Accuracy of Küchemann’s prediction for the locus of aerodynamic centres on swept wings compared to an inviscid panel method

Abstract

AbstractThe locus of aerodynamic centres of a finite wing is the collection of all spanwise section aerodynamic centres, and depends on aspect ratio, wing sweep and planform shape. This locus is of great importance in the positioning of vortex elements in lifting-line theory. Traditionally, these vortex elements are placed along the quarter-chord of a wing, leading to inaccurate predictions of aerodynamic coefficients for swept wings due to the discontinuity in the line of vorticity at the wing root. An analytical solution was presented by Küchemann in 1956 to determine the locus of aerodynamic centres as a function of sweep. While experimental studies have been performed to visualise this locus, no large amount of data is available to fully evaluate the accuracy of Küchemann’s analytical solution. In the present study, a numerical approach is taken using a high-order panel method for inviscid, incompressible flow to calculate the locus of aerodynamic centres for elliptic wings over a wide range of sweep angles, aspect ratios and profile thicknesses. An inviscid panel method is chosen over full CFD solutions because of their ability to isolate the inviscid phenomena. Küchemann’s prediction is compared to this numerical data. The root mean square error is calculated for each wing in a broad design space to determine the accuracy of Küchemann’s theory. It is shown to be remarkably accurate over the range of cases studied, with the root mean square error staying below 4% for all wings with aft sweep and aspect ratios higher than $R_A=5$ . The actual difference between Küchemann’s prediction and numerical data is lower than that for the majority of the span for many of the wing designs considered, with the RMS error being skewed by the results at the tip. Results demonstrate that Küchemann’s analytical equations can be used as an accurate approximation for the locus of aerodynamic centres and could be used in modern numerical lifting-line algorithms*.