A study on the extension of a VOF/PLIC based method to a curvilinear co-ordinate system

The conventional volume-of-fluid method has the potential to deal with large free surface deformation on a fixed Cartesian grid. However, when free-surface flows are within or over complex geometries of industrial relevance, such as free-surface flows over offshore oil platforms, it is advantageous to extend the method originally written in Cartesian forms into non-Cartesian forms. In the present study, an algorithm similar to the algorithm described by Rudman in 1997 is proposed for use with curvilinear co-ordinates. This extension results in the ability to model complex geometries which could not be modelled using the original algorithm. Excellent agreement between the solutions obtained on both orthogonal and non-orthogonal meshes is achieved, although in general the L ₁ error, based on the exact solution, on the non-orthogonal mesh is slightly higher than that on the orthogonal mesh. The extended fluid flow solving capacity of the present method is demonstrated through its application to a non-orthogonal Rayleigh–Taylor instability problem.     

A finite element solution of the time-dependent incompressible Navier–Stokes equations using a modified velocity correction method

A finite element solution of the two-dimensional incompressible Navier–Stokes equations has been developed. The present method is a modified velocity correction approach. First an intermediate velocity is calculated, and then this is corrected by the pressure gradient which is the solution of a Poisson equation derived from the continuity equation. The novelty, in this paper, is that a second-order Runge–Kutta method for time integration has been used. Discretization in space is carried out by the Galerkin weighted residual method. The solution is in terms of primitive variables, which are approximated by polynomial basis functions defined on three-noded, isoparametric triangular elements. To demonstrate the present method, two examples are provided. Results from the first example, the driven cavity flow problem, are compared with previous works. Results from the second example, uniform flow past a cylinder, are compared with experimental data.