38 



KENNARD'S DERIVATION OP THE INTEGRAL EQUATION 



A simple, physical derivation of the integral equation [9^] has been 

 given by Dr. E.H. Kennard. This will now be presented. 



Imagine the body replaced by fluid at rest. Let U be the velocity 

 on the body. Then the field of flow consists of the superposition of the uni- 

 form (unit) flow and the flow due to a vortex sheet of density U. 



Now subtract the uniform flow. There remains the flow due to the 

 vortex sheet alone, uniform inside the space originally occupied by the body, 

 of unit magnitude. 



A vortex ring of strength Uds produces at an axial point distant z 

 from its plane the velocity 



V = 



y^Uds 



2(y 2 +z 2 ) 3/2 



where y is the radius of the ring. Let s be the distance of a point on the 

 body measured along the generator from the forward end, in a meridian plane. 

 The axial and radial coordinates will then be functions x(s), y(s). The ve- 

 locity due to the sheet at a point t on the axis will then be 



[ p m yfU) ds = 1 



Jo 2 r 3 



where r 2 =[x(s)-t] + y^s) and P Is the total length of a generator. The 

 equivalence of this equation with [9^] is evident. 



A FIRST APPROXIMATION 



If we again make use of the polar transformation x - t = y(x) cot 9, 

 [9^] becomes 



r U(x) sin 2 gd0 _ 1 [95] 



Jo 2 sin[(9-y(x)] 



When x = t, 9 = -?-. For an elongated body the integrand In [9^] peaks sharply 

 in the neighborhood of x = t, so that a good approximation is obtained when 

 U(x) is replaced by U(t) for the entire range of integration. Also, y(x) will 

 be small except near the ends of the body so that the approximation 



sin [0 - y(x)]= sin 6 cos y(x) = sin 6 cos y(t) 



