Up until this course, the lift on an airplane wing was probably explained to you

Question

Up until this course, the lift on an airplane wing was probably explained to you in terms of Bernoulli's equation. That explanation is not wrong but the logic is circular. Aerodynamicists know that lift is due to circulation, that is, non-zero net vorticity in a control volume that includes the air flowing over the wing. For the following 3 configurations, compute the aerodynamic circulation (what the hook calls T) two ways. First, compute it directly by evaluating the contour integral. Second, compute the vorticity and integrate it over the surface contained by the contour. Before doing either, sketch a few streamlines so you understand the flow. u(x, y) = 2U(xy/H^2)i - U(y^2/H^2)j over a contour defined by a rectangle of height H and length L with its lower left corner at the origin. u(.r. y) = Axi - Ayj over the same contour as part a. This is stagnation point flow. u(r) (K/r)e_theta over a rectangle of length 2L and height 2H centered at the origin. Recall that c_theta is the unit vector in the theta direction so u(r) describes a potential vortex. We commented in class that an ideal vortex lupus zero vorticity except at the center. In a physical vortex, such as that formed as water goes down a drain, the discontinuity in vorticity at the center is smoothed by viscous forces, but most of the flow can be treated as inviscid.

Explanation / Answer

let us assume stress is uniform across the width and the presence of shear stress does not afftect the distribution of bending stress.

let

s be the value of complementary shear stress and hence the transverse shear stress at a distance Y0 from neutral axis.

z be the width of cross section at y0

A be area of cross section

y bar be the distance of centroid of A

Szdelta X = integral of delta F *delta A

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