Simulation of two-fluid flows by the least-squares finite element method using a continuum surface tension model

In this paper a numerical procedure for simulating two- uid ows is presented. This procedure is based on the Volume of Fluid (VOF) method proposed by Hirt and Nichols1 and the Continuum Surface Force (CSF) model developed by Brackbill et al.2 In the VOF method uids of di erent properties are identi ed through the use of a continuous eld variable (colour function). The colour function assigns a unique constant (colour) to each uid. The interfaces between di erent uids are distinct due to sharp gradients of the colour function. The evolution of the interfaces is captured by solving the convective equation of the colour function. The CSF model is used as a means to treat surface tension e ect at the interfaces. Here a modi ed version of the CSF model, proposed by Jacqmin,3 is used to calculate the tension force. In the modi ed version, the force term is obtained by calculating the divergence of a stress tensor de ned by the gradient of the colour function. In its analytical form, this stress formulation is equivalent to the original CSF model.2 Numerically, however, the use of the stress formulation has some advantages over the original CSF model, as it bypasses the di culty in approximating the curvatures of the interfaces. The least-squares nite element method (LSFEM)4 is used to discretize the governing equation systems. The LSFEM has proven to be e ective in solving incompressible Navier{Stokes equations and pure convection equations, making it an ideal candidate for the present applications. The LSFEM handles all the equations in a uni ed manner without any additional special treatment such as upwinding or arti cial dissipation. Various bench mark tests have been carried out for both two-dimensional planar and axisymmetric ows, including a dam breaking, oscillating and stationary bubbles and a conical liquid sheet in a pressure swirl atomizer.