A Very High-Order (VHO) Flux Reconstruction (FR) Formulation for the Numerical Simulation of the 1D Two-Fluid Four-Equation Model in Pipelines

Resumo

The numerical modeling of multiphase and multicomponent fluid flow in pipelines is essential for addressing various industrial challenges. In the oil industry, accurate knowledge of key flow parameters, such as velocities, pressures, and void fractions, is crucial for managing flow assurance issues. However, simulating flow inside oil pipelines remains a challenging task, as these ducts can extend for thousands of meters. Therefore, mathematical model simplifications are necessary to make the computational cost of CFD simulations feasible. The primary simplification is treating the flow as one-dimensional (1D), as implemented in commercial software such as ALFAsim, LedaFlow, and OLGA. However, these software packages still rely on first-order schemes to handle the advective terms of their governing equations. Among the available mathematical models for simulating 1D two-phase flow in pipelines, the two-fluid model is one of the main alternatives and can even be formulated as a particular case of the more general multi-field model commonly used in commercial packages. The main objective of this study is to develop and implement a very high-order (VHO) scheme in the two-fluid four-equation model, specifically using the Flux Reconstruction (FR) method. The FR method is implemented with a hierarchical limiting scheme for p-type adaptation, and two different correction functions are used and evaluated, leading to two variations of the method that transition from Nodal Discontinuous Galerkin (NDG) to Spectral Difference (SD) methods. The pressure correction equation is derived from mass conservation, and a semi-implicit pressure-based SIMPLE-like method is used for coupling. The pressure equation is solved using the Two-Point Flux Approximation (TPFA) finite volume technique with a staggered grid arrangement. Different test cases are analyzed to evaluate the performance of the FR method. The results demonstrate promising accuracy and are consistent with those reported in the literature, highlighting the potential of the FR method for advancing flow simulations in pipelines.