Conformal contour dynamics in bounded domains

A contour dynamics algorithm is presented for vortex patches in unbounded domains and in simply connected bounded domains. It is based on conformal mapping and spectral analysis. The inside and outside of a vortex patch are analytically mapped onto the inside and outside of the unit circles of two different complex planes. The flow field is determined by matching the inner and outer flows on the patch boundary. Following the Legras and Zeitlin conformal dynamics concept, the time evolution of the patch boundary is expressed by means of the time derivatives of the mapping functions. The presence of a bounding wall, which can be permeable and movable, is considered. The geometry and dynamics of the patch and the flow velocity on the bounding wall are represented by Fourier series; by assuming their coefficients as control parameters, the proposed formulation can be appealing for optimization and control purposes. Two numerical examples of the proposed technique are presented.