Lifting line theory for wing-in-ground effect in proximity to a free surface

Although it has some limitations in applications, the classical Prandtl lifting line theory remains a standard methodology for evaluating lifting problems in free space. It is of theoretical interest in revealing lifting mechanisms. It is therefore, interesting to generalize the classical lifting line theory to cases more general than just the free space problem. In this article, we present the Prandtl lifting line theory for wing-in-ground effect (WIG) near a free surface. While, the fundamental methodology being similar to the classical lifting line theory, it turns out that the difficulty lies in finding the three-dimensional Green's function for the system of horseshoe vortices operating over the deformable free surface. Linear free surface boundary conditions are applied to deal with the two-dimensional lifting problem solved by the singularity distribution method and the three-dimensional correction found by placing a system of horseshoe vortices on the wing. This approach was validated against published data. Excellent agreement is found among results obtained from this study, experiments and numerical simulations. Extensive numerical examples are carried out to show the features of lift coefficients in the vicinity of a free surface. As expected, the free surface can be represented by a rigid wall for the case of high velocity. Finally, the free surface effect on WIG is discussed.