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

Abstract

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.

Nonlinear lifting theory for unsteady WIG in proximity to incident water waves. Part 1: Two-dimension

“…The first kind addresses the unsteady lifting problem for a wing moving above a still free surface, and the second kind handles the influence of water waves on WIG. The first kind of problem is connected with the unsteady two-dimensional lifting theory [ 10 -12 ] and the steady WIG in the presence of a still free surface which has been studied by Barber [ 8 ] and Zong et al [ 9 ]. For the steady WIG in the presence of a free surface, an important conclusion they obtained is that the freesurface elevation owing to the passage of the wing is rather small and the influence of the free surface on the wing with a high advancing velocity can be regarded as a rigid wall.…”

Nonlinear lifting theory for unsteady WIG in proximity to incident water waves. Part 2: Three-dimension

“…It can be obviously observed from Figs. 8 -10 that the time-averaged lift coefficient and lift-to-drag ratio increase with the aspect ratio, and the time-averaged induced drag coefficient is declining as the aspect ratio increases as expected in the steady case of a wing advancing in proximity to a still free surface [ 7 ]. As for the amplitudes of the lift, induced drag and lift-to-drag ratio, they all go up with the aspect ratio.…”

“…For the purpose of having this approach be validated against the published data, we investigate the lift coefficient for the steady WIG using the unsteady nonlinear lifting surface theory, and the lift coefficients for the results from experiment measured by Ahmed and Sharma [ 30 ], lifting line theory investigated by Zong et al [ 7 ] and present method are plotted in Fig. 15 (ULS denotes the results calculated by the present approach -unsteady lifting surface theory; EX denotes the results obtained by experiment, and LL represents the results calculated by the lifting line theory).…”

“…15. Comparison of the results by the present method with experimental data [ 26 ] and the lifting line theory [ 7 ].…”

Modeling of Finite-span Ram Wings Moving Above Water at Finite Froude Numbers

“…Calculations by Barber (2007) for wings in moderate ground effect showed that dominant water deformations were induced by tip vortices. Also, considering moderate and distant ground effects, Zong et al (2012) estimated water surface deformations and accompanying effects on wing aerodynamics with help of a modified classic lifting line theory.…”