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A mathematical model of steady-state cavitation in Diesel injectors
S. Martynov, D. Mason, M. Heikal, S. SazhinInternal Engine Combustion GroupSchool of EngineeringUniversity of Brighton
A mathematical model of steady-state cavitation in Diesel injectors
Structure
Introduction Phenomenon of cavitation Objectives Mathematical model of cavitation flow Model implementation into PHOENICS Test cases Results Conclusions Acknowledgements
A mathematical model of steady-state cavitation in Diesel injectors
Introduction
Cavitation in the hydraulic, lubrication and fuel injection systems of automotive vehicle.
Cavitation effects: noise and vibration, rise in the hydraulic resistance, erosion wearing, improved spray breakup
A mathematical model of steady-state cavitation in Diesel injectors
Introduction
Effects of cavitation are described via the boundary conditions at the nozzle outlet: injection velocity, effective flow area, and velocity fluctuations.
A mathematical model of steady-state cavitation in Diesel injectors
Phenomenon of cavitation
Hydrodynamic cavitation - process of growth and collapse of bubbles in liquid as a result of reduction in static pressure below a critical (saturation) pressure.
Similarity criteria:
l
lUD
Revpp
ppCN
2
21
A mathematical model of steady-state cavitation in Diesel injectors
Phenomenon of cavitation
Cavitation starts from the bubble nuclei Similarity at macro-level (Arcoumanis et al, 2000) Scale effects prevent similarity at micro-level
Real-size nozzle (Ø =0.176mm) Scaled-up model (20:1) Re = 12 600; CN = 5.5
A mathematical model of steady-state cavitation in Diesel injectors
Objectives of study
Development of a scalable model for the hydrodynamic cavitation
Validation of the model against measurements of cavitation flows in Diesel injectors
A mathematical model of steady-state cavitation in Diesel injectors
Mathematical model of cavitation flow Simplified bubble-dynamics theory
bubbles of initial radius Ro and fixed concentration n
pppp
dt
dRv
l
v
sgn3
2
A mathematical model of steady-state cavitation in Diesel injectors
Mathematical model of cavitation flow
The homogeneous-mixture approach. Conservation equations for the mixture:
0~
~~
~~
j
j
x
u
t
k
kij
i
j
j
iT
ljij
jii
x
u
x
u
x
u
xx
p
x
uu
t
u~
~
3
2~
~
~
~1~Re
1~
~
2
1~
~~~
~~~
initial and boundary conditions; turbulent viscosity model; closure equations for properties.
A mathematical model of steady-state cavitation in Diesel injectors
Mathematical model of cavitation flow
R – radius of bubbles (m);n – number density (1/m3 liquid)
Rliquid
bubble
vapour
Volume fraction of vapour:
334
334
1 Rn
Rn
A mathematical model of steady-state cavitation in Diesel injectors
Mathematical model of cavitation flow
Void fraction transport equation:
ppsignpCNfCx
u
t vk
k
~)(~
~~
1
3/1~ nLC – cavitation rate constant
3/23/1)1()( f
Properties of the mixture:
lv )1(
lv )1( ),,,( constvlvl
L – hydrodynamic length scale
A mathematical model of steady-state cavitation in Diesel injectors
Model implementation into PHOENICS PHOENICS versions 2.2.1 and 3.6 Steady-state flows Collocated body-fitted grids CCM solver with compressibility factor Up-winding applied to densities in
approximations for the mass fluxes Mass fraction transport equation was
solved using the standard procedure Super-bee scheme applied to the mass
fraction equation for better resolution of steep density gradients
Turbulence model – RNG k-
A mathematical model of steady-state cavitation in Diesel injectors
Test cases – steady-state cavitation in rectangular nozzles
Roosen et al (1996):
Tap water, 20oC L=1mm, H=0.28mm, W=0.2mm, rin=0.03mm
Winklhofer, et al (2001):
Diesel fuel, 30oC L=1mm, H=0.30mm, W=0.3mm, rin=0.02mm
Measurements: Images of
cavitation Inlet/ outlet
pressures Pressure fields Velocity fields Mass flow rates
A mathematical model of steady-state cavitation in Diesel injectors
Results – Cavitation flow of water )(m104 314 n
Photograph and visualised velocity field of cavitating flow (Roosen et al, 1996) in comparison with the results of computations by the model.CN = 2.87
A mathematical model of steady-state cavitation in Diesel injectors
Results – Cavitation flow of water )(m104 314 n
Photograph of cavitating flow (Roosen et al, 1996) in comparison with the results of computations of the vapour field.
Effect of cavitation number CN = 6.27
A mathematical model of steady-state cavitation in Diesel injectors
Scalable model of cavitation flow
n L3=idem: model for n Ro/L=idem: Ro / L → 0
j
ij
ij
jii
xx
p
x
uu
t
u~
~
Re
1~
~
2
1~
~~~
~~~
ppsignpCNfCx
u
t vk
k
~)(~
~~
1
Momentum conservation:
VF transport equation:
Similarity conditions: Re=idem CN=idem
A mathematical model of steady-state cavitation in Diesel injectors
Scalable model of cavitation flow
pv – pmin = maximum tension in liquid;
pv = vapour pressure;
n* = liquid-specific number density parameter.
Number density of cavitation bubbles versus liquid tension.
2/3min
*
v
v
p
ppnn
)m(102 310*
n
A mathematical model of steady-state cavitation in Diesel injectors
Effect of shear stresses on cavitation flow
Flowing liquid (Joseph, 1995):
Static liquid:
= maximal rate of strain, 1/s;= dynamic viscosity of liquid, Pa s;= turbulent viscosity, Pa s;= adjustable coefficient.
maxiiS
tCt
= maximal rate of strain, 1/s;= dynamic viscosity of liquid, Pa s;= turbulent viscosity, Pa s;= adjustable coefficient.
maxiiS
tCtmax12 ii
tt
vcr
SC
ppp
vcr ppp
vii pSp max2
Effect of turbulent shear stresses:
A mathematical model of steady-state cavitation in Diesel injectors
Results – cavitation flow of Diesel fuel
Measured (top, Winklhofer et al, 2001) and predicted (bottom) liquid-vapour fields.
10 );(m102 318 tCn
max12 iit
tvcr SCpp
Distributions of static pressure and critical pressure along the nozzle.
CN = 1.86
A mathematical model of steady-state cavitation in Diesel injectors
Conclusions A homogeneous-mixture model of
cavitation with a transport equation for the volume fraction of vapour has been developed
An equation for the concentration of bubble nuclei has been derived based on the assumption about the hydrodynamic similarity of cavitation flows.
Effect of shear stresses on the cavitation pressure threshold has been studied
The model has been implemented in PHOENICS code and applied for analysis of cavitation flows in nozzles
A mathematical model of steady-state cavitation in Diesel injectors
Acknowledgements
PHOENICS support team
European Regional Development Fund (INTERREG Project “Les Sprays” – Ref 162/025/247)
Ricardo Consulting Engineers UK
A mathematical model of steady-state cavitation in Diesel injectors
Thank You