Induced drag calculation of tapered wings - revisiting Hoerner's approach with AVL and VSPAERO

Abstract

Background – Induced drag is calculated from C_D,i = C_L²/(pi*A*e). Clearly, induced drag can be reduced by increasing the aspect ratio, A. For elliptical wings, e = 1. Tapered wings have e < 1, even at best taper ratio, lambda = c_t/c_r of 0.45 (NASA Report 921) respectively 0.357 (Hoerner 1965). With Hoerner's f(lambda) it is possible to calculate e = 1/(1+f(lambda)*A). More is possible. --- Purpose – Calculate span efficiency factors (e) of straight (unswept) wings for various aspect ratios (A) and taper ratios (0 < lambda < 1) using computational fluid dynamics (CFD). Use the notation from Hoerner. From CFD results calculate e and continue to calculate f(lambda) = (1-e)/(A*e). Users of the results can calculate back to the span efficiency factor with e = 1/(1+f(lambda)*A). --- Methodology – Wings are modeled in AVL and VSPAERO with VLM and Panel Method (NACA 0010) with different aspect ratios and taper ratios. At a selected angle of attack, the lift coefficient and the induced drag coefficient are calculated. From the CFD results the span efficiency factor, e and f(lambda) are calculated. f(lambda) is plotted in comparison to Hoerner's (theoretical) curve. --- Findings – Span efficiency factor, e decreases with increasing aspect ratio, A. The panel method from VSPAERO shows curves approaching Hoerner's curve with increasing aspect ratio. Calculated values of f(lamda) are larger resulting in lower span efficiency factor (e). The Vortex Lattice Method (VLM) offered by AVL also shows curves approaching Hoerner's curve with increasing aspect ratio. However, calculated values of f(lamda) are smaller resulting in a larger span efficiency factor (e). The VLM of VSPAERO shows curves f(lamda) of similar magnitude as Hoerner, but without clear tendencies. Shown is only the result for A = 10. --- Practical Implications – Hoerner's curve seems to be a helpful and simple means to calculate the span efficiency factor (e). To assume that f(lambda) is independent of aspect ratio may be a simplification. A hypothesis is that Hoerner's curve represents values for an aspect ratio approaching infinity. --- Originality – Results calculated with AVL were published by Budziak (2015) and with VSPAERO by Mariën (2021), both supervised by Scholz. This is the first presentation at a conference.