Sst + SP.

Sst = σstκδn.

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 4 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

VOF for mass transfer cavitation model

Liquid volume fraction α ∈ [0, 1].

ρ = αρg + (1− α)ρl,

µ = αµg + (1− α)µl,

Transport equation for the vapor volume fraction

∂α

∂t+∇ · (αU) +∇ · (α(1− α)Ur) = Sα

Sα = − mρv

Mass and momentum equations for the mixture

∇ ·U = 0,∂ρU∂t

+∇ · (ρU⊗U) = −∇p+ µ∇2U + ρg − Sst + SP.

Sst = σstκδn.

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 4 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

VOF for mass transfer cavitation model

Liquid volume fraction α ∈ [0, 1].

ρ = αρg + (1− α)ρl,

µ = αµg + (1− α)µl,

Transport equation for the vapor volume fraction

∂α

∂t+∇ · (αU) +∇ · (α(1− α)Ur) = Sα

Sα = − mρv

Mass and momentum equations for the mixture

∇ ·U = 0,∂ρU∂t

+∇ · (ρU⊗U) = −∇p+ µ∇2U + ρg − Sst + SP.

Sst = σstκδn.

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 4 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

VOF for mass transfer cavitation model

Liquid volume fraction α ∈ [0, 1].

ρ = αρg + (1− α)ρl,

µ = αµg + (1− α)µl,

Transport equation for the vapor volume fraction

∂α

∂t+∇ · (αU) +∇ · (α(1− α)Ur) = Sα

Sα = − mρv

Mass and momentum equations for the mixture

∇ ·U = 0,∂ρU∂t

+∇ · (ρU⊗U) = −∇p+ µ∇2U + ρg − Sst + SP.

Sst = σstκδn.

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 4 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

VOF for mass transfer cavitation model

Liquid volume fraction α ∈ [0, 1].

ρ = αρg + (1− α)ρl,

µ = αµg + (1− α)µl,

Transport equation for the vapor volume fraction

∂α

∂t+∇ · (αU) +∇ · (α(1− α)Ur) = Sα

Sα = − mρv

Mass and momentum equations for the mixture

∇ ·U = 0,∂ρU∂t

+∇ · (ρU⊗U) = −∇p+ µ∇2U + ρg − Sst + SP.

Sst = σstκδn.Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 4 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

LPT for bubble cavitation model

Particle P : position xP, diameter DP , velocity UP and density ρP .

dxP

dt= UP,

mPdUP

dt=∑

F.

Two-way coupling:

SP =1

Vcell∆t

∑P

mP ((Up)tout − (Up)tin)

Rayleigh-Plesset equation:

pB − p∞ρ

= Rd2R

dt2+

32

(dR

dt

)2

+4µR

dR

dt+

2σstρR

.

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 5 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

LPT for bubble cavitation model

Particle P : position xP, diameter DP , velocity UP and density ρP .

dxP

dt= UP,

mPdUP

dt=∑

F.

Two-way coupling:

SP =1

Vcell∆t

∑P

mP ((Up)tout − (Up)tin)

Rayleigh-Plesset equation:

pB − p∞ρ

= Rd2R

dt2+

32

(dR

dt

)2

+4µR

dR

dt+

2σstρR

.

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 5 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

LPT for bubble cavitation model

Particle P : position xP, diameter DP , velocity UP and density ρP .

dxP

dt= UP,

mPdUP

dt=∑

F.

Two-way coupling:

SP =1

Vcell∆t

∑P

mP ((Up)tout − (Up)tin)

Rayleigh-Plesset equation:

pB − p∞ρ

= Rd2R

dt2+

32

(dR

dt

)2

+4µR

dR

dt+

2σstρR

.

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 5 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

0 1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16 17

18 19 20 21 22 23 24 25 26

27 28 29 30 31 32 33 34 35

36 37 38 39 40 41 42 43 44

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 6 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

0 1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16 17

18 19 20 21 22 23 24 25 26

27 28 29 30 31 32 33 34 35

36 37 38 39 40 41 42 43 44

Mark the cells with α < 0.95(i.e. at least 5% of gas.)

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 6 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

0 1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16 17

18 19 20 21 22 23 24 25 26

27 28 29 30 31 32 33 34 35

36 37 38 39 40 41 42 43 44

Mark the cells with α < 0.95(i.e. at least 5% of gas.)

Cell Number Bubble ID

10 0

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 6 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

0 1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16 17

18 19 20 21 22 23 24 25 26

27 28 29 30 31 32 33 34 35

36 37 38 39 40 41 42 43 44

Mark the cells with α < 0.95(i.e. at least 5% of gas.)

Cell Number Bubble ID

10 0

11 0

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 6 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

0 1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16 17

18 19 20 21 22 23 24 25 26

27 28 29 30 31 32 33 34 35

36 37 38 39 40 41 42 43 44

Mark the cells with α < 0.95(i.e. at least 5% of gas.)

Cell Number Bubble ID

10 0

11 0

12 0

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 6 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

0 1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16 17

18 19 20 21 22 23 24 25 26

27 28 29 30 31 32 33 34 35

36 37 38 39 40 41 42 43 44

Mark the cells with α < 0.95(i.e. at least 5% of gas.)

Cell Number Bubble ID

10 0

11 0

12 0

15 1

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 6 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

0 1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16 17

18 19 20 21 22 23 24 25 26

27 28 29 30 31 32 33 34 35

36 37 38 39 40 41 42 43 44

Mark the cells with α < 0.95(i.e. at least 5% of gas.)

Cell Number Bubble ID

10 0

11 0

12 0

15 1

16 1

19 0

20 0

21 0

22 0

24 1

25 1

29 0

30 0

31 0

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 6 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

0 1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16 17

18 19 20 21 22 23 24 25 26

27 28 29 30 31 32 33 34 35

36 37 38 39 40 41 42 43 44

VP =

ZΩ

(1− α) dΩ =X

cell@ID

(1− α)Vcell@ID

Mark the cells with α < 0.95(i.e. at least 5% of gas.)

Cell Number Bubble ID

10 0

11 0

12 0

15 1

16 1

19 0

20 0

21 0

22 0

24 1

25 1

29 0

30 0

31 0

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 6 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

0 1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16 17

18 19 20 21 22 23 24 25 26

27 28 29 30 31 32 33 34 35

36 37 38 39 40 41 42 43 44

VP =

ZΩ

(1− α) dΩ =X

cell@ID

(1− α)Vcell@ID

xP =1

VP

Xcell@ID

(1− α) xcell@ID

Mark the cells with α < 0.95(i.e. at least 5% of gas.)

Cell Number Bubble ID

10 0

11 0

12 0

15 1

16 1

19 0

20 0

21 0

22 0

24 1

25 1

29 0

30 0

31 0

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 6 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

0 1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16 17

18 19 20 21 22 23 24 25 26

27 28 29 30 31 32 33 34 35

36 37 38 39 40 41 42 43 44

VP =

ZΩ

(1− α) dΩ =X

cell@ID

(1− α)Vcell@ID

xP =1

VP

Xcell@ID

(1− α) xcell@ID

VP < 10%VLAG?

Mark the cells with α < 0.95(i.e. at least 5% of gas.)

Cell Number Bubble ID

10 0

11 0

12 0

15 1

16 1

19 0

20 0

21 0

22 0

24 1

25 1

29 0

30 0

31 0

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 6 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

0 1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16 17

18 19 20 21 22 23 24 25 26

27 28 29 30 31 32 33 34 35

36 37 38 39 40 41 42 43 44

Equivalent spherical bubble determined from the irregular vaporstructure

DP = (6πVP )1/3

Create the point particle P (DP ,xP,UP) for the LPT simulation

Remove the bubble from the VOF simulation : α set to 1

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 7 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

Test case 1

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 8 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

Test case 1

Figure: VOF simulation

Figure: Andersson R. and Andersson B., Modeling the breakup of particles inturbulent flows, AIChE Journal 2006.

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 9 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

Test case 1

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 10 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

Test case 2

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 11 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

Test case 2

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 12 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

Test case 3

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 13 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

Test case 3

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 14 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

Test case 3

0

100

200

300

400

500

600

0 50 100 150 200 250 300 350 400

Pdf

Particle diameter [µm]

Particle size distribution

αlim=0.9αlim=0.95αlim=0.99

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 15 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

Test case 3

0 1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16 17

18 19 20 21 22 23 24 25 26

27 28 29 30 31 32 33 34 35

36 37 38 39 40 41 42 43 44

Mark the cells with α < 0.95 0.99

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 16 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

Test case 3

0 1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16 17

18 19 20 21 22 23 24 25 26

27 28 29 30 31 32 33 34 35

36 37 38 39 40 41 42 43 44

More small bubbles are captured (The”isolated” cells with α ∈ [0.95, 0.99[ arenot neglected anymore)

Mark the cells with α < 0.95 0.99

Cell Number Bubble ID

0 0

4 1

8 2

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 16 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

Test case 3

0 1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16 17

18 19 20 21 22 23 24 25 26

27 28 29 30 31 32 33 34 35

36 37 38 39 40 41 42 43 44

More small bubbles are captured (The”isolated” cells with α ∈ [0.95, 0.99[ arenot neglected anymore)

Mark the cells with α < 0.95 0.99

Cell Number Bubble ID

0 0

4 1

8 2

10 3

11 3

12 3

14 4

15 4

16 4

19 3

20 3

21 3

22 3

23

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 16 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

Test case 3

0 1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16 17

18 19 20 21 22 23 24 25 26

27 28 29 30 31 32 33 34 35

36 37 38 39 40 41 42 43 44

More small bubbles are captured (The”isolated” cells with α ∈ [0.95, 0.99[ arenot neglected anymore)

Mark the cells with α < 0.95 0.99

Cell Number Bubble ID

0 0

4 1

8 2

10 3

11 3

12 3

14 /4 3

15 /4 3

16 /4 3

19 3

20 3

21 3

22 3

23 3

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 16 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

Test case 3

0 1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16 17

18 19 20 21 22 23 24 25 26

27 28 29 30 31 32 33 34 35

36 37 38 39 40 41 42 43 44

More small bubbles are captured (The”isolated” cells with α ∈ [0.95, 0.99[ arenot neglected anymore)

Less medium and large bubbles arecaptured (The ”connecting” cells withα ∈ [0.95, 0.99[ merge bubbles and thecondition VP < 10%VLAG isn’t satisfiedanymore)

Mark the cells with α < 0.95 0.99

Cell Number Bubble ID

0 0

4 1

8 2

10 3

11 3

12 3

14 /4 3

15 /4 3

16 /4 3

19 3

20 3

21 3

22 3

23 3

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 16 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

Implementation of a multi-scale model.

VOF model complemented with a two-way coupling LPT model.Identification of the small bubbles suitable for LPT approach.

Model applied to simple cases of air bubbles breaking up under theimpact of a water jet.

Ability to capture and track the small bubbles which a pure VOF modelfails to describe.Particles size and distribution depend on the parameters introduced.

Future work : test on a cavitating case.

Include mass transfer with the VOF approach

Include the description of the small bubbles dynamics with the LPTapproach

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 17 / 17

Introduction Multi-scale model Bubbles identification Results Conclusions

Implementation of a multi-scale model.

VOF model complemented with a two-way coupling LPT model.Identification of the small bubbles suitable for LPT approach.

Model applied to simple cases of air bubbles breaking up under theimpact of a water jet.

Ability to capture and track the small bubbles which a pure VOF modelfails to describe.Particles size and distribution depend on the parameters introduced.

Future work : test on a cavitating case.

Include mass transfer with the VOF approach

Include the description of the small bubbles dynamics with the LPTapproach

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-06-15 17 / 17