A Modified Method of Fundamental Solutions for Potential Flow Problems

This chapter describes an application of the recently proposed Modified Method of Fundamental Solutions (MMFS) to the potential flow problems. The solution in two dimensional Cartesian coordinates is represented in terms of the fundamental solution of the Laplace equation together with the first order polynomial augmentation. The collocation is used for determination of the expansion coefficients. This novel method does not require fictitious boundary as the conventional Method of Fundamental Solutions (MFS). The source and collocation points thus coincide on the physical boundary of the system. The desingularised value of the fundamental solution in case of the coincidence of the collocation and source points is determined directly as the average value of the fundamental solution on the boundary in the vicinity of the source point. The respective values of the derivatives of the fundamental solution in the coordinate directions, as required in potential flow calculations, are calculated indirectly from the considerations of the constant potential field. The normal on the boundary is calculated by parametrisation of its length and use of the cubic radial basis functions with the second order polynomial augmentation. The components of the normal are calculated in an analytical way. A numerical example of potential flow around two dimensional circular region is shown. The results with the MMFS are compared with the results of the classical 1 Laboratory for Multiphase Processes, University of Nova Gorica, Vipavska 13, SI-5000 Nova Gorica, Slovenia.