Coupling of VOF with LPT to improve cavitation modeling elia Vallier - - PowerPoint PPT Presentation

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Introduction Multi-scale model Bubbles identification Results Conclusions Coupling of VOF with LPT to improve cavitation modeling elia Vallier 1 Aur Johan Revstedt 1 akan Nilsson 2 H 1 Fluid Mechanics/Energy Sciences, LTH Lund

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Introduction Multi-scale model Bubbles identification Results Conclusions

Coupling of VOF with LPT to improve cavitation modeling

Aur´ elia Vallier 1 Johan Revstedt 1 H˚ akan Nilsson 2

1 Fluid Mechanics/Energy Sciences, LTH Lund University, Sweden 2 Applied Mechanics/Fluid Dynamics, Chalmers University of Technology, Sweden

2011-11-16

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 1 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions

Experiments

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 2 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions

Experiments Numerical simulations

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 2 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions

Experiments Numerical simulations Improvement Model the small bubbles which are present after the breakup of the attached cavity.

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 2 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions

Mass transfer cavitation model based on VOF (Volume Of Fluid) Bubbles larger than the grid size High vapor volume fraction Irregular structures: need to describe the interface

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 3 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions

Mass transfer cavitation model based on VOF (Volume Of Fluid) Bubbles larger than the grid size High vapor volume fraction Irregular structures: need to describe the interface Bubble cavitation model based on LPT (Lagrangian Particle Tracking) Bubbles smaller than the grid size Low vapor volume fraction Shape can be considered spherical

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 3 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions

Mass transfer cavitation model based on VOF (Volume Of Fluid) Bubbles larger than the grid size High vapor volume fraction Irregular structures: need to describe the interface Bubble cavitation model based on LPT (Lagrangian Particle Tracking) Bubbles smaller than the grid size Low vapor volume fraction Shape can be considered spherical Coupling of Eulerian and Lagrangian models : multi-scale model

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 3 / 21

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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,

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 4 / 21

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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

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 4 / 21

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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 = − ˙ m ρg

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 4 / 21

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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 = − ˙ m ρg Mass and momentum equations for the mixture ∇ · U = 1 ρl − 1 ρg

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

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 4 / 21

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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 = − ˙ m ρg Mass and momentum equations for the mixture ∇ · U = 1 ρl − 1 ρg

m, ∂ρU ∂t + ∇ · (ρU ⊗ U) = −∇p + µ∇2U + ρg − Sst + SP. Sst = σstκδn , n = ∇α |∇α| , κ = ∇n.

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 4 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions Results with the mass transfer cavitation model Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 5 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions LPT for the bubble cavitation model

Particle P: position xP, diameter DP , velocity UP and density ρP . dxP dt = UP, mP dUP dt =

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 6 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions LPT for the bubble cavitation model

Particle P: position xP, diameter DP , velocity UP and density ρP . dxP dt = UP, mP dUP dt =

Two-way coupling: SP = −1 Vcell∆t

mP ((Up)tout − (Up)tin)

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 6 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions LPT for the bubble cavitation model

Particle P: position xP, diameter DP , velocity UP and density ρP . dxP dt = UP, mP dUP dt =

Two-way coupling: SP = −1 Vcell∆t

mP ((Up)tout − (Up)tin) Four-way coupling: collision and coalescence.

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 6 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions LPT for the bubble cavitation model

Particle P: position xP, diameter DP , velocity UP and density ρP . dxP dt = UP, mP dUP dt =

Two-way coupling: SP = −1 Vcell∆t

mP ((Up)tout − (Up)tin) Four-way coupling: collision and coalescence. Bubble dynamics (Rayleigh-Plesset equation): pB − p∞ ρ = Rd2R dt2 + 3 2 dR dt 2 + 4µ R dR dt + 2σst ρR .

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 6 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions Results with the bubble cavitation model

Injection of bubbles, initial radius 50 µm.

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 7 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions Results with the bubble cavitation model Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 8 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions Results with the bubble cavitation model

Mass transfer cavitation model based on VOF (Volume Of Fluid)

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 9 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions Results with the bubble cavitation model

Mass transfer cavitation model based on VOF (Volume Of Fluid) Bubble cavitation model based on LPT (Lagrangian Particle Tracking)

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 9 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions Results with the bubble cavitation model

Mass transfer cavitation model based on VOF (Volume Of Fluid) Bubble cavitation model based on LPT (Lagrangian Particle Tracking)

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 2011-11-16 9 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions

Eulerian grid, h Lagrangian grid, hLAG = 4h

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 2011-11-16 10 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions

Eulerian grid, h Lagrangian grid, hLAG = 4h

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 2011-11-16 10 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions

Eulerian grid, h Lagrangian grid, hLAG = 4h

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

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 10 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions

Eulerian grid, h Lagrangian grid, hLAG = 4h

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 11

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 10 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions

Eulerian grid, h Lagrangian grid, hLAG = 4h

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 11 12

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 10 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions

Eulerian grid, h Lagrangian grid, hLAG = 4h

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 11 12 15 1

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 10 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions

Eulerian grid, h Lagrangian grid, hLAG = 4h

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 11 12 15 1 16 1 19 20 21 22 24 1 25 1 29 30 31

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 10 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions

Eulerian grid, h Lagrangian grid, hLAG = 4h

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 =

(1 − α) dΩ =

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 11 12 15 1 16 1 19 20 21 22 24 1 25 1 29 30 31

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 10 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions

Eulerian grid, h Lagrangian grid, hLAG = 4h

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 =

(1 − α) dΩ =

cell@ID

(1 − α) Vcell@ID xP = 1 VP

cell@ID

(1 − α) xcell@ID

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

Cell Number Bubble ID 10 11 12 15 1 16 1 19 20 21 22 24 1 25 1 29 30 31

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 10 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions

Eulerian grid, h Lagrangian grid, hLAG = 4h

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 =

(1 − α) dΩ =

cell@ID

(1 − α) Vcell@ID xP = 1 VP

cell@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 11 12 15 1 16 1 19 20 21 22 24 1 25 1 29 30 31

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 10 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions

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 vapor structure DP = ( 6 πVP )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 2011-11-16 11 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions Test case 1 Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 12 / 21

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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 in turbulent flows, AIChE Journal 2006.

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 13 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions Test case 1 Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 14 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions Test case 2 Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 15 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions Test case 2 Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 16 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions Test case 3 Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 17 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions Test case 3 Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 18 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions Test case 3 100 200 300 400 500 600 50 100 150 200 250 300 350 400 Pdf Particle diameter [µm] Particle size distribution αlim=0.9 αlim=0.95 αlim=0.99 Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 19 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions Test case 3

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 2011-11-16 20 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions Test case 3

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[ are not neglected anymore)

Mark the cells with α < 0.95 0.99

Cell Number Bubble ID 4 1 8 2

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 20 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions Test case 3

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[ are not neglected anymore)

Mark the cells with α < 0.95 0.99

Cell Number Bubble ID 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 2011-11-16 20 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions Test case 3

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[ are not neglected anymore)

Mark the cells with α < 0.95 0.99

Cell Number Bubble ID 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 2011-11-16 20 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions Test case 3

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[ are not neglected anymore) Less medium and large bubbles are captured (The ”connecting” cells with α ∈ [0.95, 0.99[ merge bubbles and the condition VP < 10%VLAG isn’t satisfied anymore)

Mark the cells with α < 0.95 0.99

Cell Number Bubble ID 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 2011-11-16 20 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions

Implementation of a multi-scale model.

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

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

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

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 21 / 21

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Introduction Multi-scale model Bubbles identification Results Conclusions

Implementation of a multi-scale model.

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

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

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

Onging work : test on a cavitating case of the complete model. Include mass transfer with the VOF approach. Include the description of the small bubbles dynamics with the LPT approach. Convert a particle back to the VOF approach when

it comes near an isosurface α = 0.5 or it is not suitable for LPT anymore (coalescence)

Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 21 / 21