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).
4
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
