Lifting-line theory for arbitrarily shaped wings

This paper presents a new method for solving the classical lifting-line equation. The spanwise lift (or circulation) distribution is to be determined when the spanwise chord distribution and spanwise twist distribution are resolved separately and arbitrarily into their Fourier-series representations. An infinite set of equations for the resulting circulation Fourier-series coefficients is obtained in terms of the Fourier coefficients for the wing-chord shape and the twist distribution. When the equations are expressed in matrix form, the contributions of planform shape and twist are readily perceived. The set of equations can be truncated arbitrarily for the first N number of unknown coefficients, and easily solved by various software packages. Convergence properties can then be readily established by examining changes associated with changes in levels of truncation. A five-by-five (N = 5) truncation system is Found to be sufficient for a wide variety of problems, Several example problems are considered, and the results are compared. The new method converges faster and is more accurate for the same level of truncation. Moreover, in comparison with other methods, it is comprehensive, rigorous, and lends itself to insight regarding the interactions of lift distribution, planform shape, and twist.