Finite Element-Fictitious Boundary Methods (FEM-FBM) for time- dependent multiphase flow problems ― Application to Sedimentation Benchmarks

This contribution presents new numerical simulation techniques using a Finite Element approach coupled with the Fictitious Boundary Method (FEM-FBM) for non-stationary multiphase flow configurations in 3D. The fluid solution is computed by a Finite Element multigrid solver, which has been realized in the open source CFD package FEATFLOW, while complex dynamic or static geometrical features of the flow domain as well as solid particles, which interact with the surrounding fluid, are treated by the Fictitious Boundary Method. This approach allows the use of structured and unstructured computational meshes which can be static or adaptively aligned by dynamic grid deformation methods. Numerical results for this workshop's ‘sphere sedimentation’ test case are provided. The results show that the presented method can accurately handle the 3D particulate flow situations under consideration. Due to the high parallel efficiency of the FEATFLOW software the method can be used for large-scale problems. In the field of Computational Fluid Dynamics one of the main research topics is the study of (rigid) particulate flows. The simulation and prediction of particulate flows has various important applications in mechanical, chemical or medical engineering. In the last few decades various basic techniques for the simulation of particulate fluid-solid flows have been developed [1], [2], [3]. The most distinctive feature of the techniques is the general approach to solving the fluid flow governed by the Navier-Stokes equations. The most common choices for particulate flows are the Finite Element Method [1] or the Lattice-Boltzmann Method [3], [4]. Our contribution uses the FEM with the FEATFLOW software package as solver [5]. The Finite Element based method can be further distinguished into Arbitrary Langrangian-Eulerian (ALE) and Eulerian approaches. In an ALE scheme both fluid and solid equations are combined in a single coupled variational equation which requires remeshing when the old computational mesh becomes too distorted and the flow field is then projection onto the new mesh. Thus the positions of the solids and the nodes of the mesh are updated explicitly while the velocities are implicitly determined. An Eulerian approach relies on the concept of a ‘fictitious domain’ [6] which is used to represent the solid phase inside of the fluid domain. An advantage of the Eulerian methods is that they allow for a fixed mesh being used so that the need for remeshing can be eliminated. The method presented in this work belongs to the Eulerian methods and is a variation of the Fictitious Domain method termed (Multigrid) Fictitious Boundary Method (FBM) [7]. In our FBM the knowledge of the fluid-solid boundary is of key importance, because it is used to calculate the hydrodynamic forces acting on the solid in a given time step using a volume integral formulation. The basic methods that have been developed several 13 Workshop on Two-Phase Flow Predictions Halle (Saale), Germany, 17. – 20. September 2012 years ago have evolved over time and they are constantly improved with regard to the complexity of flow situations they can handle, the number of solids involved in the simulation and in terms of the computational efficiency of the solver. The multigrid FEM-FBM is no exception and since its early days it has been extended to 3D, adopted to handle arbitrary geometries and was efficiently mapped to parallel hardware which enables the method to utilize the potential of highly parallel architectures and to solve large-scale problems [8]. When a method is further developed care should be taken that not only the computational efficiency is improved, but also the accuracy of the method should at least retain its original quality or preferably improve as well. This is why standardized benchmarking is important and why we subject the current state of the FEM-FBM to the ‘sphere sedimentation’ benchmark defined by ten Cate et al. [9]. In the remainder of this work we present the underlying equations as well as the structure of our fluid-solid solver and we conclude by showing the numerical results of the ‘sphere sedimentation’ benchmark. DESCRIPTION OF THE PHYSICAL MODEL In our numerical studies of particle motion in a fluid, we assume that the fluids are immiscible and Newtonian and the particles are rigid bodies. We refer to the whole computational domain as ΩT=Ωf⋃Ωp, where Ωf is the fluid domain and Ωp the particle domain. The motion of an incompressible fluid with density ρf and viscosity ν is governed by the Navier-Stokes equations in the domain Ωf,   T t u u u t u f , 0 0 , 0                       (1) where σ is the stress tensor of the fluid phase defined by:    . T f u u pI         (2) The motion of N rigid particles in a fluid is described by the Newton-Euler equations with the translational velocity Ui and the angular velocity ωi:     , , i i i i i i col i i i i T I t I F F g M t U M               (3) where Mi is the mass of the i-th particle; ΔM the mass difference between the fluid and solid occupying the same volume; Ii the moment of inertia tensor; Fi the hydrodynamic force and Fcol the collision forces arising from particle-particle collisions; Ti is the torque about the center of gravity of the i-th particle. The hydrodynamic force Fi and the torque Ti can be calculated using a surface integral formulation which has the following form:                     i i i i i i i i i d n X X T d n F   , (4) 13 Workshop on Two-Phase Flow Predictions Halle (Saale), Germany, 17. – 20. September 2012 In the FEM-FBM a single grid is used to cover the whole domain ΩT, the particles are defined on the grid by using an indicator function αi(x). This indicator function allows performing the fluid calculation on the whole domain.            i T i i x for x for x \ 0 , 1  (5) A two-way coupling between fluid and solid phase is achieved by applying the forces Fi and Ti to the particles and by imposing the resulting particle velocity as a velocity boundary condition on the fluid:                                        