Application of Osher and PRICE-C schemes to solve compressible isothermal two-fluid models of two-phase flow.

*(English)*Zbl 1290.76108Summary: Two path-conservative schemes, namely Osher and PRICE-C schemes have been used to solve isothermal compressible two-phase flows using four-equation model. Path-Conservative Osher (PC-Osher) scheme is an upwind method using the full eigenstructure of the system but PRICE-C scheme is a central method, in which using the full eigenstructure of the system is not necessary. Different two-phase flow problems are solved using these schemes and their results are compared. The numerical efficiency of two schemes and their abilities in the simulation of near single phase flows are also examined. The extension of these schemes to the second order of accuracy is performed using the well-known TVD-MUSCL-Hancock (TMH) method. The results show that for the same level of accuracy, the PC-Osher is more efficient than the PRICE-C scheme in view of computational time. However, the PC-Osher scheme fails to predict near single phase flows compared to the PRICE-C scheme. The results also show that the second order extension of both schemes is less diffusive on the sonic waves while they show small amplitude oscillations on the volume fraction waves.

##### MSC:

76M20 | Finite difference methods applied to problems in fluid mechanics |

76T99 | Multiphase and multicomponent flows |

76N15 | Gas dynamics (general theory) |

##### Keywords:

two-fluid model; four-equation model; path-conservative; Osher scheme; PRICE-C scheme; TVD-MUSCL-hancock
PDF
BibTeX
Cite

\textit{Y. Shekari} and \textit{E. Hajidavalloo}, Comput. Fluids 86, 363--379 (2013; Zbl 1290.76108)

Full Text:
DOI

##### References:

[1] | Ishii, M.; Hibiki, T., Thermo-fluid dynamics of two-phase flow, (2005), Springer West Lafayette · Zbl 1189.76629 |

[2] | Evje, S.; Flatten, T., Hybrid flux-splitting schemes for a common two-fluid model, J Comput Phys, 192, 175-210, (2003) · Zbl 1032.76696 |

[3] | Andrianov, N.; Warnecke, G., The Riemann problem for the Baer-Nunziato two-phase flow model, J Comput Phys, 195, 434-464, (2004) · Zbl 1115.76414 |

[4] | Schwendeman, D. W.; Wahle, C. W.; Kapila, A. K., The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow, J Comput Phys, 212, 490-526, (2006) · Zbl 1161.76531 |

[5] | Deledicque, V.; Papalexandris, M. V., An exact Riemann solver for compressible two-phase flow models containing non-conservative products, J Comput Phys, 222, 217-245, (2007) · Zbl 1216.76044 |

[6] | Loilier, P., Numerical simulation of two-phase gas-liquid flows in inclined and vertical pipelines, (2006), Cranfield University London |

[7] | Omgba-Essama, C., Numerical modelling of transient gas-liquid flows (application to stratified and slug flow regimes), (2004), Cranfield University London |

[8] | Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, J Comput Phys, 43, 357-372, (1981) · Zbl 0474.65066 |

[9] | Osher, S.; Solomon, F., Upwind difference schemes for hyperbolic conservation laws, Math Comput, 38, 339-374, (1982) · Zbl 0483.65055 |

[10] | Toumi, I.; Kumbaro, A., An approximate linearized Riemann solver for a two-fluid model, J Comput Phys, 124, 286-300, (1995) · Zbl 0847.76056 |

[11] | Paillere, H.; Corre, C.; García Cascales, J. R., On the extension of the AUSM scheme to compressible two-fluid models, Comput Fluids, 32, 891-916, (2003) · Zbl 1040.76044 |

[12] | Karni, S.; Kirr, E.; Kurganov, A.; Petrova, G., Compressible two-phase flows by central and upwind schemes, Math Modell Numer Anal, 38, 477-493, (2004) · Zbl 1079.76045 |

[13] | Saurel, R.; Abgrall, R., A multiphase Godunov method for compressible multifluid and multiphase flows, J Comput Phys, 150, 425-467, (1999) · Zbl 0937.76053 |

[14] | DalMaso, G.; LeFloch, P.; Murat, F., Definition and weak stability of nonconservative products, J Math Pure Appl, 74, 483-548, (1995) · Zbl 0853.35068 |

[15] | Castro, M.; LeFloch, P.; Muñoz-Ruiz, M. L.; Parés, C., Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes, J Comput Phys, 227, 8107-8129, (2008) · Zbl 1176.76084 |

[16] | Pares, C., Numerical methods for nonconservative hyperbolic systems: a theoretical framework, SIAM J Numer Anal, 44, 300-321, (2006) · Zbl 1130.65089 |

[17] | Munkejord, S. T.; Evje, S.; Flatten, T., A musta scheme for a non-conservative two-fluid model, SIAM J Sci Comput, 56, 2587-2622, (2009) · Zbl 1387.76106 |

[18] | Castro, C. E.; Toro, E. F., A Riemann solver and upwind methods for a two-phase flow model in non-conservative form, Int J Numer Meth Fluids, 50, 275-307, (2006) · Zbl 1086.76046 |

[19] | Tian, B.; Toro, E. F.; Castro, C. E., A path-conservative method for a five-equation model of two-phase flow with an HLLC-type Riemann solver, Computer and fluids., 46, 122-132, (2011) · Zbl 1433.76166 |

[20] | Tokareva, S. A.; Toro, E. F., HLLC-type Riemann solver for the Baer-Nunziato equations of compressible two-phase flow, J Comput Phys, 229, 3573-3604, (2010) · Zbl 1391.76440 |

[21] | Dumbser, M.; Toro, E. F., A simple extension of the osher Riemann solver to non-conservative hyperbolic systems, J Sci Comput, 48, 70-88, (2011) · Zbl 1220.65110 |

[22] | Dumbser, M.; Toro, E. F., On universal osher-type schemes for general nonlinear hyperbolic conservation laws, Commun Comput Phys, 10, 635-671, (2011) · Zbl 1373.76125 |

[23] | Pitman, E. B.; Le, L., A two-fluid model for avalanche and debris flows, Philos Trans Roy Soc A: Math Phys Eng Sci, 363, 1573-1601, (2005) · Zbl 1152.86302 |

[24] | Baer, M. R.; Nunziato, J. W., A two phase mixture theory for the deflagration to detonation transition (DDT) in reactive granular material, Int J Multiph Flow, 12, 861-889, (1986) · Zbl 0609.76114 |

[25] | Munkejord, S. T., Comparison of roe-type methods for solving the two-fluid model with and without pressure relaxation, Comput Fluids, 36, 1061-1080, (2007) · Zbl 1194.76161 |

[26] | Canestrelli, A.; Siviglia, A.; Dumbser, M.; Toro, E. F., Well-balanced high-order centred schemes for non-conservative hyperbolic systems. applications to shallow water equations with fixed and mobile bed, Adv Water Resour, 32, 834-844, (2009) |

[27] | Ansari, M. R.; Daramizadeh, A., Slug type hydrodynamic instability analysis using a five equations hyperbolic two-pressure, two-fluid model, Ocean Eng, 52, 1-12, (2012) |

[28] | Ransom, V.; Hicks, D. L., Hyperbolic two-pressure model for two-phase flow, J Comput Phys, 53, 124-151, (1984) · Zbl 0537.76070 |

[29] | Stuhmiller, J. H., The influence of interfacial pressure forces on the character of two-phase flow model equations, J Comput Phys, 3, 551-560, (1977) · Zbl 0368.76085 |

[30] | Bestion, D., The physical closure laws in the CATHARE code, Nucl Eng Des, 124, 229-245, (1990) |

[31] | Munkejord, S. T., Analysis of the two-fluid model and the drift-flux model for numerical calculation of two-phase flow, (2005), Norwegian University of Science and Technology Trondheim |

[32] | Bruce Stewart, H.; Wendroff, B., Two-phase flow: models and methods, J Comput Phys, 56, 363-409, (1984) · Zbl 0596.76103 |

[33] | Dinh T, Nourgaliev R, Theofanous T. Understanding the ill-posed two-fluid model. In: Proceedings of the 10th international topical meeting on nuclear reactor thermal-hydraulics (NURETH’03); 2003. |

[34] | Soo, S. L., Multiphase fluid dynamics, (1990), Science Press |

[35] | Evje, S.; Flatten, T., Hybrid central-upwind schemes for numerical resolution of two-phase flows, J Comput Phys, 192, 175-210, (2003) |

[36] | Toro, E. F., Riemann solvers and numerical methods for fluid dynamics, (2005), Springer Manchester |

[37] | Engquist, B.; Osher, S., One sided difference approximations for nonlinear conservation laws, Math Comput, 36, 321-351, (1981) · Zbl 0469.65067 |

[38] | Godunov, S. K., Finite difference methods for the computation of discontinuous solutions of the equations of fluid dynamics, Math USSR, 47, 271-306, (1959) · Zbl 0171.46204 |

[39] | van Leer, B., Towards the ultimate conservative difference scheme. V. A second-order sequel to godunov’s method, J Comput Phys, 32, 101-136, (1979) · Zbl 1364.65223 |

[40] | Abgrall, R., How to prevent pressure oscillations in multicomponent flow calculations: a quasi conservative approach, J Comput Phys, 125, 150-160, (1994) · Zbl 0847.76060 |

[41] | Ransom, V., Numerical benchmark tests, (1987), Hemisphere Publishing Corporation Berlin |

[42] | Coquel, F.; El Amine, K.; Godlewski, E.; Perthame, B.; Rascle, P., A numerical method using upwind schemes for the resolution of two-phase flows, J Comput Phys, 136, 272-288, (1997) · Zbl 0893.76052 |

[43] | Trapp, J. A.; Riemke, R. A., A nearly-implicit hydrodynamic numerical scheme for two-phase flows, J Comput Phys, 66, 62-82, (1986) · Zbl 0622.76110 |

[44] | Toumi, I., An upwind numerical method for two-fluid two-phase flow models, Nucl Sci Eng, 123, 147-168, (1996) |

[45] | Tiselj, I.; Petelin, S., Modelling of two-phase flow with second-order accurate scheme, J Comput Phys, 136, 503-521, (1997) · Zbl 0918.76050 |

[46] | Cortes, J. L.; Debussche, A.; Toumi, I., A density perturbation method to study the eigenstructure of two-phase flow equation systems, J Comput Phys, 147, 463-484, (1998) · Zbl 0917.76047 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.