Unsteady lift for the Wagner problem in the presence of additional leading/trailing edge vortices

Abstract

This study amends the inviscid Wagner lift model for starting flow at relatively large angles of attack to account for the influence of additional leading edge and trailing edge vortices. Two methods are provided for starting flow of a flat plate. The first method is a modified Wagner function, which assumes a planar trajectory of the trailing edge vortex sheet accounting for a temporal offset from the original Wagner function given release of leading edge vortices and a concentrated starting point vortex at the initiation of motion. The second method idealizes the trailing edge sheet as a series of discrete vortices released sequentially. The models presented are shown to be in good agreement with high-fidelity simulations. Through the present theory, a vortex force line map is generated, which clearly indicates lift enhancing and reducing directions and, when coupled with streamlines, allows one to qualitatively interpret the effect of the sign and position of vortices on the lift and to identify the origins of lift oscillations and peaks. It is concluded that leading edge vortices close to the leading edge elevate the Wagner lift curve while a strong leading edge vortex convected to the trailing edge is detrimental to lift production by inducing a strong trailing edge vortex moving in the lift reducing direction. The vortex force line map can be employed to understand the effect of the different vortices in other situations and may be used to improve vortex control to enhance or reduce the lift.