Thin Airfoil Theory Questions and Answers

Answer: A

The theory models the airfoil as a sheet of vorticity distributed along the mean camber line, with the sheet strength varying from leading to trailing edge. Because thickness is discarded, only camber and incidence survive, and the whole problem reduces to finding the one distribution that makes the camber line a streamline.

Answer: B

The camber line must be a streamline, so the component of velocity normal to it vanishes at every point. Writing that condition as the sum of the free-stream contribution and the velocity induced by the vortex sheet gives the fundamental integral equation of the theory, from which the sheet strength follows.

Answer: C

Because the camber line lies close to the chord for a thin, slightly cambered section, the condition is transferred to the chord line to first order. This linearisation is what makes the problem tractable analytically, and it is also the first assumption to fail as camber or incidence is increased.

Answer: D

A finite sheet strength at the trailing edge would imply a finite load there and flow turning the sharp corner at infinite speed. Requiring the strength to vanish there picks out the one circulation that leaves the flow smoothly, which is precisely the Kutta condition and what makes the lift determinate.

Answer: A

Setting the chordwise position to half the chord times one minus the cosine of an angle maps the chord onto the range zero to pi, with zero at the leading edge and pi at the trailing edge. The transformation turns the singular integral into standard Glauert integrals that can be evaluated term by term.

Answer: B

The strength is written as a leading term carrying a cotangent of half the angle, which supplies the correct inverse-square-root singularity at the leading edge, plus a sine series whose terms all vanish at the trailing edge. The form therefore satisfies the Kutta condition automatically whatever the coefficients turn out to be.

Answer: C

The leading coefficient is the angle of attack less the mean of the camber-line slope taken over the chord. It is therefore the only coefficient that changes when the airfoil is rotated, and everything about how lift responds to incidence follows from it. Thickness plays no part anywhere in the theory.

Answer: D

Each higher coefficient is a Fourier cosine average of the camber-line slope and contains no reference to incidence. They are fixed once the camber line is chosen and do not change as the airfoil is rotated, which is exactly why the moment about the quarter chord is independent of angle of attack.

Answer: A

The result is two pi times the difference between the geometric angle and the zero-lift angle. Camber does not change the slope of the lift curve; it merely shifts the curve sideways by altering the zero-lift angle. A symmetric section is the special case in which that shift is zero.

Answer: B

The slope comes out as two pi per radian regardless of the camber line, because camber affects only the constant part of the solution and not its dependence on incidence. Real sections measure a little below this because of the boundary layer, and thickness raises it very slightly, but two pi remains the standard reference.