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Kinks on elliptical convergent shock waves in hypersonic flow

Published online by Cambridge University Press:  12 December 2023

Dongxian Si Open the ORCID record for Dongxian Si [Opens in a new window]  and
Zhufei Li Open the ORCID record for Zhufei Li [Opens in a new window]
Dongxian Si
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei 230026, PR China
Zhufei Li*
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei 230026, PR China
*
†Email address for correspondence: lizhufei@ustc.edu.cn
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Abstract

Kinks commonly appear on the convergent shock surface when an internal conical flow deviates from the axisymmetric state. In this paper, the formation mechanisms of kinks on internal conical shocks (ICSs) generated by elliptical ring wedges with typical entry aspect ratios ($AR{\rm s}$) in a Mach 6 flow are revealed using a theoretical method, in which the spatial evolution of the three-dimensional elliptical ICS is converted into a temporal evolution of a two-dimensional elliptical moving shock (EMS) using the hypersonic equivalence principle. To simultaneously track the shock front of the EMS and the disturbances propagating along it, a front-disturbance tracking method (FDTM) based on geometrical shock dynamics is proposed. It is found that the shock–compression disturbances from the same family initially near the major axis catch up with the disturbance initially emitted from the major axis to form kinks on the EMS. The equivalent kink formation positions predicted by the FDTM always lag behind the real kink formation positions on the elliptical ICS because the applicability of the hypersonic equivalence principle decays as the shock strengthens along the incoming flow direction. The accuracy of the equivalent kink formation positions predicted by the FDTM gradually declines with the reduction in $AR$, but it can be significantly improved for all $AR{\rm s}$ after a modification of the equivalent relationship using the shock angle in the major plane of the elliptical ICS, which provides a new way to solve the kinks on the elliptical ICS.

JFM classification

Compressible Flows: Hypersonic Flow Compressible Flows: Shock waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 977 , 25 December 2023 , A12
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

Akbar, R., Schwendeman, D.W., Shepherd, J.E., Williams, R.L. & Thomas, G.O. 1995 Wave shaping channels for gaseous detonations. In Shock Waves@ Marseille IV: Shock Structure and Kinematics, Blast Waves and Detonations (ed. R. Brun & L.Z. Dumitrescu), pp. 465–470. Springer.CrossRefGoogle Scholar
Anderson, J.D. 2019 Hypersonic and High Temperature Gas Dynamics. AIAA.CrossRefGoogle Scholar
Apazidis, N. 2003 Focusing of strong shocks in an elliptic cavity. Shock Waves 13, 91–101.CrossRefGoogle Scholar
Ben-Dor, G. 2007 Shock Wave Reflection Phenomena, 2nd edn. Springer.Google Scholar
Best, J.P. 1991 A generalisation of the theory of geometrical shock dynamics. Shock Waves 1, 251–273.CrossRefGoogle Scholar
Best, J.P. 1993 Accounting for transverse flow in the theory of geometrical shock dynamics. Proc. R. Soc. Lond. A 442 (1916), 585–598.Google Scholar
Chester, W. 1954 CXLV. The quasi-cylindrical shock tube. Lond. Edinb. Dubl. Philos. Mag. J. Sci. 45 (371), 1293–1301.CrossRefGoogle Scholar
Chisnell, R.F. 1957 The motion of a shock wave in a channel, with applications to cylindrical and spherical shock waves. J. Fluid Mech. 2 (3), 286–298.CrossRefGoogle Scholar
Courant, R. & Friedrichs, K.O. 1999 Supersonic Flow and Shock Waves. Springer Science & Business Media.Google Scholar
De Moura, C.A. & Kubrusly, C.S. 2013 The courant–friedrichs–lewy (CFL) condition. Appl. Math. Comput. 10 (12), 363–371.Google Scholar
Denet, B., Biamino, L., Lodato, G., Vervisch, L. & Clavin, P. 2015 Model equation for the dynamics of wrinkled shockwaves: comparison with DNS and experiments. Combust. Sci. Technol. 187 (1–2), 296–323.CrossRefGoogle Scholar
Ferri, A. 1946 Application of the method of characteristics to supersonic rotational flow. Tech. Rep.Google Scholar
Gottlieb, S. & Shu, C.W. 1998 Total variation diminishing Runge–Kutta schemes. Math. Comput. 67 (221), 73–85.CrossRefGoogle Scholar
Guan, X.-K., Bai, C.-Y., Lin, J. & Wu, Z.-N. 2020 Mach reflection promoted by an upstream shock wave. J. Fluid Mech. 903, A44.CrossRefGoogle Scholar
Guderley, K.G. 1942 Starke kugelige und zylindrische verdichtungsstosse in der nahe des kugelmitterpunktes bnw. der zylinderachse. Luftfahrtforschung 19, 302.Google Scholar
Han, Z.-Y. & Yin, X.-Z. 1993 Shock Dynamics. Springer Science & Business Media.CrossRefGoogle Scholar
Henshaw, W.D., Smyth, N.F. & Schwendeman, D.W. 1986 Numerical shock propagation using geometrical shock dynamics. J. Fluid Mech. 171, 519–545.CrossRefGoogle Scholar
Hornung, H.G., Pullin, D.I. & Ponchaut, N.F. 2008 On the question of universality of imploding shock waves. Acta Mechanica 201, 31–35.CrossRefGoogle Scholar
Huynh, H.T. 1993 Accurate monotone cubic interpolation. SIAM J. Numer. Anal. 30 (1), 57–100.CrossRefGoogle Scholar
Isakova, N.P., Kraiko, A.N., Pyankov, K.S. & Tillyayeva, N.I. 2012 The amplification of weak shock waves in axisymmetric supersonic flow and their reflection from an axis of symmetry. J. Appl. Math. Mech. 76 (4), 451–465.CrossRefGoogle Scholar
Ji, J.-Z., Li, Z.-F., Zhang, E.-L., Si, D.-X & Yang, J.-M. 2022 Intensification of non-uniformity in convergent near-conical hypersonic flow. J. Fluid Mech. 931, A8.CrossRefGoogle Scholar
Katko, B.J., Chavez, R., Liu, H., Lawlor, B., McGuire, C., Zheng, L., Zanteson, J. & Eliasson, V. 2020 Experimental and numerical study of blast-structure interaction. In Structures Congress 2020 (ed. J.G. Soules), pp. 105–118. American Society of Civil Engineers.CrossRefGoogle Scholar
Lodato, G., Vervisch, L. & Clavin, P. 2016 Direct numerical simulation of shock wavy-wall interaction: analysis of cellular shock structures and flow patterns. J. Fluid Mech. 789, 221–258.CrossRefGoogle Scholar
Lodato, G., Vervisch, L. & Clavin, P. 2017 Numerical study of smoothly perturbed shocks in the Newtonian limit. Flow Turbul. Combust. 99, 887–908.CrossRefGoogle Scholar
Mohan, J.A. & Skews, B.W. 2013 Three-dimensional supersonic internal flows. Shock Waves 23, 513–524.CrossRefGoogle Scholar
Mölder, S. 1967 Internal, axisymmetric, conical flow. AIAA J. 5 (7), 1252–1255.CrossRefGoogle Scholar
Mölder, S., Gulamhussein, A., Timofeev, E. & Voinovich, P. 1997 Focusing of conical shocks at the centre-line of symmetry. Paper 5601. In Proceedings of the 21st International Symposium on Shock Waves (ed. A.F.P. Houwing & A. Paull), vol. 126. Panther Publishing.Google Scholar
Mostert, W., Pullin, D.I., Samtaney, R. & Wheatley, V. 2016 Converging cylindrical magnetohydrodynamic shock collapse onto a power-law-varying line current. J. Fluid Mech. 793, 414–443.CrossRefGoogle Scholar
Mostert, W., Pullin, D.I., Samtaney, R. & Wheatley, V. 2017 Geometrical shock dynamics for magnetohydrodynamic fast shocks. J. Fluid Mech. 811, R2.CrossRefGoogle Scholar
Mostert, W., Pullin, D.I., Samtaney, R. & Wheatley, V. 2018 a Singularity formation on perturbed planar shock waves. J. Fluid Mech. 846, 536–562.CrossRefGoogle Scholar
Mostert, W., Pullin, D.I., Samtaney, R. & Wheatley, V. 2018 b Spontaneous singularity formation in converging cylindrical shock waves. Phys. Rev. Fluids 3 (7), 071401.CrossRefGoogle Scholar
Ndebele, B.B., Skews, B.W. & Paton, R.T. 2017 On the propagation of curved shockwaves using geometric shock dynamics. In 30th International Symposium on Shock Waves 2: ISSW30-Volume 2 (ed. G. Ben-Dor, O. Sadot & O. Igra), pp. 1505–1510. Springer.CrossRefGoogle Scholar
von Neumann, J. 1943 Oblique reflection of shocks. Bureau of Ordinance, Explosives Research Report.Google Scholar
Noumir, Y., Le Guilcher, A., Lardjane, N., Monneau, R. & Sarrazin, A. 2015 A fast-marching like algorithm for geometrical shock dynamics. J. Comput. Phys. 284, 206–229.CrossRefGoogle Scholar
Prasad, P 1995 Formation and propagation of singularities on a nonlinear wavefront and a shock front. J. Indian Inst. Sci. 75 (5), 537.Google Scholar
Pullin, D.I., Mostert, W., Wheatley, V. & Samtaney, R. 2014 Converging cylindrical shocks in ideal magnetohydrodynamics. Phys. Fluids 26 (9), 097103.CrossRefGoogle Scholar
Qiu, S., Liu, K. & Eliasson, V. 2016 Parallel implementation of geometrical shock dynamics for two dimensional converging shock waves. Comput. Phys. Commun. 207, 186–192.CrossRefGoogle Scholar
Ridoux, J., Lardjane, N., Monasse, L. & Coulouvrat, F. 2018 Comparison of geometrical shock dynamics and kinematic models for shock-wave propagation. Shock Waves 28 (2), 401–416.CrossRefGoogle Scholar
Ridoux, J., Lardjane, N., Monasse, L. & Coulouvrat, F. 2019 Beyond the limitation of geometrical shock dynamics for diffraction over wedges. Shock Waves 29 (6), 833–855.CrossRefGoogle Scholar
Rylov, A.I. 1990 On the impossibility of regular reflection of a steady-state shock wave from the axis of symmetry. J. Appl. Math. Mech. 54 (2), 201–203.CrossRefGoogle Scholar
Schwendeman, D.W. 1988 Numerical shock propagation in non-uniform media. J. Fluid Mech. 188, 383–410.CrossRefGoogle Scholar
Schwendeman, D.W. 1993 A new numerical method for shock wave propagation based on geometrical shock dynamics. Proc. R. Soc. Lond. A 441 (1912), 331–341.Google Scholar
Schwendeman, D.W. 1999 A higher-order Godunov method for the hyperbolic equations modelling shock dynamics. Proc. R. Soc. Lond. A 455 (1984), 1215–1233.CrossRefGoogle Scholar
Schwendeman, D.W. & Whitham, G.B. 1987 On converging shock waves. Proc. R. Soc. Lond. A 413 (1845), 297–311.Google Scholar
Shen, N., Pullin, D.I., Samtaney, R. & Wheatley, V. 2021 Evolution of a shock generated by an impulsively accelerated, sinusoidal piston. J. Fluid Mech. 907, A35.CrossRefGoogle Scholar
Shoesmith, B., Mölder, S., Ogawa, H. & Timofeev, E. 2017 Shock reflection in axisymmetric internal flows. In International Conference on RailNewcastle Talks (ed. K. Kontis), pp. 355–366. Springer.CrossRefGoogle Scholar
Takayama, K, Kleine, H. & Grönig, H 1987 An experimental investigation of the stability of converging cylindrical shock waves in air. Exp. Fluids 5, 315–322.CrossRefGoogle Scholar
Tan, L.-H., Ren, Y.-X. & Wu, Z.-N. 2006 Analytical and numerical study of the near flow field and shape of the mach stem in steady flows. J. Fluid Mech. 546, 341–362.CrossRefGoogle Scholar
Wang, Z.-H. 2019 DSMC implementing generalized hypersonic equivalence principle for viscous flows. In 31st International Symposium on Rarefied Gas Dynamics (ed. Y.H. Zhang, D.R. Emerson, D. Lockerby & L. Wu). AIP Publishing.CrossRefGoogle Scholar
Watanabe, M. & Takayama, K. 1991 Stability of converging cylindrical shock waves. Shock Waves 1, 149–160.CrossRefGoogle Scholar
Whitham, G.B. 1957 A new approach to problems of shock dynamics. Part 1. Two-dimensional problems. J. Fluid Mech. 2 (2), 145–171.CrossRefGoogle Scholar
Whitham, G.B. 1958 On the propagation of shock waves through regions of non-uniform area or flow. J. Fluid Mech. 4 (4), 337–360.CrossRefGoogle Scholar
Whitham, G.B. 2011 Linear and Nonlinear Waves. John Wiley & Sons.Google Scholar
Xiang, G.-X., Wang, C, Teng, H.-H. & Jiang, Z.-L. 2016 Investigations of three-dimensional shock/shock interactions over symmetrical intersecting wedges. AIAA J. 54 (5), 1472–1481.CrossRefGoogle Scholar
Xie, P., Han, Z.-Y. & Takayama, K. 2005 A study of the interaction between two triple points. Shock Waves 14, 29–36.CrossRefGoogle Scholar
Yang, Y., Teng, H.-H., Jiang, Z.-L. & Takayama, K. 2012 Numerical investigation on three-dimensional shock wave reflection over two perpendicularly intersecting wedges. Shock Waves 22, 151–159.CrossRefGoogle Scholar
You, Y.-C. 2011 An overview of the advantages and concerns of hypersonic inward turning inlets. In 17th AIAA International Space Planes and Hypersonic Systems and Technologies Conference (ed. A. Storch, M. Bynum, J. Liu & M. Gruber), p. 2269, AIAA.CrossRefGoogle Scholar
You, Y.-C., Zhu, C.-X. & Guo, J.-L. 2009 Dual waverider concept for the integration of hypersonic inward-turning inlet and airframe forebody. In 16th AIAA/DLR/DGLR International Space Planes and Hypersonic Systems and Technologies Conference (ed. L. Serre), p. 7421, AIAA.CrossRefGoogle Scholar
Zhang, E.-L., Li, Z.-F., Ji, J.-Z., Si, D.-X. & Yang, J.-M. 2021 Converging near-elliptic shock waves. J. Fluid Mech. 909, A2.CrossRefGoogle Scholar
Zucrow, M.J. & Hoffman, J.D. 1997 Gas Dynamics Vol. II: Multimedia Flow. John Wiley & Sons.Google Scholar
Zuo, F.-Y. & Mölder, S. 2019 Hypersonic wavecatcher intakes and variable-geometry turbine based combined cycle engines. Prog. Aerosp. Sci. 106, 108–144.CrossRefGoogle Scholar