The Generalized Image Model—An Application to the Design of Ducted Propellers

The problem of the potential flow in the vicinity of two distinct bodies A and B, subject to a given inflow, is addressed. The flow field around B is analyzed by distributing singularities on its boundary. It is shown that in the presence of body A those singularities correspond to the modified source and dipole which satisfy the boundary condition on A, including the Kutta condition at its trailing edge. The generalized images of the singularities on B with respect to body A are defined as the difference between the modified and the free space sources or dipoles. The generalized image method is then applied to the design of ducted propellers. The propeller is modeled with lifting lines and the flow around it is analyzed as if the propeller were open. The duct is accounted for by taking as inflow to the lifting lines the flow inside the duct in the absence of the propeller and by pairing each of the lifting line vortex horseshoes I . . I with its generalized image with respect to the duct. The generalized image is determined by evaluating the flow around the duct in the presence of each of the vortex horseshoes by employing a potential based boundary element method. The optimum circulation distribution of the propeller inside the duct is determined by employing a discrete nonlinear optimization technique in order to maximize the propulsive unit efficiency. The effect of the main parameters of the problem on the optimum propeller circulation and the propulsive efficiency is shown with some examples.