Arclength numerical continuation in free-boundary flow

Using Helmholtz's wake model, we reduce the study of the free boundary flow past an obstacle consisting of an arc of circle to the investigation of a Hammerstein nonlinear integral equation depending on a real parameter λ. The papers dedicated to this problem investigated the case λ>0 which corresponds to a convex obstacle with respect to the incoming fluid. Herein, we apply for the first time in the literature the arclength continuation method for the case λ<0 corresponding to a concave arc of a circle. For λ>0 the existence and the uniqueness of the solution was demonstrated, but for λ<0, depending on its value compared to the one of a turning point, the integral equation has either no solution or two distinct solutions corresponding to two different obstacles. We numerically calculate the free lines, the velocity field and the stream lines. A diagram of the drag coefficient versus the arc measure for both convex and concave obstacles suggests us to draw some conclusions concerning the optimization of the blades of a vertical axis (Savonius) wind turbine.