Abstract
The work is devoted to the application of Runge–Kutta discontinuous Galerkin (RKDG) method for solving Baer–Nunziato hyperbolic model for nonequilibrium two-phase flows. The approach is based on the application of the simple WENO limiter directly to the conservative variables. Mathematical model and the corresponding numerical algorithm are described. The results of numerical simulations for 1D and 2D tests are presented and discussed.
References
[1] N. Andrianov and G. Warnecke, The Riemann problem for the Baer–Nunziato two-phase flow model. J. Comput. Phys. 195 (2004), No. 2, 434–464.10.1016/j.jcp.2003.10.006Suche in Google Scholar
[2] M. Baer and J. Nunziato, A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. Int. J. Multiph. Flow 12 (1986), 861–889.10.1016/0301-9322(86)90033-9Suche in Google Scholar
[3] D. Balsara, M. Dumbser, and R. Abgrall, Multidimensional HLLC Riemann solver for unstructured meshes with application to Euler and MHD flows. J. Comput. Phys. 261 (2014), 172–208.10.1016/j.jcp.2013.12.029Suche in Google Scholar
[4] P. Barton, D. Drikakis, E. Romenski, and V. A. Titarev, Exact and approximate solutions of Riemann problems in non-linear elasticity. J. Comput. Phys. 228 (2009), No. 18, 7046–7068.10.1016/j.jcp.2009.06.014Suche in Google Scholar
[5] M. Castro, P. LeFloch, M. Munoz-Ruiz, and C. Parés, Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes. J. Comput. Phys. 227 (2008), No. 17, 8107–8129.10.1016/j.jcp.2008.05.012Suche in Google Scholar
[6] B. Cockburn and C.-W. Shu, The Runge–Kutta local projection-discontinuous-Galerkin finite element method for scalar conservation laws. ESAIM Math. Model. Numer. Anal. 25 (1991), No. 3, 337–361.10.2514/6.1988-3797Suche in Google Scholar
[7] G. Dal Maso, P. Le Floch, and F. Murat, Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74 (1995), No. 6, 483–548.Suche in Google Scholar
[8] F. Daude, R. Berry, and P. Galon, A finite-volume method for compressible non-equilibrium two-phase flows in networks of elastic pipelines using the Baer–Nunziato model. Comput. Methods in Appl. Mech. Engrg. 354 (2019), 820–849.10.1016/j.cma.2019.06.010Suche in Google Scholar
[9] D. Drew and S. Passman, Theory of Multicomponent Fluids. Springer, New York, 2014.Suche in Google Scholar
[10] M. Dumbser, O. Zanotti, R. Loubere, and S. Diot, A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws. J. Comput. Phys. 278 (2014), 47–75.10.1016/j.jcp.2014.08.009Suche in Google Scholar
[11] M. Dumbser, A. Hidalgo, and O. Zanotti, High order space-time adaptive ADER-WENO finite volume schemes for non-conservative hyperbolic systems. Comput. Methods in Appl. Mech. Engrg. 268 (2014), 359–387.10.1016/j.cma.2013.09.022Suche in Google Scholar
[12] M. Dumbser and E. Toro, A simple extension of the Osher Riemann solver to non-conservative hyperbolic systems. J. Sci. Comput. 48 (2011), 70–88.10.1007/s10915-010-9400-3Suche in Google Scholar
[13] N. Favrie, S. Gavrilyuk, and R. Saurel, Solid-fluid diffuse interface model in cases of extreme deformations. J. Comput. Phys. 228 (2009), No. 16, 6037–6077.10.1016/j.jcp.2009.05.015Suche in Google Scholar
[14] E. Franquet and V. Perrier, Runge–Kutta discontinuous Galerkin method for the approximation of Baer and Nunziato type multiphase models. J. Comput. Phys. 231 (2012), 4096–4141.10.1016/j.jcp.2012.02.002Suche in Google Scholar
[15] H. de Frahan, S. Varadan, and E. Johnsen, A new limiting procedure for discontinuous Galerkin methods applied to compressible multiphase flows with shocks and interfaces. J. Comput. Phys. 280 (2015), No. C., 489–509.10.1016/j.jcp.2014.09.030Suche in Google Scholar
[16] F. Fraysse, C. Redondo, G.Rubio, and E. Valero, Upwind methods for the Baer–Nunziato equations and higher-order reconstruction using artificial viscosity. J. Comput. Phys. 326 (2016), 805–827.10.1016/j.jcp.2016.09.017Suche in Google Scholar
[17] V. Galepova, V. Lukin, I. Marchevsky, and I. Fufaev, Comparative study of WENO and Hermite WENO limiters for gas flows numerical simulations using the RKDG method. Keldysh Institute preprints 131 (2017).10.20948/prepr-2017-131Suche in Google Scholar
[18] A. Kapila, S. Son, J. Bdzil, and R. Menikoff, Two-phase modeling of DDT: Structure of the velocity-relaxation zone. Phys. Fluids 9 (1997), No. 12, 3885–3897.10.1063/1.869488Suche in Google Scholar
[19] A. Kapila, R. Menikoff, J. Bdzil, S. Son, and S. Stewart, Two-phase modeling of deflagration-to-detonation transition in granular materials: Reduced equations. Phys. Fluids 13 (2001), No. 10, 3002–3024.10.1063/1.1398042Suche in Google Scholar
[20] L. Krivodonova, Limiters for high-order discontinuous Galerkin methods. J. Comput. Phys. 226 (2007), No. 1, 879–896.10.1016/j.jcp.2007.05.011Suche in Google Scholar
[21] L. Krivodonova, J. Xin, J.-F. Remacle, N. Chevaugeon, and J. Flaherty, Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Appl. Numer. Math. 48 (2004), No. 3, 323–338.10.1016/j.apnum.2003.11.002Suche in Google Scholar
[22] C. Michoski, C. Dawson, E. Kubatko, and D. Wirasaet, A comparison of artificial viscosity, limiters, and filters, for high order discontinuous Galerkin solutions in nonlinear settings. J. Sci. Comput. 66 (2016), No. 1., 406–434.10.1007/s10915-015-0027-2Suche in Google Scholar
[23] R. Moura, R. Affonso, A. Silva, and M. Ortega, Diffusion-based limiters for discontinuous Galerkin methods-part I: one-dimensional equations. In: 22nd Int. Congress of Mechanical Engineering. Ribeirão Preto, SP, Brazil, 2013, 14 p.Suche in Google Scholar
[24] A. Murrone and H. Guillard, A five-equation reduced model for compressible two phase flow problems. J. Comput. Phys. 202 (2005), No. 2, 664–698.10.1016/j.jcp.2004.07.019Suche in Google Scholar
[25] R. Nigmatulin, Dynamics of Multiphase Media. Hemisphere, New York, 1990.Suche in Google Scholar
[26] C. Parés, Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J. Numer. Anal. 44 (2006), No. 1, 300–321.10.1137/050628052Suche in Google Scholar
[27] P.-O. Persson and J. Peraire, Sub-cell shock capturing for discontinuous Galerkin methods. In: 44th AIAA Aerospace Sciences Meeting and Exhibit. Reno, Nevada, 2006, AIAA 2006-112, 13 p.10.2514/6.2006-112Suche in Google Scholar
[28] S. Rhebergen, O. Bokhove, and J. van der Vegt, Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations. J. Comput. Phys. 227 (2008), No. 3, 1887–1922.10.1016/j.jcp.2007.10.007Suche in Google Scholar
[29] R. Saurel and R. Abgrall, A simple method for compressible multifluid flows. SIAM J. Sci. Comput. 21 (1999), No. 3, 1115–1145.10.1137/S1064827597323749Suche in Google Scholar
[30] S. Tokareva and E. Toro, HLLC-type Riemann solver for the Baer–Nunziato equations of compressible two-phase flow. J. Comput. Phys. 229 (2010), No. 10., 3573–3604.10.1007/978-3-642-17884-9_10Suche in Google Scholar
[31] O. Zanotti and M. Dumbser, Eflcient conservative ADER schemes based on WENO reconstruction and space-time predictor in primitive variables. Comput. Astrophys. 3 (2016), No. 1.10.1186/s40668-015-0014-xSuche in Google Scholar PubMed PubMed Central
[32] X. Zhong and C.-W. Shu, A simple weighted essentially nonoscillatory limiter for Runge–Kutta discontinuous Galerkin methods. J. Comput. Phys. 232 (2013), No. 1, 397–415.10.1016/j.jcp.2012.08.028Suche in Google Scholar
[33] J. Zhu, X. Zhong, C.-W. Shu, and J. Qiu, Runge–Kutta discontinuous Galerkin method with a simple and compact Hermite WENO limiter. Commun. Comput. Phys. 19 (2016), No. 4, 944–969.10.4208/cicp.070215.200715aSuche in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Runge–Kutta discontinuous Galerkin method for Baer–Nunziato model with ‘simple WENO’ limiting of conservative variables
- Influence of unbroken clouds stochastic structure on the solar radiation transfer with results of Monte Carlo simulation
- Computation of periodic solutions to models of infectious disease dynamics and immune response
- Modelling of hydrogen-air supersonic mixing and combustion in near-wall region
- Analysis of the predictability of stratospheric variability and climate indices based on seasonal retrospective forecasts of the INM RAS climate model
Artikel in diesem Heft
- Frontmatter
- Runge–Kutta discontinuous Galerkin method for Baer–Nunziato model with ‘simple WENO’ limiting of conservative variables
- Influence of unbroken clouds stochastic structure on the solar radiation transfer with results of Monte Carlo simulation
- Computation of periodic solutions to models of infectious disease dynamics and immune response
- Modelling of hydrogen-air supersonic mixing and combustion in near-wall region
- Analysis of the predictability of stratospheric variability and climate indices based on seasonal retrospective forecasts of the INM RAS climate model