ISSN 2658–5782
DOI 10.21662
Electronic Scientific Journal


© Институт механики
им. Р.Р. Мавлютова
УФИЦ РАН

Яндекс.Метрика

Bolotnova R.Kh., Faizullina E.A. Review of investigations for various modes formation of extremely expanding cryogenic liquids jets. Multiphase Systems. 20 (2025) 1. 12–19 (in Russian).
2025. Vol. 20. Issue 1, Pp. 12–19
URL: http://mfs.uimech.org/mfs2025.1.003,en
DOI: 10.21662/mfs2025.1.003
Review of investigations for various modes formation of extremely expanding cryogenic liquids jets
R.Kh. Bolotnova🖂, E.A. Faizullina
Mavlyutov Institute of Mechanics of UFRC RAS

Abstract

The review considers of investigations related to the experimental and theoretical study of formation an extremely expanding jets of cryogenic liquids when sprayed from a thin cylindrical channel into a vacuum chamber, caused by explosive boiling and pressure increase due to the avalanche growth of vapor microbubbles with the formation of fine droplets with a large opening jet angle. The relevance of such research follows from the importance of environmental protection for the safety of space exploration and is aimed at achieving reliable operation, efficiency and cost reduction of spacecraft engines using safe fuel cells such as a combination of liquid oxygen and hydrogen or methane. The importance of analyzing the distribution of fluid phases in a vacuum chamber, the diameter, concentration, and temperature of microdroplets is to determine the optimal modes that lead to successful ignition in order to intensify the rocket engine’s thrust. The scientific newness lies in the originality of the proposed approaches in the theoretical and numerical study of the designated problem based on the development of spatial models of a multiphase gas-vapor-liquid medium based on the laws of conservation of mass, momentum and energy of each phase in accordance with one- or two speeds and with two-temperature approach in a three-dimensional formulation, taking into account interfacial resistance, contact heat exchange and energy and mass exchange processes of evaporation and condensation. To describe the thermodynamic properties of the cryogenic liquid under study, it is important to construct wide-range equations of state of liquid and vapor in an analytical form. Numerical studies of the evolution of the shape and structure of an emerging vapor-liquid boiling cryogenicjet are aimed at obtaining detailed information on the problem under study and can be used as recommendations for the development of small rocket engines and devices powered by reactive thrust in a vacuum atmosphere and cryogenic temperatures.

Keywords

formation of expanding jets of cryogenic liquids,
experimental, theoretical and numerical studies

Article outline

Purpose. Analysis of experimental and theoretical studies of the formation of extremely expanding jets of cryogenic liquids.

Methodology. Experimental studies of cryogenic superheated liquids using as an example nitrogen were carried out on a test bench, where spray patterns and angles were determined using high-speed shadow graphics and the dependence of droplet velocities and diameters on injection conditions was obtained, which provided a comprehensive database for verifying numerical models and further numerical studies. Numerical modeling and investigation of formation process of expanding jet of liquid nitrogen in a vacuum space, implemented on the basis of multiphase models of gas- vapor-liquid mixtures that take into account nonequilibrium evaporation and condensation processes using modern methods of numerical implementation of the created model, namely, the development of new solvers in the environment of the OpenFOAM software package with the ability to simulate multi-scale temporary processes of sudden boiling and spraying of cryogenic liquid with formating an expanding jet in a vacuum chamber.

Findings. The nonequilibrium mass exchange processes of evaporation and condensation, taking into account contact heat exchange, that occur when a jet of liquid nitrogen boils when it flows out of a thin nozzle into a vacuum atmosphere, are considered. Estimates of the accuracy of the developed numerical method, which implements the proposed model based on the OpenFOAM software, are given. Numerical studies have shown the formation of a bubble flow region in the near zone at the outlet of the nozzle. It is shown that as the jet moves away from the nozzle, it enters the steam-drop flow mode, which dominates the bubble flow during the jet development. The influence of the degree of overheating on the spray angle of the jet, the formation and development of vortex zones during the transition from the bubble flow mode to the steam drip mode is analyzed with a valuation of the monodispersity level of the steam drip flow. A comparative analysis of numerical calculations with experimental points for the values of mass velocities, jet opening angles, and photographs of simulated experiments showed their satisfactory agreement.

Value. The review analyzes experimental and theoretical work related to the study of the dynamics of jet stream formation when cryogenic liquids flow through a nozzle from a high-pressure chamber into a vacuum atmosphere, due to an avalanche increase in the number of vapor microbubbles and the formation of jets with different ranges of opening angles, depending on the design of the experimental installation, initial conditions, degree of overheating of the working fluid, and its thermophysical properties. The importance of the results the considered studies is due to the possibility of their application in improving technological processes accompanied by sudden boiling and outflow of gas-vapor-saturated liquids from channels and pipes used in many areas of modern energy, the oil and gas industry, medicine, as well as in rocket technology, which use spraying methods at cryogenic temperatures. In particular, the use of liquid nitrogen as a working fluid is necessary in the study of gas-dynamic processes in cryogenic wind tunnels, for cooling superconducting magnets, and for spraying liquid nitrogen in medical practice for cryoablation with high-precision speed and temperature control.

References

  1. Rees A., Araneo L., Salzmann H., Kurudzija E., Suslov D., Lamanna G., Sender J., Oschwald M. Investigation of velocity and droplet size distributions of flash boiling LN2-Jets with phase doppler anemometry // In: 29th ILASS-Europe Conference, 2019. Paris, France. 8 p.
    https://elib.dlr.de/132832/1/Rees_ILASS2019.pdf
  2. Rees A., Salzmann H., Sender J., Oschwald M. Investigation of flashing LN2-Jets in terms of spray morphology, droplet size and velocity distributions // In: 8th EUCASS Conference, 2019. Madrid, Spain. Pp. 1–13.
    DOI: 10.13009/EUCASS2019-418
  3. Rees A., Araneo L., Salzmann H., Lamanna G., Sender J., Oschwald M. Droplet velocity and diameter distributions in flash boiling liquid nitrogen jets by means of phase doppler diagnostics // Experiments in Fluids Volume. 2020. V. 61, No. 182. 18 p.
    DOI: 10.1007/s00348-020-03020-7
  4. Нигматулин Р.И. Динамика многофазных сред // М.: Наука, 1987. Ч. 1. 464 c., Ч. 2. 360 с.
    Nigmatulin R.I. Dynamics of Multiphase Media. Vol.1, 2. Hemisphere, N.Y. 1990.
  5. Нигматулин Б.И., Сопленков К.И. Исследование нестационарного истечения вскипающей жидкости из каналов в термодинамически неравновесном приближении // Теплофизика высоких температур. 1980. Т. 18, № 1. C. 118–131.
    Nigmatulin B.I., Soplenkov K.I. Unsteady-state outflow of a boiling liquid from channels in a thermodynamically nonequilibriun approximation // High Temperature. 1980. V. 18, No. 1. Pp. 118–131 (in Russian).
    MathNet: tvt10023
  6. Губайдуллин А.А. Введение в волновую динамику газожидкостных сред // ТюмГНГУ, 2006. 86 с.
    Gubaidullin A.A. Introduction to the wave dynamics of gas-liquid media // TSU, 2006. 86 p. (in Russian).
  7. Ивашнев О.Е. Самоподдерживающиеся ударные волны в неравновесно кипящей жидкости: Автореф. дис. доктора физ.-мат. наук: 01.02.05. М., 2009. 40 с.
    Ivashnev O.E. Self-sustaining shock waves in nonequilibrium boiling liquid: abstract of thesis. dis. Doctor of Physics and Mathematics Sciences: 01.02.05. M., 2009. 40 p. (in Russian)
    http://mech.math.msu.su/~snark/files/vak/ark0.pdf
  8. Шагапов В.Ш., Ялаев А.В. Объемное вскипание жидкости, содержащей газовые зародыши // Теоретические основы химической технологии. 2012. Т. 46, № 4. С. 420–431.
    Shagapov V.Sh., Yalaev A.V. Volumetric boiling of a liquid containing gas nuclei // Theoretical Foundations of chemical Technology. 2012. V. 46, No. 4. Pp. 420–431 (in Russian).
    EDN: ozlcyx
  9. Xiufang Liu, Rong Xue, Yixiao Ruan, Liang Chen, Xingqun Zhang, Yu Hou. Effects of injection pressure difference on droplet size distribution and spray cone angle in spray cooling of liquid nitrogen // Cryogenics. 2017. No. 83. Pp. 57–63.
    DOI: 10.1016/j.cryogenics.2017.01.011
  10. Prashant Srivastava, Amitesh Kumar. Characterization of performance of multihole nozzle in cryospray // Cryobiology. 2020. No. 96. Pp. 197–206.
    DOI: 10.1016/j.cryobiol.2020.06.008
  11. Edwards A.R., O’Brien T.P. Studies of phenomena connected with the depressurization of water reactors // Journal of the British Nuclear Energy Society. 1970. V. 9, No. 2. Pp. 125–135.
    https://senior.app.box.com/BNES-VOL9-2
  12. Решетников А.В., Мажейко Н.А., Беглецов В.Н., и др. Динамика пульсаций при взрывном вскипании струй перегретой жидкости // Письма в журнал технической физики. 2007. Т. 33, № 17. C. 31–37.
    EDN: rcuvfj
    Reshetnikov A.V., Mazheyko N.A., Begletsov V.N., et al. Pulsation dynamics during explosive boil-up of overheated water jets // Technical Physics Letters. 2007. V. 33, No. 9. Pp. 732–734.
    DOI: 10.1134/S1063785007090052
  13. Алексеев М.В., Лежнин С.И., Прибатурин Н.А., Сорокин А.Л. Генерация ударно-волновых и вихревых структур при истечении струи вскипающей воды // Теплофизика и аэромеханика. 2014. № 6. С. 795–798.
    EDN: taojtx
    Alekseev M.V., Lezhnin S.I., Pribaturin N.A., Sorokin A.L. Generation of shockwave and vortex structures at the outflow of a boiling water jet // Thermophysics and Aeromechanics. 2014. V. 21, No. 6. Pp. 763–766.
    DOI: 10.1134/S0869864314060122
  14. Weaver D.S. et al. Loading of steam fenerator tubes during main steam line breaks // Ottawa, Canada. 2015. 171 p.
  15. Лепешинский И.А., Решетников В.А., Антоновский И.В. и др. Смеситель с двухфазным рабочим телом // Материалы XI Международной конференции по неравновесным процессам в соплах и струях. Москва: МАИ, 2016. C. 93–95.
    Lepeshinsky I.A., Reshetnikov V.A., Antonovsky I.V., et al. A mixer with a two-phase working fluid // Proceedings of the XI International Conference on Nonequilibrium Processes in Nozzles and Jets. Moscow: MAI, 2016. Pp. 93–95 (in Russian).
    EDN: uuvcaj
  16. Решетников А.В., Бусов К.А., Мажейко Н.А., Скоков В.Н., Коверда В.П. Переходные режимы вскипания струй перегретой воды // Теплофизика и аэромеханика. 2012. Т. 19, № 3. С. 359–367.
    EDN: piggdv
    Reshetnikov A.V., Busov K.A., Mazheyko N.A., Skokov V.N., Koverda V.P. Transient behavior of superheated water jets boiling // Thermophysics and Aeromechanics. 2012. V. 19, No. 2. Pp. 329–336.
    DOI: 10.1134/S0869864312020151
  17. Ширшов Я.Н., Нерсесян Д.А., Сысоев Н.Н., Иванов И.Э., Знаменская И.А. Оптические исследования динамики развития водяной струи высокого давления // Матер. XI Междунар. конф. по неравновесным процессам в соплах и струях. М.: МАИ, 2016. C. 196–198.
    Shirshov Ya.N., Nersesyan D.A., Sysoev N.N., Ivanov I.E., Znamenskaya I.A., Optical study for dynamics of high-pressure water jet // Proc. XI Int. Conf. for Inequilibrium Processes in Nozzles and Jets, MAI, Moscow. 2016. Pp. 196–198 (in Russian).
    EDN: xcxlwn
  18. Костюк В.В., Фирсов В.П. Теплообмен и гидродинамика в криогенных двигательных установках // Москва: Наука, 2015. 319 с.
    Kostyuk V.V., Firsov V.P. Heat transfer and hydrodynamics in cryogenic propulsion systems. Moscow : Nauka Publ., 2015. 319 p. (in Russian).
  19. Reitz R.D. A photographic study of flash-boiling atomization // Aerosol Science and Technology. 1990. V. 12, No. 3. Pp. 561–569.
    DOI: 10.1080/02786829008959370
  20. Simões-Moreira J.R., Vieira M.M., Angelo E. Highly expanded flashing liquid jets // Journal of Thermophysics and Heat Transfer. 2002. V. 16, No. 3. Pp. 415–424.
    DOI: 10.2514/2.6695
  21. De Rosa M., Sender J., Zimmermann H., Oschwald M. Cryogenic spray ignition at high altitude conditions // In 42nd AIAA/ASME/SAE/ASEE JPC. Sacramento, California. July. 2006.
    DOI: 10.2514/6.2006-4539
  22. Lamanna G., Kamoun H., Weigand B., Manfletti C., Rees A., Sender J., Oschwald M. and Steelant J. Flashing behavior of rocket engine propellants // Atomization and Sprays. 2015. V. 25, No. 10. Pp. 837–856.
    DOI: 10.1615/AtomizSpr.2015010398
  23. Cleary V., Bowen P. and Witlox H. Flashing liquid jets and two-phase droplet dispersion: I. Experiments for derivation of droplet atomisation correlations // Journal of Hazardous Materials. 2007. V. 142, No. 3. Pp. 786–796.
    DOI: 10.1016/j.jhazmat.2006.06.125
  24. Нигматулин Р.И., Болотнова Р.Х. Широкодиапазонное уравнение состояния воды и пара. Упрощенная форма // Теплофизика высоких температур. 2011. Т. 49, № 2. С. 310–313.
    EDN: nefzrf
    Nigmatulin R.I., Bolotnova R.Kh. Wide-range equation of state for water and steam: Simplified form // High Temperature. 2011. V. 49, No. 2. P. 303–306.
    DOI: 10.1134/S0018151X11020106
  25. Нигматулин Р.И., Болотнова Р.Х. Широкодиапазонные уравнения состояния бензола и тетрадекана в упрощенной форме // Теплофизика высоких температур. 2017. Т. 55, № 2. С. 206–215.
    DOI: 10.7868/S004036441701015X
    Nigmatulin R.I., Bolotnova R.Kh. Simplified wide-range equations of state for benzene and tetradecane // High Temperature. 2017. V. 55, No. 2, Pp. 199–208.
    DOI: 10.1134/S0018151X17010151
  26. Болотнова Р.Х., Коробчинская В.А. Пространственное моделирование процесса формирования струи вскипающей воды при истечении из тонкого сопла // Теплофизика и аэромеханика. 2017. Т. 24, № 5. С. 783–794.
    EDN: zmwkff
    Bolotnova R.Kh., Korobchinskaya V.A. Boiling water jet outflow from a thin nozzle: spatial modeling // Thermophysics and Aeromechanics. 2017. V. 24, No. 5. Pp. 761–772.
    DOI: 10.1134/S0869864317050110
  27. Болотнова Р.Х., Бузина В.А. (Коробчинская В.А.), Галимзянов М.Н., Шагапов В.Ш. Гидродинамические особенности процессов истечения вскипающей жидкости // Теплофизика и аэромеханика. 2012. № 6. С. 719–730.
    Bolotnova R.H., Buzina V.A. (Korobchinskaya V.A.), Galimzyanov M.N., Shagapov V.Sh. Hydrodynamic features of boiling liquid outflow processes // Thermophysics and Aeromechanics. 2012. No. 6. Pp. 719–730 (in Russian).
    EDN: pigavj
  28. Болотнова Р.Х., Гайнуллина Э.Ф. Особенности формирования полой струи водяного пара сверхкритических параметров состояния, истекающего через тонкое сопло // Теплофизика и Аэромеханика. 2018. Т. 25, № 5. C. 783–789.
    EDN: yursot
    Bolotnova R.Kh., Gainullina E.F. Supercritical steam out-flow through a thin nozzle: forming a hollow jet // Thermophysics and Aeromechanics. 2018. V. 25. No. 5. Pp. 751–757.
    DOI: 10.1134/S0869864318050116
  29. Болотнова Р.Х., Гайнуллина Э.Ф., Коробчинская В.А., Файзуллина Э.А. Моделирование пространственных динамических процессов в водных пенах и вскипающих струях // Уфимская осенняя математическая школа. 2020. Т. 2. С. 180–182.
    Bolotnova R.Kh., Gainullina E.F., Korobchinskaya V.A., Fayzullina E.A. Modeling of spatial dynamic processes in water foams and boiling jets // Ufa Autumn Mathematical School. 2020. V. 2. Pp. 180–182 (in Russian).
    EDN: xuffya
  30. Bolotnova R.Kh., Korobchinskaya V.A., Faizullina E.A. Analysis the dynamic formation of a vapor supersonic jet under outflow from thin nozzle // Journal of Physics: Conference Series 2021. V. 2103. 012219.
    DOI: 10.1088/1742-6596/2103/1/012219
  31. Болотнова Р.Х., Коробчинская В.А. Моделирование динамики струи при истечении через тонкое сопло водного флюида, находящегося в сверхкритическом состоянии // Теплофизика и Аэромеханика. 2022. Т. 29, № 3. C. 361–370.
    EDN: ppyoyf
    Bolotnova R.Kh., Korobchinskaya V.A. Modeling the dynamics of supercritical water jet from a thin nozzle // Thermophysics and Aeromechanics. 2022. V. 29, No. 3. Pp. 347–355.
    DOI: 10.1134/S0869864322030039
  32. Алексеев М.В., Лежнин С.И., Прибатурин Н.А., Сорокин А.Л. Генерация ударноволновых и вихревых структур при истечении струи вскипающей воды // Теплофизика и аэромеханика. 2014. Т. 21, № 6. С. 795–798.
    EDN: taojtx
    Alekseev M.V., Lezhnin S.I., Pribaturin N.A., Sorokin A.L. Generation of shock wave and vortex structures at the outflow of a boiling water jet // Thermophysics and Aeromechanics. 2014. V. 21, No. 6. Pp. 763–766.
    DOI: 10.1134/S0869864314060122
  33. Алексеев М.В., Вожаков И.С., Лежнин С.И., Прибатурин Н.А. Волновые процессы при истечении водяного теплоносителя со сверхкритическими начальными параметрами // Теплофизика и аэромеханика. 2017. Т. 24, № 5. С. 821–824.
    EDN: zmwkgt
    Alekseev M.V., Vozhakov I.S., Lezhnin S.I., Pribaturin N.A. Wave processes at outflow of water coolant with initial supercritical parameters // Thermophysics and Aeromechanics. 2017. V. 24, No. 5. Pp. 799–802.
    DOI: 10.1134/S0869864317050158
  34. Lee H.S., Merte H. Spherical vapor bubble growth in uniformly superheated liquids // International Journal of Heat and Mass Transfer. 1996. V. 39, No. 12. Pp. 2427–2447.
    DOI: 10.1016/0017-9310(95)00342-8
  35. Robinson A., Judd R. The dynamics of spherical bubble growth // International Journal of Heat and Mass Transfer. 2004. V. 47, No. 23. Pp. 5101–5113.
    DOI: 10.1016/j.ijheatmasstransfer.2004.05.023
  36. Sher E., Bar-Kohany T., Rashkovan A. Flash-boiling atomization // Progress in Energy and Combustion Science. 2008. V. 34, No. 4. Pp. 417–439.
    DOI: 10.1016/j.pecs.2007.05.001
  37. Prosperetti A. Vapor bubbles // Annual Review of Fluid Mechanics. 2017. V. 49, Pp. 221–248.
    DOI: 10.1146/annurev-fluid-010816-060221
  38. Lee J., Madabhushi R., Fotache C., Gopalakrishnan S., Schmidt D. Flashing flow of superheated jet fuel // Proceedings of the Combustion Institute. 2009. V. 32, No. 2. Pp. 3215–3222.
    DOI: 10.1016/j.proci.2008.06.153
  39. Navarro-Martinez S. Large eddy simulation of spray atomization with a probability density function method // International Journal of Multiphase Flow. 2014. V. 63. Pp. 11–22.
    DOI: 10.1016/j.ijmultiphaseflow.2014.02.013
  40. Karathanassis I.K., Koukouvinis P., Gavaises M. Comparative evaluation of phase-change mechanisms for the prediction of flashing flows // International Journal of Multiphase Flow. 2017. V. 95. Pp. 257–270.
    DOI: 10.1016/j.ijmultiphaseflow.2017.06.006
  41. Gärtner J.W., Rees A., Kronenburg A., Sender J., Oschwald M., Loureiro D. Large eddy simulation of flashing cryogenic liquid with a compressible volume of fluid solver // ILASS–Europe 2019, 29th Conference on Liquid Atomization and Spray Systems, 2–4 September 2019, Paris, France. 8 p.
    https://elib.dlr.de/132833/1/ILASS%202019%20Large%20Eddy%20Rees.pdf
  42. Gärtner J.W., Kronenburg A., Rees A., Sender J., Oschwald M., Lamanna G. Numerical and experimental analysis of flashing cryogenic nitrogen // International Journal of Multiphase Flow. 2020. V. 130. 103360.
    DOI: 10.1016/j.ijmultiphaseflow.2020.103360
  43. Calay R., Holdo A. Modelling the dispersion of flashing jets using CFD // Journal of Hazardous Materials. 2008. V. 154, Issues 1–3. Pp. 1198–1209.
    DOI: 10.1016/j.jhazmat.2007.11.053
  44. Loureiro D.D., Reutzsch J., Kronenburg A., Weigand B., Vogiatzaki K. Primary breakup regimes for cryogenic flash atomization // International Journal of Multiphase Flow. 2020. V. 132. 103405.
    DOI: 10.1016/j.ijmultiphaseflow.2020.103405
  45. Сычев В.В., Вассерман А.А., Козлов А.Д., Спиридонов Г.А., Цымарный В.А. Термодинамические свойства азота. Москва: Издательство стандартов, 1977. 352 c.
    Sychev V.V., Wasserman A.A., Kozlov A.D., Spiridonov G.A., Tsymarny‘V.A. Thermodynamic properties of nitrogen. Moscow: Publishing House of Standards, 1977. 352 p. (in Russian).
    https://nauca.ru/ref/Термодинамические-свойства-азота-ГСССД.pdf
  46. Bolotnova R.Kh., Gainullina E.F., Korobchinskaya V.A. Equation of state for liquid and gaseous nitrogen in cryogenic temperature range // Lobachevskii Journal of Mathematics. 2023. V. 44, No. 5. Pp. 1587–1592.
    DOI: 10.1134/S1995080223050116
  47. Болотнова Р.Х., Гайнуллина Э.Ф., Файзуллина Э.А. Аналитическое уравнение состояния азота для жидкой и газовой фаз // Теплофизика и аэромеханика. 2024. № 6. С. 1187–1194.
    Bolotnova R.Kh., Gainullina E.F., Fayzullina E.A. Analytical equation of state for liquid and gaseous nitrogen // Thermophysics and Aeromechanics. 2024. No. 6. Pp. 1187–1194.
    EDN: xqfadp
  48. Болотнова Р.Х., Коробчинская В.А., Гайнуллина Э.Ф.Моделирование процесса истечения жидкого азота через коническое сопло в вакуумную камеру // Письма в журнал технической физики. 2023. Т. 49, № 24. С. 46–49.
    DOI: 10.61011/PJTF.2023.24.56872.107A
    Bolotnova R.Kh., Korobchinskaya V.A., Gainullina E.F. Modeling the process of liquid nitrogen outflow through a conical nozzle into vacuum chamber // Technical Physics Letters. 2023. V. 49, No. 12. Pp. 108–111.
    https://journals.ioffe.ru/articles/57601
  49. Bolotnova R.Kh., Korobchinskaya V.A., Gainullina E.F. Modeling the dynamics of a boiling liquid nitrogen jet under cryogenic temperatures // Lobachevskii Journal of Mathematics. 2023. V. 44, No. 5. Pp. 1579–1586.
    DOI: 10.1134/S1995080223050104
  50. Коробчинская В.А. Влияние начальной степени перегрева на эволюцию струи жидкого азота при истечении в вакуумную камеру // Вестник Башкирского университета. 2024. Т. 29, № 1. С. 19–24.
    Korobchinskaya V.A. The influence of the initial degree of overheating on the evolution of a jet of liquid nitrogen when flowing into a vacuum chamber // Bulletin of the Bashkir University. 2024. T. 29, No. 1. Pp. 19–24 (in Russian).
    DOI: 10.33184/bulletin-bsu-2024.1.4
  51. Болотнова Р.Х., Коробчинская В.А., Гайнуллина Э.Ф. Влияние начальных условий в камере низкого давления на степень расширения вскипающей струи жидкого азота // Письма в журнал технической физики. 2024. Т. 50, № 23. C. 23–26.
    Bolotnova R.Kh., Korobchinskaya V.A., Gainullina E.F. Influence of initial conditions in a low-pressure chamber on the degree of expansion of a boiling jet of liquid nitrogen // Technical Physics Letters. 2024. V. 50, No. 23. Pp. 23–26 (in Russian).
    DOI: 10.61011/PJTF.2024.23.59393.6455k