Numerical Solvers for Transient Two-Phase Flow

Abstract

Certain numerical methods have been well developed for solving one-dimensional two-phase flow (e.g. gas and liquid) problems in the literatures during the last two decades. Based on the existing methods, the present work compares the computational efficiency, accuracy, and robustness of various numerical schemes by predicting the numerical solutions of fluid properties for a specific case to find the proper numerical method.

One of the numerical schemes introduced in this work is a practical, semi-implicit upwind method used for fluid flow simulations in different flow patterns,stratified flow and slug flow. This method implements the iterative and non-iterative schemes using a two-fluid model that consists of sets of non-hyperbolic equations. A numerical error term is applied in the pressure equation to maintain the volume balance of the two-phase flow model. If the temperature varies, the discretised energy equations use similar error terms as in the pressure equation. In some cases, the small values of the numerical errors are negligible and do not influence the numerical results. These errors are, however, important factors to consider when maintaining the stability and robustness of the above numerical schemes for strong non-linear cases. The computational efficiency ofthe non-iterative scheme, where the inner iterations are deactivated, is better than the iterative scheme.

Different grid arrangements are compared with respect to computational accuracy and efficiency. A staggered structured grid implements the same semi-implicit upwind method as in the non-iterative scheme; the non-staggered grid arrangement uses an existing flux-splitting scheme (Evje and Flåtten, 2003) as a reference.

All the above schemes produce numerical solutions with a single precision that normally satisfy the requirements of computational accuracy of industrial two-phase pipe flows. However, if one pursues a higher-order accuracy scheme, e.g. a Roe-averaged algorithm, the governing equations should be strictly a hyperbolic system of partial differential equations, which is achieved by introducing the nonviscous force terms in the two-fluid model (LeVeque, 2002).By properly incorporating the non-conservative terms in the formulation of the numerical fluxes, the capability of the Roe-averaged algorithm is demonstrated by capturing shock waves.

Results from the present research include the following. A one-dimensional scheme that solves a system of discretised equations with the staggered semi-implicit upwind method is presented and validated for its computational efficiencyand robustness. This scheme can be widely used in the industry with sufficient accuracy. The other first-order semi-implicit numerical schemes producestable numerical results, especially in the dynamic cases of two-phase flow, except when the gas phase nearly disappears or appears in pipes. The Roe-averaged algorithm is recommended due to the high-resolution numerical results obtained, but at the costs of computational time and effort.