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Numerical investigation of a single-mode chemically reacting Richtmyer-Meshkov instability

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Abstract

We report on high-resolution, numerical simulations of a single-mode, chemically reacting, Richtmyer-Meshkov (RM) instability, at different interface thicknesses. The gases on either side of the diffuse interface were Hydrogen (H\(_2)\) and Oxygen (O\(_2)\), with a pre-shock Atwood number \((A_t\equiv \frac{\rho _{h}-\rho _{l}}{\rho _{h}+\rho _{l} })\) of \(\sim \) 0.5. An incident shock with a Mach number of 1.2 is allowed to traverse from the light (H\(_2)\) to the heavy (O\(_2)\) medium in the 2D numerical shock tube. The simulations were performed using the astrophysical FLASH code developed at the University of Chicago, with extensive modifications implemented by the authors to describe detailed H\(_2\)–O\(_2\) chemistry, temperature-dependent specific heats, and multi-species equation of state. The interface thickness was systematically varied in the simulations to study the effect of the total mass of fuel burnt and heat added on the hydrodynamic instability growth rates. In the absence of an incident shock, burning results in the formation of so-called combustion waves, which spontaneously trigger RM and Rayleigh-Taylor like instability growth of the interface. We are able to obtain the resulting growth rates of an imposed sinusoidal perturbation, and compare them with the predictions of an impulsive model, with simple modifications to account for the finite thickness of the interface, density changes due to heat addition, and compression of the material line due to the combustion wave. When additionally an incident shock is present, we observe complex interactions between the shock and the aforementioned combustion waves, resulting in significant non-planar distortions of each. When the unstable interface is subjected to a reshock, significant mixing enhancement is observed, accompanied by a dramatic increase in combustion product formation, and combustion efficiency.

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References

  1. Khokhlov, A.M., Oran, E.S., Thomas, G.O.: Numerical simulation of deflagration-to-detonation transition: the role of shock-flame interactions in turbulent flames. Combust. Flame 117(1), 323–339 (1999)

    Article  Google Scholar 

  2. Oran, E.S., Khokhlov, A.M.: Deflagrations, hot spots, and the transition to detonation. Phil. Trans. R. Soc. Lond. Ser. A 357(1764), 3539–3551 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. Nishioka, M., Law, C.K.: A numerical study of ignition in the supersonic hydrogen/air laminar mixing layer. Combust. Flame 108(1), 199–219 (1997)

    Article  Google Scholar 

  4. Takita, K., Niioka, T.: Numerical simulation of a counterflow diffusion flame in supersonic airflow. In: Symp. Int. Combust. vol. 26, pp. 2877–2883. Elsevier (1996)

  5. Fujimori, T., Murayama, M., Sato, J., Kobayashi, H., Hasegawa, S., Niioka, T.: Improvement of flameholding characteristics by incident shock waves in supersonic flow. Int. J. Energ. Mater. Chem. Propul. 5(1–6), 330–339 (2002)

    Google Scholar 

  6. Richtmyer, R.D.: Taylor instability in shock acceleration of compressible fluids. Commun. Pure Appl. Math. 13(2), 297–319 (1960)

    Article  MathSciNet  Google Scholar 

  7. Meshkov, E.E.: Instability of the interface of two gases accelerated by a shock wave. Fluid Dyn. 4(5), 101–104 (1969)

    Article  MathSciNet  Google Scholar 

  8. Mikaelian, K.O.: Freeze-out and the effect of compressibility in the Richtmyer-Meshkov instability. Phys. Fluids 6, 356–368 (1994)

    Article  MATH  Google Scholar 

  9. Dimonte, G., Ramaprabhu, P.: Simulations and model of the nonlinear Richtmyer-Meshkov instability. Phys. Fluids 22, 014104 (2010)

    Article  Google Scholar 

  10. Brouillette, M.: The Richtmyer-Meshkov instability. Annu. Rev. Fluid Mech. 34(1), 445–468 (2002)

    Article  MathSciNet  Google Scholar 

  11. Meyer, K.A., Blewett, P.J.: Numerical investigation of the stability of a shock-accelerated interface between two fluids. Phys. Fluids 15(5), 753–759 (1972)

    Article  Google Scholar 

  12. Brouillette, M., Sturtevant, B.: Experiments on the Richtmyer-Meshkov instability: single-scale perturbations on a continuous interface. J. Fluid Mech. 263, 271–292 (1994)

    Article  Google Scholar 

  13. Markstein, G.H.: Nonsteady Flame Propagations. Mac-Millan, New York (1964)

    Google Scholar 

  14. Picone, J.M., Oran, E.S., Boris, J.P., Young, T.R.: Theory of vorticity generation by shock wave and flame interactions. NRL Memo. Rep. 5366 (1984)

  15. Gui, M., Fan, C., Dong, G., Ye, J.: Interaction of a reflected shock from a concave wall with a flame distorted by an incident shock. Shock Waves 18, 487–494 (2009)

    Article  MATH  Google Scholar 

  16. Ju, Y., Shimano, A., Inoue, O.: Vorticity generation and flame distortion induced by shock flame interaction. In: Twenty seventh Symp. Int. Combust. pp. 735–741 (1998)

  17. Batley, G.A., McIntosh, A.C., Brindley, J.: Baroclinic distortion of laminar flames. Proc. R. Soc. Lond. Ser. A 452(1945), 199–221 (1996)

    Article  MATH  Google Scholar 

  18. Teng, H.H., Jiang, Z.L., Hu, Z.M.: Detonation initiation developing from the Richtmyer-Meshkov instability. Acta Mech. Sin. 23(4), 343–349 (2007)

    Article  MATH  Google Scholar 

  19. Haehn, N., Ranjan, D., Weber, C., Oakley, J., Rothamer, D., Bonazza, R.: Reacting shock bubble interaction. Combust. Flame 159(3), 1339–1350 (2012)

    Article  Google Scholar 

  20. Massa, L., Jha, P.: Linear analysis of the Richtmyer-Meshkov instability in shock-flame interactions. Phys. Fluids 24(5), 056101 (2012)

    Article  Google Scholar 

  21. Khokhlov, A.M., Oran, E.S., Chtchelkanova, A.Y., Wheeler, J.C.: Interaction of a shock with a sinusoidally perturbed flame. Combust. Flame 117(1), 99–116 (1999)

    Article  Google Scholar 

  22. Fryxell, B., Olson, K., Ricker, P., Timmes, F.X., Zingale, M., Lamb, D.Q., MacNeice, P., Rosner, R., Truran, J.W., Tufo, H.: Flash: An adaptive mesh hydrodynamics code for modeling astrophysical thermonuclear flashes. Astrophys. J. Suppl. Ser. 131, 273–334 (2000)

    Article  Google Scholar 

  23. Attal, N., Ramaprabhu, P., Hossain, J., Karkhanis, V., Uddin, M., Gord, J.R., Roy, S.: Development and validation of a chemical reaction solver coupled to the flash code for combustion applications. Comput. Fluids 107, 59–76 (2015)

    Article  MathSciNet  Google Scholar 

  24. Colella, P., Woodward, P.R.: The piecewise parabolic method (PPM) for gas-dynamical simulations. J. Comput. Phys. 54(1), 174–201 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  25. Smith, G.P., Golden, D.M., Frenklach, M., Moriarty, N.W., Eiteneer, B., Goldenberg, M., Bowman, C.T., Hanson, R.K., Song, S., Gardiner Jr, W.C., Lissianski, V.V., Qin, Z.: Gri-mech 3.0 (1999). http://www.me.berkeley.edu/gri_mech/

  26. Billet, G.: Improvement of convective concentration fluxes in a one step reactive flow solver. J. Comput. Phys. 204(1), 319–352 (2005)

    Article  MATH  Google Scholar 

  27. Miller, J.A., Mitchell, R.E., Smooke, M.D., Kee, R.J.: Toward a comprehensive chemical kinetic mechanism for the oxidation of acetylene: comparison of model predictions with results from flame and shock tube experiments. In: Symp. Int. Combust. Proc. vol. 19, pp. 181–196. Elsevier (1982)

  28. Bader, G., Deuflhard, P.: A semi-implicit mid-point rule for stiff systems of ordinary differential equations. Numer. Math. 41(3), 373–398 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  29. Grinstein, F.F., Rider, W.J., Margolin, L.G.: Implicit Large Eddy Simulation. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  30. Thornber, B., Mosedale, A., Drikakis, D.: On the Implicit Large Eddy Simulations of homogeneous decaying turbulence. J. Comput. Phys. 226(2), 1902–1929 (2007)

    Article  MATH  Google Scholar 

  31. Domaradzki, J.A., Xiao, Z., Smolarkiewicz, P.K.: Effective eddy viscosities in Implicit Large Eddy Simulations of turbulent flows. Phys. Fluids 15(12), 3890–3893 (2003)

    Article  Google Scholar 

  32. Aspden, A., Nikiforakis, N., Dalziel, S., Bell, J.: Analysis of implicit LES methods. Commun. Appl. Math. Comput. Sci. 3(1), 103–126 (2009)

    Article  MathSciNet  Google Scholar 

  33. Thornber, B., Drikakis, D.: Implicit Large Eddy Simulation of a deep cavity using high-resolution methods. AIAA J. 46(10), 2634–2645 (2008)

    Article  Google Scholar 

  34. Thornber, B., Drikakis, D., Youngs, D.L., Williams, R.J.R.: The influence of initial conditions on turbulent mixing due to Richtmyer-Meshkov instability. J. Fluid Mech. 654, 99–139 (2010)

    Article  MATH  Google Scholar 

  35. Rana, Z.A., Thornber, B., Drikakis, D.: Transverse jet injection into a supersonic turbulent cross-flow. Phys. Fluids 23(4), 046103 (2011)

    Article  Google Scholar 

  36. Youngs, D.L.: Numerical simulation of mixing by Rayleigh-Taylor and Richtmyer-Meshkov instabilities. Laser Particle Beams 12(04), 725–750 (1994)

    Article  Google Scholar 

  37. Gowardhan, A.A., Grinstein, F.F.: Numerical simulation of Richtmyer-Meshkov instabilities in shocked gas curtains. J. Turbul. 12(43), 1–24 (2011)

    MathSciNet  Google Scholar 

  38. Ranjan, D., Niederhaus, J.H.J., Oakley, J.G., Anderson, M.H., Bonazza, R., Greenough, J.A.: Shock-bubble interactions: features of divergent shock-refraction geometry observed in experiments and simulations. Phys. Fluids 20(3), 036101 (2008)

    Article  Google Scholar 

  39. Dimonte, G., Youngs, D.L., Dimits, A., Weber, S., Marinak, M., Wunsch, S., Garasi, C., Robinson, A., Andrews, M.J., Ramaprabhu, P., Calder, A.C., Fryxell, B., Biello, J., Dursi, L., MacNeice, P., Olson, K., Ricker, P., Rosner, R., Timmes, F., Tufo, H., Young, Y.-N., Zingale, M.: A comparative study of the turbulent Rayleigh-Taylor instability using high-resolution three-dimensional numerical simulations: the alpha-group collaboration. Phys. Fluids 16(5), 1668–1693 (2004)

    Article  Google Scholar 

  40. Ramaprabhu, P., Karkhanis, V., Lawrie, A.G.W.: The Rayleigh-Taylor instability driven by an accel-decel-accel profile. Phys. Fluids 25(11), 115104 (2013)

    Article  Google Scholar 

  41. CERFACS: Adiabatic flame temperature calculator (2012). http://elearning.cerfacs.fr/combustion/tools/adiabaticflametemperature/index.php

  42. Duff, R.E., Harlow, F.H., Hirt, C.W.: Effects of diffusion on interface instability between gases. Phys. Fluids 5(4), 417–425 (2004)

    Article  Google Scholar 

  43. Miles, A.R.: Bubble merger model for the nonlinear Rayleigh-Taylor instability driven by a strong blast wave. Phys. Plasmas 11(11), 5140–5155 (2004)

    Article  Google Scholar 

  44. Miles, A.R., Braun, D.G., Edwards, M.J., Robey, H.F., Drake, R.P., Leibrandt, D.R.: Numerical simulation of supernova-relevant laser-driven hydro experiments on OMEGA. Phys. Plasmas 11(7), 3631–3645 (2004)

    Article  Google Scholar 

  45. Kuranz, C.C.: Blast-wave-driven, multidimensional Rayleigh-Taylor instability experiments. PhD thesis, The University of Michigan (2009)

  46. Bourguignon, E., Johnson, M.R., Kostiuk, L.W.: The use of a closed-loop wind tunnel for measuring the combustion efficiency of flames in a cross flow. Combust. Flame 119(3), 319–334 (1999)

    Article  Google Scholar 

  47. Glendinning, S.G., Weber, S.V., Bell, P., DaSilva, L.B., Dixit, S.N., Henesian, M.A., Kania, D.R., Kilkenny, J.D., Powell, H.T., Wallace, R.J., Wegner, P.J., Knauer, J.P., Verdon, C.P.: Laser-driven planar Rayleigh-Taylor instability experiments. Phys. Rev. Lett. 69(8), 1201–1204 (1992)

    Article  Google Scholar 

  48. Fujioka, S., Sunahara, A., Nishihara, K., Ohnishi, N., Johzaki, T., Shiraga, H., Shigemori, K., Nakai, M., Ikegawa, T., Murakami, M., Nagai, K., Norimatsu, T., Azechi, H., Yamanaka, T.: Suppression of the Rayleigh-Taylor instability due to self-radiation in a multiablation target. Phys. Rev. Lett. 92(19), 195001 (2004)

    Article  Google Scholar 

  49. Fujioka, S., Shiraga, H., Nishikino, M., Shigemori, K., Sunahara, A., Nakai, M., Azechi, H., Nishihara, K., Yamanaka, T.: First observation of density profile in directly laser-driven polystyrene targets for ablative Rayleigh-Taylor instability research. Phys. Plasmas 10(12), 4784–4789 (2003)

    Article  Google Scholar 

  50. Pawley, C.J., Gerber, K., Lehmberg, R.H., McLean, E.A., Mostovych, A.N., Obenschain, S.P., Sethian, J.D., Serlin, V., Stamper, J.A., Sullivan, C.A., Bodner, S.E., Colombant, D., Dahlburg, J.P., Schmitt, A.J., Gardner, J.H., Brown, C., Seely, J.F., Lehecka, T., Aglitskiy, Y., Deniz, A.V., Chan, Y., Metzler, N., Klapisch, M.: Measurements of laser-imprinted perturbations and Rayleigh-Taylor growth with the Nike KrF laser. Phys. Plasmas 4(5), 1969–1977 (1997)

    Article  Google Scholar 

  51. Cole, A.J., Kilkenny, J.D., Rumsby, P.T., Evans, R.G., Hooker, C.J., Key, M.H.: Measurement of Rayleigh-Taylor instability in a laser-accelerated target. Nature (London) 299, 329–331 (1982)

    Article  Google Scholar 

  52. Kuranz, C.C., Drake, R.P., Harding, E.C., Grosskopf, M.J., Robey, H.F., Remington, B.A., Edwards, M.J., Miles, A.R., Perry, T.S., Blue, B.E., Plewa, T., Hearn, N.C., Knauer, J.P., Arnett, D., Leibrandt, D.R.: Two-dimensional blast-wave-driven Rayleigh-Taylor instability: experiment and simulation. Astrophys. J. 696(1), 749–759 (2009)

    Article  Google Scholar 

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Acknowledgments

This work was supported in part by the (U.S.) Department of Energy under contract no. DE-AC52-06NA2-5396. N.A. was partially supported by a Scholarship grant from the Energy Production & Infrastructure Center (EPIC) at UNC Charlotte. FLASH was developed by the DOE-sponsored ASC/Alliance Center for Astrophysical Thermonuclear Flashes at the University of Chicago. The authors wish to thank Hilda Varshochi for helpful discussions and assistance in preparing the manuscript.

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  1. Department of Mechanical Engineering and Engineering Science, University of North Carolina at Charlotte, 9201 University City Blvd, Charlotte, North Carolina, 28223, USA

    N. Attal & P. Ramaprabhu

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  1. N. Attal
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  2. P. Ramaprabhu
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Correspondence to P. Ramaprabhu.

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Communicated by R. Bonazza.

This paper is based on work that was presented at the 29th International Symposium on Shock Waves, Madison, Wisconsin, USA, July 14–19, 2013.

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Attal, N., Ramaprabhu, P. Numerical investigation of a single-mode chemically reacting Richtmyer-Meshkov instability. Shock Waves 25, 307–328 (2015). https://doi.org/10.1007/s00193-015-0571-6

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