The Use of Continuous and Discrete Markers for Solving Hydrodynamic Problems with Movable Interface Boundaries

Algorithms and results of continuous and discrete markers methods application to the problem of heavy viscous fluid flows calculations with free boundaries are described. The three-dimensional non-stationary Navier-Stokes equations and finite element method are used in variational formulation. An interface capturing method is implemented using fixed-grid with a continuous marker-function and (alternatively) discrete Lagrangian markers. The marker-functions have jumps at the interfaces and the interface boundary is detected as a level surface with intermediate value of marker-function. In numerical solutions, such way to detect interface boundary may result in the conservation laws violation even if conservative numerical methods are used. In order to prevent such effects in our algorithms the procedures of marker-function antidiffusion and conservation laws correction are introduced. In addition, in algorithms an immediate removal of possible monotonicity violations is used. Another implemented way to capture moving interfaces is based on the use of discrete Lagrangian markers. In addition to classic variant of the Marker And Cell (MAC) method, the algorithms of creation of new markers and removal of old markers at input and output boundaries are used. It allows us to consider the problems with open boundaries at long times. The improved interpolation at interface boundaries is described. The methods of continuous and discrete markers are used to simulate some set of incompressible viscous fluid flows with moving free boundaries and variable topology of solution region (joining and separation of parts of solution region). The solutions for following flows are presented: (1) falling water drop into water basin, (2) the flow of water from floor to floor through the hole, (3) the collapse of a water column and the oscillations of a fluid in a closed basin, (4) a fountain and a puddle from a vertical jet, and (5) fall of horizontal jets into the pool with water.