대한수학회지 (Journal of the Korean Mathematical Society)

  • 제54권2호
  • /
  • Pages.663-673
  • /
  • 2017
  • /
  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

MULTI-HARMONIC MODELS FOR BUBBLE EVOLUTION IN THE RAYLEIGH-TAYLOR INSTABILITY

  • Choi, Sujin (Department of Mathematics Gangneung-Wonju National University) ;
  • Sohn, Sung-Ik (Department of Mathematics Gangneung-Wonju National University)
  • 투고 : 2016.04.03
  • 발행 : 2017.03.01
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

We consider the multi-harmonic model for the bubble evolution in the Rayleigh-Taylor instability in two and three dimensions. We extend the multi-harmonic model in two dimensions to a high-order and present a new class of steady-state solutions of the bubble motion. The growth rate of the bubble is expressed by a continuous family of two free parameters. The critical point in the family of solutions is identified as a saddle point and is chosen as the physically significant solution. We also present the multi-harmonic model in the cylindrical geometry and find the steady-state solution of the axisymmetric bubble. Validity and limitation of the model are also discussed.

키워드

참고문헌

  1. S. I. Abarzhi, Stable steady flows in Rayleigh-Taylor instability, Phys. Rev. Lett. 81 (1998), 337-340. https://doi.org/10.1103/PhysRevLett.81.337
  2. S. I. Abarzhi, Review of theoretical modelling approaches of Rayleigh-Taylor instabilities and turbulent mixing, Phil. Trans. R. Soc. A 368 (2010), 1809-1828. https://doi.org/10.1098/rsta.2010.0020
  3. S. I. Abarzhi, J. Glimm, and A.-D. Lin, Dynamics of two-dimensional Rayleigh-Taylor bubbles for fluids with a finite density contrast, Phys. Fluids 15 (2003), no. 8, 2190-2197. https://doi.org/10.1063/1.1583732
  4. S. I. Abarzhi, K. Nishihara, and J. Glimm, Rayleigh-Taylor and Richtmyer-Meshkov instabilities for fluids with a finite density ratio, Phys. Lett. A 317 (2003), 470-476. https://doi.org/10.1016/j.physleta.2003.09.013
  5. S. I. Abarzhi, K. Nishihara, and R. Rosner, Multiscale character of the nonlinear coherent dynamics in the Rayleigh-Taylor instability, Phys. Rev. E 73 (2006), 036310. https://doi.org/10.1103/PhysRevE.73.036310
  6. G. Birkhoff and D. Carter, Rising plane bubbles, J. Math. Mech. 6 (1957), 769-779.
  7. Y. G. Cao, H. Z. Guo, Z. F. Zhang, Z. H. Sun, and W. K. Chow, Effect of viscosity on the growth of Rayleigh-Taylor instability, J. Phys. A 44 (2011), no. 27, 275501, 8 pp.
  8. P. R. Garabedian, On steady-state bubbles generated by Taylor instability, Proc. R. Soc. London Ser. A 241 (1957), 423-431. https://doi.org/10.1098/rspa.1957.0137
  9. J. Glimm, D. H. Sharp, T. Kaman, and H. Lim, New directions for Rayleigh-Taylor mixing, Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 371 (2013), no. 2003, 20120183, 19 pp.
  10. V. N. Goncharov, Analytical model of nonlinear, single-mode, classical Rayleigh-Taylor instability at arbitrary Atwood numbers, Phys. Rev. Lett. 88 (2002), 134502. https://doi.org/10.1103/PhysRevLett.88.134502
  11. J. Hecht, U. Alon, and D. Shvarts, Potential flow models of Rayleigh-Taylor and Richtmyer-Meshkov bubble fronts, Phys. Fluids 6 (1994), 4019-4030. https://doi.org/10.1063/1.868391
  12. H. Jin, X. F. Liu, T. Lu, B. Cheng, J. Glimm, and D. H. Sharp, Rayleigh-Taylor mixing rates for compressible flow, Phys. Fluids 17 (2005), 024104. https://doi.org/10.1063/1.1843155
  13. D. Layzer, On the instability of superposed fluids in a gravitational field, Astrophys. J. 122 (1955), 1-12. https://doi.org/10.1086/146048
  14. K. O. Mikaelian, Explicit expressions for the evolution of single-mode Rayleigh-Taylor and Richtmyer-Meshkov instabilities at arbitrary Atwood numbers, Phys. Rev. E 67 (2003), 026319. https://doi.org/10.1103/PhysRevE.67.026319
  15. K. O. Mikaelian, Limitations and failures of the Layzer model for hydrodynamic instabilities, Phys. Rev. E 78 (2008), 015303. https://doi.org/10.1103/PhysRevE.78.015303
  16. P. Ramaprabhu and G. Dimonte, Single-mode dynamics of the Rayleigh-Taylor instability at any density ratio, Phys. Rev. E 71 (2005), 036314. https://doi.org/10.1103/PhysRevE.71.036314
  17. L. Rayleigh, Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density, Proc. London Math. Soc. 14 (1883), 170-177.
  18. B. Rollin and M. J. Andrews, Mathematical model of Rayleigh-Taylor and Richtmyer-Meshkov instabilities for viscoelastic fluids, Phys. Rev. E 83 (2011), 046317. https://doi.org/10.1103/PhysRevE.83.046317
  19. D. Sharp, An overview of Rayleigh-Taylor instability, Phys. D 12 (1984), 3-10. https://doi.org/10.1016/0167-2789(84)90510-4
  20. S.-I. Sohn, Vortex model and simulations for Rayleigh-Taylor and Richtmyer-Meshkov instabilities, Phys. Rev. E 69 (2004), 036703. https://doi.org/10.1103/PhysRevE.69.036703
  21. S.-I. Sohn, Density dependence of a Zufiria-type model for Rayleigh-Taylor and Richtmyer-Meshkov bubble fronts, Phys. Rev. E 70 (2004), 045301.
  22. S.-I. Sohn, Effects of surface tension and viscosity on the growth rates of Rayleigh-Taylor and Richtmyer-Meshkov instabilities, Phys. Rev. E 80 (2009), 055302. https://doi.org/10.1103/PhysRevE.80.055302
  23. S.-I. Sohn, Asymptotic bubble evolutions of the Rayleigh-Taylor instability, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), no. 11, 4017-4022. https://doi.org/10.1016/j.cnsns.2012.03.006
  24. J. P. Wilkinson and J. W. Jacobs, Experimental study of the single-mode three-dimensional Rayleigh-Taylor instability, Phys. Fluids 19 (2007), 124102. https://doi.org/10.1063/1.2813548
  25. Q. Zhang, Analytical solutions of Layzer-type approach to unstable interfacial fluid mixing, Phys. Rev. Lett. 81 (1998), 3391-3394. https://doi.org/10.1103/PhysRevLett.81.3391
  26. Y. B. Zudin, Analytical solution of the problem of the rise of a Taylor bubble, Phys. Fluids. 25 (2013), 053302. https://doi.org/10.1063/1.4803878