3D Anisotropic Adaptive Meshing and Stabilised Finite Element Methods for Multiphase Flows at Low and High Reynolds Number

Abstract : This paper presents a stabilized finite element method for the solution of incompressible multiphase flow problems in three dimensions using a level set method with anisotropic adaptive meshing. A recently developed stabilized finite element solver which draws upon features of solving general fluid-structure interactions is presented. The proposed method is developed in the context of the monolithic formulation and the Immersed Volume Method. Such strategy gives rise to an extra stress tensor in the Navier-Stokes equations coming from the presence of the structure (e.g. rigid or elastic) in the fluid. The distinctive feature of the Variational MultiScale approach is not only the decomposition for both the velocity and the pressure fields into coarse/resolved scales and fine/unresolved scales but also the possible efficient enrichment of the extra constraint. This choice of decomposition is shown to be favorable for simulating multiphase flows at low or high Reynolds number. The interface between the phases is resolved using a convected level set approach developed. This approach enables first to restrict convection resolution to the neighbourhood of the interface and second to replace the reinitialisation steps by an advective reinitialisation. This enables an efficient resolution and accurate computations of flows even with large density and viscosity differences. The level set function is discretized using a stabilized upwind Petrov-Galerkin method and can be coupled to a direct anisotropic mesh adaptation process enhancing the interface representation. Therefore, we propose to build a metric field directly at the nodes of the mesh for a direct use in the meshing tools. In addition, we show that we obtain an optimal stretching factor field by solving an optimization problem under the constraint of a fixed number of edges in the mesh. The capability of the resultant algorithm is demonstrated with three dimensional time-dependent numerical examples such as: the complex fluid buckling phenomena, the water waves propagations, and the rigid bodies motion in incompressible flows.