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Second-order curved shock theory

Published online by Cambridge University Press:  27 March 2020

Chongguang Shi Open the ORCID record for Chongguang Shi [Opens in a new window] ,
Weiqiang Han ,
Ralf Deiterding Open the ORCID record for Ralf Deiterding [Opens in a new window] ,
Chengxiang Zhu Open the ORCID record for Chengxiang Zhu [Opens in a new window]  and
Yancheng You Open the ORCID record for Yancheng You [Opens in a new window]
Chongguang Shi
Affiliation:
School of Aerospace Engineering, Xiamen University, Xiamen, Fujian, 361005, China
Weiqiang Han
Affiliation:
School of Aerospace Engineering, Xiamen University, Xiamen, Fujian, 361005, China
Ralf Deiterding
Affiliation:
School of Engineering, University of Southampton, SouthamptonSO16 7QF, UK
Chengxiang Zhu
Affiliation:
School of Aerospace Engineering, Xiamen University, Xiamen, Fujian, 361005, China
Yancheng You*
Affiliation:
School of Aerospace Engineering, Xiamen University, Xiamen, Fujian, 361005, China
*
†Email address for correspondence: [email protected]
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Abstract

Second-order curved shock theory is developed and applied to planar and axisymmetric curved shock flow fields. Explicit equations are given in an influence coefficient format, relating the second-order gradients of pre-shock and post-shock flow parameters to shock curvature gradients. Two types of applications are demonstrated. First, the post-shock flow fields behind known curved shocks are solved using the second-order curved shock equations. Compared with the first-order curved shock equations, the second-order equations give better agreement with solutions obtained using the method of characteristics. Second, the second-order theory is applied to capture the curved shock shape with limited flow field information. In terms of the residual sum of squares of the curved shock, the second-order curved shock equations give a value one order of magnitude better than those given by the Rankine–Hugoniot equations and the first-order equations. This improved accuracy makes the second-order theory a good candidate for solving shock capture problems in computational fluid dynamics algorithms.

JFM classification

Aerodynamics: High-speed flow Compressible Flows: Shock waves Compressible Flows: Gas dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 891 , 25 May 2020 , A21
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Christopher, A. M. 2008 Experiments on the mechanism of inducing transition between regular and Mach reflection. Phys. Fluids 20, 126103.Google Scholar
Crocco, L. 1937 Singolarita della corrente gassosa iperacustica nellinterno di una prora a diedro. Congresso dell Unione Matematica Italiana 15, 597–615.Google Scholar
Emanuel, G. 1988 Shock wave derivatives. Phys. Fluids 31, 3625–3633.CrossRefGoogle Scholar
Emanuel, G. 2012 Shock Wave Dynamics: Derivatives and Related Topics. CRC Press.CrossRefGoogle Scholar
Emanuel, G. 2018 Analytical extension of curved shock theory. Shock Waves 28, 417–425.CrossRefGoogle Scholar
Filippi, A. A. & Skews, B. W. 2018 Streamlines behind curved shock waves in axisymmetric flow fields. Shock Waves 28, 785–793.CrossRefGoogle Scholar
Gerber, N. & Bartos, J. M. 1960 Calculation of flow-variable gradients behind curved shock waves. J. Aerosp. Sci. 27, 958–959.Google Scholar
Hayes, W. D. & Probstein, R. F. 1959 Hypersonic Flow Theory. Academic Press.Google Scholar
Hornung, H. G. 1998 Gradients at a curved shock in reacting flow. Shock Waves 8 (1), 11–21.CrossRefGoogle Scholar
Ivanov, M. S. 1995 Hysteresis effect in stationary reflection of shock waves. Phys. Fluids 7, 685–687.CrossRefGoogle Scholar
Kanwal, R. P. 1959 On curved shock waves in three-dimensional gas flows. Q. Appl. Maths 16, 361–372.CrossRefGoogle Scholar
Lin, C. C. & Rubinov, S. I. 1948 On the flow behind curved shocks. Stud. Appl. Maths 27, 105–129.CrossRefGoogle Scholar
Mölder, S. 1972 Polar streamline directions at the triple point of Mach interaction of shock waves. Can. Aeronaut. Space Inst. Trans. 5 (2), 88–89.Google Scholar
Mölder, S. 2015 Curved shock theory. Shock Waves 26 (4), 1–17.Google Scholar
Mölder, S. 2017a Flow behind concave shock waves. Shock Waves 27 (5), 721–730.CrossRefGoogle Scholar
Mölder, S. 2017b Reflection of curved shock waves. Shock Waves 27 (1), 1–22.CrossRefGoogle Scholar
Moretti, G. & Abbett, M. 1966 A time-dependent computational method for blunt body flows. AIAA J. 4, 736–741.CrossRefGoogle Scholar
Moretti, G. & Salas, M. D. 1970 Numerical analysis of viscous one-dimensional flows. J. Comput. Phys. 5, 487–506.CrossRefGoogle Scholar
NACA1952 Equations, tables, and charts for compressible flow. NACA Tech. Rep. 1135. Ames Aeronautical Laboratory.Google Scholar
Pant, J. C. 1971 Reflection of a curved shock from a straight rigid boundary. Phys. Fluids 14, 534–538.CrossRefGoogle Scholar
Thomas, T. Y. 1947 On curved shock waves. J. Math. Phys. 26 (14), 62–68.CrossRefGoogle Scholar
Thomas, T. Y. 1949 The consistency relations for shock waves. Stud. Appl. Maths 28, 62–90.Google Scholar
Truesdell, C. 1952 On curved shocks in steady plane flow of an ideal fluid. J. Aerosp. Sci. 19, 826–828.Google Scholar
Uskov, V. N. & Mostovykh, P. S. 2016 The flow gradients in the vicinity of a shock wave for a thermodynamically imperfect gas. Shock Waves 26 (6), 693–708.CrossRefGoogle Scholar
von Neumann, J. 1943 Oblique reflection of shock waves. In John von Neumann Collected Works. Pergamon Press.Google Scholar
Wang, H. 2017 A specially curved wedge for eliminating wedge angle effect in unsteady shock reflection. Phys. Fluids 29, 086103.CrossRefGoogle Scholar