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Ideal Gas Outflow from a Cylindrical or Spherical Source into a Vacuum

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An Erratum to this article was published on 01 September 2019

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Abstract

The solutions of initial and boundary value problems of the outflow of an ideal (inviscid and non-heat-conducting) gas from cylindrical and spherical sources into a vacuum are obtained. Time is measured from the moment, when the source is turned on; at this moment the source is surrounded by a vacuum. The entropy, flow rate, and the Mach number of the gas outflowing from the source are given, together with the source radius; the Mach number can be greater of or equal to unity. If the source radius is greater than zero, then the flow domain in the “radial coordinate–time” plane consists of the stationary source flow and adjoining non-self-similar centered expansion wave consisting of C−-characteristics. The stationary flow is described by the known formulas, while the expansion wave is calculated by the method of characteristics. The calculations by this method confirm the earlier obtained laws for large values of the radial coordinate. The interface between the vacuum and the expansion wave is the straight trajectory of particles and, at the same time, a unique rectilinear C−-characteristic. For the source of zero radius (“pointwise” source) the velocity, density, and speed of sound of the outflowing gas are infinite. The gas velocity remains infinite everywhere, while the density and speed of sound become zero for any non-zero values of the radial coordinate. For the pointwise source the problem of outflow into a vacuum is self-similar. In the plane of the “self-similar” velocity and speed of sound its solution is given by three singular points of a differential equation in these variables. At one of these points the self-similar velocity is infinite, the self-similar speed of sound is zero, and the self-similar independent variable varies from zero to infinity, with the exception of the extreme values.

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Change history

  • 03 October 2019

    After publication of the paper, the authors realized that the affiliation of the fourth author (Tiberiu Esanu) was given incorrectly. Its correct version appears above. Moreover, a second funding grant was missed in the acknowledgements. We give their complete correct version below.

References

  1. A. N. Kraiko, “On the Free Expansion of an Ideal Gas,” Fluid Dynamics 28 (4), 553 (1993).

    Article  ADS  MathSciNet  Google Scholar 

  2. A. N. Kraiko, “The Asymptotic Features of the Unsteady Expansion of an Ideal Gas into a Vacuum,” J. Appl. Math. Mech. 58 (4), 645 (1994).

    Article  MathSciNet  Google Scholar 

  3. Kh. F. Valiyev and A. N. Kraiko, “Self-Similar Problems on Compression of Ideal Gas and its Dispersing from a Point,” J. Appl.Math. Mech. 79 (3), 237 (2015).

    Article  MathSciNet  Google Scholar 

  4. Kh. F. Valiyev and A. N. Kraiko, “The Dispersion of an Ideal Gas from a Point into a Void. A New Model of the Big Bang and the Expansion of the Universe,” J. Appl.Math. Mech. 79 (6), 556 (2015).

    Article  MathSciNet  Google Scholar 

  5. Kh. F. Valiyev and A. N. Kraiko, “Compression of Ideal Gas into a Point, its Finite Mass Dispersion from the Point into the Vacuum, the Big Bang and the Universe Expansion,” in: A. L. Mikhailov (ed.), Extreme States of Substance, Detonation. Shock Waves / Proc. Intern. Conf. XVII Khariton’s Topical Scientific Readings (Sarov, 2015), p. 621.

    Google Scholar 

  6. A. N. Kraiko and Kh. F. Valiyev, “The NewModel of the Big Bang and theUniverse Expansion.A Comparison with Modern Observational Data and Cosmological Theories,” in: V. Fomin (ed.), AIP Conf. Proc. V. 1770 (Int. Conf. Methods of Aerophysical Research (ICMAR 2016): Proc. 18th Int. Conf. Methods of Aerophysical Research (American Institute of Physics, 2016), p. 020002–1.

    Google Scholar 

  7. A. N. Kraiko and Kh. F. Valiyev, “A Model of the Big Bang and the Universe Expansion with the Dispersion to the Void of a Gas Compressed Almost in a Point’. A Comparison with Observational Data and Modern Cosmological Theories,” in: A.L.Mikhailov (ed.), Extreme States of Substance, Detonation. ShockWaves / Proc. Intern. Conf. XVII Khariton’s Topical Scientific Readings (Sarov, 2015), p. 12.

    Google Scholar 

  8. B.L. Rozhdestvenskii and N.N. Yanenko, Systems of Quasilinear Equations and Their Applications to Gasdynamics [in Russian] (Nauka, Moscow, 1978).

    Google Scholar 

  9. A. N. Kraiko, Theoretical Gasdynamics: Classics and State of the Art [in Russian] (Torus Press, Moscow, 2010).

    Google Scholar 

  10. A. N. Kraiko and V. V. Shelomovskii, “Free Expansion of Two-Dimensional Ideal Gas Jets,” Prikl. Mat. Mekh. 44 (2), 271 (1980)

    Google Scholar 

  11. A.N. Kraiko, Gasdynamics. Selected Works, Vol. 1 [in Russian] (Fizmatlit, Moscow, 2005), p. 345.]

    Google Scholar 

  12. L. D. Ladyzhenskii, “On Certain Properties of Hypersonic Flows,” Tr. TsAGI 778 (1960).

    Google Scholar 

  13. L. D. Ladyzhenskii, “Gas Flows with Large Supersonic Velocities,” Dokl. Akad. Nauk SSSR 134 (2), 296 (1960).

    MathSciNet  Google Scholar 

  14. V. N. Gusev and L. D. Ladyzhenskii, “Gasdynamic Calculation of Shock Tubes and Hypersonic Nozzles under the Conditions of Equilibrium Air Dissociation and Ionization,” Tr. TsAGI 779 (1960).

  15. L. D. Ladyzhenskii, “Analysis of the Equations of Hypersonic Flows and Solution of the Cauchy Problem,” Prikl. Mat. Mekh. 26 (2), 289 (1962).

    MathSciNet  Google Scholar 

  16. L. D. Ladyzhenskii, “Hypersonic Flows in Nozzles,” Prikl. Mat. Mekh. 29 (1), 99 (1965).

    Google Scholar 

  17. L. D. Ladyzhenskii, Three-Dimensional Hypersonic Gas Flows [in Russian] (Mashinostroenie, Moscow, 1968).

    Google Scholar 

  18. C. Greifinger and J. D. Cole, “Expansion of a FiniteMass of Gas into Vacuum,” AIAA J. 3 (6), 1200 (1965).

    ADS  Google Scholar 

  19. Kh. F. Valiyev and A. N. Kraiko, “Non-Self-Similar Flow with a Shock Wave Reflected from the Center of Symmetry and New Self-Similar Solutions with Two Reflected Shocks,” Comp. Math. Math. Phys. 53 (3), 350 (2013).

    Article  Google Scholar 

  20. L. I. Sedov, Similarity and Dimensional Methods (CRC Press, Boca Raton, 1993).

    Google Scholar 

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Authors and Affiliations

  1. Baranov Central Institute of Aviation Motors (CIAM), ul. Aviamotornaya 2, Moscow, 111116, Russia

    Kh. F. Valiev & A. N. Kraiko

  2. Moscow Institute of Sciences and Technology, Institutskii per. 9, Dolgoprudnyi, Moscow oblast, 141701, Russia

    A. N. Kraiko

Authors
  1. Kh. F. Valiev
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  2. A. N. Kraiko
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Correspondence to A. N. Kraiko.

Additional information

Original Russian Text © Kh.F. Valiev, A.N. Kraiko, 2018, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 2018, No. 5, pp. 17–28.

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Valiev, K.F., Kraiko, A.N. Ideal Gas Outflow from a Cylindrical or Spherical Source into a Vacuum. Fluid Dyn 53, 596–607 (2018). https://doi.org/10.1134/S0015462818050154

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