LPT with random walk model And interPhaseChangeFoam with non

LPT with random walk model And interPhaseChangeFoam with non-homogeneous nuclei distribution.

Aurélia Vallier Division of Fluid Mechanics Department of Energy Sciences Lund Institute of Technology, Lund Sweden 1 Lund university / Energy Sciences / Fluid Mechanics

INTRODUCTION

Cavitation in hydro-turbines - Travelling bubble - Attached cavitation - Sheet cavitation - Cloud cavitation - Supercavitation - Vortex cavitation → interPhaseChangeFoam (based on Sauer model)

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OUTLINE

• Sauer model • Nuclei distribution • Sauer model with a non-uniform nuclei distribution

3 Lund university / Energy Sciences / Fluid Mechanics

SAUER MODEL Pressure drop

Cavitation number

σ=

p∞− p v 1/ 2ρu 2∞

4 n0 . πR 3 3 α= 4 1+ n0 . πR 3 3

Nuclei

Nuclei density n0 Nuclei radius R

Two phase flow

Interface tracking Transport equation Mass transfert

Volume Of Fluid

Turbulence

Turbulence model

RANS (2D) LES(3D)

Bubble dynamics

Rayleght-Plesset equation

Lund university / Energy Sciences / Fluid Mechanics

= v l 1− =v l 1− ∂ ˙  =m ∇ .  U ∂t l



dR 2 p  R− p ∞ = dt 3 l

4

SAUER MODEL - Achievements The mechanisms of the unsteady behavior are well simulated.

. Attached cavity t=0.01 s

. Re-entrant jet

t=0.025 s

. Cloud transported downstream

t=0.075 s

. Vapor shedding

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SAUER MODEL - Limitations The region with low void fraction is not reproduced correctly.

. Transition between the attached cavity and the cloud of vapor

. The break-off with many small structures

. Compressibility → collapse, shock wave, erosion

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SAUER MODEL Pressure drop

Cavitation number

σ=

p∞− p v 1/ 2ρu 2∞

4 n0 . πR 3 3 α= 4 1+ n0 . πR 3 3

Nuclei

Nuclei density n0 Nuclei radius R

Two phase flow

Interface tracking Transport equation Mass transfert

Volume Of Fluid

Turbulence

Turbulence model

RANS (2D) + RandomWalk LES(3D)

Bubble dynamics

Rayleght-Plesset equation

Lund university / Energy Sciences / Fluid Mechanics

= v l 1− =v l 1− ∂ ˙  =m ∇ .  U ∂t l



dR 2 p  R− p ∞ = dt 3 l

7

SAUER MODEL Pressure drop

Cavitation number

σ=

p∞− p v 1/ 2ρu 2∞

4 n0 . πR 3 3 α= 4 1+ n0 . πR 3 3

Nuclei

Nuclei density n0 Nuclei radius R

Two phase flow

Interface tracking Transport equation Mass transfert

Volume Of Fluid

Turbulence

Turbulence model

RANS (2D) + RandomWalk LES(3D)

Bubble dynamics

Rayleght-Plesset equation

Lund university / Energy Sciences / Fluid Mechanics

= v l 1− =v l 1− ∂ ˙  =m ∇ .  U ∂t l



dR 2 p  R− p ∞ = dt 3 l

8

SAUER MODEL Pressure drop

Cavitation number

σ=

Nuclei

Nuclei density n0 Nuclei radius R

Two phase flow

Interface tracking Transport equation Mass transfert

Turbulence

Bubble dynamics

Turbulence model

Volume Of Fluid

=v l 1− =v l 1− ∂  = m˙ ∇ .  U ∂t l

RANS (2D) + RandomWalk LES(3D)

→ numerical simulations? Rayleght-Plesset equation

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1/ 2ρu 2∞

4 n0 . πR 3 3 α= 4 1+ n0 . πR 3 3

Nuclei distribution? Assumption: uniform Experiments : non uniform

p∞− p v



dR 2 p  R− p∞ = dt 3 l

9

OUTLINE

• Sauer model • Nuclei distribution • Sauer model with a non-uniform nuclei distribution

10 Lund university / Energy Sciences / Fluid Mechanics

NUCLEI DISTRIBUTION - LPT Mean flow data (Converged, steady state solution – RANS)

U

Particles (Lagrangian Particle Tracking LPT)

P

x p , Dp ,U p ,mp

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NUCLEI DISTRIBUTION - LPT Mean flow data (Converged, steady state solution – RANS)

U

Particles (Lagrangian Particle Tracking LPT)

P

x p , Dp ,U p ,mp

{ 12 Lund university / Energy Sciences / Fluid Mechanics

NUCLEI DISTRIBUTION - LPT Mean flow data (Converged, steady state solution – RANS)

U

Particles (Lagrangian Particle Tracking LPT)

P

x p , Dp ,U p ,mp

{ Turbulence inhomogeneities Lund university / Energy Sciences / Fluid Mechanics

Random Walk model

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NUCLEI DISTRIBUTION - Random walk model U ⇒ U =U U fluct U

fluct

=  2k /3

ue , t e , l e

N(0, 1)

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NUCLEI DISTRIBUTION - Random walk model U ⇒ U =U U fluct U

fluct

=  2k /3

ue , t e , l e

N(0, 1) Eddy life time

0.63 3/ 2 l e C  k / t e= = ue  2k /3

le  Particle transit time t tr =− P ln 1−  P∣U −U P∣ Interaction time

t inter =min t e , t tr 

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NUCLEI DISTRIBUTION – Problem Set Up Wall U=0 m/s ∇p=0 Inlet Ux=8m/s ∇p=0

Outlet ∇U=0 p=0 Pa

Injection 500 particles / time step Dp=1 to 50 μm ρ=1000 (hard particle) Up=0 Solution method LPT one way coupling with/without Random Walk Average on 50 flow-through periods Lund university / Energy Sciences / Fluid Mechanics

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NUCLEI DISTRIBUTION - Results

Dp=40 μm

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NUCLEI DISTRIBUTION - Results

Dp=1 μm

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NUCLEI DISTRIBUTION - Results

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NUCLEI DISTRIBUTION - Results

Dp=50 μm Lund university / Energy Sciences / Fluid Mechanics

SAUER MODEL Pressure drop

Cavitation number

σ=

Nuclei

Nuclei density n0 Nuclei radius R

Two phase flow

Interface tracking Transport equation Mass transfert

Nuclei distribution?

Assumption: uniform Experiment: non uniform Turbulence model LPTTurbulence simulations: non uniform

p∞− p v 1/ 2ρu 2∞

4 n0 . πR 3 3 α= 4 1+ n0 . πR 3 3

Volume Of Fluid

= v l 1− =v l 1− ∂ ˙  =m ∇ .  U ∂t l

RANS (2D) + RandomWalk LES(3D)

Bubble dynamics of Sauer model Rayleght-Plesset → sensitivity to a equation non uniform nuclei content?



dR 2 p  R− p ∞ = dt 3 l

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OUTLINE

• Sauer model • Nuclei distribution • Sauer model with a non-uniform nuclei distribution

22 Lund university / Energy Sciences / Fluid Mechanics

NON UNIFORM NUCLEI CONTENT • InterPhaseChangeFoam n0=1e8 • Nuclei density Smaller nuclei density N=1e2 or 1e4 Large nuclei density n0=1e8 Thickness of the layer =0.5, 1, 2 or 4 mm • MyInterPhaseChangeFoam - volScalarField n0nonunif - funkySetField (1e-6 or 1e-4 in the domain, and 1 in the layer) - n0 → n0nonunif . n0 Lund university / Energy Sciences / Fluid Mechanics

NON UNIFORM NUCLEI CONTENT

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NON UNIFORM NUCLEI CONTENT Uniform N =

t=0.02s

t=0.03s

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t=0.05s

t=0.06s

t=0.07s

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CONCLUSIONS • Performances/limitations of Sauer model • Non uniform nuclei content → inception, → thickness and velocity of the re-entrant jet, → shape of the cavitating structures and volume of vapor.

Lund university / Energy Sciences / Fluid Mechanics