An efficient numerical scheme for modeling two-phase bubbly homogeneous air-water mixtures

This paper focuses on the formulation and assessment of a second-order accurate Finite Volume (FV) shock-capturing scheme for modeling two-phase water hammer flows using the single-equivalent fluid approximation. The FV formulation of the proposed scheme ensures that mass and momentum are conserved. For achieving a second-order rate of convergence for smooth transients (i.e., flows that do not present discontinuities), a second-order boundary condition is implemented using virtual cells and the theory of Riemann invariants, which is similar to that used for the Method of Characteristics (MOC). Since the two-phase flow governing equations when using the single-equivalent fluid approximation are the same as the one-phase water hammer equations (with exception that the pressure-wave celerity is constant in the latter case), and because analytical solutions are available for the latter case, the numerical efficiency of the proposed model is tested using the one-phase water hammer equations with constant pressure-wave celerity. The validity of the single-equivalent fluid approximation and the proposed scheme herein are verified with laboratory experiments. For one-phase transient flows, numerical tests were performed for smooth and strong transients. For smooth transients, the results show that the efficiency of the proposed scheme is highly superior to the fixed-grid MOC scheme with space-line interpolation and another second-order FV scheme. For one-phase strong transient flows, the results show that the efficiency of the proposed scheme is highly superior to the MOC scheme, and significantly superior to the other FV scheme for coarse grids. For fine grids, the accuracy of the proposed scheme converges to that of the other FV scheme. For two-phase water hammer flows, the results show good agreement between experimental data and the results of numerical simulations.