Abstract
In this paper we propose a class of linearly implicit numerical schemes for a two-phase flow model, allowing for violation of the CFL-criterion for all waves. Based on the Weakly Implixit Mixture Flux (WIMF) approach [SIAM J. Sci. Comput., 26 (2005), pp. 1449–1484], we here develop an extension denoted as Strongly Implicit Mixture Flux (SIMF). Whereas the WIMF schemes are restricted by a weak CFL condition which relates time steps to the fluid velocity, the SIMF schemes are able to break the CFL conditions corresponding to both the sonic and advective velocities. The schemes possess some desirable features compared to current industrial pressure-based codes. They allow for sequential updating of the momentum and mass variables on a nonstaggered grid by solving two sparse linear systems. The schemes are conservative in all convective fluxes and consistency between the mass variables and pressure is formally maintained. Numerical experiments are presented to shed light on the inherent differences between the WIMF and SIMF families of schemes. In particular, we demonstrate that the WIMF scheme is able to give an exact resolution of a moving contact discontinuity. The SIMF schemes do not possess the “exact resolution” property of WIMF, however, the ability to take larger time steps can be exploited so that more efficient calculations can be made when accurate resolution of sharp fronts is not essential, e.g. to calculate steady state solutions.
Similar content being viewed by others
References
Barre, F. et al. (1990). The cathare code strategy and assessment Nucl. Eng. Des. 124, 257–284
Bendiksen, K. H., Malnes, D., Moe, R., and Nuland, S. (1991). The dynamic two-fluid model OLGA: Theory and application. SPE Prod. Eng. 6, 171–180
Coquel, F., El Amine, K., Godlewski, E., Perthame, B., and Rascle, P. (1997). A numerical method using upwind schemes for the resolution of two-phase flows. J. Comput. Phys. 136, 272–288
Cortes, J., Debussche, A., and Toumi, I. (1998). A density perturbation method to study the eigenstructure of two-phase flow equation systems. J. Comput. Phys. 147, 463–484
Evje, S., and Fjelde, K. K. (2003). On a rough ausm scheme for a one-dimensional two-phase flow model. Comput. Fluids 32, 1497–1530
Evje, S., and Flåtten, T. (2003). Hybrid flux-splitting schemes for a common two-fluid model. J. Comput. Phys. 192, 175–210
Evje, S., and Flåtten, T. (2005). Weakly implicit numerical schemes for a two-fluid model. SIAM J. Sci. Comput. 26, 1449–1484
Evje, S., and Flåtten, T. (2005). Hybrid central-upwind schemes for numerical resolution of two-phase flows. ESAIM-Math. Model. Num. 39, 253–274
Larsen, M., Hustvedt, E., Hedne, P., and Straume, T. Petra: A novel computer code for simulation of slug flow, in SPE Annual Technical Conference and Exhibition, SPE 38841, p. 1–12, October 1997
Lorentzen, R. J., and Fjelde, K. K. (2005). Use of slopelimiter techniques in traditional numerical methods for multi-phase flow in pipelines and wells. Int. J. Numer. Meth. Fluids 48, 723–745
Niu, Y.-Y. (2001). Advection upwinding splitting method to solve a compressible two-fluid model. Int. J. Numer. Meth. Fluids 36, 351–371
Paillère, H., Corre, C., and Cascales, J. R. G (2003). On the extension of the AUSM+ scheme to compressible two-fluid models Comput. Fluids 32, 891–916
Ransom, V. H. (1987). Numerical bencmark tests, Multiphase Sci. Tech. 3, 465–473
Tadmor, E. (1984). Numerical viscosity and the entropy condition for conservative difference schemes. Math. Comput. 43, 369–381
Toumi, I., and Caruge, D. (1998). An Implicit Second-Order Numerical Method for Three-Dimensional Two-Phase Flow Calculations. Nucl. Sci. Eng. 130, 213–225
Toumi, I., and Kumbaro, A. (1996). An approximate linearized riemann solver for a two-fluid model. J. Comput. Phys. 124, 286–300
Trapp, J. A., and Riemke, R. A. (1986). A nearly-implicit hydrodynamic numerical scheme for two-phase flows. J. Comput Phys. 66, 62–82
Wada, Y., and Liou, M.-S. (1997). An accurate and robust flux splitting scheme for shock and contact discontinuities. SIAM J. Sci. Comput. 18, 633–657
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Evje, S., Flåtten, T. CFL-Violating Numerical Schemes for a Two-Fluid Model. J Sci Comput 29, 83–114 (2006). https://doi.org/10.1007/s10915-005-9000-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-005-9000-9