To put this all together for an excel formula we must

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Unformatted text preview: owski Airfoil 1 Start with the map f(z) = z + , and look at the images of some circles. In general, we z look at a circle whose center is offset a little to the right of a point on the y-axis passing through -1 That is, Center = iµ + œ 2 2 Radius = µ + (1+œ) Parametric equations for this circle are: x = œ + P cos t y = µ + P sin t 0 ≤ t ≤ 2π 2 2 where P = µ + (1+œ) 2 Its image under f(z) = z + z–/|z| has parametric equations 2 2 2 x = (œ + Pcos t)[1 + 1/[œ + µ + P + 2P(œ cos t + µ sin t)]] 2 2 2 y = (µ + Psin t)[1 - 1/[œ + µ + P + 2P(œ cos t + µ sin t)]] let us fix µ = 0.2 and keep œ as a parameter. This gives P= 0.04 + (1+œ) 2 49 2 2 x = œ + Pcos t [1 + 1/[œ + 0.04 + P + 2P(œ cos t + µ sin t)]] 2 2 y = 0.02 + Psin t [1 - 1/[œ + 0.04 + P + 2P(œ cos t + µ sin t)]] Setting them up for Excel gives, with the parameter denoted by k: x= (k+(0.04+(1+k)^2)^.5*cos(t))*(1+1/(k^2+0.08+(1+k)^2+ 2*(0.04+(1+k)^2)^.5*(k*cos(t)+0.2*sin(t)))) y= (0.2+(0.04+(1+k)^2)^.5*sin(t))*(1-1/(k^2+0.08+(1+k)^2+ 2*(0.04+(1+k)^2)^.5*(k*cos(t)+0.2*sin(t)))) And here are the plots: k = 0, 0.033, 0.067, 0.1 Each of the curves is a different airfoil, with the degenerate one corresponding to k = 0, the image of the circle passing through (-1, 0) and (1, 0) with center (0, 0.2)....
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This document was uploaded on 03/20/2014 for the course MATH 144 at Hofstra University.

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