Verification of a Higher-Order Finite Difference Scheme for the One-Dimensional Two-Fluid Model

• Published in: The Journal of Computational Multiphase Flows Vol. 5; no. 2; pp. 139 - 155
• Main Authors: Fullmer, William D., de Bertodano, Martin A. Lopez, Zhang, Xiaoying
• Format: Journal Article
• Language: English
• Published: London, England SAGE Publications 01.06.2013
• Subjects: Computational fluid dynamics; Contact; Convergence; Discontinuity; Faucets; Mathematical analysis; Mathematical models; Voids; Strong stability preserving; Flux limiter; One-dimensional two-fluid model; Higher-order accuracy; Code verification
• ISSN: 1757-482X; 1757-4838
• DOI: 10.1260/1757-482X.5.2.139

The one-dimensional two-fluid model is widely acknowledged as the most detailed and accurate macroscopic formulation model of the thermo-fluid dynamics in nuclear reactor safety analysis. Currently the prevailing one-dimensional thermal hydraulics codes are only first-order accurate. The benefit of first-order schemes is numerical viscosity, which serves as a regularization mechanism for many otherwise ill-posed two-fluid models. However, excessive diffusion in regions of large gradients leads to poor resolution of phenomena related to void wave propagation. In this work, a higher-order shock capturing method is applied to the basic equations for incompressible and isothermal flow of the one-dimensional two-fluid model. The higher-order accuracy is gained by a strong stability preserving multi-step scheme for the time discretization and a minmod flux limiter scheme for the convection terms. Additionally the use of a staggered grid allows for several second-order centered terms, when available. The continuity equations are first tested by manipulating the two-fluid model into a pair of linear wave equations and tested for smooth and discontinuous initial data. The two-fluid model is benchmarked with the water faucet problem. With the higher-order method, the ill-posed nature of the governing equations presents severe challenges due to a growing void fraction jump in the solution. Therefore the initial and boundary conditions of the problem are modified in order to eliminate a large counter-current flow pattern that develops. With the modified water faucet problem the numerical models behave well and allow a convergence study. Using the L1 norm of the liquid fraction, it is verified that the first and higher-order numerical schemes converge to the quasi-analytical solution at a rate of O(1/2) and O(2/3), respectively. It is also shown that the growing void jump is a contact discontinuity, i.e. it is a linearly degenerate wave. The sub-linear convergence rates are in exact agreement with the theory for a contact discontinuity.