Advertisement

Acta Mechanica Sinica

, Volume 34, Issue 5, pp 855–870 | Cite as

Numerical study on the turbulent mixing of planar shock-accelerated triangular heavy gases interface

  • Wei-Gang Zeng
  • Jian-Hua Pan
  • Yu-Xin Ren
  • Yu-Tao Sun
Research Paper
  • 125 Downloads

Abstract

The interaction of a planar shock wave with a triangle-shaped sulfur hexafluoride (\(\mathrm{SF_6}\)) cylinder surrounded by air is numerically studied using a high resolution finite volume method with minimum dispersion and controllable dissipation reconstruction. The vortex dynamics of the Richtmyer–Meshkov instability and the turbulent mixing induced by the Kelvin–Helmholtz instability are discussed. A modified reconstruction model is proposed to predict the circulation for the shock triangular gas–cylinder interaction flow. Several typical stages leading the shock-driven inhomogeneity flow to turbulent mixing transition are demonstrated. Both the decoupled length scales and the broadened inertial range of the turbulent kinetic energy spectrum in late time manifest the turbulent mixing transition for the present case. The analysis of variable-density energy transfer indicates that the flow structures with high wavenumbers inside the Kelvin–Helmholtz vortices can gain energy from the mean flow in total. Consequently, small scale flow structures are generated therein by means of nonlinear interactions. Furthermore, the occasional “pairing” between a vortex and its neighboring vortex will trigger the merging process of vortices and, finally, create a large turbulent mixing zone.

Keywords

Richtmyer–Meshkov instability Kelvin–Helmholtz instability Length scale Turbulent mixing 

Notes

Acknowledgements

The project was supported by the National Natural Science Foundation of China (Grant U1430235) and the National Key Research and Development Program of China (Grant 2016YFA0401200).

References

  1. 1.
    Lindl, J.: Development of the indirect-drive approach to inertial confinement fusion and the target physics basis for ignition and gain. Phys. Plasmas 2, 3933–4024 (1995)CrossRefGoogle Scholar
  2. 2.
    Arnett, D.: The role of mixing in astrophysics. Astrophys. J. Suppl. Ser. 127, 213–217 (2000)CrossRefGoogle Scholar
  3. 3.
    Yang, J., Kubota, T., Zukoski, E.E.: Applications of shock-induced mixing to supersonic combustion. AIAA J. 31, 854–862 (1993)CrossRefGoogle Scholar
  4. 4.
    Richtmyer, R.D.: Taylor instability in shock acceleration of compressible fluids. Commun. Pure Appl. Math. 13, 297–319 (1960)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Meshkov, E.E.: Instability of the interface of two gases accelerated by a shock wave. Fluid Dyn. 4, 101–104 (1969)CrossRefGoogle Scholar
  6. 6.
    Zhang, Q., Sohn, S.I.: Nonlinear theory of unstable fluid mixing driven by shock wave. Phys. Fluids 9, 1106–1124 (1997)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Vandenboomgaerde, M., Gauthier, S., Mgler, C.: Nonlinear regime of a multimode Richtmyer–Meshkov instability: a simplified perturbation theory. Phys. Fluids 14, 1111–1122 (2002)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Matsuoka, C., Nishihara, K., Fukuda, Y.: Nonlinear evolution of an interface in the Richtmyer–Meshkov instability. Phys. Rev. E 67, 036301 (2003)CrossRefGoogle Scholar
  9. 9.
    Sohn, S.I.: Simple potential-flow model of Rayleigh–Taylor and Richtmyer–Meshkov instabilities for all density ratios. Phys. Rev. E 67, 026301 (2003)CrossRefGoogle Scholar
  10. 10.
    Picone, J.M., Boris, J.P.: Vorticity generation by shock propagation through bubbles in a gas. J. Fluid Mech. 189, 23–51 (1988)CrossRefGoogle Scholar
  11. 11.
    Yang, J., Kubota, T., Zukoski, E.E.: A model for characterization of a vortex pair formed by shock passage over a light-gas inhomogeneity. J. Fluid Mech. 258, 217–244 (1994)CrossRefGoogle Scholar
  12. 12.
    Samtaney, R., Zabusky, N.J.: Circulation deposition on shock-accelerated planar and curved density-stratified interfaces: models and scaling laws. J. Fluid Mech. 269, 45–78 (1994)CrossRefGoogle Scholar
  13. 13.
    Niederhaus, J.H.J., Greenough, J.A., Oakley, J.G., et al.: A computational parameter study for the three-dimensional shock-bubble interaction. J. Fluid Mech. 594, 85–124 (2008)CrossRefGoogle Scholar
  14. 14.
    Jacobs, J.W., Krivets, V.V.: Experiments on the late-time development of single-mode Richtmyer–Meshkov instability. Phys. Fluids 17, 034105 (2005)CrossRefGoogle Scholar
  15. 15.
    Wang, X., Yang, D., Wu, J., et al.: Interaction of a weak shock wave with a discontinuous heavy-gas cylinder. Phys. Fluids 27, 064104 (2015)CrossRefGoogle Scholar
  16. 16.
    Tritschler, V.K., Olson, B.J., Lele, S.K., et al.: On the Richtmyer–Meshkov instability evolving from a deterministic multimode planar interface. J. Fluid Mech. 755, 429–462 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Thornber, B., Drikakis, D., Youngs, D.L., et al.: The influence of initial conditions on turbulent mixing due to Richtmyer–Meshkov instability. J. Fluid Mech. 654, 99–139 (2010)CrossRefGoogle Scholar
  18. 18.
    Hill, D.J., Pantano, C., Pullin, D.I.: Large-eddy simulation and multiscale modelling of a Richtmyer–Meshkov instability with reshock. J. Fluid Mech. 557, 29–61 (2006)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Thornber, B., Drikakis, D., Youngs, D.L., et al.: Growth of a Richtmyer–Meshkov turbulent layer after reshock. Phys. Fluids 23, 095107 (2011)CrossRefGoogle Scholar
  20. 20.
    Thornber, B., Zhou, Y.: Numerical simulations of the two-dimensional multimode Richtmyer–Meshkov instability. Phys. Plasm. 22, 032309 (2015)CrossRefGoogle Scholar
  21. 21.
    Olson, B.J., Greenough, J.A.: Comparison of two- and three-dimensional simulations of miscible Richtmyer–Meshkov instability with multimode initial conditions. Phys. Fluids 26, 101702 (2014)CrossRefGoogle Scholar
  22. 22.
    Mizuno, Y.: Spectra of energy transport in turbulent channel flows for moderate Reynolds numbers. J. Fluid Mech. 805, 171–187 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Cook, A.W., Zhou, Y.: Energy transfer in Rayleigh–Taylor instability. Phys. Rev. E 66, 026312 (2002)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Thornber, B., Zhou, Y.: Energy transfer in the Richtmyer–Meshkov instability. Phys. Rev. E 86, 056302 (2012)CrossRefGoogle Scholar
  25. 25.
    Zhou, Q., Huang, Y.X., Lu, Z.M., et al.: Scale-to-scale energy and enstrophy transport in two-dimensional Rayleigh–Taylor turbulence. J. Fluid Mech. 786, 294–308 (2016)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Wang, J., Yang, Y., Shi, Y., et al.: Cascade of kinetic energy in three-dimensional compressible turbulence. Phys. Rev. Lett. 110, 214505 (2013)CrossRefGoogle Scholar
  27. 27.
    Liu, H., Xiao, Z.: Scale-to-scale energy transfer in mixing flow induced by the Richtmyer–Meshkov instability. Phys. Rev. E 93, 053112 (2016)CrossRefGoogle Scholar
  28. 28.
    Thornber, B., Griffond, J., Poujade, O., et al.: Late-time growth rate, mixing, and anisotropy in the multimode narrowband Richtmyer–Meshkov instability: the \(\theta \)-group collaboration. Phys. Fluids 29, 105107 (2017)CrossRefGoogle Scholar
  29. 29.
    Mohaghar, M., Carter, J., Musci, B., et al.: Evaluation of turbulent mixing transition in a shock-driven variable-density flow. J. Fluid Mech. 831, 779–825 (2017)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Thornber, B.: Impact of domain size and statistical errors in simulations of homogeneous decaying turbulence and the Richtmyer–Meshkov instability. Phys. Fluids 28, 045106 (2016)CrossRefGoogle Scholar
  31. 31.
    Zhou, Y., Cabot, W.H., Thornber, B.: Asymptotic behavior of the mixed mass in Rayleigh–Taylor and Richtmyer–Meshkov instability induced flows. Phys. Plasmas 23, 052712 (2016)CrossRefGoogle Scholar
  32. 32.
    Guan, B., Zhai, Z., Si, T., et al.: Manipulation of three-dimensional Richtmyer–Meshkov instability by initial interfacial principal curvatures. Phys. Fluids 29, 032106 (2017)CrossRefGoogle Scholar
  33. 33.
    Zhai, Z., Dong, P., Si, T., et al.: The Richtmyer–Meshkov instability of a V shaped air/helium interface subjected to a weak shock. Phys. Fluids 28, 082104 (2016)CrossRefGoogle Scholar
  34. 34.
    Zhai, Z., Li, W., Si, T., et al.: Refraction of cylindrical converging shock wave at an air/helium gaseous interface. Phys. Fluids 29, 016102 (2017)CrossRefGoogle Scholar
  35. 35.
    Zhai, Z., Liang, Y., Liu, L., et al.: Interaction of rippled shock wave with flat fast-slow interface. Phys. Fluids 30, 046104 (2018)CrossRefGoogle Scholar
  36. 36.
    Zhu, Y., Yu, L., Pan, J., et al.: Jet formation of \({\rm SF_6}\) bubble induced by incident and reflected shock waves. Phys. Fluids 29, 126105 (2017)CrossRefGoogle Scholar
  37. 37.
    Ou, J., Ding, J., Luo, X., et al.: Effects of Atwood number on shock focusing in shock-cylinder interaction. Exp. Fluids 59, 29 (2018)CrossRefGoogle Scholar
  38. 38.
    Ding, J., Si, T., Chen, M., et al.: On the interaction of a planar shock with a three-dimensional light gas cylinder. J. Fluid Mech. 828, 289–317 (2017)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Liang, Y., Ding, J., Zhai, Z., et al.: Interaction of cylindrically converging diffracted shock with uniform interface. Phys. Fluids 29, 086101 (2017)CrossRefGoogle Scholar
  40. 40.
    Ding, J., Si, T., Yang, J., et al.: Measurement of a Richtmyer–Meshkov instability at an air-\({\rm SF_6}\) interface in a semiannular shock tube. Phys. Rev. Lett. 119, 014501 (2017)CrossRefGoogle Scholar
  41. 41.
    Zou, L., Liao, S., Liu, C., et al.: Aspect ratio effect on shock-accelerated elliptic gas cylinders. Phys. Fluids 28, 036101 (2016)CrossRefGoogle Scholar
  42. 42.
    Zhou, Y.: Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. I. Phys. Reports 720–722, 1–136 (2017)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Zhou, Y.: Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. II. Phys. Reports 723–725, 1–160 (2017)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Abgrall, R.: How to prevent pressure oscillations in multicomponent flow calculations: a quasi conservative approach. J. Comput. Phys. 125, 150–160 (1996)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Shankar, S.K., Lele, S.K.: Numerical investigation of turbulence in reshocked Richtmyer–Meshkov unstable curtain of dense gas. Shock Waves 24, 79–95 (2014)CrossRefGoogle Scholar
  46. 46.
    Wilke, C.R.: A viscosity equation for gas mixtures. J. Chem. Phys. 18, 517–519 (1950)CrossRefGoogle Scholar
  47. 47.
    Giordano, J., Burtschell, Y.: Richtmyer–Meshkov instability induced by shock-bubble interaction: numerical and analytical studies with experimental validation. Phys. Fluids 18, 036102 (2006)CrossRefGoogle Scholar
  48. 48.
    Ramshaw, J.D.: Self-consistent effective binary diffusion in multicomponent gas mixtures. J. Non-Equilib. Thermodyn. 15, 295–300 (1990)CrossRefGoogle Scholar
  49. 49.
    Johnsen, E., Ham, F.: Preventing numerical errors generated by interface-capturing schemes in compressible multi-material flows. J. Comput. Phys. 231, 5705–5717 (2012)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Wang, Q.J., Ren, Y.X., Sun, Z.S., et al.: Low dispersion finite volume scheme based on reconstruction with minimized dispersion and controllable dissipation. Sci. China-Phys. Mech. Astron. 56, 423–431 (2013)CrossRefGoogle Scholar
  51. 51.
    Toro, E.F., Spruce, M., Speares, W.: Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 4, 25–34 (1994)CrossRefGoogle Scholar
  52. 52.
    Johnsen, E., Colonius, T.: Implementation of WENO schemes in compressible multicomponent flow problems. J. Comput. Phys. 219, 715–732 (2006)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Jiang, G.S., Shu, C.W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Luo, X., Wang, M., Si, T., et al.: On the interaction of a planar shock with an \(\text{ SF }_ {6}\) polygon. J. Fluid Mech. 773, 366–394 (2015)CrossRefGoogle Scholar
  55. 55.
    Wang, M., Si, T., Luo, X.: Generation of polygonal gas interfaces by soap film for Richtmyer–Meshkov instability study. Exp. Fluids 54, 1427 (2013)CrossRefGoogle Scholar
  56. 56.
    Matsumoto, Y., Hoshino, M.: Onset of turbulence induced by a Kelvin–Helmholtz vortex. Geophys. Res. Lett. 31, L02807 (2004)CrossRefGoogle Scholar
  57. 57.
    Zhou, Y., Robey, H.F., Buckingham, A.C.: Onset of turbulence in accelerated high-Reynolds-number flow. Phys. Rev. E 67, 056305 (2003)CrossRefGoogle Scholar
  58. 58.
    Zhou, Y., Remington, B.A., Robey, H.F., et al.: Progress in understanding turbulent mixing induced by Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Phys. Plasmas 10, 1883–1896 (2003)CrossRefGoogle Scholar
  59. 59.
    Reilly, D., McFarland, J., Mohaghar, M., et al.: The effects of initial conditions and circulation deposition on the inclined-interface reshocked Richtmyer–Meshkov instability. Exp. Fluids 56, 168 (2015)CrossRefGoogle Scholar
  60. 60.
    Claude, M., Serge, G.: Two-dimensional Navier–Stocks simulations of gaseous mixtures induced by Richtmyer–Meshkov instability. Phys. Fluids 7, 1783–1798 (2000)zbMATHGoogle Scholar
  61. 61.
    Dimotakis, P.E.: The mixing transition in turbulent flows. J. Fluid Mech. 409, 69–98 (2000)MathSciNetCrossRefGoogle Scholar
  62. 62.
    Weber, C.R., Haehn, N.S., Oakley, J.G., et al.: An experimental investigation of the turbulent mixing transition in the Richtmyer–Meshkov instability. J. Fluid Mech. 748, 457–487 (2014)CrossRefGoogle Scholar
  63. 63.
    Ranjan, D., Oakley, J., Bonazza, R.: Shock-bubble interactions. Annu. Rev. Fluid Mech. 43, 117–140 (2011)MathSciNetCrossRefGoogle Scholar
  64. 64.
    Whitham, G.B.: A new approach to problems of shock dynamics Part I Two-dimensional problems. J. Fluid Mech. 2, 145–171 (1957)MathSciNetCrossRefGoogle Scholar
  65. 65.
    Rahmani, M., Lawrence, G.A., Seymour, B.R.: The effect of Reynolds number on mixing in Kelvin–Helmholtz billows. J. Fluid Mech. 759, 612–641 (2014)CrossRefGoogle Scholar
  66. 66.
    Corcos, G.M., Sherman, F.S.: Vorticity concentration and the dynamics of unstable free shear layers. J. Fluid Mech. 73, 241–264 (1976)CrossRefGoogle Scholar
  67. 67.
    Mashayek, A., Peltier, W.R.: The zooof secondary instabilities precursory to stratified shear flow transition. Part 1: Shear aligned convection, pairing, and braid instabilities. J. Fluid Mech. 708, 5–44 (2012)MathSciNetCrossRefGoogle Scholar
  68. 68.
    Mashayek, A., Peltier, W.R.: The zooof secondary instabilities precursory to stratified shear flow transition. Part 2: The influence of stratification. J. Fluid Mech. 708, 45–70 (2012)MathSciNetCrossRefGoogle Scholar
  69. 69.
    Rikanati, A., Alon, U., Shvarts, D.: Vortex-merger statistical-mechanics model for the late time self-similar evolution of the Kelvin–Helmholtz instability. Phys. Fluids 15, 3776–3785 (2003)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Wei-Gang Zeng
    • 1
  • Jian-Hua Pan
    • 1
  • Yu-Xin Ren
    • 1
  • Yu-Tao Sun
    • 2
  1. 1.Department of Engineering Mechanics, School of AerospaceTsinghua UniversityBeijingChina
  2. 2.Institute of Applied Physics and Computational MathematicsBeijingChina

Personalised recommendations