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MODELLING OF A VORTEX R I NG FLOW AT HIGH REYNOLDS NUMBERS

Sprays: modelling versus experimentation UK-Israel Workshop Quenns Hotel, Brighton UK July 16-18, 2007. MODELLING OF A VORTEX R I NG FLOW AT HIGH REYNOLDS NUMBERS. F. KAPLANSKI 1 , Y. FUKUMOTO 2 , S. SAZHIN 3 1 Tallinn University of Technology, Estonia 2 Kyushu University, Japan

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MODELLING OF A VORTEX R I NG FLOW AT HIGH REYNOLDS NUMBERS

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  1. Sprays: modelling versus experimentation UK-Israel Workshop Quenns Hotel, Brighton UK July 16-18, 2007 MODELLING OF A VORTEX RING FLOW AT HIGH REYNOLDS NUMBERS F. KAPLANSKI1, Y. FUKUMOTO2, S. SAZHIN3 1Tallinn University of Technology, Estonia 2Kyushu University, Japan 3The University of Brighton, UK

  2. Virtual image of a vortex ring flow Force acts impulsively

  3. Schematic view of a vortex ring

  4. Outline • Basic solution • Ring properties: translational velocity, energy, circulation, streamfunction • Effect of the Reynolds number • Turbulent vortex ring • Applications: ‘optimal’ vortex ring formation, vortex ring-like structuresin GDI engines

  5. Formulation of the problem , ring-to-core radius

  6. Basicsolution

  7. Limiting cases short times identical to the vorticity distributionobtained using the asymptotic analysis for rings at the initial stage (Chi-Tzung Wang et al., 1994) long times identical to the vorticity distribution obtained for ringsat the decaying stage (Phillip’s,1956) The solution at the initial stage has the Gaussian form.

  8. The Fourier- Hankel integral transform

  9. Kinetic energy and translationalspeed

  10. Limit These equations are identical withthose reported by Safffman (1970); (see Fukumoto & Kaplanski, 2007,submitted for publication).

  11. Limit This equation is identical withthe one reported by Rott & Cantwell (1993); (see Kaplanski & Rudi,2005).

  12. Streamfunction Limit Lamb, 1932 Circulation

  13. Comparison of the results (speed)

  14. Comparison of the results (kinetic energy)

  15. Effect of the Reynolds number (initial stage)Numerical simulations (S.Stanaway, B.J. Cantwell and P.R. Spalart, NASA Technical Memorandum 101041 (1988)) versus analytical results Fukumoto & Kaplanski, 2007, submitted for publication.

  16. Second Saffman’s formula for where , k and k´ are tunable constants .

  17. Comparison of the results Fukumoto&Kaplanski, 2007, submitted for publication.

  18. Comparison the results with experimental data(see Weigand & Gharib,1997)

  19. Turbulent and laminar vortex rings produced by an impulsive force (Glezer & Coles, 1990). Initial Reynolds number (a) 27000, (b) 7500.

  20. In a gross sense the overall mean field in a turbulent flow tends to behave somewhat like a very viscous constant Reynolds number flow. Brian J. Cantwell (Introduction to Symmetry Analysis, Cambridge Texts in Applied Mathematics, 2002)

  21. Arbitrary scales The solution exists when:

  22. Limit The solution exists when

  23. ‘Turbulent‘ scales- Phillip’s self-similar solution (cf. the Lugovtsov model (1970))

  24. Translational speed of the turbulent vortex ring ( Afanasyev et. al., 2004)

  25. Effect ofTimelines of the turbulent vortex ring for I/r=1

  26. Timelines of the laminar vortex ring for

  27. Entrainment diagrams fora turbulent ring Calculations Glezer&Coles, 1990

  28. Entrainment diagram for a laminar ring

  29. Schematic view of vortex ring generator (Gharib et al.,1998 )

  30. Formation stage(Gharib et al.,1998 ) L/D<4 ‘optimal’ ring Elim»0.3, Glim »2.0 L/D»4 L/D>4

  31. Estimate of the ‘formation’ number(kinematic approach, Shusser&Gharib, 2000) Equation for t at the pinch–off, based on the slug model where Criterion for the pinch-off

  32. Results B=0.6907,b =0.6775, Norbury’s data B=0.6350,b =0.7071, our data Norbury’s invariants give t=2.5 Our invariants give t=4.85

  33. Comparison of the results with experimental data (Cater et al., 2004) for Re=2000. Fukumoto&Kaplanski, 2007, submitted for publication.

  34. Comparison of the results with experimental data (Cater et al., 2004) for Re=2000. Fukumoto&Kaplanski, 2007, submitted for publication.

  35. Conclusions • A new vortex ring model, valid in the entire range of times, has been suggested. The modelagrees with earlier reported models for the initial and decaying stages of vortex ring development and experimental results. It can be considered as a viscous analog to the Norbury family of rings. • The model is shown to be useful for modelling high-Reynolds-number ring flows and turbulent vortex rings. It predicts the ‘formation number’ L/D for ‘optimal’ vortex rings.

  36. Acknowledgements • The authors are grateful to the Japan Society for the Promotion of Science(JSPS), the Estonian Science Foudation (ETF 6832), and the Engineering and Physical Sciences Research Council ( EPSRC, Project EP/E047912/1), UK,for financial support.

  37. Thank for your attention.

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