Transonic shock and supersonic shock in the regular reflection of a planar shock

Sheng, Wancheng; Yin, Gan

doi:10.1007/s00033-008-8003-4

Cited by 7 publications

(5 citation statements)

References 9 publications

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“…We recall that many theoretical attempts have been made to understand the transition criterion from the regular reflection to the simple or even the double Mach reflection, but it is very difficult. The earliest contribution seems due to [40] and later on it was refined in [13], [47] and references therein. Comprehensive descriptions can be found in [8].…”

Section: Interaction Of Pure Planar Shock Wavesmentioning

confidence: 99%

Accuracy of the Adaptive GRP Scheme and the Simulation of 2-D Riemann Problems for Compressible Euler Equations

Han

Tang

2011

Commun. comput. phys.

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Abstract. The adaptive generalized Riemann problem (GRP) scheme for 2-D compressible fluid flows has been proposed in [25, Han, Li and Tang, An adaptive GRP scheme for compressible fluid flows, J. Comput. Phys., 229 (2010Phys., 229 ( ), 1448Phys., 229 ( -1466 and it displays the capability in overcoming difficulties such as the start-up error for a single shock, and the numerical instability of the almost stationary shock. In this paper, we will provide the accuracy study and particularly show the performance in simulating 2-D complex wave configurations formulated with the 2-D Riemann problems for compressible Euler equations. For this purpose, we will first review the GRP scheme briefly when combined with the adaptive moving mesh technique and consider the accuracy of the adaptive GRP scheme via the comparison with the explicit formulae of analytic solutions of planar rarefaction waves, planar shock waves, the collapse problem of a wedge-shaped dam and the spiral formation problem. Then we simulate the full set of wave configurations in the 2-D four-wave Riemann problems for compressible Euler equations [60, Zhang and Zheng, Conjecture on the structure of solutions of the Riemann problem for 2-D gas dynamics systems, SIAM J. Math. Anal., 21 (1990), 593-630], including the interactions of strong shocks (shock reflections), vortex-vortex and shock-vortex etc. This study combines the theoretical results with the numerical simulations, and thus demonstrates what Ami Harten observed "for computational scientists there are two kinds of truth: the truth that you prove, and the truth you see when you compute" [33, Lax, Computational Fluid Dynamics, J. Sci. Comput., 31 (2007), 185-193].

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Section: Interaction Of Pure Planar Shock Wavesmentioning

confidence: 99%

Accuracy of the Adaptive GRP Scheme and the Simulation of 2-D Riemann Problems for Compressible Euler Equations

Han

Tang

2011

Commun. comput. phys.

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“…3). The following theorem was rigorously shown in Chang-Chen [34] (also see Sheng-Yin [187], Bleakney-Taub [20], Neumann [170,171]). Theorem 6.1 (Local theory).…”

Section: Shock Reflection-diffraction and Self-similar Solutionsmentioning

confidence: 99%

“…This sonic conjecture is stronger than the detachment one. In fact, the regime between the angles θ s and θ d is very narrow and is only fraction of a degree apart; see Sheng-Yin [187].…”

Section: Shock Reflection-diffraction and Self-similar Solutionsmentioning

confidence: 99%

Multidimensional Conservation Laws: Overview, Problems, and Perspective

Chen

2011

Nonlinear Conservation Laws and Applications

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Abstract. Some of recent important developments are overviewed, several longstanding open problems are discussed, and a perspective is presented for the mathematical theory of multidimensional conservation laws. Some basic features and phenomena of multidimensional hyperbolic conservation laws are revealed, and some samples of multidimensional systems/models and related important problems are presented and analyzed with emphasis on the prototypes that have been solved or may be expected to be solved rigorously at least for some cases. In particular, multidimensional steady supersonic problems and transonic problems, shock reflection-diffraction problems, and related effective nonlinear approaches are analyzed. A theory of divergence-measure vector fields and related analytical frameworks for the analysis of entropy solutions are discussed.

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“…The following theorem was rigorously shown in Chang-Chen [18] (also see Sheng-Yin [127], Bleakney-Taub [12], Neumann [142,143]). Theorem 5.1 (Local theory).…”

Section: Local Theory and Von Neumann's Conjectures For Regular Refle...mentioning

confidence: 99%

“…In fact, the regime between the angles θ s and θ d is very narrow and is only fractions of a degree apart; see Fig. 5 from Sheng-Yin [127].…”

Section: Local Theory and Von Neumann's Conjectures For Regular Refle...mentioning

confidence: 99%

Shock reflection-diffraction phenomena and multidimensional conservation laws

Feldman¹

2009

Hyperbolic Problems: Theory, Numerics and Applications

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When a plane shock hits a wedge head on, it experiences a reflectiondiffraction process, and then a self-similar reflected shock moves outward as the original shock moves forward in time. The complexity of reflection-diffraction configurations was first reported by Ernst Mach in 1878, and experimental, computational, and asymptotic analysis has shown that various patterns of shock reflection-diffraction configurations may occur, including regular reflection and Mach reflection. In this paper we start with various shock reflectiondiffraction phenomena, their fundamental scientific issues, and their theoretical roles as building blocks and asymptotic attractors of general solutions in the mathematical theory of multidimensional hyperbolic systems of conservation laws. Then we describe how the global problem of shock reflection-diffraction by a wedge can be formulated as a free boundary problem for nonlinear conservation laws of mixed-composite hyperbolic-elliptic type. Finally we discuss some recent developments in attacking the shock reflection-diffraction problem, including the existence, stability, and regularity of global regular reflectiondiffraction solutions. The approach includes techniques to handle free boundary problems, degenerate elliptic equations, and corner singularities, which is highly motivated by experimental, computational, and asymptotic results.Further trends and open problems in this direction are also addressed.

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Transonic shock and supersonic shock in the regular reflection of a planar shock

Cited by 7 publications

References 9 publications

Accuracy of the Adaptive GRP Scheme and the Simulation of 2-D Riemann Problems for Compressible Euler Equations

Accuracy of the Adaptive GRP Scheme and the Simulation of 2-D Riemann Problems for Compressible Euler Equations

Multidimensional Conservation Laws: Overview, Problems, and Perspective

Shock reflection-diffraction phenomena and multidimensional conservation laws

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