Hostname: page-component-669899f699-rg895 Total loading time: 0 Render date: 2025-04-29T19:43:43.302Z Has data issue: true hasContentIssue false
  • English
  • Français

On a supersonic-sonic patch arising from the two-dimensional Riemann problem of the compressible Euler equations

Part of: Equations and systems of special type Partial differential equations Transonic flows Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  18 September 2024

Yanbo Hu Open the ORCID record for Yanbo Hu [Opens in a new window]  and
Guodong Wang
Yanbo Hu*
Affiliation:
Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou, 310023, PR China ([email protected])
Guodong Wang
Affiliation:
School of Mathematics and Physics, Anhui Jianzhu University, Hefei, 230601, PR China ([email protected])
*
*Corresponding author
Get access Rights & Permissions [Opens in a new window]

Abstract

We are interested in the two-dimensional four-constant Riemann problem to the isentropic compressible Euler equations. In terms of the self-similar variables, the governing system is of nonlinear mixed-type and the solution configuration typically contains transonic and small-scale structures. We construct a supersonic-sonic patch along a pseudo-streamline from the supersonic part to a sonic point. This kind of patch appears frequently in the two-dimensional Riemann problem and is a building block for constructing a global solution. To overcome the difficulty caused by the sonic degeneracy, we apply the characteristic decomposition technique to handle the problem in a partial hodograph plane. We establish a regular supersonic solution for the original problem by showing the global one-to-one property of the partial hodograph transformation. The uniform regularity of the solution and the regularity of an associated sonic curve are also discussed.

Keywords

Compressible Euler equationstwo-dimensional Riemann problemsonic curvesupersonic-sonic solutionpartial hodograph transformation

MSC classification

Primary: 35Q31: Euler equations
Secondary: 35Q35: PDEs in connection with fluid mechanics 35M33: Initial-boundary value problems for systems of mixed type 35A09: Classical solutions 76H05: Transonic flows
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 40
Check for updates
Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Bae, M., Chen, G. Q. and Feldman, M.. Regularity of solutions to regular shock reflection for potential flow. Invent. Math. 175 (2009), 505–543.Google Scholar
Canic, S., Keyfitz, B. L. and Kim, E. H.. A free boundary problem for a quasi-linear degenerate elliptic equation: regular reflection of weak shocks. Comm. Pure Appl. Math. 55 (2002), 71–92.Google Scholar
Chen, G. Q., Deng, X. M. and Xiang, W.. Shock diffraction by convex cornered wedges for the nonlinear wave system. Arch. Ration. Mech. Anal. 211 (2014), 61–112.Google Scholar
Chen, G. Q. and Feldman, M.. Global solutions of shock reflection by large-angle wedges for potential flow. Ann. Math. 171 (2010), 1067–1182.Google Scholar
Chen, S. X.. Mach configuration in pseudo-stationary compressible flow. J. Amer. Math. Soc. 21 (2008), 63–100.Google Scholar
Chen, X. and Zheng, Y. X.. The direct approach to the interaction of rarefaction waves of the two-dimensional Euler equations. Indiana Univ. Math. J. 59 (2010), 231–256.Google Scholar
Cole, J. D. and Cook, L. P.. Transonic Aerodynamics, North-Holland Series in Applied Mathematics and Mechanics (Amsterdam: Elsevier, 1986).Google Scholar
Courant, R. and Friedrichs, K.. Supersonic Flow and Shock Waves (New York: Interscience, 1948).Google Scholar
Elling, V. and Liu, T. P.. Supersonic flow onto a solid wedge. Comm. Pure Appl. Math. 61 (2008), 1347–1448.Google Scholar
Glimm, G., Ji, X., Li, J., Li, X., Zhang, P., Zhang, T. and Zheng, Y.. Transonic shock formation in a rarefaction Riemann problem for the 2-D compressible Euler equations. SIAM J. Appl. Math. 69 (2008), 720–742.Google Scholar
Hu, Y. B.. On a supersonic-sonic patch in the 3-D steady axisymmetric transonic flows. SIAM J. Math. Anal. 54 (2022), 1515–1542.Google Scholar
Hu, Y. B. and Chen, J. J.. Sonic-supersonic solutions to a mixed-type boundary value problem for the two-dimensional full Euler equations. SIAM J. Math. Anal. 53 (2021), 1579–1629.Google Scholar
Hu, Y. B. and Li, J. Q.. Sonic-supersonic solutions for the two-dimensional steady full Euler equations. Arch. Ration. Mech. Anal. 235 (2020), 1819–1871.Google Scholar
Hu, Y. B. and Li, J. Q.. On a supersonic-sonic patch arising from the Frankl problem in transonic flows. Commun. Pure Appl. Anal. 20 (2021), 2643–2663.Google Scholar
Hu, Y. B. and Li, T.. An improved regularity result of semi-hyperbolic patch problems for the 2-D isentropic Euler equations. J. Math. Anal. Appl. 467 (2018), 1174–1193.Google Scholar
Hu, Y. B. and Li, T.. Sonic-supersonic solutions for the two-dimensional self-similar full Euler equations. Kinet. Rel. Mod. 12 (2019), 1197–1228.Google Scholar
Hu, Y. B. and Wang, G. D.. Semi-hyperbolic patches of solutions to the two-dimensional nonlinear wave system for Chaplygin gases. J. Differ. Equ. 257 (2014), 1579–1590.Google Scholar
Kuz'min, A.. Boundary Value Problems for Transonic Flow (West Sussex: John Wiley and Sons, 2002).Google Scholar
Lai, G.. Global solutions to a class of two-dimensional Riemann problems for the Euler equations with a general equation of state. Indiana Univ. Math. J. 68 (2019), 1409–1464.Google Scholar
Lai, G.. Global nonisentropic rotational supersonic flows in a semi-infinite divergent duct. SIAM J. Math. Anal. 52 (2020), 5121–5154.Google Scholar
Lai, G.. Global continuous sonic-supersonic flows in two-dimensional semi-infinite divergent ducts. J. Math. Fluid Mech. 23 (2021), 1–30.Google Scholar
Lai, G. and Sheng, W. C.. Simple waves for two-dimensional compressible pseudo-steady Euler system. Appl. Math. Mech. Engl. Ed. 31 (2010), 827–838.Google Scholar
Lai, G. and Sheng, W. C.. Centered wave bubbles with sonic boundary of pseudosteady Guderley Mach reflection configurations in gas dynamics. J. Math. Pure Appl. 104 (2015), 179–206.Google Scholar
Lai, G. and Sheng, W. C.. Two-dimensional pseudosteady flows around a sharp corner. Arch. Ration. Mech. Anal. 241 (2021), 805–884.Google Scholar
Levine, L. E.. The expansion of a wedge of gas into a vacuum. Proc. Camb. Philol. Soc. 64 (1968), 1151–1163.Google Scholar
Li, J. Q.. On the two-dimensional gas expansion for compressible Euler equations. SIAM J. Appl. Math. 62 (2002), 831–852.Google Scholar
Li, J. Q., Sheng, W. C., Zhang, T. and Zheng, Y. X.. Two-dimensional Riemann problems: from scalar conservation laws to compressible Euler equations. Acta Math. Sci. Ser. B 29 (2009), 777–802.Google Scholar
Li, J. Q., Yang, Z. C. and Zheng, Y. X.. Characteristic decompositions and interactions of rarefaction waves of 2-D Euler equations. J. Differ. Equ. 250 (2011), 782–798.Google Scholar
Li, J. Q., Zhang, T. and Yang, S. L.. The Two–Dimensional Riemann Problem in Gas Dynamics (Harlow: Longman, 1998).Google Scholar
Li, J. Q., Zhang, T. and Zheng, Y. X.. Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations. Comm. Math. Phys. 267 (2006), 1–12.Google Scholar
Li, J. Q. and Zheng, Y. X.. Interaction of rarefaction waves of the two-dimensional self-similar Euler equations. Arch. Rat. Mech. Anal. 193 (2009), 623–657.Google Scholar
Li, J. Q. and Zheng, Y. X.. Interaction of four rarefaction waves in the bi-symmetric class of the two-dimensional Euler equations. Comm. Math. Phys. 296 (2010), 303–321.Google Scholar
Li, M. J. and Zheng, Y. X.. Semi-hyperbolic patches of solutions of the two-dimensional Euler equations. Arch. Rat. Mech. Anal. 201 (2011), 1069–1096.Google Scholar
Li, T. T. and Yu, W. C., Boundary Value Problem for Quasilinear Hyperbolic Systems (Duke University, 1985).Google Scholar
Sheng, W. C., Wang, G. D. and Zhang, T.. Critical transonic shock and supersonic bubble in oblique rarefaction wave reflection along a compressive corner. SIAM J. Appl. Math. 70 (2010), 3140–3155.Google Scholar
Sheng, W. C. and You, S. K.. Interaction of a centered simple wave and a planar rarefaction wave of the two-dimensional Euler equations for pseudo-steady compressible flow. J. Math. Pures Appl. 114 (2018), 29–50.Google Scholar
Song, K., Wang, Q. and Zheng, Y. X.. The regularity of semihyperbolic patches near sonic lines for the 2-D Euler system in gas dynamics. SIAM J. Math. Anal. 47 (2015), 2200–2219.Google Scholar
Song, K. and Zheng, Y. X.. Semi-hyperbolic patches of solutions of the pressure gradient system. Discrete Contin. Dyn. Syst. 24 (2009), 1365–1380.Google Scholar
Suchkov, V. A.. Flow into a vacuum along an oblique wall. J. Appl. Math. Mech. 27 (1963), 1132–1134.Google Scholar
Wang, C. P. and Xin, Z. P.. On sonic curves of smooth subsonic-sonic and transonic flows. SIAM J. Math. Anal. 48 (2016), 2414–2453.Google Scholar
Wang, C. P. and Xin, Z. P.. Smooth transonic flows of Meyer type in de Laval nozzles. Arch. Ration. Mech. Anal. 232 (2019), 1597–1647.Google Scholar
Wang, C. P. and Xin, Z. P.. Regular subsonic-sonic flows in general nozzles. Adv. Math. 380 (2021), 107578.Google Scholar
Zhang, T. and Zheng, Y. X.. Conjecture on the structure of solution of the Riemann problem for two-dimensional gas dynamics systems. SIAM J. Math. Anal. 21 (1990), 593–630.Google Scholar
Zhang, T. Y. and Zheng, Y. X.. Sonic-supersonic solutions for the steady Euler equations. Indiana Univ. Math. J. 63 (2014), 1785–1817.Google Scholar
Zhang, T. Y. and Zheng, Y. X.. Existence of classical sonic-supersonic solutions for the pseudo steady Euler equations (in Chinese). Sci. Sinica Math. 47 (2017), 1–18.Google Scholar
Zheng, Y. X.. Systems of Conservation Laws: Two-Dimensional Riemann Problems (Boston: Birkhauser, 2001).Google Scholar
Zheng, Y. X.. Two-dimensional regular shock reflection for the pressure gradient system of conservation laws. Acta Math. Appl. Sin. Eng. Ser. 22 (2006), 177–210.Google Scholar