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UBC Theses and Dissertations

Multi-element thin airfoil theory Watt, George Donald


A linearized, two-dimensional, potential flow analysis of multi-element airfoil configurations is attempted. Arguments are presented that suggest the accuracy of the linearized theory should be as good as or better than that of the well-known one-element thin airfoil theory--despite the large mean line curvature common to multi-element configurations. The results obtained in this thesis tend to support this expectation. Good success is achieved in developing a general two-element tandem airfoil linearized theory. The effects of incidence, leading or trailing edge flap deflection, and camber on the overall and localized lifts and moments are summarized in integrals which are the tandem airfoil versions of the historical Munk integrals. Analytical solutions for the forces on tandem NACA airfoils are then obtained from these integrals. The expressions for the forces reduce, when the two airfoil elements come together, to the familiar one-element thin airfoil theory formulas for an airfoil with a simple flap. All the forces are calculated on a small hand-held computer and the results are compared with exact potential flow theory. The overall force results for the tandem airfoil incidence problem are particularly simple and, in fact, this simplicity allows the solution of the incidence problem for an arbitrary number of in-line airfoils. Although thickness has no effect on tandem thin airfoil theory forces, it does affect pressures. The general thickness distribution analysis for tandem airfoils, an exceedingly simple analysis involving only elementary functions, is presented. A similar type of analysis enables the design of tandem airfoil camber lines. Examples are given. Modelling the effects of overlap of a staggered two-element airfoil configuration is a much more complicated analysis than that of tandem airfoil elements. Nevertheless, substantial success is achieved. This includes the development of the equivalent Munk integrals that give the overall and localized forces on airfoil elements of arbitrary chord length, arbitrarily positioned relative to one another (providing the chord lines are approximately parallel to each other and to the flow at infinity--the linearization requirement). Although analytical solutions are obtained for the overall forces of the incidence and flap deflection problems, they are in terms of parameters which can only be obtained through a trial and error solution of the staggered parallel slit conformal mapping.

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