A B-spline based higher order panel method is developed for the analysis of the motion of bodies in ideal fluid, either of infinite extent or with free boundary surface. Both the geometry and the potential are represented by B-splines, thus increasing the accuracy of geometry and flow representation to any higher order. Numerical experiments show that the new B-spline based higher order panel method is robust and can handle the thin trailing edge and tip region flow and the sharp corner flows, where the geometric singularity and the hydrodynamic singularity render difficulties to get resolutions by the low order panel method with finite number of panels.
The motion of bodies in the inviscid flow has been successfully analyzed by the so-called low-order panel method where the strength of singularities is assumed constant within each discretized surface panel, since the pioneering work of Hess and Smith (1964). A large number of different panel methods have been developed for a variety of applications (See for example, Hess 1975), leading recently to application to the analysis of steady performance of marine propellers (Hess and Valarezo, 1985). In the works of Hess and his followers, the integral equations based on the Green's third identity are formulated where the unknowns of the equations are sources and/or dipoles with the Kernels expressing the induced velocities due to associated singularities on the body surface. Morino (1974) later introduced another panel method where the primary known is the potential itself and the Kernels are the induced potentials due to sources and dipoles on the body surface.
The motion of the bodies floating on the free surface of the inviscid fluid can also be formulated similarly to Hess and Smith, with the Kelvin-type Green function replacing the Rankine-type Green function.