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An oscillation free shock-capturing method for compressible van der Waals supercritical fluid flows. (English) Zbl 1375.76083

Summary: Numerical solutions of the Euler equations using real gas equations of state (EOS) often exhibit serious inaccuracies. The focus here is the van der Waals EOS and its variants (often used in supercritical fluid computations). The problems are not related to a lack of convexity of the EOS since the EOS are considered in their domain of convexity at any mesh point and at any time. The difficulties appear as soon as a density discontinuity is present with the rest of the fluid in mechanical equilibrium and typically result in spurious pressure and velocity oscillations. This is reminiscent of well-known pressure oscillations occurring with ideal gas mixtures when a mass fraction discontinuity is present, which can be interpreted as a discontinuity in the EOS parameters. We are concerned with pressure oscillations that appear just for a single fluid each time a density discontinuity is present. The combination of density in a nonlinear fashion in the EOS with diffusion by the numerical method results in violation of mechanical equilibrium conditions which are not easy to eliminate, even under grid refinement. A cure to this problem is developed in the present paper for the van der Waals EOS based on previous ideas. A special extra field and its corresponding evolution equation is added to the flow model. This new field separates the evolution of the nonlinear part of the density in the EOS and produce oscillation free solutions. The extra equation being nonconservative the behavior of two established numerical schemes on shocks computation is studied and compared to exact reference solutions that are available in the present context. The analysis shows that shock conditions of the nonconservative equation have important consequence on the results. Last, multidimensional computations of a supercritical gas jet is performed to illustrate the benefits of the present method, compared to conventional flow solvers.

MSC:

76L05 Shock waves and blast waves in fluid mechanics
76N15 Gas dynamics (general theory)
76M12 Finite volume methods applied to problems in fluid mechanics

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[1] Lee, E.; Hornig, H.; Kury, J., Adiabatic Expansion of High Explosive Detonation Products (1968), California Univ., Livermore. Lawrence Radiation Lab., Tech. rep.
[2] Schmitt, T.; Selle, L.; Ruiz, A.; Cuenot, B., Large-eddy simulation of supercritical-pressure round jets, AIAA J., 48, 9, 2133-2144 (2010)
[3] Terashima, H.; Kawai, S.; Yamanishi, N., High-resolution numerical method for supercritical flows with large density variations, AIAA J., 49, 12, 2658-2672 (2011)
[4] Terashima, H.; Koshi, M., Approach for simulating gas-liquid-like flows under supercritical pressures using a high-order central differencing scheme, J. Comput. Phys., 231, 20, 6907-6923 (2012) · Zbl 1351.76291
[5] Terashima, H.; Kawai, S.; Koshi, M., Consistent numerical diffusion terms for simulating compressible multicomponent flows, Comput. Fluids, 88, 484-495 (2013) · Zbl 1391.76497
[6] Abgrall, R., How to prevent pressure oscillations in multicomponent flow calculations: a quasi conservative approach, J. Comput. Phys., 125, 1, 150-160 (1996) · Zbl 0847.76060
[7] Shyue, K.-M., A fluid-mixture type algorithm for compressible multicomponent flow with van der Waals equation of state, J. Comput. Phys., 156, 1, 43-88 (1999) · Zbl 0957.76039
[8] Saurel, R.; Franquet, E.; Daniel, E.; Le Metayer, O., A relaxation-projection method for compressible flows. Part I: the numerical equation of state for the Euler equations, J. Comput. Phys., 223, 2, 822-845 (2007) · Zbl 1183.76840
[9] Gallouët, T.; Hérard, J.-M.; Seguin, N., A hybrid scheme to compute contact discontinuities in one-dimensional Euler systems, Math. Model. Numer. Anal., 36, 6, 1133-1159 (2002) · Zbl 1137.65419
[10] Lee, B.; Toro, E.; Castro, C.; Nikiforakis, N., Adaptive Osher-type scheme for the Euler equations with highly nonlinear equations of state, J. Comput. Phys., 246, 1, 165-183 (2013) · Zbl 1349.76354
[11] Karni, S., Multicomponent flow calculations by a consistent primitive algorithm, J. Comput. Phys., 112, 1, 31-43 (1994) · Zbl 0811.76044
[12] Toro, E.; Castro, C.; Lee, B., A novel numerical flux for the 3D Euler equations with general equation of state, J. Comput. Phys., 303, 80-94 (2015) · Zbl 1349.76398
[13] Shyue, K.-M., An efficient shock-capturing algorithm for compressible multicomponent problems, J. Comput. Phys., 142, 1, 208-242 (1998) · Zbl 0934.76062
[14] Saurel, R.; Abgrall, R., A simple method for compressible multifluid flows, SIAM J. Sci. Comput., 21, 3, 1115-1145 (1999) · Zbl 0957.76057
[15] Shyue, K.-M., A fluid-mixture type algorithm for compressible multicomponent flow with Mie-Grüneisen equation of state, J. Comput. Phys., 171, 2, 678-707 (2001) · Zbl 1047.76573
[16] Saurel, R.; Abgrall, R., A multiphase Godunov method for compressible multifluid and multiphase flows, J. Comput. Phys., 150, 2, 425-467 (1999) · Zbl 0937.76053
[17] Kapila, A.; Menikoff, R.; Bdzil, J.; Son, S.; Stewart, D., Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations, Phys. Fluids, 13, 10, 3002-3024 (2001) · Zbl 1184.76268
[18] Allaire, G.; Clerc, S.; Kokh, S., A five-equation model for the simulation of interfaces between compressible fluids, J. Comput. Phys., 181, 2, 577-616 (2002) · Zbl 1169.76407
[19] Massoni, J.; Saurel, R.; Nkonga, B.; Abgrall, R., Proposition de méthodes et modèles eulériens pour les problèmes à interfaces entre fluides compressibles en présence de transfert de chaleur: some models and eulerian methods for interface problems between compressible fluids with heat transfer, Int. J. Heat Mass Transf., 45, 6, 1287-1307 (2002) · Zbl 1121.76378
[20] Murrone, A.; Guillard, H., A five equation reduced model for compressible two phase flow problems, J. Comput. Phys., 202, 2, 664-698 (2005) · Zbl 1061.76083
[21] Saurel, R.; Petitpas, F.; Berry, R., Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixtures, J. Comput. Phys., 228, 5, 1678-1712 (2009) · Zbl 1409.76105
[22] Harten, A.; Lax, P.; van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25, 1, 35-61 (1983) · Zbl 0565.65051
[23] Roe, P., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43, 2, 357-372 (1981) · Zbl 0474.65066
[24] Toumi, I., A weak formulation for Roe’s approximate Riemann solver, J. Comput. Phys., 102, 2, 360-373 (1992) · Zbl 0783.65068
[25] Ambrose, D.; Tsonopoulos, C., Vapor-liquid critical properties of elements and compounds. 2. Normal Alekenes, J. Chem. Eng. Data, 40, 531-546 (1995)
[28] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S., Uniformly high order accurate essentially non-oscillatory schemes, III, J. Comput. Phys., 71, 2, 231-303 (1987) · Zbl 0652.65067
[29] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77, 2, 439-471 (1988) · Zbl 0653.65072
[30] Loubere, R.; Dumbser, M.; Diot, S., A new family of high order unstructured MOOD and ADER finite volume schemes for multidimensional systems of hyperbolic conservation laws, Commun. Comput. Phys., 16, 3, 718-763 (2014) · Zbl 1373.76137
[31] Shukla, R. K.; Pantano, C.; Freund, J. B., An interface capturing method for the simulation of multi-phase compressible flows, J. Comput. Phys., 229, 19, 7411-7439 (2010) · Zbl 1425.76289
[32] Tiwari, A.; Freund, J. B.; Pantano, C., A diffuse interface model with immiscibility preservation, J. Comput. Phys., 252, 290-309 (2013) · Zbl 1349.76395
[33] Shyue, K.-M.; Xiao, F., An Eulerian interface sharpening algorithm for compressible two-phase flow: the algebraic THINC approach, J. Comput. Phys., 268, 326-354 (2014) · Zbl 1349.76388
[34] Lax, P.; Wendroff, B., Systems of conservation laws, Commun. Pure Appl. Math., 13, 2, 217-237 (1960) · Zbl 0152.44802
[35] Guardone, A.; Vigevano, L., Roe linearization for the van der Waals gas, J. Comput. Phys., 175, 1, 50-78 (2002) · Zbl 1039.76038
[36] Cinnella, P., Roe-type schemes for dense gas flow computations, Comput. Fluids, 35, 10, 1264-1281 (2006) · Zbl 1177.76216
[37] Lax, P., Hyperbolic Systems of Conservation Laws (1987), SIAM
[38] Dafermos, C., Hyperbolic Conservation Laws in Continuum Physics (2005), Springer · Zbl 1078.35001
[39] Hayes, B.; LeFloch, P., Nonclassical shocks and kinetic relations: strictly hyperbolic systems, SIAM J. Math. Anal., 31, 5, 941-991 (2000) · Zbl 0953.35095
[40] DalMaso, G.; LeFloch, P.; Murat, F., Definition and weak stability of nonconservative products, J. Math. Pures Appl., 74, 6, 483-548 (1995) · Zbl 0853.35068
[41] Volpert, A., The space BV and quasilinear equations, Math. USSR, 2, 257-267 (1967)
[42] Abgrall, R.; Karni, S., A comment on the computation of non-conservative products, J. Comput. Phys., 229, 8, 2759-2763 (2010) · Zbl 1188.65134
[43] Toro, E.; Spruce, M.; Spears, W., Restoration of the contact surface in the HLL-Riemann solver, Shock Waves, 4, 1, 25-34 (1994) · Zbl 0811.76053
[44] Saurel, R.; Chinnayya, A.; Renaud, F., Thermodynamic analysis and numerical resolution of a turbulent - fully ionized plasma flow model, Shock Waves, 13, 4, 283-297 (2003) · Zbl 1063.76114
[45] Castro, M.; LeFloch, P.; Muñoz-Ruiz, M.; Parés, C., Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes, J. Comput. Phys., 227, 17, 8107-8129 (2008) · Zbl 1176.76084
[46] Abgrall, R.; Nkonga, B.; Saurel, R., Efficient numerical approximation of compressible multi-material flow for unstructured meshes, Comput. Fluids, 32, 4, 571-605 (2003) · Zbl 1084.76543
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