A mathematical model of steady-state cavitation in Diesel injectors

1
A mathematical model of steady-state cavitation
in Diesel injectors

• S. Martynov, D. Mason, M. Heikal, S. Sazhin
• Internal Engine Combustion Group
• School of Engineering
• University of Brighton

2
Structure

• Introduction
• Phenomenon of cavitation
• Objectives
• Mathematical model of cavitation flow
• Model implementation into PHOENICS
• Test cases
• Results
• Conclusions
• Acknowledgements

3
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

4
Introduction

• Effects of cavitation are described via the
  boundary conditions at the nozzle outlet
• injection velocity,
• effective flow area, and
• velocity fluctuations.

5
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

6
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 (201) Re
12 600 CN 5.5
7
Objectives of study

• Development of a scalable model for the
  hydrodynamic cavitation
• Validation of the model against measurements of
  cavitation flows in Diesel injectors

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Mathematical model of cavitation flow

• Simplified bubble-dynamics theory
• bubbles of initial radius Ro and
• fixed concentration n

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Mathematical model of cavitation flow

• The homogeneous-mixture approach.
• Conservation equations for the mixture

• initial and boundary conditions
• turbulent viscosity model
• closure equations for properties.

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Mathematical model of cavitation flow

• Volume fraction of vapour

R radius of bubbles (m) n number density
(1/m3 liquid)
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Mathematical model of cavitation flow

• Properties of the mixture

• Void fraction transport equation

cavitation rate constant
hydrodynamic length scale
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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-e

13
Test cases steady-state cavitation in
rectangular nozzles

• Roosen et al (1996)
• Tap water, 20oC L1mm, H0.28mm, W0.2mm,
  rin0.03mm
• Winklhofer, et al (2001)
• Diesel fuel, 30oC L1mm, H0.30mm, W0.3mm,
  rin0.02mm

• Measurements
• Images of cavitation
• Inlet/ outlet pressures
• Pressure fields
• Velocity fields
• Mass flow rates

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Results Cavitation flow of water
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
15
Results Cavitation flow of water

• Effect of cavitation number

CN 6.27
Photograph of cavitating flow (Roosen et al,
1996) in comparison with the results of
computations of the vapour field.
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Scalable model of cavitation flow

• Similarity conditions
• Reidem
• CNidem

• n L3idem model for n
• Ro/Lidem Ro / L ? 0

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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.
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Effect of shear stresses on cavitation flow
Static liquid
Flowing liquid (Joseph, 1995)
Effect of turbulent shear stresses
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Results cavitation flow of Diesel fuel
CN 1.86
Distributions of static pressure and critical
pressure along the nozzle.

• Measured (top, Winklhofer et al, 2001) and
  predicted (bottom) liquid-vapour fields.

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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

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Acknowledgements

• PHOENICS support team
• European Regional Development Fund (INTERREG
  Project Les Sprays Ref 162/025/247)
• Ricardo Consulting Engineers UK

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Thank You ?