Accuracy of high-order, discrete approximations to the lifting-line equation

Abstract

The accuracy of several numerical schemes for solving the lifting-line equation is investigated. Circulation is represented on discrete elements using polynomials of varying degree, and a novel scheme is introduced based on a discontinuous representation that permits arbitrary polynomial degrees to be used. Satisfying the Helmholtz theorems at inter-element boundaries penalises the discontinuities in the circulation distribution, which helps ensure the solution converges towards the correct, continuous behaviour as the number of elements increases. It is found that the singular vorticity at the wing tips drives the leading-order error of the solution. With constant panel widths, numerical schemes exhibit suboptimal accuracy irrespective of the basis degree; however, driving the width of the tip panel to zero at a rate faster than the domain average enables improved accuracy to be recovered for the quadratic-strength elements. In all cases considered, higher-order circulation elements exhibit higher accuracy than their lower-order counterparts for the same total degrees of freedom in the solution. It is also found that the discontinuous quadratic elements are more accurate than their continuous counterparts while also being more flexible for geometric representation.