Numerical modelling of hyperbolic partial differential equations for two-phase flows

Abstract

This thesis aims to develop numerical tools for hyperbolic partial differential equations which can be applied to several two-phase flow problems in applied sciences research fields such as energy, environment and oil industry. Although there are enormous demands of engineering twophase flow simulations, there is no established mathematical and computational tools which can simulate a wide variety of two-phase flow problems. Here we present a numerical method to get a solution for a system of equations for two phase flow. This solution is valid for a special case of initial condition called the Riemann problem. The system consists of three hyperbolic conservation laws including gas mass balance, liquid mass balance and total momentum balance. This thesis focuses on the extension of the Weighted Average Flux (WAF) scheme for the numerical simulation of two-phase gas-liquid flow by imposing velocity equilibrium and without mechanical equilibrium of the transient drift-flux model. The model becomes a hyperbolic system of conservation laws with realistic closure relations where both phases are strongly coupled during their motion. Exploiting this, the drift-flux model discretization, simulation and investigation become very fast, simple and robust. The efficiency of the WAF scheme as being a second order in space and time without data reconstruction have been demonstrated in the published literature for compressible single-phase flows. However, the scheme is rarely applied to compressible two-phase flows. Based on a recent and complete exact Riemann solver for the drift-flux model, the model is numerically solved by the WAF scheme. The numerical algorithm accuracy and ability are validated through different published test cases. It is shown that the proposed scheme can be effectively employed to simulate two-phase flows involving discontinuities such as shocks and interfaces. The proposed WAF scheme is also compared with other numerical methods. Simulation results show appropriate agreement of WAF scheme even with the exact solution. Comparisons of the presented simulations demonstrate that the behaviour of WAF scheme is encouraging, more accurate and fast than other numerical methods