Lifting Line with Various Mollifications: Theory and Application to an Elliptical Wing

Abstract

Building on the well-known Prandtl lifting line, this work proposes an extended framework for mollified lifting lines, which is valid for multidimensional Gaussian smoothings. The smoothing length scale can be taken as constant or variable across the span. In this framework, a slender lifting device and its wake are represented by a mollified vortex tube and a mollified vortex sheet. Although the resulting formulation is similar to the original one, it is shown that a sectional mean downwash must be used in the integrodifferential equation from which the circulation distribution is obtained. Interestingly, it is also shown that this equation is unaffected by the presence of mollification in the streamwise direction. As an application, the case of the elliptical wing is revisited with several smoothing configurations. The results are consistent with the original singular lifting line, and the peculiarities of the circulation and downwash distributions are investigated. A uniform downwash is also recovered for a specific configuration. Finally, the constant and the variable regularizations are found to yield similar results. Overall, the proposed method provides accurate results that can be compared with those from numerical methods, such as the actuator line model, immersed lifting lines, or other similar methods.