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Application of Hybrid Riemann Solvers Based on HLLC and HLL for Simulation of Flows with Gas-Dynamic Discontinuities

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Abstract

The application of hybrid approximate Riemann solvers based on standard HLLC and HLL solvers is discussed. Three different hybrid solvers are considered. The first hybrid solver (rHLLC-HLL) uses a weighted sum of HLLC and HLL so that HLLC is applied in the direction normal to the shock wave, while HLL is applied in the direction along the wave. The second hybrid solver (HLLC-ADC) uses the weighted sum of HLLC and HLL, applying as weights the pressure function at the centers of the left and right cells. The third hybrid solver (HLLC-HLL) computes inviscid fluxes using HLL inside shock waves and HLLC in the other areas of the flow. The faces within the shock waves are determined by a shock-wave indicator based on the reconstructed pressure values to the left and to the right of the face. Several tests have been carried out showing that hybrid solvers prevent the emergence of carbuncle and reduce oscillations on shock waves.

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REFERENCES

  1. Yu. V. Tunik, “Problems of numerical modeling on the basis of some modifications of the Godunov’s scheme,” Fiz.-Khim. Kinet. Gaz. Din. 19 (1) (2018). https://doi.org/10.33257/PhChGD.19.1.701

  2. Y. V. Tunik, “Problems of numerical simulation based on some modifications of Godunov’s scheme,” Fluid Dyn. 57, S75–S83 (2022). https://doi.org/10.1134/S0015462822601334

    Article  MathSciNet  Google Scholar 

  3. A. V. Karpov and E. A. Vasil’ev, “Numerical simulation of the outflow of an overexpanded gas jet from a short axisymmetric nozzle,” Vestn. VolGU, Ser. 1: Prikl. Mat. 9, 81–88 (2005).

    Google Scholar 

  4. R. MacCormack, “The carbuncle CFD problem,” in Proc. 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Orlando, Fla., Jan. 4–7, 2011 (American Inst. of Aeronautics and Astronautics, Reston, Va., 2011). https://doi.org/10.2514/6.2011-381

  5. A. V. Rodionov, “Artificial viscosity to cure the carbuncle phenomenon: The three-dimensional case,” J. Comput. Phys. 361, 50–55 (2018). https://doi.org/10.1016/j.jcp.2018.02.001

    Article  MathSciNet  Google Scholar 

  6. A. V. Rodionov, “The artificial viscosity for suppressing the "carbuncle” type numerical instability in 3d problem simulations,” Vopr. At. Nauki Tekh., Ser.: Mat. Model. Fiz. Protsessov, No. 3, 44–51 (2018).

    Google Scholar 

  7. I. Yu. Tagirova and A. V. Rodionov, “Application of artificial viscosity for suppressing the carbuncle phenomenon in Godunov-type schemes,” Math. Models Comput. Simul. 8, 249–262 (2015). https://doi.org/10.1134/S2070048216030091

    Article  Google Scholar 

  8. D. R. Ismagilov and R. V. Sidel’nikov, “Features of numerical simulation of hypersonic flow around simple bodies,” Vestn. Kontserna PVO “Almaz-Antei,” No. 2, 49–54 (2015). https://doi.org/10.38013/2542-0542-2015-2-49-54

  9. N. S. Smirnova, “Comparison of flux splitting schemes for numerical solution of the compressible Euler equations,” Tr. MFTI, Mat. 10 (1), 122–141 (2018).

    Google Scholar 

  10. E. V. Kolesnik, E. M. Smirnov, and A. A. Smirnovskii, “Numerical solution of a 3D problem on a supersonic viscous gas flow past a plate-cylindrical body junction at M = 2.95,” Mat. Model. Fiz. Protsessov 12 (2), 7–22 (2019). https://doi.org/10.18721/JPM.12201

    Article  Google Scholar 

  11. Z. Shen, W. Yan, and G. Yuan, “A robust HLLC-type Riemann solver for strong shock,” J. Comput. Phys. 309, 185–206 (2016). https://doi.org/10.1016/j.jcp.2016.01.001

    Article  MathSciNet  Google Scholar 

  12. S. Simon and J. C. Mandal, “A cure for numerical shock instability in HLLC Riemann solver using antidiffusion control,” Comput. Fluids 174, 144–166 (2018). https://doi.org/10.1016/j.compfluid.2018.07.001

    Article  MathSciNet  Google Scholar 

  13. H. Nishikawa and K. Kitamura, “Very simple, carbuncle-free, boundary-layer-resolving, rotated-hybrid Riemann solvers,” J. Comput. Phys. 227, 2560–2581 (2008). https://doi.org/10.1016/j.jcp.2007.11.003

    Article  MathSciNet  Google Scholar 

  14. A. Harten, “High resolution schemes for hyperbolic conservation laws,” J. Comput. Phys. 49, 357–393 (1983).https://doi.org/10.1016/0021-9991(83)90136-5

    Article  MathSciNet  Google Scholar 

  15. E. F. Toro, M. Spruce, and W. Speares, “Restoration of the contact surface in the HLL-Riemann solver,” Shock Waves 4, 25–34 (1994). https://doi.org/10.1007/BF01414629

    Article  Google Scholar 

  16. P. Batten, M. A. Leschziner, and U. C. Goldberg, “Average-state Jacobians and implicit methods for compressible viscous and turbulent flows,” J. Comput. Phys. 137, 38–78 (1997). https://doi.org/10.1006/jcph.1997.5793

    Article  MathSciNet  Google Scholar 

  17. E. F. Toro and V. A. Titarev, “MUSTA fluxes for systems of conservation laws,” J. Comput. Phys. 216, 403–429 (2006). https://doi.org/10.1016/j.jcp.2005.12.012

    Article  MathSciNet  Google Scholar 

  18. V. A. Titarev, E. Romenski, and E. F. Toro, “MUSTA-type upwind fluxes for non-linear elasticity,” Int. J. Numer. Methods Eng. 73, 897–926 (2008). https://doi.org/10.1002/nme.2096

    Article  MathSciNet  Google Scholar 

  19. R. Dematte, V. A. Titarev, G. I. Montecinos, and E. F. Toro, “ADER methods for hyperbolic equations with a time-reconstruction solver for the generalized Riemann problem: The scalar case,” Commun. Appl. Math. Comput. 2, 369–402 (2020). https://doi.org/10.1007/s42967-019-00040-x

    Article  MathSciNet  Google Scholar 

  20. A. A. Shershnev, A. N. Kudryavtsev, A. V. Kashkovsky, G. V. Shoev, S. P. Borisov, T. Yu. Shkredov, D. P. Po- levshchikov, A. A. Korolev, D. V. Khotyanovsky, and Yu. V. Kratova, “A numerical code for a wide range of compressible flows on hybrid computational architectures,” Supercomput. Front. Innovations 9 (4), 85–99 (2022). https://doi.org/10.14529/jsfi220408

    Article  Google Scholar 

  21. TarnavskiiG. A. and A. V. Aliev, “Specific features of high-speed flight aerodynamics: computer simulation of hypersonic flow around the head of an object,” Vychisl. Metody Programm. 9, 371–394 (2008).

    Google Scholar 

  22. A. V. Rodionov, Development of Methods and Programs for Numerical Simulation of Nonequilibrium Supersonic Flows in Application to Aerospace and Astrophysical Problems, Doctoral Dissertation in Mathematics and Physics (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, 2019).

  23. B. van Leer, “Towards the ultimate conservative difference scheme. V. A secondorder sequel to Godunov’s method,” J. Comput. Phys. 32, 101–136 (1979). https://doi.org/10.1016/0021-9991(79)90145-1

    Article  Google Scholar 

  24. D. W. Levy, K. G. Powell, and B. van Leer, “Use of a rotated Riemann solver for the two-dimensional Euler equations,” J. Comput. Phys. 106, 201–214 (1993). https://doi.org/10.1016/S0021-9991(83)71103-4

    Article  Google Scholar 

  25. X. Ren Yu, “A robust shock-capturing scheme based on rotated Riemann solvers,” Comput. Fluids 32, 1379–1403 (2003). https://doi.org/10.1016/S0045-7930(02)00114-7

    Article  Google Scholar 

  26. R. Sanders, E. Morano, and M.-C. Druguet, “Multidimensional dissipation for upwind schemes: Stability and applications to gas dynamics,” J. Comput. Phys. 145, 511–537 (1998). https://doi.org/10.1006/jcph.1998.6047

    Article  MathSciNet  Google Scholar 

  27. A. N. Semenov, M. K. Berezkina, and I. V. Krasovskaya, “Classification of shock wave reflections from a wedge. Part 1: Boundaries and domains of existence for different types of reflections,” Tech. Phys. 54, 491–496 (2009). https://doi.org/10.1134/S1063784209040082

    Article  Google Scholar 

  28. A. N. Semenov, M. K. Berezkina, and I. V. Krasovskaya, “Classification of shock wave reflections from a wedge. Part 2: Experimental and numerical simulations of different types of Mach reflections,” Tech. Phys. 54, 497–503 (2009).

    Article  Google Scholar 

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ACKNOWLEDGMENTS

The author is grateful to T.Yu. Shkredov for suggesting the idea of the study, Prof. A.N. Kudryavtsev for a fruitful discussion, and Dr. A.A. Shershnev for technical support in the implementation of the hybrid solvers in HyCFS. Numerical calculations were performed using the resources of the computer center of Novosibirsk State University and the Institute of Theoretical and Applied Mechanics.

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The research was carried out within the state assignment of Ministry of Science and Higher Education of the Russian Federation.

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  1. Khristianovich Institute of Theoretical and Applied Mechanics, 630090, Novosibirsk, Russia

    G. V. Shoev

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Translated by M. Shmatikov

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Shoev, G.V. Application of Hybrid Riemann Solvers Based on HLLC and HLL for Simulation of Flows with Gas-Dynamic Discontinuities. Vestnik St.Petersb. Univ.Math. 56, 598–610 (2023). https://doi.org/10.1134/S1063454123040155

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