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f03_sp - Fluids Lecture 3 Notes 1 Thin-Airfoil Analysis...

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Fluids – Lecture 3 Notes 1. Thin-Airfoil Analysis Problem (continued) Reading: Anderson 4.8 Cambered airfoil case We now consider the case where the camberline Z ( x ) is nonzero. The general thin airfoil equation, which is a statement of flow tangency on the camberline, was derived previously. 1 π γ ( θ ) sin θ dθ dZ = V α (for 0 < θ o < π ) (1) 2 π 0 cos θ cos θ o dx For an arbitrary camberline shape Z ( x ), the slope dZ/dx varies along the chord, and in the equation it is negated and shifted by the constant α . Let us consider this combination to be some general function of θ o . dZ α f ( θ o ) dx For the purpose of computation, any such function can be conveniently represented or ap- proximated by a Fourier cosine series , N f ( θ o ) = A 0 A n cos o n =1 which is illustrated in the figure. The negative sign in front of the sum could be absorbed into all the A n coefficients, but is left outside for later algebraic simplicity. f ) 1 cos θ cos cos A 0 A 1 A 2 A 3 ... The overall summation can be made arbitrarily close to a known f ( θ o ) by making N suffi- ciently large (i.e. using sufficiently many terms). The required coefficients A 0 , A 1 , . . . A N are computed one by one using Fourier analysis , which is the evaluation of the following integrals. π 1 A 0 = f ( θ ) π 0 π 2 A 1 = f ( θ ) cos θ dθ π 0 π 2 A 2 = f ( θ ) cos 2 θ dθ π 0 . . . π 2 A N = f ( θ ) cos Nθ dθ π 0 1
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For the particular f ( θ o ) used here, these integrals become 1 π dZ A 0 = α π 0 dx 2 π dZ A n = cos nθ dθ ( n = 1 , 2 , . . . ) π 0 dx
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