proposal for an erosion model from cavitating flow simulations

Proposal for an erosion model from cavitating flow simulations

L. Krumenacker a , R. Fortes-Patellaa , A. Archerb

(a) LEGI, Grenoble - (b) EDF R&D, Chatou

June 5th and 6th, Grenoble

L. Krumenacker , R. Fortes-Patella , A. Archer SOCIETE HYDROTECHNIQUE DE FRANCE - June 5th and 6th, Grenoble 1 / 25

Objective of the study

CFD Simulations

• Vref = 20m.s−1

• Lref = 0.1m

• time step ≈ 10 µs

• dimension of a cell ≈ 1 mm

Figure: TUD profil studied in Darmstadt University of Technology

Modelisation

⇓ ? ⇓

Pitting

Figure: An example of pit profile ob-served on an aluminum sample surface[Fortes et al., 2012]

• Dimension of a pit ≈ 1 µm


Objective of the study

CFD Simulations

Modelisation

U

Pitting area

EWave

dPdt

Hypothetis: Spherical bubbles collapseand pressure waves damage the surface

• ∆pwave ≈ GPa

• δt ≈ 1-10 ns

Pitting


CFD code

Table of contents

1 CFD codeCavitationg flow simulationsTwo phases in CFD equationsModelisation of the air contentInterface impact on the equation

2 Bubbles’distributionBarotropic law correctionBubbles initialisationBubbles mobilities and implosionsSpherical bubbles’dynamic

3 ResultsGeometries and simulation conditionsEPFL results

4 Conclusion and Perspectives


CFD code Cavitationg flow simulations

Cavitating flow simulations

Hypothesis

• URANS with an Homogeneous approach

• Turbulence’s model k − ε RNG

• Reboud’s correction for turbulent viscosity (n=10) [Reboud, 2001]

• The Navier-Stokes equations for an homogeneous model [Ishii et al., 2006]:Conservation of mass:

∂ρm

∂t+∂ρmum,l

∂xl= 0

Conservation of momentum

∂ρmum,i

∂t+∂ρmum,ium,l

∂xl= −

∂pm

∂xi+∂σm,il − τm,il

∂xl+

��∑

k

Mk,i

• Barotropic law [Delannoy and Kueny, 1990]:

ρm =ρl + ρv

2+ρl − ρv

2sin

(pm − psat

c2min

2

ρl − ρv

) 0

200

400

600

800

1000

1000 1500 2000 2500 3000 3500 4000

de

nsity (

kg

/m3

)

pressure (Pa)


CFD code Two phases in CFD equations

Two phases in CFD equations

void ration

αgaz =Vgaz

Vmailleand αliq =

Vliq

Vmaille= (1− αgaz )

for the homogeneous model, we know a relation between the mixed fluid and eachphase with the volume ratio αk :

• Density: ρm = αgazρgaz + αliqρliq

• Pressure: pm = αgazpgaz + αliqpliq

Figure: Bubbles representation in a mesh

• The void ration is represented asbubbles in a mechanicalequilibrium.

• In a cell, each bubble has thesame radius Rm, the samepressure of air.

• Number of bubbles

Nbubbles

Vcell=

3αgaz

4πR3m


CFD code Modelisation of the air content

Air contents consideration

Liquid phase

• Density: ρliq = ρwater + ρair liq ≈ ρwater

Gaseous phase

• Density: ρgaz = ρvap + ρairgaz• Pressure: pgaz = pvap + pairgaz

pvap = psat(T ) et ρvap =psat

rvapT

new density variable

ρm = αgaz (ρvap + ρair ) + αliqρliq

Conservation of mass

∂ρm

∂t+∂ρmum,l

∂xl= 0


CFD code Modelisation of the air content

Modelisation of the air contents

• The total air density (liquid + gaz) is supposed constant in all the mesh Cair0(kg .m−3).

• The total air density is repartitioned between the liquid phase and the gaseousphase.

mair

Vmaille= Cair0 = αgazρairgaz + (1− αgaz ) ρairliq

• We suppose that density of air in the liquid phase and the pressure of air in thegaz phase can be related by Henry’s law [Sander, 1999]

Cair = Hpairgaz

Air characteristics in a cell

pairgaz =Cair0(

αgaz

rairT+ H (1− αgaz )

) et ρairgaz =pairgaz

rairT


CFD code Interface impact on the equation

Interface impact on the equations

For spherical bubbles of radius Rm and with a surface tension σ, the interface term ofthe momentum equation can be written as:∑

k

Mk,i =∂

∂xi

(αgaz

2σ

Rm

)

new pressure variable

pm = pm − αgaz2σ

Rm

Conservation of the momentum

∂ρmum,i

∂t+∂ρmum,ium,l

∂xl= −

∂

∂xipm +

∂σm,il − τm,il∂xl


Bubbles’distribution

Table of contents






Bubbles’distribution Barotropic law correction

Barotropic law correction

Original barotropic law

ρm =ρl + ρv

2+ρl − ρv

2sin

(pm − psat

c2min

2

ρl − ρv

)

αgaz =ρliq − ρm

ρliq − ρvap

Modified barotropic law

αgaz =1

2

[1 − sin

(pm − psat

c2min

2

ρliq − ρgaz0

)]ρm = αgaz (ρair + ρvap) + (1 − αgaz ) ρliq

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

100

1e-06 1e-05 0.0001 0.001 0.01 0.1 1

density -

rela

tive g

ap (

%)

alpha liquide

Cair0= 3ppm Cair0= 8ppm Cair0=19ppm

Pressure in each phase

The new variable definition and the mechanical equilibrium state of the bubbles give:

pm = pliq = pgaz −2σ

Rm


Bubbles’distribution Bubbles initialisation

Bubbles Radius

In each cells

Rm =2σ

pair + pvap − pliq

Air contents and bubble Radius

(a) Measured size dis-tribution functions forsmall bubbles in threedifferent water tunnels[Brennen, 2005]

1e+08

1e+09

1e+10

1e+11

1e+12

1e+13

1 10 100 1000

Num

ber

density d

istr

ibution function N

(R)

(m-4

)

Radius (um)

(b) for this model

Figure: Size distribution functions

0

20

40

60

80

100

120

140

160

180

200

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Radiu

s (

um

)

alpha_gaz

Cair0= 3ppm Cair0= 8ppm Cair0=19ppm

Figure: Distribution of radius for different aircontents


Bubbles’distribution Bubbles mobilities and implosions

Bubbles mobilities and collapses

In the cavitating sheet

• high void ratio - spherical bubbletheory non possible

•dp

dtlow

• Henry’s law respected

Periphery of cavitation volume

• bubble theory accepted

•dp

dthigh

• air mass transfer at the interfaceneglected

U p

Poche de cavitationcavitation sheet


Bubbles’distribution Spherical bubbles’dynamic

Spherical bubbles’dynamic

Keller’s equations for the bubbledynamics: [Blake and al., 1994]

• equation of state in the bubble

pvap = psat

pair = pair0

(R0

R

)3κ

• momentum equation at the bubbleinterface:

pR = pvap + pair −2σ

R−

4µ

RR

air + vapor

waterp

p + p

R0

inf

air vap

• momentum equation in the liquid[1−

1

c∞R

]RR +

3

2

[1−

1

3c∞R

]R2 =

[1 +

1

c∞R

]pR − p∞

ρ∞+

R

ρ∞c∞

d (pR − p∞)

dt



Bubble behaviour

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4

R/R

0

t/TR

0

1

2

3

4

5

6

7

8

0 0.5 1 1.5 2 2.5 3 3.5 4

Delta P

/Pin

f

t/TR

Figure: Bulle R0 = 10−4 m, pair0 = 3000 Pa, pinfty = 50000 Pa, pressure wave at 2 R0 from thebubble center.



vapor volume periphery

Pressure gap

dpliq

dt=��∂pliq

∂t+ ui

∂pliq

∂xi

U

Pitting area

EWave

dPdt

• One bubble represents the implosion of all the bubbles in the same cell.

• The keller equations allows us to determinate the pressure wave emits during thebubble collapse depending on the wall distance.

Pits and energy [Fortes-Patella et al., 2013]

Ewave =4πr2pmax (r)2

ρc∞δt = βVpit

temporary concession

• For the moment each bubble collapses individually and there is no collectiveeffect.

• We multiply the pit’s volume made by one bubble by the number of bubbles persecond who leave the cell.


Results

Table of contents






Results Geometries and simulation conditions

Geometries and simulation conditions

• Boundary conditions: Velocity for the inlet, Static pressure for the outlet

• Check control on the cavitation parameter at the inlet

σcav =p − psat12ρliqV

2ref

Figure: hydrofoil EPFL - NACA 65012 6 degres of incidence Vref = 15m.s−1 σcavin = 1.5 -

experimental studies by [Pereira et al., 1998]


Results EPFL results

0

5

10

15

20

25

30

35

0 20 40 60 80 100

Vd

(u

m3

/mm

2/s

)

corde (%)

(a) experimentation

0

20

40

60

80

100

120

140

0 20 40 60 80 100

Vd

(u

m3

/mm

2/s

)

corde %

(b) simulation

Figure: volume damage rate for the EPFL profil with an incidence of 4 degres - Vref = 20m.s−1

σin = 1.1


Results EPFL results

Resultat 6 degres

0

10

20

30

40

50

60

70

0 20 40 60 80 100V

d (

um

3/m

m2/s

)

corde (%)

(a) experimentation

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 20 40 60 80 100

Vd (

um

3/m

m2/s

)

corde %

(b) simulation


σin = 1.86

0

5

10

15

20

25

30

35

40

45

0 20 40 60 80 100

Vd (

um

3/m

m2/s

)

corde (%)

(a) experimentation

0

5

10

15

20

25

30

35

0 20 40 60 80 100

Vd (

um

3/m

m2/s

)

corde %

(b) simulation


σin = 1.5L. Krumenacker , R. Fortes-Patella , A. Archer SOCIETE HYDROTECHNIQUE DE FRANCE - June 5th and 6th, Grenoble 20 / 25

Conclusion and Perspectives

Table of contents







Conclusion

• This method has the advantage of taking into account:the superficial tension of the bubblesthe concentration of gaz

• give a pressure load on the wall

Perspectives

• Consideration of the bubbles cloudeffect

• Solid Response to the pressure load.

• New experimental data available

• calibration and validation of themethod

Figure: Simulation of a diaphragm[Mimouni et al., 2009] Figure: Simulation of the diaphragm



Thanks for your attention


References

References I

Blake and al. (1994).Bubble dynamics: Some things we did not know 10 years ago.Bubble Dynamics and interface phenomena.

Brennen (2005).Fundamentals of multiphase flows.Cambridge University Press.

Delannoy and Kueny (1990).Two phase flow approach in unsteady cavitation modelling.Cavitation and Multiphase Flow Forum, ASME-FED vol 98.

Fortes, R., Archer, A., and Flageul, C. (2012).Numerical and experimental investigations on cavitation erosion.The 26th IAHR Symposium on Hydraulic Machinery and Systems.

Fortes-Patella, Choffat, Reboud, and c, A. (2013).Mass loss simulation in cavitation erosion: Fatigue criterion approach.WEAR.

Ishii, Yoon, and Revankar (2006).Choking flow modeling with mechanical and thermal non-equilibrium.Int. Journal of Heat and Mass Transfer.


References

References II

Mimouni, Lavieville, and Archer (2009).Modelling and computation of cavitation with a two-phase flow approachdownstream an orifice.The 13th International Topical Meeting on Nuclear Reactor Thermal Hydraulics(NURETH-13),Kanazawa City, Ishikawa Prefecture, Japan.

Pereira, Avellan, and Dupont (1998).Prediction of cavitation erosion: an energy approach.Journal of Fluid Engineering, Transactions of the ASME.

Reboud (2001).Prevision de l’erosion de cavitation: couplage ecoulement et materiau. partie i:aspect materiau.Rapport Final - commande EDF-R&D N P41/C03332.

Sander (1999).Compilation of henry’s law constants for inorganic and organic species ofpotential importance in environmental chemistry.Air Chemistry Department Max-Planck Institute of Chemistry.


proposal for an erosion model from cavitating flow simulations

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