Stabilized Space-time FEM and its Applications to Free-surface Flows

Fluid flow simulations that involve deforming domains, in the presence of one or more moving boundaries or fluidfluid interfaces, continue to present unique challenges, and form one of the frontiers of computational science. Freesurface flows in particular involve the motion of the fluid interface which is unknown at the outset of the simulation. Thus, both the domain and the flow field are parts of the solution. The requirement of accurate and robust tracking of the domain deformation, together with the need for proper representation of artificial flow boundaries and interface effects, are some of the difficult issues encountered in free-surface flow simulations. Two dramatically different approaches, that of interfacetracking and interface-capturing, have emerged, and both have their proponents. An interface-tracking method always places computational nodes at the moving interface and adjusts the computational mesh to the movement of those nodes. An interface-capturing method lets the computational mesh be stationary, and simply records which computational cells, or elements, are filled with fluid, empty, or contain the interface. Each method presents advantages and disadvantages, and neither can be discounted. In the interface-tracking methods, robust algorithms for placing and moving the nodes at the interface, as well as the necessary adjustments to the computational mesh away from the interface, must be developed, especially in three dimensions. In the interface-capturing methods, steps need to be taken to increase the accuracy with which the position of the interface can be predicted. We present a set of methods, based on the interface-tracking approach, which is being used in simulations of free-surface flows in dams and channels. The Deformable-Spatial-Domain/Stabilized SpaceTime (DSD/SST) finite element formulation was first introduced in 1) and 2), and was applied to many classes of flow problems involving moving boundaries and interfaces (see e.g. 3)). In space-time methods, the stabilized finite element formulations of the governing equations are written over the space-time domain of the problem. Consequently, changes in the shape of the spatial domain due to the motion of the boundaries and interfaces are taken into account automatically. This approach has been successfully used to solve sloshing problems 4), flows past a surface-piercing cylinder 5), as well as other classes of deforming-domain problems 6),7). In Section 2, the governing equations are reviewed. The stabilized finite element formulations are presented in Section 3. In Section 4, a simulation of 3D flow past spillway of a dam is presented, and concluding remarks are provided in Section 5.