How Does Thin Airfoil Theory Calculate Lift and Moment Coefficients?

[caliguy]

a thin airfoil at a geometric angle of attack (alpha) in a uniform stream of inviscid incompressible fluid has a parabolic mean camber line described by

z(x)=4z[(x/c)-(x/c)^2]

where z is the maximum camber. Use thin airfoil theory to calculate the following:

1. Lift coefficient
2. pitching moment coefficient about leading edge
3. lift-curve slope

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I've tried to solve this, but it gives me a hint. The hint is to first non-dimensionalize the equation by c which is the chord length. If I do this I get:

z/c=(4z/c)[(x/c^2)-(x/c^2)^2]

I don't think this is right...
If I don't nondimensionalize it I get 4z[(1/c)-(2x/c^2)] after doing the derivative to solve for the lift coefficient. Any ideas? this is where I get stuck

The main thing that I am having trouble on is how do i exactly nondimensionalize it. I just can't simply divide everything by c correct? By the way, C is in units of length just as Z is and x

 

[Valkarie]

is in units of length in the original equation.Answer:1. Lift coefficient:The lift coefficient is given by C_L = 2π α, where α is the angle of attack. This can be calculated from the equation for the camber line, which is given by z(x) = 4z[(x/c)-(x/c)^2]. Taking the derivative of this equation yields C_L = 8πz/c^2.2. Pitching moment coefficient about leading edge:The pitching moment coefficient about the leading edge is given by C_m = -4πz/c^2. This can be calculated by taking the second derivative of the camber line equation and multiplying it by x. 3. Lift-curve slope:The lift-curve slope is given by dC_L/dα = 2π. This is a constant and does not depend on the parameters of the airfoil.