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A simple extension of Roe’s scheme for multi-component real gas flows. (English) Zbl 1452.76205
Summary: This paper proposes a numerical algorithm to model multi-component real gas flows in local thermodynamic equilibrium (LTE). The approach is based on the solution of the Euler equations for the mixtures which is coupled with the mass conservation equation for each component of fluid. This algorithm is based on the extension of Roe’s scheme for real gases introduced in [the first author et al., J. Comput. Phys. 329, 16–28 (2017; Zbl 1406.76056)], which is robust, efficient and simpler than the other extensions. The Euler equations for the mixture are solved whereas fluid components are updated using a classical upwind approximation. As the solution evolves, the developed method preserves the positivity of the mass fractions of all components. The proposed technique does not directly involve the thermodynamics derivatives at the average state, which avoids assumptions and approximations in the calculation of these derivatives. The pressure oscillation problem appearing in the case of flow fields including material contact has been also discussed. The numerical experiments show the accuracy and robustness of the method in various cases.
76N15 Gas dynamics, general
76T30 Three or more component flows
76M20 Finite difference methods applied to problems in fluid mechanics
Full Text: DOI
[1] Steger, J. L.; Warming, R. F., Flux vector splitting of the inviscid gas dynamic equations with application to finite difference methods, J. Comput. Phys., 40, 263-293 (1981) · Zbl 0468.76066
[2] Van Leer, B., Flux vector splitting for the Euler equations, (Lecture Notes in Physics, vol. 170 (1982))
[3] Roe, P. L., Approximate Riemann solvers, parameter vectors and difference schemes, J. Comput. Phys., 43, 357-372 (1981) · Zbl 0474.65066
[4] Harten, A.; Lax, P. D.; van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25, 1, 35-61 (1983) · Zbl 0565.65051
[5] Toro, E. F.; Spruce, M.; Speares, W., Restoration of the Contact Surface in the HLL-Riemann Solver (1992), Department of Aerospace Science, College of Aeronautics, Cranfield Institute of Technology: Department of Aerospace Science, College of Aeronautics, Cranfield Institute of Technology UK, Technical Report CoA-9204 · Zbl 0811.76053
[6] Toro, E. F.; Spruce, M.; Speares, W., Restoration of the contact surface in the HLL-Riemann solver, Shock Waves, 4, 25-34 (1994) · Zbl 0811.76053
[7] Toro, E. F.; Vázquez-Cendón, M. E., Flux splitting schemes for the Euler equations, Comput. Fluids, 70, 1-12 (2012) · Zbl 1365.76243
[8] Liou, M. S.; Stefen, C. J., A new flux splitting scheme, J. Comput. Phys., 107, 1, 23-39 (1993) · Zbl 0779.76056
[9] Liou, M. S., A sequel to AUSM: \(AUSM^+\), J. Comput. Phys., 129, 364-382 (1996) · Zbl 0870.76049
[10] Liou, M. S., A sequel to AUSM, part II: \(AUSM^+\)-up for all speeds, J. Comput. Phys., 214, 137-170 (2006) · Zbl 1137.76344
[11] Nagdewe, S. P.; Shevare, G. R.; Kim, Heuy-Dong, Study on the numerical schemes for hypersonic flow simulation, Shock Waves, 19, 5, 433-442 (2009) · Zbl 1255.76084
[12] Coelho, R. M.L.; Lage, P. L.C.; Silva Telles, A., A comparison of hyperbolic solvers for ideal and real gas flows, Braz. J. Chem. Eng., 23, 3, 301-318 (2006)
[13] Glaister, P., An approximate linearised Riemann solver for the Euler equations for real gases, J. Comput. Phys., 74, 382-408 (1988) · Zbl 0632.76079
[14] Liou, M.-S.; Leer, B. V.; Shuen, J-S., Splitting of inviscid fluxes for real gases, J. Comput. Phys., 87, 1-24 (1990) · Zbl 0687.76074
[15] Harten, A.; Hyman, J. M., Self-adjusting grid methods for one-dimensional hyperbolic conservation laws, J. Comput. Phys., 50, 2 (1983) · Zbl 0565.65049
[16] Dubroca, B., Positively conservative Roe’s matrix for Euler equations, Numerical Analysis, C. R. Acad. Sci. Paris, 329, S6rie I, 827-832 (1999) · Zbl 0957.76049
[17] Vinokur, M.; Montagné, J.-L., Generalized flux-vector splitting and Roe average for an equilibrium real gas, J. Comput. Phys., 89, 276-300 (1990) · Zbl 0701.76072
[18] Grossman, B.; Walters, R.-W., Analysis of the flux-split algorithms for Euler’s equations with real gases, AIAA J., 27, 524-531 (1989) · Zbl 0666.76094
[19] Abgrall, R., An extension of Roe’s upwind scheme to algebraic equilibrium real gas models, Comput. Fluids, 19, 2, 171-182 (1991) · Zbl 0721.76061
[20] Larrouturou, B., How to preserve the mass fractions positive when computing compressible multi-component flows, J. Comput. Phys., 95, 1, 59-84 (1991) · Zbl 0725.76090
[21] Arabi, A.; Trépanier, J.-Y.; Camarero, R., A simple extension of Roe’s scheme for real gases, J. Comput. Phys., 329, 16-28 (2017) · Zbl 1406.76056
[22] Martin, A.; Reggio, M.; Trépanier, J.-Y., Numerical solution of axisymmetric multi-species compressible gas flow: towards improved circuit breaker simulation, Int. J. Comput. Fluid Dyn., 22, 4, 259-271 (2008) · Zbl 1184.76747
[23] Godin, D.; Trépanier, J.-Y., A robust and efficient method for the computation of equilibrium composition in gaseous mixtures, Plasma Chem. Plasma Process., 24, 447-473 (2004)
[24] Barth, T. J.; Frederickson, P. O., Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction, \((28{}^{t h}\) AIAA Aerospace Sciences Meeting (1990)), 90-0013
[25] Haider, F.; Croisille, J-P.; Courbet, B., Stability analysis of the cell centered finite-volume MUSCL method on unstructured grids, Numer. Math., 113, 4, 555-600 (2009) · Zbl 1179.65117
[26] Abgrall, R.; Karni, S., Computations of compressible multifluids, J. Comput. Phys., 169, 594-623 (2001) · Zbl 1033.76029
[27] Ton, V. T., Improved shock-capturing methods for multicomponent and reacting flows, J. Comput. Phys., 128, 237-253 (1996) · Zbl 0860.76060
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