Numerical simulation of cavitating channel flows ...

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inter-phase momentum exchange terms: Continuity: Momentum: (plus: Energy, Turbulence & EOS for each fluid). • Bubble growth based on Rayleigh-Plesset eq.
Numerical simulation of cavitating channel flows including non-condensable gases effects Abstract: L3.00010

Michele Battistoni1,2, Sibendu Som1, Douglas E. Longman1 1Argonne

National Laboratory, Lemont (IL) USA

2University

of Perugia, ITALY

66th Annual Meeting of the APS Division of Fluid Dynamics November 24-26, 2013 – Pittsburgh (PA), USA

Motivation  Cavitation occurs in fuel injector nozzles (e.g. direct injection engines)  It has a range of implications for jet atomization, spray dynamics and surface wear  The typical size of fuel nozzles have made quantitative observations extremely difficult for quite long time (scaled-up nozzles)  Recent advancements: quantitative high resolution experimental data provide unique test bench for CFD multi-phase models.

1. Giannadakis et. al., J. Fluid Mech. 2008.

 Demand for quantitative data about void fraction for modeling validation purposes. 2. Battistoni et. al., SAE 2012-01-1267

2

X-ray nozzle data(1)

non cavit.

cavitating

100 mm

nozzle Advanced Photon Source (APS) at Argonne National Laboratory • •

flat-field time-average x-ray images (1.5 mm) focused beam microprobe x-ray radiographic projection (5 mm)

1.

 Conditions & Fluid data: # Inlet p 1 10.6 bar 2 10.6 bar

Outlet p 0.87 bar 0.87 bar

T 25 °C 25 °C

Re 1.58x104 1.58x104

CN 11.2 11.2

notes std. fuel degassed fuel

Fluid

Density

Viscosity

Saturation Pressure

gasoline surrog.

781.8 kg/m3

9.35x10-4 Pa.s

640 Pa

Duke D., Kastengren A. L. , Tilocco Z., Swantek A. B., Powell C. F., Atomization and Sprays (2013) 3

Conservation equations  Mixture Model with VOF 𝜕𝜌 + 𝛻 ∙ 𝜌𝑣 = 0 𝜕𝑡 Momentum: 𝜕𝜌𝑣 + 𝛻 ∙ 𝜌𝑣 𝑣 = −𝛻𝑝 + 𝛻 ∙ 𝜏 + 𝜌𝑓 𝜕𝑡 𝜕𝜌𝑌𝑖 Species: + 𝛻 ∙ 𝜌𝑌𝑖 𝑣 = 𝛻 ∙ 𝜌𝐷𝑖 𝛻𝑌𝑖 + 𝑆𝑖 𝜕𝑡

𝑛

Continuity:

mixture density:

𝜌=

volume & mass fractions related through: void fraction:

𝛼𝑖 𝜌𝑖

𝑖=1

𝛼𝑖 𝜌𝑖 = 𝑌𝑖 𝜌

𝛼𝑔 =

𝑌𝑔 𝜌𝑔 𝑌𝑖 𝜌𝑖

(plus: Energy, Turbulence & EOS)

 Multi-Fluid model

inter-phase mass exchange terms: Γij n

Continuity:

𝜕𝛼𝑖 𝜌𝑖 + 𝛻 ∙ 𝛼𝑖 𝜌𝑖 𝑣𝑖 = 𝜕𝑡

Momentum:

𝜕𝛼𝑖 𝜌𝑖 𝑣𝑖 + 𝛻 ∙ 𝛼𝑖 𝜌𝑖 𝑣𝑖 𝑣𝑖 = −𝛼𝑖 𝛻𝑝 + 𝛻 ∙ 𝛼𝑖 𝜏𝑖 + 𝛼𝑖 𝜌𝑖 𝑓𝑖 + 𝜕𝑡

inter-phase momentum exchange terms: Mij

𝑗=1 Γij 𝑗≠𝑖

(plus: Energy, Turbulence & EOS for each fluid)

Cavitation model (mass transfer): • Homogeneous Relaxation Model (HRM)

(1)

• Bubble growth based on Rayleigh-Plesset eq. (2)

n

n

𝑗=1 Mij + 𝑣𝑖

𝑗=1 Γij 𝑗≠𝑖

𝑗≠𝑖

𝑑𝑌𝑣 𝑌𝑣 − 𝑌𝑣 = 𝑑𝑡 Θ

R=

2 𝑝𝑠𝑎𝑡 − 𝑝𝑒𝑓𝑓 − 𝑅𝑅 3 𝜌𝑙

1.

Schmidt, D.P., Gopalakrishnan, S. and Jasak, H., Int. J. of Multiphase Flow 36 (2010) 284–292.

2.

Lord Rayleigh, Philosophical Magazine 34, 94–98, (1917).

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Numerical setup Fluid model Components Compressibility Cavitation model Turbulence model Time integration Spatial discretiz. Time step [s] Physical time simulated Code

Mixture model (VOF) Multi-Fluid 3 components: 1. liquid, 2. vapor, 3. air compressible HRM

R-P

RANS - standard k-e Euler 1st order 2nd order in the order of 1.0E-9 s (CFL limited) 1.0E-3 s CONVERGE (CSI) FIRE (AVL)

Grid Unstructured hexahedral with fixed embedding and AMR: base size = 160 mm min size = 20, 10, 5 mm Time = 0.3E-3 s

5

Example: flow development (start-up) s

(void fraction)

6

Example: flow development (start-up) s

(void fraction)

7

Radiographic Void Projections Validation projection plane

nozzle wall air or vapor liquid

line of sight

Schematic of radiographic projection (1)

EXP (std. fuel)

Mixture model

Multi-Fluid model

• Y3 = 2E-5 by mass is a typical value for a liquid exposed to atmospheric pressure 1.

Duke D., Kastengren A. L. , Tilocco Z., Swantek A. B., Powell C. F., Atomization and Sprays (2013)

8

Grid sensitivity and Validation Mass flow rate [g/s] Multi-Fluid model 5.46 5.36 5.82

Min. cell size[mm] 18 9 Experimental value 0.35

Experimental data EXP.

total void fraction ag

0.3

Multi-Fluid - 20 mm FIRE - 18 microns Multi-Fluid - 10 mm FIRE - 9 microns

0.25 0.2 0.15 0.1 0.05 0 0

0.1

0.2

0.3

0.4

0.5 x/L

0.6

0.7

0.8

0.9

1

(*) Multi-Fluid model results 9

Gaseous fraction distributions with std. fuel Y = 2E-5 by mass 3

Contour plots on cut plane(*) Volume fractions

0

1

Subscripts: 1 = liquid 2 = vapor 3 = air g = 2+3 (gases)

air+vapor fraction ag cavitation vapor fraction a2

expansion of dissolved air air fraction a3

(*) Multi-Fluid model results 10

Pressure [Pa]

Pressure [Pa]

Pressure profiles with std. fuel 10

6

10

5

10

4

10

3

10

2

10

1

10

0

6

10

5

10

4

10

3

10

2

10

1

10

0

axis

wall

Saturation pressure

0.0

10

wall

0.2

0.4

x/L

0.6

0.8

1.0

axis

Saturation pressure

0.0

0.2

0.4

x/L

0.6

0.8

1.0

11

Effect of non-condensable gases(*) Subscripts: 1 = liquid 2 = vapor 3 = air g = 2+3 (gases)

Hypothesis: The void in the middle observed in experiments and simulations is due to the non-condensable gases

• Decreasing Y3, cavitation increases, while void in the middle decreases • Increasing Y3, cavitation is inhibited and void is increased

(*) Mixture model results 12

Effect of non-condensable gases: validation std. fuel

degassed fuel

Y3 = 2E-5 by mass

Y3 = 2E-7 by mass

2.50

-4

x 10

2.50

void in the path [m] -4

x 10 2.5

2.00

2.00 2

2 1.50

1.50

1.5 1

1.00

1.00

1

0.5 0

0.50

-0.25

EXP.

0.00

0.50

-0.25

0.25

Mixture model

Test with degassed fuel confirmed that the void in the middle was due to air expansion and not vapor production

EXP.

0.00

0.25

0

Mixture model 13

Conclusions  Cavitation model predictions assessed against quantitative x-ray data (APS at Argonne National Laboratory).  Effects of non-condensable gases:

 Void cloud in the channel center is due to air expansion.  Accounting for the compressibility of gaseous species is essential.  Without dissolved gas, the centerline void fraction cannot be explained with a cavitation model alone.  Conversely, when dissolved gas is included in the fuel, the results correlate well with the x-ray measurements. 14