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 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
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
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
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
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
Introduction Multi-scale model Bubbles identification Results Conclusions Bubble cavitation model based on Mass transfer cavitation model based on VOF (Volume Of Fluid) LPT (Lagrangian Particle Tracking) Bubbles larger than the grid size Bubbles smaller than the grid size High vapor volume fraction Low vapor volume fraction Irregular structures: need to Shape can be considered spherical describe the interface Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 3 / 21
Introduction Multi-scale model Bubbles identification Results Conclusions Bubble cavitation model based on Mass transfer cavitation model based on VOF (Volume Of Fluid) LPT (Lagrangian Particle Tracking) Bubbles larger than the grid size Bubbles smaller than the grid size High vapor volume fraction Low vapor volume fraction Irregular structures: need to Shape can be considered spherical describe the interface 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
Introduction Multi-scale model Bubbles identification Results Conclusions VOF for mass transfer cavitation model Liquid volume fraction α ∈ [0 , 1] . ρ = αρ g + (1 − α ) ρ l , µ = αµ g + (1 − α ) µ l , Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 4 / 21
Introduction Multi-scale model Bubbles identification Results Conclusions VOF for mass transfer cavitation model Liquid volume fraction α ∈ [0 , 1] . ρ = αρ g + (1 − α ) ρ l , µ = αµ g + (1 − α ) µ l , Transport equation for the vapor volume fraction ∂α ∂t + ∇ · ( α U ) + ∇ · ( α (1 − α ) U r ) = S α Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 4 / 21
Introduction Multi-scale model Bubbles identification Results Conclusions VOF for mass transfer cavitation model Liquid volume fraction α ∈ [0 , 1] . ρ = αρ g + (1 − α ) ρ l , µ = αµ g + (1 − α ) µ l , Transport equation for the vapor volume fraction ∂α ∂t + ∇ · ( α U ) + ∇ · ( α (1 − α ) U r ) = S α S α = − ˙ m ρ g Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 4 / 21
Introduction Multi-scale model Bubbles identification Results Conclusions VOF for mass transfer cavitation model Liquid volume fraction α ∈ [0 , 1] . ρ = αρ g + (1 − α ) ρ l , µ = αµ g + (1 − α ) µ l , Transport equation for the vapor volume fraction ∂α ∂t + ∇ · ( α U ) + ∇ · ( α (1 − α ) U r ) = S α S α = − ˙ m ρ g Mass and momentum equations for the mixture � 1 − 1 � ∇ · U = m, ˙ ρ l ρ g ∂ρ U + ∇ · ( ρ U ⊗ U ) = −∇ p + µ ∇ 2 U + ρ g − S st + S P . ∂t Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 4 / 21
Introduction Multi-scale model Bubbles identification Results Conclusions VOF for mass transfer cavitation model Liquid volume fraction α ∈ [0 , 1] . ρ = αρ g + (1 − α ) ρ l , µ = αµ g + (1 − α ) µ l , Transport equation for the vapor volume fraction ∂α ∂t + ∇ · ( α U ) + ∇ · ( α (1 − α ) U r ) = S α S α = − ˙ m ρ g Mass and momentum equations for the mixture � 1 − 1 � ∇ · U = m, ˙ ρ l ρ g ∂ρ U + ∇ · ( ρ U ⊗ U ) = −∇ p + µ ∇ 2 U + ρ g − S st + S P . ∂t S st = σ st κδ n , n = ∇ α |∇ α | , κ = ∇ n . Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 4 / 21
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
Introduction Multi-scale model Bubbles identification Results Conclusions LPT for the bubble cavitation model Particle P : position x P , diameter D P , velocity U P and density ρ P . d x P = U P , dt d U P � m P = F . dt Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 6 / 21
Introduction Multi-scale model Bubbles identification Results Conclusions LPT for the bubble cavitation model Particle P : position x P , diameter D P , velocity U P and density ρ P . d x P = U P , dt d U P � m P = F . dt Two-way coupling: − 1 � S P = m P (( U p ) t out − ( U p ) t in ) V cell ∆ t P Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 6 / 21
Introduction Multi-scale model Bubbles identification Results Conclusions LPT for the bubble cavitation model Particle P : position x P , diameter D P , velocity U P and density ρ P . d x P = U P , dt d U P � m P = F . dt Two-way coupling: − 1 � S P = m P (( U p ) t out − ( U p ) t in ) V cell ∆ t P Four-way coupling: collision and coalescence. Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 6 / 21
Introduction Multi-scale model Bubbles identification Results Conclusions LPT for the bubble cavitation model Particle P : position x P , diameter D P , velocity U P and density ρ P . d x P = U P , dt d U P � m P = F . dt Two-way coupling: − 1 � S P = m P (( U p ) t out − ( U p ) t in ) V cell ∆ t P Four-way coupling: collision and coalescence. Bubble dynamics (Rayleigh-Plesset equation): � 2 = Rd 2 R p B − p ∞ dt 2 + 3 � dR + 4 µ dt + 2 σ st dR ρR . ρ 2 dt R Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 6 / 21
Introduction Multi-scale model Bubbles identification Results Conclusions Results with the bubble cavitation model Injection of bubbles, initial radius 50 µ m. Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 7 / 21
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
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
Introduction Multi-scale model Bubbles identification Results Conclusions Results with the bubble cavitation model Bubble cavitation model based on Mass transfer cavitation model based on VOF (Volume Of Fluid) LPT (Lagrangian Particle Tracking) Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 9 / 21
Introduction Multi-scale model Bubbles identification Results Conclusions Results with the bubble cavitation model Bubble cavitation model based on Mass transfer cavitation model based on VOF (Volume Of Fluid) 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
Introduction Multi-scale model Bubbles identification Results Conclusions Eulerian grid, h Lagrangian grid, h LAG = 4 h 36 37 38 39 40 41 42 43 44 27 28 29 30 31 32 33 34 35 18 19 20 21 22 23 24 25 26 9 10 11 12 13 14 15 16 17 0 1 2 3 4 5 6 7 8 Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 10 / 21
Introduction Multi-scale model Bubbles identification Results Conclusions Eulerian grid, h Mark the cells with α < 0 . 95 Lagrangian grid, h LAG = 4 h (i.e. at least 5 % of gas.) 36 37 38 39 40 41 42 43 44 27 28 29 30 31 32 33 34 35 18 19 20 21 22 23 24 25 26 9 10 11 12 13 14 15 16 17 0 1 2 3 4 5 6 7 8 Aurelia Vallier Coupling of VOF with LPT to improve cavitation modeling 2011-11-16 10 / 21
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