A robust and accurate approach to computing compressible multiphase flow: Stratified flow model and AUSM\(^+\)-up scheme.

*(English)*Zbl 1192.76030
J. Comput. Phys. 225, No. 1, 840-873 (2007); erratum ibid. 227, No. 10; 5360 (2008).

Summary: We propose a new approach to compute compressible multifluid equations. Firstly, a single-pressure compressible multifluid model based on the stratified flow model is proposed. The stratified flow model, which defines different fluids in separated regions, is shown to be amenable to the finite volume method. We can apply the conservation law to each subregion and obtain a set of balance equations. Secondly, the AUSM\(^{+}\) scheme, which is originally designed for the compressible gas flow, is extended to solve compressible liquid flows. By introducing additional dissipation terms into the numerical flux, the new scheme, called AUSM\(^{+}\)-up, can be applied to both liquid and gas flows. Thirdly, the contribution to the numerical flux due to interactions between different phases is taken into account and solved by the exact Riemann solver. We will show that the proposed approach yields an accurate and robust method for computing compressible multiphase flows involving discontinuities, such as shock waves and fluid interfaces. Several one-dimensional test problems are used to demonstrate the capability of our method, including the Ransom’s water faucet problem and the air-water shock tube problem. Finally, several two dimensional problems will show the capability to capture enormous details and complicated wave patterns in flows having large disparities in the fluid density and velocities, such as interactions between water shock wave and air bubble, between air shock wave and water column(s), and underwater explosion.

##### MSC:

76M12 | Finite volume methods applied to problems in fluid mechanics |

76T10 | Liquid-gas two-phase flows, bubbly flows |

76N15 | Gas dynamics (general theory) |

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\textit{C.-H. Chang} and \textit{M.-S. Liou}, J. Comput. Phys. 225, No. 1, 840--873 (2007; Zbl 1192.76030)

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##### References:

[1] | Buyevich, Y.A., Statistical hydrodynamics for dispersed system, physical background and general equations, J. fluid mech., 49, 489-507, (1971) · Zbl 0225.76053 |

[2] | Ishii, M., Thermo-fluid dynamic theory of two-phase flow, (1975), Eyrolles Paris · Zbl 0325.76135 |

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

[4] | I. Toumi, A. Kumbaro, H. Paillère, Approximate Riemann solvers and flux vector splitting schemes for two-phase flow, Lecture series 1999-03, von Karman Institute for Fluid Dynamics, 1999. |

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

[6] | Drew, D.; Cheng, L.; Lahey, J.R.T., The analysis of virtual mass effects in two-phase flow, Int. J. multiphase flow, 5, 233-242, (1979) · Zbl 0434.76078 |

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

[8] | Chang, C.-H.; Liou, M.-S., A new approach to the simulation of compressible multifluid flows with AUSM^{+} scheme, AIAA paper, 03-4107, (2003) |

[9] | C.-H. Chang, M.-S. Liou, Simulation of multifluid multiphase flows with AUSM^{+}-up scheme, in: Third Internation Conference of Computational Fluid Dynamics, Toronto, Canada, 2004. |

[10] | Abgrall, R.; Saurel, R., Discrete equations for physical and numerical compressible multiphase mixtures, J. comput. phys., 186, 361-396, (2003) · Zbl 1072.76594 |

[11] | Saurel, R.; Gavrilyuk, S.; Renaud, F., A multiphase model with internal degrees of freedom: application to shock – bubble interaction, J. fluid mech., 495, 283-321, (2003) · Zbl 1080.76062 |

[12] | LeMétayer, O.; Massoni, J.; Saurel, R., Modeling evaporation fronts with reactive Riemann solvers, J. comput. phys., 205, 567-610, (2005) · Zbl 1088.76051 |

[13] | Van Leer, B., Flux vector splitting for the Euler equations, () · Zbl 0555.76014 |

[14] | M.-S. Liou, Ten years in the making - AUSM-family, AIAA paper 2001-2521, in: 15th Computational Fluid Dynamics Conference Proceedings, 2001. |

[15] | Liou, M.-S., A sequel to AUSM: AUSM^{+}, J. comput. phys., 129, 364-382, (1996) · Zbl 0870.76049 |

[16] | Liou, M.-S.; Edwards, J.R., Numerical speed of sound and its application to schemes for all speeds, AIAA paper, 99-3268, (1999) |

[17] | F. Harlow, A. Amsden, Fluid dynamics, Technical Report LA-4700, Las Alamos National Laboratory, 1971. · Zbl 0221.76011 |

[18] | M.-S. Liou, A further development of the AUSM^{+} scheme toward robust accurate solution for all speed, AIAA paper 2003-4116, in: 16th Computational Fluid Dynamics Conference Proceedings, 2003. |

[19] | Liou, M.-S., A sequel to AUSM, part II: AUSM^{+}-up for all speeds, J. comput. phys., 214, 137-170, (2006) · Zbl 1137.76344 |

[20] | Edwards, J.R.; Liou, M.-S., Low-diffusion flux-splitting methods for flows at all speeds, AIAA journal, 36, 9, 1610-1617, (1998) |

[21] | Wallis, G., One-dimensional two-phase flow, (1964), McGraw-Hill New York |

[22] | Hancox, W.; Ferch, R.; Liu, W.; Nieman, R., One-dimensional models for transient gas – liquid flows inducts, Int. J. multiphase flow, 6, 25-40, (1980) · Zbl 0422.76066 |

[23] | Toumi, I., An upwind numerical method for two-fluid two-phase flow models, Nuclear sci. eng., 123, 147-168, (1996) |

[24] | Godunov, S.K., Numerical solution of multidimensional gas dynamics problems, (1976), Nauka Moscow |

[25] | M.-S. Liou, L. Nguyen, C.-H. Chang, S. Sushchikh, R. Nourgaliev, T. Theofanous, Hyperbolicity, discontinuities, and numerics of two-fluid models, Technical report, Springer, in: 4th International Conference on Computational Fluid Dynamics, 2006. |

[26] | Paillère, H.; Corre, C.; Cascales, J.R.G., On the extension of the AUSM^{+} scheme to compressible two-fluid models, Comput. fluids, 32, 891-916, (2003) · Zbl 1040.76044 |

[27] | Chakravarthy, S.R.; Osher, S., Computing with high-resolution upwind schemes for hyperbolic equation, Lect. appl. math., 22, 57-86, (1985) |

[28] | Ransom, V.H., Numerical benchmark tests, () |

[29] | Theofanous, T.; Li, G.; Dinh, T., Aerobreakup in rarefied supersonic gas flows, J. fluids eng., 126, 516-527, (2004) |

[30] | Hankin, R.K.S., The Euler equations for multiphase compressible flow in conservation form: simulation of shock – bubble interactions, J. comput. phys., 172, 2, 808-826, (2001) · Zbl 1028.76050 |

[31] | Saurel, R.; Abgrall, R., A simple method for compressible multifluid flows, SIAM J. sci. comput., 21, 3, 1115-1145, (1999) · Zbl 0957.76057 |

[32] | Toro, E.F., Riemann solvers and numerical methods for fluid dynamics: a practical introduction, (1997), Springer-Verlag Telos · Zbl 0888.76001 |

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