The Ghost Point Method for Wall Boundary Conditions

Large-stencil finite difference methods are used for CAA problems because they are generally less dispersive and less dissipative, and they tend to be more isotropic. In addition, they yield numerical wave speeds that are nearly equal to the wave speeds of the original partial differential equations. On the other hand, large-stencil or high – order methods support spurious numerical waves. These waves are contaminants or pollutants of the numerical solution. Some of these spurious waves have very short wavelengths (grid-to-grid oscillations) and propagate with ultrafast speeds. They have been referred to as parasite waves. Other spurious waves are spatially damped. Their presence is, therefore, confined locally in space. To ensure a high-quality
computational aeroacoustics solution, it is important that these spurious waves are not excited in the computation.

For inviscid flows, the well-known boundary condition at a solid wall is that the velocity component normal to the wall is zero. This condition is sufficient for the determination of a unique solution to the Euler equations. When a large-stencil finite difference scheme is used, the order of the difference equations is higher than that of the Euler equations. Thus, the zero normal velocity boundary condition is insufficient to define a unique solution. Extraneous conditions must be imposed. Unfortunately, these extraneous conditions would often lead to the generation of spurious numerical waves as mentioned before. The net result is the degradation of the quality of the numerical solution. The degradation may be divided into three types. First, if an acoustic wave is incident on a solid wall, perfect reflection (e. g., reflected wave amplitude equal to incident wave amplitude) may not be obtained. Second, parasite waves may be reflected off the solid surface. Third, a numerical boundary layer can form adjacent to the solid wall surface by the spatially damped spurious numerical waves.