ChapterV.PartIV - Chapter V Lifting Line Theory Part IV...

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Chapter V. Lifting Line Theory Part IV. Elliptically Loaded Wings In the previous handout titled “Chapter V. Part III: Forces on the Airfoil Section at Point P on the Lifting Line” we derived the following equation for the distribution of circulation Γ (y) along the span of a wing. ( 29 ( 29 ( 29 dy dy d y y V y c V y y b y b y L Γ - + Γ + = + = - = = 0 0 0 0 0 1 4 1 ) ( π π α α (1) We also derived an expression for the downwash velocity: ( 29 dy dy d y y w b y b y Γ - = + = - = 0 1 4 1 π (2) In this section, we study the solution of equation (1) for a special case – an elliptically loaded wing. For this case, the loading is given by the equation of an ellipse: 1 2 2 0 = + Γ Γ b y (3) Note that we call ‘b’ the semi-span (i.e. half of the wing span) in our notes. Anderson calls ‘b’ the full span. These discrepancies will, hopefully, add to your general confusion about the course material. The quantity Γ 0 is the maximum circulation that will occur at mid-span, at y=0. This distribution may be visualized for one-half of a wing as follows: Γ (y) y/b Γ 0 We will use this assumed solution in equation (2) and see what happens. To facilitate this, let us do a simple transformation:
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π θ θ θ θ π = - 0 0 cos cos cos d θ cos b y = (4) Equation (3) then yields: ( 29 θ θ θ d d dy dy d y cos sin 0 0 Γ = Γ = Γ Γ = Γ (5) Also, in equations (1) and (2) ( 29 0 0 cos cos θ θ - = - b y y (6) Plugging (5) and (6) into equation (2), and using the following identity from the integral tables: (7) Equation (2) becomes: b w 4 0 Γ - = (8) In other words, for an elliptically loaded wing, the downwash velocity is a constant along the span ! The negative sign simply indicates that this velocity will be directed
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