A study of induced drag and spanwise lift distribution for three-dimensional inviscid flow over a wing

TLDR: In this article, the authors used a wake integral analysis to estimate the induced drag of an untwisted, finite rectangular wing (NACA 0012, AR = 6.7) using an Euler-based computational fluid dynamic solver.

Abstract: The purpose of this study was to validate an approach to estimating the induced drag on a finite wing by using a wake integral analysis. The long-term goal is related to developing an aerodynamic-structural systems integrated design methodology for wings through the use of a transpiration boundary condition to control the spanwise lift distribution throughout a typical aircraft mission so as to minimize lift–induced drag. The short term goal addressed by this study is to develop a methodology to extract accurate and robust calculations of the induced drag from second order numerical solutions. Numerical results for an untwisted, finite rectangular wing (NACA 0012, AR = 6.7) using no flap deflections are compared against theoretical lifting line predictions. The numerical approach used an Euler-based computational fluid dynamic (CFD) solver. An in-house lifting line code was used to predict the theoretical reference values. By dividing the wing into twenty span-wise sections and using a surface integral of pressure at each section, a span-wise lift distribution was extracted from the CFD solution. Under flow conditions representing subsonic and transonic flows (Mach 0.3 – 0.7) at small angles of attack, the comparison between the predicted numerical and lifting-line spanwise lift distributions show good agreement with a maximum deviation of only 2.4% over the wing span. The induced drag was extracted from the downstream wake using a wake integral technique referred to as Trefftz plane analysis. This approach was attempted because (1) there are known inherent inaccuracies associated with using the more common surface integral method for calculating the drag of a wing, and (2) the wake integral approach directly isolates the induced drag from other drag (viscous and wake) components. The predictions for induced drag based on surface integration, wake integration and lifting line methods are compared. The numerical induced drag results show a dependency on the downstream location of the Trefftz plane. Near wake and compressible flow corrections were applied to improve the induced drag predictions by wake integration. The wake integration approach is susceptible to artificial dissipation due to the numerical flow grid used, which provides an error that increases as the position of the Trefftz plane moves further downstream. Attempts to estimate the extent of this effect and to correct for it are discussed. The numerical solution of the Euler equations demonstrates successful implementation of the wake integral method via a Trefftz Plane analysis of the induced drag. The study details an initial effort to identify and to quantify the numerical uncertainties associated with the simulation and, specifically, the induced drag prediction.