Aspects of Prandtl’s Lifting-Line Theory and Wake Roll-Up

Aspects of Prandtl’s Lifting-Line theory are addressed, such as the aerodynamic characteristics of various distributions of circulation. Closed-form expressions for induced upwash distributions, lift, drag, bending moment, adverse/proverse yawing moment and center-ofvorticity are derived for prescribed distributions of the circulation. The distributions include the classical elliptic distribution and bell-shaped distributions. For a specified wing planform, solving Prandtl’s integro-differential equation, analytically, or numerically, is more or less standard. Prandtl’s methodology for minimization of induced drag for given lift and spanintegrated section-bending-moment, requires finding the circulation distribution for specified even-order polynomial distributions of the upwash. The minimization involving the rootbending-moment requires a linear distribution of the upwash to be inverted into the associated circulation distribution. Determining circulation distribution for given upwash distribution, i.e., the inverted Biot-Savart relation, is less known than determining the upwash distribution for given vortex distribution, i.e., the Biot-Savart relation for vortex distributions. For that purpose, in the present paper the inverted relation: given the upwash, find the circulation has been (re-)derived. This inverted Biot-Savart relation gives for linear upwash distribution a closed-form non-polynomial circulation distribution. In the present paper, for each circulation distribution considered, the upwash is determined along the span as well as along the extensions of the span from the tips outwards. This provides insight in the singular behavior of the upwash distribution in the tip region and the behavior of the roll-up of the wake vortex sheet.