f04_sp - Fluids Lecture 4 Notes 1 Thin Airfoil Theory...

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parenleftbigg parenrightbigg parenleftbigg parenrightbigg negationslash Fluids – Lecture 4 Notes 1. Thin Airfoil Theory Application: Analysis Example Reading: Anderson 4.8, 4.9 Analysis Example Airfoil camberline definition Consider a thin airfoil with a simple parabolic-arc camberline, with a maximum camber height εc . x Z ( x ) = 4 ε x 1 c The camberline slope is then a linear function in x , or a cosine function in θ . dZ x = 4 ε 1 2 = 4 ε cos θ o dx c x x dZ dx Z dZ dx θ 0 c c π c ε Fourier coefficient calculation Substituting the above dZ/dx into the general expressions for the Fourier coefficients gives integraldisplay integraldisplay π 1 π dZ 1 A 0 = α = α 4 ε cos θ dθ π 0 dx π 0 integraldisplay integraldisplay π 2 π dZ 2 A n = cos nθ dθ = 4 ε cos θ cos nθ dθ π 0 dx π 0 The integral in the A 0 expression easily evaluates to zero. The integral in the A n expression can be evaluated by using the orthogonality property of the cosine functions. integraldisplay π π (if n = m = 0) cos cos mθ dθ = π/ 2 (if n = m = 0) negationslash 0 0 (if n = m ) For our case we have m = 1, and then set n = 1 , 2 , 3 . . . to evaluate A 1 , A 2 , A 3 , . . . . The final
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