American Institute of Mathematical Sciences

Advanced
April  2000, 6(2): 419-430. doi: 10.3934/dcds.2000.6.419

On the 2-D Riemann problem for the compressible Euler equations II. Interaction of contact discontinuities

Tung Chang 1, , Gui-Qiang Chen 2, and  Shuli Yang 3,

1. 

Institute of Mathematics, Academia Sinica, Beijing 100080, China
2. 

Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730
3. 

Institute of Applied Mathematics, Academia Sinica, Beijing 100080, China

Received  April 1999 Revised  October 1999 Published  January 2000

We are concerned with the Riemann problem for the two-dimensional compressible Euler equations in gas dynamics. This paper is a continuation of our program (see [CY1,CY2]) in studying the interaction of nonlinear waves in the Riemann problem. The central point in this issue is the dynamical interaction of shock waves, centered rarefaction waves, and contact discontinuities that connect two neighboring constant initial states in the quadrants. In this paper we focus mainly on the interaction of contact discontinuities, which consists of two genuinely different cases. For each case, the structure of the Riemann solution is analyzed by using the method of characteristics, and the corresponding numerical solution is illustrated via contour plots by using the upwind averaging scheme that is second-order in the smooth region of the solution developed in [CY1]. For one case, the four contact discontinuities role up and generate a vortex, and the density monotonically decreases to zero at the center of the vortex along the stream curves. For the other, two shock waves are formed and, in the subsonic region between two shock waves, a new kind of nonlinear hyperbolic waves (called smoothed Delta-shock waves) is observed.
Keywords: rarefaction waves, 2-D Riemann problem, shock waves, interaction of waves, vortices, contact discontinuities, upwind averaging schemes., transonic flow, Euler equations.
Mathematics Subject Classification: Primary: 76N10, 35L65, 35L67; Secondary: 65M0.
Citation: Tung Chang, Gui-Qiang Chen, Shuli Yang. On the 2-D Riemann problem for the compressible Euler equations II. Interaction of contact discontinuities. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 419-430. doi: 10.3934/dcds.2000.6.419
[1]

Tung Chang, Gui-Qiang Chen, Shuli Yang. On the 2-D Riemann problem for the compressible Euler equations I. Interaction of shocks and rarefaction waves. Discrete and Continuous Dynamical Systems, 1995, 1 (4) : 555-584. doi: 10.3934/dcds.1995.1.555
[2]

Feimin Huang, Yi Wang, Tong Yang. Fluid dynamic limit to the Riemann Solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity. Kinetic and Related Models, 2010, 3 (4) : 685-728. doi: 10.3934/krm.2010.3.685
[3]

Hualin Zheng. Stability of a superposition of shock waves with contact discontinuities for the Jin-Xin relaxation system. Kinetic and Related Models, 2015, 8 (3) : 559-585. doi: 10.3934/krm.2015.8.559
[4]

Min Ding, Hairong Yuan. Stability of transonic jets with strong rarefaction waves for two-dimensional steady compressible Euler system. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2911-2943. doi: 10.3934/dcds.2018125
[5]

Xiao-Biao Lin, Stephen Schecter. Traveling waves and shock waves. Discrete and Continuous Dynamical Systems, 2004, 10 (4) : i-ii. doi: 10.3934/dcds.2004.10.4i
[6]

Dongbing Zha. Remarks on nonlinear elastic waves in the radial symmetry in 2-D. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 4051-4062. doi: 10.3934/dcds.2016.36.4051
[7]

Feng Xie. Nonlinear stability of combination of viscous contact wave with rarefaction waves for a 1D radiation hydrodynamics model. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 1075-1100. doi: 10.3934/dcdsb.2012.17.1075
[8]

Christian Rohde, Wenjun Wang, Feng Xie. Hyperbolic-hyperbolic relaxation limit for a 1D compressible radiation hydrodynamics model: superposition of rarefaction and contact waves. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2145-2171. doi: 10.3934/cpaa.2013.12.2145
[9]

Yanbo Hu, Tong Li. The regularity of a degenerate Goursat problem for the 2-D isothermal Euler equations. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3317-3336. doi: 10.3934/cpaa.2019149
[10]

James K. Knowles. On shock waves in solids. Discrete and Continuous Dynamical Systems - B, 2007, 7 (3) : 573-580. doi: 10.3934/dcdsb.2007.7.573
[11]

José R. Quintero. Nonlinear stability of solitary waves for a 2-d Benney--Luke equation. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 203-218. doi: 10.3934/dcds.2005.13.203
[12]

Yuri Gaididei, Anders Rønne Rasmussen, Peter Leth Christiansen, Mads Peter Sørensen. Oscillating nonlinear acoustic shock waves. Evolution Equations and Control Theory, 2016, 5 (3) : 367-381. doi: 10.3934/eect.2016009
[13]

Tong Yang, Huijiang Zhao. Asymptotics toward strong rarefaction waves for $2\times 2$ systems of viscous conservation laws. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 251-282. doi: 10.3934/dcds.2005.12.251
[14]

Frederike Kissling, Christian Rohde. The computation of nonclassical shock waves with a heterogeneous multiscale method. Networks and Heterogeneous Media, 2010, 5 (3) : 661-674. doi: 10.3934/nhm.2010.5.661
[15]

Martina Chirilus-Bruckner, Guido Schneider. Interaction of oscillatory packets of water waves. Conference Publications, 2015, 2015 (special) : 267-275. doi: 10.3934/proc.2015.0267
[16]

Masashi Ohnawa. Convergence rates towards the traveling waves for a model system of radiating gas with discontinuities. Kinetic and Related Models, 2012, 5 (4) : 857-872. doi: 10.3934/krm.2012.5.857
[17]

Bingkang Huang, Lusheng Wang, Qinghua Xiao. Global nonlinear stability of rarefaction waves for compressible Navier-Stokes equations with temperature and density dependent transport coefficients. Kinetic and Related Models, 2016, 9 (3) : 469-514. doi: 10.3934/krm.2016004
[18]

Gero Friesecke, Karsten Matthies. Geometric solitary waves in a 2D mass-spring lattice. Discrete and Continuous Dynamical Systems - B, 2003, 3 (1) : 105-114. doi: 10.3934/dcdsb.2003.3.105
[19]

Michiel Bertsch, Masayasu Mimura, Tohru Wakasa. Modeling contact inhibition of growth: Traveling waves. Networks and Heterogeneous Media, 2013, 8 (1) : 131-147. doi: 10.3934/nhm.2013.8.131
[20]

Iryna Egorova, Johanna Michor, Gerald Teschl. Rarefaction waves for the Toda equation via nonlinear steepest descent. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 2007-2028. doi: 10.3934/dcds.2018081

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (213)
  • HTML views (0)
  • Cited by (22)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy