The harmonic polynomial cell method for moving bodies immersed in a Cartesian background grid

In numerical simulations with moving bodies, and often with complex geometries, generation of high-quality body-fitted grids is a cumbersome and time-consuming task. An alternative is to use a fixed (Cartesian) background grid, and allow the body to move freely over this. The challenge in such methods is to transfer the body-boundary conditions of the moving body to fixed grid nodes in a rational manner. In this paper, an Immersed Boundary Method (IBM) is proposed to simulate potential flow about a moving body on a Cartesian background grid. The recently developed Harmonic Polynomial Method, proven both accurate and computationally efficient, is used to represent the velocity potential in the fluid. The body-boundary conditions are interpolated by using ghost nodes inside the body with mirror interpolation points in the fluid. The method is first tested for a fixed cylinder in oscillatory flow to determine the accuracy of the proposed IBM, before considering the equivalent case of an oscillating cylinder in still fluid. Finally, a steadily-advancing cylinder is studied, which is considered as the most challenging case with respect to spurious pressure oscillations. These are known to be a challenge in many IBMs, and special attention is therefore devoted to this aspect.