A hybrid finite‐element–volume‐of‐fluid method for simulating free surface flows and interfaces

SUMMARY A numerical technique is developed for the simulation of free surface flows and interfaces. This technique combines the strength of the finite element method (FEM) in calculating the field variables for a deforming boundary and the versatility of the volume-of-fluid (VOF) technique in advection of the fluid interfaces. The advantage of the VOF technique is that it allows the simulation of interfaces with large deformations, including surface merging and breaking. However, its disadvantage is that in solving the flow equations, it cannot resolve interfaces smaller than the cell size, since information on the subgrid scale is lost. Therefore the accuracy of the interface reconstruction and the treatment of the boundary conditions (i.e. viscous stresses and surface tension forces) become grid-size-dependent. On the other hand, the FEM with deforming interface mesh allows accurate implementation of the boundary conditions, but it cannot handle large surface deformations occurring in breaking and merging of liquid regions. Combining the two methods into a hybrid FEM-VOF method eliminates the major shortcomings of both. The outcome is a technique which can handle large surface deformations with accurate treatment of the boundary conditions. For illustration, two computational examples are presented, namely the instability and break-up of a capillary jet and the coalescence collision of two liquid drops. Free surface flows and interfaces between two immiscible fluids or materials with different phases are observed in many natural and industrial processes. Various numerical techniques have been developed to simulate these flows. However, owing to the complexity of the problem, each technique is tailored to a particular category of flows. For instance, boundary integral techniques'" are mainly used for simulating inviscid irrotational flows and the limiting case of zero Reynolds number. Finite element methods (FEMs) and finite difference methods (FDMs) are potentially applicable to generalized Navier-Stokes equations; however, they have to be coupled with a technique to track the advecting fluid boundaries and interfaces. The difficulty in the interface tracking is inherently related to the complexity of its topology. Therefore techniques which can handle small surface deformations fail when applied to large interface distortions. For simulation of the former category of flows (small surface deformations) the FEM is more popular. Here the fluid boundary is described by a set of fixed4 or def~nning~-'~ meshes, the location of which is obtained by either an iterative procedure or the Lagrangian movement of the interface nodes. This results in the simultaneous calculation of the position of the free surface and the field variables at the new nodal positions. Boundary-fitted orthogonal c~-ordinates*~-'~ and Lagrangian technique^'^>'^ are also used to follow the advecting liquid interfaces. These techniques are confronted with difficulties when applied to large surface deformations, surface breaking and merging. CCC 0271-2091/95/121363-18 0 1995 by John Wiley & Sons, Ltd.