DOI: 10.17586/2226-1494-2017-17-5-920-928


M. P. Bulat, I. A. Volobuev, Волков К.Н., V. A. Pronin

Read the full article 
Article in Russian

For citation: Bulat M.P., Volobuev I.A., Volkov K.N., Pronin V.A. Numerical simulation of regular and Mach reflection of shock wave from the wall. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2017, vol. 17, no. 5, pp. 920–928 (in Russian). doi: 10.17586/2226-1494-2017-17-5-920-928


A numerical simulation of the shock wave reflection from a plane wall is carried out. Depending on the input parameters, a regular (two-wave configuration) or Mach (three-wave configuration) reflection is observed. A finite volume method and high-order difference schemes are used for discretization of the Euler equations describing the flow of an inviscid compressible gas. The application of weighted essentially non-oscillatory (WENO) schemes of high accuracy order realized in different forms on unstructured meshes is demonstrated. The calculated shock-wave configuration is compared with the data available in the literature. It is shown that the WENO-scheme of the fourth order accuracy in characteristic version gives the possibility to reproduce much more details than the scheme of the third order with the absence of solution oscillations characteristic of TDV (Total Variation Diminishing) schemes and component-wise WENO schemes. The criteria for the accuracy of numerical calculations related to the location of shock-wave structures are discussed. Recommendations on the practical application of high order difference schemes on unstructured grids are given.

Keywords: shock wave, reflection, supersonic flow, numerical simulation, WENO scheme

Acknowledgements. This work was financially supported by the Ministry of Education and Science of the Russian Federation (agreement No 14.578.21.0203, unique identifier of applied scientific research RFMEFI57816X0203).

 1.     Adrianov A.L., Starykh A.L., Uskov V.N. Interference of Stationary Gasdynamic Discontinuities. Novosibirsk, Nauka Publ., 1995, 180 p. (In Russian)
2.     Ben-Dor G. Shock Wave Reflection Phenomena. New York, Springer-Verlag, 1991, 307 p.
3.     Ivanov M.S., Kudryavtsev A.N., Nikiforov S.B., Khotyanovskii D.V. Transition between regular and Mach reflection of shock waves: new numerical and experimental results. Aeromekhanika i Gazovaya Dinamika, 2002, no. 3, pp. 3–12.(In Russian)
4.     Tarnavsky G.A. Shock waves in real gases with different specific heat ratios ahead of and behind shock fronts.Vychislitel'nye Metody i Programmirovanie, 2003, vol. 3, no. 1, pp. 222–236. (In Russian)
5.     Tarnavsky G.A. Nonuniqueness of shockwave structures in real gases: the Mach and/or regular reflection.Vychislitel'nye Metody i Programmirovanie, 2003, vol. 4, no. 1, pp. 258–277. (In Russian)
6.     Colella P. Multidimensional upwind methods for hyperbolic conservation laws. Journal of Computational Physics, 1990,vol. 87,no.1,pp. 171–200.doi: 10.1016/0021-9991(90)90233-q
7.     Woodward P.R., Colella P. The numerical simulation of two-dimentional fluid flow with strong shocks. Journal of Computational Physics, 1984,vol. 54,no.1,pp. 115–173.doi: 10.1016/0021-9991(84)90142-6
8.     Correa L., Lima G.A.B., Candezano M.A.C., Braun M.P.S., Oishi C.M., Navarro H.A., Ferreira V.G. A C2-continuous high-resolution upwind convection scheme. International Journal for Numerical Methods in Fluids, 2013, vol. 72, no. 12, pp. 1263–1285. doi: 10.1007/978-3-642-60543-7_17
9.     Suresh A., Huynh H.T. Accurate monotonicity-preserving schemes with Runge–Kutta time stepping. Journal of Computational Physics, 1997,vol. 136,no.1,pp. 83–99.doi: 10.1006/jcph.1997.5745
10.  Balsara D., Shu C.-W. Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. Journal of Computational Physics, 2000,vol. 160,no.2,pp. 405–452.doi: 10.1006/jcph.2000.6443
11.  Castro M., Costa B., Don W.-S. High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws. Journal of Computational Physics, 2011, vol. 230, no. 5, pp. 1766–1792. doi: 10.1016/
12.  Clain S., Diot S., Loubere R. A high-order finite volume method for hyperbolic systems: multi-dimensional optimal order detection (MOOD). Journal of Computational Physics, 2011, vol. 230, no. 10, pp. 4028–4050. doi: 10.21914/anziamj.v57i0.9038
13.  Liu Y., Shu C.-W., Tadmor E., Zhang M. Non-oscillatory hierarchical reconstruction for central and finite volume schemes. Communications in Computational Physics, 2007,vol. 2,no.5,pp. 933–963.
14.  Shi J., Zhang Y.-T., Shu C.-W. Resolution of high order WENO schemes for complicated WENO structures. Journal of Computational Physics, 2003,vol. 186,no.2,pp. 690–696.doi: 10.1016/s0021-9991(03)00094-9
15.  Hu C.Q., Shu C.-W. Weighted essentially non-oscillatory schemes on triangular meshes. Journal of Computational Physics, 1999,vol. 150,no.1,pp. 97–127.doi: 10.1006/jcph.1998.6165
16.  Shi J., Hu C., Shu C.-W. A technique of treating negative weights in WENO schemes. Journal of Computational Physics, 2002,vol. 175,no.1,pp. 108–127.doi: 10.1006/jcph.2001.6892
17.  Wang Z.J., Zhang L., Liu Y. Spectral (finite) volume method for conservation laws on unstructured grids. IV. Extension to two-dimensional systems. Journal of Computational Physics, 2004, vol. 194, no. 2, pp. 716–741. doi: 10.1016/
18.  Cockburn B., Shu C.-W. The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM Journal on Numerical Analysis, 1998, vol. 35, no. 6, pp. 2440–2463. doi: 10.1137/s0036142997316712
19.  Galanin M.P., Savenkov E.B., Tokareva S.A. The solution of gas dynamics problems with shock waves using Runge-Kutta discontinous Galerkin method. Matematicheskoe Modelirovanie, 2008, vol. 20, no. 11, pp. 55–66. (In Russian)
20.  Volkov K.N., Emel'yanov V.N. Computational Technologies in the Problems of Fluid and Gas Mechanics. Moscow, Fizmatlit Publ., 2012, 468 p. (In Russian)
21.  Volkov K.N., Deryugin Yu.N., Emel'yanov V.N., Kozelkov A.S., Karpenko A.G., Teterina I.V. Methods of Acceleration of Gas Dynamic Calculations on Unstructured Grids. Moscow, Fizmatlit Publ., 2013, 536 p. (In Russian)
22.  Yee H.C., Warming R.F., Harten A. Application of TVD schemes for the Euler equations of fas dynamics. Lectures in Applied Mathematics, 1983,vol. 22,pp. 357–377.
23.  Ekaterinaris J.A. High-order accurate, low-numerical diffusion methods for aerodynamics. Progress in Aerospace Sciences, 2005, vol. 41, no. 3–4, pp. 192–300. doi: 10.1016/j.paerosci.2005.03.003
24.  Gottlieb S., Shu C.-W. Total variation diminishing Runge–Kutta schemes. Mathematics of Computation of the American Mathematical Society, 1998, vol. 67, no. 221, pp. 73–85. doi: 10.1090/s0025-5718-98-00913-2
25.  Vorozhtsov E.V. Application of Lagrange-Burmann expansions for the numerical integration of the inviscid gas equations. Vychislitel'nye Metody i Programmirovanie,2011, vol. 12, no. 1, pp.348–361.(In Russian)
Bona C., Bona-Casas C., Terradas J. Linear high-resolution schemes for hyperbolic conservation laws: TVB numerical evidence. Journal of Computational Physics, 2009, vol. 228, no. 6, pp. 2266–2281. doi: 10.1016/

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License
Copyright 2001-2019 ©
Scientific and Technical Journal
of Information Technologies, Mechanics and Optics.
All rights reserved.