A Coupled Pressure Based Solution Algorithm Based on the Volume-of-fluid Approach for Two or More Immiscible Fluids

Abstract. The purpose of this study is to describe a coupled pressure based solution algorithm to model interface capturing problems for multiphase flows where neither of the phases can be regarded as dominant. As to a segregated approach using an iterative solution scheme consisting of successive solutions of the transport equation for each phase, the phase for which the equation is solved first dominates the phases for which the transport equations are solved later. This is the result of the requirement of a bounded solution for the phase fraction variables. A cell which is already completely filled with one phase cannot contain any more fluid of any other phase. In the present study, a coupled solution approach of the equations to describe the transport of the phases and the pressure equation is presented. This coupled approach avoids the predescribed numerically induced predominance of a single phase which may occur if a common segregated algorithm is used. The system is described by the continuity equation, the equations to model the transport of the volume fractions αi and the volumetrically averaged momentum equation. By analogy with the derivation of the pressure equation in PISO or SIMPLE, the discretised momentum balance is inserted in the equations to describe the transport of the N phases. While the volume fractions of (N − 1) phases are gained by solving the transport equations, the volume fraction of the N -th phase is calculated by applying the closure equation. In contrast to segregated approaches, the equations for pressure and volume fractions are now solved simultaneously in one single step. Therefore the state variables can implicitly depend on other state variables. To achieve a strong coupling between the volume fractions and the pressure, the pressure is treated implicitly in the equations for the volume fractions and the volume fractions are equally treated implicitly in the pressure equation. The coupled approach with the additional implicit dependencies leads to a matrix which is larger and has more non-zero entries than the matrices of a segregated approach. Therefore the effort to solve the equation system increases significantly. The system is solved applying the block matrix implementation by Jasak.1