Application of fast poisson solver with immersed interface method for velocity poisson equation in the presence of body

Abstract

In the vortex particle method, the Biot--Savart law is used to calculate induced velocity of vortex particles. However, the computational cost is proportional to the square of the number of the particles. The vortex-in-cell method and the fast Poisson solver were developed. There are two types of Poisson equations, namely the vector potential Poisson equation and velocity Poisson equation. It is difficult for the vector potential Poisson equation to impose velocity boundary condition as vector potential in 3D. Velocity boundary condition can be directly applied to velocity Poisson equation. However, in the presence of bodies, domain transformation is not sufficient to reflect body. Therefore, fast Poisson solver cannot be applied. In this thesis, velocity Poisson equation was selected to apply velocity boundary condition directly. The vortex-in-cell code was implemented in Fortran 90. In addition, the immersed interface method was applied to consider body without domain transformation in Cartesian grid for the velocity Poisson equation. Hence, the vortex-in-cell method based on the velocity Poisson equation with a fast Poisson solver is suggested. In this thesis, the induced velocity of increasing random vortex particles in 2D Cartesian grid with and without cylinder inside the domain was calculated and the accuracy and the efficiency were compared to those of the solution obtained using the Biot--Savart law. The computations validated that the vortex-in-cell and fast Poisson solver are applied to unbounded domain without transformation, and this shortens the computation by solving with the Biot―Savart law.