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A mathematical model of steady-state cavitation in Diesel injectors

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. Structure. Introduction Phenomenon of cavitation Objectives Mathematical model of cavitation flow

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A mathematical model of steady-state cavitation in Diesel injectors

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  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 A mathematical model of steady-state cavitation in Diesel injectors

  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 A mathematical model of steady-state cavitation in Diesel injectors

  4. 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

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

  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 (20:1) Re = 12 600; CN = 5.5 A mathematical model of steady-state cavitation in Diesel injectors

  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 A mathematical model of steady-state cavitation in Diesel injectors

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

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

  10. R Mathematical model of cavitation flow • Volume fraction of vapour: R – radius of bubbles (m); n – number density (1/m3 liquid) A mathematical model of steady-state cavitation in Diesel injectors

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

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

  13. 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

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

  15. Results – Cavitation flow of water CN = 6.27 • Effect of cavitation number Photograph of cavitating flow (Roosen et al, 1996) in comparison with the results of computations of the vapour field. A mathematical model of steady-state cavitation in Diesel injectors

  16. Momentum conservation: VF transport equation: Scalable model of cavitation flow • nL3=idem: model for n • Ro/L=idem: Ro/ L→ 0 Similarity conditions: • Re=idem • CN=idem A mathematical model of steady-state cavitation in Diesel injectors

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

  18. max max S S = maximal rate of strain, 1/s; = maximal rate of strain, 1/s; ii ii m m = dynamic viscosity of liquid, Pa s; = dynamic viscosity of liquid, Pa s; m m = turbulent viscosity, Pa s; = turbulent viscosity, Pa s; t t = adjustable coefficient. = adjustable coefficient. C C t t Effect of shear stresses on cavitation flow Static liquid: Flowing liquid (Joseph, 1995): Effect of turbulent shear stresses: A mathematical model of steady-state cavitation in Diesel injectors

  19. Results – cavitation flow of Diesel fuel CN = 1.86 Measured (top, Winklhofer et al, 2001) and predicted (bottom) liquid-vapour fields. Distributions of static pressure and critical pressure along the nozzle. A mathematical model of steady-state cavitation in Diesel injectors

  20. 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

  21. 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

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

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