Conformal Mapping to Calculate Force Induced by Attached Potential Flow around Arbitrary Unsteady Simply Connected Planar Regions

A current topic of interest in naval hydrodynamics is maneuvering and control of non-axisymmetric underwater vehicles; i.e. the forces and moments acting on submerged vehicles with hulls that are not bodies of revolution. One approach is conformal mapping from the flow past a cylinder, which constitutes mapping from an axisymmetric case. This approach takes cross sections of the hull and finds complex maps from the unit circle to the hull contour that are conformal outside the hull. The force on the hull can be found by integrating an unsteady version of Bernoulli’s pressure equation, which formulates the explicit dependence of the pressure on the temporal and spatial derivatives of the complex velocity potential. The force integral can be computed by Cauchy’s Integral Formula from complex analysis. The evaluation of this integral is complicated because the conformal map is kept in a general form so that the hull shape may remain unspecified. However, the integral possesses a rather beautiful derivative structure that yields a simple algebraic expression for the force due to the attached crossflow.