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Unformatted text preview: 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 ow tangency on the camberline, was derived previously. 1 ( ) sin d dZ = V (for 0 < o < ) (1) 2 0 cos cos 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 . Let us consider this combination to be some general function of o . dZ f ( o ) dx For the purpose of computation, any such function can be conveniently represented or ap- proximated by a Fourier cosine series , N f ( o ) = A 0 A n cos n 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 coecients, but is left outside for later algebraic simplicity. f () 1 cos cos 2 cos 3 A 0 A 1 A 2 A 3 ... The overall summation can be made arbitrarily close to a known f ( o ) by making N su- ciently large (i.e. using suciently many terms). The required coecients A , A 1 , . . . A N are computed one by one using Fourier analysis , which is the evaluation of the following integrals. 1 A 0 = f ( ) d 0 2 A 1 = f ( ) cos d 0 2 A 2 = f ( ) cos 2 d 0 ....
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This note was uploaded on 01/28/2012 for the course AERO 16.01 taught by Professor Markdrela during the Fall '05 term at MIT.
- Fall '05