A 2009 combined experimental and three-dimensional two-phase simulation study of the transient spray from a swirl (hollow-cone) injector (Fuchimoto, Yanase, Mizushima and Senda, Fluid Dynamics Research 41:045503) found that many vortex rings are created successively early in the injection while only one ring survives later; that spray tip penetration grows as a power law in time at early times and logarithmically at later times; and that air entrainment into the spray core peaks near the sole surviving ring. The authors accounted for the difference in the tip-penetration time dependence between the two periods only qualitatively, by the change of the vortex rings being created or surviving, leaving open the quantitative mechanism behind these observations. The problem is to establish a quantitative mechanistic account of the vortex-ring population dynamics in hollow-cone swirl sprays - why successive early rings collapse while the final ring grows and survives, how this evolution causally produces the observed change of the tip-penetration growth law, and what controls the surviving ring's lifetime. This mechanistic core remains open in the literature to date.

Background

A pressure-swirl (hollow-cone) injector discharges liquid as a thin conical sheet that ruptures into droplets, producing a transient, droplet-laden hollow-cone spray of the kind used in gasoline direct-injection engines. Two phenomenological descriptors of such transient sprays are the spray tip penetration, the distance of the foremost spray front from the nozzle as a function of time, and the air entrainment, the rate at which ambient air is drawn into the spray. In such sprays the gas-phase motion organizes into vortex rings - toroidal rolls of circulating air, here generated by the momentum the droplets transfer to the air - which govern how far and how fast the spray develops and how well it mixes. Simulations of these flows typically couple an Eulerian description of the air (partial-differential equations solved on a grid) with a Lagrangian discrete-droplet model that tracks individual droplets; two-way coupling means the drag and reaction forces exchange momentum between phases. The droplet population is shaped by primary atomization (breakup of the liquid sheet at the nozzle, which sets the initial droplet sizes) and secondary atomization (further breakup of droplets by aerodynamic forces). Because resolving primary atomization is difficult, studies commonly prescribe an empirical inlet droplet-size distribution such as the Nukiyama-Tanasawa distribution instead of computing it; droplet coalescence (merging of colliding droplets) is often neglected. Vortical structures are extracted from simulated velocity fields using local criteria such as the QQ-criterion (a point belongs to a vortex when the rotation rate exceeds the strain rate), the λ2\lambda_2-criterion (a negative second eigenvalue of the symmetric part of the velocity gradient), or the Δ\Delta-criterion (a test that the discriminant of the characteristic polynomial of the velocity-gradient tensor be positive, indicating a complex-conjugate eigenvalue pair and hence local swirling flow), and the choice among them can affect which structures are found. Against this background, Fuchimoto, Yanase, Mizushima and Senda ('Dynamics of vortex rings in the spray from a swirl injector', Fluid Dynamics Research 41 (2009) 045503) developed an original three-dimensional two-phase simulation code with two-way momentum coupling and secondary atomization, validated its tip penetration against their own experiments, and reported the ring-population phenomenology summarized above: successive early vortex rings, a single late survivor whose translational velocity decays logarithmically before the ring collapses and disappears, maximum entrainment near that sole ring, and a spectral analysis showing coarse and fine wavy deformation of the ring alternating in time. The simulation prescribed a Nukiyama-Tanasawa inlet droplet-size distribution, taking, as the earlier conference version of the study states, the droplet and gas motion after primary breakup as the object of computation, and neglected droplet coalescence; these are documented modelling choices of that study. Prior and adjacent literature had established single spray-induced vortex rings experimentally and modelled them with idealized viscous vortex-ring theory (for example the model line of Kaplanski, Sazhin and co-workers), and had characterized spray-induced vortical structures experimentally, but did not explain the population dynamics of successive rings in hollow-cone sprays or derive the penetration growth laws from them.

Problem Statement

For a transient hollow-cone swirl spray injected into quiescent air (the pressure-swirl injector configuration of Fuchimoto, Yanase, Mizushima and Senda, Fluid Dynamics Research 41 (2009) 045503, simulated there with two-way coupled Eulerian air and Lagrangian droplets plus secondary atomization and validated against experiment), establish a quantitative mechanistic account of the spray's vortex-ring dynamics with a single coherent objective: to explain, and predict from first principles, how the vortex-ring population governs the macroscopic development of the spray. Concretely: (a) identify the physical processes and energy budget by which the many vortex rings formed successively in the early injection stage collapse while the final ring grows and survives - i.e. a stability/interaction mechanism for the successive rings, including what distinguishes the surviving ring; and (b) derive, rather than merely fit, the observed transition of spray tip penetration from early power-law growth in time to later logarithmic growth - a transition the 2009 study accounted for only qualitatively, by the change of the vortex rings being created or surviving - from this evolving vortical-structure dynamics, and determine the conditions controlling the surviving ring's translational-velocity decay, its ultimate collapse, and its lifetime. Sub-questions (a) and (b) are scientifically linked: (a) supplies the mechanism and (b) its macroscopic signature and test, and the account is to yield quantitative predictions - growth-law exponents and transition time, surviving-ring lifetime and trajectory, entrainment rates - that can be compared with experimental measurement.

References

  1. Tetsuya Fuchimoto, Shinichiro Yanase, Jiro Mizushima, Jiro Senda, 'Dynamics of vortex rings in the spray from a swirl injector', Fluid Dynamics Research 41 (2009) 045503, DOI 10.1088/0169-5983/41/4/045503, https://doi.org/10.1088/0169-5983/41/4/045503
  2. Tetsuya Fuchimoto, Shinichiro Yanase, 'Analysis of Vortical Structures in the Spray of Swirl Injector', Proceedings of the JSME Chugoku-Shikoku Branch Conference, 2005, 43.315, DOI 10.1299/jsmecs.2005.43.315, https://www.jstage.jst.go.jp/article/jsmecs/2005.43/0/2005.43_315/_article/-char/ja/
  3. Hu Ma, Xiaosong Wu, Feng Feng, Dong Wang, Chenglong Yang, Changfei Zhuo, 'An experimental study on fuel spray-induced vortex-like structures', Experimental Thermal and Fluid Science 57 (2014) 335-343, DOI 10.1016/J.EXPTHERMFLUSCI.2014.05.013, https://doi.org/10.1016/J.EXPTHERMFLUSCI.2014.05.013
  4. Jan Jedelský, Milan Malý, Noé Pinto del Corral, Graham Wigley, Lada Janáčková, Miroslav Jícha, 'Air–liquid interactions in a pressure-swirl spray', International Journal of Heat and Mass Transfer 121 (2018) 788-804, DOI 10.1016/J.IJHEATMASSTRANSFER.2018.01.003, https://doi.org/10.1016/J.IJHEATMASSTRANSFER.2018.01.003
  5. F. Kaplanski, S. S. Sazhin, S. Begg, Y. Fukumoto, M. Heikal, 'Dynamics of vortex rings and spray-induced vortex ring-like structures', European Journal of Mechanics - B/Fluids 29 (2010) 208-216, DOI 10.1016/j.euromechflu.2010.01.002, https://doi.org/10.1016/j.euromechflu.2010.01.002
  6. Hao Wu, Fujun Zhang, Zhenyu Zhang, Zhenyao Guo, Weizheng Zhang, Hongli Gao, 'On the role of vortex-ring formation in influencing air-assisted spray characteristics of n-heptane', Fuel 266 (2020) 117044, DOI 10.1016/j.fuel.2020.117044, https://doi.org/10.1016/j.fuel.2020.117044