"Missing" boundary conditions? Discretize first, substitute next, and combine later

A simple approach exists to prevent the need for constructing boundary conditions in situations where they are not explicitly supplied by the original analytical formulation of the problem. An example is the Poisson equation for the pressure in calculations of incompressible flow. Other examples are the streamfunction-vorticity formulation where no condition for the vorticity is present, and ADI methods where boundary conditions for the intermediate timesteps must be provided. In short, this approach can be described as follows: first discretize the equations of motion, next substitute the original boundary conditions (for the velocity), and finally combine the discrete equations (e.g., to a modified Poisson equation).

On Local Relaxation Methods and Their Application to Convection-Diffusion Equations

This paper discusses local relaxation (LR) methods which can be regarded as generalizations of the successive overrelaxation (SOR) method. The difference is that within an LR method the relaxation factor is allowed to vary from equation to equation. A number of existing methods are found to be in fact special LR methods. Moreover, based on SOR theory, a new LR method is developed. The performance of LR methods is illustrated by applying them to central difference approximations of convection-diffusion equations. It is found that equations with small diffusion coefficients can be handled without difficulty. For equations with strongly varying coefficients, and for nonlinear equations, a properly selected LR method can be significantly more efficient than the optimum SOR method. As a special example, a 16 × 16 driven cavity problem for a Reynolds number of 10^6 can be solved in just a few seconds on a modern computer.

The Simulation of Violent Free-Surface Dynamics at Sea and in Space

Non-linear free-surface phenomena, such as extreme waves and violent sloshing, and their impact on the dynamics response of the containing vessels have long been subjects that could only be studied with experimental methods. Nowadays, computational CFD tools can make a significant contribution to these flow problems. The paper starts with a short overview of the most popular numerical methods to simulate highly non-linear free-surface phenomena, with emphasis on Navier–Stokes methods. Thereafter, it describes the development efforts made in the maritime SAFE-FLOW project and the micro-gravity SloshSat FLEVO project. In particular, the improved Volume-of-Fluid (iVOF) free-surface simulation method ComFlo is presented. Examples of violent fluid dynamics in both application areas are presented. In all cases experimental data are available to validate the outcome of the calculations