The Kutta-Joukouski Theorem

It is shown in Section 4.7 that an integration of the pressure distribution on a lifting cylinder leads to the result that the lift per unit span is proportional to the circu­lation around that cylinder. The Kutta-Joukouski Theorem[20] states that for any body of arbitrary cross section (e. g., an airfoil), the lift per unit span L’ = рУмГ, where Г is taken around any closed path enclosing the body. The proof of the theorem is beyond the scope of this book. However, the theorem was demonstrated to be true for a right circular cylinder by integration of the static pressure acting on the surface of a body, and it is shown later to be true for an airfoil shape.

The importance of this theorem is that it provides an alternative way to calcu­late the lift force on a lifting body. Instead of calculating the velocity magnitude at a point on the body surface, then using the Bernoulli Equation to evaluate the pressure there, and finally integrating to determine the force, the theorem states that the lift force can be found simply by calculating the circulation around the body. As previously mentioned, it often is easier to calculate the circulation than it is to deter­mine the pressure distribution. Of course, if the pressure magnitude at a point or the pressure distribution on the body surface is required, the theorem is of no help because it speaks only of the net force.

This theorem is the basis for the so-called circulation theory of lift. This is a math­ematical method of calculating lift that convenient for many inviscid-flow problems. However, remember that the lift (and drag) on a body is physically generated by the pressure (and shear-stress) distribution over the surface. The oncoming flow adjusts to accommodate the presence of the lifting body and, in so doing, sets up a velocity and pressure field such that the circulation around the lifting body is nonzero.

To fix ideas regarding lift and circulation, imagine a two-dimensional wing installed at an angle of attack in a wind tunnel. Also imagine that there is a suitable instrument available that measures the flow velocity (i. e., magnitude and direction) at numerous points around the wing. The particular points of interest are located along a closed path in a vertical plane aligned with the oncoming stream. Make the measurement, form the vector-dot product V • ds at each measurement station, and sum around the closed path. This calculation of circulation yields a positive quantity that is equal to L7pVTO. Thus, the lift (i. e., physically, the net pressure force acting upward on the wing) is exhibited as a circulation around the wing. Recall an ana­logy in Chapter 3 in which a measurement of drag was carried out by evaluating the momentum loss in the wake. There, the drag was due physically to the pressure and
shear forces acting on the body surface and the drag was exhibited as a momentum loss.

Remember that the presence of a circulation around a body does not imply any fluid particles rotating about it. It simply means that the flow above and below the lifting body is higher and lower average velocities than the zero-lift value, respectively.

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