Computation of the Helmholtz-Kirchhoff and reentrant jet flows using Fourier series

Numerical algorithms are presented for two classical free boundary problems for ideal flow past an obstacle: Helmhotz-Kirchhoff and reentrant jet flows. The Levi-Civita representation of the log-hodograph function is used in each case to derive non-linear integral equations for the boundary correspondence between the obstacle and the parameter domain. The integral equations are solved by a method of successive conjugation implemented with the fast Fourier transform. For the reentrant jet flow an additional non-linear system must be solved to update certain flow parameters at each iteration. Several examples are computed for polygonal and curvilinear obstacles. A convergence result is given for the Helmhotz-Kirchhoff flow iteration.