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A model for the trajectory of the transverse detonation resulting from re-initiation of a diffracted detonation

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Abstract

A semi-analytical model is presented for predicting the trajectory of the transverse detonation during the re-initiation process of a diffracted cellular detonation wave from a channel. Numerical simulations based on the two-dimensional reactive Euler equations with a detailed hydrogen/oxygen chemistry model were first performed to observe key characteristics of cellular detonation wave diffraction and to obtain required input parameters for the model construction. The present numerical observations indicate that the transverse detonation stems from a location on the expansion wave front, and the horizontal distance from this initial location to the channel exit can be scaled by a constant multiplied by the detonation cell width for large deviation angles of the channel. The velocity of the transverse detonation basically equals the Chapman–Jouguet detonation wave velocity consisting of two orthogonal components: the expansion velocity of the diffracted wave front and the relative velocity to the diffracted wave front. The shape of the decoupled wave front is not affected by local explosion and thus can be predicted by the Chester–Chisnell–Whitham theory. Based on these numerical observations and the Chester–Chisnell–Whitham theory, a semi-analytical model is constructed to predict the wave trajectories as well as the distances of the wall reflection point for various deviation angles and initial pressures. The model prediction agrees with the corresponding numerical results. The model result shows that the distance of the wall reflection point varies from 15 to 20 multiples of the cell width with a minimal dependence on deviation angle, independent of the initial pressure. The trajectory calculated by the model is self-similar and determined by the horizontal distance of the initial location. The dimensionless trajectories divided by the horizontal distance are coincident for different initial pressures.

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References

  1. Zeldovich, Y.B.: An experimental investigation of spherical detonation of gases. Sov. Phys. Tech. Phys. 1, 1689–1713 (1956)

    Google Scholar 

  2. Edwards, D., Thomas, G., Nettleton, M.: The diffraction of a planar detonation wave at an abrupt area change. J. Fluid Mech. 95(1), 79–96 (1979). https://doi.org/10.1017/S002211207900135X

    Article  Google Scholar 

  3. Mitrovanov, V.: The diffraction of multifront detonation waves. Sov. Phys. Dokl. 9, 1055 (1964)

    Google Scholar 

  4. Knystautas, R., Lee, J., Guirao, C.: The critical tube diameter for detonation failure in hydrocarbon–air mixtures. Combust. Flame 48, 63–83 (1982). https://doi.org/10.1016/0010-2180(82)90116-X

    Article  Google Scholar 

  5. Murray, S., Lee, J.: On the transformation of planar detonation to cylindrical detonation. Combust. Flame 52, 269–289 (1983). https://doi.org/10.1016/0010-2180(83)90138-4

    Article  Google Scholar 

  6. Arienti, M., Shepherd, J.: A numerical study of detonation diffraction. J. Fluid Mech. 529, 117–146 (2005). https://doi.org/10.1017/S0022112005003319

    Article  MathSciNet  MATH  Google Scholar 

  7. Pintgen, F., Shepherd, J.: Detonation diffraction in gases. Combust. Flame 156(3), 665–677 (2009). https://doi.org/10.1016/j.combustflame.2008.09.008

    Article  Google Scholar 

  8. Gallier, S., Le Palud, F., Pintgen, F., Mével, R., Shepherd, J.: Detonation wave diffraction in H\(_2\)–O\(_2\)–Ar mixtures. Proc. Combust. Inst. 36(2), 2781–2789 (2017). https://doi.org/10.1016/j.proci.2016.06.090

    Article  Google Scholar 

  9. Lee, J.H.S.: The Detonation Phenomenon. Cambridge University Press, Cambridge (2008). https://doi.org/10.1017/CBO9780511754708

    Book  Google Scholar 

  10. Mehrjoo, N., Zhang, B., Portaro, R., Ng, H., Lee, J.: Response of critical tube diameter phenomenon to small perturbations for gaseous detonations. Shock Waves 24(2), 219–229 (2014). https://doi.org/10.1007/s00193-013-0491-2

    Article  Google Scholar 

  11. Mehrjoo, N., Gao, Y., Kiyanda, C.B., Ng, H.D., Lee, J.H.: Effects of porous walled tubes on detonation transmission into unconfined space. Proc. Combust. Inst. 35(2), 1981–1987 (2015). https://doi.org/10.1016/j.proci.2014.06.031

    Article  Google Scholar 

  12. Xu, H., Mi, X., Kiyanda, C.B., Ng, H.D., Lee, J.H., Yao, C.: The role of cellular instability on the critical tube diameter problem for unstable gaseous detonations. Proc. Combust. Inst. 37(3), 3545–3553 (2019). https://doi.org/10.1016/j.proci.2018.05.133

    Article  Google Scholar 

  13. Whitham, G.: A new approach to problems of shock dynamics Part I Two-dimensional problems. J. Fluid Mech. 2(2), 145–171 (1957). https://doi.org/10.1017/S002211205700004X

    Article  MathSciNet  MATH  Google Scholar 

  14. Bartlmä, F., Schröder, K.: The diffraction of a plane detonation wave at a convex corner. Combust. Flame 66(3), 237–248 (1986). https://doi.org/10.1016/0010-2180(86)90137-9

    Article  Google Scholar 

  15. Thomas, G., Edwards, D., Lee, J., Knystautas, R., Moen, I., Wei, Y.: Detonation diffraction by divergent channels. Dyn. Explos. Prog. Astronaut. Aeronaut. 106, 144–154 (1986). https://doi.org/10.2514/5.9781600865800.0144.0154

    Article  Google Scholar 

  16. Khasainov, B., Presles, H.N., Desbordes, D., Demontis, P., Vidal, P.: Detonation diffraction from circular tubes to cones. Shock Waves 14(3), 187–192 (2005). https://doi.org/10.1007/s00193-005-0262-9

    Article  Google Scholar 

  17. Nagura, Y., Kasahara, J., Sugiyama, Y., Matsuo, A.: Comprehensive visualization of detonation–diffraction structures and sizes in unstable and stable mixtures. Proc. Combust. Inst. 34(2), 1949–1956 (2013). https://doi.org/10.1016/j.proci.2012.07.078

    Article  Google Scholar 

  18. Nagura, Y., Kasahara, J., Matsuo, A.: Multi-frame visualization for detonation wave diffraction. Shock Waves 26(5), 645–656 (2016). https://doi.org/10.1007/s00193-016-0663-y

    Article  Google Scholar 

  19. Kasahara, J., Kawasaki, A.: Critical condition for detonation diffraction with stable and unstable mixtures. 26th International Colloquium on the Dynamics of Explosions and Reactive Systems, Boston, MA, Paper 963 (2017)

  20. Deiterding, R.: Parallel adaptive simulation of multi-dimensional detonation structures. PhD Thesis, Brandenburgischen Technischen Universität Cottbus, Germany (2003)

  21. Deiterding, R.: A parallel adaptive method for simulating shock-induced combustion with detailed chemical kinetics in complex domains. Comput. Struct. 87(11), 769–783 (2009). https://doi.org/10.1016/j.compstruc.2008.11.007

    Article  Google Scholar 

  22. Berger, M.J.: Adaptive mesh refinement for hyperbolic partial differential equations. PhD Thesis, Stanford University, California (1982)

  23. Yuan, X., Zhou, J., Lin, Z., Cai, X.: Adaptive simulations of detonation propagation in 90-degree bent tubes. Int. J. Hydrog. Energy 41(40), 18,259–18,272 (2016). https://doi.org/10.1016/j.ijhydene.2016.07.130

    Article  Google Scholar 

  24. Yuan, X., Zhou, J., Lin, Z., Cai, X.: Numerical study of detonation diffraction through 90-degree curved channels to expansion area. Int. J. Hydrog. Energy 42(10), 7045–7059 (2017). https://doi.org/10.1016/j.ijhydene.2017.01.206

    Article  Google Scholar 

  25. Cai, X., Deiterding, R., Liang, J., Sun, M., Mahmoudi, Y.: Diffusion and mixing effects in hot jet initiation and propagation of hydrogen detonations. J. Fluid Mech. 836, 324–351 (2018). https://doi.org/10.1017/jfm.2017.770

    Article  MathSciNet  MATH  Google Scholar 

  26. Miao, S., Zhou, J., Lin, Z., Cai, X., Liu, S.: Numerical study on thermodynamic efficiency and stability of oblique detonation waves. AIAA J. 56(8), 3112–3122 (2018). https://doi.org/10.2514/1.J056887

    Article  Google Scholar 

  27. Peng, H., Huang, Y., Deiterding, R., Luan, Z., Xing, F., You, Y.: Effects of jet in crossflow on flame acceleration and deflagration to detonation transition in methane–oxygen mixture. Combust. Flame 198, 69–80 (2018). https://doi.org/10.1016/j.combustflame.2018.08.023

    Article  Google Scholar 

  28. Yuan, X., Zhou, J., Mi, X., Ng, H.D.: Numerical study of cellular detonation wave reflection over a cylindrical concave wedge. Combust. Flame 202, 179–194 (2019). https://doi.org/10.1016/j.combustflame.2019.01.018

    Article  Google Scholar 

  29. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, vol. 14. Springer, Berlin (1991). https://doi.org/10.1007/978-3-662-09947-6

    Book  MATH  Google Scholar 

  30. Westbrook, C.K.: Chemical kinetics of hydrocarbon oxidation in gaseous detonations. Combust. Flame 46, 191–210 (1982). https://doi.org/10.1016/0010-2180(82)90015-3

    Article  Google Scholar 

  31. Taylor, B., Kessler, D., Gamezo, V., Oran, E.: Numerical simulations of hydrogen detonations with detailed chemical kinetics. Proc. Combust. Inst. 34(2), 2009–2016 (2013). https://doi.org/10.1016/j.proci.2012.05.045

    Article  Google Scholar 

  32. Shi, L., Shen, H., Zhang, P., Zhang, D., Wen, C.: Assessment of vibrational non-equilibrium effect on detonation cell size. Combust. Sci. Technol. 189(5), 841–853 (2017). https://doi.org/10.1080/00102202.2016.1260561

    Article  Google Scholar 

  33. Smirnov, N., Betelin, V., Shagaliev, R., Nikitin, V., Belyakov, I., Deryuguin, Y.N., Aksenov, S., Korchazhkin, D.: Hydrogen fuel rocket engines simulation using LOGOS code. Int. J. Hydrog. Energy 39(20), 10748–10756 (2014). https://doi.org/10.1016/j.ijhydene.2014.04.150

    Article  Google Scholar 

  34. Smirnov, N., Betelin, V., Nikitin, V., Stamov, L., Altoukhov, D.: Accumulation of errors in numerical simulations of chemically reacting gas dynamics. Acta Astronaut. 117, 338–355 (2015). https://doi.org/10.1016/j.actaastro.2015.08.013

    Article  Google Scholar 

  35. Chester, W.: CXLV. The quasi-cylindrical shock tube. Lond. Edinb. Dublin Philos. Mag. J. Sci. 45(371), 1293–1301 (1954). https://doi.org/10.1080/14786441208561138

    Article  MathSciNet  MATH  Google Scholar 

  36. Chisnell, R.: The motion of a shock wave in a channel, with applications to cylindrical and spherical shock waves. J. Fluid Mech. 2(3), 286–298 (1957). https://doi.org/10.1017/S0022112057000130

    Article  MathSciNet  MATH  Google Scholar 

  37. Skews, B.W.: The shape of a diffracting shock wave. J. Fluid Mech. 29(2), 297–304 (1967). https://doi.org/10.1017/S0022112067000825

    Article  Google Scholar 

  38. Best, J.: A generalisation of the theory of geometrical shock dynamics. Shock Waves 1(4), 251–273 (1991). https://doi.org/10.1007/BF01418882

    Article  MATH  Google Scholar 

  39. Li, H., Ben-Dor, G.: A modified CCW theory for detonation waves. Combust. Flame 113(1–2), 1–12 (1998). https://doi.org/10.1016/S0010-2180(97)00136-3

    Article  Google Scholar 

  40. Ridoux, J., Lardjane, N., Monasse, L., Coulouvrat, F.: Comparison of geometrical shock dynamics and kinematic models for shock-wave propagation. Shock Waves 28(2), 401–416 (2018). https://doi.org/10.1007/s00193-017-0748-2

    Article  Google Scholar 

  41. Peace, J., Lu, F.: On the propagation of decaying planar shock and blast waves through non-uniform channels. Shock Waves (2018). https://doi.org/10.1007/s00193-018-0818-0

    Article  Google Scholar 

  42. Ridoux, J., Lardjane, N., Monasse, L., Coulouvrat, F.: Beyond the limitation of geometrical shock dynamics for diffraction over wedges. Shock Waves (2019). https://doi.org/10.1007/s00193-018-00885-w

    Article  Google Scholar 

  43. Nettleton, M.: Shock attenuation in a ‘gradual’ area expansion. J. Fluid Mech. 60(2), 209–223 (1973). https://doi.org/10.1017/S0022112073000121

    Article  Google Scholar 

  44. Sloan, S., Nettleton, M.: A model for the axial decay of a shock wave in a large abrupt area change. J. Fluid Mech. 71(4), 769–784 (1975). https://doi.org/10.1017/S0022112075002844

    Article  MATH  Google Scholar 

  45. Sloan, S., Nettleton, M.: A model for the decay of a wall shock in a large abrupt area change. J. Fluid Mech. 88(2), 259–272 (1978). https://doi.org/10.1017/S0022112078002098

    Article  Google Scholar 

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Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grant Nos. 91441201 and 51776220 and the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors thank A. J. Higgins and the anonymous reviewers whose comments have greatly improved the quality of this manuscript.

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Authors and Affiliations

  1. Science and Technology on Scramjet Laboratory, National University of Defense Technology, Changsha, 410073, China

    X. Q. Yuan & J. Zhou

  2. Department of Mechanical Engineering, McGill University, Montreal, QC, H3A 2K6, Canada

    X. C. Mi

  3. Department of Mechanical, Industrial and Aerospace Engineering, Concordia University, Montreal, QC, H3G 1M8, Canada

    H. D. Ng

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  1. X. Q. Yuan
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Correspondence to X. Q. Yuan.

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Communicated by N. Smirnov.

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Yuan, X.Q., Mi, X.C., Ng, H.D. et al. A model for the trajectory of the transverse detonation resulting from re-initiation of a diffracted detonation. Shock Waves 30, 13–27 (2020). https://doi.org/10.1007/s00193-019-00904-4

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