Free streamline detachment with surface tension: A conformal mapping-based numerical solution

The problem of the detachment of a free-streamline from a smoothly shaped body is considered taking into account the effect of surface tension. The fluid is assumed to be inviscid and incompressible, and the flow is assumed to be irrotational and stationary in the body-fixed frame but corresponds to a progressive capillary waves in the freestream frame. We employ the Brillouin-Villat criterion to determine the position of the free-streamline detachment. Using the integral hodograph method, we derive expressions for the complex velocity and the derivative of the complex potential, both defined in an auxiliary parameter plane. A system of singular integral equations, formulated in terms of the velocity magnitude along the free surface and the slope of the body, is derived by applying the dynamic and kinematic boundary conditions. The numerical method is highly efficient, as it only requires the evaluation of these two nonsingular functions. Numerical results demonstrate the effect of surface tension on the free-streamline detachment, the drag force and the free surface shape for a wide range of Weber numbers. It is shown that the surface tension generates capillary waves due to the distortion of the magnitude of the inflow velocity at the detachment point.