{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,17]],"date-time":"2026-05-17T11:38:19Z","timestamp":1779017899277,"version":"3.51.4"},"reference-count":76,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,3,2]],"date-time":"2024-03-02T00:00:00Z","timestamp":1709337600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["52176134"],"award-info":[{"award-number":["52176134"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["9610601"],"award-info":[{"award-number":["9610601"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"APRC-CityU New Research Initiatives\/Infrastructure Support from Central of City University of Hong Kong","award":["52176134"],"award-info":[{"award-number":["52176134"]}]},{"name":"APRC-CityU New Research Initiatives\/Infrastructure Support from Central of City University of Hong Kong","award":["9610601"],"award-info":[{"award-number":["9610601"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This study computationally investigates small-scale flickering buoyant diffusion flames in externally swirling flows and focuses on identifying and characterizing various distinct dynamical behaviors of the flames. To explore the impact of finite rate chemistry on flame flicker, especially in sufficiently strong swirling flows, a one-step reaction mechanism is utilized for investigation. By adjusting the external swirling flow conditions (the intensity R and the inlet angle \u03b1), six flame modes in distinct dynamical behaviors were computationally identified in both physical and phase spaces. These modes, including the flickering flame, oscillating flame, steady flame, lifted flame, spiral flame, and flame with a vortex bubble, were analyzed from the perspective of vortex dynamics. The numerical investigation provides relatively comprehensive information on these flames. Under the weakly swirling condition, the flames retain flickering (the periodic pinch-off of the flame) and are axisymmetric, while the frequency nonlinearly increases with the swirling intensity. A relatively high swirling intensity can cause the disappearance of the flame pinch-off, as the toroidal vortex sheds around either the tip or the downstream of the flame. The flicker vanishes, but the flame retains axisymmetric in a small amplitude oscillation or a steady stay. A sufficiently high swirling intensity causes a small Damk\u00f6hler number, leading to the lift-off of the flame (the local extinction occurs at the flame base). Under the same swirling intensity but large swirling angles, the asymmetric modes of the spiral and vortex bubble flames were likely to occur. With R and \u03b1 increasing, these flames exhibit axisymmetric and asymmetric patterns, and their dynamical behaviors become more complex. To feature the vortical flows in flames, the phase portraits are established based on the velocity information of six positions along the axis of the flame, and the dynamical behaviors of various flames are presented and compared in the phase space. Observing the phase portraits and their differences in distinct modes could help identify the dynamical behaviors of flames and understand complex phenomena.<\/jats:p>","DOI":"10.3390\/sym16030292","type":"journal-article","created":{"date-parts":[[2024,3,4]],"date-time":"2024-03-04T05:31:12Z","timestamp":1709530272000},"page":"292","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Dynamical Behavior of Small-Scale Buoyant Diffusion Flames in Externally Swirling Flows"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6070-2715","authenticated-orcid":false,"given":"Tao","family":"Yang","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4794-5496","authenticated-orcid":false,"given":"Yuan","family":"Ma","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Peng","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, City University of Hong Kong, Kowloon Tong, Kowloon, Hong Kong"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1016\/S0082-0784(67)80172-3","article-title":"The fire whirl","volume":"11","author":"Emmons","year":"1967","journal-title":"Proc. Combust. Inst."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1146\/annurev.fl.21.010189.000533","article-title":"Turbulent diffusion flames","volume":"21","author":"Bilger","year":"1989","journal-title":"Annu. Rev. Fluid Mech."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1146\/annurev-fluid-122316-045209","article-title":"Fire whirls","volume":"50","author":"Tohidi","year":"2018","journal-title":"Annu. Rev. Fluid Mech."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"116870","DOI":"10.1016\/j.energy.2019.116870","article-title":"Entropy and flame transfer function analysis of a hydrogen-fueled diffusion flame in a longitudinal combustor","volume":"194","author":"Sun","year":"2020","journal-title":"Energy"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/j.proci.2020.11.006","article-title":"Combustion dynamics of large-scale wildfires","volume":"38","author":"Liu","year":"2021","journal-title":"Proc. Combust. Inst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"560","DOI":"10.52571\/PTQ.v16.n32.2019.578_Periodico32_pgs_560_571.pdf","article-title":"New parameters for the forest biomass waste ecofirewood manufacturing process optimization","volume":"16","author":"Ruoso","year":"2019","journal-title":"Peri\u00f3dico Tch\u00ea Qu\u00edmica"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"31467","DOI":"10.1016\/j.ijhydene.2021.07.005","article-title":"Review on hydrogen safety issues: Incident statistics, hydrogen diffusion, and detonation process","volume":"46","author":"Yang","year":"2021","journal-title":"Int. J. Hydrogen Energy"},{"key":"ref_8","first-page":"27","article-title":"The flicker of luminous flames","volume":"1\u20132","author":"Chamberlin","year":"1948","journal-title":"Proc. Combust. Inst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"765","DOI":"10.1016\/S0082-0784(53)80100-1","article-title":"Diffusion flames","volume":"4","author":"Barr","year":"1953","journal-title":"Proc. Combust. Inst."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"677","DOI":"10.1016\/S0082-0784(89)80075-X","article-title":"Buoyant diffusion flames","volume":"22","author":"Chen","year":"1989","journal-title":"Proc. Combust. Inst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1981","DOI":"10.1016\/S0082-0784(00)80604-9","article-title":"Flickering frequencies of diffusion flames observed under various gravity fields","volume":"28","author":"Sato","year":"2000","journal-title":"Proc. Combust. Inst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1017\/S002211209900508X","article-title":"The flickering candle: Transition to a global oscillation in a thermal plume","volume":"390","author":"Maxworthy","year":"1999","journal-title":"J. Fluid Mech."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1989","DOI":"10.1016\/S0082-0784(00)80605-0","article-title":"Combustion-induced buoyancy effects of an axisymmetric reactive plume","volume":"28","author":"Jiang","year":"2000","journal-title":"Proc. Combust. Inst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"997","DOI":"10.1017\/jfm.2016.358","article-title":"Diffusion-flame flickering as a hydrodynamic global mode","volume":"798","author":"Coenen","year":"2016","journal-title":"J. Fluid Mech."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"426","DOI":"10.1016\/j.combustflame.2018.02.011","article-title":"On the critical conditions for pool-fire puffing","volume":"192","author":"Coenen","year":"2018","journal-title":"Combust. Flame"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"A26","DOI":"10.1017\/jfm.2020.271","article-title":"Scaling of the puffing Strouhal number for buoyant jets and plumes","volume":"895","author":"Wimer","year":"2020","journal-title":"J. Fluid Mech."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"124104","DOI":"10.1063\/5.0073347","article-title":"Oscillations and collective behavior in convective flows","volume":"33","author":"Gergely","year":"2021","journal-title":"Phys. Fluids"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"L111501","DOI":"10.1103\/PhysRevFluids.7.L111501","article-title":"Puffing frequency of interacting buoyant plumes","volume":"7","author":"Patil","year":"2022","journal-title":"Phys. Rev. Fluids"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3141","DOI":"10.1016\/j.proci.2020.06.203","article-title":"Dynamic self-tuning, flickering and shedding in buoyant droplet diffusion flames under acoustic excitation","volume":"38","author":"Pandey","year":"2021","journal-title":"Proc. Combust. Inst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"105216","DOI":"10.1063\/5.0095714","article-title":"Vortex shedding controlled combustion of the wake flame of an n-heptane wetted porous sphere","volume":"12","author":"Zhang","year":"2022","journal-title":"AIP Adv."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"112002","DOI":"10.1016\/j.combustflame.2022.112002","article-title":"Insight into flickering\/shedding in buoyant droplet-diffusion flame during interaction with vortex","volume":"240","author":"Thirumalaikumaran","year":"2022","journal-title":"Combust. Flame"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1016\/0010-2180(95)00051-7","article-title":"Premixed and diffusion flames in a centrifuge","volume":"102","author":"Durox","year":"1995","journal-title":"Combust. Flame"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1089","DOI":"10.1016\/j.ijheatmasstransfer.2018.01.072","article-title":"Effects of co-flow and equivalence ratio on flickering in partially premixed flame","volume":"121","author":"Fujisawa","year":"2018","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1016\/S0082-0784(85)80522-1","article-title":"Visible structure of buoyant diffusion flames","volume":"20","author":"Zukoski","year":"1985","journal-title":"Proc. Combust. Inst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1829","DOI":"10.1016\/S0082-0784(88)80417-X","article-title":"The infinite candle and its stability\u2014A paradigm for flickering diffusion flames","volume":"21","author":"Buckmaster","year":"1988","journal-title":"Proc. Combust. Inst."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/0010-2180(93)90090-P","article-title":"Experiments on the periodic instability of buoyant plumes and pool fires","volume":"93","author":"Cetegen","year":"1993","journal-title":"Combust. Flame"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/S0010-2180(00)00154-1","article-title":"Diffusion flames and their flickering motions related with Froude numbers under various gravity levels","volume":"123","author":"Sato","year":"2000","journal-title":"Combust. Flame"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"043603","DOI":"10.1063\/1.2191851","article-title":"A numerical investigation of flame liftoff, stabilization, and blowout","volume":"18","author":"Briones","year":"2006","journal-title":"Phys. Fluids"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"087124","DOI":"10.1063\/5.0097496","article-title":"Velocity reconstruction in puffing pool fires with physics-informed neural networks","volume":"34","author":"Sitte","year":"2022","journal-title":"Phys. Fluids"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1156","DOI":"10.1017\/jfm.2018.707","article-title":"A vortex-dynamical scaling theory for flickering buoyant diffusion flames","volume":"855","author":"Xia","year":"2018","journal-title":"J. Fluid Mech."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1017\/S0022112097008410","article-title":"A universal time scale for vortex ring formation","volume":"360","author":"Gharib","year":"1998","journal-title":"J. Fluid Mech."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"781","DOI":"10.1007\/s00162-023-00671-0","article-title":"Faster flicker of buoyant diffusion flames by weakly rotatory flows","volume":"37","author":"Yang","year":"2023","journal-title":"Theor. Comput. Fluid Dyn."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"8164","DOI":"10.1021\/jp901487e","article-title":"Oscillation and synchronization in the combustion of candles","volume":"113","author":"Kitahata","year":"2009","journal-title":"J. Phys. Chem. A"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"015119","DOI":"10.1063\/1.5078674","article-title":"Role of buoyancy-driven vortices in inducing different modes of coupled behaviour in candle-flame oscillators","volume":"9","author":"Dange","year":"2019","journal-title":"AIP Adv."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"JTST0028","DOI":"10.1299\/jtst.2020jtst0028","article-title":"Similarity of dynamic behavior of buoyant single and twin jet-flame (s)","volume":"15","author":"Bunkwang","year":"2020","journal-title":"J. Therm. Sci. Technol."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"024218","DOI":"10.1103\/PhysRevE.104.024218","article-title":"Effect of gravity on synchronization of two coupled buoyancy-induced turbulent flames","volume":"104","author":"Tokami","year":"2021","journal-title":"Phys. Rev. E"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"109924","DOI":"10.1016\/j.expthermflusci.2019.109924","article-title":"Synchronization of dual diffusion flame in co-flow","volume":"110","author":"Fujisawa","year":"2020","journal-title":"Exp. Therm Fluid Sci."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.combustflame.2018.12.003","article-title":"On the influence of distance between two jets on flickering diffusion flames","volume":"201","author":"Liu","year":"2019","journal-title":"Combust. Flame"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"36145","DOI":"10.1038\/srep36145","article-title":"Synchronization in flickering of three-coupled candle flames","volume":"6","author":"Okamoto","year":"2016","journal-title":"Sci. Rep."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1893","DOI":"10.1016\/j.proci.2022.07.011","article-title":"Vortex interaction in triple flickering buoyant diffusion flames","volume":"39","author":"Yang","year":"2022","journal-title":"Proc. Combust. Inst."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"112526","DOI":"10.1016\/j.combustflame.2022.112526","article-title":"Dynamical mode recognition of triple flickering buoyant diffusion flames in Wasserstein space","volume":"248","author":"Chi","year":"2023","journal-title":"Combust. Flame"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"16994","DOI":"10.1038\/srep16994","article-title":"Arrays of coupled chemical oscillators","volume":"5","author":"Forrester","year":"2015","journal-title":"Sci. Rep."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"2067","DOI":"10.1016\/j.proci.2020.08.015","article-title":"Vortex formation and frequency tuning of periodically-excited jet diffusion flames","volume":"38","author":"Xia","year":"2021","journal-title":"Proc. Combust. Inst."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"112268","DOI":"10.1016\/j.combustflame.2022.112268","article-title":"Experimental investigation of the helical mode in a stratified swirling flame","volume":"244","author":"Wang","year":"2022","journal-title":"Combust. Flame"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"2859","DOI":"10.1063\/1.1308510","article-title":"The fluid mechanics of fire whirls: An inviscid model","volume":"12","author":"Battaglia","year":"2000","journal-title":"Phys. Fluids"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"2503","DOI":"10.1016\/j.proci.2014.05.110","article-title":"Buoyant pool fires under imposed circulations before the formation of fire whirls","volume":"35","author":"Lei","year":"2015","journal-title":"Proc. Combust. Inst."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.combustflame.2019.03.032","article-title":"Observed dependence of characteristics of liquid-pool fires on swirl magnitude","volume":"205","author":"Coenen","year":"2019","journal-title":"Combust. Flame"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"3705","DOI":"10.1016\/j.proci.2022.07.085","article-title":"An experimental study of the blue whirl onset","volume":"39","author":"Yang","year":"2022","journal-title":"Proc. Combust. Inst."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"112435","DOI":"10.1016\/j.combustflame.2022.112435","article-title":"Effect of circulation on flame heights over liquid fuel pools","volume":"246","author":"Ju","year":"2022","journal-title":"Combust. Flame"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1146\/annurev-fluid-010313-141300","article-title":"Dynamics of swirling flames","volume":"46","author":"Candel","year":"2014","journal-title":"Annu. Rev. Fluid Mech."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"2599","DOI":"10.1016\/j.proci.2006.07.109","article-title":"The prediction of flame heights and flame shapes of small fire whirls","volume":"31","author":"Chuah","year":"2007","journal-title":"Proc. Combust. Inst."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1753","DOI":"10.1016\/j.ces.2006.11.012","article-title":"Flame flickering frequency on a rotating Bunsen burner","volume":"62","author":"Gotoda","year":"2007","journal-title":"Chem. Eng. Sci."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"5423","DOI":"10.1016\/j.ijheatmasstransfer.2009.06.035","article-title":"Nonlinear analysis on dynamic behavior of buoyancy-induced flame oscillation under swirling flow","volume":"52","author":"Gotoda","year":"2009","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"3149","DOI":"10.1016\/j.proci.2016.06.072","article-title":"Experimental investigation on flame patterns of buoyant diffusion flame in a large range of imposed circulations","volume":"36","author":"Lei","year":"2017","journal-title":"Proc. Combust. Inst."},{"key":"ref_55","first-page":"1","article-title":"Fire dynamics simulator user\u2019s guide","volume":"1019","author":"McGrattan","year":"2013","journal-title":"NIST Spec. Publ."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"053202","DOI":"10.1103\/PhysRevFluids.4.053202","article-title":"Vortex-dynamical interpretation of anti-phase and in-phase flickering of dual buoyant diffusion flames","volume":"4","author":"Yang","year":"2019","journal-title":"Phys. Rev. Fluids"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1146\/annurev-fluid-010814-014711","article-title":"Ignition, liftoff, and extinction of gaseous diffusion flames","volume":"47","author":"Vera","year":"2015","journal-title":"Annu. Rev. Fluid Mech."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/0377-0265(77)90006-9","article-title":"A laboratory model of dust devil vortices","volume":"1","author":"Mullen","year":"1977","journal-title":"Dyn. Atmos. Ocean."},{"key":"ref_59","unstructured":"Kolb, E.J. (2018). An Experimental Approach to the Blue Whirl, University of California."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0360-1285(84)90118-7","article-title":"Chemical kinetic modeling of hydrocarbon combustion","volume":"10","author":"Westbrook","year":"1984","journal-title":"Prog. Energy Combust. Sci."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1016\/j.fuel.2015.09.068","article-title":"Momentum-and buoyancy-driven laminar methane diffusion flame shapes and radiation characteristics at sub-atmospheric pressures","volume":"163","author":"Fang","year":"2016","journal-title":"Fuel"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"1695","DOI":"10.1016\/S0082-0784(06)80198-0","article-title":"An experimental investigation of the pulsation frequency of flames","volume":"24","author":"Hamins","year":"1992","journal-title":"Proc. Combust. Inst."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"2581","DOI":"10.1016\/j.proci.2014.05.080","article-title":"Effect of gravity on puffing phenomenon of liquid pool fires","volume":"35","author":"Abe","year":"2015","journal-title":"Proc. Combust. Inst."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"484","DOI":"10.1016\/j.combustflame.2008.11.017","article-title":"Effects of fuel type and equivalence ratios on the flickering of triple flames","volume":"156","author":"Sahu","year":"2009","journal-title":"Combust. Flame"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1016\/j.fuel.2008.09.013","article-title":"Fuel effects on diffusion flames at elevated pressures","volume":"88","author":"Darabkhani","year":"2009","journal-title":"Fuel"},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"1404","DOI":"10.2514\/2.1462","article-title":"Functional dependence of flame flicker on gravitational level","volume":"39","author":"Bahadori","year":"2001","journal-title":"AIAA J."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"4971","DOI":"10.1016\/j.proci.2020.07.086","article-title":"Instability transition of a jet diffusion flame in quiescent environment","volume":"38","author":"Zhang","year":"2021","journal-title":"Proc. Combust. Inst."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"546","DOI":"10.1007\/s003480050415","article-title":"Experiments on the instability modes of buoyant diffusion flames and effects of ambient atmosphere on the instabilities","volume":"28","author":"Cetegen","year":"2000","journal-title":"Exp. Fluids"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"105113","DOI":"10.1063\/1.4825283","article-title":"The formation of turbulent vortex rings by synthetic jets","volume":"25","author":"Lawson","year":"2013","journal-title":"Phys. Fluids"},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"115101","DOI":"10.1063\/1.4935011","article-title":"Far-field momentum flux of high-frequency axisymmetric synthetic jets","volume":"27","author":"Xia","year":"2015","journal-title":"Phys. Fluids"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1146\/annurev.fl.04.010172.001211","article-title":"Vortex breakdown","volume":"4","author":"Hall","year":"1972","journal-title":"Annu. Rev. Fluid Mech."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1016\/S0360-1285(00)00022-8","article-title":"Vortex breakdown: A review","volume":"27","year":"2001","journal-title":"Prog. Energy Combust. Sci."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"eaba0827","DOI":"10.1126\/sciadv.aba0827","article-title":"The structure of the blue whirl revealed","volume":"6","author":"Chung","year":"2020","journal-title":"Sci. Adv."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"9457","DOI":"10.1073\/pnas.1605860113","article-title":"From fire whirls to blue whirls and combustion with reduced pollution","volume":"113","author":"Xiao","year":"2016","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/j.combustflame.2019.03.043","article-title":"Conditions for formation of the blue whirl","volume":"205","author":"Hu","year":"2019","journal-title":"Combust. Flame"},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"2041","DOI":"10.1016\/j.proci.2020.06.327","article-title":"Numerical description of axisymmetric blue whirls over liquid-fuel pools","volume":"38","author":"Carpio","year":"2021","journal-title":"Proc. Combust. Inst."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/3\/292\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:08:21Z","timestamp":1760105301000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/3\/292"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,2]]},"references-count":76,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,3]]}},"alternative-id":["sym16030292"],"URL":"https:\/\/doi.org\/10.3390\/sym16030292","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,3,2]]}}}