{\7iix')dx'=-C 2a^(x')dx'=-2U,y(0) [10] 



Jo J-c ax 



THE THIN AIRFOIL THEORY AND THE JUNCTURE CONDITION 



Equation [Id] requires that the sum of the distributed source strengths be zero or, 

 equivalently, that the body be closed. Equation [9a] has the form of the fundamental integral 

 equation of the thin airfoil theory^ where m(x') would there by replaced by (in Glauert's nota- 

 tion) Jiix') which is the strength of the distributed vorticity and where f(x) depends upon the 

 shape of the airfoil camber line and the airfoil angle of attack. That problem of the thin air- 

 foil theory which is entirely equivalent to the present problem would be stated: To find a dis- 

 tribution of vorticity in the interval < x < I such that the streamline shape in the vicinity of 

 that interval coincides with the shape of the airfoil camber line and such that the lift on the 

 airfoil 



(^pU^Ck{x')dx'^ 



is equal to a certain prescribed value. 



Now it is a well-known result of both the exact and thin airfoil theories that the airfoil 

 shape and attitude being known, a flow may be found such that the airfoil lift assumes any 

 prescribed value. Thus the thin airfoil problem stated above and, equivalently, the linearized 

 cavitation problem (Equations [9a] and [10]) always have a solution. This result is somewhat 

 paradoxical, since according to our physical experience the lift on an airfoil of particular 

 shape and attitude is quite well determined, and for a given cavitating body there seems to 

 exist a nonarbitrary correspondence between cavity length and cavitation number. In the case 

 of the airfoil the paradoxical result is due entirely to an oversimplification of a physical na- 

 ture (it is not due to the linearization) and is resolved by invoking the Joukowsky condition 

 that the flow leave the trailing edge smoothly, which in the thin airfoil theory takes the form 

 of a further specification that the vorticity strength vanish at the airfoil trailing edge. 



In the case of the cavitating body the paradoxical result is due entirely to mathemat- 

 ical oversimplification introduced with the linearizing assumptions. That the linearized form 

 of a cavitation type boundary value problem may have a solution although the solution does 

 not exist for the exact form of the same problem is easily demonstrated by introduction of the 

 following problem: To find a closed body, symmetric with respect to the flow direction, and 

 of such a shape that the pressure is everywhere constant on the body surface. Now it follows 

 from Drillouin's Paradox that such a body does not exist, and yet it is easily shown that a 

 linearized solution exists and that, in fact, elliptic cylinders are the bodies sought. To speak 

 very loosely, the linearized theory produces some approximate solutions which almost, but 



