Physical Properties of Maskell's Induced Drag Integral

This study investigates the physical properties of Maskell's induced drag integral, which is commonly used in the wake survey analysis method to predict induced drag of airplane models. This study also examines the reasons as to why Maskell's induced drag integral performs better than the well-known induced drag integral from the classic lifting line theory. Although the induced drag integral from the classic lifting line theory can be derived from Maskell's induced drag integral with a planar wake approximation, it does not necessarily mean that the integrand of Maskell's induced drag integral is identical to the induced drag distribution of the lifting line theory. For elliptically loaded wings, the induced drag distribution (from the lifting line theory) along the wing span is proportional to the local circulation strength F(t/). Maskell's integrand is proportional to the first moment of the slope of circulation, ydT/dy. Because of this physical property of Maskell's induced drag integral, the impacts on the induced drag calculation by the inaccurate lift distribution near the model symmetry plane (y ~ 0) are minimized. In general, it is difficult to obtain an accurate lift distribution of the model near its symmetry plane, because of the interference from the model's fuselage and support struts. Some of the wake survey results of a high-speed model test at the Boeing Transonic Wind Tunnel are presented to demonstrate the superiority of Maskell's induced drag integral over the induced drag integral from the lifting line theory. flow variables measured at a downstream survey station. The problem with the kinetic energy form of induced drag, which is derived from the integral momentum equations, is that the integral must be evaluated over the entire downstream cross section of the wind tunnel. Once Maskell [1] demonstrated that an integral formulation for induced drag can be obtained from flow variables measured within the model wake region, the practical application of quantitative wake surveys became possible [2,3,4,5]. After Maskell introduced his new induced drag integral, however, the clean physical meaning behind the original kinetic energy form was lost. In this brief paper we investigate the physical properties of Maskell's induced drag integral, and we seek the reasons as to why Maskell's induced drag integral performs better than that from the lifting line theory. It is worth noting that the well-known induced drag formula can be derived from Maskell's induced drag integral by applying the planar wake approximation [6,7]. To demonstrate the superiority of Maskell's induced drag integral over the induced drag integral from the lifting line theory, a few selected results from a highspeed model test (for the study of the effects of flap trap fairings on the airplane drag) at Boeing Transonic Wind Tunnel are used. Because this paper focuses on the properties of Maskell's integral, a detailed explanation of the experimental setup, including the dataacquisition and -reduction systems, are not discussed. Introduction Maskell's Integrand Generally, the forces acting upon a model can be derived by applying the integral form of the momentum equations to a control volume containing the model. Model drag and lift can be written as integrals of the Copyright ©2001 by The Boeing Company. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. Maskell's induced drag integral wA (1) which is commonly used in wake survey analysis method [1,2,3,4,5,6,7,8] can be derived from the kiAmerican Institute of Aeronautics and Astronautics c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. netic energy form of induced drag [1,9,10] (v +w)dydz (2) Here, we assumed the x— and z—axes to be aligned with the undisturbed freestream direction and lift direction, respectively. The remaining y—axis is normal to both x— and z—axes. 52 and WA represent transverse plane located downstream of model and model wake region on it, respectively (Figure 1). Wake Survey • Region Fig. 1 Control volume and coordinate system. For the calculation of Maskell's scalar function ^(y, z), a Poisson equation , dy dz (3) should be solved over the entire transverse plane (S2) with the boundary condition V = 0 (4) along the tunnel intersection [1,9,10] (see Figure 1). f represents the x-component of the vorticity vector, which is defined as (5) and £ = 0 outside of the model wake region WAFor the case of a half model test (including a full model test case without yaw), the boundary condition, ip = 0, will be also satisfied at the model symmetry plane (y — 0) [7]For the "free-air" case (without interference of tunnel walls) the boundary condition (4) should be evaluated at y + z -»> oo, and Maskell's scalar function, T/>, can be expressed using a Green's function in the following analytical form [6,7,8]. 1 It is apparent that the induced drag integral in kinetic energy form (which is given by equation (2)) has a clear physical meaning by itself: induced drag is determined by the total amount of kinetic energy loss that has been consumed to generate cross-flow velocities in the wake field. On the other hand, it is difficult to see clear physical meanings in Maskell's induced drag integral (equation (1)). In particular, when the integrand of Maskell's equation is plotted along the wake span, it is not clear what the integrand signifies. For the purpose of examining the physical properties of the integrand of Maskell's induced drag integral, we define "Maskell's integrand" as (7) ZWA denotes an integral region of z (for a given value of y) on the model wake WA> Then Maskell's induced drag integral can be simplified to -r JyL diM(y)dy (8) where yi and y# represent the minimum and maximum values of y on the model wake WA, respectively (see Figure 2).