Skip to main content
×
Home

A Two-Stage Fourth-Order Gas-Kinetic Scheme for Compressible Multicomponent Flows

  • Liang Pan (a1), Junxia Cheng (a1), Shuanghu Wang (a1) and Kun Xu (a2)
Abstract
Abstract

With the use of temporal derivative of flux function, a two-stage temporal discretization has been recently proposed in the design of fourth-order schemes based on the generalized Riemann problem (GRP) [21] and gas-kinetic scheme (GKS) [28]. In this paper, the fourth-order gas-kinetic scheme will be extended to solve the compressible multicomponent flow equations, where the two-stage temporal discretization and fifth-order WENO reconstruction will be used in the construction of the scheme. Based on the simplified two-species BGK model [41], the coupled Euler equations for individual species will be solved. Each component has its individual gas distribution function and the equilibrium states for each component are coupled by the physical requirements of total momentum and energy conservation in particle collisions. The second-order flux function is used to achieve the fourth-order temporal accuracy, and the robustness is as good as the second-order schemes. At the same time, the source terms, such as the gravitational force and the chemical reaction, will be explicitly included in the two-stage temporal discretization through their temporal derivatives. Many numerical tests from the shock-bubble interaction to ZND detonative waves are presented to validate the current approach.

Copyright
Corresponding author
*Corresponding author. Email addresses: panliangjlu@sina.com (L. Pan), cheng_junxia@iapcm.ac.cn (J. X. Cheng), wang_shuanghu@iapcm.ac.cn (S. H. Wang), makxu@ust.hk (K. Xu)
References
Hide All
[1] Abgrall R., How to prevent pressure oscillations in multicomponent flow calculations, Aquasi conservative approach. J. Comput. Phys. 125 (1996) 150160.
[2] Abgrall R., Karni S., Computations of compressible multifluids, J. Comput. Phys. 169 (2001) 594623.
[3] Andries P., Aoki K., and Perthame B., A Consistent BGK-type Model for Gas Mixtures, Journal of Statistical Physics, 106 (2002) 9931018.
[4] Bourlioux A., Numerical Study of Unsteady Detonations, Ph.D. thesis, Princeton University, (1991).
[5] Cockburn B., Shu C. W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Mathematics of Computation, 52 (1989), 411435.
[6] Fedkiw R.P., Aslam T., Merriman B., Osher S., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the Ghost Fluid Method), J. Comput. Phys. 152 (1999) 457492.
[7] Fickett W., Davis W. C., Detonation, University of California Press, Berkeley, (1979).
[8] Fickett W., Wood W.W., Flow calculations for pulsating one-dimensional detonations, Phys. Fluids 9 (1966) 903.
[9] Galera S., Maire P. H., Breil J., A two-dimensional unstructured cell-centered multi-material ALE scheme using VOF interface reconstruction, J. Comput. Phys. 229 (2010) 57555787.
[10] Gottlieb S., Shu C. W., Total variation diminishing runge-kutta schemes, Mathematics of computation, 67 (1998) 7385.
[11] Guo Z.L., Xu K., Wang R.J., Discrete unified gas kinetic scheme for all Knudsen number flows: Low-speed isothermal case, Physical Review E, 88 (2013) 033305.
[12] Harten A., Engquist B., Osher S. and Chakravarthy S. R., Uniformly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys. 71 (1987) 231303.
[13] Haas J.F., Strurtevant B., Interactions of a shock waves with cylindrical and spherical gas inhomogeneites. J. Fluid Mech. 181 (1987) 4176.
[14] Jiang S., Ni G.X., A γ-model BGK scheme for compressible multifluids, Int. J. Numer. Meth. Fluid 46 (2004) 163182
[15] Jiang S., Ni G.X., A second order γ-model BGK scheme for multifluids compressible flows. Applied numerical mathematics 57 (2007) 597608.
[16] Jiang G.S., Shu C.W., Efficient implementation of Weighted ENO schemes, J. Comput. Phys. 126 (1996) 202228.
[17] Karni S., Multicomponent flow calculations by a consistent premitive algorithm, J. Comput. Phys 128 (1994) 237253.
[18] Kotelnikov A.D., Montgomery D.C., A Kinetic Method for Computing Inhomogeneous Fluid Behavior, J. Comput. Phys. 134 (1997) 364388.
[19] Kucharik M., Garimella R.V., Schofield S.P., Shashkov M.J., A comparative study of interface reconstruction methods for multi-material ALE simulations, J. Comput. Phys. 229 (2010) 24322452.
[20] Larrouturou B., How to Preserve the Mass Fraction Positive When Computing Compressible Multi-component Flow, J. Comput. Phys. 95 (1991) 5984.
[21] Li J., Du Z., A two-stage fourth order time-accurate discretization for Lax-Wendroff type flow solvers I. hyperbolic conservation laws, SIAM J. Sci. Computing, 38 (2016) 30463069.
[22] Li Q., Xu K., Fu S., A high-order gas-kinetic Navier-Stokes flow solver, J. Comput. Phys. 229 (2010) 67156731.
[23] Lian Y. S., Xu K., A Gas-Kinetic Scheme for Multimaterial Flows and Its Application in Chemical Reactions, J. Comput. Phys. 163 (2000) 349375.
[24] Liu N., Tang H.Z., A high-order accurate gas-kinetic scheme for one- and two-dimensional flow simulation, Commun. Comput. Phys. 15 (2014) 911943.
[25] Luo J., Xu K., A high-order multidimensional gas-kinetic scheme for hydrodynamic equations, SCIENCE CHINA Technological Sciences, 56 (2013) 23702384.
[26] Mieussens L., On the asymptotic preserving property of the unified gas-kinetic scheme for the diffusion limit of linear kinetic models, J. Comput. Phys. 253 (2013) 138156.
[27] Nonomura T., Morizawa S., Terashima H., Obayashi S., Fujii K., Numerical (error) issues on compressible multicomponent flows using a high-order differencing scheme: Weighted compact nonlinear scheme, J. Comput.Phys. 231 (2012) 31813210.
[28] Pan L., Xu K., Li Q., Li J., An efficient and accurate two-stage fourth-order gas-kinetic scheme for the Navier-Stokes equations, J. Comput. Phys. 326 (2016) 197221.
[29] Ren X.D., Xu K., Shyy W., Gu C.W., A multi-dimensional high-order discontinuous Galerkin-method based on gas kinetic theory for viscous flow computations, J. Comput. Phys. 292 (2015) 176193.
[30] Osher S., Fedkiw R.P., Level set methods: An overview and some recent results, J. Comput. Phys. 169 (2001) 463502.
[31] Quirk J.J., Karni S., On the dynamics of a shock bubble interaction, J. Fluid Mech. 318 (1996) 129163.
[32] Reed W.H., Hill T.R., Triangular mesh methods for the neutron transport equation, Technical Report LA-UR-73-479, 1973, Los Alamos Scientific Laboratory, Los Alamos.
[33] Shyue K.M., F. Xiao An Eulerian interface sharpening algorithm for compressible two-phase flow: The algebraic THINC approach, J. Comput. Phys. 231 (2012) 4304C4323
[34] So K.K., Hu X.Y., Adams N.A., Anti-diffusion interface sharpening technique for two-phase compressible flow simulations, J. Fluid Mechanics, 269, (1994), 4578.
[35] Saurel R., Abgrall R., A simple method for compressible multifluid flows, SIAM J. Sci. Comput. 21 (1999) 11151145.
[36] Saurel R., Abgrall R., A multiphase Godunov method for compressible multifluid and multiphase flows, J. Comput. Phys. 150 (1999) 425467.
[37] Shi J., Zhang Y. T., Shu C.W., Resolution of high order WENO schemes for complicated flow structures, J. Comput. Phys. 186 (2003) 690696.
[38] Terashima H., Tryggvason G., A front-tracking/ghost-fluid method for fluid interfaces in compressible flows, J. Comput. Phys. 228 (2009) 40124037.
[39] Terashima H., Tryggvason G., A front-tracking method with projected interface conditions for compressible multi-fluid flows, Computers & Fluids 39 (2010) 18041814.
[40] Wang R., Unified Gas-kinetic Scheme for the Study of Non-equilibrium Flows, PhD theisi, HKUST, (2015).
[41] Xu K., BGK-based scheme formulticomponent flow calculations, J. Comput. Phys. 134 (1997) 122133.
[42] Xu K., Direct modeling for computational fluid dynamics: construction and application of unfied gas kinetic schemes, World Scientific (2015).
[43] Xu K., A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method, J. Comput. Phys. 171 (2001) 289335.
[44] Xu K., Huang J., A unified gas-kinetic scheme for continuum and rarefied flows, J. Comput. Phys. 229 (2010) 77477764.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 27 *
Loading metrics...

Abstract views

Total abstract views: 90 *
Loading metrics...

* Views captured on Cambridge Core between 28th July 2017 - 18th November 2017. This data will be updated every 24 hours.