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Interface-Capturing Method for Calculating Transport Equations for a Multicomponent Heterogeneous System on Fixed Eulerian Grids

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Abstract

In this paper, we consider a new numerical method for solving the transport equations for a multicomponent heterogeneous system on fixed Eulerian grids. The system consists of an arbitrary number of components. Any two components are separated by a boundary (interface). Each component is determined by a characteristic function, i.e., a volume fraction that is transported in a specified velocity field and determines the spatial instantaneous component distribution. A feature of this system is that its solution requires two conditions to be met. Firstly, the volume fraction of each component should be in the range [0, 1], and, secondly, any partial sum of volume fractions should not exceed unity. To ensure these conditions, we introduce special characteristic functions instead of volume fractions and propose solving transport equations with respect to them. It is proved that the fulfillment of these conditions is ensured when using this approach. In this case, the method is compatible with various TVD schemes (MINMOD, Van Leer, Van Albada, and Superbee) and interface-sharpening methods (Limited downwind, THINC, Anti-diffusion, and Artificial compression). The method is verified by calculating a number of test problems using all of these schemes. The numerical results show the accuracy and reliability of the proposed method.

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REFERENCES

  1. S. Galera, P. H. Maire, and J. Breil, “A two-dimensional unstructured cell-centered multi-material ALE scheme using VOF interface reconstruction,” J. Comput. Phys. 229, 5755–5787 (2010).

    Article  MathSciNet  Google Scholar 

  2. P. H. Maire, R. Abgrall, J. Breil, and J. Ovadia, “A cell-centered lagrangian scheme for two-dimensional compressible flow problems,” SIAM J. Sci. Comput. 29, 1781–1824 (2007).

    Article  MathSciNet  Google Scholar 

  3. J. Glimm, X. L. Li, Y. J. Liu, Z. L. Xu, and N. Zhao, “Conservative front tracking with improved accuracy,” SIAM J. Numer. Anal. 41, 1926–1947 (2003).

    Article  MathSciNet  Google Scholar 

  4. H. Terashima and G. Tryggvason, “A front-tracking/ghost-fluid method for fluid interfaces in compressible flows,” J. Comput. Phys. 228, 4012–4037 (2009).

    Article  Google Scholar 

  5. W. Mulder, S. Osher, and J. A. Sethian, “Computing interface motion in compressible gas dynamics,” J. Comput. Phys. 100, 209–228 (1992).

    Article  MathSciNet  Google Scholar 

  6. R. Abgrall, “How to prevent pressure oscillations in multicomponent flow calculations: a quasi-conservative approach,” J. Comput. Phys. 125, 150–160 (1996).

    Article  MathSciNet  Google Scholar 

  7. R. Saurel and R. Abgrall, “A multiphase Godunov method for compressible multifluid and multiphase flows,” J. Comput. Phys. 150 (2), 425–467 (1999).

    Article  MathSciNet  Google Scholar 

  8. I. Menshov and P. Zakharov, “On the composite Riemann problem for multi-material fluid flows,” Int. J. Numer. Methods Fluids 76, 109–127 (2014).

    Article  MathSciNet  Google Scholar 

  9. G. Allaire, S. Clerc, and S. Kokh, “A five-equation model for the simulation of interfaces between compressible fluids,” J. Comput. Phys. 181, 577–616 (2002).

    Article  MathSciNet  Google Scholar 

  10. S. Jaouen and F. Lagoutière, “Numerical transport of an arbitrary number of components,” Comput. Methods Appl. Math. 196, 3127–3140 (2007).

    MathSciNet  MATH  Google Scholar 

  11. B. Després and F. Lagoutière, “Contact discontinuity capturing schemes for linear advection, compressible gas dynamics,” J. Sci. Comput. 16, 479–524 (2001).

    Article  MathSciNet  Google Scholar 

  12. F. Xiao, Y. Honma, and T. Kono, “A simple algebraic interface capturing scheme using hyperbolic tangent function,” Int. J. Numer. Mech. Fluids 48, 1023–1040 (2005).

    Article  Google Scholar 

  13. K. K. So, X. Y. Hu, and N. A. Adams, “Anti-diffusion method for interface steepening in two-phase incompressible flow,” J. Comput. Phys. 230, 5155–5177 (2011).

    Article  MathSciNet  Google Scholar 

  14. K. K. So, X. Y. Hu, and N. A. Adams, “Anti-diffusion interface sharpening technique for two-phase compressible flow simulations,” J. Comput. Phys. 231, 4304–4323 (2012).

    Article  MathSciNet  Google Scholar 

  15. A. Harten, “The artificial compression method for computation of shocks and contact discontinuities, I: Single conservation laws,” Commun. Pure Appl. Math 30, 611–638 (1977).

    Article  MathSciNet  Google Scholar 

  16. A. Harten, “The artificial compression method for computation of shocks and contact discontinuities, III: Self-adjusting hybrid schemes,” Math. Comput. 32, 363–389 (1978).

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Moscow State University, 119991, Moscow, Russia

    Ch. Zhang & I. S. Menshov

  2. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, 125047, Moscow, Russia

    I. S. Menshov

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  1. Ch. Zhang
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  2. I. S. Menshov
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Correspondence to Ch. Zhang or I. S. Menshov.

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Translated by A. Ivanov

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Zhang, C., Menshov, I.S. Interface-Capturing Method for Calculating Transport Equations for a Multicomponent Heterogeneous System on Fixed Eulerian Grids. Math Models Comput Simul 11, 973–987 (2019). https://doi.org/10.1134/S2070048219060012

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  • DOI: https://doi.org/10.1134/S2070048219060012

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