Balanced-force numerical method for two phase flow at the onset of instability

Abstract

The Interface Capturing method, which is a finite volume method as formulated by Queutey and Vissoneau for free surface immiscible, incompressible multiphase flows employs a collocated arrangement of unknowns
and achieves a discrete force balance for the case when the interface coincides with the faces of the control volumes in the computational domain. This constraint limits the applicability of the method. Furthermore,
the authors do not provide the exact formulation of the operators involved in the pressure velocity coupling. In the present research, a balanced-force numerical method is formulated, applicable for an interface that
neither has to coincide nor be aligned with the faces of the control volumes. The approach consists of the reconstruction of the values of the flow variables at the interface based on the interface jump conditions, with which the limit values of the normal derivatives at the interface are calculated. Furthermore, the construction of the operators of the discrete system is delineated to achieve a discrete force balance, by incorporating the reconstructed flow variables and employing a discretization which complies with the interface jump conditions. It is sufficient for a stationary discrete formulation to comply with the differential equation and the interface jump conditions. However, to apply this approach to solve unsteady flow problems the influence of the reformulated operators on the stability properties of the system should also be investigated.
The properties of the individual operators are analyzed as well as their behaviour when they are embedded in the complete solver algorithm. Results are shown for both steady and unsteady test cases and compared with numerical results obtained with OpenFOAM. The resulting framework avoids the occurrence of
spurious velocities as it discretely complies with the interface conditions.