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A study of streamline geometries in subsonic and supersonic regions of compressible flow fields

Published online by Cambridge University Press:  20 May 2024

Zhong Wei Tian Open the ORCID record for Zhong Wei Tian [Opens in a new window]  and
Kai Cui
Zhong Wei Tian*
Affiliation:
Institute of Mechanics, Chinese Academy of Sciences, 100190 Beijing, PR China
Kai Cui
Affiliation:
Institute of Mechanics, Chinese Academy of Sciences, 100190 Beijing, PR China
*
†Email address for correspondence: tzw@imech.ac.cn
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Abstract

The geometrical properties of streamlines, such as the curvatures, directions and positions, are studied in steady inviscid compressible flow fields via differential geometry theories and conservation laws. The influences of the streamline geometries on the flow speeds and pressures are also identified and discussed. By transforming the streamlines to fill the domain and satisfy the boundary conditions, a unified geometry-based solver, the streamline transformation method, is proposed for both subsonic and supersonic regions. The governing equations and boundary conditions along streamlines and shock waves are also derived. This method is verified by numerical results of three typical flow fields, including the subsonic channel flow, the supersonic downstream of attached shock waves and especially the subsonic/supersonic downstream of detached bow shock waves. Both two-dimensional planar and axisymmetric flow fields are considered. Compared with the results from computational fluid dynamics, good agreements are achieved by this method, while fewer computational resources, by an order of magnitude, are consumed. Features of these flow fields are also analysed from a geometrical perspective, such as flow speeds and pressures deviated by the wall curvatures, and three-dimensional effects in the after-shock flow fields. For a hyperbolic-shaped bow shock wave, the stand-off distances and the transitions from subsonic to supersonic regions are also discussed. As indicated by the accuracy, efficiency and applicability in a wide range of flow speeds, the streamline transformation method would be a potential candidate for the theoretical analysis and inverse design of high-speed flow fields, especially where the subsonic regions exist downstream of strong shock waves.

JFM classification

Aerodynamics: High-speed flow Compressible Flows: Shock waves Mathematical Foundations: Computational methods
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 987 , 25 May 2024 , A26
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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References

Ambrosio, A. & Wortman, A. 1962 Stagnation-point shock-detachment distance for flow around spheres and cylinders in air. J. Aerosp. Sci. 29 (7), 875–875.Google Scholar
Azevedo, D.J. 1989 Analytical prediction of shock patterns in a high-speed wedge bounded duct. PhD thesis, Department of Mechanical and Aeronautical Engineering, State University, Buffalo, NY.Google Scholar
Azevedo, D.J. & Liu, C.S. 1993 Engineering approach to the prediction of shock patterns in bounded high-speed flows. AIAA J. 31 (1), 83–90.CrossRefGoogle Scholar
Bai, C.-Y. & Wu, Z.-N. 2017 Size and shape of shock waves and slipline for Mach reflection in steady flow. J. Fluid Mech. 818, 116–140.CrossRefGoogle Scholar
Bai, C.-Y. & Wu, Z.-N. 2021 A study of the dependence of the Mach stem height on the trailing edge height. Fluids 6 (9), 313.CrossRefGoogle Scholar
Bai, C.-Y. & Wu, Z.-N. 2022 Type IV shock interaction with a two-branch structured transonic jet. J. Fluid Mech. 941, A45.CrossRefGoogle Scholar
Billig, F.S. 1967 Shock-wave shapes around spherical-and cylindrical-nosed bodies. J. Spacecr. Rockets 4 (6), 822–823.CrossRefGoogle Scholar
Cheng, J., Yang, K., Zheng, X., Shi, C., Zhu, C. & You, Y. 2022 Analytical model for predicting the length scale of shock/boundary layer interaction with curvature. Phys. Fluids 34 (11), 111701.CrossRefGoogle Scholar
Chern, S.S., Chen, W.H. & Lam, K.S. 2000 Lectures on Differential Geometry, 1st edn. Series on University Mathematics, vol. 1. World Scientific.Google Scholar
Chester, W. 1954 The quasi-cylindrical shock tube. Phil. Mag. 45 (371), 1293–1301.CrossRefGoogle Scholar
Chester, W. 1960 The propagation of shock waves along ducts of varying cross section. In Advances in Applied Mechanics, vol. 6, pp. 119–152. Elsevier.CrossRefGoogle Scholar
Chisnell, R.F. 1955 The normal motion of a shock wave through a non-uniform one-dimensional medium. Proc. R. Soc. Lond. A Math. Phys. Sci. 232 (1190), 350–370.Google Scholar
Chisnell, R.F. & Yousaf, M. 1982 The effect of the overtaking disturbance on a shock wave moving in a non-uniform medium. J. Fluid Mech. 120, 523–533.CrossRefGoogle Scholar
Crocco, L. 1937 Singolarita della corrente gassosa iperacustica nellinterno di una prora adiedro. L'Aerotecnica 17 (6), 519–534.Google Scholar
Edney, B. 1968 a Anomalous heat transfer and pressure distributions on blunt bodies at hypersonic speeds in the presence of an impinging shock. Tech. Rep. FFA–115, 4480948.Google Scholar
Edney, B.E. 1968 b Effects of shock impingement on the heat transfer around blunt bodies. AIAA J. 6 (1), 15–21.CrossRefGoogle Scholar
Emanuel, G. & Liu, M.-S. 1988 Shock wave derivatives. Phys. Fluids 31 (12), 3625–3633.CrossRefGoogle Scholar
Emanuel, G. & Mölder, S. 2022 Three-dimensional curved shock theory. Shock Waves 32 (2), 129–146.CrossRefGoogle Scholar
Filippi, A.A. & Skews, B.W. 2018 Streamlines behind curved shock waves in axisymmetric flow fields. Shock Waves 28 (4), 785–793.CrossRefGoogle Scholar
Gao, B. & Wu, Z.N. 2010 A study of the flow structure for Mach reflection in steady supersonic flow. J. Fluid Mech. 656, 29–50.CrossRefGoogle Scholar
Gerber, N. & Bartos, J.M. 1960 Calculation of flow-variable gradients behind curved shock waves. J. Aerosp. Sci. 27 (12), 958–959.CrossRefGoogle Scholar
Grasso, F., Purpura, C., Chanetz, B. & Délery, J. 2003 Type III and type IV shock/shock interferences: theoretical and experimental aspects. Aerosp. Sci. Technol. 7 (2), 93–106.CrossRefGoogle Scholar
Guan, X.-K., Bai, C.-Y. & Wu, Z.-N. 2020 Double solution and influence of secondary waves on transition criteria for shock interference in pre-Mach reflection with two incident shock waves. J. Fluid Mech. 887, A22.CrossRefGoogle Scholar
Han, Z.-Y. & Yin, X.-Z. 2001 Geometrical shock dynamics. In Handbook of Shock Waves, pp. 485–552. Elsevier.CrossRefGoogle Scholar
Henshaw, W.D., Smyth, N.F. & Schwendeman, D.W. 1986 Numerical shock propagation using geometrical shock dynamics. J. Fluid Mech. 171, 519.CrossRefGoogle Scholar
Hida, K. 1953 An approximate study on the detached shock wave in front of a circular cylinder and a sphere. J. Phys. Soc. Japan 8 (6), 740–745.CrossRefGoogle Scholar
Lewis, T.S. & Sirovich, L. 1981 Approximate and exact numerical computation of supersonic flow over an airfoil. J. Fluid Mech. 112, 265–282.CrossRefGoogle Scholar
Li, H. & Ben-Dor, G. 1997 A parametric study of Mach reflection in steady flows. J. Fluid Mech. 341, 101–125.CrossRefGoogle Scholar
Li, H., Ben-Dor, G. & Han, Z.Y. 1994 Modification of Whitham's ray shock theory for analyzing the reflection of weak shock waves over small wedge angles. Shock Waves 4 (1), 41–45.CrossRefGoogle Scholar
Lighthill, M.J. 1957 Dynamics of a dissociating gas. Part I. Equilibrium flow. J. Fluid Mech. 2 (1), 1–32.CrossRefGoogle Scholar
Lin, J., Bai, C.-Y. & Wu, Z.-N. 2019 Study of asymmetrical shock wave reflection in steady supersonic flow. J. Fluid Mech. 864, 848–875.CrossRefGoogle Scholar
Mach, E. 1878 Uber den verlauf von Funkenwellen in der Ebene und im Raume. Sitz. ber. Akad. Wiss. Wien. 78, 819–838.Google Scholar
Moeckel, M.E. 1921 Approximate method for predicting form and location of detached shock waves ahead of plane or axially symmetric bodies. Tech. Rep. NACA-TN-1921.Google Scholar
Mölder, S. 2012 Curved aerodynamic shock waves. PhD thesis, McGill University, Montreal.Google Scholar
Mölder, S. 2016 Curved shock theory. Shock Waves 26 (4), 337–353.CrossRefGoogle Scholar
Mölder, S. 2017 Flow behind concave shock waves. Shock Waves 27 (5), 721–730.CrossRefGoogle Scholar
Mouton, C.A. 2006 Transition between regular reflection and mach reflection in the dual-solution domain. PhD thesis, California Institute of Technology, Pasadena, CA.Google Scholar
Shi, C., Han, W., Deiterding, R., Zhu, C. & You, Y. 2020 Second-order curved shock theory. J. Fluid Mech. 891, A21.CrossRefGoogle Scholar
Shi, C., You, Y., Zheng, X. & Zhu, C. 2023 Analytical model for curved-shock Mach reflection. Phys. Fluids 35 (3), 031702.CrossRefGoogle Scholar
Shi, C., Zhu, C., You, Y. & Zhu, G. 2021 Method of curved-shock characteristics with application to inverse design of supersonic flowfields. J. Fluid Mech. 920, A36.CrossRefGoogle Scholar
Shoesmith, B. & Timofeev, E. 2021 Modelling of Mach reflections in internal axisymmetric steady supersonic flow. Shock Waves 31 (8), 945–957.CrossRefGoogle Scholar
Sinclair, J. & Cui, X. 2017 A theoretical approximation of the shock standoff distance for supersonic flows around a circular cylinder. Phys. Fluids 29 (2), 026102.CrossRefGoogle Scholar
Surujhlal, D. & Skews, B.W. 2018 Two-dimensional supersonic flow over concave surfaces. Shock Waves 28 (6), 1199–1205.CrossRefGoogle Scholar
Tan, L.-H., Ren, Y.-X. & Wu, Z.-N. 2005 Analytical and numerical study of the near flow field and shape of the Mach stem in steady flows. J. Fluid Mech. 546, 341.CrossRefGoogle Scholar
Tao, Y., Liu, W., Fan, X., Xiong, B., Yu, J. & Sun, M. 2017 A study of the asymmetric shock reflection configurations in steady flows. J. Fluid Mech. 825, 1–15.CrossRefGoogle Scholar
Thomas, T.Y. 1949 The consistency relations for shock waves. J. Math. Phys. 28 (2), 62–90.CrossRefGoogle Scholar
Turkel, E. 1987 Preconditioned methods for solving the incompressible and low speed compressible equations. J. Comput. Phys. 72 (2), 277–298.CrossRefGoogle Scholar
von Neumann, J. 1943 Oblique reflection of shock. Tech. Rep. 12. Navy Department, Bureau of Ordinance, Washington, DC.Google Scholar
von Neumann, J. 1945 Refraction, intersection and reflection of shock waves. Tech. Rep. Navy Department, Bureau of Ordinance, Washington, DC.Google Scholar
Whitham, G.B. 1957 A new approach to problems of shock dynamics. Part I. 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.CrossRefGoogle Scholar
Whitham, G.B. 1959 A new approach to problems of shock dynamics. Part 2. Three-dimensional problems. J. Fluid Mech. 5 (3), 369–386.CrossRefGoogle Scholar
Xiang, G., Wang, C., Teng, H., Yang, Y. & Jiang, Z. 2016 Study on Mach stems induced by interaction of planar shock waves on two intersecting wedges. Acta Mechanica Sin. 32 (3), 362–368.CrossRefGoogle Scholar
Yao, Y., Li, S.G. & Wu, Z.N. 2013 Shock reflection in the presence of an upstream expansion wave and a downstream shock wave. J. Fluid Mech. 735, 61–90.CrossRefGoogle Scholar
Zelalem, A.W., Timofeev, E. & Molder, S. 2018 Estimation of shock stand-off distance using the curved shock theory and its validation via numerical modelling. In Progress in Canadian Mechanical Engineering. York University Libraries.CrossRefGoogle Scholar
Zhang, T., Xu, K., Shi, C., Zhu, C. & You, Y. 2023 Reflection and transition of planar curved shock waves. J. Fluid Mech. 959, A11.CrossRefGoogle Scholar
Supplementary material: File

Tian and Cui supplementary movie 1

Mach numbers (red) and streamline curvatures (green) varying with the iteration steps, where the black grids are composed of the streamlines and orthogonal lines. The iteration starts with the mass-averaged solution from quasi-one-dimensional relations. It ends with the actual solution similar to CFD results, where the effects of streamline curvatures on the flow properties are reflected accordingly.
Download Tian and Cui supplementary movie 1(File)
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Supplementary material: File

Tian and Cui supplementary material 2

Tian and Cui supplementary material
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