Numerical Study of Tip Vortex Cavitation

SHF Workshop : Hydraulic Machinery, Cavitation.

3-4 June 2015, Nantes (CETIM), France

Dr. Jean Decaix* and Pr. Cecile Munch, Univ. of Applied Sciences and Arts - WesternSwitzerland Valais, Sion, SwitzerlandPr. Guillaume Balarac, Univ. Grenoble Alpes, LEGI, CNRS, F38000, Grenoble, FranceDr. Matthieu Dreyer and Pr. Mohamed Farhat, Ecole Polytechnique Federale de Lausanne,Laboratory for Hydraulic Machines, Lausanne, Switzerland

*jean.decaix@hevs.ch

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CONTENTS

1. CONTEXT

2. OBJECTIVES

3. TEST CASE

4. MODELLING

5. NUMERICAL SET UP

6. RESULTS

7. CONCLUSION

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CONTEXT

HYDRONET 2

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TIP VORTEX

Tip vortex in Kaplan turbine

• Promotes cavitation.

• Flow topology of the tip-leakageflow ?

• Gap width influence ?

• Vortex control ?

• Scale up from the model to theprototype ?

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OBJECTIVES

COUPLING EXPERIMENTAL AND NUMERICAL INVESTIGATION

• To investigate the gap width influence.

• To investigate the flow topology in the gap.

• To investigate the cavitation influence.

Induced vortex

Tip-leakage vortex

Tip-separation vortex

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TEST CASE : NACA0009

Main characteristics

• Chord length : c = 0.1 m.

• Inlet velocity : U∞ = 10 m/s.

• Reynolds number : Rec = 106.

• Cross section : 1.5c × 1.5c .

• Gap width : 0.1c .

• Blade incidence : 10.

• Cavitation number : σexpe = 2.1, σnum = 1.3.

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MODELLING : Turbulence

Reynolds Averaged Navier-Stokes (RANS) equationsContinuity and momentum equations :

∂ρ

∂t+

∂ρui∂xi

= 0

∂ρui∂t

+∂ρuiuj∂xj

= −∂p

∂xi+

∂ (σij + τij)

∂xj

With :

σij = ρν

(∂ui∂xj

+∂uj∂xi

)

; τij = −ρu′

i u′

j = ρνt

(∂ui∂xj

+∂uj∂xi

)

νt computed using the k − ω SST model.

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MODELLING : Cavitation

Homogeneous Relaxation Model (HRM)Mixture density defined as :

ρ = αLρL + (1− αL) ρV

Transport equation for αL :

∂αL

∂t+ uj

∂αL

∂xj= mv +mc

With (Kunz’s model [1]) :

mv =ρ

ρL

Cv

t∞

min (p − pvap , 0)

0.5 ρLU2∞

mc =ρ

ρL

Cc

t∞α2L

max (p − pvap , 0)

max (p − pvap , 0.01pvap)

[1] R. F. Kunz et al., ”A preconditioned Navier - Stokes method for two-phase Flows with application to cavitation prediction”,Comput. Fluids, vol. 29, pp. 849-875, 2000.

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NUMERICAL DOMAIN

Flow direction

Gap

Computational domain

• Length L = 8c : 2c upstreamand 5c downstream.

• Section : 1.5c × 1.5c .

Structured mesh

• 2 millions of nodes.

• 30 nodes in the gap.

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NUMERICAL SET UPOpenFOAM 2.1.0 : interPhaseChangeFoam algorithm

Boundary conditions

• No slip wall.

• Inlet : normal velocity u∞ = 10 m/s.

• Outlet : gradient free.

• Pressure reference sets in one cell.

Numerical Schemes

• Time scheme : second order (”backward”) with ∆ t = 10−5 s.

• Convective scheme : second order based on the Total Variation Diminishing (TVD)approach (”limitedLinear” and ”vanLeer”).

• Laplacian scheme : second order (”linear corrected”).

• Sharp interface is preseved introducing a counter-gradient in the transport equationfor αL.

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VORTEX CORE POSITIONwithout cavitation

x/c = 1x/c = 1.2

x /c= 1.5y

x

z

Pitchwise positionx/c y/c (Expe) y/c (RANS)1 0.14 0.131.2 0.18 0.181.5 0.30 0.22

Spanwise positionx/c z/c (Expe) z/c (RANS)1 0.12 0.101.2 0.13 0.131.5 0.16 0.15

More detailled results in :

1. J. Decaix, G. Balarac, M. Dreyer, M. Farhat, and C. Munch, 2015, ”RANS and LES simulations of the tip leakage vortex fordifferent gap widths”, Journal of Turbulence, Volume 16, Issue 4, pp. 309-341

2. M. Dreyer, J. Decaix, C. Munch, M. Farhat, 2014, ”Mind the gap : a new insight into the tip leakage vortex usingstereo-PIV”, Experiments in Fluids, Volume 55, Issue 11

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AXIAL VORTICITYwithout cavitation

Position x/c = 1

Experiment RANS Computation

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VELOCITY FIELD

Dimensionless streamwise velocity component in the mid-plane

Withoutcavitation

Withcavitation

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CAVITATING TIP VORTEX

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VORTEX CORE ?

Q =1

2

(

||Ω||2 − ||S||2)

x/c = -0.3

x/c = -0.1x/c = 0.1

x/c = 0.3x/c = 0.5

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VORTICITY EQUATION

∂~ω

∂t+ (~u.∇)~ω = (~ω.∇)~u − ~ω∇.~u + ν∇2~ω + νt∇

2~ω +1

ρ2∇ρ ∧∇p

︸ ︷︷ ︸

RHS

Cross plane at x/c = 0

Without cavitation With cavitation

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CONCLUSION

Non-cavitating tip vortex is captured accurately compared to theexperiment.

Vorticity at the vortex core is under-estimated

⇒ Pressure drop at the vortex core is also under-estimated.⇒ Cavitating computations are performed at a lower cavitation number

(σnum = 1.3 instead of σexpe = 2.1).

Cavitating tip vortex shows a qualitative agreement with the experiment.

Vortex core identification depends on the criterion chosen : Maximum of the void fraction. Maximum of the Q-criterion.

The production of streamwise vorticity is negative in case of cavitation.

A detail analysis is expected in an uncoming paper.

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THANK YOU FOR YOUR

ATTENTION

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