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. 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

3. 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

4. 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.

5. 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:

6. 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

7. 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

8. A mathematical model of steady-state cavitation in Diesel injectors Mathematical model of cavitation flow Simplified bubble-dynamics theory bubbles of initial radius R o and fixed concentration n

9. A mathematical model of steady-state cavitation in Diesel injectors 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.

10. A mathematical model of steady-state cavitation in Diesel injectors Mathematical model of cavitation flow R – radius of bubbles (m); n – number density (1/m 3 liquid) R Volume fraction of vapour:

11. A mathematical model of steady-state cavitation in Diesel injectors Mathematical model of cavitation flow Void fraction transport equation: – cavitation rate constant Properties of the mixture: – hydrodynamic length scale

12. 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-

13. A mathematical model of steady-state cavitation in Diesel injectors Test cases – steady-state c avitation in rectangular nozzles Roosen et al (1996): Tap water, 20 o C L=1mm, H=0.28mm, W=0.2mm, r in =0.03mm Winklhofer, et al (2001): Diesel fuel, 30 o C L=1mm, H=0.30mm, W=0.3mm, r in =0.02mm Measurements: Images of cavitation Inlet/ outlet pressures Pressure fields Velocity fields Mass flow rates

14. A mathematical model of steady-state cavitation in Diesel injectors 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. A mathematical model of steady-state cavitation in Diesel injectors Results – Cavitation flow of water 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

16. A mathematical model of steady-state cavitation in Diesel injectors Scalable model of cavitation flow n L 3 =idem: model for n R o /L=idem: R o / L 0 Momentum conservation:VF transport equation: Similarity conditions: Re=idem CN=idem

17. A mathematical model of steady-state cavitation in Diesel injectors Scalable model of cavitation flow p v – p min = maximum tension in liquid; p v = vapour pressure; n * = liquid-specific number density parameter. Number density of cavitation bubbles versus liquid tension.

18. 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. max ii S t C t = maximal rate of strain, 1/s; = dynamic viscosity of liquid, Pa s; = turbulent viscosity, Pa s; = adjustable coefficient. max ii S t C t Effect of turbulent shear stresses:

19. 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. Distributions of static pressure and critical pressure along the nozzle. CN = 1.86

20. 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

21. 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

22. A mathematical model of steady-state cavitation in Diesel injectors Thank You